{A More General Iterable Condition \\ Ensuring $\aleph_1$ is not Collapsed}
Saharon Shelah \thanks {\null\newline I would like to thank Alice Leonhardt for the beautiful typing.\newline Publication 311.} \endthanks
Sh:b
311.tex
%&tex
% 311, processed by citealice (2002-10-18) on Wed Feb 26 19:40:04 CST 2003

\ifx\shlhetal\undefinedcontrolsequence\let\shlhetal\relax\fi
% To: "Saharon Shelah's Office" <shlhetal@math.huji.ac.il>,   Andrzej Roslanowski <roslanow@member.ams.org>
% Subject: sh311 - entire paper
% Date: Wed, 26 Feb 2003 14:35:28 -0500 (EST)
% From: Alice Leonhardt <leonhard@math.rutgers.edu>
% Mime-Version: 1.0
% Content-Description: revisions
% X-sliced-and-diced-by: 'savemail' 1.3b, Feb 2003


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\def\coth{\qopname@{coth}}
\def\csc{\qopname@{csc}}
\def\deg{\qopname@{deg}}
\def\det{\qopnamewl@{det}}
\def\dim{\qopname@{dim}}
\def\exp{\qopname@{exp}}
\def\gcd{\qopnamewl@{gcd}}
\def\hom{\qopname@{hom}}
\def\inf{\qopnamewl@{inf}}
\def\injlim{\qopnamewl@{inj\,lim}}
\def\ker{\qopname@{ker}}
\def\lg{\qopname@{lg}}
\def\lim{\qopnamewl@{lim}}
\def\liminf{\qopnamewl@{lim\,inf}}
\def\limsup{\qopnamewl@{lim\,sup}}
\def\ln{\qopname@{ln}}
\def\log{\qopname@{log}}
\def\max{\qopnamewl@{max}}
\def\min{\qopnamewl@{min}}
\def\Pr{\qopnamewl@{Pr}}
\def\projlim{\qopnamewl@{proj\,lim}}
\def\sec{\qopname@{sec}}
\def\sin{\qopname@{sin}}
\def\sinh{\qopname@{sinh}}
\def\sup{\qopnamewl@{sup}}
\def\tan{\qopname@{tan}}
\def\tanh{\qopname@{tanh}}
\def\varinjlim{\mathop{\vtop{\ialign{##\crcr
 \hfil\rm lim\hfil\crcr\noalign{\nointerlineskip}\rightarrowfill\crcr
 \noalign{\nointerlineskip\kern-\ex@}\crcr}}}}
\def\varprojlim{\mathop{\vtop{\ialign{##\crcr
 \hfil\rm lim\hfil\crcr\noalign{\nointerlineskip}\leftarrowfill\crcr
 \noalign{\nointerlineskip\kern-\ex@}\crcr}}}}
\def\varliminf{\mathop{\underline{\vrule height\z@ depth.2exwidth\z@
 \hbox{\rm lim}}}}
\def\varlimsup{\mathop{\overline{\hbox{\rm lim}}}}
\newdimen\buffer@
\buffer@\fontdimen13 \tenex
\newdimen\buffer
\buffer\buffer@
\def\ChangeBuffer#1{\fontdimen13 \tenex#1\relax
 \global\buffer\fontdimen13 \tenex}
\def\ResetBuffer{\fontdimen13 \tenex\buffer@\global\buffer\buffer@}
\def\shave#1{\mathop{\hbox{$\m@th\fontdimen13 \tenex\z@                     %1
 \displaystyle{#1}$}}\fontdimen13 \tenex\buffer}
\def\topshave#1{\mathop{\topsmash{\displaystyle{#1}}\vphantom{\shave{#1}}}}
\def\botshave#1{\mathop{\botsmash{\displaystyle{#1}}\vphantom{\shave{#1}}}}
\message{multilevel sub/superscripts,}
\Invalid@\\
\def\Let@{\relax\iffalse{\fi\let\\=\cr\iffalse}\fi}
\Invalid@\vspace
\def\vspace@{\def\vspace##1{\crcr\noalign{\vskip##1\relax}}}
\def\multilimits@{\bgroup\vspace@\Let@
 \baselineskip\fontdimen10 \scriptfont\tw@
 \advance\baselineskip\fontdimen12 \scriptfont\tw@
 \lineskip\thr@@\fontdimen8 \scriptfont\thr@@
 \lineskiplimit\lineskip
 \vbox\bgroup\ialign\bgroup\hfil$\m@th\scriptstyle{##}$\hfil\crcr}
\def\Sb{_\multilimits@}
\def\endSb{\crcr\egroup\egroup\egroup}
\def\Sp{^\multilimits@}
\let\endSp\endSb
\def\spreadlines#1{\RIfMIfI@\onlydmatherr@\spreadlines\else
 \openup#1\relax\fi\else\onlydmatherr@\spreadlines\fi}
\def\Mathstrut@{\copy\Mathstrutbox@}
\newbox\Mathstrutbox@
\setbox\Mathstrutbox@\null
\setboxz@h{$\m@th($}
\ht\Mathstrutbox@\ht\z@
\dp\Mathstrutbox@\dp\z@
\message{matrices,}
\newdimen\spreadmlines@
\def\spreadmatrixlines#1{\RIfMIfI@
 \onlydmatherr@\spreadmatrixlines\else
 \spreadmlines@#1\relax\fi\else\onlydmatherr@\spreadmatrixlines\fi}
\def\matrix{\null\,\vcenter\bgroup\Let@\vspace@
 \normalbaselines\openup\spreadmlines@\ialign
 \bgroup\hfil$\m@th##$\hfil&&\quad\hfil$\m@th##$\hfil\crcr
 \Mathstrut@\crcr\noalign{\kern-\baselineskip}}
\def\endmatrix{\crcr\Mathstrut@\crcr\noalign{\kern-\baselineskip}\egroup
 \egroup\,}
\def\format{\crcr\egroup\iffalse{\fi\ifnum`}=0 \fi\format@}
\newtoks\hashtoks@
\hashtoks@{#}
\def\format@#1\\{\def\preamble@{#1}%
 \def\l{$\m@th\the\hashtoks@$\hfil}%
 \def\c{\hfil$\m@th\the\hashtoks@$\hfil}%
 \def\r{\hfil$\m@th\the\hashtoks@$}%
 \edef\preamble@@{\preamble@}\ifnum`{=0 \fi\iffalse}\fi
 \ialign\bgroup\span\preamble@@\crcr}
\def\smallmatrix{\null\,\vcenter\bgroup\vspace@\Let@
 \baselineskip9\ex@\lineskip\ex@
 \ialign\bgroup\hfil$\m@th\scriptstyle{##}$\hfil&&\thickspace\hfil
 $\m@th\scriptstyle{##}$\hfil\crcr}
\def\endsmallmatrix{\crcr\egroup\egroup\,}
\def\pmatrix{\left(\matrix}
\def\endpmatrix{\endmatrix\right)}
\def\bmatrix{\left[\matrix}
\def\endbmatrix{\endmatrix\right]}
\def\vmatrix{\left|\matrix}
\def\endvmatrix{\endmatrix\right|}
\def\Vmatrix{\left\|\matrix}
\def\endVmatrix{\endmatrix\right\|}
\def\hdots{\mathinner{\ldotp\ldotp\ldotp}}
\newmuskip\dotsspace@
\dotsspace@1.5mu
\def\strip@#1 {#1}
\def\spacehdots#1\for#2{\multispan{#2}\xleaders
 \hbox{$\m@th\mkern\strip@#1 \dotsspace@.\mkern\strip@#1 \dotsspace@$}\hfill}
\def\hdotsfor#1{\spacehdots\@ne\for{#1}}
\def\multispan@#1{\omit\mscount#1\unskip\loop\ifnum\mscount>\@ne\sp@n\repeat}
\def\spaceinnerhdots#1\for#2\after#3{\multispan@{\strip@#2 }#3\xleaders
 \hbox{$\m@th\mkern\strip@#1 \dotsspace@.\mkern\strip@#1 \dotsspace@$}\hfill}
\def\innerhdotsfor#1\after#2{\spaceinnerhdots\@ne\for#1\after{#2}}
\def\cases{\bgroup\spreadmlines@\jot\left\{\,\matrix\format\l&\quad\l\\}
\def\endcases{\endmatrix\right.\egroup}
\message{multiline displays,}
\newif\ifinany@
\newif\ifinalign@
\newif\ifingather@
\def\strut@{\copy\strutbox@}
\newbox\strutbox@
\setbox\strutbox@\hbox{\vrule height8\p@ depth3\p@ width\z@}
\def\topaligned{\null\,\vtop\aligned@}
\def\botaligned{\null\,\vbox\aligned@}
\def\aligned{\null\,\vcenter\aligned@}
\def\aligned@{\bgroup\vspace@\Let@
 \ifinany@\else\openup\jot\fi\ialign
 \bgroup\hfil\strut@$\m@th\displaystyle{##}$&
 $\m@th\displaystyle{{}##}$\hfil\crcr}
\def\endaligned{\crcr\egroup\egroup}
\let\endtopaligned\endaligned
\let\endbotaligned\endaligned
\def\alignedat#1{\null\,\vcenter\bgroup\doat@{#1}\vspace@\Let@
 \ifinany@\else\openup\jot\fi\ialign\bgroup\span\preamble@@\crcr}
\newcount\atcount@
\def\doat@#1{\toks@{\hfil\strut@$\m@th
 \displaystyle{\the\hashtoks@}$&$\m@th\displaystyle
 {{}\the\hashtoks@}$\hfil}%                                                 %1
 \atcount@#1\relax\advance\atcount@\m@ne                                    %2
 \loop\ifnum\atcount@>\z@\toks@=\expandafter{\the\toks@&\hfil$\m@th
 \displaystyle{\the\hashtoks@}$&$\m@th
 \displaystyle{{}\the\hashtoks@}$\hfil}\advance
  \atcount@\m@ne\repeat                                                     %3
 \xdef\preamble@{\the\toks@}\xdef\preamble@@{\preamble@}}
\let\endalignedat\endaligned
\def\gathered{\null\,\vcenter\bgroup\vspace@\Let@
 \ifinany@\else\openup\jot\fi\ialign
 \bgroup\hfil\strut@$\m@th\displaystyle{##}$\hfil\crcr}
\def\endgathered{\crcr\egroup\egroup}
\newif\iftagsleft@
\def\TagsOnLeft{\global\tagsleft@true}
\def\TagsOnRight{\global\tagsleft@false}
\TagsOnLeft
\newif\ifmathtags@
\def\TagsAsMath{\global\mathtags@true}
\def\TagsAsText{\global\mathtags@false}
\TagsAsText
\def\tagform@#1{\hbox{\rm(\ignorespaces#1\unskip)}}
\def\thetag{\leavevmode\tagform@}
\def\tag#1$${\iftagsleft@\leqno\else\eqno\fi                                %1
 \maketag@#1\maketag@                                                       %2
 $$}                                                                        %3
\def\maketag@{\FN@\maketag@@}
\def\maketag@@{\ifx\next"\expandafter\maketag@@@\else\expandafter\maketag@@@@
 \fi}
\def\maketag@@@"#1"#2\maketag@{\hbox{\rm#1}}                                %1
\def\maketag@@@@#1\maketag@{\ifmathtags@\tagform@{$\m@th#1$}\else
 \tagform@{#1}\fi}
\interdisplaylinepenalty\@M
\def\allowdisplaybreaks{\RIfMIfI@
 \onlydmatherr@\allowdisplaybreaks\else
 \interdisplaylinepenalty\z@\fi\else\onlydmatherr@\allowdisplaybreaks\fi}
\Invalid@\allowdisplaybreak
\Invalid@\displaybreak
\Invalid@\intertext
\def\allowdisplaybreak@{\def\allowdisplaybreak{\crcr\noalign{\allowbreak}}}
\def\displaybreak@{\def\displaybreak{\crcr\noalign{\break}}}
\def\intertext@{\def\intertext##1{\crcr\noalign{%
 \penalty\postdisplaypenalty \vskip\belowdisplayskip
 \vbox{\normalbaselines\noindent##1}%
 \penalty\predisplaypenalty \vskip\abovedisplayskip}}}
\newskip\centering@
\centering@\z@ plus\@m\p@
\def\align{\relax\ifingather@\DN@{\csname align (in
  \string\gather)\endcsname}\else
 \ifmmode\ifinner\DN@{\onlydmatherr@\align}\else
  \let\next@\align@\fi
 \else\DN@{\onlydmatherr@\align}\fi\fi\next@}
\newhelp\andhelp@
{An extra & here is so disastrous that you should probably exit^^J
and fix things up.}
\newif\iftag@
\newcount\and@
\def\align@{\inalign@true\inany@true
 \vspace@\allowdisplaybreak@\displaybreak@\intertext@
 \def\tag{\global\tag@true\ifnum\and@=\z@\DN@{&&}\else
          \DN@{&}\fi\next@}%
 \iftagsleft@\DN@{\csname align \endcsname}\else
  \DN@{\csname align \space\endcsname}\fi\next@}
\def\Tag@{\iftag@\else\errhelp\andhelp@\err@{Extra & on this line}\fi}
\newdimen\lwidth@
\newdimen\rwidth@
\newdimen\maxlwidth@
\newdimen\maxrwidth@
\newdimen\totwidth@
\def\measure@#1\endalign{\lwidth@\z@\rwidth@\z@\maxlwidth@\z@\maxrwidth@\z@
 \global\and@\z@                                                            %1
 \setbox@ne\vbox                                                            %2
  {\everycr{\noalign{\global\tag@false\global\and@\z@}}\Let@                %3
  \halign{\setboxz@h{$\m@th\displaystyle{\@lign##}$}%                       %4
   \global\lwidth@\wdz@                                                     %5
   \ifdim\lwidth@>\maxlwidth@\global\maxlwidth@\lwidth@\fi                  %6
   \global\advance\and@\@ne                                                 %7
   &\setboxz@h{$\m@th\displaystyle{{}\@lign##}$}\global\rwidth@\wdz@        %8
   \ifdim\rwidth@>\maxrwidth@\global\maxrwidth@\rwidth@\fi                  %9
   \global\advance\and@\@ne                                                %10
   &\Tag@
   \eat@{##}\crcr#1\crcr}}%                                                %11
 \totwidth@\maxlwidth@\advance\totwidth@\maxrwidth@}                       %12
\def\displ@y@{\global\dt@ptrue\openup\jot
 \everycr{\noalign{\global\tag@false\global\and@\z@\ifdt@p\global\dt@pfalse
 \vskip-\lineskiplimit\vskip\normallineskiplimit\else
 \penalty\interdisplaylinepenalty\fi}}}
\def\black@#1{\noalign{\ifdim#1>\displaywidth
 \dimen@\prevdepth\nointerlineskip                                          %1
 \vskip-\ht\strutbox@\vskip-\dp\strutbox@                                   %2
 \vbox{\noindent\hbox to#1{\strut@\hfill}}%                                 %3
 \prevdepth\dimen@                                                          %4
 \fi}}
\expandafter\def\csname align \space\endcsname#1\endalign
 {\measure@#1\endalign\global\and@\z@                                       %1
 \ifingather@\everycr{\noalign{\global\and@\z@}}\else\displ@y@\fi           %2
 \Let@\tabskip\centering@                                                   %3
 \halign to\displaywidth
  {\hfil\strut@\setboxz@h{$\m@th\displaystyle{\@lign##}$}%                  %4
  \global\lwidth@\wdz@\boxz@\global\advance\and@\@ne                        %5
  \tabskip\z@skip                                                           %6
  &\setboxz@h{$\m@th\displaystyle{{}\@lign##}$}%                            %7
  \global\rwidth@\wdz@\boxz@\hfill\global\advance\and@\@ne                  %8
  \tabskip\centering@                                                       %9
  &\setboxz@h{\@lign\strut@\maketag@##\maketag@}%                          %10
  \dimen@\displaywidth\advance\dimen@-\totwidth@
  \divide\dimen@\tw@\advance\dimen@\maxrwidth@\advance\dimen@-\rwidth@     %11
  \ifdim\dimen@<\tw@\wdz@\llap{\vtop{\normalbaselines\null\boxz@}}%        %12
  \else\llap{\boxz@}\fi                                                    %13
  \tabskip\z@skip                                                          %14
  \crcr#1\crcr                                                             %15
  \black@\totwidth@}}                                                      %16
\newdimen\lineht@
\expandafter\def\csname align \endcsname#1\endalign{\measure@#1\endalign
 \global\and@\z@
 \ifdim\totwidth@>\displaywidth\let\displaywidth@\totwidth@\else
  \let\displaywidth@\displaywidth\fi                                        %1
 \ifingather@\everycr{\noalign{\global\and@\z@}}\else\displ@y@\fi
 \Let@\tabskip\centering@\halign to\displaywidth
  {\hfil\strut@\setboxz@h{$\m@th\displaystyle{\@lign##}$}%
  \global\lwidth@\wdz@\global\lineht@\ht\z@                                 %2
  \boxz@\global\advance\and@\@ne
  \tabskip\z@skip&\setboxz@h{$\m@th\displaystyle{{}\@lign##}$}%
  \global\rwidth@\wdz@\ifdim\ht\z@>\lineht@\global\lineht@\ht\z@\fi         %3
  \boxz@\hfil\global\advance\and@\@ne
  \tabskip\centering@&\kern-\displaywidth@                                  %4
  \setboxz@h{\@lign\strut@\maketag@##\maketag@}%
  \dimen@\displaywidth\advance\dimen@-\totwidth@
  \divide\dimen@\tw@\advance\dimen@\maxlwidth@\advance\dimen@-\lwidth@
  \ifdim\dimen@<\tw@\wdz@
   \rlap{\vbox{\normalbaselines\boxz@\vbox to\lineht@{}}}\else
   \rlap{\boxz@}\fi
  \tabskip\displaywidth@\crcr#1\crcr\black@\totwidth@}}
\expandafter\def\csname align (in \string\gather)\endcsname
  #1\endalign{\vcenter{\align@#1\endalign}}
\Invalid@\endalign
\newif\ifxat@
\def\alignat{\RIfMIfI@\DN@{\onlydmatherr@\alignat}\else
 \DN@{\csname alignat \endcsname}\fi\else
 \DN@{\onlydmatherr@\alignat}\fi\next@}
\newif\ifmeasuring@
\newbox\savealignat@
\expandafter\def\csname alignat \endcsname#1#2\endalignat                   %1
 {\inany@true\xat@false
 \def\tag{\global\tag@true\count@#1\relax\multiply\count@\tw@
  \xdef\tag@{}\loop\ifnum\count@>\and@\xdef\tag@{&\tag@}\advance\count@\m@ne
  \repeat\tag@}%
 \vspace@\allowdisplaybreak@\displaybreak@\intertext@
 \displ@y@\measuring@true                                                   %2
 \setbox\savealignat@\hbox{$\m@th\displaystyle\Let@
  \attag@{#1}%                                                              %3
  \vbox{\halign{\span\preamble@@\crcr#2\crcr}}$}%
 \measuring@false                                                           %4
 \Let@\attag@{#1}%                                                          %5
 \tabskip\centering@\halign to\displaywidth
  {\span\preamble@@\crcr#2\crcr                                             %6
  \black@{\wd\savealignat@}}}                                               %7
\Invalid@\endalignat
\def\xalignat{\RIfMIfI@
 \DN@{\onlydmatherr@\xalignat}\else
 \DN@{\csname xalignat \endcsname}\fi\else
 \DN@{\onlydmatherr@\xalignat}\fi\next@}
\expandafter\def\csname xalignat \endcsname#1#2\endxalignat
 {\inany@true\xat@true
 \def\tag{\global\tag@true\def\tag@{}\count@#1\relax\multiply\count@\tw@
  \loop\ifnum\count@>\and@\xdef\tag@{&\tag@}\advance\count@\m@ne\repeat\tag@}%
 \vspace@\allowdisplaybreak@\displaybreak@\intertext@
 \displ@y@\measuring@true\setbox\savealignat@\hbox{$\m@th\displaystyle\Let@
 \attag@{#1}\vbox{\halign{\span\preamble@@\crcr#2\crcr}}$}%
 \measuring@false\Let@
 \attag@{#1}\tabskip\centering@\halign to\displaywidth
 {\span\preamble@@\crcr#2\crcr\black@{\wd\savealignat@}}}
\def\attag@#1{\let\Maketag@\maketag@\let\TAG@\Tag@                          %1
 \let\Tag@=0\let\maketag@=0%                                                %2
 \ifmeasuring@\def\llap@##1{\setboxz@h{##1}\hbox to\tw@\wdz@{}}%
  \def\rlap@##1{\setboxz@h{##1}\hbox to\tw@\wdz@{}}\else
  \let\llap@\llap\let\rlap@\rlap\fi                                         %3
 \toks@{\hfil\strut@$\m@th\displaystyle{\@lign\the\hashtoks@}$\tabskip\z@skip
  \global\advance\and@\@ne&$\m@th\displaystyle{{}\@lign\the\hashtoks@}$\hfil
  \ifxat@\tabskip\centering@\fi\global\advance\and@\@ne}%                   %4
 \iftagsleft@
  \toks@@{\tabskip\centering@&\Tag@\kern-\displaywidth
   \rlap@{\@lign\maketag@\the\hashtoks@\maketag@}%
   \global\advance\and@\@ne\tabskip\displaywidth}\else
  \toks@@{\tabskip\centering@&\Tag@\llap@{\@lign\maketag@
   \the\hashtoks@\maketag@}\global\advance\and@\@ne\tabskip\z@skip}\fi      %5
 \atcount@#1\relax\advance\atcount@\m@ne
 \loop\ifnum\atcount@>\z@
 \toks@=\expandafter{\the\toks@&\hfil$\m@th\displaystyle{\@lign
  \the\hashtoks@}$\global\advance\and@\@ne
  \tabskip\z@skip&$\m@th\displaystyle{{}\@lign\the\hashtoks@}$\hfil\ifxat@
  \tabskip\centering@\fi\global\advance\and@\@ne}\advance\atcount@\m@ne
 \repeat                                                                    %6
 \xdef\preamble@{\the\toks@\the\toks@@}%                                    %7
 \xdef\preamble@@{\preamble@}%                                              %8
 \let\maketag@\Maketag@\let\Tag@\TAG@}                                      %9
\Invalid@\endxalignat
\def\xxalignat{\RIfMIfI@
 \DN@{\onlydmatherr@\xxalignat}\else\DN@{\csname xxalignat
  \endcsname}\fi\else
 \DN@{\onlydmatherr@\xxalignat}\fi\next@}
\expandafter\def\csname xxalignat \endcsname#1#2\endxxalignat{\inany@true
 \vspace@\allowdisplaybreak@\displaybreak@\intertext@
 \displ@y\setbox\savealignat@\hbox{$\m@th\displaystyle\Let@
 \xxattag@{#1}\vbox{\halign{\span\preamble@@\crcr#2\crcr}}$}%
 \Let@\xxattag@{#1}\tabskip\z@skip\halign to\displaywidth
 {\span\preamble@@\crcr#2\crcr\black@{\wd\savealignat@}}}
\def\xxattag@#1{\toks@{\tabskip\z@skip\hfil\strut@
 $\m@th\displaystyle{\the\hashtoks@}$&%
 $\m@th\displaystyle{{}\the\hashtoks@}$\hfil\tabskip\centering@&}%
 \atcount@#1\relax\advance\atcount@\m@ne\loop\ifnum\atcount@>\z@
 \toks@=\expandafter{\the\toks@&\hfil$\m@th\displaystyle{\the\hashtoks@}$%
  \tabskip\z@skip&$\m@th\displaystyle{{}\the\hashtoks@}$\hfil
  \tabskip\centering@}\advance\atcount@\m@ne\repeat
 \xdef\preamble@{\the\toks@\tabskip\z@skip}\xdef\preamble@@{\preamble@}}
\Invalid@\endxxalignat
\newdimen\gwidth@
\newdimen\gmaxwidth@
\def\gmeasure@#1\endgather{\gwidth@\z@\gmaxwidth@\z@\setbox@ne\vbox{\Let@
 \halign{\setboxz@h{$\m@th\displaystyle{##}$}\global\gwidth@\wdz@
 \ifdim\gwidth@>\gmaxwidth@\global\gmaxwidth@\gwidth@\fi
 &\eat@{##}\crcr#1\crcr}}}
\def\gather{\RIfMIfI@\DN@{\onlydmatherr@\gather}\else
 \ingather@true\inany@true\def\tag{&}%
 \vspace@\allowdisplaybreak@\displaybreak@\intertext@
 \displ@y\Let@
 \iftagsleft@\DN@{\csname gather \endcsname}\else
  \DN@{\csname gather \space\endcsname}\fi\fi
 \else\DN@{\onlydmatherr@\gather}\fi\next@}
\expandafter\def\csname gather \space\endcsname#1\endgather
 {\gmeasure@#1\endgather\tabskip\centering@
 \halign to\displaywidth{\hfil\strut@\setboxz@h{$\m@th\displaystyle{##}$}%
 \global\gwidth@\wdz@\boxz@\hfil&
 \setboxz@h{\strut@{\maketag@##\maketag@}}%
 \dimen@\displaywidth\advance\dimen@-\gwidth@
 \ifdim\dimen@>\tw@\wdz@\llap{\boxz@}\else
 \llap{\vtop{\normalbaselines\null\boxz@}}\fi
 \tabskip\z@skip\crcr#1\crcr\black@\gmaxwidth@}}
\newdimen\glineht@
\expandafter\def\csname gather \endcsname#1\endgather{\gmeasure@#1\endgather
 \ifdim\gmaxwidth@>\displaywidth\let\gdisplaywidth@\gmaxwidth@\else
 \let\gdisplaywidth@\displaywidth\fi\tabskip\centering@\halign to\displaywidth
 {\hfil\strut@\setboxz@h{$\m@th\displaystyle{##}$}%
 \global\gwidth@\wdz@\global\glineht@\ht\z@\boxz@\hfil&\kern-\gdisplaywidth@
 \setboxz@h{\strut@{\maketag@##\maketag@}}%
 \dimen@\displaywidth\advance\dimen@-\gwidth@
 \ifdim\dimen@>\tw@\wdz@\rlap{\boxz@}\else
 \rlap{\vbox{\normalbaselines\boxz@\vbox to\glineht@{}}}\fi
 \tabskip\gdisplaywidth@\crcr#1\crcr\black@\gmaxwidth@}}
\newif\ifctagsplit@
\def\CenteredTagsOnSplits{\global\ctagsplit@true}
\def\TopOrBottomTagsOnSplits{\global\ctagsplit@false}
\TopOrBottomTagsOnSplits
\def\split{\relax\ifinany@\let\next@\insplit@\else
 \ifmmode\ifinner\def\next@{\onlydmatherr@\split}\else
 \let\next@\outsplit@\fi\else
 \def\next@{\onlydmatherr@\split}\fi\fi\next@}
\def\insplit@{\global\setbox\z@\vbox\bgroup\vspace@\Let@\ialign\bgroup
 \hfil\strut@$\m@th\displaystyle{##}$&$\m@th\displaystyle{{}##}$\hfill\crcr}
\def\endsplit{\crcr\egroup\egroup\iftagsleft@\expandafter\lendsplit@\else
 \expandafter\rendsplit@\fi}
\def\rendsplit@{\global\setbox9 \vbox
 {\unvcopy\z@\global\setbox8 \lastbox\unskip}%                              %1
 \setbox@ne\hbox{\unhcopy8 \unskip\global\setbox\tw@\lastbox
 \unskip\global\setbox\thr@@\lastbox}%                                      %2
 \global\setbox7 \hbox{\unhbox\tw@\unskip}%                                 %3
 \ifinalign@\ifctagsplit@                                                   %4
  \gdef\split@{\hbox to\wd\thr@@{}&
   \vcenter{\vbox{\moveleft\wd\thr@@\boxz@}}}%                              %5
 \else\gdef\split@{&\vbox{\moveleft\wd\thr@@\box9}\crcr
  \box\thr@@&\box7}\fi                                                      %6
 \else                                                                      %7
  \ifctagsplit@\gdef\split@{\vcenter{\boxz@}}\else
  \gdef\split@{\box9\crcr\hbox{\box\thr@@\box7}}\fi
 \fi
 \split@}                                                                   %8
\def\lendsplit@{\global\setbox9\vtop{\unvcopy\z@}%                          %1
 \setbox@ne\vbox{\unvcopy\z@\global\setbox8\lastbox}%                       %2
 \setbox@ne\hbox{\unhcopy8\unskip\setbox\tw@\lastbox
  \unskip\global\setbox\thr@@\lastbox}%                                     %3
 \ifinalign@\ifctagsplit@                                                   %4
  \gdef\split@{\hbox to\wd\thr@@{}&
  \vcenter{\vbox{\moveleft\wd\thr@@\box9}}}%                                %5
  \else                                                                     %6
  \gdef\split@{\hbox to\wd\thr@@{}&\vbox{\moveleft\wd\thr@@\box9}}\fi
 \else
  \ifctagsplit@\gdef\split@{\vcenter{\box9}}\else
  \gdef\split@{\box9}\fi
 \fi\split@}
\def\outsplit@#1$${\align\insplit@#1\endalign$$}
\newdimen\multlinegap@
\multlinegap@1em
\newdimen\multlinetaggap@
\multlinetaggap@1em
\def\MultlineGap#1{\global\multlinegap@#1\relax}
\def\multlinegap#1{\RIfMIfI@\onlydmatherr@\multlinegap\else
 \multlinegap@#1\relax\fi\else\onlydmatherr@\multlinegap\fi}
\def\nomultlinegap{\multlinegap{\z@}}
\def\multline{\RIfMIfI@
 \DN@{\onlydmatherr@\multline}\else
 \DN@{\multline@}\fi\else
 \DN@{\onlydmatherr@\multline}\fi\next@}
\newif\iftagin@
\def\tagin@#1{\tagin@false\in@\tag{#1}\ifin@\tagin@true\fi}
\def\multline@#1$${\inany@true\vspace@\allowdisplaybreak@\displaybreak@
 \tagin@{#1}\iftagsleft@\DN@{\multline@l#1$$}\else
 \DN@{\multline@r#1$$}\fi\next@}
\newdimen\mwidth@
\def\rmmeasure@#1\endmultline{%
 \def\shoveleft##1{##1}\def\shoveright##1{##1}%                             %1
 \setbox@ne\vbox{\Let@\halign{\setboxz@h
  {$\m@th\@lign\displaystyle{}##$}\global\mwidth@\wdz@
  \crcr#1\crcr}}}
\newdimen\mlineht@
\newif\ifzerocr@
\newif\ifonecr@
\def\lmmeasure@#1\endmultline{\global\zerocr@true\global\onecr@false
 \everycr{\noalign{\ifonecr@\global\onecr@false\fi
  \ifzerocr@\global\zerocr@false\global\onecr@true\fi}}%                    %1
  \def\shoveleft##1{##1}\def\shoveright##1{##1}%
 \setbox@ne\vbox{\Let@\halign{\setboxz@h
  {$\m@th\@lign\displaystyle{}##$}\ifonecr@\global\mwidth@\wdz@
  \global\mlineht@\ht\z@\fi\crcr#1\crcr}}}
\newbox\mtagbox@
\newdimen\ltwidth@
\newdimen\rtwidth@
\def\multline@l#1$${\iftagin@\DN@{\lmultline@@#1$$}\else
 \DN@{\setbox\mtagbox@\null\ltwidth@\z@\rtwidth@\z@
  \lmultline@@@#1$$}\fi\next@}
\def\lmultline@@#1\endmultline\tag#2$${%
 \setbox\mtagbox@\hbox{\maketag@#2\maketag@}%                               %1
 \lmmeasure@#1\endmultline\dimen@\mwidth@\advance\dimen@\wd\mtagbox@
 \advance\dimen@\multlinetaggap@                                            %2
 \ifdim\dimen@>\displaywidth\ltwidth@\z@\else\ltwidth@\wd\mtagbox@\fi       %3
 \lmultline@@@#1\endmultline$$}
\def\lmultline@@@{\displ@y
 \def\shoveright##1{##1\hfilneg\hskip\multlinegap@}%
 \def\shoveleft##1{\setboxz@h{$\m@th\displaystyle{}##1$}%
  \setbox@ne\hbox{$\m@th\displaystyle##1$}%
  \hfilneg
  \iftagin@
   \ifdim\ltwidth@>\z@\hskip\ltwidth@\hskip\multlinetaggap@\fi
  \else\hskip\multlinegap@\fi\hskip.5\wd@ne\hskip-.5\wdz@##1}%              %1
  \halign\bgroup\Let@\hbox to\displaywidth
   {\strut@$\m@th\displaystyle\hfil{}##\hfil$}\crcr
   \hfilneg                                                                 %2
   \iftagin@                                                                %3
    \ifdim\ltwidth@>\z@                                                     %4
     \box\mtagbox@\hskip\multlinetaggap@                                    %5
    \else
     \rlap{\vbox{\normalbaselines\hbox{\strut@\box\mtagbox@}%
     \vbox to\mlineht@{}}}\fi                                               %6
   \else\hskip\multlinegap@\fi}                                             %7
\def\multline@r#1$${\iftagin@\DN@{\rmultline@@#1$$}\else
 \DN@{\setbox\mtagbox@\null\ltwidth@\z@\rtwidth@\z@
  \rmultline@@@#1$$}\fi\next@}
\def\rmultline@@#1\endmultline\tag#2$${\ltwidth@\z@
 \setbox\mtagbox@\hbox{\maketag@#2\maketag@}%
 \rmmeasure@#1\endmultline\dimen@\mwidth@\advance\dimen@\wd\mtagbox@
 \advance\dimen@\multlinetaggap@
 \ifdim\dimen@>\displaywidth\rtwidth@\z@\else\rtwidth@\wd\mtagbox@\fi
 \rmultline@@@#1\endmultline$$}
\def\rmultline@@@{\displ@y
 \def\shoveright##1{##1\hfilneg\iftagin@\ifdim\rtwidth@>\z@
  \hskip\rtwidth@\hskip\multlinetaggap@\fi\else\hskip\multlinegap@\fi}%
 \def\shoveleft##1{\setboxz@h{$\m@th\displaystyle{}##1$}%
  \setbox@ne\hbox{$\m@th\displaystyle##1$}%
  \hfilneg\hskip\multlinegap@\hskip.5\wd@ne\hskip-.5\wdz@##1}%
 \halign\bgroup\Let@\hbox to\displaywidth
  {\strut@$\m@th\displaystyle\hfil{}##\hfil$}\crcr
 \hfilneg\hskip\multlinegap@}
\def\endmultline{\iftagsleft@\expandafter\lendmultline@\else
 \expandafter\rendmultline@\fi}
\def\lendmultline@{\hfilneg\hskip\multlinegap@\crcr\egroup}
\def\rendmultline@{\iftagin@                                                %1
 \ifdim\rtwidth@>\z@                                                        %2
  \hskip\multlinetaggap@\box\mtagbox@                                       %3
 \else\llap{\vtop{\normalbaselines\null\hbox{\strut@\box\mtagbox@}}}\fi     %4
 \else\hskip\multlinegap@\fi                                                %5
 \hfilneg\crcr\egroup}
\def\bmod{\mskip-\medmuskip\mkern5mu\mathbin{\fam\z@ mod}\penalty900
 \mkern5mu\mskip-\medmuskip}
\def\pmod#1{\allowbreak\ifinner\mkern8mu\else\mkern18mu\fi
 ({\fam\z@ mod}\,\,#1)}
\def\pod#1{\allowbreak\ifinner\mkern8mu\else\mkern18mu\fi(#1)}
\def\mod#1{\allowbreak\ifinner\mkern12mu\else\mkern18mu\fi{\fam\z@ mod}\,\,#1}
\message{continued fractions,}
\newcount\cfraccount@
\def\cfrac{\bgroup\bgroup\advance\cfraccount@\@ne\strut
 \iffalse{\fi\def\\{\over\displaystyle}\iffalse}\fi}
\def\lcfrac{\bgroup\bgroup\advance\cfraccount@\@ne\strut
 \iffalse{\fi\def\\{\hfill\over\displaystyle}\iffalse}\fi}
\def\rcfrac{\bgroup\bgroup\advance\cfraccount@\@ne\strut\hfill
 \iffalse{\fi\def\\{\over\displaystyle}\iffalse}\fi}
\def\gloop@#1\repeat{\gdef\body{#1}\iterate}
\def\endcfrac{\gloop@\ifnum\cfraccount@>\z@\global\advance\cfraccount@\m@ne
 \egroup\hskip-\nulldelimiterspace\egroup\repeat}
\message{compound symbols,}
\def\binrel@#1{\setboxz@h{\thinmuskip0mu
  \medmuskip\m@ne mu\thickmuskip\@ne mu$#1\m@th$}%
 \setbox@ne\hbox{\thinmuskip0mu\medmuskip\m@ne mu\thickmuskip
  \@ne mu${}#1{}\m@th$}%
 \setbox\tw@\hbox{\hskip\wd@ne\hskip-\wdz@}}
\def\overset#1\to#2{\binrel@{#2}\ifdim\wd\tw@<\z@
 \mathbin{\mathop{\kern\z@#2}\limits^{#1}}\else\ifdim\wd\tw@>\z@
 \mathrel{\mathop{\kern\z@#2}\limits^{#1}}\else
 {\mathop{\kern\z@#2}\limits^{#1}}{}\fi\fi}
\def\underset#1\to#2{\binrel@{#2}\ifdim\wd\tw@<\z@
 \mathbin{\mathop{\kern\z@#2}\limits_{#1}}\else\ifdim\wd\tw@>\z@
 \mathrel{\mathop{\kern\z@#2}\limits_{#1}}\else
 {\mathop{\kern\z@#2}\limits_{#1}}{}\fi\fi}
\def\oversetbrace#1\to#2{\overbrace{#2}^{#1}}
\def\undersetbrace#1\to#2{\underbrace{#2}_{#1}}
\def\sideset#1\and#2\to#3{%
 \setbox@ne\hbox{$\dsize{\vphantom{#3}}#1{#3}\m@th$}%
 \setbox\tw@\hbox{$\dsize{#3}#2\m@th$}%
 \hskip\wd@ne\hskip-\wd\tw@\mathop{\hskip\wd\tw@\hskip-\wd@ne
  {\vphantom{#3}}#1{#3}#2}}
\def\rightarrowfill@#1{\setboxz@h{$#1-\m@th$}\ht\z@\z@
  $#1\m@th\copy\z@\mkern-6mu\cleaders
  \hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
  \mkern-6mu\mathord\rightarrow$}
\def\leftarrowfill@#1{\setboxz@h{$#1-\m@th$}\ht\z@\z@
  $#1\m@th\mathord\leftarrow\mkern-6mu\cleaders
  \hbox{$#1\mkern-2mu\copy\z@\mkern-2mu$}\hfill
  \mkern-6mu\box\z@$}
\def\leftrightarrowfill@#1{\setboxz@h{$#1-\m@th$}\ht\z@\z@
  $#1\m@th\mathord\leftarrow\mkern-6mu\cleaders
  \hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
  \mkern-6mu\mathord\rightarrow$}
\def\overrightarrow{\mathpalette\overrightarrow@}
\def\overrightarrow@#1#2{\vbox{\ialign{##\crcr\rightarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}
\let\overarrow\overrightarrow
\def\overleftarrow{\mathpalette\overleftarrow@}
\def\overleftarrow@#1#2{\vbox{\ialign{##\crcr\leftarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}
\def\overleftrightarrow{\mathpalette\overleftrightarrow@}
\def\overleftrightarrow@#1#2{\vbox{\ialign{##\crcr\leftrightarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}
\def\underrightarrow{\mathpalette\underrightarrow@}
\def\underrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil$\crcr
 \noalign{\nointerlineskip}\rightarrowfill@#1\crcr}}}
\let\underarrow\underrightarrow
\def\underleftarrow{\mathpalette\underleftarrow@}
\def\underleftarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil$\crcr
 \noalign{\nointerlineskip}\leftarrowfill@#1\crcr}}}
\def\underleftrightarrow{\mathpalette\underleftrightarrow@}
\def\underleftrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil$\crcr
 \noalign{\nointerlineskip}\leftrightarrowfill@#1\crcr}}}
\message{various kinds of dots,}
\let\DOTSI\relax
\let\DOTSB\relax
\let\DOTSX\relax
\newif\ifmath@
{\uccode`7=`\\ \uccode`8=`m \uccode`9=`a \uccode`0=`t \uccode`!=`h
 \uppercase{\gdef\math@#1#2#3#4#5#6\math@{\global\math@false\ifx 7#1\ifx 8#2%
 \ifx 9#3\ifx 0#4\ifx !#5\xdef\meaning@{#6}\global\math@true\fi\fi\fi\fi\fi}}}
\newif\ifmathch@
{\uccode`7=`c \uccode`8=`h \uccode`9=`\"
 \uppercase{\gdef\mathch@#1#2#3#4#5#6\mathch@{\global\mathch@false
  \ifx 7#1\ifx 8#2\ifx 9#5\global\mathch@true\xdef\meaning@{9#6}\fi\fi\fi}}}
\newcount\classnum@
\def\getmathch@#1.#2\getmathch@{\classnum@#1 \divide\classnum@4096
 \ifcase\number\classnum@\or\or\gdef\thedots@{\dotsb@}\or
 \gdef\thedots@{\dotsb@}\fi}
\newif\ifmathbin@
{\uccode`4=`b \uccode`5=`i \uccode`6=`n
 \uppercase{\gdef\mathbin@#1#2#3{\relaxnext@
  \DNii@##1\mathbin@{\ifx\space@\next\global\mathbin@true\fi}%
 \global\mathbin@false\DN@##1\mathbin@{}%
 \ifx 4#1\ifx 5#2\ifx 6#3\DN@{\FN@\nextii@}\fi\fi\fi\next@}}}
\newif\ifmathrel@
{\uccode`4=`r \uccode`5=`e \uccode`6=`l
 \uppercase{\gdef\mathrel@#1#2#3{\relaxnext@
  \DNii@##1\mathrel@{\ifx\space@\next\global\mathrel@true\fi}%
 \global\mathrel@false\DN@##1\mathrel@{}%
 \ifx 4#1\ifx 5#2\ifx 6#3\DN@{\FN@\nextii@}\fi\fi\fi\next@}}}
\newif\ifmacro@
{\uccode`5=`m \uccode`6=`a \uccode`7=`c
 \uppercase{\gdef\macro@#1#2#3#4\macro@{\global\macro@false
  \ifx 5#1\ifx 6#2\ifx 7#3\global\macro@true
  \xdef\meaning@{\macro@@#4\macro@@}\fi\fi\fi}}}
\def\macro@@#1->#2\macro@@{#2}
\newif\ifDOTS@
\newcount\DOTSCASE@
{\uccode`6=`\\ \uccode`7=`D \uccode`8=`O \uccode`9=`T \uccode`0=`S
 \uppercase{\gdef\DOTS@#1#2#3#4#5{\global\DOTS@false\DN@##1\DOTS@{}%
  \ifx 6#1\ifx 7#2\ifx 8#3\ifx 9#4\ifx 0#5\let\next@\DOTS@@\fi\fi\fi\fi\fi
  \next@}}}
{\uccode`3=`B \uccode`4=`I \uccode`5=`X
 \uppercase{\gdef\DOTS@@#1{\relaxnext@
  \DNii@##1\DOTS@{\ifx\space@\next\global\DOTS@true\fi}%
  \DN@{\FN@\nextii@}%
  \ifx 3#1\global\DOTSCASE@\z@\else
  \ifx 4#1\global\DOTSCASE@\@ne\else
  \ifx 5#1\global\DOTSCASE@\tw@\else\DN@##1\DOTS@{}%
  \fi\fi\fi\next@}}}
\newif\ifnot@
{\uccode`5=`\\ \uccode`6=`n \uccode`7=`o \uccode`8=`t
 \uppercase{\gdef\not@#1#2#3#4{\relaxnext@
  \DNii@##1\not@{\ifx\space@\next\global\not@true\fi}%
 \global\not@false\DN@##1\not@{}%
 \ifx 5#1\ifx 6#2\ifx 7#3\ifx 8#4\DN@{\FN@\nextii@}\fi\fi\fi
 \fi\next@}}}
\newif\ifkeybin@
\def\keybin@{\keybin@true
 \ifx\next+\else\ifx\next=\else\ifx\next<\else\ifx\next>\else\ifx\next-\else
 \ifx\next*\else\ifx\next:\else\keybin@false\fi\fi\fi\fi\fi\fi\fi}
\def\dots{\RIfM@\expandafter\mdots@\else\expandafter\tdots@\fi}
\def\tdots@{\unskip\relaxnext@
 \DN@{$\m@th\mathinner{\ldotp\ldotp\ldotp}\,
   \ifx\next,\,$\else\ifx\next.\,$\else\ifx\next;\,$\else\ifx\next:\,$\else
   \ifx\next?\,$\else\ifx\next!\,$\else$ \fi\fi\fi\fi\fi\fi}%
 \ \FN@\next@}
\def\mdots@{\FN@\mdots@@}
\def\mdots@@{\gdef\thedots@{\dotso@}%                                       %1
 \ifx\next\boldkey\gdef\thedots@\boldkey{\boldkeydots@}\else                %2
 \ifx\next\boldsymbol\gdef\thedots@\boldsymbol{\boldsymboldots@}\else       %3
 \ifx,\next\gdef\thedots@{\dotsc}%                                          %4
 \else\ifx\not\next\gdef\thedots@{\dotsb@}%                                 %5
 \else\keybin@
 \ifkeybin@\gdef\thedots@{\dotsb@}%                                         %6
 \else\xdef\meaning@{\meaning\next..........}\xdef\meaning@@{\meaning@}%    %7
  \expandafter\math@\meaning@\math@
  \ifmath@
   \expandafter\mathch@\meaning@\mathch@
   \ifmathch@\expandafter\getmathch@\meaning@\getmathch@\fi                 %8
  \else\expandafter\macro@\meaning@@\macro@                                 %9
  \ifmacro@                                                                %10
   \expandafter\not@\meaning@\not@\ifnot@\gdef\thedots@{\dotsb@}%          %11
  \else\expandafter\DOTS@\meaning@\DOTS@
  \ifDOTS@
   \ifcase\number\DOTSCASE@\gdef\thedots@{\dotsb@}%
    \or\gdef\thedots@{\dotsi}\else\fi                                      %12
  \else\expandafter\math@\meaning@\math@                                   %13
  \ifmath@\expandafter\mathbin@\meaning@\mathbin@
  \ifmathbin@\gdef\thedots@{\dotsb@}%                                      %14
  \else\expandafter\mathrel@\meaning@\mathrel@
  \ifmathrel@\gdef\thedots@{\dotsb@}%                                      %15
  \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi
 \thedots@}
\def\plainldots@{\mathinner{\ldotp\ldotp\ldotp}}
\def\plaincdots@{\mathinner{\cdotp\cdotp\cdotp}}
\def\dotsi{\!\plaincdots@}
\let\dotsb@\plaincdots@
\newif\ifextra@
\newif\ifrightdelim@
\def\rightdelim@{\global\rightdelim@true                                    %1
 \ifx\next)\else                                                            %2
 \ifx\next]\else
 \ifx\next\rbrack\else
 \ifx\next\}\else
 \ifx\next\rbrace\else
 \ifx\next\rangle\else
 \ifx\next\rceil\else
 \ifx\next\rfloor\else
 \ifx\next\rgroup\else
 \ifx\next\rmoustache\else
 \ifx\next\right\else
 \ifx\next\bigr\else
 \ifx\next\biggr\else
 \ifx\next\Bigr\else                                                        %3
 \ifx\next\Biggr\else\global\rightdelim@false
 \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}
\def\extra@{%
 \global\extra@false\rightdelim@\ifrightdelim@\global\extra@true            %1
 \else\ifx\next$\global\extra@true                                          %2
 \else\xdef\meaning@{\meaning\next..........}%                              %3
 \expandafter\macro@\meaning@\macro@\ifmacro@                               %4
 \expandafter\DOTS@\meaning@\DOTS@
 \ifDOTS@
 \ifnum\DOTSCASE@=\tw@\global\extra@true                                    %5
 \fi\fi\fi\fi\fi}
\newif\ifbold@
\def\dotso@{\relaxnext@
 \ifbold@
  \let\next\delayed@
  \DNii@{\extra@\plainldots@\ifextra@\,\fi}%
 \else
  \DNii@{\DN@{\extra@\plainldots@\ifextra@\,\fi}\FN@\next@}%
 \fi
 \nextii@}
\def\extrap@#1{%
 \ifx\next,\DN@{#1\,}\else
 \ifx\next;\DN@{#1\,}\else
 \ifx\next.\DN@{#1\,}\else\extra@
 \ifextra@\DN@{#1\,}\else
 \let\next@#1\fi\fi\fi\fi\next@}
\def\ldots{\DN@{\extrap@\plainldots@}%
 \FN@\next@}
\def\cdots{\DN@{\extrap@\plaincdots@}%
 \FN@\next@}
\let\dotso\ldots
\let\dotsb\cdots
\let\dotsm\dotsb
\def\dotsc{\relaxnext@
 \DN@{\ifx\next;\plainldots@\,\else
  \ifx\next.\plainldots@\,\else\extra@\plainldots@
  \ifextra@\,\fi\fi\fi}%
 \FN@\next@}
\def\cdot{\mathchar"2201 }
\def\longrightarrow{\DOTSB\relbar\joinrel\rightarrow}
\def\Longrightarrow{\DOTSB\Relbar\joinrel\Rightarrow}
\def\longleftarrow{\DOTSB\leftarrow\joinrel\relbar}
\def\Longleftarrow{\DOTSB\Leftarrow\joinrel\Relbar}
\def\longleftrightarrow{\DOTSB\leftarrow\joinrel\rightarrow}
\def\Longleftrightarrow{\DOTSB\Leftarrow\joinrel\Rightarrow}
\def\mapsto{\DOTSB\mapstochar\rightarrow}
\def\longmapsto{\DOTSB\mapstochar\longrightarrow}
\def\hookrightarrow{\DOTSB\lhook\joinrel\rightarrow}
\def\hookleftarrow{\DOTSB\leftarrow\joinrel\rhook}
\def\doteq{\DOTSB\buildrel\textstyle.\over=}
\def\iff{\DOTSB\;\Longleftrightarrow\;}
\message{special superscripts,}
\def\dddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@
 \hbox{\rm...}\vss}}}}
\def\ddddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@
 \hbox{\rm....}\vss}}}}
\def\sphat{^{\mathchoice{}{}%
 {\,\,\botsmash{\hbox{\lower4\ex@\hbox{$\m@th\widehat{\null}$}}}}%
 {\,\botsmash{\hbox{\lower3\ex@\hbox{$\m@th\hat{\null}$}}}}}}
\def\spcheck{^\vee}
\def\sptilde{^\sim}
\def\spacute{^{\!\botsmash{\hbox{\lower\@ne ex\hbox{\'{}}}}}}
\def\spgrave{^{\mathchoice{}{}{}{\!}%
 \botsmash{\hbox{\lower\@ne ex\hbox{\`{}}}}}}
\def\spdot{^{\hbox{\raise\ex@\hbox{\rm.}}}}
\def\spddot{^{\hbox{\raise\ex@\hbox{\rm..}}}}
\def\spdddot{^{\hbox{\raise\ex@\hbox{\rm...}}}}
\def\spddddot{^{\hbox{\raise\ex@\hbox{\rm....}}}}
\def\spbreve{^{\!\botsmash{\hbox{\lower4\ex@\hbox{\u{}}}}}}
\def\spbar{^-}
\def\spvec{^\rightarrow}
\message{\string\text,}
\def\textonlyfont@#1#2{\def#1{\RIfM@
 \Err@{Use \string#1\space only in text}\else#2\fi}}
\textonlyfont@\rm\tenrm
\textonlyfont@\it\tenit
\textonlyfont@\sl\tensl
\textonlyfont@\bf\tenbf
\def\oldnos#1{\RIfM@{\mathcode`\,="013B \fam\@ne#1}\else
 \leavevmode\hbox{$\m@th\mathcode`\,="013B \fam\@ne#1$}\fi}
\def\text{\RIfM@\expandafter\text@\else\expandafter\text@@\fi}
\def\text@@#1{\leavevmode\hbox{#1}}
\def\mathhexbox@#1#2#3{\text{$\m@th\mathchar"#1#2#3$}}
\def\dag{{\mathhexbox@279}}
\def\ddag{{\mathhexbox@27A}}
\def\S{{\mathhexbox@278}}
\def\P{{\mathhexbox@27B}}
\newif\iffirstchoice@
\firstchoice@true
\def\text@#1{\mathchoice
 {\hbox{\everymath{\displaystyle}\def\textfonti{\the\textfont\@ne}%
  \def\textfontii{\the\textfont\tw@}\textdef@@ T#1}}
 {\hbox{\firstchoice@false
  \everymath{\textstyle}\def\textfonti{\the\textfont\@ne}%
  \def\textfontii{\the\textfont\tw@}\textdef@@ T#1}}
 {\hbox{\firstchoice@false
  \everymath{\scriptstyle}\def\textfonti{\the\scriptfont\@ne}%
  \def\textfontii{\the\scriptfont\tw@}\textdef@@ S\rm#1}}
 {\hbox{\firstchoice@false
  \everymath{\scriptscriptstyle}\def\textfonti
  {\the\scriptscriptfont\@ne}%
  \def\textfontii{\the\scriptscriptfont\tw@}\textdef@@ s\rm#1}}}
\def\textdef@@#1{\textdef@#1\rm\textdef@#1\bf\textdef@#1\sl\textdef@#1\it}
\def\rmfam{0}
\def\textdef@#1#2{%
 \DN@{\csname\expandafter\eat@\string#2fam\endcsname}%
 \if S#1\edef#2{\the\scriptfont\next@\relax}%
 \else\if s#1\edef#2{\the\scriptscriptfont\next@\relax}%
 \else\edef#2{\the\textfont\next@\relax}\fi\fi}
\scriptfont\itfam\tenit \scriptscriptfont\itfam\tenit
\scriptfont\slfam\tensl \scriptscriptfont\slfam\tensl
\newif\iftopfolded@
\newif\ifbotfolded@
\def\topfoldedtext{\topfolded@true\botfolded@false\foldedtext@}
\def\botfoldedtext{\botfolded@true\topfolded@false\foldedtext@}
\def\foldedtext{\topfolded@false\botfolded@false\foldedtext@}
\Invalid@\foldedwidth
\def\foldedtext@{\relaxnext@
 \DN@{\ifx\next\foldedwidth\let\next@\nextii@\else
  \DN@{\nextii@\foldedwidth{.3\hsize}}\fi\next@}%
 \DNii@\foldedwidth##1##2{\setbox\z@\vbox
  {\normalbaselines\hsize##1\relax
  \tolerance1600 \noindent\ignorespaces##2}\ifbotfolded@\boxz@\else
  \iftopfolded@\vtop{\unvbox\z@}\else\vcenter{\boxz@}\fi\fi}%
 \FN@\next@}
\message{math font commands,}
\def\bold{\RIfM@\expandafter\bold@\else
 \expandafter\nonmatherr@\expandafter\bold\fi}
\def\bold@#1{{\bold@@{#1}}}
\def\bold@@#1{\fam\bffam\relax#1}
\def\slanted{\RIfM@\expandafter\slanted@\else
 \expandafter\nonmatherr@\expandafter\slanted\fi}
\def\slanted@#1{{\slanted@@{#1}}}
\def\slanted@@#1{\fam\slfam\relax#1}
\def\roman{\RIfM@\expandafter\roman@\else
 \expandafter\nonmatherr@\expandafter\roman\fi}
\def\roman@#1{{\roman@@{#1}}}
\def\roman@@#1{\fam\rmfam\relax#1}
\def\italic{\RIfM@\expandafter\italic@\else
 \expandafter\nonmatherr@\expandafter\italic\fi}
\def\italic@#1{{\italic@@{#1}}}
\def\italic@@#1{\fam\itfam\relax#1}
\def\Cal{\RIfM@\expandafter\Cal@\else
 \expandafter\nonmatherr@\expandafter\Cal\fi}
\def\Cal@#1{{\Cal@@{#1}}}
\def\Cal@@#1{\noaccents@\fam\tw@#1}
\mathchardef\Gamma="0000
\mathchardef\Delta="0001
\mathchardef\Theta="0002
\mathchardef\Lambda="0003
\mathchardef\Xi="0004
\mathchardef\Pi="0005
\mathchardef\Sigma="0006
\mathchardef\Upsilon="0007
\mathchardef\Phi="0008
\mathchardef\Psi="0009
\mathchardef\Omega="000A
\mathchardef\varGamma="0100
\mathchardef\varDelta="0101
\mathchardef\varTheta="0102
\mathchardef\varLambda="0103
\mathchardef\varXi="0104
\mathchardef\varPi="0105
\mathchardef\varSigma="0106
\mathchardef\varUpsilon="0107
\mathchardef\varPhi="0108
\mathchardef\varPsi="0109
\mathchardef\varOmega="010A
\let\alloc@@\alloc@
\def\hexnumber@#1{\ifcase#1 0\or 1\or 2\or 3\or 4\or 5\or 6\or 7\or 8\or
 9\or A\or B\or C\or D\or E\or F\fi}
\def\loadmsam{%
 \font@\tenmsa=msam10
 \font@\sevenmsa=msam7
 \font@\fivemsa=msam5
 \alloc@@8\fam\chardef\sixt@@n\msafam
 \textfont\msafam=\tenmsa
 \scriptfont\msafam=\sevenmsa
 \scriptscriptfont\msafam=\fivemsa
 \edef\next{\hexnumber@\msafam}%
 \mathchardef\dabar@"0\next39
 \edef\dashrightarrow{\mathrel{\dabar@\dabar@\mathchar"0\next4B}}%
 \edef\dashleftarrow{\mathrel{\mathchar"0\next4C\dabar@\dabar@}}%
 \let\dasharrow\dashrightarrow
 \edef\ulcorner{\delimiter"4\next70\next70 }%
 \edef\urcorner{\delimiter"5\next71\next71 }%
 \edef\llcorner{\delimiter"4\next78\next78 }%
 \edef\lrcorner{\delimiter"5\next79\next79 }%
 \edef\yen{{\noexpand\mathhexbox@\next55}}%
 \edef\checkmark{{\noexpand\mathhexbox@\next58}}%
 \edef\circledR{{\noexpand\mathhexbox@\next72}}%
 \edef\maltese{{\noexpand\mathhexbox@\next7A}}%
 \global\let\loadmsam\empty}%
\def\loadmsbm{%
 \font@\tenmsb=msbm10 \font@\sevenmsb=msbm7 \font@\fivemsb=msbm5
 \alloc@@8\fam\chardef\sixt@@n\msbfam
 \textfont\msbfam=\tenmsb
 \scriptfont\msbfam=\sevenmsb \scriptscriptfont\msbfam=\fivemsb
 \global\let\loadmsbm\empty
 }
\def\widehat#1{\ifx\undefined\msbfam \DN@{362}%
  \else \setboxz@h{$\m@th#1$}%
    \edef\next@{\ifdim\wdz@>\tw@ em%
        \hexnumber@\msbfam 5B%
      \else 362\fi}\fi
  \mathaccent"0\next@{#1}}
\def\widetilde#1{\ifx\undefined\msbfam \DN@{365}%
  \else \setboxz@h{$\m@th#1$}%
    \edef\next@{\ifdim\wdz@>\tw@ em%
        \hexnumber@\msbfam 5D%
      \else 365\fi}\fi
  \mathaccent"0\next@{#1}}
\message{\string\newsymbol,}
\def\newsymbol#1#2#3#4#5{\define#1{}%
  \count@#2\relax \advance\count@\m@ne % to push case 0 to the \else clause
 \ifcase\count@
   \ifx\undefined\msafam\loadmsam\fi \let\next@\msafam
 \or \ifx\undefined\msbfam\loadmsbm\fi \let\next@\msbfam
 \else  \Err@{\Invalid@@\string\newsymbol}\let\next@\tw@\fi
 \mathchardef#1="#3\hexnumber@\next@#4#5\space}
\def\UseAMSsymbols{\loadmsam\loadmsbm \input amssym.tex }
\def\Bbb{\RIfM@\expandafter\Bbb@\else
 \expandafter\nonmatherr@\expandafter\Bbb\fi}
\def\Bbb@#1{{\Bbb@@{#1}}}
\def\Bbb@@#1{\noaccents@\fam\msbfam\relax#1}
\message{bold Greek and bold symbols,}
\def\loadbold{%
 \font@\tencmmib=cmmib10 \font@\sevencmmib=cmmib7 \font@\fivecmmib=cmmib5
 \skewchar\tencmmib'177 \skewchar\sevencmmib'177 \skewchar\fivecmmib'177
 \alloc@@8\fam\chardef\sixt@@n\cmmibfam
 \textfont\cmmibfam\tencmmib
 \scriptfont\cmmibfam\sevencmmib \scriptscriptfont\cmmibfam\fivecmmib
 \font@\tencmbsy=cmbsy10 \font@\sevencmbsy=cmbsy7 \font@\fivecmbsy=cmbsy5
 \skewchar\tencmbsy'60 \skewchar\sevencmbsy'60 \skewchar\fivecmbsy'60
 \alloc@@8\fam\chardef\sixt@@n\cmbsyfam
 \textfont\cmbsyfam\tencmbsy
 \scriptfont\cmbsyfam\sevencmbsy \scriptscriptfont\cmbsyfam\fivecmbsy
 \let\loadbold\empty
}
\def\boldnotloaded#1{\Err@{\ifcase#1\or First\else Second\fi
       bold symbol font not loaded}}
\def\mathchari@#1#2#3{\ifx\undefined\cmmibfam
    \boldnotloaded@\@ne
  \else\mathchar"#1\hexnumber@\cmmibfam#2#3\space \fi}
\def\mathcharii@#1#2#3{\ifx\undefined\cmbsyfam
    \boldnotloaded\tw@
  \else \mathchar"#1\hexnumber@\cmbsyfam#2#3\space\fi}
\edef\bffam@{\hexnumber@\bffam}
\def\boldkey#1{\ifcat\noexpand#1A%
  \ifx\undefined\cmmibfam \boldnotloaded\@ne
  \else {\fam\cmmibfam#1}\fi
 \else
 \ifx#1!\mathchar"5\bffam@21 \else
 \ifx#1(\mathchar"4\bffam@28 \else\ifx#1)\mathchar"5\bffam@29 \else
 \ifx#1+\mathchar"2\bffam@2B \else\ifx#1:\mathchar"3\bffam@3A \else
 \ifx#1;\mathchar"6\bffam@3B \else\ifx#1=\mathchar"3\bffam@3D \else
 \ifx#1?\mathchar"5\bffam@3F \else\ifx#1[\mathchar"4\bffam@5B \else
 \ifx#1]\mathchar"5\bffam@5D \else
 \ifx#1,\mathchari@63B \else
 \ifx#1-\mathcharii@200 \else
 \ifx#1.\mathchari@03A \else
 \ifx#1/\mathchari@03D \else
 \ifx#1<\mathchari@33C \else
 \ifx#1>\mathchari@33E \else
 \ifx#1*\mathcharii@203 \else
 \ifx#1|\mathcharii@06A \else
 \ifx#10\bold0\else\ifx#11\bold1\else\ifx#12\bold2\else\ifx#13\bold3\else
 \ifx#14\bold4\else\ifx#15\bold5\else\ifx#16\bold6\else\ifx#17\bold7\else
 \ifx#18\bold8\else\ifx#19\bold9\else
  \Err@{\string\boldkey\space can't be used with #1}%
 \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi
 \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}
\def\boldsymbol#1{%
 \DN@{\Err@{You can't use \string\boldsymbol\space with \string#1}#1}%
 \ifcat\noexpand#1A%
   \let\next@\relax
   \ifx\undefined\cmmibfam \boldnotloaded\@ne
   \else {\fam\cmmibfam#1}\fi
 \else
  \xdef\meaning@{\meaning#1.........}%
  \expandafter\math@\meaning@\math@
  \ifmath@
   \expandafter\mathch@\meaning@\mathch@
   \ifmathch@
    \expandafter\boldsymbol@@\meaning@\boldsymbol@@
   \fi
  \else
   \expandafter\macro@\meaning@\macro@
   \expandafter\delim@\meaning@\delim@
   \ifdelim@
    \expandafter\delim@@\meaning@\delim@@
   \else
    \boldsymbol@{#1}%
   \fi
  \fi
 \fi
 \next@}
\def\mathhexboxii@#1#2{\ifx\undefined\cmbsyfam
    \boldnotloaded\tw@
  \else \mathhexbox@{\hexnumber@\cmbsyfam}{#1}{#2}\fi}
\def\boldsymbol@#1{\let\next@\relax\let\next#1%
 \ifx\next\cdot\mathcharii@201 \else
 \ifx\next\prime{{\null\mathcharii@030 \null}}\else
 \ifx\next\lbrack\mathchar"4\bffam@5B \else
 \ifx\next\rbrack\mathchar"5\bffam@5D \else
 \ifx\next\{\mathcharii@466 \else
 \ifx\next\lbrace\mathcharii@466 \else
 \ifx\next\}\mathcharii@567 \else
 \ifx\next\rbrace\mathcharii@567 \else
 \ifx\next\surd{{\mathcharii@170}}\else
 \ifx\next\S{{\mathhexboxii@78}}\else
 \ifx\next\P{{\mathhexboxii@7B}}\else
 \ifx\next\dag{{\mathhexboxii@79}}\else
 \ifx\next\ddag{{\mathhexboxii@7A}}\else
 \DN@{\Err@{You can't use \string\boldsymbol\space with \string#1}#1}%
 \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}
\def\boldsymbol@@#1.#2\boldsymbol@@{\classnum@#1 \count@@@\classnum@        %1
 \divide\classnum@4096 \count@\classnum@                                    %2
 \multiply\count@4096 \advance\count@@@-\count@ \count@@\count@@@           %3
 \divide\count@@@\@cclvi \count@\count@@                                    %4
 \multiply\count@@@\@cclvi \advance\count@@-\count@@@                       %5
 \divide\count@@@\@cclvi                                                    %6
 \multiply\classnum@4096 \advance\classnum@\count@@                         %7
 \ifnum\count@@@=\z@                                                        %8
  \count@"\bffam@ \multiply\count@\@cclvi
  \advance\classnum@\count@
  \DN@{\mathchar\number\classnum@}%
 \else
  \ifnum\count@@@=\@ne                                                      %9
   \ifx\undefined\cmmibfam \DN@{\boldnotloaded\@ne}%
   \else \count@\cmmibfam \multiply\count@\@cclvi
     \advance\classnum@\count@
     \DN@{\mathchar\number\classnum@}\fi
  \else
   \ifnum\count@@@=\tw@                                                    %10
     \ifx\undefined\cmbsyfam
       \DN@{\boldnotloaded\tw@}%
     \else
       \count@\cmbsyfam \multiply\count@\@cclvi
       \advance\classnum@\count@
       \DN@{\mathchar\number\classnum@}%
     \fi
  \fi
 \fi
\fi}
\newif\ifdelim@
\newcount\delimcount@
{\uccode`6=`\\ \uccode`7=`d \uccode`8=`e \uccode`9=`l
 \uppercase{\gdef\delim@#1#2#3#4#5\delim@
  {\delim@false\ifx 6#1\ifx 7#2\ifx 8#3\ifx 9#4\delim@true
   \xdef\meaning@{#5}\fi\fi\fi\fi}}}
\def\delim@@#1"#2#3#4#5#6\delim@@{\if#32%
\let\next@\relax
 \ifx\undefined\cmbsyfam \boldnotloaded\@ne
 \else \mathcharii@#2#4#5\space \fi\fi}
\def\vert{\delimiter"026A30C }
\def\Vert{\delimiter"026B30D }
\let\|\Vert
\def\backslash{\delimiter"026E30F }
\def\boldkeydots@#1{\bold@true\let\next=#1\let\delayed@=#1\mdots@@
 \boldkey#1\bold@false}  % = required!
\def\boldsymboldots@#1{\bold@true\let\next#1\let\delayed@#1\mdots@@
 \boldsymbol#1\bold@false}
\message{Euler fonts,}
\def\loadeufm{\loadmathfont{eufm}}
\def\loadeufb{\loadmathfont{eufb}}
\def\loadeusm{\loadmathfont{eusm}}
\def\loadeusb{\loadmathfont{eusb}}
\def\loadeurm{\loadmathfont{eurm}}
\def\loadeurb{\loadmathfont{eurb}}
\def\frak{\mathfont@\frak}
\def\frakfam{\eufmfam}
\let\goth\frak
\def\loadmathfont#1{% 
   \expandafter\font@\csname ten#1\endcsname=#110
   \expandafter\font@\csname seven#1\endcsname=#17
   \expandafter\font@\csname five#1\endcsname=#15
   \edef\next{\noexpand\alloc@@8\fam\chardef\sixt@@n
     \expandafter\noexpand\csname#1fam\endcsname}%
   \next
   \textfont\csname#1fam\endcsname \csname ten#1\endcsname
   \scriptfont\csname#1fam\endcsname \csname seven#1\endcsname
   \scriptscriptfont\csname#1fam\endcsname \csname five#1\endcsname
   \expandafter\def\csname #1\expandafter\endcsname\expandafter{%
      \expandafter\mathfont@\csname#1\endcsname}%
 \expandafter\gdef\csname load#1\endcsname{}%
}
\def\mathfont@#1{\RIfM@\expandafter\mathfont@@\expandafter#1\else
  \expandafter\nonmatherr@\expandafter#1\fi}
\def\mathfont@@#1#2{{\mathfont@@@#1{#2}}}
\def\mathfont@@@#1#2{\noaccents@
   \fam\csname\expandafter\eat@\string#1fam\endcsname
   \relax#2}
\message{math accents,}
\def\accentclass@{7}
\def\noaccents@{\def\accentclass@{0}}
\def\makeacc@#1#2{\def#1{\mathaccent"\accentclass@#2 }}
\makeacc@\hat{05E}
\makeacc@\check{014}
\makeacc@\tilde{07E}
\makeacc@\acute{013}
\makeacc@\grave{012}
\makeacc@\dot{05F}
\makeacc@\ddot{07F}
\makeacc@\breve{015}
\makeacc@\bar{016}
\def\vec{\mathaccent"017E }
\newcount\skewcharcount@
\newcount\familycount@
\def\theskewchar@{\familycount@\@ne
 \global\skewcharcount@\the\skewchar\textfont\@ne                           %1
 \ifnum\fam>\m@ne\ifnum\fam<16
  \global\familycount@\the\fam\relax
  \global\skewcharcount@\the\skewchar\textfont\the\fam\relax\fi\fi          %2
 \ifnum\skewcharcount@>\m@ne
  \ifnum\skewcharcount@<128
  \multiply\familycount@256
  \global\advance\skewcharcount@\familycount@
  \global\advance\skewcharcount@28672
  \mathchar\skewcharcount@\else
  \global\skewcharcount@\m@ne\fi\else
 \global\skewcharcount@\m@ne\fi}                                            %3
\newcount\pointcount@
\def\getpoints@#1.#2\getpoints@{\pointcount@#1 }
\newdimen\accentdimen@
\newcount\accentmu@
\def\dimentomu@{\multiply\accentdimen@ 100
 \expandafter\getpoints@\the\accentdimen@\getpoints@
 \multiply\pointcount@18
 \divide\pointcount@\@m
 \global\accentmu@\pointcount@}
\def\Makeacc@#1#2{\def#1{\RIfM@\DN@{\mathaccent@
 {"\accentclass@#2 }}\else\DN@{\nonmatherr@{#1}}\fi\next@}}
\def\unbracefonts@{\let\Cal@\Cal@@\let\roman@\roman@@\let\bold@\bold@@
 \let\slanted@\slanted@@}
\def\mathaccent@#1#2{\ifnum\fam=\m@ne\xdef\thefam@{1}\else
 \xdef\thefam@{\the\fam}\fi                                                 %1
 \accentdimen@\z@                                                           %2
 \setboxz@h{\unbracefonts@$\m@th\fam\thefam@\relax#2$}%                     %3
 \ifdim\accentdimen@=\z@\DN@{\mathaccent#1{#2}}%                            %4
  \setbox@ne\hbox{\unbracefonts@$\m@th\fam\thefam@\relax#2\theskewchar@$}% %5a
  \setbox\tw@\hbox{$\m@th\ifnum\skewcharcount@=\m@ne\else
   \mathchar\skewcharcount@\fi$}%                                          %5b
  \global\accentdimen@\wd@ne\global\advance\accentdimen@-\wdz@
  \global\advance\accentdimen@-\wd\tw@                                     %5c
  \global\multiply\accentdimen@\tw@
  \dimentomu@\global\advance\accentmu@\@ne                                 %5d
 \else\DN@{{\mathaccent#1{#2\mkern\accentmu@ mu}%
    \mkern-\accentmu@ mu}{}}\fi                                             %6
 \next@}\Makeacc@\Hat{05E}
\Makeacc@\Check{014}
\Makeacc@\Tilde{07E}
\Makeacc@\Acute{013}
\Makeacc@\Grave{012}
\Makeacc@\Dot{05F}
\Makeacc@\Ddot{07F}
\Makeacc@\Breve{015}
\Makeacc@\Bar{016}
\def\Vec{\RIfM@\DN@{\mathaccent@{"017E }}\else
 \DN@{\nonmatherr@\Vec}\fi\next@}
\def\accentedsymbol#1#2{\csname newbox\expandafter\endcsname
  \csname\expandafter\eat@\string#1@box\endcsname
 \expandafter\setbox\csname\expandafter\eat@
  \string#1@box\endcsname\hbox{$\m@th#2$}\define
  #1{\copy\csname\expandafter\eat@\string#1@box\endcsname{}}}
\message{roots,}
\def\sqrt#1{\radical"270370 {#1}}
\let\underline@\underline
\let\overline@\overline
\def\underline#1{\underline@{#1}}
\def\overline#1{\overline@{#1}}
\Invalid@\leftroot
\Invalid@\uproot
\newcount\uproot@
\newcount\leftroot@
\def\root{\relaxnext@
  \DN@{\ifx\next\uproot\let\next@\nextii@\else
   \ifx\next\leftroot\let\next@\nextiii@\else
   \let\next@\plainroot@\fi\fi\next@}%
  \DNii@\uproot##1{\uproot@##1\relax\FN@\nextiv@}%
  \def\nextiv@{\ifx\next\space@\DN@. {\FN@\nextv@}\else
   \DN@.{\FN@\nextv@}\fi\next@.}%
  \def\nextv@{\ifx\next\leftroot\let\next@\nextvi@\else
   \let\next@\plainroot@\fi\next@}%
  \def\nextvi@\leftroot##1{\leftroot@##1\relax\plainroot@}%
   \def\nextiii@\leftroot##1{\leftroot@##1\relax\FN@\nextvii@}%
  \def\nextvii@{\ifx\next\space@
   \DN@. {\FN@\nextviii@}\else
   \DN@.{\FN@\nextviii@}\fi\next@.}%
  \def\nextviii@{\ifx\next\uproot\let\next@\nextix@\else
   \let\next@\plainroot@\fi\next@}%
  \def\nextix@\uproot##1{\uproot@##1\relax\plainroot@}%
  \bgroup\uproot@\z@\leftroot@\z@\FN@\next@}
\def\plainroot@#1\of#2{\setbox\rootbox\hbox{$\m@th\scriptscriptstyle{#1}$}%
 \mathchoice{\r@@t\displaystyle{#2}}{\r@@t\textstyle{#2}}
 {\r@@t\scriptstyle{#2}}{\r@@t\scriptscriptstyle{#2}}\egroup}
\def\r@@t#1#2{\setboxz@h{$\m@th#1\sqrt{#2}$}%
 \dimen@\ht\z@\advance\dimen@-\dp\z@
 \setbox@ne\hbox{$\m@th#1\mskip\uproot@ mu$}\advance\dimen@ 1.667\wd@ne
 \mkern-\leftroot@ mu\mkern5mu\raise.6\dimen@\copy\rootbox
 \mkern-10mu\mkern\leftroot@ mu\boxz@}
\def\boxed#1{\setboxz@h{$\m@th\displaystyle{#1}$}\dimen@.4\ex@
 \advance\dimen@3\ex@\advance\dimen@\dp\z@
 \hbox{\lower\dimen@\hbox{%
 \vbox{\hrule height.4\ex@
 \hbox{\vrule width.4\ex@\hskip3\ex@\vbox{\vskip3\ex@\boxz@\vskip3\ex@}%
 \hskip3\ex@\vrule width.4\ex@}\hrule height.4\ex@}%
 }}}
\message{commutative diagrams,}
\let\ampersand@\relax
\newdimen\minaw@
\minaw@11.11128\ex@
\newdimen\minCDaw@
\minCDaw@2.5pc
\def\minCDarrowwidth#1{\RIfMIfI@\onlydmatherr@\minCDarrowwidth
 \else\minCDaw@#1\relax\fi\else\onlydmatherr@\minCDarrowwidth\fi}
\newif\ifCD@
\def\CD{\bgroup\vspace@\relax\iffalse{\fi\let\ampersand@&\iffalse}\fi
 \CD@true\vcenter\bgroup\Let@\tabskip\z@skip\baselineskip20\ex@
 \lineskip3\ex@\lineskiplimit3\ex@\halign\bgroup
 &\hfill$\m@th##$\hfill\crcr}
\def\endCD{\crcr\egroup\egroup\egroup}
\newdimen\bigaw@
\atdef@>#1>#2>{\ampersand@                                                  %1
 \setboxz@h{$\m@th\ssize\;{#1}\;\;$}%                                       %2
 \setbox@ne\hbox{$\m@th\ssize\;{#2}\;\;$}%                                  %3
 \setbox\tw@\hbox{$\m@th#2$}%                                               %4
 \ifCD@\global\bigaw@\minCDaw@\else\global\bigaw@\minaw@\fi                 %5
 \ifdim\wdz@>\bigaw@\global\bigaw@\wdz@\fi
 \ifdim\wd@ne>\bigaw@\global\bigaw@\wd@ne\fi                                %6
 \ifCD@\enskip\fi                                                           %7
 \ifdim\wd\tw@>\z@
  \mathrel{\mathop{\hbox to\bigaw@{\rightarrowfill@\displaystyle}}%
    \limits^{#1}_{#2}}%                                                     %8
 \else\mathrel{\mathop{\hbox to\bigaw@{\rightarrowfill@\displaystyle}}%
    \limits^{#1}}\fi                                                        %9
 \ifCD@\enskip\fi                                                          %10
 \ampersand@}                                                              %11
\atdef@<#1<#2<{\ampersand@\setboxz@h{$\m@th\ssize\;\;{#1}\;$}%
 \setbox@ne\hbox{$\m@th\ssize\;\;{#2}\;$}\setbox\tw@\hbox{$\m@th#2$}%
 \ifCD@\global\bigaw@\minCDaw@\else\global\bigaw@\minaw@\fi
 \ifdim\wdz@>\bigaw@\global\bigaw@\wdz@\fi
 \ifdim\wd@ne>\bigaw@\global\bigaw@\wd@ne\fi
 \ifCD@\enskip\fi
 \ifdim\wd\tw@>\z@
  \mathrel{\mathop{\hbox to\bigaw@{\leftarrowfill@\displaystyle}}%
       \limits^{#1}_{#2}}\else
  \mathrel{\mathop{\hbox to\bigaw@{\leftarrowfill@\displaystyle}}%
       \limits^{#1}}\fi
 \ifCD@\enskip\fi\ampersand@}
\begingroup
 \catcode`\~=\active \lccode`\~=`\@
 \lowercase{%
  \global\atdef@)#1)#2){~>#1>#2>}
  \global\atdef@(#1(#2({~<#1<#2<}}
\endgroup
\atdef@ A#1A#2A{\llap{$\m@th\vcenter{\hbox
 {$\ssize#1$}}$}\Big\uparrow\rlap{$\m@th\vcenter{\hbox{$\ssize#2$}}$}&&}
\atdef@ V#1V#2V{\llap{$\m@th\vcenter{\hbox
 {$\ssize#1$}}$}\Big\downarrow\rlap{$\m@th\vcenter{\hbox{$\ssize#2$}}$}&&}
\atdef@={&\enskip\mathrel
 {\vbox{\hrule width\minCDaw@\vskip3\ex@\hrule width
 \minCDaw@}}\enskip&}
\atdef@|{\Big\Vert&&}
\atdef@\vert{\Big\Vert&&}
\def\pretend#1\haswidth#2{\setboxz@h{$\m@th\scriptstyle{#2}$}\hbox
 to\wdz@{\hfill$\m@th\scriptstyle{#1}$\hfill}}
\message{poor man's bold,}
\def\pmb{\RIfM@\expandafter\mathpalette\expandafter\pmb@\else
 \expandafter\pmb@@\fi}
\def\pmb@@#1{\leavevmode\setboxz@h{#1}%
   \dimen@-\wdz@
   \kern-.5\ex@\copy\z@
   \kern\dimen@\kern.25\ex@\raise.4\ex@\copy\z@
   \kern\dimen@\kern.25\ex@\box\z@
}
\def\binrel@@#1{\ifdim\wd2<\z@\mathbin{#1}\else\ifdim\wd\tw@>\z@
 \mathrel{#1}\else{#1}\fi\fi}
\newdimen\pmbraise@
%      Note: because of the use of \mathpalette, if \pmb@ is
%      applied to a single math italic character (or a single
%      character from some other slanted math font), the italic
%      correction will be added.  This will cause subscripts
%      to fall too far away from the character in some
%      cases, e.g., $\pmb{T}_1$ or $\pmb{\Cal T}_1$.
\def\pmb@#1#2{\setbox\thr@@\hbox{$\m@th#1{#2}$}%
 \setbox4\hbox{$\m@th#1\mkern.5mu$}\pmbraise@\wd4\relax
 \binrel@{#2}%
 \dimen@-\wd\thr@@
   \binrel@@{%
   \mkern-.8mu\copy\thr@@
   \kern\dimen@\mkern.4mu\raise\pmbraise@\copy\thr@@
   \kern\dimen@\mkern.4mu\box\thr@@
}}
\def\documentstyle#1{\W@{}\input #1.sty\relax}
\message{syntax check,}
\font\dummyft@=dummy
\fontdimen1 \dummyft@=\z@
\fontdimen2 \dummyft@=\z@
\fontdimen3 \dummyft@=\z@
\fontdimen4 \dummyft@=\z@
\fontdimen5 \dummyft@=\z@
\fontdimen6 \dummyft@=\z@
\fontdimen7 \dummyft@=\z@
\fontdimen8 \dummyft@=\z@
\fontdimen9 \dummyft@=\z@
\fontdimen10 \dummyft@=\z@
\fontdimen11 \dummyft@=\z@
\fontdimen12 \dummyft@=\z@
\fontdimen13 \dummyft@=\z@
\fontdimen14 \dummyft@=\z@
\fontdimen15 \dummyft@=\z@
\fontdimen16 \dummyft@=\z@
\fontdimen17 \dummyft@=\z@
\fontdimen18 \dummyft@=\z@
\fontdimen19 \dummyft@=\z@
\fontdimen20 \dummyft@=\z@
\fontdimen21 \dummyft@=\z@
\fontdimen22 \dummyft@=\z@
\def\fontlist@{\\{\tenrm}\\{\sevenrm}\\{\fiverm}\\{\teni}\\{\seveni}%
 \\{\fivei}\\{\tensy}\\{\sevensy}\\{\fivesy}\\{\tenex}\\{\tenbf}\\{\sevenbf}%
 \\{\fivebf}\\{\tensl}\\{\tenit}}
\def\font@#1=#2 {\rightappend@#1\to\fontlist@\font#1=#2 }
\def\dodummy@{{\def\\##1{\global\let##1\dummyft@}\fontlist@}}
\def\nopages@{\output{\setbox\z@\box\@cclv \deadcycles\z@}%
 \alloc@5\toks\toksdef\@cclvi\output}
\let\galleys\nopages@
\newif\ifsyntax@
\newcount\countxviii@
\def\syntax{\syntax@true\dodummy@\countxviii@\count18
 \loop\ifnum\countxviii@>\m@ne\textfont\countxviii@=\dummyft@
 \scriptfont\countxviii@=\dummyft@\scriptscriptfont\countxviii@=\dummyft@
 \advance\countxviii@\m@ne\repeat                                           %1
 \dummyft@\tracinglostchars\z@\nopages@\frenchspacing\hbadness\@M}
\def\first@#1#2\end{#1}
\def\printoptions{\W@{Do you want S(yntax check),
  G(alleys) or P(ages)?}%
 \message{Type S, G or P, followed by <return>: }%
 \begingroup % to localize the following change to \endlinechar:
 \endlinechar\m@ne % to prevent a space or \par in \ans@ from ^^M
 \read\m@ne to\ans@
%  Define \ans@ to uppercase itself, and default to P if the user
%  just pressed <return>.
 \edef\ans@{\uppercase{\def\noexpand\ans@{%
   \expandafter\first@\ans@ P\end}}}%
%  Cast the new definition of \ans@ outside the group
 \expandafter\endgroup\ans@
 \if\ans@ P% fine, no action needs to be taken
 \else \if\ans@ S\syntax
 \else \if\ans@ G\galleys
 \else\message{? Unknown option: \ans@; using the `pages' option.}%
 \fi\fi\fi}
\def\alloc@#1#2#3#4#5{\global\advance\count1#1by\@ne
 \ch@ck#1#4#2\allocationnumber=\count1#1
 \global#3#5=\allocationnumber
 \ifalloc@\wlog{\string#5=\string#2\the\allocationnumber}\fi}
\def\document{\def\alloclist@{}\def\fontlist@{}}
\let\enddocument\bye
\let\plainproclaim\proclaim
\let\plainfootnote\footnote
\let\proclaim\undefined
\let\footnote\undefined
\let\=\undefined
\let\>\undefined
\let\cal\undefined
\let\mit\undefined
\let\oldstyle\undefined
\catcode`\@=\active
\message{... finished}

%  *** end including amstex.tex *** 
% % \input mathdefs
%  *** start including mathdefs.tex *** 
\expandafter\ifx\csname mathdefs.tex\endcsname\relax
  \expandafter\gdef\csname mathdefs.tex\endcsname{}
\else \message{Hey!  Apparently you were trying to
  \string\input{mathdefs.tex} twice.   This does not make sense.} 
\errmessage{Please edit your file (probably \jobname.tex) and remove
any duplicate ``\string\input'' lines}\endinput\fi


%mathdefs.tex v1.3.2

%%% Changes from v1.0: footnote macros, warning for duplicated tags,
%%%   control sequences \( and \verbatimtags.
%%% From v1.2: \pretags, redefinition of \( using \ifinner, multi-part
%%%   equation numbering, control sequences \[, \references, and
%%%   \resetbracket. 
%%% From v1.3: \rm in \lastpart; write root of multi-part tag to .tgs 

%See file texdefs.doc for documentation.

\catcode`\X=12\catcode`\@=11

%Minor control sequences:
\def\n@wcount{\alloc@0\count\countdef\insc@unt}
\def\n@wwrite{\alloc@7\write\chardef\sixt@@n}
\def\n@wread{\alloc@6\read\chardef\sixt@@n}
\def\r@s@t{\relax}\def\v@idline{\par}\def\@mputate#1/{#1}
\def\l@c@l#1X{\firstpart.#1}\def\gl@b@l#1X{#1}\def\t@d@l#1X{{}}

%Creation of tag families and output of assignments and citations:
\def\crossrefs#1{\ifx\all#1\let\tr@ce=\all\else\def\tr@ce{#1,}\fi
   \n@wwrite\cit@tionsout\openout\cit@tionsout=\jobname.cit 
   \write\cit@tionsout{\tr@ce}\expandafter\setfl@gs\tr@ce,}
\def\setfl@gs#1,{\def\@{#1}\ifx\@\empty\let\next=\relax
   \else\let\next=\setfl@gs\expandafter\xdef
   \csname#1tr@cetrue\endcsname{}\fi\next}
\def\m@ketag#1#2{\expandafter\n@wcount\csname#2tagno\endcsname
     \csname#2tagno\endcsname=0\let\tail=\all\xdef\all{\tail#2,}
   \ifx#1\l@c@l\let\tail=\r@s@t\xdef\r@s@t{\csname#2tagno\endcsname=0\tail}\fi
   \expandafter\gdef\csname#2cite\endcsname##1{\expandafter
     \ifx\csname#2tag##1\endcsname\relax?\else\csname#2tag##1\endcsname\fi
     \expandafter\ifx\csname#2tr@cetrue\endcsname\relax\else
     \write\cit@tionsout{#2tag ##1 cited on page \folio.}\fi}
   \expandafter\gdef\csname#2page\endcsname##1{\expandafter
     \ifx\csname#2page##1\endcsname\relax?\else\csname#2page##1\endcsname\fi
     \expandafter\ifx\csname#2tr@cetrue\endcsname\relax\else
     \write\cit@tionsout{#2tag ##1 cited on page \folio.}\fi}
   \expandafter\gdef\csname#2tag\endcsname##1{\expandafter
      \ifx\csname#2check##1\endcsname\relax
      \expandafter\xdef\csname#2check##1\endcsname{}%
      \else\immediate\write16{Warning: #2tag ##1 used more than once.}\fi
      \multit@g{#1}{#2}##1/X%
      \write\t@gsout{#2tag ##1 assigned number \csname#2tag##1\endcsname\space
      on page \number\count0.}%
   \csname#2tag##1\endcsname}}


\def\multit@g#1#2#3/#4X{\def\t@mp{#4}\ifx\t@mp\empty%
      \global\advance\csname#2tagno\endcsname by 1 
      \expandafter\xdef\csname#2tag#3\endcsname
      {#1\number\csname#2tagno\endcsnameX}%
   \else\expandafter\ifx\csname#2last#3\endcsname\relax
      \expandafter\n@wcount\csname#2last#3\endcsname
      \global\advance\csname#2tagno\endcsname by 1 
      \expandafter\xdef\csname#2tag#3\endcsname
      {#1\number\csname#2tagno\endcsnameX}
      \write\t@gsout{#2tag #3 assigned number \csname#2tag#3\endcsname\space
      on page \number\count0.}\fi
   \global\advance\csname#2last#3\endcsname by 1
   \def\t@mp{\expandafter\xdef\csname#2tag#3/}%
   \expandafter\t@mp\@mputate#4\endcsname
   {\csname#2tag#3\endcsname\lastpart{\csname#2last#3\endcsname}}\fi}
\def\t@gs#1{\def\all{}\m@ketag#1e\m@ketag#1s\m@ketag\t@d@l p
\let\realscite\scite
\let\realstag\stag
   \m@ketag\gl@b@l r \n@wread\t@gsin
   \openin\t@gsin=\jobname.tgs \re@der \closein\t@gsin
   \n@wwrite\t@gsout\openout\t@gsout=\jobname.tgs }
\outer\def\localtags{\t@gs\l@c@l}
\outer\def\globaltags{\t@gs\gl@b@l}
\outer\def\newlocaltag#1{\m@ketag\l@c@l{#1}}
\outer\def\newglobaltag#1{\m@ketag\gl@b@l{#1}}

%Reading in tag information:
\newif\ifpr@ 
\def\m@kecs #1tag #2 assigned number #3 on page #4.%
   {\expandafter\gdef\csname#1tag#2\endcsname{#3}
   \expandafter\gdef\csname#1page#2\endcsname{#4}
   \ifpr@\expandafter\xdef\csname#1check#2\endcsname{}\fi}
\def\re@der{\ifeof\t@gsin\let\next=\relax\else
   \read\t@gsin to\t@gline\ifx\t@gline\v@idline\else
   \expandafter\m@kecs \t@gline\fi\let \next=\re@der\fi\next}
\def\pretags#1{\pr@true\pret@gs#1,,}
\def\pret@gs#1,{\def\@{#1}\ifx\@\empty\let\n@xtfile=\relax
   \else\let\n@xtfile=\pret@gs \openin\t@gsin=#1.tgs \message{#1} \re@der 
   \closein\t@gsin\fi \n@xtfile}

%Sections and subsections; local numbering:
\newcount\sectno\sectno=0\newcount\subsectno\subsectno=0
\newif\ifultr@local \def\ultralocal{\ultr@localtrue}
\def\firstpart{\number\sectno}
\def\lastpart#1{\ifcase#1 \or a\or b\or c\or d\or e\or f\or g\or h\or 
   i\or k\or l\or m\or n\or o\or p\or q\or r\or s\or t\or u\or v\or w\or 
   x\or y\or z \fi}
\def\closeup{\vskip-\bigskipamount}
\def\resetall{\global\advance\sectno by 1\subsectno=0
   \gdef\firstpart{\number\sectno}\r@s@t}
\def\resetsub{\global\advance\subsectno by 1
   \gdef\firstpart{\number\sectno.\number\subsectno}\r@s@t}
\def\newsection#1\par{\resetall\vskip0pt plus.3\vsize\penalty-250
   \vskip0pt plus-.3\vsize\bigskip\bigskip
   \message{#1}\leftline{\bf#1}\nobreak\bigskip}
\def\subsection#1\par{\ifultr@local\resetsub\fi
   \vskip0pt plus.2\vsize\penalty-250\vskip0pt plus-.2\vsize
   \bigskip\smallskip\message{#1}\leftline{\bf#1}\nobreak\medskip}

%jj tags:
% On Andrzej's request:  we want to be able 
% to show tags as in noverbatim, with verbatim in the margin,
% and cites as in verbatim, with nonverbatim in the margin
% mg -- July 2000

\newdimen\marginshift

\newdimen\margindelta
\newdimen\marginmax
\newdimen\marginmin

\def\margininit{       
\marginmax=3 true cm                  % how much room, approximately
				      
\margindelta=0.1 true cm              % distance between entries
\marginmin=0.1true cm                 % where will leftmost entry be
\marginshift=\marginmin
}    % we cannot execute this right now, since 
     % there may be a \magnification coming later in the 
     % main file.   So we call \margininit at the end of 
     % alice2jlem

\def\t@gsjj#1,{\def\@{#1}\ifx\@\empty\let\next=\relax\else\let\next=\t@gsjj
   \def\@@{p}\ifx\@\@@\else
   \expandafter\gdef\csname#1cite\endcsname##1{\citejj{##1}}
   \expandafter\gdef\csname#1page\endcsname##1{?}
   \expandafter\gdef\csname#1tag\endcsname##1{\tagjj{##1}}\fi\fi\next}
\newif\ifshowstuffinmargin
\showstuffinmarginfalse
\def\jjtags{\ifx\shlhetal\relax 
      % so this is a public version --> no-op 
  \else
\ifx\shlhetal\undefinedcontrolseq
      % again, this is a public version --> no-op 
\else
\showstuffinmargintrue
\ifx\all\relax\else\expandafter\t@gsjj\all,\fi\fi \fi
}

% \def\zeigentag#1{\def\xxy{#1}\show\xxy\show\realstag\realstag#1\mginpar{\hbox{$\langle$}#1\hbox{$\rangle$}}}
% \def\zeigencite#1{\hbox{$\langle$}#1\hbox{$\rangle$}\mginpar{\realscite#1}}


\def\tagjj#1{\realstag{#1}\mginpar{\zeigen{#1}}}
% \def\citejj#1{\zeigen{#1}\mginpar{\rechnen{#1}}}
\def\citejj#1{\rechnen{#1}\mginpar{\zeigen{#1}}}     % modified Sep 02, saharon's suggestion

% \def\rechnen#1{rechnen(\expandafter\ifx\csname stag#1\endcsname\relax ??\else
%                            \csname stag#1\endcsname\fi)endrech }
\def\rechnen#1{\expandafter\ifx\csname stag#1\endcsname\relax ??\else
                           \csname stag#1\endcsname\fi}

% \def\zeigentag#1{//#1--\csname stag#1\endcsname::}

\newdimen\theight



\def\marginfont{\sevenrm}



\def\trymarginbox#1{\setbox0=\hbox{\marginfont\hskip\marginshift #1}%
		\global\marginshift\wd0 
		\global\advance\marginshift\margindelta}


\def \mginpar#1{%
\ifvmode\setbox0\hbox to \hsize{\hfill\rlap{\marginfont\quad#1}}%
\ht0 0cm
\dp0 0cm
\box0\vskip-\baselineskip
\else 
             \vadjust{\trymarginbox{#1}%
		\ifdim\marginshift>\marginmax \global\marginshift\marginmin
			\trymarginbox{#1}%
                \fi
             \theight=\ht0
             \advance\theight by \dp0    \advance\theight by \lineskip
             \kern -\theight \vbox to \theight{\rightline{\rlap{\box0}}%
\vss}}\fi}


% \def\mginpar#1{mg-#1-mg }



%Verbatim tags:
\def\t@gsoff#1,{\def\@{#1}\ifx\@\empty\let\next=\relax\else\let\next=\t@gsoff
   \def\@@{p}\ifx\@\@@\else
   \expandafter\gdef\csname#1cite\endcsname##1{\zeigen{##1}}
   \expandafter\gdef\csname#1page\endcsname##1{?}
   \expandafter\gdef\csname#1tag\endcsname##1{\zeigen{##1}}\fi\fi\next}
\def\verbatimtags{\showstuffinmarginfalse
\ifx\all\relax\else\expandafter\t@gsoff\all,\fi}
%%% \def\zeigen#1{zeigen(\hbox{$\langle$}#1\hbox{$\rangle$})endz }
\def\zeigen#1{\hbox{$\langle$}#1\hbox{$\rangle$}}

% % \def\margincite#1{\ifshowstuffinmargin\mginpar{\rechnen{#1}}\fi}
%  changed, april 2003, mg: we now have always the verbatim tag in the margin
\def\margincite#1{\ifshowstuffinmargin\mginpar{\zeigen{#1}}\fi}

\def\margintag#1{\ifshowstuffinmargin\mginpar{\zeigen{#1}}\fi}

\def\marginplain#1{\ifshowstuffinmargin\mginpar{{#1}}\fi}
\def\marginbf#1{\marginplain{{\bf \ \ #1}}}


%Equation numbering:
\def\(#1){\edef\dot@g{\ifmmode\ifinner(\hbox{\noexpand\etag{#1}})
   \else\noexpand\eqno(\hbox{\noexpand\etag{#1}})\fi
   \else(\noexpand\ecite{#1})\fi}\dot@g}

%Reference numbering:
\newif\ifbr@ck
\def\eat#1{}
\def\[#1]{\br@cktrue[\br@cket#1'X]}
\def\br@cket#1'#2X{\def\temp{#2}\ifx\temp\empty\let\next\eat
   \else\let\next\br@cket\fi
   \ifbr@ck\br@ckfalse\br@ck@t#1,X\else\br@cktrue#1\fi\next#2X}
\def\br@ck@t#1,#2X{\def\temp{#2}\ifx\temp\empty\let\neext\eat
   \else\let\neext\br@ck@t\def\temp{,}\fi
   \def\teemp{#1}\ifx\teemp\empty\else\rcite{#1}\fi\temp\neext#2X}
\def\resetbr@cket{\gdef\[##1]{[\rtag{##1}]}}
\def\references{\resetbr@cket\newsection References\par}

%Footnotes:
\newtoks\symb@ls\newtoks\s@mb@ls\newtoks\p@gelist\n@wcount\ftn@mber
    \ftn@mber=1\newif\ifftn@mbers\ftn@mbersfalse\newif\ifbyp@ge\byp@gefalse
\def\defm@rk{\ifftn@mbers\n@mberm@rk\else\symb@lm@rk\fi}
\def\n@mberm@rk{\xdef\m@rk{{\the\ftn@mber}}%
    \global\advance\ftn@mber by 1 }
\def\rot@te#1{\let\temp=#1\global#1=\expandafter\r@t@te\the\temp,X}
\def\r@t@te#1,#2X{{#2#1}\xdef\m@rk{{#1}}}
\def\b@@st#1{{$^{#1}$}}\def\str@p#1{#1}
\def\symb@lm@rk{\ifbyp@ge\rot@te\p@gelist\ifnum\expandafter\str@p\m@rk=1 
    \s@mb@ls=\symb@ls\fi\write\f@nsout{\number\count0}\fi \rot@te\s@mb@ls}
\def\byp@ge{\byp@getrue\n@wwrite\f@nsin\openin\f@nsin=\jobname.fns 
    \n@wcount\currentp@ge\currentp@ge=0\p@gelist={0}
    \re@dfns\closein\f@nsin\rot@te\p@gelist
    \n@wread\f@nsout\openout\f@nsout=\jobname.fns }
\def\m@kelist#1X#2{{#1,#2}}
\def\re@dfns{\ifeof\f@nsin\let\next=\relax\else\read\f@nsin to \f@nline
    \ifx\f@nline\v@idline\else\let\t@mplist=\p@gelist
    \ifnum\currentp@ge=\f@nline
    \global\p@gelist=\expandafter\m@kelist\the\t@mplistX0
    \else\currentp@ge=\f@nline
    \global\p@gelist=\expandafter\m@kelist\the\t@mplistX1\fi\fi
    \let\next=\re@dfns\fi\next}
\def\symbols#1{\symb@ls={#1}\s@mb@ls=\symb@ls} 
\def\bigsymbol{\textstyle}
\symbols{\bigsymbol\ast,\dagger,\ddagger,\sharp,\flat,\natural,\star}
\def\ftnumbers{\ftn@mberstrue} \def\ftsymbols{\ftn@mbersfalse}
\def\paginal{\byp@ge} \def\resetftnumbers{\ftn@mber=1}
\def\ftnote#1{\defm@rk\expandafter\expandafter\expandafter\footnote
    \expandafter\b@@st\m@rk{#1}}

%Miscellaneous macros:
\long\def\jump#1\endjump{}
\def\ssum{\mathop{\lower .1em\hbox{$\textstyle\Sigma$}}\nolimits}
\def\down#1{_{{}_{\scriptstyle #1}}}
\def\qed{\nobreak\kern 1em \vrule height .5em width .5em depth 0em}
\def\newneq{\hbox{\rlap{\hbox to 1\wd9{\hss$=$\hss}}\raise .1em 
   \hbox to 1\wd9{\hss$\scriptscriptstyle/$\hss}}}
\def\subsetne{\setbox9 = \hbox{$\subset$}\mathrel{\hbox{\rlap
   {\lower .4em \newneq}\raise .13em \hbox{$\subset$}}}}
\def\supsetne{\setbox9 = \hbox{$\subset$}\mathrel{\hbox{\rlap
   {\lower .4em \newneq}\raise .13em \hbox{$\supset$}}}}

%Blackboard bold:
\def\vbar{\mathchoice{\vrule height6.3ptdepth-.5ptwidth.8pt\kern-.8pt}
   {\vrule height6.3ptdepth-.5ptwidth.8pt\kern-.8pt}
   {\vrule height4.1ptdepth-.35ptwidth.6pt\kern-.6pt}
   {\vrule height3.1ptdepth-.25ptwidth.5pt\kern-.5pt}}
\def\f@dge{\mathchoice{}{}{\mkern.5mu}{\mkern.8mu}}
\def\b@c#1#2{{\rm \mkern#2mu\vbar\mkern-#2mu#1}}
\def\b@b#1{{\rm I\mkern-3.5mu #1}}
\def\b@a#1#2{{\rm #1\mkern-#2mu\f@dge #1}}
\def\bb#1{{\count4=`#1 \advance\count4by-64 \ifcase\count4\or\b@a A{11.5}\or
   \b@b B\or\b@c C{5}\or\b@b D\or\b@b E\or\b@b F \or\b@c G{5}\or\b@b H\or
   \b@b I\or\b@c J{3}\or\b@b K\or\b@b L \or\b@b M\or\b@b N\or\b@c O{5} \or
   \b@b P\or\b@c Q{5}\or\b@b R\or\b@a S{8}\or\b@a T{10.5}\or\b@c U{5}\or
   \b@a V{12}\or\b@a W{16.5}\or\b@a X{11}\or\b@a Y{11.7}\or\b@a Z{7.5}\fi}}

\catcode`\X=11 \catcode`\@=12


% Sep 2003, mg:

% % \input 300stuff
%  *** start including 300stuff.tex *** 

% definitions needed to process the 300x papers  (plus 88 etc) 

\let\thischap\jobname



\def\partof#1{\csname returnthe#1part\endcsname}
\def\chapof#1{\csname returnthe#1chap\endcsname}

\def\setchapter#1,#2,#3.{% 
  \expandafter\def\csname returnthe#1part\endcsname{#2}%
  \expandafter\def\csname returnthe#1chap\endcsname{#3}%
}

\setchapter 300a,A,I.
\setchapter 300b,A,II.
\setchapter 300c,A,III.
\setchapter 300d,A,IV.
\setchapter 300e,A,V.
\setchapter 300f,A,VI.
\setchapter 300g,A,VII.
\setchapter   88,B,I.
\setchapter  600,B,II.
\setchapter  705,B,III.


\def\cprefix#1{%     \cprefix{300b} generates A.II.  or II.  or "nothing"
% \leavevmode\vrule width 2cm height 1cm depth 0cm
\edef\theotherpart{\partof{#1}}\edef\theotherchap{\chapof{#1}}%
\ifx\theotherpart\thispart
   \ifx\theotherchap\thischap % nothing
    \else % same part, different chap
     \theotherchap%
    \fi
   \else % different part
     \theotherpart.\theotherchap\fi}


 % \sectioncite[\S4]{300b}  ->  II.\S4   or A.II.\S4
\def\sectioncite[#1]#2{%
     \cprefix{#2}#1}

\def\chaptercite#1{Chapter \cprefix{#1}}

% now define \thispart
\edef\thispart{\partof{\thischap}}
\edef\thischap{\chapof{\thischap}}


%  *** end including 300stuff.tex *** 

\def\spuriousreset{}

%  *** end including mathdefs.tex *** 
% % \input citeadd
%  *** start including citeadd.tex *** 
%   citeadd -- a few additions for 
% files from alice that were procesed with "citealice"

\expandafter\ifx\csname citeadd.tex\endcsname\relax
\expandafter\gdef\csname citeadd.tex\endcsname{}
\else \message{Hey!  Apparently you were trying to
\string\input{citeadd.tex} twice.   This does not make sense.} 
\errmessage{Please edit your file (probably \jobname.tex) and remove
any duplicate ``\string\input'' lines}\endinput\fi





\def\sciteu{\sciteerror{undefined}}
\def\sciteuphantom{\complainaboutcitation{undefined}}

\def\scitet{\sciteerror{ambiguous}}
\def\scitetphantom{\complainaboutcitation{ambiguous}}




\def\sciteerror#1#2{{\mathortextbf{\scite{#2}}}\complainaboutcitation{#1}{#2}}
\def\mathortextbf#1{\hbox{\bf #1}}
\def\complainaboutcitation#1#2{%
\vadjust{\line{\llap{---$\!\!>$ }\qquad scite$\{$#2$\}$ #1\hfil}}}





%  *** end including citeadd.tex *** 
\sectno=-1   % start with sect 0
\localtags
\jjtags
\NoBlackBoxes
\define\mr{\medskip\roster}
\define\sn{\smallskip\noindent}
\define\mn{\medskip\noindent}
\define\bn{\bigskip\noindent}
\define\ub{\underbar}
\define\wilog{\text{without loss of generality}}
\define\ermn{\endroster\medskip\noindent}
\define\dbca{\dsize\bigcap}
\define\dbcu{\dsize\bigcup}
\define\nl{\newline}
\magnification=\magstep 1
\documentstyle{amsppt}
% % \input alice2000
%  *** start including alice2000.tex *** 
% This file should be inputted whenever we use amsppt.sty and 
% old tex.  
%  Here we redefine \subjclass (use 1991 instead of 2000, otherwise 
% the following definition comes directly from 
%% 
%%              `amsppt.sty', generated 
%% on <1997/2/2> with the docstrip utility (2.2i).
%% 
%% The original source files were:
%% 
%% amsppt.doc 
%%% ====================================================================
%%% @AMSTeX-style-file{
%%%   filename  = "amsppt.sty",
%%%   version   = "2.1h",
%%%   date      = "1997/02/02",
%%%   time      = "09:27:44 EST",
%%%   checksum  = "56844 3264 16617 137829",
%%%   author    = "American Mathematical Society",
%%%   address   = "PO Box 6248, Providence, RI 02940-6248, USA",
%%%   telephone = "401-455-4080 or (in the USA) 800-321-4AMS",

{    % the braces make the catcode-change local. 
\catcode`@11

\ifx\alicetwothousandloaded@\relax
  \endinput\else\global\let\alicetwothousandloaded@\relax\fi


\gdef\subjclass{\let\savedef@\subjclass
 \def\subjclass##1\endsubjclass{\let\subjclass\savedef@
   \toks@{\def\usualspace{{\rm\enspace}}\eightpoint}%
   \toks@@{##1\unskip.}%
   \edef\thesubjclass@{\the\toks@
     \frills@{{\noexpand\rm2000 {\noexpand\it Mathematics Subject
       Classification}.\noexpand\enspace}}%
     \the\toks@@}}%
  \nofrillscheck\subjclass}
} 

%  *** end including alice2000.tex *** 
% % \input alice2jlem
%  *** start including alice2jlem.tex *** 
%% # Keywords  Input file to be used for texing Alice's files

\expandafter\ifx\csname alice2jlem.tex\endcsname\relax
  \expandafter\xdef\csname alice2jlem.tex\endcsname{\the\catcode`@}
\else \message{Hey!  Apparently you were trying to
\string\input{alice2jlem.tex}  twice.   This does not make sense.}
\errmessage{Please edit your file (probably \jobname.tex) and remove
any duplicate ``\string\input'' lines}\endinput\fi



% % \input bib4plain
%  *** start including bib4plain.tex *** 
\expandafter\ifx\csname bib4plain.tex\endcsname\relax
  \expandafter\gdef\csname bib4plain.tex\endcsname{}
\else \message{Hey!  Apparently you were trying to \string\input
  bib4plain.tex twice.   This does not make sense.}
\errmessage{Please edit your file (probably \jobname.tex) and remove
any duplicate ``\string\input'' lines}\endinput\fi


%  This file should be inputted if you want to use 
%  bibtex fom within plain TeX. 
\def\makeatletter{\catcode`\@11 }      % Not really need for standard
\def\makeatother{\catcode`\@12 }       % bibtex files, but these commands
\def\renewcommand{\newcommand}	       % are used in our literal-unsrt.bst
				       % and other files. 
%  The above lines were written by Martin.Goldstern@tuwien.ac.at
%  The rest is just the file btxmac.tex. 
%  (except for one small change below, regarding undefined citations)

%% @texfile{
%%   author = "Karl Berry and Oren Patashnik",
%%   version = "0.99j",
%%   date = "14 Mar 1992",
%%   filename = "btxmac.tex",
%%   address = "Please use electronic mail",
%%   checksum = "834    4503   33061",
%%   email = "opbibtex@cs.stanford.edu",
%%   codetable = "ISO/ASCII",
%%   supported = "yes",
%%   docstring = "Defines macros that make BibTeX work with plain TeX",
%% }
% BibTeX-for-TeX macros, version 0.99j, for BibTeX 0.99c, TeX 3.0 or later.
% Copyright (C) 1990--92 by Karl Berry and Oren Patashnik; all rights reserved.
% You may copy this file provided: that it's accompanied by the
% "BibTeXing" document, whose text is contained in the file `btxdoc.tex';
% that any documentation you write for these macros also gives a
% reference for "BibTeXing"; and that either you make absolutely no
% changes to your copy, or if you do make changes, (1) you name the file
% something other than `btxmac.tex' and you remove all occurrences of
% `btxmac.tex' from the file, (2) you put, somewhere in the first twenty
% lines of the file, your name, along with an electronic address at which
% others who might use the file may reach you, and (3) you remove each
% occurrence of Oren's name and electronic address from this file.  These
% restrictions help ensure that all standard versions of these macros are
% identical, and that Oren doesn't get deluged with inappropriate e-mail.
%
% This file, btxmac.tex, contains TeX macros that allow BibTeX, a
% bibliography program that was originally designed for use with LaTeX,
% to work with plain TeX.  Please report any bugs (outright goofs,
% improvable macros, misfeatures, or unclear documentation) to Oren
% Patashnik (opbibtex@cs.stanford.edu).  These macros will become frozen
% shortly after BibTeX version 1.00 is released.
%
% AMS-TEX WARNING: We tried very hard, for version .99i of these macros,
% to make them compatible with AmS-TeX.  We succeeded to the extent
% that, if you use one of the standard bibliography styles, you probably
% won't notice any problems with version 0.99i of btxmac.tex.  But
% ultimately we failed, in that the inherent incompatibilities between
% plain TeX and AmS-TeX kept making these macros break, for certain
% inputs or certain styles.  Examples:  (1) AmS-TeX treats at-signs as
% special, in ways that plain TeX and LaTeX don't, so that, for example,
% you can't have any `@' characters in an argument to the \cite command,
% the way you can in TeX or LaTeX; (2) AmS-TeX decided that plain TeX's
% and LaTeX's macron-accent control sequence `\=' should be undefined;
% so you'll need to define `\=' to be `\B' to get the xampl.bib example
% suggested below to work with AmS-TeX; (3) AmS-TeX redefines the tie
% character `~' of plain TeX, and AmS-TeX's `amsppt' style redefines
% plain TeX's `\nobreak' macro, so that if you use an author-date style
% like `apalike' and you have a multiple-author reference for which the
% author-date style automatically produces a citation in the text like
% `(Jones et~al., 1992)' you will throw AmS-TeX's `amsppt' style into
% an infinite loop, exceeding its input stack size.  In practice, such
% incompatibilities surface infrequently; but it is now clear to us that
% it's not worth the effort (perhaps it's not even possible) to make the
% btxmac.tex macros robust when used both with plain TeX and Ams-TeX.
% If the BibTeX/AmS-TeX results attainable with the current btxmac.tex
% macros are sufficient, fine.  But if there's a demand for more robust
% BibTeX/AmS-TeX behavior, then someone who's very familiar with the
% AmS-TeX package should probably make an amsbtxmc.tex version of the
% macros (remembering to follow the copyright restrictions above).
% Until then, if you're an AmS-TeX user, or a LaTeX or plain TeX user
% sharing files with an AmS-TeX user, beware.
% END OF AMS-TEX WARNING.
%
% To use these macros you should be familiar with how BibTeX interacts
% with LaTeX, since BibTeX's interaction with TeX is very similar; that
% interaction is explained in the LaTeX manual.  It also helps to
% have read "BibTeXing", the documentation that accompanies BibTeX.
% Then, if you want, you should redefine any of the macros that begin
% with `\bbl' or `\biblabel' or `\print' that you need to get formatting
% different from the default (the default settings are designed to
% accompany a bibliography style like BibTeX's standard style `plain').
% The macros you might want to change are described briefly a few
% paragraphs hence.  [To get started without reading any documentation,
% try running the nine-line .tex file below through TeX and BibTeX.
% Remember the general scheme: Running (La)TeX writes information on
% the .aux (auxiliary) file; then running BibTeX reads information from
% the .aux, .bst (style), and .bib (database) files, and writes
% information (the bibliography) on a .bbl file; then running (La)TeX
% incorporates the bibliography; then running (La)TeX once more fixes
% the remaining forward references into the bibliography.  Thus, to get
% everything incorporated into your output, you'll have to run (La)TeX,
% BibTeX, (La)TeX, (La)TeX.  (Standup, sitdown, fight, fight, fight.)]
%
% These macros can stand alone or they can be \input into a macro
% package, like Eplain, that is sufficiently compatible with plain TeX.
% To use these macros to format the 0.99 version of the xampl.bib file
% that's distributed with BibTeX (that version of the file has no
% self-identification), you'll need to define \mbox, which is a LaTeX
% command, to be \hbox, as in the example below.
%
% Here's a nine-line plain TeX file for trying out btxmac.tex; of course
% you'll have to remove the comment characters at the beginning of each
% line, and, depending on your system, you might have to take steps so
% that BibTeX can "see" the files xampl.bib and plain.bst (BibTeX will
% give you two empty-field warning messages that you should ignore).
%
%     \def\mbox#1{\leavevmode\hbox{#1}}
%     \input btxmac
%     \noindent This cites Aamport's gnominious article~\cite{article-full}.
%     \medskip
%     \leftline{\bf References}
%     \nocite{*}   % put all database entries into the reference list
%     \bibliography{xampl}   % specify the database files; here, just xampl.bib
%     \bibliographystyle{plain}   % specify plain.bst as the style file
%     \bye
%
%
%   HISTORY
%
% Karl Berry wrote the original version of these macros in 1989 and
% 1990, for use in his `Eplain' package.  Oren Patashnik modified them
% slightly in July 1990, as part of the official BibTeX distribution.
%
%    1-Aug-90  Version 0.99a, not released to the general public.
%   14-Aug-90  0.99b, first general release.
%   26-Aug-90  0.99c, made \@undefinedmessage work with other macro packages.
%    6-Sep-90  0.99d, allowed for general formatting of bibliography labels,
%                     for general formatting of (in-text) citations, and for
%                     changing certain catcodes while reading the .aux file.
%   14-Nov-90  0.99e, changed the way \@setletters works, made some \new...'s
%                     non-outer, and changed the way Eplain reads this file.
%   12-Dec-90  0.99f, made \@resetnumerals change the `,' and `.' catcodes; and
%                     added \biblabelextrahang, \@getoptionalarg, and \bblsc.
%   11-Mar-91  0.99g, made a few minor changes required by the way Eplain reads
%                     this file, but no functional changes.
%   24-Apr-91  0.99h, inhibited the reading and writing of the .aux file if it
%                     isn't used or if the \noauxfile macro is defined, and
%                     removed some .aux-file-opening detritus; printed the
%                     cite-key of undefined citations in \tt font; changed the
%                     catcode of `_' inside \cite; and called \@resetnumerals
%                     from inside a group.
%   29-Feb-92  0.99i, made these macros semi-compatible with AmS-TeX; removed
%                     \@resetnumerals, \@setletters, \@tokstostring, and
%                     friends; changed the way \cite handles catcodes; changed
%                     \@getoptionalarg, and had \bibitem and \newcommand use
%                     it; added \@futurenonspacelet and (to facilitate the use
%                     of multiple reference lists) \bblfilebasename; changed
%                     \biblabelprint to use the new macros \biblabelprecontents
%                     and \biblabelpostcontents, and to, by default, right-
%                     justify numeric labels; and renamed \biblabelextrahang to
%                     the more descriptive \biblabelextraspace.
%   14-Mar-92  0.99j, made 0.99i's use of `\\' local to btxmac.tex.
%
%
% The LaTeX-related commands defined in this file include (a) the four
% commands that a user types (\bibliography, \bibliographystyle, \cite,
% and \nocite); (b) the three commands that BibTeX looks for in the .aux
% file (\bibdata, \bibstyle, and \citation---there is a fourth command
% that BibTeX looks for, but that command is related to LaTeX's \include
% facility, so these macros ignore that command); and (c) a LaTeX
% command (\newcommand) that's written by one of the four standard
% bibliography styles (alpha).  The definitions here are not exactly the
% same as the corresponding LaTeX definitions (those eight LaTeX
% definitions depend on a significant fraction of LaTeX itself).  But
% the only substantial differences are with \newcommand, which here,
% without complaining, lets you redefine a preexisting control sequence
% (in LaTeX, \newcommand won't let you redefine a preexisting command),
% and which here doesn't make the control sequences it defines \long (in
% LaTeX, that happens automatically); there may also be other minor
% differences.  To summarize: Unless you know what you're doing, you
% shouldn't define any control sequences with these eight names:
%
% \bibdata
% \bibliography
% \bibliographystyle
% \bibstyle
% \citation
% \cite
% \newcommand
% \nocite
%
% There are three other commands written by one or more of the four
% standard (plain, abbrv, alpha, unsrt) or four semistandard (acm,
% apalike, ieeetr, siam) bibliography styles; they take effect only
% within the bibliography, and are redefinable, as explained later:
%
% \em
% \newblock
% \sc
%
% There's one control sequence you might want to use (but not redefine)
% in redefining \biblabelprint:
%
% \biblabelwidth
%
% There are fifteen other control sequences (explained later in more detail)
% that the macros of this file will use if you define them---you should
% define them after the \input btxmac command but before the \bibliography
% command.  The first six begin with `\bbl' and affect fonts, spacing,
% perhaps other characteristics of the bibliography, and which .bbl files
% get read; the next five begin with `\biblabel' and determine how labels
% are formatted in the bibliography; and the last four begin with `\print'
% and determine how the in-text citations are formatted:
%
% \bblem
% \bblfilebasename
% \bblhook
% \bblnewblock
% \bblrm
% \bblsc
% \biblabelcontents
% \biblabelprecontents
% \biblabelprint
% \biblabelpostcontents
% \biblabelextraspace
% \printbetweencitations
% \printcitefinish
% \printcitenote
% \printcitestart
%
% If it's defined before the \input btxmac command, the control sequence
% below inhibits the reading and writing of the .aux file(s), and the
% issuing of related warning messages.  Any definition will do.  This
% feature might help when you're working on draft stages of a document:
%
% \noauxfile
%
%
% Here's another control sequence (it's described later) that you
% probably won't want to redefine unless you are writing another macro
% package; if you do redefine it, however, do it before the \input btxmac
% command (and notice that it has an `@' in its name):
%
% \@undefinedmessage
%
% Any other control sequence in this file that might conflict with
% something you've defined will have an `@' in its name, so such conflicts
% are unlikely; but if you're worried about a specific control sequence
% name, do a text search of this file to look for it.
%
%
% So to start things off we turn `@' into a letter (category code 11),
% keeping track of the old category code for future restoration.
% (Simply resetting it to 12 when we leave these macros is
% insufficient.)  The use of `\cite' as a temporary control sequence is
% a kludge, but it's a reasonably simple way to accomplish what we need
% without possibly overwriting something (without an `@' in its name)
% that might already be defined.
%
\edef\cite{\the\catcode`@}%
\catcode`@ = 11
\let\@oldatcatcode = \cite
\chardef\@letter = 11
\chardef\@other = 12
%
%
% Next come some things that will be useful later.
%
% Make an outer definition into an inner one (due to Chris Thompson).
% The arguments should be the control sequence to be defined, and the
% new of the \outer control sequence, as characters; the control
% sequence #1 is defined to be just the same as \csname#2\endcsname, but
% not \outer.  For example, \@innerdef\innernewcount{newcount} would
% define \innernewcount to be a non-outer version of \newcount.
%
\def\@innerdef#1#2{\edef#1{\expandafter\noexpand\csname #2\endcsname}}%
%
% We use \@innerdef to make some of our allocations local, because
% Eplain includes our code inside a conditional.  We put @'s in the
% names to minimize the (already small) chance of conflicts.
%
\@innerdef\@innernewcount{newcount}%
\@innerdef\@innernewdimen{newdimen}%
\@innerdef\@innernewif{newif}%
\@innerdef\@innernewwrite{newwrite}%
%
%
% Swallow one parameter.
%
\def\@gobble#1{}%
%
%
% Use TeX 3.0's \inputlineno to get the line number, for better error
% messages, but if we're using an old version of TeX, don't do anything.
%
\ifx\inputlineno\@undefined
   \let\@linenumber = \empty % Pre-3.0.
\else
   \def\@linenumber{\the\inputlineno:\space}%
\fi
%
%
% The following macro \@futurenonspacelet (from the TeXbook) behaves
% essentially like \futurelet except that it discards any implicit or
% explicit space tokens that intervene before a nonspace is scanned:
%
\def\@futurenonspacelet#1{\def\cs{#1}%
   \afterassignment\@stepone\let\@nexttoken=
}%
\begingroup % The grouping here avoids stepping on an outside use of `\\'.
\def\\{\global\let\@stoken= }%
\\ % now \@stoken is a space token (\\ is a control symbol, so that
   % space after it is seen).
\endgroup
\def\@stepone{\expandafter\futurelet\cs\@steptwo}%
\def\@steptwo{\expandafter\ifx\cs\@stoken\let\@@next=\@stepthree
   \else\let\@@next=\@nexttoken\fi \@@next}%
\def\@stepthree{\afterassignment\@stepone\let\@@next= }%
%
%
% \@getoptionalarg\CS gets an optional argument from the input, enclosed
% in brackets, then expands \CS.  We set \@optionalarg to \empty if we
% don't find one, otherwise to the text of the argument.  This assumes
% the brackets don't have a funny category code.
%
\def\@getoptionalarg#1{%
   \let\@optionaltemp = #1%
   \let\@optionalnext = \relax
   \@futurenonspacelet\@optionalnext\@bracketcheck
}%
%
% The \expandafter's in this macro let us avoid the use of \aftergroup,
% which is somewhat more expensive.
%
\def\@bracketcheck{%
   \ifx [\@optionalnext
      \expandafter\@@getoptionalarg
   \else
      \let\@optionalarg = \empty
      % We can't do the \temp after the \fi, because then the \temp gets
      % in the way of reading the optional argument from the input, if
      % we do have one.
      \expandafter\@optionaltemp
   \fi
}%
%
\def\@@getoptionalarg[#1]{%
   \def\@optionalarg{#1}%
   \@optionaltemp
}%
%
%
% From LaTeX.
%
\def\@nnil{\@nil}%
\def\@fornoop#1\@@#2#3{}%
%
\def\@for#1:=#2\do#3{%
   \edef\@fortmp{#2}%
   \ifx\@fortmp\empty \else
      \expandafter\@forloop#2,\@nil,\@nil\@@#1{#3}%
   \fi
}%
%
\def\@forloop#1,#2,#3\@@#4#5{\def#4{#1}\ifx #4\@nnil \else
       #5\def#4{#2}\ifx #4\@nnil \else#5\@iforloop #3\@@#4{#5}\fi\fi
}%
%
\def\@iforloop#1,#2\@@#3#4{\def#3{#1}\ifx #3\@nnil
       \let\@nextwhile=\@fornoop \else
      #4\relax\let\@nextwhile=\@iforloop\fi\@nextwhile#2\@@#3{#4}%
}%
%
%
% This macro tests if a file \jobname.#1 exists, and sets \if@fileexists
% appropriately.  If an optional argument is given, it is used as the
% root part of the filename instead of \jobname.
%
\@innernewif\if@fileexists
%
\def\@testfileexistence{\@getoptionalarg\@finishtestfileexistence}%
\def\@finishtestfileexistence#1{%
   \begingroup
      \def\extension{#1}%
      \immediate\openin0 =
         \ifx\@optionalarg\empty\jobname\else\@optionalarg\fi
         \ifx\extension\empty \else .#1\fi
         \space
      \ifeof 0
         \global\@fileexistsfalse
      \else
         \global\@fileexiststrue
      \fi
      \immediate\closein0
   \endgroup
}%
%
%
%% [[[start of BibTeX-specific stuff]]]
%
% Now come the four main LaTeX commands and their associated .aux
% commands.  Just as in LaTeX, \bibliographystyle defines the BibTeX
% style name (.bst file, that is), and \bibliography defines the
% database (.bib) file(s).  The corresponding .aux-file commands are
% \bibstyle and \bibdata, which are there only for BibTeX's (but not
% LaTeX's) use.
%
\def\bibliographystyle#1{%
   \@readauxfile
   \@writeaux{\string\bibstyle{#1}}%
}%
\let\bibstyle = \@gobble
%
% As well as writing the \bibdata command to tell BibTeX which .bib
% files to read, we read the .bbl file that BibTeX (or a person,
% conceivably) has produced.  We use \bblfilebasename as the root of the
% filename to read; this defaults to \jobname.
%
\let\bblfilebasename = \jobname
\def\bibliography#1{%
   \@readauxfile
   \@writeaux{\string\bibdata{#1}}%
   \@testfileexistence[\bblfilebasename]{bbl}%
   \if@fileexists
      % We just output a non-discardable item (the `whatsit' with the
      % \bibdata command).  This means that the glue that will be
      % inserted next (\parskip or \baselineskip, most likely) will be a
      % legal breakpoint.  Most likely, this is after some kind of
      % heading, where we don't want to allow a page break.  So:
      \nobreak
      \@readbblfile
   \fi
}%
\let\bibdata = \@gobble
%
% The \nocite{label,label,...} command writes its argument to \@auxfile,
% unless instructed not to, but produces no text in the document.  Both
% the \nocite and \cite commands produce \citation commands in the .aux file.
%
\def\nocite#1{%
   \@readauxfile
   \@writeaux{\string\citation{#1}}%
}%
%
\@innernewif\if@notfirstcitation
%
% \cite[note]{label,label,...} produces the citations for the labels as
% well.  If the optional argument `note' is present, it's added after
% the labels.  Since \cite calls \nocite to do its .aux-file writing,
% \cite doesn't need to call \@readauxfile (\nocite does).
%
\def\cite{\@getoptionalarg\@cite}%
%
% Typeset the citations for the labels in #1, followed by the note, if
% it exists.  To change the citation's format in the text, redefine one
% or more `\print...' macros, whose defaults appear later in this file.
%
\def\@cite#1{%
   % Remember the optional argument, in case one of the macros we call
   % below ends up looking for an optional argument itself.  For
   % example, if a \cite[note] triggers reading the .aux file, then the
   % [note] would be clobbered, since \@testfileexistence looks for an
   % optional arg.
   \let\@citenotetext = \@optionalarg
   % Start printing the text, beginning with a left bracket by default.
   \printcitestart
   % It's complicated, but because \nocite puts a `whatsit' onto the list,
   % \nocite should follow \printcitestart.  It's conceivable, but very
   % unlikely, that this `whatsit' will cause a problem (glue that doesn't
   % disappear when you want it to is the most likely symptom), requiring
   % a change either to \printcitestart or to the label that the .bst file
   % produces.
   \nocite{#1}%
   \@notfirstcitationfalse
   \@for \@citation :=#1\do
   {%
      \expandafter\@onecitation\@citation\@@
   }%
   \ifx\empty\@citenotetext\else
      \printcitenote{\@citenotetext}%
   \fi
   \printcitefinish
}%
%
\newif\ifweareinprivate
\weareinprivatetrue
\ifx\shlhetal\undefinedcontrolseq\weareinprivatefalse\fi
\ifx\shlhetal\relax\weareinprivatefalse\fi
\def\@onecitation#1\@@{%
   \if@notfirstcitation
      \printbetweencitations
   \fi
   %
   \expandafter \ifx \csname\@citelabel{#1}\endcsname \relax
      \if@citewarning
         \message{\@linenumber Undefined citation `#1'.}%
      \fi
      % Give it a dummy definition:
     \ifweareinprivate
      \expandafter\gdef\csname\@citelabel{#1}\endcsname{%
% Change: marginal remark added, goldstrn@math.huji.ac.il, 
% goldstern@tuwien.ac.at, May 1996 mg
%  !!! change !!!
\strut 
\vadjust{\vskip-\dp\strutbox
\vbox to 0pt{\vss\parindent0cm \leftskip=\hsize 
\advance\leftskip3mm
\advance\hsize 4cm\strut\openup-4pt 
\rightskip 0cm plus 1cm minus 0.5cm ?  #1 ?\strut}}
         {\tt
            \escapechar = -1
            \nobreak\hskip0pt\pfeilsw%\special{ps:100 0 0 setrgbcolor }%
            \expandafter\string\csname#1\endcsname
                     %\special{ps:0 0 0 setrgbcolor }
             \pfeilso
            \nobreak\hskip0pt
         }%
      }%
     \else  % ifweareinprivate = false
      \expandafter\gdef\csname\@citelabel{#1}\endcsname{%
            {\tt\expandafter\string\csname#1\endcsname}
      }%
     \fi  % ifweareinprivate
   \fi
   % Now produce the text, whether it was undefined or not.
   \csname\@citelabel{#1}\endcsname
   \@notfirstcitationtrue
}%
%
% Given a label `foo', the macro `\b@foo' is supposed to
% hold the text that should be produced.
%
\def\@citelabel#1{b@#1}%
%
% So, how does a citation label get defined?  When we read the .bbl file
% (below), a \bibitem writes out a \@citedef command.  And when we read
% the \@citedef, we define \@citelabel{#1}, where #1 is the user's
% label.
%
\def\@citedef#1#2{\expandafter\gdef\csname\@citelabel{#1}\endcsname{#2}}%
%
%
% Reading the .bbl file also produces the typeset bibliography.  Please
% notice, however, that we do not produce the title for the references
% (e.g., `References'), as LaTeX does.  The formatting and spacing of
% that title, whether it should go into the headline, and so on, are all
% things determined by your format.  We cannot know those things in
% advance.  If you wish, you can define \bblhook to produce the title.
% Or just do it before the \bibliography command.
%
\def\@readbblfile{%
   % Define a counter to tell us which item number we are on, unless
   % we've already defined it (because the document has more than one
   % bibliography).
   \ifx\@itemnum\@undefined
      \@innernewcount\@itemnum
   \fi
   %
   \begingroup
      \def\begin##1##2{%
         % ##1 is just `thebibliography'.
         % ##2 is the widest label.
         % We set (new dimen) \biblabelwidth based on the widest label
         \setbox0 = \hbox{\biblabelcontents{##2}}%
         \biblabelwidth = \wd0
      }%
      \def\end##1{}% ##1 is `thebibliography' again.
      %
      % Here we have two possibilities:
      % \bibitem[typesetlabel]{citationlabel}
      % \bibitem{citationlabel}
      % If we have the second of these, the citations are numbered, starting
      % from one; we use our own count register \@itemnum for this.
      %
      \@itemnum = 0
      \def\bibitem{\@getoptionalarg\@bibitem}%
      \def\@bibitem{%
         \ifx\@optionalarg\empty
            \expandafter\@numberedbibitem
         \else
            \expandafter\@alphabibitem
         \fi
      }%
      \def\@alphabibitem##1{%
         % Need \xdef here for various reasons.
         \expandafter \xdef\csname\@citelabel{##1}\endcsname {\@optionalarg}%
         % Left-justify alpha labels, unless \biblabel{pre,post}contents
         % are already defined.
         \ifx\biblabelprecontents\@undefined
            \let\biblabelprecontents = \relax
         \fi
         \ifx\biblabelpostcontents\@undefined
            \let\biblabelpostcontents = \hss
         \fi
         \@finishbibitem{##1}%
      }%
      %
      \def\@numberedbibitem##1{%
         \advance\@itemnum by 1
         \expandafter \xdef\csname\@citelabel{##1}\endcsname{\number\@itemnum}%
         % Right-justify numeric labels, unless \biblabel{pre,post}contents
         % are already defined.
         \ifx\biblabelprecontents\@undefined
            \let\biblabelprecontents = \hss
         \fi
         \ifx\biblabelpostcontents\@undefined
            \let\biblabelpostcontents = \relax
         \fi
         \@finishbibitem{##1}%
      }%
      %
      \def\@finishbibitem##1{%
         \biblabelprint{\csname\@citelabel{##1}\endcsname}%
         \@writeaux{\string\@citedef{##1}{\csname\@citelabel{##1}\endcsname}}%
         \ignorespaces
      }%
      %
      % Do the printing (we're producing the bibliography, remember).
      %
      \let\em = \bblem
      \let\newblock = \bblnewblock
      \let\sc = \bblsc
      % Punctuation won't affect spacing;
      \frenchspacing
      % the penalties below are from LaTeX's [article,book,report].sty;
      \clubpenalty = 4000 \widowpenalty = 4000
      % the next two values come from LaTeX's \sloppy command;
      \tolerance = 10000 \hfuzz = .5pt
      \everypar = {\hangindent = \biblabelwidth
                      \advance\hangindent by \biblabelextraspace}%
      \bblrm
      % the \parskip is a guess at what looks good;
      \parskip = 1.5ex plus .5ex minus .5ex
      % and the space between label and text comes from LaTeX's \labelsep.
      \biblabelextraspace = .5em
      \bblhook
      %
      \input \bblfilebasename.bbl
   \endgroup
}%
%
% The widest label's width is useful for redefining \biblabelprint;
% you redefine \biblabelwidth, in effect, by redefining the
% \biblabelcontents macro that appears below.  And \biblabelextraspace,
% which is redefinable inside \bblhook, is added to \biblabelwidth to
% determine the amount of hanging indentation.
%
\@innernewdimen\biblabelwidth
\@innernewdimen\biblabelextraspace
%
% Now come the main macros that are related to the printing of the
% bibliography.  Since you might want to redefine them, they are given
% default definitions outside of \@readbblfile.
%
% The first one controls the printing of a bibliography entry's label.
% If you change it, make sure that it starts with something like
% \noindent or \indent or \leavevmode that puts TeX into horizontal mode
% (even if the label itself is empty); otherwise, the hanging
% indentation will get messed up in certain circumstances.
%
\def\biblabelprint#1{%
   \noindent
   \hbox to \biblabelwidth{%
      \biblabelprecontents
      \biblabelcontents{#1}%
      \biblabelpostcontents
   }%
   \kern\biblabelextraspace
}%
%
% If you are using numeric labels, and you want them left-justified
% (numeric labels by default are right-justified), do something like:
%     \def\biblabelprecontents{\relax}
%     \def\biblabelpostcontents{\hss}
%
% By default the labels are typeset in \bblrm, and enclosed in brackets.
%
\def\biblabelcontents#1{{\bblrm [#1]}}%
%
% The main text, too, is typeset using \bblrm, which is \rm by default.
%
\def\bblrm{\rm}%
%
% Emphasis for producing, e.g., titles, is done with \it by default.
%
\def\bblem{\it}%
%
% Some styles use a caps-and-small-caps font for author names.  LaTeX
% defines an \sc command but plain TeX doesn't, so we need one here.
% The definition below doesn't load the font unless it's needed, but it
% tries to load only the 10pt version, because it might not exist at
% other point sizes.
%
\def\bblsc{\ifx\@scfont\@undefined
              \font\@scfont = cmcsc10
           \fi
           \@scfont
}%
%
% The major parts of an entry are separated with \bblnewblock.  The
% numbers below are taken from LaTeX's `article' style.
%
\def\bblnewblock{\hskip .11em plus .33em minus .07em }%
%
% Here's where you stick any other bibliography-formatting goodies, or
% redefine the values above.
%
\let\bblhook = \empty
%
%
% Here are the four default definitions for formatting the in-text
% citations.  These are what you redefine (after your \input btxmac but
% before your \bibliography) to get parens instead of brackets, or
% superscripts, or footnotes, or whatever.
%
\def\printcitestart{[}%         left bracket
\def\printcitefinish{]}%        right bracket
\def\printbetweencitations{, }% comma, space
\def\printcitenote#1{, #1}%     comma, space, note (if it exists)
%
% That scheme is pretty flexible.  For example you could use
%     \def\printcitestart{\unskip $^\bgroup}
%     \def\printcitefinish{\egroup$}
%     \def\printbetweencitations{,}
%     \def\printcitenote#1{\hbox{\sevenrm\space (#1)}}
%     \font\eighttt = cmtt8
%     \scriptfont\ttfam = \eighttt
% to get superscripted in-text citations.  (The scriptfont stuff
% exists only to print an undefined citation; it's in cmtt8 because
% there is no cmtt7.)  To get something radically different, however,
% you'll have to define your own \cite command.
%
% When we read `\citation' from the .aux file, it means nothing.
%
\let\citation = \@gobble
%
%
% Now comes the stuff for dealing with LaTeX's \newcommand.  As
% mentioned earlier, this \newcommand will redefine a preexisting
% command; that's different from how LaTeX's \newcommand behaves.
%
\@innernewcount\@numparams
%
% \newcommand{\foo}[n]{text} defines the control sequence \foo to have
% n parameters, and replacement text `text'.
%
\def\newcommand#1{%
   \def\@commandname{#1}%
   \@getoptionalarg\@continuenewcommand
}%
%
% Figure out if this definition has parameters.
%
\def\@continuenewcommand{%
   % If no optional argument, we have zero parameters.  Otherwise, we
   % have that many.
   \@numparams = \ifx\@optionalarg\empty 0\else\@optionalarg \fi \relax
   \@newcommand
}%
%
% \@numparams is how many arguments this command has.  The name of the
% command is \@commandname.  The replacement text for the new macro is #1.
%
\def\@newcommand#1{%
   \def\@startdef{\expandafter\edef\@commandname}%
   \ifnum\@numparams=0
      \let\@paramdef = \empty
   \else
      \ifnum\@numparams>9
         \errmessage{\the\@numparams\space is too many parameters}%
      \else
         \ifnum\@numparams<0
            \errmessage{\the\@numparams\space is too few parameters}%
         \else
            \edef\@paramdef{%
               % This is disgusting, but \loop doesn't work inside \edef,
               % because \body isn't defined.
               \ifcase\@numparams
                  \empty  No arguments.
               \or ####1%
               \or ####1####2%
               \or ####1####2####3%
               \or ####1####2####3####4%
               \or ####1####2####3####4####5%
               \or ####1####2####3####4####5####6%
               \or ####1####2####3####4####5####6####7%
               \or ####1####2####3####4####5####6####7####8%
               \or ####1####2####3####4####5####6####7####8####9%
               \fi
            }%
         \fi
      \fi
   \fi
   \expandafter\@startdef\@paramdef{#1}%
}%
%
%% [[[end of BibTeX-specific stuff]]]
%
%
% Names of references (arguments given in the \cite and \nocite
% commands) and file names (arguments given in the \bibliography and
% \bibliographystyle commands) are recorded in \jobname.aux, called the
% \@auxfile in these macros.  Here's how they get read in.
%
\def\@readauxfile{%
   \if@auxfiledone \else % remember: \@auxfiledonetrue if \noauxfile is defined
      \global\@auxfiledonetrue
      \@testfileexistence{aux}%
      \if@fileexists
         \begingroup
            % Because we might be in horizontal mode when \@readauxfile
            % is called (if it's in response to a \cite or \nocite), we
            % want to ignore all the would-be spaces at the ends of
            % lines in the aux file.  Fortunately, it's highly unlikely
            % an end-of-line might actually be desired.
            % And because we don't change the category code of anything
            % but @, primitives like \gdef can't be used to define labels
            % in the aux file.  The solution adopted by btxmac.tex is to
            % write `\@citedef{LABEL}{DEFINITION}' to the aux file, and
            % use \csname on LABEL.
            \endlinechar = -1
            \catcode`@ = 11
            \input \jobname.aux
         \endgroup
      \else
         \message{\@undefinedmessage}%
         \global\@citewarningfalse
      \fi
      \immediate\openout\@auxfile = \jobname.aux
   \fi
}%
%
% The \@readauxfile macro does all that work the first time it's called.
% Since it's called once for every \cite, \nocite, \bibliography, and
% \bibliographystyle command that the user issues, we need to remember
% whether the work's been done.  It's considered done if we're not to do
% it---that is, if \noauxfile is defined.
%
\newif\if@auxfiledone
\ifx\noauxfile\@undefined \else \@auxfiledonetrue\fi
%
% It's conceivable you'd want to change how other characters are read;
% to do that, change their category code before doing \input btxmac.
%
%
% After reading the .aux file, \@readauxfile opens it for writing.
% The \@writeaux macro does the actual writing (as long as
% \noauxfile is undefined).
%
\@innernewwrite\@auxfile
\def\@writeaux#1{\ifx\noauxfile\@undefined \write\@auxfile{#1}\fi}%
%
%
% A macro package that uses btxmac.tex might define
% \@undefinedmessage (before doing an \input btxmac).
%
\ifx\@undefinedmessage\@undefined
   \def\@undefinedmessage{No .aux file; I won't give you warnings about
                          undefined citations.}%
\fi
%
% Even if citations are undefined, we want to complain only if
% \@citewarningtrue.  The default is to set \@citewarningtrue unless
% \noauxfile is defined.  Again, a macro package that uses
% btxmac.tex might want to redefine this.
%
\@innernewif\if@citewarning
\ifx\noauxfile\@undefined \@citewarningtrue\fi
%
%
% Finally, before leaving we restore @'s old category code.
%
\catcode`@ = \@oldatcatcode



\def\pfeilso{\leavevmode
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           \vrule width 1pt height8.3pt depth 0pt\relax
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\def\pfeilnw{\leavevmode
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\def\pfeilsw{ \leavevmode 
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\def\pfeilno{ \leavevmode 
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      }


%  *** end including bib4plain.tex *** 
  % This will define \cite and make sure it works as in latex


\def\widestnumber#1#2{}
  % Our amstex-ppt style does not know about \widestnumber




\def\citewarning#1{\ifx\shlhetal\relax 
   % normal mode, do not write anything
    \else
   % private mode
    \par{#1}\par
    \fi
}




\def\rm{\fam0 \tenrm}


\def\fakesubhead#1\endsubhead{\bigskip\noindent{\bf#1}\par}


% % \input rsfs
%  *** start including rsfs.tex *** 

% # Keywords: Script or Calligraphic (Caligraphic) letters with the RSFS Font

% The story so far:    July 1998 -- Saharon would like to have a
% ``nicer'' calligraphic font. In particualr, the leters S and P in
% the usual calligraphic font do not look ``special'' enough. 
% 
% I found out that ``rsfs'' (``Ralph Smith Formal Script'') may be
% what he wants.   I installed the mf file, the .tfm file, as well as
% a few pk files in ~/TeX/rsfs.    Let's hope that this is enough.
% Using amstex, all you have to do is to \input rsfs.tex 
% Files prepared with citealice willdothis automatically. 
%
%  Note:  for some reason xdvi calls MakeTeXpk, then Maketexpk
%  complains about wrong resolution, but still writes commands to
%  missfont.log...  
%



% we redefine a macro inside amstex's \Cal command , so that it calls
% our nice font ``rsfs'' rather than the usual calligraphic font. 
% Note thisworks for amstex only.   
% In plain tex, would have to add definitions of \Cal
% in latex... we should insteaduse mathrsfs.sty
% 



\font\textrsfs=rsfs10
\font\scriptrsfs=rsfs7
\font\scriptscriptrsfs=rsfs5


\newfam\rsfsfam
\textfont\rsfsfam=\textrsfs
\scriptfont\rsfsfam=\scriptrsfs
\scriptscriptfont\rsfsfam=\scriptscriptrsfs


\edef\oldcatcodeofat{\the\catcode`\@}
\catcode`\@11

\def\Cal@@#1{\noaccents@ \fam \rsfsfam #1}

\catcode`\@\oldcatcodeofat

%  *** end including rsfs.tex *** 

\expandafter\ifx \csname margininit\endcsname \relax\else\margininit\fi



\long\def\red#1\endred{}
\long\def\green#1\endgreen{}
\long\def\blue#1\endblue{}

\def\endred{ \unmatched endred! }
\def\endgreen{ \unmatched endgreen! }
\def\endblue{ \unmatched endblue! }

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 \let\outerhead\head
 \def\head{\innerhead}
 \let\innerhead\outerhead

 \let\outersubhead\subhead
 \def\subhead{\innersubhead}
 \let\innersubhead\outersubhead

 \let\outersubsubhead\subsubhead
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\def\demo#1{\medskip\noindent{\it #1.\/}}
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\def\remark#1{\medskip\noindent{\it #1.\/}}
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%  *** end including alice2jlem.tex *** 
\pageheight{8.5truein}
\topmatter
\title{{\it A More General Iterable Condition \\
Ensuring $\aleph_1$ is not Collapsed}} \endtitle
\rightheadtext{General Iterable Condition}
\author {Saharon Shelah 
\thanks {\null\newline
I would like to thank Alice Leonhardt for the beautiful typing.\newline
Publication 311.} \endthanks} \endauthor
\affil{Institute of Mathematics\\
 The Hebrew University\\
 91 904 Jerusalem, Israel
 \medskip
 Rutgers University\\
 Mathematics Department\\
 New Brunswick, NJ  USA} \endaffil

\mn
\abstract   In a self-contained way, we deal with revised countable
support iterated forcing for the reals; improve theorems on
preservation of a property UP, weaker than semi proper, and hopefully
improve the presentation.  We continue \cite[Ch.X,XI]{Sh:b} (or see
\cite[Ch.X,XI]{Sh:f}), and 
Gitik Shelah \cite{GiSh:191} and \cite[Ch.XIII,XIV]{Sh:f} and 
particularly Ch.XV; concerning ``no new reals" see lately
Larson Shelah \cite{LrSh:746}.  In particular, we fulfill some promises from
\cite{Sh:f} and give a more streamlined version.  \endabstract
\endtopmatter
\document  
 
\newpage

\head {Annotated Content} \endhead  \resetall 
 % \resetall 
\bn
\S1 \ub{Preliminaries}, p.5
\mr
\item "{{}}"  [We agree that forcing notion $\Bbb P$ has actually also pure
$(\le_{pr})$ quasi-order and very pure $(<_{vpr})$ quasi-order.  For a
$\lessdot$-increasing sequence $\bar{\Bbb Q}$ of forcing notions we 
define what is a $\bar{\Bbb Q}$-named and a $\bar{\Bbb Q}$-named 
$[j,\alpha)$-ordinal.  Then we define $\kappa-Sp_e(W)$-iterations 
(revised support of size $< \kappa$, including the
case $\kappa$ inaccessible) with finite apure support, countable pure support
(the revised version) and Easton or W-Easton very pure support, similar to
\cite[XIV]{Sh:f} and prove its basic properties (this is done by simultaneous
induction).]
\endroster
\bn
\S2 \ub{Trees of Models}, p.31
\mr
\item "{{}}"  [We quote the basic definitions and theorems concerning
trees with $\omega$ levels tagged by ideal and partition theorems.]
\endroster
\bn
\S3 \ub{Ideals and Partial Orders}, p.36
\mr
\item "{{}}" [We can replace the families $\Bbb I$ of ideals by corresponding
partial orders or quasi orders (we ``ignore" the distinction).  This is
essentially equivalent (for ``some $\lambda$-complete $\Bbb I$" with ``for 
some $\lambda$-complete ${\Cal L}$")
but the ${\Cal L}$'s have better ``pullback" from forcing extensions, so we
can replace ${\underset\tilde {}\to{\Cal L}}$ in a forcing extension
of $\bold V$ by ${\Cal L}'$ in $\bold V$ 
preserving ${\underset\tilde {}\to{\Cal L}} \le_{RK} 
{\Cal L}'$ and preserving the amount of completeness we have, so a similar 
situation holds for a set of ideals; in the cases we have in mind here 
increasing those sets $\Bbb I$ or ${\Cal L}$ do not matter.]
\endroster
\bn
\S4 \ub{UP Reintroduced}, p.42
\mr
\item "{{}}" [We define when $\bar N$ is an $\Bbb I$-tagged tree of 
models, when
it is $\Bbb I$-suitable, or $(\Bbb I,\bold W)$-suitable, and when it is
strictly or $\lambda$-strictly, etc., where $\Bbb I$ is a family of
ideals.   Similarly we define $N$ is 
$\lambda$-strictly $(\Bbb I,\bold S,\bold W)$-suitable; i.e. can serve as 
$N_{\langle \rangle}$ and prove some basic properties.  Such models will
fulfill here the role that ``any countable $N \prec ({\Cal H}(\chi),
\in)$" fulfills in theorems on semi-proper iteration.  Lastly, we define 
when a forcing notion $\Bbb P$ satisfies $UP^\ell(\Bbb I,\bold W)$ for 
$\ell = 0,1,2$, and here
$UP^1,UP^2$ replace $\bold W$-proper, $\bold W$-semi-proper, where $\bold W$
is a stationary subset of $\omega_1$.  All those properties imply 
that forcing by $\Bbb P$ does not collapse $\aleph_1$, preserve 
the stationarity of $\bold W$
and even of any stationary subset of it.  They are all relatives of 
``semi-properness" for strictly $(\Bbb I,\bold W)$-suitable models, they
speak on $\Bbb I$-tagged trees of countable models.]
\endroster
\bn
\S5 \ub{An Iteration Theorem for $UP^1$}, p.53 
\mr
\item "{{}}" [We prove that satisfying $UP^1(\bold W)$, i.e. satisfying it 
for some family $\Bbb I$ of ideals, complete enough, is preserved by 
$\aleph_1-Sp_6(W)$-iterations, $\bar{\Bbb Q} = \langle \Bbb P_i,
{\underset\tilde {}\to {\Bbb Q}_i}:i < \alpha \rangle$; that is if each 
${\underset\tilde {}\to {\Bbb Q}_i}$ is like that then $P_\alpha =
\aleph_1-Sp_6(W)$-Lim$(\bar{\Bbb Q})$ is like that, provided some 
mild condition holds (say $\Bbb Q_i$ is \nl
$UP^1(\Bbb I_i,\bold W),{\underset\tilde {}\to {\Bbb I}_i}$
is ${\underset\tilde {}\to \kappa_i}$-complete, $\Bbb P_i$ satisfies the
$\kappa_i$-c.c.; we can even make ${\underset\tilde {}\to {\Bbb I}_i},
{\underset\tilde {}\to \kappa_i}$ to be just $\Bbb P_{i+1}$-names, 
see there).  
The proof is more similar to the proofs of preservation of properness and 
semi-properness \ub{than} with the proofs in \cite[XI]{Sh:b}, 
(=\cite[XI]{Sh:f}), \cite{GiSh:191}, \cite[XV]{Sh:f}, and hopefully more 
transparent.  The proof will be non-trivially shorter if we use just 
the particular case of the revised countable support (i.e.,
$\le_{\text{vpr}}$ is equality and $\le_{\text{pr}}$ is $\le$).  
We give a sufficient
condition for $\alpha$ 
not being collapsed by $\Bbb P_\alpha$ e.g. $\alpha$ is
strongly inaccessible, $\beta < \alpha \Rightarrow |\Bbb P_\beta| 
< \alpha$ and:
$W$ stationary in $\alpha$ (so $\alpha$ is Mahlo) or
$\le_{\text{vpr}}$ is equality
and the iteration is suitable enough.  Lastly, if e.g. $\alpha$ is the first
strongly inaccessible, $i < \alpha \Rightarrow |\Bbb P_i| < \alpha$ we give a
sufficient condition for $\alpha$ not being collapsed.]
\endroster
\bn
\S6 \ub{Preservation of $UP^0$}, p.72
\mr
\item "{{}}" [Here we make the condition more similar to semi-proper
iteration, that is the demand is that for suitable models $N$ (one on which
``the right trees grow") above each $p \in \Bbb Q \cap N$ there is an 
$(N,\Bbb Q)$-semi-generic $q$.  There is some price though. \nl
[?] However, if $\Bbb Q$ satisfies UP$^0$ and the $\kappa$-c.c., then 
$\Bbb Q * {\underset\tilde {}\to {\text{Levy}}}(\aleph_1,< \kappa)$ 
is appropriate in our iteration.]
\endroster  
\bn
\S7 \ub{No New Reals} - replacements for completeness, p.91
\mr
\item "{{}}"  [Here we deal with the parallel of $``\Bbb Q$ 
add no new real because it is \nl
$\bold W$-complete for some stationary $\bold W \subseteq \omega_1"$.]
\endroster
\bn
\S8 \ub{Examples}, p.98
\mr
\item "{{}}"  [We show that various forcing notions fall under our
context.  In particular (variants of) Namba forcing, shooting a club through
a stationary $S \subseteq \{\delta < \lambda:\text{cf}(\delta) = \aleph_0\}$
where $\lambda = \text{ cf}(\lambda) > \aleph_0$, and prove that the 
older condition from \cite{Sh:f} implies the present one.]
\endroster
\bn
\S9 \ub{Reflection in $[\omega_2]^{\aleph_0}$}, p.104
\mr
\item "{{}}"  [We answer a question of Jech, on the consistency of
$2^{\aleph_0} = \aleph_2 + {\Cal D}_{\omega_1}$ 
is $\aleph_2$-saturated + every
stationary subset of $[\omega_2]^{\aleph_0}$ reflects and 
there is a special projectively stationary subset of 
$[\omega_2]^{\aleph_0}$.]
\endroster
\bn
\S10  \ub{Mixing finitary norms and ideals}, p.110
\mr
\item "{{}}"  [We consider a common generalization of creature forcing
(see \cite{RoSh:470}) and relatives of Namba forcing.]
\endroster
\newpage

\head {\S1 Preliminaries} \endhead  \resetall \sectno=1
 % \resetall 
\bigskip

\definition{\stag{1.1} Definition/Notation}  1) A forcing notion 
here, $\Bbb P$, is a nonempty set (abusing notation, it too is denoted by
$\Bbb P$) and three partial orders $\le_0,\le_1,\le_2$ 
(more exactly quasi-orders and $\le^*_\ell = \le^{\Bbb P}_\ell$)
 and a minimal element $\emptyset_{\Bbb P} \in \Bbb P$ (so
$\emptyset_{\Bbb P} \le_\ell p$ for $p \in \Bbb P$) and for 
$\ell = 0,1$ we have $[p \le_\ell q \rightarrow p \le_{\ell +1} q]$.  
We call $p \in \Bbb P$ very pure
if $\emptyset_{\Bbb P} \le_0 p$ 
and we call $q$ a very pure extension of $p$ if $p \le_0 q$.
We call $p \in \Bbb P$ pure if $\emptyset_{\Bbb P} \le_1 p$ and 
we call $q$ a pure extension of $p$ if $p \le_1 q$.  Let $\le$ 
be $\le_2$, let $\le_{\text{pr}}$ be $\le_1$ and $\le_{\text{vp}}$ 
be $\le_0$.

We call $\Bbb P$ $\kappa$-vp-complete if: for 
any $<_{\text{vpr}}$-increasing sequence $\langle p_\alpha:
\alpha < \delta \rangle,\delta < \kappa$ with $p_0 = 
\emptyset_{\Bbb P}$ there is a $\le_{text{vpr}}$-upper bound 
$p$.  We define vp-$\kappa$-complete similarly waiving $p_0 =
\emptyset_{\Bbb P}$.
We define $\kappa-\le_\ell$-complete and $\le_\ell-\kappa$-complete
similarly.

The forcing relation, of course, refers to the 
partial order $\le$.  We denote
forcing notions by $\Bbb P,\Bbb Q,\Bbb R$.  
Let $\Bbb P^1 \subseteq \Bbb P^2$ mean
$p \in \Bbb P^1 \Rightarrow p \in \Bbb P^2,\le^{{\Bbb P}^1}_\ell 
= \le^{{\Bbb P}^2}_\ell \restriction
\Bbb P^1$ for $\ell = 0,1,2$ and let $\Bbb P^1 \subseteq_{\text{ic}}
\Bbb P^2$ means $\Bbb P^1 \subseteq \Bbb P^2$ and for $\ell \le 2$, if 
$p,q \in \Bbb P^1$ are $\le_\ell$-incompatible in $\Bbb P^1$, then they
are $\le_\ell$-incompatible in $\Bbb P^2$.  Let $\Bbb P^1 \lessdot
\Bbb P^2$ means $\Bbb P^1 \subseteq_{\text{ic}} \Bbb P^2 \and 
(\Bbb P^1,\le) \lessdot (\Bbb P^2,\le)$. \newline
2) $\bar{\Bbb P}$ denotes a $\lessdot$-increasing 
sequence of forcing notions.
$\bar{\Bbb Q}$ denotes a sequence of the form 
$\langle \Bbb P_i,{\underset\tilde {}\to {\Bbb Q}_i}:i <
\alpha \rangle$ such that 
$\langle \Bbb P_i:i < \alpha)$ is a $\lessdot$-increasing sequence.
Usually ${\underset\tilde {}\to {\Bbb Q}_i}$ is a $\Bbb P_i$-name, 
$\Vdash_{{\Bbb P}_i} ``\Bbb P_{i+1}/\Bbb P_i \cong 
{\underset\tilde {}\to {\Bbb Q}_i}"$. \newline
3) Convention: If $\bar{\Bbb Q} = \langle \Bbb P_i,
{\underset\tilde {}\to {\Bbb Q}_i}:i <
\alpha \rangle,\Bbb P_i$ is $\lessdot$-increasing, we 
may write $\bar{\Bbb Q}$ instead
of $\langle \Bbb P_i:i < \alpha \rangle$. \newline
4) For a forcing notion $\Bbb P$ (as in part (1)) we 
define $\hat{\Bbb P}$:
\mr
\item "{$(a)$}"   the set of elements of $\hat{\Bbb P}$ is 
$$
\align
\biggl\{A:&A \subseteq \Bbb P,\text{ and for some } p \in A \text{ (called a 
witness) we have} \\
  &(i) \quad (\forall q \in A)(\exists r)[q \le r \in A \and p 
\le_{\text{vpr}} r] \\
  &(ii) \quad \text{there is an upper bound } p^* \in \Bbb P \text{ of } A
\text{ such that } p \le_{\text{vpr}} p^* \\
  &\qquad \text{ moreover } (\forall p' \in A)(p \le_{\text{vpr}} p' 
\Rightarrow p' \le_{vpr} p^*) \biggr\}
\endalign
$$
(we call $p^*$ an outer witness for 
$A$ or for $A \in \hat{\Bbb P}$ if clause (ii) hold), and
\sn
\item "{$(b)$}"   $\hat{\Bbb P}$ is ordered by: $A \le B$ iff: 
$A=B$ or $A = \emptyset$ or for some $q \in B,
(\forall p \in A)(p \le q)$ and we call $q$ a witness to $A \le B$
\sn
\item "{$(c)$}"  we define the order $\le_\ell$ on 
$\hat{\Bbb P}$ by: $A \le_\ell B$ \underbar{iff}
$A \le B$ and $A \ne B$ implies that for every witnesses $p$ for $A$
and every witness $q$ for $B$ we have $p \in A \and q \in B \and (\forall p')
(p \le_{\text{vpr}} p' \in A \rightarrow p' \le_\ell q)$; we call
such a pair $(p,q)$ a witness for $A \le_\ell B$. [See \scite{1.8}(5)]
\sn
\item "{$(d)$}"  we stipulate sometime $\emptyset \le_\ell A$ for
every $A \in \hat{\Bbb P}$ \ub{or} $\emptyset = \emptyset_{\Bbb P}$
[Saharon].
\ermn
5) CC$(\Bbb P)$ is the minimal regular uncountable cardinal $\theta$
such that the $\Bbb P$ satisfies the $\theta$-c.c.  We may add
\mr
\item "{$(iii)$}"  if $p' \in A$ satisfies clause $(i) + (ii)$ then
there is $p'' \in A$ such that $p' \le_{text{vpr}} p'' \and p
\le_{\text{vpr}} p''$.
\endroster
\enddefinition
\bigskip

\demo{\stag{1.1A} Observation}:  1) For any forcing notion $\Bbb P$, (as in
\scite{1.1}(1), of course), also $\hat{\Bbb P}$ is a forcing notion 
(in particular $\le^{\hat{\Bbb P}_\ell}$ is a quasi order for $\ell
\le 2$) and $\Bbb P \subseteq_{\text{ic}} \hat{\Bbb P}$ and $\Bbb P 
\lessdot \hat{\Bbb P}$
and $\Bbb P$ is $\le_{\text{vpr}}$-dense in 
$\hat{\Bbb P}$ when we identify $p$ and $\{p\}$. \nl
2) If $A_i \le_\ell B$ for $i < i^*$ then $B \le_{\text{vpr}} B^+$
where $B^+ = \dbcu_{i < i^*} A_i \cup B$. \nl
3) If $\ell \in \{0,1,2\}$ and $(\Bbb P,\le_\ell)$ is $\theta$-complete (i.e.,
an increasing sequence of length $< \theta$ has an upper bound) then so 
is $(\hat{\Bbb P},\le_\ell)$.
\enddemo
\bigskip

\demo{Proof}  1) Check. \nl
2) Easy. \nl
3) If $\delta < \theta,\langle A_i:i < \delta \rangle$ is
$\le_\ell$-increasing let $(p_i,q_i)$ witness $A_i \in \hat{\Bbb P}$.  
If $\langle p_i:i < \delta \rangle$ is eventually constant then 
$\langle A_i:i < \delta \rangle$ is eventually constant and
$A_j$ for $j$ large enough can serve.  If not, without loss of generality 
$(\forall i < \delta)p_i \ne p_{i+1}$ and let $(p_i,q_i)$
witness $A_i \le_\ell A_{i+1}$.  Clearly $\langle q_i:i < \delta \rangle$ has
a $\le_\ell$-upper bound in $\Bbb P$, call it $q$.  Now $\{q\}$ is 
as required.  \hfill$\square_{\scite{1.1A}}$\margincite{1.1A}
\enddemo
\bigskip

\definition{\stag{1.2} Definition}  Let $MAC(\Bbb P)$ be the set of maximal
antichains of the forcing notion $\Bbb P$.
\enddefinition
\bigskip

\remark{\stag{1.3} Remark}  1) Note: $|MAC(\Bbb P)| \le 2^{|\Bbb
P|},\Bbb P$ satisfies the
$|\Bbb P|^+$-c.c. and if $\Bbb P$ satisfies the $\lambda$-c.c. then
$|MAC(\Bbb P)| \le |\Bbb P|^{< \lambda}$. \newline
2) Note:
\mr
\item "{$(*)$}"  if 
$\Bbb Q$ is a forcing notion, $\lambda = \lambda^{< \lambda}
> |\Bbb Q| + \aleph_0,\Vdash_\Bbb Q ``(\forall \mu < \lambda)
\mu^{\aleph_0} < \lambda"$
and $\Bbb Q' = \Bbb Q * \text{ Levy}(\aleph_1,< \lambda)$ \ub{then}
$|MAC(\Bbb Q')| = |\Bbb Q'| =
\lambda$.
\endroster
\endremark
\bigskip

\demo{\stag{1.4} Notation}  Car is the class of cardinals. \newline
IRCar is the class of (infinite) regular cardinals. \newline
RCar = IRCar $\cup \{1\}$. \newline
URCar is the class of uncountable regular cardinals. \newline
${\Cal D}^{cb}_\lambda$ is the filter of co-bounded subsets of $\lambda$.
\newline
${\Cal D}_\lambda$ is the club filter on $\lambda$ for $\lambda$
regular uncountable. \nl
$\eta^- = \eta \restriction (\ell g(\eta) - 1)$ for a finite sequence $\eta$
of length $>0$.
\enddemo
\bigskip

\demo{\stag{1.5} Notation}  1) ${\Cal H}(\chi)$ is the family of sets with
transitive closure of power $< \chi$; let $<^*_\chi$ be a well ordering of
${\Cal H}(\chi)$. \nl
2) Let $W$ be a function from the set of strongly inaccessible cardinals to
$\{0,1,\frac 12\}$; if $\alpha \notin \text{ Dom}(W)$ we understand
$W(\alpha) = 0$ and let $\alpha \in W$ means $W(\alpha) = 1$. 
\enddemo
\bigskip

\definition{\stag{1.6} Definition}  1) Assume $\bar{\Bbb P}$ is a 
$\lessdot$-increasing sequence of forcing notions. \newline
Let

$$
\align
\text{Gen}^r(\bar{\Bbb P}) =: 
\biggl\{ G:&\text{ for some (set) forcing notion }
\Bbb P^* \text{ we have } \dsize \bigwedge_{i < \alpha} \Bbb P_i \lessdot
\Bbb P^* \\
  &\text{ and for some } G^* \subseteq 
\Bbb P^* \text{ generic over } \bold V \text{ we have } \\
  &\,G = G^* \cap \dsize \bigcup_{i < \alpha} \Bbb P_i \biggr\}.
\endalign
$$
\mn
2) If $\bar{\Bbb Q} = \langle \Bbb P_i:i < \alpha \rangle$ 
or $\bar{\Bbb Q} = \langle  
\Bbb P_i,{\underset\tilde {}\to {\Bbb Q}_i}:i < \alpha \rangle$ where
$\Bbb P_i$ is a $\lessdot$-increasing we define a 
$\bar{\Bbb Q}$-name $\underset\tilde {}\to \tau$
almost as we define $(\dsize \bigcup_{i < \alpha} 
\Bbb P_i)$-names, but we do not
use maximal antichains of $\dsize \bigcup_{i < \alpha} \Bbb P_i$, that is:
\medskip
\roster
\item "{$(*)$}"  $\underset\tilde {}\to \tau$ is a function, 
Dom$(\underset\tilde {}\to \tau) \subseteq \dsize \bigcup_{i < \alpha}
\Bbb P_i$ and for every $G \in \text{ Gen}^r(\bar{\Bbb Q}),
\underset\tilde {}\to \tau[G]$ is defined iff 
Dom$(\underset\tilde {}\to \tau)
\cap G \ne \emptyset$ and then $\underset\tilde {}\to \tau[G] \in
\bold V[G]$
[from where ``every $G \ldots$" is taken?  E.g., $\bold V$ is countable, $G$ any set
from the true universe] and $\underset\tilde {}\to \tau[G]$ is definable with
parameters from $\bold V$ and the parameter $\dbcu_{i < \alpha} \Bbb P_i \cap G$ 
(so $\underset\tilde {}\to \tau$ is really a first-order formula with the 
variable $\dbcu_{i < \alpha} \Bbb P_i \cap G$ and parameters from $\bold V$).
\endroster
\medskip

\noindent
Now $\Vdash_{\bar{\Bbb Q}}$ has a natural meaning. \newline
3) For $p \in \bar{\Bbb Q}$ (i.e. $p \in \dsize \bigcup_{i < \alpha}
\Bbb P_i$) and
$\bar{\Bbb Q}$-names ${\underset\tilde {}\to \tau_0},\dotsc,
{\underset\tilde {}\to \tau_{n-1}}$ we let $\{{\underset\tilde {}\to \tau_0},
\dotsc,{\underset\tilde {}\to \tau_{n-1}}\}$ be the name for the set that
contains exactly those ${\underset\tilde {}\to \tau_i}[\bar{\Bbb Q}]$ 
that are
defined.  We let $p \Vdash ``\underset\tilde {}\to \tau = x"$ if for every
$G$ such that $p \in G \in \text{ Gen}^r(\bar{\Bbb Q})$ we have
$\underset\tilde {}\to \tau[G] = x$. 
If $\beta < \alpha$ and $G_\beta \subseteq \Bbb P_\beta$, we let
$\underset\tilde {}\to \tau[G_\beta] = x$ means that for some 
$p \in G_\beta$ we have $p \Vdash_{\bar{\Bbb Q}} 
``\underset\tilde {}\to \tau = x"$, so possibly no $p \in \dbcu_{i <
\alpha} \Bbb P_i$ forces a value to $\underset\tilde {}\to \tau$ and
no such $p$ forces $\tau$ is not definable. \newline
4) We say a $\bar{\Bbb Q}$-name $\underset\tilde {}\to x$ is full
\ub{if} $\underset\tilde {}\to x[G]$ is well defined for every $G \in
\text{ Gen} {}^r(\bar{\Bbb Q})$. \nl
5) A simple $\bar{\Bbb Q}$-named$^1$ \, $[j,\beta)$-ordinal 
$\underset\tilde {}\to \zeta$ 
is a $\bar{\Bbb Q}$-name $\underset\tilde {}\to \zeta$ such that: if
$G \in \text{ Gen}^r(\bar{\Bbb Q})$ and
$\underset\tilde {}\to \zeta[G] = \xi$ then $j \le \xi < \beta$ and
$(\exists p \in G \cap \Bbb P_{\xi \cap \alpha}) p \Vdash_{\bar{\Bbb Q}}
``\underset\tilde {}\to \zeta = \xi"$ (where $\alpha = 
\ell g(\bar{\Bbb Q})$);
however, we allowed $\underset\tilde {}\to \zeta[G]$ to be undefined.
If we omit ``$[j,\beta)$" we mean $[0,\ell g(\bar{\Bbb Q})) = 
[0,\alpha)$.  If we
omit ``simple", we mean 
replacing $(\exists p \in G \cap \Bbb P_{\xi \cap \alpha})$
by $(\exists p \in G \cap \Bbb P_{(\xi + 1) \cap \alpha})$ (this is
used in \cite[Ch.X,\S1]{Sh:f}, we shall only remark on it here). \nl
6) A simple $\bar{\Bbb Q}$-named$^2$ \, $[j,\beta)$-ordinal
$\underset\tilde {}\to \zeta$ is a simple $\bar{\Bbb Q}$-named$^2$ \, 
$[j,\beta)$-ordinal of depth $\Upsilon$ for some ordinal $\Upsilon$, where
this is defined below by induction on $\Upsilon$.  In all cases it is a
$\bar{\Bbb Q}$-name of an ordinal from the interval $[j,\beta)$ so may 
be undefined, i.e., we allow non full such names.
\enddefinition
\bn
\ub{Case 1}:  $\Upsilon = 0$.

This is an ordinal $\in [j,\beta)$, or is ``undefined" (in the full
case this is forbidden).
\bn
\ub{Case 2}:  $\Upsilon > 0$.

For some $\gamma < \ell g(\bar{\Bbb Q}) \cap \beta$ and maximal antichain
${\Cal I} = \{p_\varepsilon:\varepsilon < \varepsilon^*\}$ of 
$\Bbb P_\gamma$, for each $i < i^*$ there is a sequence
$\langle {\underset\tilde {}\to \zeta_\varepsilon}:
\varepsilon < \varepsilon^* \rangle$
such that ${\underset\tilde {}\to \zeta_\varepsilon}$ is a simple 
$\bar{\Bbb Q}$-named$^2$ \, $[\text{Max}\{j,\gamma\},\beta)$-ordinal of depth
$\Upsilon_\varepsilon < \Upsilon$ and: $\underset\tilde {}\to \zeta
[G_\xi] = \xi$ iff $\xi \ge \gamma$ and for some $\varepsilon$ we have
$p_\varepsilon \in G_\xi \cap \Bbb P_\gamma$ and 
${\underset\tilde {}\to \zeta_\varepsilon}[G_\xi] = \xi$ 
(including the case: not defined).  If we omit ``$[j,\beta)$" we mean
$[0,\ell g(\bar{\Bbb Q})) = [0,\alpha)$. \nl
7)  If we omit ``simple" in (6) we mean that in case 2,
${\underset\tilde {}\to \zeta_i}$ is a not necessarily simple
$\bar{\Bbb Q}$-name$^r$ and ${\Cal I}
\subseteq \Bbb P_{\gamma +1}$.
\nl
8) We say $\bar{\Bbb P}$ is $W$-continuous or $(\bar{\Bbb P},W)$ is
continuous when for every $\delta \in W \cap \ell g(\bar{\Bbb P})$
when if $(\forall i < \delta)$ [density $(\Bbb P_i) < \delta$, or just
$\Bbb P_i$ satisfies the cf$(\delta)$-c.c.], 
\ub{then} $\Bbb P_\delta = \dbcu_{i < \delta}\Bbb P_i$.  
We say $\Bbb P$ is $W$-smooth or $(\bar{\Bbb P},W)$ is smooth
if $\delta \in W \Rightarrow \Bbb P_\delta = \dbcu_{i < \delta} \Bbb P_i$.
We say $\underset\tilde {}\to \zeta$ is a simple$^\ell \,
(\bar{\Bbb Q},W)$-named
$[j,\beta)$-ordinal \ub{if}
it is a simple$^\ell \, \bar{\Bbb Q}$-named ordinal 
and $\delta \in W \cap (\ell g(\bar{\Bbb Q}) +1) 
\Rightarrow (\exists \alpha < \delta)(\Vdash
\underset\tilde {}\to \zeta \notin [\alpha,\delta))$. 
\bigskip

\proclaim{\stag{1.6A} Claim}  1) Assume that $\bar Q$ is $W$-continuous.
If $\underset\tilde {}\to \zeta$ is a simple
$\bar{\Bbb Q}$-named$^2 \, [0,\alpha)$-ordinal, $\gamma \in W$ is regular
and $\beta < \gamma$ implies density$(\Bbb P_\beta) < \gamma$ or just
($\Bbb P_\beta$ satisfies the ${\text{\rm cf\/}}(\alpha))$-c.c., 
\ub{then} for some $\beta < \gamma,\underset\tilde {}\to \zeta$ is 
a simple $\bar{\Bbb Q} \restriction
\beta$-named$^2 \, [0,\beta)$-ordinal. \nl
2) If $\bar{\Bbb Q}$ is $W$-continuous and $\gamma \in W \Rightarrow
\gamma$ regular and $\underset\tilde {}\to \zeta$ is a simple 
$\bar{\Bbb Q}$-named$^2 [0,\alpha)$-ordinal \ub{then} $\underset\tilde
{}\to \zeta$ is a simple $(\bar{\Bbb
Q},W)$-named$^2[0,\alpha)$-ordinal. \nl
3) If $\underset\tilde {}\to \zeta$ is a simple $\bar{\Bbb
Q}$-named$^2[0,\alpha)$-ordinal \ub{then} there is a full simple
$\bar{\Bbb Q}$-named$^2[0,\alpha)$-ordinal ${\underset\tilde {}\to
\zeta'}$ such that $\Vdash_{\bar{\Bbb Q}}$ ``if $\underset\tilde {}\to
\zeta$ is well defined then it is equal to $\zeta'$".
\endproclaim
\bigskip

\demo{Proof}  By induction on the depth of $\underset\tilde {}\to \zeta$.
\hfill$\square_{\scite{1.6A}}$\margincite{1.6A}
\enddemo
\bigskip

\remark{\stag{1.7} Remark}  1) We can restrict in the definition of
Gen$^r(\Bbb Q)$ to $\Bbb P^*$ in some class $K$, and get a $K$-variant
of our notions. \newline
2) Note: even if in \scite{1.6}(1) we ask Dom$(\underset\tilde {}\to \tau)$
to be a maximal antichain of $\dbcu_{i < \delta} \Bbb P_i$
 it will not be meaningful as in the appropriate
$\Bbb P_\delta$, we have $\dsize \bigwedge_{i < \delta} \Bbb P_i \lessdot
\Bbb P_\delta$ but not necessarily $\dbcu_{i < \delta} \Bbb P_i
\lessdot P_\delta$ hence it will not in general be a maximal antichain. \nl
3) Note that in the simple case we wrote 
$\Bbb P_{\xi \cap \alpha}$ not $\Bbb P_{(\xi + 1) \cap \alpha}$.
Compare this the remark \cite[Ch.XIV,1.1B]{Sh:f}.  Here in the main
case we use full simple $\bar{\Bbb Q}$-named$^2$ ordinals, 
though we shall remark on the affect of the non-simple case;
as a result we will not have a general associativity law, but
the definition of Sp$_3 - \text{ Lim}_\kappa
(\bar{\Bbb Q})$ will be somewhat simplified.  
As said earlier, we can interchange
decisions on this matter.  Of course, also \cite[Ch.XV]{Sh:f} can be 
represented with this iteration. \nl
4) The ``name$^1$" is necessary for the $\kappa > \aleph_1$ case, but
``name$^2$" is preferable for $\kappa = \aleph_1$, so we could have
concentrated on name$^1$ for $\kappa > \aleph_1$, name$^2$ for $\kappa
= \aleph_1$, but actually we concentrated on simple, name$^2$ for
$\kappa = \aleph_1$; see \scite{1.13}(B) below.
\endremark
\bigskip

\demo{\stag{1.8} Fact}  1) For $\bar{\Bbb P} = \langle \Bbb P_i:
i < \ell g(\bar{\Bbb P})
\rangle$, a $\lessdot$-increasing sequence of forcing notions, $\ell \in
\{1,2\}$ and simple $\bar{\Bbb P}$-named$^\ell$ \, $[j,\beta)$-ordinal 
$\underset\tilde {}\to \zeta$ and $p \in
\dsize \bigcup_{i < \alpha} \Bbb P_i$ there 
are $\xi,q$ and $q_1$ such that $p \le
q \in \dsize \bigcup_{i < \ell g(\bar{\Bbb P})}
\Bbb P_i$ and: either $q \Vdash_{\bar{\Bbb P}}
``q_1 \in \underset\tilde {}\to G",q_1 \in \Bbb P_\xi,\xi < \alpha,[p \in
\Bbb P_\xi \Rightarrow q = q_1]$ and $q_1 \Vdash_{\bar{\Bbb P}} 
``\underset\tilde {}\to \zeta 
= \xi"$ or $q \Vdash_{\bar{\Bbb P}} 
``\underset\tilde {}\to \zeta$ is not defined"
(and even $p \Vdash_{\bar{\Bbb P}} 
``\underset\tilde {}\to \zeta$ is not defined").
\newline
2) For $\bar{\Bbb P}$ and $\ell \in \{1,2\}$ as above, and simple 
$\bar{\Bbb P}$-named$^\ell$ \, $[j,\beta)$-ordinals 
$\underset\tilde {}\to \zeta,\underset\tilde {}\to \xi$, also
max$\{\underset\tilde {}\to \zeta,\underset\tilde {}\to \xi\}$ and
min$\{\underset\tilde {}\to \zeta,\underset\tilde {}\to \xi\}$ are simple
$\bar{\Bbb Q}$-named$^\ell$ \, $[j,\beta)$-ordinals (naturally
defined, so max $\{\underset\tilde {}\to \zeta,
\underset\tilde {}\to \xi\}[G]$
is defined iff a $\underset\tilde {}\to \zeta[G],
\underset\tilde {}\to \xi[G]$ are defined, and 
min$\{\underset\tilde {}\to \zeta,\underset\tilde {}\to \xi\}
[G]$ is defined iff $\underset\tilde {}\to \zeta[G]$ is defined or
$\underset\tilde {}\to \xi[G]$ is defined). 
If $\underset\tilde {}\to \zeta,\underset\tilde {}\to \xi$ are full
then so are max$\{\underset\tilde {}\to \zeta,\underset\tilde {}\to
\xi\}$ and min$\{\underset\tilde {}\to \zeta,\underset\tilde {}\to \xi\}$.
\newline
3) For $\bar{\Bbb P}$ and $\ell$ as above, $n < \omega$ and simple 
$\bar{\Bbb P}$-named$^\ell$ ordinals
${\underset\tilde {}\to \xi_1},\dotsc,{\underset\tilde {}\to \xi_n}$ and
$p \in \dsize \bigcup_{i < \ell g(\bar{\Bbb P})} \Bbb P_i$ there are 
$\zeta < \alpha$ and
$q \in \Bbb P_\zeta$ such that, first: $p\le q$ or at least $q
\Vdash_{{\Bbb P}_\zeta} 
``p \in \Bbb P_i/{\underset\tilde {}\to G_{{\Bbb P}_\zeta}}$ 
for some $i < \ell g
(\bar{\Bbb P})"$ and second: for some $\ell \in \{1,\dotsc,n\}$ we have $q 
\Vdash_{\bar{\Bbb P}} ``\zeta = 
{\underset\tilde {}\to \xi_\ell} = \text{max}\{{\underset\tilde {}\to \xi_1},
\dotsc,{\underset\tilde {}\to \xi_n}\}$" or $q \Vdash_{\bar{\Bbb P}} ``\text{ max}
\{{\underset\tilde {}\to \xi_1},\dotsc,{\underset\tilde {}\to \xi_n}\}$ not
defined".  Similarly for min. \newline
4) The same holds for simple $(\bar{\Bbb Q},W)$-names$^\ell$ and we 
can omit simple. \nl
5) If ${\underset\tilde {}\to \zeta_i}$ is a simple $\bar{\Bbb Q}$-named$^1 
[\beta_i,\gamma_i)$-ordinal for $i < i^*$ \ub{then} 
\footnote{this seems lacking for ``name$^2$".} 
$\sup\{{\underset\tilde {}\to \zeta_i}:i < i^*\}$ 
is simple $\bar{\Bbb Q}$-named$^1 \, [{\underset i {}\to {\text{Min}}} \, \beta_i,
{\underset i {}\to {\text{sup}}} \gamma_i)$-ordinal. \nl
6) Similarly for $\{{\underset \tilde {}\to \zeta_i}:i\}$ and when we
omit ``simple". \nl
7A)  A simple $\bar{\Bbb P}$-name$^\ell[j,\beta)$-ordinal
\mr
\item "{$(a)$}"  $\zeta$ is a $\bar{\Bbb P}$-named$^\ell[j,\beta)$-ordinal;
\sn
\item "{$(b)$}"  if $j_2 \le j_1 < \beta_1 \le \beta_2$ then any
[simple] $\bar{\Bbb P}$-named $[j_1,\beta_1)$-ordinal is a [simple]
$\bar{\Bbb P}$-named$^\ell [j_2,\beta_2)$-ordinal;
\sn
\item "{$(c)$}"  a [simple] $\bar{\Bbb P}$-named$^2 [j,\beta)$ ordinal
is a [simple] $\bar{\Bbb P}$-named$^1 [j,\beta)$-ordinal
\sn
\item "{$(d)$}"  if $\beta \le \alpha' \le \ell g(\bar{\Bbb P})$ then
any [simple] $\bar{\Bbb P}$-named$^\ell [j,\beta)$-ordinal is a
$(\bar{\Bbb P} \restriction \alpha')$-named$^\ell [j,\beta$)-ordinal.
\endroster
\enddemo
\bigskip

\demo{Proof}  Straight.
\enddemo
\bigskip

\demo{\stag{1.9} Discussion}  We have in defining our iteration 
several possible variants, some of our particular choices are not 
important: we can
make it like revised countable support as in \cite[Ch.X,\S1]{Sh:f} or like
$\aleph_1$-RS in \cite[Ch.XIV,\S1]{Sh:f}, or as in \cite[Ch.XIV,\S2]{Sh:f}
(as here); for most uses $\kappa = \aleph_1$ and we could restrict ourselves
to $\le_{\text{pr}} = \le_{\text{vpr}}$ as equality; but in 
\cite{GoSh:511} we need the three partial orders.

So below we have finite support for non-pure, countable for pure and Easton
for very pure.
\enddemo
\bigskip

\definition{\stag{1.10} Definition}  1) For a forcing notion $\Bbb P$ (as in 
\scite{1.1}) let $\Bbb P^{[\text{pr}]}$ 
be defined like $\Bbb P$ except that we make
$\emptyset_{\Bbb P} \le_{\text{pr}} p$ for every $p \in \Bbb P$. \newline
2) For a forcing notion $\Bbb P$ 
(as in \scite{1.1}) let $\Bbb P^{[\text{vp}]}$ be defined 
like $\Bbb P$ except that we make $\emptyset_{\Bbb P} \le_{\text{vp}} p$
(and $\emptyset \le_{\text{pr}} \Bbb P$, of course) for every $p \in \Bbb P$.
\enddefinition
\bigskip

\demo{\stag{1.11} Fact}  1) For a forcing notion $\Bbb P$ and 
$x \in \{\text{pr,vp}\},
\Bbb P^{[x]}$ is also a 
forcing notion, and they are equivalent as forcing notions. \nl
2) For $x \in \{\text{pr,vp}^r\}$ the operations $\Bbb P \mapsto
\hat{\Bbb P}$ and $\Bbb P \mapsto \Bbb P^{[x]}$ commute. \nl
3) If $(x_1,x_2,x_3) \in
\{(\text{pr,pr,pr),(vpr,pr,vpr),(vpr,vpr,pr)),(vpr,vpr,vpr)}\}$
then $\Bbb P^{[x_3]} = (\Bbb P^{[x_1][x_2]}$. \nl
4) $\theta$-completeness is preserved in the natural cases.
\enddemo
\bigskip

\demo{\stag{1.12} Discussion}  1) Why do we 
bother with $\Bbb P^{[\text{pr}]},\Bbb P^{[\text{vpr}]}$?  If
in the iteration defined below we use only $Q^{[\text{pr}]}_i,
{\underset\tilde {}\to {\Bbb Q}^{[\text{vp}]}_i}$, we get a variant 
of the definition
without the need to repeat it.  We may want that: if $\ell g(\bar{\Bbb Q}) = 
\lambda$ inaccessible and $i < \kappa \Rightarrow |\Bbb P_i| < \lambda$ then
$\dsize \bigcup_{i < \lambda} \Bbb P_i = \Bbb P_\lambda$ (here as done in 
\cite[Ch.XIV,\S2]{Sh:f} we can just impose it). \nl
Some other restrictions are for simplicity only. \nl
2) Below the case $e=6$ is the main one. [Saharon]
\enddemo
\bigskip

\definition{\stag{1.13} Definition/Claim}  We define and prove the
following by induction on $\alpha$.  Below $\kappa = \aleph_1,e = G$
so we can omit them (they are meaningful in \S11)
\mr
\item "{$(A)$}"  \,\,[Definition] $\quad \bar{\Bbb Q} = 
\langle \Bbb P_i,{\underset\tilde {}\to Q_i}:i < \alpha
\rangle$ is a $\kappa-\text{Sp}_e$-iteration for 
$W$ or $\kappa-\text{Sp}_e(W)$-iteration
(if $W$ is absent we mean $\{ \beta \le \alpha:\beta 
\text{ strongly inaccessible}\}$); $\alpha$ is called the length of
$\bar{\Bbb Q},\ell g(\bar{\Bbb Q})$.
\sn
\item "{$(B)$}"  \,\,[Definition] $\quad$ A simple 
$(\bar{\Bbb Q},W)$-named$_e$ ordinal
$\underset\tilde {}\to \zeta$ and $\underset\tilde {}\to \zeta \restriction
[\alpha,\beta)$. 
\sn
\item "{$(C)$}"  \,\,[Definition] $\quad$ A simple 
$(\bar{\Bbb Q},W)$-named$_e$ 
atomic condition $\underset\tilde {}\to q$
(or atomic $[j,\beta)$-condition where $j \le \beta \le \alpha$); also 
we define $\underset\tilde {}\to q \restriction \xi,
\underset\tilde {}\to q \restriction \{\varepsilon\},
\underset\tilde {}\to q \restriction [\xi,\zeta)$ for a simple 
$\bar{\Bbb Q}$-named$_e$ atomic condition 
$\underset\tilde {}\to q$ and ordinals $\varepsilon <
\alpha,\xi \le \zeta \le \alpha$ 
(or simple $\bar{\Bbb Q}$-named$_e$ ordinals 
$\underset\tilde {}\to \xi,\underset\tilde {}\to \zeta$ 
instead $\xi,\zeta$).  We may add pure/very pure
as adjectives to the condition.  
\sn
\item "{$(D)$}"  \,\,[Claim] $\quad$  Assume 
$\underset\tilde {}\to \zeta$ is a simple 
$(\bar{\Bbb Q},W)$-named$_e [j,\beta)$-ordinal.  \ub{Then} for any $\xi,
\underset\tilde {}\to \zeta \restriction \xi$ is a 
simple $(\bar{\Bbb Q},W)$-named
$[j,\text{min}\{\beta,\xi\})$-ordinal and $\Vdash_{\bar{\Bbb Q}}$ ``if
$\underset\tilde {}\to \zeta < \xi$ then $\underset\tilde {}\to \zeta =
\underset\tilde {}\to \zeta \restriction \xi$; if
$\underset\tilde {}\to \zeta \ge \xi$ or $\underset\tilde {}\to \zeta$ is
undefined, then $\underset\tilde {}\to \zeta \restriction \xi$ is undefined",
also $\underset\tilde {}\to \zeta \restriction \xi$ is a simple
$(\bar{\Bbb Q} \restriction \xi,W)$-named ordinal. \nl
Similarly $\underset\tilde {}\to \zeta \restriction [\xi_1,\xi_2)$.  If
${\underset\tilde {}\to \xi_1} \le {\underset\tilde {}\to \xi_2} \le \alpha$
are simple $(\bar{\Bbb Q},W)$-named $[\alpha_1,\alpha_2)$-ordinals (the
${\underset\tilde {}\to \xi_1} \le {\underset\tilde {}\to \xi_2}$ means
$\Vdash_{\bar{\Bbb Q}} ``{\underset\tilde {}\to \xi_1} \le
{\underset\tilde {}\to \xi_2}"$), \ub{then}
$\underset\tilde {}\to \zeta \restriction [{\underset\tilde {}\to \xi_1},
{\underset\tilde {}\to \xi_2})$ is a simple $(\bar{\Bbb Q},W)$-named
$[\alpha_1,\alpha_2)$-ordinal and $\Vdash_{\bar{\Bbb Q}}$ ``if
$\underset\tilde {}\to \zeta \in [{\underset\tilde {}\to \xi_1},
{\underset\tilde {}\to \xi_2})$, then $\underset\tilde {}\to \zeta =
\underset\tilde {}\to \zeta \restriction [{\underset\tilde {}\to \xi_2},
{\underset\tilde {}\to \xi_2})$ otherwise $\underset\tilde {}\to \zeta
\restriction [\xi_1,\xi_2)$ is undefined.  If in addition $\beta =
\text{ Min}\{\alpha,\alpha_2,\ell g(\bar{\Bbb Q})\}$ and $\beta \le \gamma,
\alpha'_1 \le \alpha_1$ then $\underset\tilde {}\to \zeta
\restriction[{\underset\tilde {}\to \xi_1},{\underset\tilde {}\to \xi_2})$ 
is a simple $(\bar{\Bbb Q} \restriction
\beta,W)$-named $[\alpha'_1,\gamma)$-ordinal.  Also if
$n < \omega$, for $\ell \in \{1,\dotsc,n\},{\underset\tilde {}\to \xi_\ell}$
is a simple $\bar{\Bbb Q}$-named $[\beta_1,\beta_2)$-ordinal then
Max$\{{\underset\tilde {}\to \xi_1},\dotsc,{\underset\tilde {}\to \xi_n}\}$
is a simple $\bar{\Bbb Q}$-named 
$[\beta_1,\beta_2)$-ordinal.  Similarly for Min.
\sn
\item "{$(E)$}"  \,\,[Claim] $\quad$ If 
$q$ is a simple $(\bar{\Bbb Q},W)$-named 
atomic $[j,\beta)$-condition, $\xi < \alpha$, then $\underset\tilde {}\to q 
\restriction \xi$ is a simple $(\bar{\Bbb Q} \restriction
\xi,W)$-named atomic 
$[j,\text{min}\{\beta,\xi\})$-condition and $\underset\tilde {}\to q 
\restriction \{\xi\}$ is a $\Bbb P_\xi$-name of a member of 
${\underset\tilde {}\to {\Bbb Q}_\xi}$ or undefined (and then 
it may be assigned the value 
$\emptyset_{{\underset\tilde {}\to {\Bbb Q}_\xi}}$, the minimal member of
${\underset\tilde {}\to {\Bbb Q}_\xi}$).  If $q$ is a simple 
$(\bar{\Bbb Q},W)$-named 
atomic condition, $\underset\tilde {}\to \xi \le \underset\tilde {}\to \zeta 
\le \alpha$ are simple $(\bar{\Bbb Q},W)$-named ordinals then 
$\underset\tilde {}\to q \restriction[\underset\tilde {}\to \xi,
\underset\tilde {}\to \zeta)$ is a simple $(\bar{\Bbb Q},W)$-named ordinal.  
Also $\underset\tilde {}\to q \restriction \{\zeta\} = 
\underset\tilde {}\to q
\restriction [\zeta,\zeta +1)$, and if $\underset\tilde {}\to q$ is a simple
$(\bar{\Bbb Q},W)$-named $[\zeta,\xi)$-ordinal, $\zeta' < \xi'$ and
$\Vdash_{\bar{\Bbb Q}} ``\underset\tilde {}\to \zeta \in
[\zeta',\xi')"$, then it is a simple $(\bar{\Bbb Q},W)$-named 
$[\zeta',\xi')$-ordinal.
Also ``pure" and ``very pure" are preserved by restriction.
\sn  
\item "{$(F)$}" \,\,[Definition] $\quad$  The $\kappa-\text{Sp}_e(W)$-limit of 
$\bar{\Bbb Q},\text{Sp}_e(W)$-Lim$_\kappa(\bar{\Bbb Q})$,
denoted by $\Bbb P_\alpha$ for $\bar{\Bbb Q}$ as in clause (A) in
particular of length $\alpha$, and $p \restriction
\xi$ and Dom$(p)$ for $p \in \text{ Sp}_e(W)$-Lim$_\kappa
(\bar{\Bbb Q}),\xi$ an ordinal $\le \alpha$; (similarly for a
simple $(\bar{\Bbb Q},W)$-named $[0,\ell g(\bar{\Bbb Q}))$ 
ordinal $\underset\tilde {}\to \xi$, etc.  
We also define $\Bbb P_{\underset\tilde {}\to \zeta}$ for
$\underset\tilde {}\to \zeta$ a $(\bar{\Bbb Q},W)$-named ordinal.
\sn
\item "{$(G)$}" \,\,[Theorem] $\quad \text{ Sp}_e(W)$-Lim$_\kappa(\bar{\Bbb
Q})$ is a forcing notion (in the sense of \scite{1.1}(1)).
\sn
\item "{$(H)$}" \,\,[Theorem] $\quad$ Assume $\beta < \alpha = 
\ell g(\bar{\Bbb Q})$ or more generally, $\underset\tilde {}\to \beta$ 
is a full simple $(\bar{\Bbb Q},W)$-named ordinal (see end of clause 
(F) above).  Then $\Bbb P_\beta \subseteq_{\text{ic}} 
\text{ Sp}_e(W)$-Lim$_\kappa(\bar{\Bbb Q})$ (so a submodel 
with the three partial orders, 
even compatibilities are preserved) and 
$[p \in \Bbb P_\beta \Rightarrow p \restriction \beta = p]$ and 
$[\Bbb P_\alpha \models ``p \le_\ell q" \Rightarrow \Bbb P_\beta \models
``p \restriction \beta \le_\ell q \restriction \beta"]$ (for $\ell = 0,1,2$,
of course) and $\Bbb P_\alpha \models ``p \restriction \beta \le_\ell
p"$.  Also $q \in \Bbb P_\beta,p \in 
\text{ Sp}_e(W)$-Lim$_\kappa(\bar{\Bbb Q})$ are compatible iff $q,p
\restriction \beta$ are compatible in 
$\Bbb P_\beta$.  In fact, if $q \in \Bbb P_\beta,
\Bbb P_\beta \models ``p \restriction \beta \le q"$ 
then $q \cup (p \restriction [\beta,\alpha))$ belongs 
to $\Bbb P_\alpha$ and is a least upper bound of $p,q$ 
and if $\Bbb P_\beta \models ``p \restriction \beta \le_\ell q"$ even a 
$\le_\ell$-least upper bound of $q$.
Hence $\Bbb P_\beta \lessdot (\text{Sp}_e(W)$-Lim$_\kappa(\bar q))$.
\sn
\item "{$(I)$}"  \,\,[Claim] $\quad$  The set of 
$p \in \Bbb P_\alpha$ such that for every $\beta < \alpha$ 
we have $\Vdash_{{\Bbb P}_\beta} ``p \restriction \{\beta\}$ is a singleton
or empty", is a dense subset of $\Bbb P_\alpha$.  Also we can replace
${\underset\tilde {}\to {\Bbb Q}_\beta}$ by $\hat{\Bbb Q}_\beta$ and 
the set of ``old" $p \in \Bbb P_\alpha$ is a dense subset of the new 
(but actually do not use this).
\sn
\item "{$(J)$}"  \,\, [Claim] $\quad$  If 
$\alpha$ is strongly inaccessible, $\zeta < \alpha
\Rightarrow |\Bbb P_\zeta| < \alpha$ or just $\zeta < \alpha
\Rightarrow CC(\Bbb P_\zeta) \le \alpha$
 and $\alpha \in W$, \ub{then} $\Bbb P_\alpha =
\dbcu_{\zeta < \alpha} \Bbb P_\zeta$.
\endroster
\enddefinition
\bn
\underbar{Proof and Definition}:  
\mr
\item "{$(A)$}"  $\bar{\Bbb Q} = \langle \Bbb P_i,
{\underset\tilde {}\to {\Bbb Q}_i}:i < \alpha
\rangle$ is a $\kappa-\text{Sp}_e(W)$-iteration if 
$\bar{\Bbb Q} \restriction \beta$ is a
$\kappa-\text{Sp}_e(W)$-iteration for 
every $\beta < \alpha$, and if $\alpha = 
\beta + 1$, \ub{then} $\Bbb P_\beta = 
\text{ Sp}_e(W)$-Lim$_\kappa(\bar{\Bbb Q} \restriction 
\beta)$ 
and ${\underset\tilde {}\to {\Bbb Q}_\beta}$ is a $\Bbb P_\beta$-name of a 
forcing notion as in \scite{1.1}(1) here.
\sn
\item "{$(B)$}"  We say $\underset\tilde {}\to \zeta$ is a simple
$(\bar{\Bbb Q},W)$-named$_e$ \, $[j,\beta)$-ordinal \ub{if}
$\underset\tilde {}\to \zeta$ is a simple $(\bar{\Bbb Q},W)$-named$^2$ \, 
$[j,\beta)$-ordinal.
\sn
\item "{$(C)$}"  We say $\underset\tilde {}\to q$ is a simple 
$(\bar{\Bbb Q},W)$-named atomic $[j,\beta)$-condition when: 
$\underset\tilde {}\to q$ is a $\bar{\Bbb Q}$-name, and for some 
$\underset\tilde {}\to \zeta = 
{\underset\tilde {}\to \zeta_{\underset\tilde {}\to q}}$, a simple
$(\bar{\Bbb Q},W)$-named $[j,\beta)$-ordinal, we have 
$\Vdash_{\bar{\Bbb Q}} ``\underset\tilde {}\to \zeta$
has a value iff $\underset\tilde {}\to q$ has, and if they have then $j \le
\underset\tilde {}\to \zeta < \text{ min}\{\beta,\ell g(\bar{\Bbb Q})\}$ and
$\underset\tilde {}\to q \in 
{\underset\tilde {}\to {\Bbb Q}_{\underset\tilde {}\to \zeta}}"$.  If we omit
``$[j,\beta)$" we mean ``$[0,\alpha)$".
Now $\underset\tilde {}\to q \restriction \xi$ will have a value iff
${\underset\tilde {}\to \zeta_{\underset\tilde {}\to q}}$ has a value $< \xi$
and then its value is the value of $\underset\tilde {}\to q$.  Lastly,
$\underset\tilde {}\to q \restriction \{\xi\}$ will have a value iff
${\underset\tilde {}\to \zeta_{\underset\tilde {}\to q}}$ has value $\xi$ and
then its value is the value of $\underset\tilde {}\to q$.  Similarly for
$\underset\tilde {}\to q \restriction [\zeta,\xi)$ and 
$\underset\tilde {}\to q \restriction \underset\tilde {}\to \xi,
\underset\tilde {}\to q \restriction \{ \underset\tilde {}\to \xi\}$,
$\underset\tilde {}\to q \restriction
[\underset\tilde {}\to \zeta,\underset\tilde {}\to \xi)$.  We say
$\underset\tilde {}\to q$ is pure if 
$\Vdash_{\bar{\Bbb Q}}$ ``for $\xi < \alpha$,
if ${\underset\tilde {}\to \zeta_q} = \xi$ and a
$\underset\tilde {}\to q \in {\underset\tilde {}\to {\Bbb Q}_\xi}$ then
${\underset\tilde {}\to {\Bbb Q}_\xi} \models 
\emptyset_{\underset\tilde {}\to {\Bbb Q}_\xi} 
\le_{\text{pr}} \underset\tilde {}\to q$".
We say $\underset\tilde {}\to q$ is very pure 
if $\Vdash_{\bar{\Bbb Q}}$ ``for 
$\xi < \alpha$, if $\underset\tilde {}\to q \in 
{\underset\tilde {}\to {\Bbb Q}_\xi}$, then ${\underset\tilde {}\to
{\Bbb Q}_\xi} \models \emptyset_{\underset\tilde {}\to {\Bbb Q}_i} 
\le_{\text{vpr}} \underset\tilde {}\to q$".
\sn
\item "{$(D),(E)$}"  Left to the reader.
\sn
\item "{$(F)$}"  We are defining 
$\text{Sp}_e(W)$-Lim$_\kappa(\bar{\Bbb Q})$ (where
$\bar{\Bbb Q} = \langle \Bbb P_\beta,
{\underset\tilde {}\to {\Bbb Q}_\beta}:\beta < \alpha
\rangle$, of course).  It is a quadruple $\Bbb P_\alpha = (\Bbb P_\alpha,\le,
\le_{\text{pr}},\le_{\text{vpr}})$ where
\sn
{\roster
\itemitem{ $(a)$ }  $\Bbb P_\alpha$ is the set of $p = \{
{\underset\tilde {}\to q_i}:i < i^*\}$ satisfying for some witness 
${\underset\tilde {}\to {\bar \zeta}}$:
\newline

$\qquad \quad (i) \quad$ each 
${\underset\tilde {}\to q_i}$ is a simple $(\bar{\Bbb Q},W)$-named
atomic condition,  \newline

$\qquad \qquad \quad$ and for every $\xi < \alpha$, we have \nl

$\qquad \qquad \qquad \Vdash_{{\Bbb P}_\xi} ``p \restriction
\{\xi\} =: \{{\underset\tilde {}\to q_i} 
\restriction \{\xi\}:i < i^*\} \cup 
\{\emptyset_{\underset\tilde {}\to {\Bbb Q}_\xi}\} \in 
{\underset\tilde {}\to {\hat{\Bbb Q}}_\xi}"$ \newline

$\qquad \quad (ii) \quad$ if $\alpha \in W$ is strongly inaccessible
$> CC(\Bbb P_i) + \kappa$ \nl  \newline

$\qquad \qquad \qquad$ for every $i < \alpha$, \ub{then} 
$i^* < \alpha$ \newline

$\qquad \quad (iii) \quad {\underset\tilde {}\to {\bar \zeta}} = \langle
{\underset\tilde {}\to \zeta_\varepsilon}:\varepsilon < j \rangle$
where $j < \kappa$ and

$\qquad \qquad \qquad$ each ${\underset\tilde {}\to \zeta_\varepsilon}$ is
a simple $(\bar{\Bbb Q},W)$-named $[0,\alpha)$-ordinal, \nl

$\qquad \qquad \qquad$ [the reader should think of
$\{{\underset\tilde {}\to \zeta_\varepsilon}:\varepsilon < j\}$ \nl

$\qquad \qquad \qquad$ as the non-very-pure support of $p$] \nl

$\qquad \quad (iv) \quad$ for every $\xi < \alpha$ we have (we may 
replace \nl

$\qquad \qquad \qquad 
\Vdash_{\bar{\Bbb Q}}$ by $\Vdash_{{\Bbb P}_\xi}$ as we use simple
names) \nl

$\qquad \qquad \qquad \Vdash_{\bar{\Bbb Q}}$ ``if $(\forall \varepsilon < j)
({\underset\tilde {}\to \zeta_\varepsilon}
[\underset\tilde {}\to G \cap \Bbb P_\xi]$ 
is $\ne \xi)$ (for example is \newline

$\qquad \qquad \qquad$ not well defined) \ub{then}
$\emptyset_{\underset\tilde {}\to {\Bbb Q}_\xi} \le_{\text{vpr}} 
p \restriction \{\xi\}$ in ${\underset\tilde {}\to {\hat{\Bbb Q}}_\xi}$"
\nl

$\qquad \quad (v)$ if $\beta < \alpha$ then $p \restriction \beta =:
\{q_i \restriction \beta:i < i^*\}$ belongs to $\Bbb P_\beta$
\nl

$\qquad \quad (vi)$ if $\alpha \in W$ is strongly inaccessible 
$> CC(\Bbb P_i) + \kappa$ \nl

$\qquad \qquad \qquad$ for every $i < \alpha$ \ub{then} for some $\beta <
\alpha$ \nl

$\qquad \qquad \qquad$ every ${\underset\tilde {}\to \zeta_i}$ is a simple
$(\bar{\Bbb Q},W)$-named $[0,\beta)$-ordinal; \nl

$\qquad \qquad \qquad$ needed, e.g., 
in \scite{si6.11} (note: this demand follows by \scite{1.6A}) \nl

$\qquad \quad (vii)$ 
\sn
\itemitem{ $(b)$ }  for $p \in 
\text{ Sp}_e(W)$-Lim$_\kappa(\bar{\Bbb Q})$ and $\xi < \ell g
(\bar{\Bbb Q})$ we let:

$$
p \restriction \xi =: \{r \restriction \xi:r \in p\}
$$

$$
p \restriction \{\xi\} =: \{r \restriction \{\xi\}:r \in p\}
$$
\newline

\noindent
$\quad$ we define similarly $p \restriction [\zeta,\xi),p \restriction \{
\underset\tilde {}\to \zeta\}, p \restriction [\underset\tilde {}\to \zeta,
\underset\tilde {}\to \xi)$.
\sn
\itemitem{ $(c)$ }  $\Bbb P_\alpha 
\models ``p^1 \le_{\text{vpr}} p^2"$ iff for every
$\xi < \alpha$ we have (letting \nl

$\quad p^\ell = \{ q^\ell_i:i < i^\ell(*)\}$ for $\ell = 1,2$): 
$$
p^2 \restriction \xi \Vdash_{{\Bbb P}_\xi}
``{\underset\tilde {}\to {\hat{\Bbb Q}}_\xi} 
\models p^1 \restriction \{ \xi \} 
\le_{\text{vpr}} p^2 \restriction \{ \xi \}"
$$
\sn
\itemitem{ $(d)$ }  $\Bbb P_\alpha 
\models ``p^1 \le_{\text{pr}} p^2"$ \underbar{iff}
\newline
\smallskip

$\qquad \quad (i) \quad$ for every 
$\xi < \ell g(\bar{\Bbb Q})$, we have \footnote{recall
\scite{1.1}(4)(d) we can omit ``$p^1 \restriction \{xi\} \ne \emptyset
\Rightarrow$"} \newline

$\qquad \qquad \qquad
p^2 \restriction \xi \Vdash_{{\Bbb P}_\xi}$ ``then $\hat{\Bbb Q}_\xi
\models p^1 \restriction \{\xi\} \le_{\text{pr}}
p^2 \restriction \{\xi\}$" \nl
\smallskip

$\qquad \quad (ii) \quad$ for some ordinal $j < \kappa$ and simple
$(\bar{\Bbb Q},W)$-named \nl
 
$\qquad \qquad \qquad [0,\alpha)$-ordinals 
${\underset\tilde {}\to \zeta_\varepsilon}$ for 
$\varepsilon < j$, for every $\xi < \ell g(\bar{\Bbb Q})$ we have: \newline

$\qquad \qquad \qquad
p^2 \restriction \xi 
\Vdash_{{\Bbb P}_\xi}$ ``if for no $\varepsilon < j$ do we have
${\underset\tilde {}\to \zeta_\varepsilon}
[{\underset\tilde {}\to G_{{\Bbb P}_\xi}}] = \xi$, then \newline

$\qquad \qquad \qquad
p^1 \restriction \{\xi\} \le_{\text{vpr}} p^2 \restriction \{\xi\}$ in
${\underset\tilde {}\to {\hat Q}_\xi}$"; we call
$\langle {\underset\tilde {}\to \zeta_\varepsilon}:\varepsilon < j \rangle$
a witness.
\sn
\itemitem{ $(e)$ }  $\Bbb P_\alpha \models p^1 \le p^2$ iff \newline
\smallskip

$\qquad \quad (i) \quad$ for every $\xi < \ell g(\bar{\Bbb Q})$ we have:
$$
p^2 \Vdash_{{\Bbb P}_\xi} ``{\underset\tilde {}\to {\hat{\Bbb Q}}_\xi} 
\models p^1 \restriction \{\xi\} \le
p^2 \restriction \{\xi\}"
$$

$\qquad \quad (ii) \quad$ as in the definition of $\le_{pr}$; 
clause (ii) \newline
\smallskip

$\qquad \quad (iii) \quad$ for some 
$n < \omega$ and simple $(\bar{\Bbb Q},W)$-named ordinals 
${\underset\tilde {}\to \xi_1},\dotsc,{\underset\tilde {}\to \xi_n}$
\nl 

$\qquad \qquad \qquad$ we have: \nl

$\qquad \qquad \qquad$ for each 
$\xi < \ell g(\bar{\Bbb Q})$ we have \nl

$\qquad \qquad \qquad \,\,p_2 \restriction \xi \Vdash_{{\Bbb P}_\xi}$ ``if
$\xi \ne {\underset\tilde {}\to \xi_\ell}
[{\underset\tilde {}\to G_{{\Bbb P}_\xi}}]$ for \nl

$\qquad \qquad \qquad \qquad \qquad \qquad \quad \ell = 1,\dotsc,n$ and \nl

$\qquad \qquad \qquad \qquad \qquad \qquad \quad 
\neg(\emptyset_{\underset\tilde {}\to {\Bbb Q}_\xi} 
\le_{\text{vp}} p^1 \restriction \{\xi\})]$  \nl

$\qquad \qquad \qquad \qquad \qquad \qquad \quad$ then: 
${\underset\tilde {}\to {\hat{\Bbb Q}}_\xi} 
\models p^1 \restriction
\{\xi\} \le_{\text{pr}} p^2 \restriction \{\xi\}"$ \nl

$\qquad \quad$ note that the truth value of $\zeta = 
{\underset\tilde {}\to \xi_\ell}$ is a $\Bbb P_\zeta$-name so this is \nl

$\qquad \quad$ well defined.  \nl

$\qquad \quad$ We then (i.e. if $(i) + (ii) + (iii)$) say: $p_1 \le p_2$ over
$\{{\underset\tilde {}\to \xi_1},\dotsc,{\underset\tilde {}\to \xi_n}\}$.
\sn
\itemitem{ $(f)$ }  Lastly, for $p \in \Bbb P_\alpha$ we let
Dom$(p) = \text{ Dom}_{\text{vp}}(p) =
\{{\underset\tilde {}\to \zeta_{\underset\tilde {}\to q}}:
\underset\tilde {}\to q \in p\}$ and Dom$_{\text{pr}}(p) = 
\{{\underset\tilde {}\to \zeta_\varepsilon}:\varepsilon < j\}$ where
$\bar \zeta = \langle {\underset\tilde {}\to \zeta_\varepsilon}:\varepsilon
< j \rangle$ is as in clause (F)(a) above (we can make it part of $p$).
\sn
\itemitem{ $(g)$ }  We still have to 
define $\Bbb P_{\underset\tilde {}\to \beta}$
for $\underset\tilde {}\to \beta$ a full simple 
$(\bar{\Bbb Q},W)$-named ordinal, it
is $\{p:p = \{{\underset\tilde {}\to q_i}:i < i^*\}$ and 
$\Vdash_{\bar{\Bbb Q}} 
``\zeta_{\underset\tilde {}\to q_i} < \underset\tilde {}\to \beta"$
that is if $\xi < \alpha$ then $\Vdash_{{\Bbb P}_\xi}$ ``if
$\zeta_{\underset\tilde {}\to q_i}[G_{{\Bbb P}_\xi}] = \xi$ and
$\underset\tilde {}\to \beta[G_{{\Bbb P}_\xi}]$ is well defined then
it is $> \xi"\}$.
\sn
\itemitem{ $(h)$ }  We call $p \in \Bbb P_\alpha$ full if it has a
witness $\langle {\underset\tilde {}\to \zeta_\varepsilon}:\varepsilon
< j \rangle$ with each ${\underset\tilde {}\to \zeta_\varepsilon}$ full.
\endroster}
\sn
\item "{$(G)$}"   Let us check Definition \scite{1.1}(1) for
$\Bbb P_\alpha =: \text{ Sp}_e(W)$-Lim$_\kappa(\bar{\Bbb Q})$: \nl
\ub{Proof of} $\le^{{\Bbb P}_\alpha}$ is a partial order. \nl
Suppose $p_0 \le p_1 \le p_2$.  Let $n^\ell,
{\underset\tilde {}\to \xi^\ell_1},\dotsc,
{\underset\tilde {}\to \xi^\ell_{n^\ell}}$ and $j^\ell \,(< \kappa)
\,\,{\underset\tilde {}\to \zeta^\ell_\varepsilon}$ 
(for $\varepsilon < j^\ell$) appear in the
definition of $p_\ell \le p_{\ell + 1}$.  Let $n = n^0 + n^1$, and

$$
{\underset\tilde {}\to \xi_i} = \cases
{\underset\tilde {}\to \xi^0_i} \quad &\text{if} \quad 1 \le i \le n^1 \\
{\underset\tilde {}\to \xi^1_{i-n}} &\text{if} \quad n^1 < i \le n^1 + n^2.
\endcases
$$

Let $j = j^0 + j^1$ and

$$
{\underset\tilde {}\to \zeta_\varepsilon} = \cases
{\underset\tilde {}\to \zeta^0_\varepsilon} \quad &\text{\underbar{if}} \quad 
\varepsilon < j^0 \\
{\underset\tilde {}\to \zeta^1_{\varepsilon-j^0}} &\text{\underbar{if}} \quad 
\varepsilon \in [j^0,j^0 + j^1).
\endcases
$$
\medskip

\noindent
Let us check the three clauses of (e) of part (D).
\bn
\ub{Clause (i)}: \nl
Let $\xi < \ell g(\bar{\Bbb Q})$ so for $\ell = 0,1$

$$
p_{\ell +1} \restriction \xi \Vdash_{{\Bbb P}_\xi} ``p_\ell \restriction \{\xi\} 
\le p_{\ell +1} \restriction \{\xi\} \text{ in } \hat{\Bbb Q}_\xi".
$$
\mn
As $\Bbb P_\xi \models 
``p_1 \restriction \xi \le p_2 \restriction \xi"$ (by the
induction hypothesis, clause (H)) clearly
$p_2 \restriction \xi$ forces both assertions.  As
${\underset\tilde {}\to {\hat{\Bbb Q}}_\xi}$ is a 
partial order (under $\le$) the conclusion follows.
\bn
\ub{Clause (ii)}: \nl
Let $\xi < \ell g(\bar{\Bbb Q})$, so similarly 
$p^2 \restriction \xi \Vdash_{{\Bbb P}_\xi}$ ``if $\xi \ne
{\underset\tilde {}\to \xi^\ell_m}$ for $m = 1,\dotsc,n^\ell$ 
and $\ell = 0,1$ (i.e., $\xi^\ell_m[{\underset\tilde {}\to G_{{\Bbb P}_\xi}}]$ 
is $\ne \xi$ or is not well defined), \ub{then} $p_0 \restriction
\{\xi\} \le_{\text{pr}} p_1 \restriction 
\{\xi\}$ in ${\underset\tilde {}\to {\hat{\Bbb Q}}_\varepsilon}$ and
$p_1 \restriction \{\xi\} \le_{\text{pr}} p_2 \restriction \{\xi\}$ in
${\underset\tilde {}\to {\hat{\Bbb Q}}_\xi}"$ from which the result
follows.
\bn
\ub{Clause (iii)}: \nl
Lastly, for $\xi < \alpha$ we have $p_2 \restriction \xi
\Vdash_{{\Bbb P}_\xi}$ ``if $\xi \notin 
\{{\underset\tilde {}\to \zeta^\ell_\varepsilon}[G_{{\Bbb P}_\zeta}]:
{\underset\tilde {}\to \zeta^\ell_\varepsilon}[G_{{\Bbb P}_\zeta}]$ well defined,
$\varepsilon < j^\ell$ and $\ell \in \{0,1\}\}$ then 
$p_0 \restriction \{ \xi \} \le_{\text{vpr}} p_1
\restriction \{ \xi\} \le_{\text{vpr}} p_2 \restriction \{ \xi\}$ hence $p_0
\restriction \{\xi\} \le_{\text{vpr}} p_2 \restriction \{\xi\}$". 
\nl
\mn
So we have proved the three conditions needed for $p_0 \le p_2$ by 
the definition above so 
really $p_0 \le p_2$ holds, so $\le$ is a partial order. \nl
\sn
\ub{Proof of} $\le_{\text{pr}}$ is a partial order. \nl

Similar proof.
\sn
\ub{Proof of} $\le_{\text{vpr}}$ is a partial order.  \nl

Similar proof just easier. \nl
\sn
\ub{Proof of} $p \le_{\text{pr}} q \Rightarrow p \le q$: 

By the definition; easy. \nl
\sn
\ub{Proof of} $p \le_{\text{vpr}} q \Rightarrow p \le_{\text{pr}} q$:

By the definition, check. \nl
So in \scite{1.1}(1) all the requirements on $\Bbb P_\alpha$ holds.
\sn
\item "{$(H),(I),(J)$}"  We leave the checking to the reader 
(actually we prove $(I)$ in the proof of \scite{1.17} below). 
\hfill$\square_{\scite{1.13}}$\margincite{1.13}
\endroster
\bigskip

\demo{\stag{1.14} Fact}  1) If $\bar{\Bbb Q}$ is a 
$\kappa-\text{Sp}_e$-iteration and for each $i < \ell g(\bar{\Bbb Q})$ 
we have it is forced (i.e., $\Vdash_{{\Bbb P}_i}$) that
$\le^{{\Bbb Q}_i}_{\text{pr}} = \le^{{\Bbb Q}_i}$ and 
$\le^{{\Bbb Q}_i}_{\text{vpr}}$ is 
equality, \ub{then} $\bar{\Bbb Q}$ is a variant of $\kappa$-RS iteration 
(as in \cite[Ch.XIV,\S1]{Sh:f}), i.e. they are the same if we use there 
only simple $\bar{\Bbb Q}$-named ordinals (or allow here non-simple 
ones so the version here is exactly as in \cite[Ch.XIV,2.6]{Sh:f}). 
\enddemo
\bigskip

\demo{Proof}  Straightforward.
\enddemo
\bigskip

\proclaim{\stag{1.15} Claim}  1) In \scite{1.13} in Definition (F),
clause (a)(iii) we can demand that each ${\underset\tilde {}\to
\zeta_\varepsilon}$ is full (simple $(\bar{\Bbb Q},W)$-named
$[0,\alpha)$-ordinal) and similarly in (d)(ii), (=(e)(ii)) and
(e)(iii). \nl
2) Suppose $\bar{\Bbb Q} = \langle \Bbb P_i,
{\underset\tilde {}\to {\Bbb Q}_i}:i < \alpha 
\rangle$ is a $\kappa-{\text{\rm Sp\/}}_e(W)$-iteration
(so $\Bbb P_\alpha = { \text{\rm Sp\/}}_e(W)$-{\text{\rm
Lim\/}}$_\kappa(\bar{\Bbb Q})$).  If $p \le q$ in 
$\Bbb P_\alpha$, \ub{then} there are $r,n < \omega$ and 
$\xi_1 < \ldots < \xi_n < \alpha$ such that:
\medskip
\roster
\item "{$(a)$}"  $r \in \Bbb P_\alpha$
\sn
\item "{$(b)$}"  $q \le r$
\sn
\item "{$(c)$}"  $p \le r$ above $\{\xi_1,\dotsc,\xi_n\}$.
\ermn
3)  If $p^1,p^2 \in \Bbb P_\alpha$ and \scite{1.13}(F)(e) holds but we 
allow $\underset\tilde {}\to n$ to be a full simple $(\bar{\Bbb
Q},W)$-named ordinal, \ub{then} $p^2 \Vdash_{{\Bbb P}_\alpha} ``p_1
\in {\underset\tilde {}\to G_{{\Bbb P}_\alpha}}"$. 
\endproclaim
\bigskip

\remark{Remark}  In fact, in \scite{1.15}(1) we can have 
$r \restriction [\xi_n,\alpha) = q \restriction [\xi_n,\alpha)$.
\endremark
\bigskip

\demo{Proof}  1) As increasing those sets $(\{{\underset\tilde {}\to
\zeta_\varepsilon}:\varepsilon < \zeta\},\{{\underset\tilde {}\to
\xi_1},\dotsc,{\underset\tilde {}\to \xi_n}\}$, respectively) cause no
harm. \nl
2) We prove this by induction on $\alpha$.
\enddemo
\bn
\underbar{Case 1}:  $\alpha = 0$. \newline
Trivial.
\bn
\underbar{Case 2}:  $\alpha = \beta + 1$.

Apply the induction 
hypothesis to $\bar{\Bbb Q} \restriction \beta,p \restriction
\beta,q \restriction \beta$ (clearly $\bar{\Bbb Q} \restriction 
\beta$ is an \nl
$\kappa-\text{Sp}_e(W)$-iteration, 
$p \restriction \beta \in \Bbb P_\beta,q \restriction 
\beta \in \Bbb P_\beta$ and $\Bbb P_\beta \models ``p \le q"$, 
by \scite{1.13}, clause (H)).  So we can find $r',m < \omega$ 
and $\{\xi'_1,\dotsc,\xi'_m\}$ such that:
\medskip
\roster
\item "{$(a)'$}"  $r' \in \Bbb P_\beta$
\sn
\item "{$(b)'$}"  $\Bbb P_\beta \models q \restriction \beta \le r'$
\sn
\item "{$(c)'$}"  $p \restriction \beta \le r'$ (in $\Bbb P_\beta$) above
$\{\xi_1,\dotsc,\xi_m\}$
\sn
\item "{$(d)'$}"  $\xi'_1 < \ldots < \xi'_m$.
\endroster
\medskip

Let $n =: m+1$ and

$$
\xi_\ell = \cases \xi'_\ell \quad &\text{if} \quad \ell \in \{1,\dotsc,m\} \\
\beta \quad &\text{if} \quad \ell = n
\endcases
$$
\mn
and lastly $r = r' \cup (q \restriction \{\beta\})$.
\bn
\underbar{Case 3}:  $\alpha$ is a limit ordinal. \nl
Let $p \le q$ (in $\Bbb P_\alpha$) above 
$\{{\underset\tilde {}\to \xi_1},\dotsc,
{\underset\tilde {}\to \xi_n}\}$.  We choose by 
induction on $\ell \le n$, the objects
$r_\ell,\beta_\ell,\xi^*_\ell$ such that:
\mr
\item "{$(\alpha)$}"  $r_\ell \in \Bbb P_{\beta_\ell}$
\sn
\item "{$(\beta)$}"  $r_\ell \le r_{\ell +1}$
\sn
\item "{$(\gamma)$}"  $q \restriction \beta_\ell \le r_\ell$
\sn
\item "{$(\delta)$}"  $\beta_\ell \le \beta_{\ell +1} < \alpha$
\sn
\item "{$(\varepsilon)$}"  $\beta_0 = 0,r_0 = \emptyset_{{\Bbb P}_0}$
\sn
\item "{$(\zeta)$}"  for $\ell \in \{1,\dotsc,n\}$ we have: either
$r_\ell \Vdash_{{\Bbb P}_{\beta_\ell}} ``{\underset\tilde {}\to \xi_\ell} = 
\xi^*_\ell"$ and $\xi^*_\ell \le \beta_\ell$ or \nl
$\beta_\ell = \beta_{\ell-1} \and r_\ell = 
r_{\ell-1}$ and $r_\ell \cup (q \restriction [\beta,\alpha))
\Vdash_{{\Bbb P}_\alpha}
``{\underset\tilde {}\to \xi_\ell}$ is not defined or is $\ge \alpha"$.
\endroster
\medskip

Carrying the definition is straight: for $i=0$ use clause $(\varepsilon)$.
For $\ell + 1 \le n$ when the second possibility of clause $(\zeta)$ fails
there is $r'$, such that $r_\ell \cup (q \restriction [\beta,\alpha)) \le 
r' \in 
\Bbb P_\alpha$, and $r' \Vdash_{{\Bbb P}_\alpha} ``{\underset\tilde {}\to 
\xi_{\ell+1}}$ is defined and is $< \alpha$", so there are 
$r'',\xi^*_{\ell +1} < \alpha$ such that
$r' \le r'' \in \Bbb P_\alpha$ and $r'' \Vdash ``{\underset\tilde {}\to 
\xi_{\ell+1}} = \xi^*_{\ell +1}$" so as ``$\xi^*_{\ell+1}$ is a simple
$\bar{\Bbb Q}$-named ordinal" 
we know that $\xi^*_{\ell +1} < \alpha$ and $r''
\restriction \xi^*_{\ell+1} \Vdash_{{\Bbb P}_{\xi^*_{\ell+1}}} ``
{\underset\tilde {}\to \xi_{\ell+1}} = \xi^*_{\ell +1}$".  Let
$\beta_{\ell+1} =: \text{ max}\{\beta_\ell,\xi^*_{\ell+1}\}$, and
$r_{\ell +1} =: r'' \restriction \beta_{\ell+1}$.  So we have carried the
induction.

We now apply 
the induction hypothesis to $\bar{\Bbb Q} \restriction \beta_n,p \restriction
\beta_n,r_n$; it is applicable as $\beta_n < \alpha$, and $\Bbb P_{\beta_n}
\models ``p \restriction \beta_n \le q \restriction \beta_n \le r_n"$.  So
there are $m < \omega,\xi_1 < \ldots < \xi_m < \beta_n$ and $r^*$ such that
$\Bbb P_{\beta_n} \models ``r_n \le r^*"$ and 
$p \le r^*$ (in $\Bbb P_{\beta_n}$)
above $\{\xi_1,\dotsc,\xi_m\}$.  Now let $r =: r^* \cup (q \restriction
[\beta_n,\alpha))$, clearly $q \le r$ and $p \le r$ above $\{\xi_1,\dotsc,
\xi_m,\beta_n\}$. \nl
3) So $p^1,p^2,\{\xi_\ell:\ell =1,\dotsc,n\}$ are given.  Assume that
$p^2 \le q_1 \in \Bbb P_\alpha$.  We can find $q_2$ such that $q_1 \le
_2 \in \Bbb P_\alpha$ and $q_2$ forces a value to $\underset\tilde
{}\to n$ say $n(*)$.  Next we can find $q_3$ such that $q_2 \le q_3
\in \Bbb P_\alpha$ and $q_3$ forces values to ${\underset\tilde {}\to
\xi_\ell} \, (\ell =1,\dotsc,n(*)\}$, say
$\xi^0_1,\dotsc,\xi^0_{n(*)}$.  Now repeating the proof of
``$\le^{{\Bbb P}_\alpha}$ is a partial order" (in clause (H) of
\scite{1.13}) with $p^1,p^2,q_3$ here standing for $p_0,p_1,p_2$ there
we let $\Bbb P_\alpha \models p^1 \le q_3$.  As we have assumed just
$p^2 \le q_1 \in \Bbb P_\alpha$ and $q_1 \le q_3$ we are done.
\hfill$\square_{\scite{1.15}}$\margincite{1.15}  
\bigskip

\citewarning{\noindent \llap{---$\!\!>$} MARTIN WARNS: Label 1.16 on next line is also used somewhere else (Perhaps should have used scite instead of stag?)}
\proclaim{\stag{1.16} Claim}  Let 
$\bar{\Bbb Q}$ be a $\kappa-{\text{\rm Sp\/}}_e(W)$-iteration
of length $\alpha$. \nl
0) If $\underset\tilde {}\to \zeta$ 
is a simple $(\bar{\Bbb Q},W)$-named ordinal
\ub{then} for some ordinal $\gamma$ we have:
$\underset\tilde {}\to \zeta$ is a simple $(\bar{\Bbb Q},W)$-named
$[0,\gamma)$-ordinal. \newline
1) 
\mr
\item "{$(i)$}"  If 
$\beta < \alpha$ and $\underset\tilde {}\to \zeta$ is a $\Bbb P_\beta$-name
of a [full] simple 
$(\bar{\Bbb Q},W)$-named $[\beta,\alpha)$-ordinal \ub{then} for some [full]
simple $(\bar{\Bbb Q},W)$-named $[\beta,\alpha)$-ordinal 
$\underset\tilde {}\to \xi$
we have
$$
\Vdash_{\bar{\Bbb Q}} 
``\underset\tilde {}\to \zeta = \underset\tilde {}\to \xi"
$$
\sn
\item "{$(ii)$}"  if $\beta \le \alpha,\beta \le \gamma_1 \le \gamma_2$
and $\underset\tilde {}\to \zeta$ is a $\Bbb P_\beta$-name of a [full]
simple $(\bar{\Bbb Q},W)$-named $[\gamma_1,\gamma_2)$-ordinal
\ub{then} for some [full] simple
$(\bar{\Bbb Q},W)$-named 
$[\gamma_1,\gamma_2)$-ordinal $\underset\tilde {}\to \xi$
we have $\Vdash_{\bar{\Bbb Q}} ``\underset\tilde {}\to \zeta =
\underset\tilde {}\to \xi"$.
\ermn
2) The same holds if we replace ``ordinal" by ``atomic condition"
(so in (ii) we should demand $\gamma_2 \le \alpha$). \newline
3) If $\alpha \le \gamma$ and $\underset\tilde {}\to \beta$ is a full simple 
$(\bar{\Bbb Q},W)$-named $[0,\alpha)$-ordinal, and for each 
$\beta < \alpha,{\underset\tilde {}\to \zeta_\beta}$ is a [full] simple
$(\bar{\Bbb Q},W)$-named $[\beta,\gamma)$-ordinal \ub{then} for some
[full] simple $(\bar{\Bbb Q},W)$-named $[0,\gamma)$-ordinal 
$\underset\tilde {}\to \xi$ we have

$$
\Vdash_{\bar{\Bbb Q}} ``\text{if } \underset\tilde {}\to \beta
[\underset\tilde {}\to G] = \beta (\text{so } \beta < \alpha) \text{ then }
\underset\tilde {}\to \xi[\underset\tilde {}\to G] = \zeta_\beta
[\underset\tilde {}\to G]".
$$
\mn
4) If $\underset\tilde {}\to \beta$ is a full
simple $(\bar{\Bbb Q},W)$-named $[0,\alpha)$-ordinal and 
for each $\beta < \alpha,{\underset\tilde {}\to p_\beta}$ is a 
$(\bar{\Bbb Q},W)$-named $[\beta,\alpha)$-atomic 
condition \underbar{then} for 
some $(\bar{\Bbb Q},W)$-named atomic condition $p$ we have

$$
\Vdash_{\bar{\Bbb Q}} ``\text{if } \underset\tilde {}\to \beta
[\underset\tilde {}\to G] = \beta \text{ then }
\underset\tilde {}\to p[\underset\tilde {}\to G] = 
{\underset\tilde {}\to p_\beta}[\underset\tilde {}\to G]".
$$
\endproclaim
\bigskip

\demo{Proof}  Easy.  \hfill$\square_{\scitet{1.16}}$\scitetphantom{1.16}\margincite{1.16}
\enddemo
\bigskip

\proclaim{\stag{1.16A} Claim}  1) Suppose $F$ is a function, \ub{then} 
for every ordinal $\alpha$ there is \nl
$\kappa-{\text{\rm Sp\/}}_e(W)$-iteration $\bar{\Bbb Q} = \langle \Bbb P_i,
{\underset\tilde {}\to {\Bbb Q}_i}:i < \alpha^\dag \rangle$, such that:
\mr
\item "{$(a)$}"  for every $i$ we have ${\underset\tilde {}\to {\Bbb Q}_i} = 
F(\bar{\Bbb Q} \restriction i)$,
\sn
\item "{$(b)$}"  $\alpha^\dag \le \alpha$
\sn
\item "{$(c)$}"  either $\alpha^\dag = \alpha$ or $F(\bar{\Bbb Q})$ is not an
${\text{\rm Sp\/}}_e(W)-{\text{\rm Lim\/}}_\kappa
(\bar{\Bbb Q})$-name of a forcing notion.
\ermn
2) Suppose $\beta < \alpha,G_\beta \subseteq \Bbb P_\beta$ is generic
over $\bold V$, \ub{then} in $\bold V[G_\beta],\bar{\Bbb Q}/G_\beta 
= \langle \Bbb P_i/G_\beta,{\underset\tilde {}\to {\Bbb Q}_i}:\beta 
\le i < \alpha \rangle$ is an $\kappa-{\text{\rm Sp\/}}_e$-iteration 
and $\kappa-{\text{\rm Sp\/}}_e(W)-{\text{\rm Lim\/}}(\bar{\Bbb Q}) = 
\Bbb P_\beta * ({\text{\rm Lim\/}}(\bar{\Bbb Q})/
{\underset\tilde {}\to G_\beta}$) (essentially; more exactly
up to equivalence) where, of course, $\Bbb P_i/G_\beta = \{p \in \Bbb P_i:p
\restriction \beta \in G_\beta\}$. \newline
3) If $\bar{\Bbb Q}$ is an $\kappa-{\text{\rm Sp\/}}_e(W)$-iteration, 
$p \in { \text{\rm Sp\/}}_e(W)-{\text{\rm Lim\/}}_\kappa
(\bar{\Bbb Q}),\Bbb P'_i = \{q \in \Bbb P_i:q \ge p 
\restriction i\}$, \nl
${\underset\tilde {}\to {\Bbb Q}'_i} = \{ p \in
{\underset\tilde {}\to {\Bbb Q}_i}:p \ge p \restriction \{i\}\}$ and
$\emptyset_{\underset\tilde {}\to {\Bbb Q}'_i} = p \restriction \{i\},
\le^{\underset\tilde {}\to {\Bbb Q}'_i}_\ell =
\le^{\underset\tilde {}\to {\Bbb Q}_i}_\ell 
\restriction {\underset\tilde {}\to {\Bbb Q}'_i}$ 
for $\ell = 0,1,2$ \ub{then} 
$\bar{\Bbb Q} = \langle \Bbb P'_i,\Bbb Q'_i:
i < \ell g(\bar{\Bbb Q}) \rangle$ is
(essentially) a $\kappa-{\text{\rm Sp\/}}_e(W)$-iteration 
(and ${\text{\rm Sp\/}}_e(W)-{\text{\rm Lim\/}}_\kappa(\bar{\Bbb Q}')$ is 
$P'_{\ell g(\bar{\Bbb Q})}$).
\endproclaim
\bigskip

\demo{Proof}  Should be clear.
\enddemo
\bigskip

\proclaim{\stag{1.17} Claim}  Suppose 
\mr
\item "{$(a)$}"  $\bar{\Bbb Q} = \langle \Bbb P_i,{\underset\tilde
{}\to {\Bbb Q}_i}:i < \alpha
\rangle$ is a $\kappa-{\text{\rm Sp\/}}_e(W)$-iteration 
(and $\Bbb P_\alpha = { \text{\rm Sp\/}}_e(W)-{\text{\rm
Lim\/}}_\kappa(\bar{\Bbb Q}))$
\sn
\item "{$(b)$}"  $\ell(*) \in \{0,1\}$
\sn
\item "{$(c)$}"  $\Vdash_{{\Bbb P}_i} ``({\underset\tilde {}\to {\Bbb
Q}_i},\le_{\ell(*)})$ is a $\theta$-complete" for each $i < \alpha$. 
\ermn
\ub{Then}: \nl
1) $(\Bbb P_\alpha,\le_{\ell(*)})$ is $\theta$-complete, i.e. 
if $\delta < \theta,\langle p_i:i < \delta \rangle$ is 
$\le_{\ell(*)}$-increasing \ub{then} it has an 
$\le_{\ell(*)}$-upper bound provided that:
\nl
$\theta \le \kappa$ \ub{or} $\ell(*) = 0 \and 
\theta \le { \text{\rm Min\/}}\{\delta:
\delta \in W \text{ is strongly inaccessible and } (\forall \beta < \delta)
(|\Bbb P_\beta| < \delta)\}$. \newline
2) Moreover for $\beta < \alpha$ we have 
$(\Bbb P_\alpha/\Bbb P_\beta,\le_{\ell(*)})$ is 
$\theta$-complete. \newline
3) In fact, we can get $\le_{\ell(*)}$-lub (provided that 
there are such lub's for each ${\underset\tilde {}\to {\Bbb Q}_i}$.
\endproclaim
\bigskip

\remark{Remark}   We deal with $\theta$-complete rather than strategically
$\theta$-complete (here and later) just for simplicity presentation, as it
does not matter much by \cite[CH.XIV,2.4]{Sh:f}.
\endremark
\bigskip

\demo{Proof}  Straightforward but we elaborate. \newline
1) So assume $\delta < \theta$ and $p_i \in \Bbb P_\alpha$ for $i < 
\delta$ and $[i < j < \delta \Rightarrow p_i \le_{\ell(*)} p_j]$.  
Now it is enough to find $p \in \Bbb P_\alpha$ such that

$$
i < \delta \Rightarrow p_i \le_{\ell(*)} p.
$$
\mn
Let $p_i = \{{\underset\tilde {}\to q^i_\gamma}:\gamma < \gamma_i\}$ 
and for each $\gamma < \gamma_i,{\underset\tilde {}\to q^i_\gamma}$ is 
a simple $(\bar{\Bbb Q},W)$-named atomic condition, say
$\Vdash_{\bar{\Bbb Q}} ``{\underset\tilde {}\to q^i_\gamma} \in 
{\underset\tilde {}\to {\Bbb Q}_{\underset\tilde {}\to
\zeta^i_\gamma}}"$, where 
${\underset\tilde {}\to \zeta^i_\gamma}$ is a simple
$(\bar{\Bbb Q},W)$-named ordinal 
(which is 
${\underset\tilde {}\to \zeta_{\underset\tilde {}\to q^i_\gamma}}$).
Now for each $\beta < \alpha$ let ${\underset\tilde {}\to <^*_\beta}$ 
be a $\Bbb P_\beta$-name of a well ordering of
${\underset\tilde {}\to {\Bbb Q}_\beta}$.  For 
each $i(*) < \delta,\gamma(*) <
\gamma_i$ let ${\underset\tilde {}\to r^{i(*)}_{\gamma(*)}}$ be the following
simple $(\bar{\Bbb Q},W)$-named atomic condition:

Let $\zeta < \alpha,G_\zeta \subseteq \Bbb P_\zeta$ generic over
$\bold V$ and
${\underset\tilde {}\to \zeta^{i(*)}_{\gamma(*)}}[G_\zeta] = \zeta$, now work
in $\bold V[G_\zeta]$, let $w_\zeta = \{i < \delta:\text{for some } \gamma <
\gamma_i \text{ we have } {\underset\tilde {}\to \zeta^i_\gamma}[G_\zeta] = 
\zeta\}$.
We let $u^\zeta_i = \{\gamma < \gamma_i:
{\underset\tilde {}\to \zeta^i_\gamma}
[G_\zeta] = \zeta\}$ for each $i \in w_\zeta$.
(As $p_i$ is $\le_{\ell(*)}$-increasing, $w_\zeta$ is an end segment of 
$\delta$ and 
$i(*) \in w_\zeta,\gamma(*) \in u^\zeta_{i(*)}$).  For $i \in w_\zeta$ let
$q^*_{i,\zeta} = (p_i \restriction \{\zeta\})[G_\zeta]$.
Now define ${\underset\tilde {}\to  r^{i(*)}_{\gamma(*)}}[G_\zeta]$ as
follows.
\mn
\underbar{Case 1}:  For some $j < \delta$ the sequence 
$\langle q^*_{i,\zeta}:i \in w_\zeta \backslash j \rangle$ is constant.

Let ${\underset\tilde {}\to r^{i(*)}_{\gamma(*)}}[G_\zeta] =
q_{\text{Min}(w_\zeta \backslash j),i}$.
\mn
\underbar{Case 2}:  Not Case 1, but for some $\ell < 2,j < \delta$
the sequence $\langle q_{i,\zeta}:i \in w_\zeta \backslash j \rangle$ 
is $\le_\ell$-increasing,
without loss of generality $\ell$ minimal (on all possible $j$) and then
${\underset\tilde {}\to r^{i(*)}_{\gamma(*)}}[G_\zeta] \in
{\underset\tilde {}\to {\Bbb Q}_\zeta}[G_\zeta]$ is the
$<^*_\zeta$-first $\le_\ell$-upper bound of
$\{{\underset\tilde {}\to q^*_{i,\zeta}}:i \in w_\zeta \backslash j\}$ where
$<^*_\zeta = {\underset\tilde {}\to <^*_\zeta}[G_\zeta]$.
It exists by \scite{1.1A}(2).  
\mn
\ub{Case 3}:  Neither Case 1 nor Case 2 ${\underset\tilde {}\to
r^{i(*)}_{\gamma(*)}}[G_\zeta]$ is $\emptyset_{\underset\tilde {}\to
{\Bbb Q}_\zeta}$. \nl
Let $p = \{ {\underset\tilde {}\to  r^{i(*)}_{\gamma(*)}}:i(*) < \delta
\text{ and } \gamma < \gamma_{i(*)}\}$.

If $\ell(*) = 0$ (i.e., $p_i$ is $\le_{\text{vpr}}$-increasing) and 
$\langle {\underset\tilde {}\to \zeta_j}:j < j^* \rangle$
witness $p_0 \in \Bbb P_\alpha$, then it witnesses 
$p \in \Bbb P_\alpha$ and easily
$i < \delta \Rightarrow p_i \le_0 p$. \nl
So assume $\ell(*)=1$, that is $p_i$ is $\le_{\text{pr}}$-increasing.  
For $i_0 < i_2 < \delta$ let 
$\{ {\underset\tilde {}\to \zeta^{i_0,i_1}_j}:j < j_{\ell_0,\ell_1}\}$ be a
witness to $p_{i_0} \le_{pr} p_{i_1}$ and let 
$\{{\underset\tilde {}\to \zeta^i_j}:j < j_i\}$  witness $p_i \in \Bbb P$.  
Letting $\kappa^-$ be the maximal cardinal $< \kappa$, clearly
$\{{\underset\tilde {}\to \zeta^{i_0,i_1}_j}:i_0 < i_1 < \delta \text{ and }
j < j_{i_0,i_1}\}$ has cardinality $\kappa^-$, so we can order it as  
$\{ {\underset\tilde {}\to \zeta^{i_0(\varepsilon),i_1(\varepsilon)}
_{j(\varepsilon)}}:\varepsilon < \kappa^-\}$ and some
$\{ \zeta^{i(\varepsilon)}_{j'(\varepsilon)}:\varepsilon < \kappa^-\}$ list
$\{ {\underset\tilde {}\to \zeta^i_j}:i < \delta,j < j_i\}$.  Now
$\{ \zeta^{i(\varepsilon)}_{j'(\varepsilon)}:
\varepsilon < \kappa^-\}$ witness $p \in \Bbb P_\alpha$ and
$\{ \zeta^{i_0(\varepsilon),i_1(\varepsilon)}_{j(\varepsilon)}:\varepsilon
< \kappa^-\}$ witness $p_i \le_1 p_\delta$ for every $i < \delta$.
\mn
2), 3)  Similar proof. \hfill$\square_{\scite{1.17}}$\margincite{1.17}
\enddemo
\bigskip

\definition{\stag{1.18} Definition}  Let $\bar{\Bbb Q} = \langle \Bbb P_i,
{\underset\tilde {}\to {\Bbb Q}_i}:i < \alpha \rangle$ be an 
$\kappa-\text{Sp}_e(W)$-iteration. \newline
1) We say $\underset\tilde {}\to y$ is a 
$(\bar{\Bbb Q},W,\underset\tilde {}\to \zeta)$-name if: $\underset\tilde {}\to y$ is
a $\Bbb P_\alpha$-name, $\underset\tilde {}\to \zeta$ is a simple $\bar{\Bbb Q}$-named 
$[0,\alpha)$-ordinal, and: if $\beta < \alpha,G_{{\Bbb P}_\alpha} \subseteq
\Bbb P_\alpha$ is generic over $\bold V$ and for some 
$r \in G_{{\Bbb P}_\alpha} \cap \Bbb P_\beta$
we have $r \Vdash_{\bar{\Bbb Q}} ``\underset\tilde {}\to \zeta = \beta"$, then
$\underset\tilde {}\to y[G_{{\Bbb P}_\alpha}] \in \bold V[G_{{\Bbb
P}_\beta}]$ is well defined and
depends only on $G_{{\Bbb P}_\alpha} \cap \Bbb P_\beta$ so we write
$\underset\tilde {}\to y[G_{{\Bbb P}_\alpha} \cap \Bbb P_\beta]$; and if
$G_{{\Bbb P}_\alpha} \subseteq \Bbb P_\alpha$ is generic over $\bold V$ and
$\underset\tilde {}\to \zeta[G_{{\Bbb P}_\alpha}]$ not well defined then 
$\underset\tilde {}\to y[G_{{\Bbb P}_\alpha}]$
is not well defined (do not arise if $\underset\tilde {}\to \zeta$ is
full). \nl
2) If $p \in \Bbb P_\alpha$ and $G_{{\Bbb P}_\alpha} \subseteq 
\Bbb P_\alpha$ is generic over $\bold V$, (or
just in Gen$^r(\bar{\Bbb Q})$), \ub{then} 
$p[G_{{\Bbb P}_\alpha}]$ is the following 
function, Dom$(p[{\underset\tilde {}\to G_{{\Bbb P}_\alpha}}]) = \{
{\underset\tilde {}\to \zeta_{\underset\tilde {}\to q}}[G_{{\Bbb P}_\alpha}]:
\underset\tilde {}\to q \in p\}$ and $(p[G_{{\Bbb P}_\alpha}])(\varepsilon) =
\{\underset\tilde {}\to q[G_{{\Bbb P}_\alpha}]:\underset\tilde {}\to q
\in p$ and ${\underset\tilde {}\to \zeta_{\underset\tilde {}\to q}}
[G_{{\Bbb P}_\alpha}] = \varepsilon\}$. 
\enddefinition
\bigskip

\proclaim{\stag{1.19} Claim}  Suppose
\mr
\item "{$(a)$}"  $\bar{\Bbb Q} = \langle \Bbb P_i,
{\underset\tilde {}\to {\Bbb Q}_i}:i < \alpha
\rangle$ is a $\kappa-{\text{\rm Sp\/}}_e(W)$-iteration.
\sn
\item "{$(b)$}"  $p \in \Bbb P_\alpha$ and $\underset\tilde {}\to \zeta$ is a
simple $(\bar{\Bbb Q},W)$-named $[0,\alpha)$-ordinal
\sn
\item "{$(c)$}"  $\underset\tilde {}\to r$ is a 
$(\bar{\Bbb Q},W,\underset\tilde {}\to \zeta)$-named member of 
$\Bbb P_\alpha/\Bbb P_{\underset\tilde {}\to \zeta}$.
\ermn
\ub{Then}: \newline
1)  There is $q \in \Bbb P_\alpha$ satisfying $p \le q$ such that:
\mr
\item "{$(*)$}"  if $\xi < \alpha,G_\xi \subseteq \Bbb P_\xi$ generic over
$\bold V$, then
{\roster
\itemitem{ $(\alpha)$ }  $\underset\tilde {}\to \zeta[G_\xi] = \xi$ \nl
implies $(p \restriction \xi)[G_\xi] = (q \restriction \xi)[G_\xi]$
and
\sn
\itemitem{ $(\beta)$ }  $\underset\tilde {}\to \zeta[G_\xi] = \xi$
implies \nl
$(q \restriction [\xi,\alpha))[G_\xi] = \underset\tilde {}\to r[G_\xi]$.
\endroster}
\ermn
2) If in addition (for any $\ell < 3)$ clause (c)$^+$ below,
\ub{then} we can in $(*)$ add $p \le_\ell q$
\mr
\item "{$(c)^+$}"  $\underset\tilde {}\to r$ is a 
$(\bar{\Bbb Q},W,\underset\tilde {}\to r)$-named member of $\Bbb P_\alpha/
\Bbb P_{\underset\tilde {}\to \zeta}$ which is $\le_\ell$-above $p 
\restriction [\underset\tilde {}\to \zeta,\alpha)$.
\endroster
\endproclaim
\bigskip

\demo{Proof}  Straightforward.
\enddemo
\bn
Central here is pure decidability.
\definition{\stag{1.25} Definition}  1) A forcing notion $\Bbb Q$ has pure
$(\theta_1,\theta_2)$-decidability if: for every $p \in \Bbb Q$ and
$\Bbb Q$-name $\underset\tilde {}\to \gamma < \theta_1$, there are 
$a \subseteq \theta_1,|a| < \theta_2$ (but $|a| > 0$) and $r \in 
\Bbb Q$ such that $p \le_{\text{pr}} r$ and 
$r \Vdash_{\Bbb Q} ``\underset\tilde {}\to \gamma \in a"$ (for $\theta_1=2$,
alternatively, $\underset\tilde {}\to \gamma$ is a truth value).  If we write
$``\le \theta_2"$ we mean $|a| \le |\theta_2|$. \nl
2) A forcing notion $\Bbb Q$ has pure $\theta$-decidability 
where $\theta$ is an
ordinal \ub{if}: for every $p \in \Bbb Q$ and 
${\underset\tilde {}\to {\Bbb Q}}$-name
$\underset\tilde {}\to \gamma < \theta$ 
there are $\gamma < \theta$ and $r \in
\Bbb Q$ such that $p \le_{\text{pr}} r$ and 
$r \Vdash_{\Bbb Q}$ ``if $\theta < \omega$
then $\underset\tilde {}\to \gamma = \gamma$ \ub{and} if 
$\theta \ge \omega$ is
a limit ordinal then $\underset\tilde {}\to \gamma < \gamma$".
\enddefinition
\bigskip

\demo{\stag{1.26} Observation}  1) If $\aleph_0 > \theta_2 > 2$ \ub{then} pure
$(\theta_2,2)$-decidability is equivalent to pure $(2,2)$-decidability. \nl
2) If $\Bbb Q$ is purely semi-proper (see \cite[X]{Sh:f} or here xxx) 
or just $\Bbb Q$ 
satisfies UP$^0(\Bbb I,\bold W)$
(see \S5) \ub{then} $\Bbb Q$ has pure $(\aleph_1,\aleph_1)$-decidability. \nl
3) If $\Bbb Q$ is purely proper, \ub{then} $\Bbb Q$ has 
$(\lambda,\aleph_1)$-decidability for every $\lambda$. \nl
4) If $\Bbb Q$ has the c.c.c. (and we let 
$\le_{\text{pr}}$ be equality if not defined), 
\ub{then} $\Bbb Q$ is purely proper.
\nl
5) If $\le_{\text{pr}} = \le$, \ub{then} $\Bbb Q$ has pure 
$(\lambda,2)$-decidability for every $\lambda$.
\enddemo
\bigskip

\demo{Proof}  Think of the definitions.
\enddemo
\bigskip

\definition{\stag{1.26A} Definition}  1) A forcingnotion $\Bbb P$ is
purely proper for $\chi$ large enough (e.g., ${\Cal P}(\Bbb P) \in
{\Cal H}(\chi)$ is enough) and $N$ is an elementary submodel of
$(H(\chi),\in)$ to which $\Bbb P$ belongs and $p \in N \cap \Bbb P$
\ub{then} three is $(N,\Bbb P)$-semi-generic satisfying $p
\le_{\text{pr}} q \in \Bbb P$, see below. \nl
2) $q$ is $(N,\Bbb P)$-generic if $q \Vdash_{\Bbb P}$ ``if
$\underset\tilde {}\to \tau$ which belongs to $N$ is a $\Bbb P$-name
of an ordinal then $\underset\tilde {}\to \tau[G_{\Bbb P}] \in N \cap
\text{ Ord}$.  \nl
3) A forcing notion $\Bbb P$ is purely semi-proper if in part 4) we
replace $(N,\Bbb P)$-generic by $(N,\Bbb P)$-semi-generic.
\nl
4) $q$ is $(N,\Bbb P)$-semi-generic if $q \Vdash_{\Bbb P}$ ``if
$\underset\tilde {}\to \tau$ which belongs to $N$, is a $\Bbb P$-name
of a countable ordinal then $\underset\tilde {}\to \tau[G_{\Bbb P}]
\in N \cap \omega_1$".
\enddefinition
\bigskip


\proclaim{\stag{1.27} Claim}  Let $\bar{\Bbb Q}$ 
be a $\kappa-{\text{\rm Sp\/}}_e(W)$-iteration.
\nl
1) The property ``$\Bbb Q$ has pure $\delta^*$-decidability and pure 
$(2,2)$-decidability" is preserved by
$\aleph_1-{\text{\rm Sp\/}}_e(W)$-iterations \ub{if} $\delta^*$ is a limit
ordinal. \nl
2) The property ``$\Bbb Q$ has pure $(2,2)$-decidablity" is preserved by
$\aleph_1-{\text{\rm Sp\/}}_e(W)$-iterations.
\endproclaim
\bigskip

\remark{\stag{1.27A} Remark}  1) This is like \cite[Ch.XIV,2.13]{Sh:f} and is
reasonable for iterations not adding reals.  For getting rid of pure
$(2,2)$-decidability at the expense of others, natural demands, see
\S5. \nl
2) Is this not suitable for name$^1$ ordinals only?  By UP help. \nl
3) See proof of \scite{si.7} for use of $\dbcu_n q_n \cup p^*$
and more cases phrase as a subclaim?
\endremark
\bigskip

\demo{Proof}  Let $\alpha = \ell g(\bar{\Bbb Q})$ and let $\le^*_\chi$ be a
well ordering of ${\Cal H}(\chi)$, let
${\underset\tilde {}\to \le^*_{\chi,\Bbb P_\beta}}$ 
be a $\Bbb P_\beta$-name of a well ordering of 
${\Cal H}(\chi)^{\bold V^{{\Bbb P}_\beta}}$.
Let $p \in \Bbb P_\alpha$ and $\underset\tilde {}\to \tau$ be
a $\Bbb P_\alpha$-name of an ordinal $< \theta,\theta \in 
\{2,\delta^*\}$ and let
${\underset\tilde {}\to {\bar \zeta}^0} = \langle
{\underset\tilde {}\to \zeta^0_\varepsilon}:\varepsilon < j \rangle$ be a
witness for $p$ (see \scite{1.13}(F), clause (a)) \wilog \, each
${\underset\tilde {}\to \zeta^0_\varepsilon}$ is full.  
For each $\varepsilon < j$ and $\xi < \alpha$ below we shall
${\underset\tilde {}\to r^0_{\varepsilon,\xi}},
{\underset\tilde {}\to \gamma^0_{\varepsilon,\xi}}$ and
${\underset\tilde {}\to {\bold t}_{\varepsilon,\xi}}$ 
such that ${\underset\tilde {}\to r^0_{\varepsilon,\xi}}$ is a 
$\Bbb P_{\xi +1}$-name of a condition with domain $\subseteq (\xi +1,\alpha), 
{\underset\tilde {}\to \gamma^0_{\varepsilon,\xi}}$ is a
$\Bbb P_{\xi +1}$-name of an ordinal $< \theta$ and 
${\underset\tilde {}\to {\bold t}_{\varepsilon,\xi}}$ is a 
$\Bbb P_{\xi +1}$-name of a truth value, satisfying the following.  
Let $\xi < \alpha,G_{{\Bbb P}_{\xi +1}} 
\subseteq \Bbb P_{\xi +1}$ be generic over
$\bold V$ and ${\underset\tilde {}\to \zeta^0_\varepsilon}[G_{\xi+1}]= \xi$:
\mr
\item "{$(*)_1$}"  if there are $r \in \Bbb P_\alpha/G_{{\Bbb
P}_{\xi+1}}$ and 
$\gamma < \theta$ satisfying (a) + (b) below then $\gamma =
{\underset\tilde {}\to \gamma^0_{\varepsilon,\xi}}
[G_{{\Bbb P}_{\xi +1}}],r =
{\underset\tilde {}\to r^0_{\varepsilon,\xi}}[G_{{\Bbb P}_{\xi+1}}]$ are
such objects, first by the fixed well ordering $<^*_\chi$ and 
${\underset\tilde {}\to {\bold t}_{\varepsilon,\zeta}}
[G_{{\Bbb P}_{\xi+1}}]$ is
truth, and does not depend on $\varepsilon$; if there are no such $\gamma,r$
we let ${\underset\tilde {}\to r^0_{\varepsilon,\zeta}}
[G_{{\Bbb P}_{\xi+1}}]$ be
the empty condition, ${\underset\tilde {}\to \gamma^0_{\varepsilon,\zeta}}
[G_{{\Bbb P}_{\xi+1}}] = 0$ and 
${\underset\tilde {}\to {\bold t}_{\varepsilon,\zeta}}
[G_{{\Bbb P}_{\xi+1}}]$ is false.
{\roster
\itemitem{ $(a)$ }  $p \le_{\text{pr}} 
r \in \Bbb P_\alpha/G_{{\Bbb P}_{\xi+1}},p \restriction
(\xi + 1) = r \restriction (\xi +1)$ and we have
$r \Vdash_{{\Bbb P}_\alpha/G_{{\Bbb P}_{\xi+1}}}$ ``if $\theta =2$ then 
$\underset\tilde {}\to \tau = \gamma$ and if $\theta \ge \aleph_0$ then
$\underset\tilde {}\to \tau < \gamma$"
\sn
\itemitem{ $(b)$ }  if $\xi_1 < \xi$ then no $r'$ satisfies (a) with $\xi_1,
G_{{\Bbb P}_{\xi+1}} \cap \Bbb P_{\xi_1 +1}$.
\endroster}
\ermn
So by \scitet{1.16}(4), there is in $\Bbb P_\alpha$ a condition 
${\underset\tilde {}\to r^0_\varepsilon}$ which is
${\underset\tilde {}\to r^0_{\varepsilon,\xi}}$ if
${\underset\tilde {}\to \zeta^0_\varepsilon}[G] = \xi,\bold t_{\varepsilon,
\zeta}[G] =$ truth, is well defined (the $\underset\tilde {}\to \beta$ there
is ${\underset\tilde {}\to \zeta^0_\varepsilon}$ here!, hence it is
full).  
So easily
$p_1 = p \cup \{{\underset\tilde {}\to r^0_\varepsilon}:\varepsilon < j\}$
belongs to $\Bbb P_\alpha$ and is a pure extension of $p$ (using
$<^*_\chi$, noting that for each $\xi + 1 \le \alpha$ and
$G_{P_{\xi+1}} \subseteq \Bbb P_{\xi+1}$ generic over $\bold V$, if
${\underset\tilde {}\to \zeta^0_{\xi+1}}[G_{P_{\xi+1}}] = \xi =
{\underset\tilde {}\to \zeta^0_{\varepsilon_1}}[G_{P_{\xi+1}}]$ then
${\underset\tilde {}\to r^0_{\varepsilon_1}}[G_{{\Bbb P}_{\xi+1}}] =
{\underset\tilde {}\to r^0_{\varepsilon_2}}[G_{{\Bbb P}_{\xi+1}}])$.
\mn
We now define $p_2 = p_1 \cup \{{\underset\tilde {}\to r^1_\varepsilon}:
\varepsilon < j\}$ where ${\underset\tilde {}\to r^1_\varepsilon}$ is an
atomic $\bar{\Bbb Q}$-named condition with
$\zeta_{\underset\tilde {}\to r^1_\varepsilon} =
{\underset\tilde {}\to \zeta_\varepsilon}$ defined as follows
\mr
\item "{$(*)$}"  if $\beta < \alpha,G_{{\Bbb P}_\beta} \subseteq \Bbb P_\beta$ generic
over $\bold V$ and ${\underset\tilde {}\to \zeta_\varepsilon}[G_{{\Bbb P}_\beta}] =
\beta$ then in $\bold V[G_{{\Bbb P}_\beta}]$ we 
have $r \in {\underset\tilde {}\to {\Bbb Q}_\beta}
[G]$ is ${\underset\tilde {}\to \le^*_{\chi,\Bbb P_\beta}}[G_{{\Bbb P}_\beta}]$-minimal
such that
{\roster 
\itemitem{ $(i)$ }  $\hat{\Bbb Q}_\beta[G_{{\Bbb P}_\beta}] \models ``p_2 \restriction
\{\beta\} \le_{\text{pr}}\{r\} \in \hat{\Bbb Q}_\beta[G_{{\Bbb P}_\beta}]"$
\sn
\itemitem{ $(ii)$ }  for some $r_1 \in \Bbb P_{\beta +1}$ and $\gamma < \theta$
we have: $r_1 \restriction \beta \in G_{{\Bbb P}_\beta}$ and $r_1 \restriction \beta
=r$  and: 
$r_1 \Vdash_{{\Bbb P}_{\beta +1}} ``\theta = 2 \and 
{\underset\tilde {}\to \gamma^0_{\varepsilon,\beta}} = \gamma$ or 
$\theta \ge \aleph_0 \and {\underset\tilde {}\to \gamma^0_{\varepsilon,\xi}} 
< \gamma$ and 
${\underset\tilde {}\to {\bold t}_{\varepsilon,\beta}} =$ truth"
or $r_1$ forces ($\Vdash_{{\Bbb P}_{\beta +1}}$) that
${\bold t}_{\varepsilon,\beta} =$ false.
\endroster}
\ermn
Let us choose now $\beta \le \alpha$ and $r_1$ with $\beta$ minimal such that
\mr
\item "{$\otimes$}"   $r_1 \in \Bbb P_\beta$ and there are $q \in \Bbb P_\alpha$ and 
$\gamma < \theta$ such
that $p_2 \le_{\text{pr}} q$ and $q \restriction \beta \le r_1$ and $r_1
\cup (q \restriction [\beta,\alpha)) \Vdash ``\theta = 2,
\underset\tilde {}\to \tau = \gamma$ or $\theta \ne 2,
\underset\tilde {}\to \tau < \gamma"$.
\ermn
There is such $\beta$ as $\beta = \alpha (= \ell g(\bar \theta))$ is O.K.
\enddemo
\bn
\ub{Case 1}:  $\beta =0$.

We are done.
\bn
\ub{Case 2}:  $\beta$ is limit.

Without loss of generality, by \scite{1.15} for some $n < \omega$ and $\xi_1
< \ldots < \xi_n < \beta$ we have: $p_2 \restriction \beta \le r_1$ above
$\{\xi_1,\dotsc,\xi_n\}$.  If $n=0$ we are done (as $\beta =0$) so
assume $n > 0$.  Let $\beta' = \xi_n +1,r' = r_1 \restriction
(\xi_n + 1)$ and there is $q'$ defined by $q' \restriction (\xi_n +1) = q
\restriction (\xi_n +1),q' \restriction (\xi_n +1,\beta)$ is $r_1 \restriction
(\xi_n +1,\beta)$ if $r_1 \restriction (\xi_n +1) \in
{\underset\tilde {}\to G_{{\Bbb P}_{\xi_n+1}}}$ and is
$r_1 \restriction (\xi_n +1,\beta)$ otherwise and lastly
$q' \restriction [\beta,\alpha) = q \restriction [\beta,\alpha)$.  Now
$\beta',r',q'$ satisfies: $r' \in \Bbb P_{\xi_n +1} = \Bbb P_{\beta'},p_2 
\le_{\text{pr}} q',q' \restriction \beta' \le r'$ and $r' \cup q \restriction
[\beta',\alpha) \Vdash_{{\Bbb P}_\alpha} ``\theta = 2,\underset\tilde {}\to \tau =
\gamma$ or $\theta \ne 2,\underset\tilde {}\to \tau < \gamma"$ and
$\beta' < \beta$.
So we get a contradiction to the choice of $\beta$. 
\bn
\ub{Case 3}:  $r_1 \Vdash ``\beta_0 \notin \{
{\underset\tilde {}\to \zeta^0_\varepsilon}:\varepsilon < j\}"$ where
$\beta = \beta_0 +1$.

The proof is similar to the one of case 2 using $\beta' = \beta_0$.
\bn
\ub{Case 4}:  None of the above.  

So by ``neither case 1 nor case 2" we have
$\beta = \beta_0 +1$, and as we can increase $r_1$ \wilog \, $r_1$ forces
$\beta_0 \in \{{\underset\tilde {}\to \zeta^0_\varepsilon}:\varepsilon 
< j\}$, so \wilog \, $r_1 \Vdash ``\beta_0 = 
{\underset\tilde {}\to \zeta^0_\varepsilon}"$ where $\varepsilon < j$.

Let $r_1 \in G_\beta \subseteq \Bbb P_\beta$ with $G_\beta$ generic
over $\bold V$; let
$G_{\beta'} = G_\beta \cap \Bbb P_{\beta'}$ for $\beta' \le \beta$.
So ${\underset\tilde {}\to \zeta^0_\varepsilon}[G_{\beta_0}] = \beta_0$. \nl
We first ask: is there $\varepsilon_1 < j$ such that
${\underset\tilde {}\to \zeta^0_{\varepsilon_1}}[G_{\beta_0}]$ is well 
defined so call it $\zeta$, (so necessarily $\zeta \le \beta_0$) and
${\underset\tilde {}\to {\bold t}_{\varepsilon_1,\zeta}}[G_\beta]$ 
is truth? \nl
If yes, then we get a contradiction to the minimality of $\beta$ as $\zeta$
can serve by the choice of $\beta_2$, so assume not.  Now considering 
${\underset\tilde {}\to r^0_{
\varepsilon,\xi}},{\underset\tilde {}\to \gamma^0_{\varepsilon,\xi}}$,
clause (b) holds and $r_1,q$ exemplifies
${\underset\tilde {}\to {\bold t}_{\varepsilon,\xi}}[G_p] =$ truth. 
\hfill$\square_{\scite{1.27}}$\margincite{1.27}
\bigskip

\remark{\stag{1.27B} Remark}  1)  You may ask why we do not use the
${\underset\tilde {}\to \zeta^*}$ defined by
${\underset\tilde {}\to \zeta^*}
[G_{\xi +1}] = \xi +1$ if ${\underset\tilde {}\to {\bold t}_{\varepsilon,
\zeta}}[G_{\xi +1}] =$ truth for some $\varepsilon < j$?  The reason is that
(as for $e=6,\kappa = \aleph_1$) this seems not to be a simple 
$\bar{\Bbb Q}$-named ordinal.
\nl
2) By the proof, if $\bar{\Bbb Q}$ is a $\kappa$-Sp$_e(W)$-iteration,
$\alpha \le \ell g(\bar{\Bbb Q}),p \in \Bbb P_\alpha$ and for each
$\beta < \alpha,{\underset\tilde {}\to {\bold t}_\beta}$ is a
$\Bbb P_\beta$-name of a truth value, ${\underset\tilde {}\to p_\beta}$ a
$\Bbb P_\beta$-name of some $p \in \Bbb P_\alpha/
{\underset\tilde {}\to G_{{\Bbb P}_\beta}}$ such that $\Vdash_{{\Bbb P}_\beta}
``p \le_{pr} {\underset\tilde {}\to p_\beta},p \restriction  \beta =
{\underset\tilde {}\to p_\beta} \restriction \beta"$ \ub{then} we can find
$q$ such that $p \le_{\text{pr}} q \in \Bbb P_\alpha$ and: if $G_\beta \subseteq
\Bbb P_\beta$ is generic over $V,\beta < \alpha,
{\underset\tilde {}\to {\bold t}_\beta}[G_\beta] =$ truth and $\gamma < \beta
\Rightarrow ``{\underset\tilde {}\to {\bold t}_\gamma}[G_\beta \cap
\Bbb P_\gamma] =$ false then $q \Vdash_{{\Bbb P}_\alpha/G} ``
{\underset\tilde {}\to {\Bbb P}_\beta} 
\in {\underset\tilde {}\to G_{{\Bbb P}_\beta}}"$.
\endremark  
\bn
We now consider some variants of the $\lambda$-c.c.
\definition{\stag{1.28} Definition}  1) We say $\Bbb P$ satisfies the local
$\underset\tilde {}\to \partial$-c.c. \ub{if} $\underset\tilde {}\to \kappa$
is a $\Bbb P$-name and $\{p \in \Bbb P:\Bbb P \restriction \{q:p \le q 
\in \Bbb P\} \text{ satisfies the }
\partial' \text{-c.c. and } p 
\Vdash_{\Bbb P} ``\underset\tilde {}\to \partial = 
\partial'" \text{ for some } \kappa'\}$ is dense in $\Bbb P$. \nl
2) We say $\Bbb P$ satisfies the local $\underset\tilde {}\to 
\partial$-c.c. purely if the set above is dense in $(\Bbb P,
\le_{\text{pr}})$. \nl
3) We say $\Bbb P$ satisfies lc.pr. $\underset\tilde {}\to \partial$-c.c. if:
\mr
\item "{$(a)$}"  $\underset\tilde {}\to \kappa$ is a 
$(\Bbb P,\le_{\text{pr}})$-name, usually of a regular cardinal of
$\bold V$ \nl
(could use just a partial function from $\Bbb P$ to cardinals such that
$\underset\tilde {}\to \kappa(p) = \kappa \wedge p \le_{\text{pr}} q
\Rightarrow \underset\tilde {}\to \kappa(q) = \kappa$, but abusing
notation we write $q \Vdash ``\underset\tilde {}\to \kappa = \kappa"$
if $\underset\tilde {}\to \kappa(q) = \kappa$)
\sn
\item "{$(b)$}"  for every $p \in \Bbb P$ for some $q,\partial$ we have
$p \le_{\text{pr}} q,q \Vdash_{({\Bbb P},\le_{\text{pr}})}
``\underset\tilde {}\to \partial = \partial"$ 
and $\Bbb P_{\ge q}$ satisfies the $\partial$-c.c. \nl
(we could use: if $\underset\tilde {}\to \kappa(p) = \kappa$ then
$\Bbb P_{\ge p}$ i.e. $(\{q:p \le q \in \Bbb P\},
\le^{\Bbb P})$ satisfies the $\kappa$-c.c.)
\ermn
4) If $\Bbb P$ satisfies the lc.pr. $\underset\tilde {}\to \partial$-c.c.
and $q \in \Bbb P$ let $\kappa^{\text{mcc}}_{\underset\tilde {}\to
\partial}(q,\Bbb P) = \partial$ means $q 
\Vdash_{({\Bbb P},\le_{\text{pr}})}$ ``$\underset\tilde {}\to \kappa =
\kappa$" and $\Bbb P_{\ge q}$ satisfies the $\kappa$-c.c. \nl
5) Let $\underset\tilde {}\to \partial^{\text{mcc}}
(\Bbb P)$ be minimal such that $\Bbb P$ satisfies
the lc.pr.  $\underset\tilde {}\to \kappa$-c.c.; that is
${\underset\tilde {}\to \partial'}(q,\Bbb P) 
= \text{ Min}\{\kappa:\Bbb P_{\ge q}$
satisfies that $\kappa$-c.c.$\}$ and 
$\underset\tilde {}\to \partial(q,\Bbb P) = \kappa$ 
if $(\forall r')(q \le_{\text{pr}} 
r \rightarrow {\underset\tilde {}\to \partial'}
(r,\Bbb P) = \kappa$) (see below) and let
$\partial^{\text{mcc}}(q,\Bbb P) = \partial$ 
mean $\kappa^{\text{mcc}}_{\underset\tilde {}\to \partial({\Bbb P})}
(q,\Bbb P) = \partial$ where $\underset\tilde {}\to \kappa = 
\underset\tilde {}\to \kappa_{\text{mcc}}(\Bbb P)$.
\enddefinition
\bigskip

\proclaim{\stag{1.29} Claim}  1) For a forcing notion 
$\Bbb P$ (as in \scite{1.1}) 
the $(\Bbb P,\le_{\text{pr}})$-name 
$\underset\tilde {}\to \partial^{\text{mcc}}(\Bbb P)$ is well
defined, so \nl
2) If $\Bbb P$ satisfies the lc.pr. 
$\underset\tilde {}\to \partial$-c.c. and $p \in \Bbb P$ 
\ub{then} for some $q$ we have $p \le_{\text{pr}} q$ and
$\partial_{\text{mcc}}(q,\Bbb P)$ is well defined.
\endproclaim
\bigskip

\demo{Proof}  Straight.
\enddemo
\bigskip

\definition{\stag{1.30} Definition}  1) We say $\Bbb Q$ has strong pr.
$({\underset\tilde {}\to \partial_1},
{\underset\tilde {}\to \partial_2})$-decidability when 
$\underset\tilde {}\to \kappa,{\underset\tilde {}\to \kappa_2}$ are
$(\Bbb Q,\le_{\text{pr}})$-names of regular cardinals of $\bold V$ 
and \ub{if} $p \in \Bbb Q,
p \Vdash_{({\Bbb Q},\le_{\text{pr}})} 
``{\underset\tilde {}\to \partial_1} = \theta_1$
and ${\underset\tilde {}\to \kappa_2} = \theta_2"$ and
${\underset\tilde {}\to \zeta_\varepsilon}$ is a $\Bbb Q$-name of an ordinal
$< \theta_1$ for $\varepsilon < \varepsilon^* < \theta_2$ \ub{then} for some
$a \subseteq \theta_1$ of cardinality $< \theta_2$ and $q$ such that
$p \le_{\text{pr}} q \in \Bbb Q$ we have $q \Vdash_{\Bbb Q} 
``{\underset\tilde {}\to \zeta_\varepsilon} \in a$ for $\varepsilon <
\varepsilon^*"$. \nl
2) We say $\Bbb Q$ has strong pr. 
$\underset\tilde {}\to \partial$-decidability
\ub{if} for any $\theta$ it has 
pr. $(\theta,\underset\tilde {}\to \kappa)$-decidability 
(i.e. each ${\underset\tilde {}\to \zeta_\varepsilon}$ is a $\Bbb Q$-name 
of an ordinal $< \theta$). \nl
3) We use ``weak" instead of ``strong" in parts (1), (2) if above we restrict
ourselves to the case $\varepsilon^*=1$. \nl
4) We let ${\underset\tilde {}\to \partial^w_\otimes}(\Bbb P)$ is the minimal
$(\Bbb P,\le_{\text{pr}})$-name $\underset\tilde {}\to \kappa$ of a regular
cardinal from $\bold V$ such that $\Bbb P$ has weak pr.
$\underset\tilde {}\to \kappa$-decidability.  Similarly
${\underset\tilde {}\to \partial^{\text{St}}_\otimes}(\Bbb P)$ for strong
pr. $\underset\tilde {}\to \kappa$-decidability.
\enddefinition
\bigskip

\demo{\stag{1.31} Note/Observation}  If $\bar{\Bbb Q}$ is 
an $\kappa-\text{Sp}_e(W)$-iteration, \ub{then} 
\mr
\item "{$(a)$}"  $\langle (\Bbb P_i,\le^{{\Bbb P}_i}_{\text{pr}}):
i \le \ell g(\bar{\Bbb Q})
\rangle$ is a $\lessdot$-increasing sequence
\sn
\item "{$(b)$}"  if $i < j \le \ell g(\bar{\Bbb Q}),q \in \Bbb P_j,
q \restriction i \le_{\text{pr}} p \in \Bbb P_i$ then $p,q$ has a 
$\le_{\text{pr}}$-lub, $p \cup q \restriction [i,j)$.
\endroster
\enddemo
\bigskip

\demo{Proof}  Check.
\enddemo
\bigskip

\proclaim{\stag{1.32} Claim}  1) If $\Bbb P$ satisfies the lc.pr.
$\underset\tilde {}\to \partial$-c.c. and $\partial =
\partial^{\text{mcc}}_{\underset\tilde {}\to \partial}(p,\Bbb P) \Rightarrow 
\Vdash_{\Bbb P} ``\partial$ is regular"
\ub{then} 
$\Bbb P$ has a strong pr $\underset\tilde {}\to \partial$-decidability. \nl
2) Let $\bar{\Bbb Q}$ be a $\kappa_1-{\text{\rm Sp\/}}_e(W)$-iteration
$e \in \{4\}$.  If $\delta \le \ell g(\bar{\Bbb Q})$ is a limit ordinal, $u$ an unbounded subset of
$\delta$, for $i \in u$ we have $\Bbb P_i$ has the strong pr.
${\underset\tilde {}\to \partial_i}$-decidability \ub{then} letting
$\underset\tilde {}\to \partial = \text{ Min}\{\partial:
\partial \text{ a regular cardinal in } \bold V$ and $\partial 
\ge {\underset\tilde {}\to \partial_i}$ for $i \in u\}$ we have
\mr
\item "{$(i)$}"  $\underset\tilde {}\to \partial$ is a $(\Bbb P_\delta,
\le_{\text{pr}})$-name of a regular cardinal of $\bold V$
\sn
\item "{$(ii)$}"  $\Bbb P_\delta$ has weak pr. 
$\underset\tilde {}\to \partial$-decidability.
\ermn
3) Similarly for 
$\Bbb P_\delta/G_{{\Bbb P}_\alpha}$ when $\alpha < \delta$ and even
$\Bbb P_\delta/\Bbb P_\alpha$ where $\underset\tilde {}\to \alpha$ is 
a simple $\bar{\Bbb Q}$-named $[0,\delta)$-ordinal. \nl
4) If $\Bbb P$ 
satisfies the strong pr $\underset\tilde {}\to \partial$-decidability
and $p \in \Bbb P,\kappa = \kappa_{\underset\tilde {}\to
\kappa}(p,\Bbb P)$ \ub{then} $p \Vdash ``\partial$ is a regular cardinal.
\endproclaim
\bigskip

\demo{Proof}  1) Trivial. \nl
2),3) Similar to the proof of \scite{1.27} [Saharon!] \nl
4) Trivial.  \hfill$\square_{\scite{1.32}}$\margincite{1.32}
\enddemo
\bn
\margintag{n.0}\ub{\stag{n.0} Convention}:  Let $\bar{\Bbb Q} = \langle \Bbb P_j,
{\underset\tilde {}\to {\Bbb Q}_i}:j \le \alpha,i < \alpha \rangle$ be a
$\kappa$-Sp$_e(W)$-iteration.
\bigskip

\proclaim{\stag{n.1} Claim}  Assume $(\bar{\Bbb Q},W)$ is smooth, (see
Definition \scite{1.6}(8)). \nl
1) If $p^* \in \Bbb P_j$ \ub{then} $\langle \Bbb P'_j,
{\underset\tilde {}\to {\Bbb Q}'_i}:j \le
\alpha,i < \alpha \rangle$ is a $\kappa_1$-Sp$_e(W)$-iteration where \nl
$\Bbb P'_j = \{p \in \Bbb P_j:p^* \restriction j \le p\}$, \nl
$\Bbb Q'_j = \{p \in {\underset\tilde {}\to {\Bbb Q}_j}:
\Vdash_{{\Bbb P}_j} ``p^* \restriction
\{j\} \le p \text{ in } \hat{\Bbb Q}_j"$.
\mn
2) If $\gamma < \alpha$ and 
$G_j \subseteq \Bbb P_\gamma$ is generic over $\bold V$ \ub{then}
$\langle \Bbb P_{\gamma +j}/G_\gamma,
{\underset\tilde {}\to {\Bbb Q}_{\gamma +i}}:j \le
\alpha - \gamma$ and $i < \alpha - \gamma \rangle$ is a
$\aleph_1-{\text{\rm Sp\/}}_e(W)$-iteration.
\endproclaim
\bigskip

\demo{Proof}  Straight.
\enddemo
\newpage

\head {\S2 Tree of Models} \endhead  \resetall \sectno=2
 % \resetall 
\bn
We present here needed information on trees and tagged trees.
On partition of tagged trees, see Rubin, Shelah
\cite{RuSh:117} and, \cite{Sh:b}, 
\cite[XI,3.5,3.5A,3.7,XV,2.6,2.6A,2.6B]{Sh:f} and
\cite[2.4,2.5,p.111-113]{Sh:136}; or the representation in
\cite[AP,\S1]{Sh:e}; on history see \cite{RuSh:117} and \cite{Sh:f}. 
\bigskip

\definition{\stag{2.1} Definition}  A tagged tree is a pair $(T,\bold I)$ 
such that:
\mr
\item  $T$ is a $\omega$-tree, which here means a nonempty set of finite
sequences of ordinals such that if $\eta \in T$ then any initial segment of
$\eta$ belongs to $T$.  $T$ is ordered by initial segments; i.e., $\eta 
\triangleleft \nu$ iff $\eta$ is a proper initial segment of $\nu$.
\sn
\item  $\bold I$ is a partial function such that for every $\eta \in
T \cap \text{ Dom}(\bold I)$: if $\bold I(\eta) = \bold I_\eta$ is 
defined then $\bold I(\eta)$ is an
ideal of subsets of some set called the domain of $\bold I_\eta$,
Dom$(\bold I_\eta)$, and \footnote{in this section it is not
unreasonable to demand equality but this is very problematic in
Definition \scite{4.1}(1), clause (d)}

$$
\text{Suc}_T(\eta) =: \{\nu:\nu \text{ is an immediate successor of } \eta
\text{ in } T\} \subseteq \text{ Dom}(\bold I_\eta),
$$
\mn
Usually $\bold I_\eta$ is $\aleph_2$-complete.
\sn
\item  For every $\eta \in T$ we have Suc$_T(\eta) \ne \emptyset$. 
\endroster
\enddefinition
\bigskip

\demo{\stag{2.2} Convention} For any tagged tree $(T,\bold I)$ we can define
$\bold I^\dag$ by:

$$
\text{Dom}(\bold I^\dag) = \{\eta:\eta \in T \cap \text{ Dom}(\bold I)
\text{ and Suc}_T(\eta) \subseteq \bold I_\eta
\text{ and  Suc}_T(\eta) \notin \bold I_\eta\}
$$
\mn
and

$$
\bold I^\dag_\eta = \bigl\{ \{\alpha:\eta \char 94 \langle \alpha \rangle \in
A\}:A \in \bold I_\eta \bigr\};
$$
\mn
we sometimes, in an abuse of notation, do not distinguish between $\bold I$
and $\bold I^\dag$; e.g. if $\bold I^\dag_\eta$ is constantly $I^*$, we write
$I^*$ instead of $\bold I$.  Also, if $\bold I = \bold I_x$, we may write
$\bold I^x_\eta$ for $\bold I_x(\eta)$.
\enddemo
\bigskip

\definition{\stag{2.3} Definition}  1) Let $\eta$ be called a splitting point
of $(T,\bold I)$ if $\bold I_\eta$ is well defined and Suc$_T(\eta) \notin
\bold I_\eta$ (normally this follows but we may ``forget" 
to decrease the domain of $\bold I$).  Let split$(T,\bold I)$ be 
the set of splitting points of
$(T,\bold I)$.  We usually consider trees where each $\omega$-branch meets 
split$(T,\bold I)$ infinitely often (see Definition \scite{2.5}(6),
i.e. are normal). \nl
2) For $\eta \in T$ let $T^{[\eta]} =: \{\nu \in T:\nu = \eta \text{ or }
\nu \triangleleft \eta \text{ or } \eta \triangleleft \nu\}$. \nl
3) For a tree $T$, let lim$(T)$ be the set of branches of $T$; i.e. all
$\omega$-sequences of ordinals, such that every finite initial segment 
of them is a member of $T:\text{lim}(T) =$ \nl
$\{s \in {}^\omega\text{Ord}:(\forall n)s \restriction n \in T\}$. 
We call them also $\omega$-branches. \newline
4) A subset $Z$ of a tree $T$ is a front if: $\eta \ne \nu \in Z$ implies
none of them is an initial segment of the other, and every $\eta \in
\text{ lim}(T)$ has an initial segment which is a member of $Z$. 
\enddefinition
\bigskip

\definition{\stag{2.4} Definition}  We now define orders between tagged trees:
\mr
\item "{$(a)$}"  $(T_1,\bold I_1) \le (T_2,\bold I_2)$ \ub{if} $T_2 \subseteq
T_1$, and split$(T_2,\bold I_2) \subseteq \text{ split}(T_1,\bold I_1)$, and
\nl
for every $\eta \in \text{ split}(T_2,\bold I_2)$ we have 
$\bold I_2(\eta) \restriction \text{ Suc}_{T_2}(\eta) = \bold I_1(\eta) 
\restriction \text{ Suc}_{T_2}(\eta)$ \nl
(where $I \restriction A = \{B:B \subseteq A \text{ and } B \in I\})$. \nl
(So every splitting point of $T_2$ is a splitting point of $T_1$, and
$\bold I_2$ is completely determined by $\bold I_1$ and 
split$(T_2,\bold I_2)$ and $T_2$.)
\sn
\item "{$(b)$}"  $(T_1,\bold I_1) \le^* (T_2,\bold I_2)$ \ub{if} 
$(T_1,\bold I_1)
\le (T_2,\bold I_2)$ and \nl
split$(T_2,\bold I_2) = \text{ split}(T_1,\bold I_1) \cap T_2$.
\sn
\item "{$(c)$}"  For any set 
$A,(T_1,\bold I_1) \le^\otimes_A (T_2,\bold I_2)$
if $T_1 \supseteq T_2$ and $\eta \in A \cap T_2 \Rightarrow \text{ Suc}_{T_2}
(\eta) = \text{ Suc}_{T_1}(\eta)$ and $\eta \in T_2 \cap
\text{ split}(T_1,\bold I_1)
\backslash A \Rightarrow \text{ Suc}_{T_2}(\eta) \ne \emptyset \text{ mod }
\bold I^1_\eta$.
\sn
\item "{$(d)$}"  In (c) we may omit the subscript $A$ when $A = T_2
\backslash \text{ split}(T_1,\bold I_1)$.
\sn
\item "{$(e)$}"  $(T_1,\bold I_1) \le^\boxtimes_\kappa (T_2,\bold I_2)$
\ub{if} $T_2 \subseteq T_1$ is a subtree and if $\nu \triangleleft
\eta \in \text{ lim}(T_2),\bold I_\nu$ is $\kappa$-complete, then for some
$k \ge \ell g(\nu),\bold I^1_\nu \le_{RK} \bold I^2_{\eta \restriction k},
\bold I^2_{\eta \restriction k}$ is 
$\kappa$-complete and $\eta \restriction k
\in \text{ split}(T_2,\bold I_2)$.  We can replace $\kappa$ and 
$\kappa$-complete by $\varphi$ and ``satisfying $\varphi$", e.g. $\in
\Bbb I$.
\sn
\item "{$(f)$}"  $(T_1,\bold I_1) \le^\kappa (T_2,\bold I_2)$
\ub{when} $T_2 \subseteq T_1$, and if $\eta \in T_2 \cap$split$(T_1,\bold
I_1)$ and $\bold I^1_\eta$ is $\kappa$-complete then 
$\eta \in \text{ split}(T_2,\bold I_2)$ and $\bold I^2_\eta = \bold
I^1_\eta$ and every $\eta \in \text{ split}(T_2,\bold I_2)$ is like
that.
\endroster
\enddefinition
\bigskip

\definition{\stag{2.5} Definition}  1) For a set $\Bbb I$ of ideals, a 
tagged tree $(T,\bold I)$ is an $\Bbb I$-tree if for every splitting point 
$\eta \in T$ we have $\bold I_\eta \in \Bbb I$ (up to an isomorphism). \nl
2) For a tagged tree $(T,\bold I)$ and set $\Bbb I$ of ideals let
$\bold I \restriction \Bbb I = \bold I \restriction \{\eta \in \text{ Dom}
(\bold I)$ and $\bold I_\eta \in \Bbb I\}$. \nl
3) Let in (2), $(T,\bold I) \restriction \Bbb I = (T,\bold I \restriction
\Bbb I)$. \nl
4) A tagged tree 
$(T,\bold I)$ is standard \ub{if} for every non-splitting point $\eta \in T,
|\text{Suc}_T(\eta)| = 1$. \newline
5) A tagged tree $(T,\bold I)$ is full \ub{if} 
every $\eta \in T$ is a splitting point. \nl
6) A tagged tree $(T,\bold I)$ is normal \ub{if} 
for every $\eta \in \text{ lim}(T)$ for
infinitely many $k < \omega$ we have $\eta \restriction k \in \text{
split}(T,\bold I)$.
\enddefinition
\bigskip

\remark{\stag{2.6} Remark}  1) Of course, the set 
lim$(T)$ is not absolute; i.e., if
$\bold V_1 \subseteq \bold V_2$ are two universes of set 
theory then in general
(lim$(T))^{{\bold V}_1}$ will be a proper 
subset of (lim$(T))^{{\bold V}_2}$. \newline
2) However, the notion of being a front is absolute: if $\bold V_1 
\models ``Z$
contains a front in $T$", \ub{then} there is a depth function $\rho:T 
\rightarrow \text{ Ord}$ 
satisfying $\eta \triangleleft \nu \and \forall k \le \ell g(\eta)
[\eta \restriction k \notin Z] \rightarrow \rho(\eta) > \rho(\nu)$.  This
function will also witness in $\bold V_2$ that $Z$ is a front. \newline
3) $Z \subseteq T$ contains a front \ub{iff} $Z$ 
meets every branch of $T$.  So if
$Z \subseteq T$ contains a front of $T$ and $T' \subseteq T$ is a subtree, 
\ub{then} $Z \cap T'$ contains a front of $T'$.  This is absolute, too.
\endremark
\bigskip

\definition{\stag{2.7} Definition}  1) An ideal $I$ is
$\lambda$-complete  \ub{if}
any union of less than $\lambda$ members of $I$ is still a member of $I$.
\newline
2) A tagged tree $(T,\bold I)$ is $\lambda$-complete if for each $\eta \in
T \cap \text{ Dom}(\bold I)$ or just $\eta \in \text{ split}(T,\bold I),$
the ideal $\bold I_\eta$ is $\lambda$-complete.
\newline
3) A family $\Bbb I$ of ideals is $\lambda$-complete \ub{if} 
each $I \in \Bbb I$ is
$\lambda$-complete.  We will only consider $\aleph_2$-complete families 
$\Bbb I$. \newline
4) A family $\Bbb I$ is called restriction-closed \ub{if}: 
$I \in \Bbb I,A \subseteq \text{ Dom}(I),A \notin I$ 
implies $I \restriction A = \{B \in I:B \subseteq 
A\}$ belongs to $\Bbb I$. \newline
5) The restriction closure of $\Bbb I$, res-c$\ell(\Bbb I)$ is 
$\{I \restriction A:I \in \Bbb I,
A \subseteq \text{ Dom}(I),A \notin I\}$. \nl
6) $I$ is $\lambda$-indecomposable \ub{if}
for every $A \subseteq \text{ Dom}(I),
A \notin I$ and $h:A \rightarrow \lambda$ there is $Y \subseteq \lambda,|Y|
< \lambda$ such that $h^{-1}(Y) \notin I$.  We say $\bold I$ 
(or we say $\Bbb I$) is
$\lambda$-indecomposable \ub{if} each 
$\bold I_\eta$ where $\eta \in \text{ split}(T,
\bold I)$ (or $I \in \Bbb I$) is $\lambda$-indecomposable. \newline
7) $I$ is strongly $\lambda$-indecomposable \ub{if} for any
$A_i \in I \,\,(i < \lambda)$ and $A \subseteq \text{ Dom}(I),
A \notin I$ we can find $B \subseteq
A$ of cardinality $< \lambda$ such that for no $i < \lambda$ does $A_i$ 
include $B$. \newline
8) Let $\Bbb I^{[\kappa]} = 
\{I \in \Bbb I:I \text{ is } \kappa \text{-complete}\}$.
\enddefinition
\bigskip

\demo{\stag{2.9} Fact}  If $\lambda$ is a regular cardinal and $I$ 
is a strongly $\lambda$-indecomposable,
\ub{then} $I$ is $\lambda$-indecomposable.
\enddemo
\bigskip

\demo{Proof}  Given 
$A,h$ as in \scite{2.7}(6), let $A_i = h^{-1}(\{j:j < i\})$ and $A'_i
= h^{-1}(\{i\})$; if
for some $i < \lambda,A_i \notin I$ we are done, 
otherwise by Definition \scite{2.7}(7)
there is $B \subseteq A,|B| < \lambda$ such that: $i < \lambda \Rightarrow
B \nsubseteq A_i$.  But as $\lambda$ is regular, $B \subseteq A,|B| <
\lambda$ and $\langle A_i:i < \lambda \rangle$ is a
$\subseteq$-increasing sequence of sets with union $A$, clearly
for some $j < \lambda,B \subseteq A_j$, contradiction.
\hfill$\square_{\scite{2.9}}$\margincite{2.9}
\enddemo
\bigskip

\definition{\stag{2.10} Definition}  For a subset of $A$ of 
(an $\omega$-tree) $T$ we define by induction on the length 
of a sequence $\eta$, res$_T(\eta,A)$
for each $\eta \in T$.  Let res$_T(\langle \rangle,A) = \langle \rangle$.
Assume res$_T(\eta,A)$ is already defined and we define res$_T(\eta \char 94
\langle \alpha \rangle,A)$ for all members $\eta \char 94 \langle \alpha
\rangle$ of Suc$_T(\eta)$.  If $\eta \in A$ then res$_T(\eta \char 94
\langle \alpha \rangle,A) = \text{ res}_T(\eta,A) \char 94 \langle 
\alpha \rangle$, and if $\eta \notin A$ then res$_T(\eta \char 94 \langle 
\alpha \rangle,A) = \text{ res}_T(\eta,A) \char 94 \langle 0 \rangle$.
If $\eta \in \text{ lim}(T)$, we let \newline
res$(\eta,A) = \dsize \bigcup_{k \in \omega} \text{ res}
(\eta \restriction k,A)$.
\enddefinition
\bigskip

\demo{\stag{2.11} Explanation}  Thus res$(T,A) =: \{\text{ res}_T(\eta,A):\eta
\in T\}$ is a tree obtained by projecting $T$; i.e., glueing 
together all members of Suc$_T(\nu)$ whenever $\nu \notin A$.
\enddemo
\bigskip

We state now (Lemma \scite{2.12} is \cite[Ch.XI,5.3;p.559]{Sh:f} and Lemma
\scite{2.13} is \nl
\cite[XV,2.6;p.738]{Sh:f} and Lemma \scite{2.14} is
\cite[XI,3.5(2); p.546-7]{Sh:f}.
\proclaim{\stag{2.12} Lemma}  Let $\lambda,\mu$ be uncountable cardinals
satisfying $\lambda^{< \mu} = \lambda$ and let $(T,\bold I)$ be a tagged
tree in which for each $\eta \in T$ either 
$|{\text{\rm Suc\/}}_T(\eta)| < \mu$ or
$\bold I(\eta)$ is $\lambda^+$-complete.  \ub{Then} for every function $H:T
\rightarrow \lambda$ there exists a subtree $T'$ of $T$ 
satisfying $(T,\bold I) \le^* (T',
\bold I)$ such that for $\eta^1,\eta^2 \in T'$ satisfying
${\text{\rm res\/}}_T(\eta^1,A) = { \text{\rm res\/}}_T(\eta^2,A)$ we
have:
\mr
\widestnumber\item {$(iii)$}
\item "{$(i)$}"  $H(\eta^1) = H(\eta^2)$
\sn
\item "{$(ii)$}"  $\eta^1 \in A \Leftrightarrow \eta^2 \in A$,
\sn
\item "{$(iii)$}"  $\text{if } \eta \in T' \cap A$, then 
${\text{\rm Suc\/}}_T(\eta) = { \text{\rm Suc\/}}_{T'}(\eta)$.
\endroster
\endproclaim
\bigskip

\demo{Proof}  See \cite[XV,2.6;p.738]{Sh:f}.
\enddemo
\bigskip

\proclaim{\stag{2.13} Lemma}  Let $\theta$ be an uncountable regular cardinal
(the main case here is $\theta = \aleph_1$).  Assume
\mr
\item "{$(\alpha)$}"   $\Bbb I$ be a family of $\theta^+$-complete
ideals,
\sn
\item "{$(\beta)$}"   $(T_0,\bold I)$ a tagged tree,
\sn
\item "{$(\gamma)$}"  $A =: \{\eta \in T:|\text{Suc}_{T_0}(\eta)| 
\le \theta\}$,
\sn
\item "{$(\delta)$}"  $[\eta \in T_0 \backslash A \Rightarrow
\bold I_\eta \in \Bbb I \and { \text{\rm Suc\/}}_{T_0}(\eta) 
\notin \bold I_\eta]$ 
\sn
\item "{$(\varepsilon$}"  $[\eta \in A \Rightarrow { \text{\rm
Suc\/}}_{T_0}(\eta) \subseteq \{ \eta \char 94
\langle i \rangle:i < \theta\}]$ 
\sn
\item "{$(\zeta)$}"   $H:T_0 \rightarrow \theta$ and
\sn
\item "{$(\eta)$}"  $\bar e = 
\langle e_\eta:\eta \in A,|{\text{\rm Suc\/}}_{T_0}(\eta)| = \theta \rangle$ 
is such that $e_\eta$ is a club of $\theta$.  
\ermn
\ub{Then} there is a club $C$ of $\theta$ 
such that: for each $\delta \in C$ there is $T_\delta \subseteq T_0$ 
satisfying:
\mr
\item "{$(a)$}"  $T_\delta$ is a tree
\sn
\item "{$(b)$}"  if $\eta \in T_\delta,|{\text{\rm
Suc\/}}_{T_0}(\eta)| < \theta$, then ${\text{\rm
Suc\/}}_{T_\delta}(\eta) 
= { \text{\rm Suc\/}}_{T_0}(\eta)$ and if ${\text{\rm Suc\/}}(\eta) = 
\theta$, then ${\text{\rm Suc\/}}_{T_\delta}(\eta) = \{ \eta \char 94 
\langle i \rangle:i < \delta\}$ and $\delta \in e_\eta$
\sn
\item "{$(c)$}"  $\eta \in T_\delta \backslash A$ implies
${\text{\rm Suc\/}}_{T_\delta}(\eta) \notin \bold I_\eta$
\sn
\item "{$(d)$}"  for every $\eta \in T_\delta$ we have $H(\eta) < \delta$.
\endroster
\endproclaim
\bigskip

\demo{Proof}  See \cite[XV,2.6]{Sh:f}.
\enddemo
\bigskip

\proclaim{\stag{2.14} Lemma}   Let $(T,\bold I)$ be a $\theta^+$-complete
$\bold I$-tree, and assume $\theta = { \text{\rm cf\/}}(\theta)$.  
If ${\text{\rm lim\/}}(T) = \dsize \bigcup_{i < \theta} B_i$
where $B_i$ is a Borel subset of ${\text{\rm lim\/}}(T)$, 
\underbar{then} for some $i <
\theta$ and $(T',\bold I')$ we have $(T,\bold I) \le^* (T',\bold I')$ and
${\text{\rm lim\/}}(T') \subseteq B_i$.
\endproclaim
\bigskip

\demo{Proof}  By \cite[XI,3.5(2);p.546-7]{Sh:f}.
\enddemo
\newpage

\head {\S3 Ideals and Partial Orders} \endhead  \resetall \sectno=3
 % \resetall 
\bigskip

\definition{\stag{3.1} Definition}  1) We call an ideal $J$ non-atomic if
$\{x\} \in J$ for every \newline
$x \in \text{ Dom}(J)$. \newline
2) We call the ideal with domain $\{0\}$, which is $\{\emptyset\}$, the
trivial ideal.
\enddefinition
\bigskip

\definition{\stag{3.2} Definition}  1) For ideals $J_1,J_2$ we say

$$
J_1 \le_{\text{RK}} J_2
$$
\mn
$\text{ Dom}(J_1)$ is such that

$$
\text{for every } A \subseteq \text{ Dom}(J_2) \text{ we have}:
A \ne \emptyset \text{ mod }
J_2 \Rightarrow h''(A) \ne \emptyset \text{ mod } J_1
$$
\mn
or equivalently,

$$
J_2 \supseteq \{h^{-1}(A):A \in J_1\}.
$$
\mn
2) For families $\Bbb I_1,\Bbb I_2$ of ideals we say $\Bbb I_1 
\le_{\text{RK}} \Bbb I_2$ if there is a function $H$ witnessing it; i.e.,
\mr
\item "{$(i)$}"  $H$ is a function from $\Bbb I_1$ into $\Bbb I_2$
\sn
\item "{$(ii)$}"  for every 
$J \in \Bbb I_1$ we have $J \le_{\text{RK}} H(J)$.
\ermn
3) For families $\Bbb I_1,\Bbb I_2$ of ideals, $\Bbb I_1 
\equiv_{\text{RK}} \Bbb I_2$ \ub{if} $\Bbb I_1 \le_{\text{RK}} \Bbb I_2 \and 
\Bbb I_2 \le_{\text{RK}} \Bbb I_1$.
\enddefinition
\bigskip

\proclaim{\stag{3.3} Claim}  1) If an ideal $J$ is not non-atomic \ub{then} 
$J \le_{\text{RK}}$ ``the trivial ideal". \newline
2) $\le_{\text{RK}}$ is a partial quasi-order (among ideals and also among
families of ideals).
\endproclaim
\bigskip

\demo{Proof}  Easy.
\enddemo
\bigskip

\definition{\stag{3.4} Definition}  1) For an (upward) directed quasi 
order \footnote{no real difference if we ask partial order or just quasi
orders; i.e., partial orders satisfy 
$x \le y \and y \le x \nRightarrow x = y$, quasi order not
necessarily; note that in the case
there is $x^* \in L$ such that $(\forall y \in L)(y \le x^*)$ gives an ideal
which is not non-atomic} $L = (B,<)$ we define an ideal id$_L$:

$$
\text{id}_L = \bigl\{ A \subseteq B:\text{for some } y \in L \text{ we have }
A \subseteq \{x \in L:\neg y \le x\} \bigr\}.
$$
\mn
Equivalently, the dual filter fil$_L$ is generated by the ``cones",
where the cone of $L$ defined by $y \in L$ is 

$$
L_y =: \{ x \in L:y \le x\}.
$$
\mn
We call such an ideal a partial order ideal.  We let Dom$(L) = 
\text{ Dom}(\text{id}_L)(= B)$, but we may use $L$ instead of Dom$(L)$
(like $\forall x \in L$). \newline
2) For a partial order $L$ let dens$(L) = \min\{|X|:X \subseteq \text{ Dom}
(L)$ is dense \footnote{also called cofinal in this context}; 
i.e. $(\forall a \in \text{ Dom}(L))(\exists b \in X)
[a \le b])\}$ (this applies also to ideals considered as $(I,\subseteq)$).
\newline
3) For a family ${\Cal L}$ of directed quasi orders let 
id$_{\Cal L} = \{\text{id}_L:L \in {\Cal L}\}$.
\enddefinition
\bigskip

\demo{\stag{3.5} Fact}  1)  id$_L$ is $\lambda$-complete \ub{iff} $L$ is 
$\lambda$-directed. \newline
2) dens$(L) = \text{ dens(id}_{(L,<)},\subseteq)$. \newline
3) If $h:L_1 \rightarrow L_2$ preserves order (i.e. $\forall x,y \in L,(x \le
y \Rightarrow h(x) \le h(y))$) and has cofinal (= dense) range (i.e. $(\forall x \in
L_2)(\exists y \in L_1)[x \le h(y)]$) \ub{then} id$_{L_2} \le_{\text{RK}}
\text{ id}_{L_1}$. \newline
4) Let $h:L_1 \rightarrow L_2$.  Now $h$ exemplifies 
id$_{L_2} \le_{\text{RK}} \text{ id}
_{L_1}$ \ub{iff} for every $x_2 \in L_2$ there is $x_1 \in L_1$ such that:
$(\forall y)[y \in L_1 \and x_1 \le_{L_1} y \Rightarrow x_2 \le_{L_2} h(y)]$;
(equivalently for every 
$y \in L_1:\neg(x_2 \le_{L_2} h(y)) \Rightarrow \neg(x_1 \le_{L_1}
y)$; note that $h$ is not necessarily order preserving). \newline
5) If $L_1 \subseteq L_2$ and
$L_1$ is a dense in $L_2$, \ub{then} id$_{L_1} \equiv_{RK} 
\text{ id}_{L_2}$. \nl
6) If $\lambda = \text{ density}(L)$ \ub{then} id$_L$ is $\lambda$-based; i.e.
$X \subseteq \text{ Dom}(\text{id}_L),X \notin \text{ id}_L \Rightarrow
(\exists Y \subseteq X)[|Y| \le \lambda \and Y \notin \text{ id}_L]$.
\enddemo
\bigskip

\demo{Proof}  Straight.  E.g. \newline
3) If $A \subseteq L_1$ and $A \notin \text{ id}_{L_1}$, then $(\forall x \in
L_1)(\exists y)(x \le_{L_1} y \in A)$ hence $(\forall x \in L_2)(\exists y,
z \in L_1)(x \le_{L_2} h(y) \and y \le_{L_1} z \in A)$ hence
$(\forall x \in L_2)(\exists z)(x \le_{L_2} z \in h''(A))$ hence
$h''(A) \notin \text{ id}_{L_2}$ (and trivially $h''(A) \subseteq L_1)$. \nl
By Defintion \scite{3.2}(1) this 
shows id$_{L_2} \le_{RK} \text{ id}_{L_1}$. \nl
4) Note: $h$ exemplifies id$_{L_2} \le \text{ id}_{L_1}$ 
\sn
iff

$$
(\forall A \subseteq L_1)(A \ne \emptyset \text{ mod id}_{L_1} 
\rightarrow h''(A) \ne \emptyset \text{ mod id}_{L_2})
$$ 
\mn
\underbar{iff}

$$
(\forall A \subseteq L_1)(\forall x_2 \in L_2)[A \ne \emptyset 
\text{ mod id}_{L_1} \rightarrow h''(A) \cap \{y \in L_2:x_2 \le_{L_2} y\}
\ne \emptyset]
$$
\medskip

\noindent
iff

$$
(\forall x_2 \in L_2)(\forall A \subseteq L_1)(A \ne \emptyset 
\text{ mod id}_{L_1} \rightarrow h''(A) \cap \{y \in L_2:x_2 \le_{L_2} y\}
\ne \emptyset)
$$
\medskip

\noindent
iff

$$
(\forall x_2 \in L_2)(\{y \in L_1:\neg(x_2 \le_{L_2} h(y))\} = \emptyset 
\text{ mod id}_{L_1}]
$$
\mn
iff

$$
(\forall x_2 \in L_2)(\exists x_1 \in L_1)(\forall y \in L_1)[\neg(x_2 
\le_{L_2} h(y)) \rightarrow \neg x_1 \le_{L_1} y]
$$
\mn
iff

$$
(\forall x_2 \in L_2)(\exists x_1 \in L_1)(\forall y \in L_1)
[x_1 \le_{L_1} y \rightarrow x_2 \le_{L_2} h(y)].
$$
\mn
5) Letting $h_1$ be the identity map on $L_1$ by part (3) we get
id$_{L_2} \le_{\text{RK}} \text{ id}_{L_1}$; choose $h_2:L_2
\rightarrow L_1$ which extends $h_1$, by part (4) we get id$_{L_1}
\le_{\text{RK}} \text{ id}_{L_2}$, together we are done. \nl
6) Easily.  \hfill$\square_{\scite{3.5}}$\margincite{3.5}
\enddemo
\bigskip

\demo{\stag{3.6} Fact}  1) For any ideal $J$ (such that (Dom$(J)) \notin J$),
letting $J_1 = \text{ id}_{(J,\subseteq)}$, we have
\mr
\widestnumber\item{(iii)}
\item "{$(i)$}"  $J_1$ is a partial order ideal
\sn
\item "{$(ii)$}"  $|\text{Dom}(J_1)| = |J| \le 2^{|\text{Dom}(J)|}$
\sn
\item "{$(iii)$}"  $J \le_{\text{RK}} J_1$
\sn
\item "{$(iv)$}"  if $J$ is $\lambda$-complete, then $(J,\subseteq)$ is
$\lambda$-directed hence $J_1$ is $\lambda$-complete
\sn
\item "{$(v)$}"  dens$(J,\subseteq) = \text{ dens}(J_1,\subseteq)$
\sn
\item "{$(vi)$}"  dens$(J,\subseteq) \le |J| \le 2^{|\text{Dom}(J)|}$.
\ermn
2) For every ideal $J$ and 
dense $X \subseteq J$ we can use id$_{(X,\subseteq)}$ and get
the same conclusions. \newline
3) For every ideal $J$ there is a directed order $L$ such that:

$$
J \le_{\text{RK}} \text{ id}_L, \text{ dens}(J) = \text{ dens}(L)
$$
\medskip

\noindent
and: for every $\lambda$ if $J$ is $\lambda$-complete, then so is
id$_L$.
\enddemo
\bigskip

\demo{Proof}  Least trivial is (1)(iii), let $h:J \rightarrow \text{ Dom}(J)$
be such that $h(A) \in (\text{Dom}(J)) \backslash A$ (exists as (Dom$(J))
\notin J$).  Let $J_1 = \text{ id}_{(J,\subseteq)}$, so $h$ is a
function from Dom$(J_1)$ into Dom$(h)$ and we shall prove that $h$
exemplifies the desired conclusion $J \le_{\text{RK}} J_1$ by
Definition \scite{3.2}(1)

Assume toward contradicion that 
$X \subseteq \text{ Dom}(J_1) = J,X \notin J_1$ and $A =: h''(X)$ belongs
to $J$.  So
$Y =: \{B \in J:\neg (A \subseteq B)\} \in \text{ id}_{(J,\subseteq)} =
J_1$ (by the definition of $J_1 = \text{ id}_{(J,\subseteq)}$) 
hence (as $X \notin J_1$)
for some $B \in X$ we have $B \notin Y$ hence by definition
$A \subseteq B$, so 
$h(B) \in h''(X) = A$ contradicting the
choice of $h(B)$ (as $A \subseteq B$). \hfill$\square_{\scite{3.6}}$\margincite{3.6}
\enddemo
\bigskip

\remark{\stag{3.7} Remark}  So we can replace a family of ideals by a
family of directed quasi orders without changing much the relevant 
invariants 
such as completeness or density as long as we do not mind adding ``larger"
ones in the appropriate sense.
\endremark
\bigskip

\demo{\stag{3.8} Conclusion}  For any family of ideals $\Bbb I$ there is 
a family of ${\Cal L}$ of quasi order such that:
\mr
\item "{$(i)$}"  $\Bbb I \le_{\text{RK}} \{\text{id}_{(L,<)}:(L,<) \in
{\Cal L}\}$
\sn
\item "{$(ii)$}"  $|{\Cal L}| \le |\Bbb I|$
\sn
\item "{$(iii)$}"  $\sup\{|L|:(L,<) \in {\Cal L}\} = \sup\{|J|:J \in 
\Bbb I\} \le \sup\{2^{\text{Dom}(J)}:J \in \Bbb I\}$
\sn
\item "{$(iv)$}"  $\sup\{\text{dens}(L,<):(L,<) \in {\Cal L}\} = \sup
\{\text{dens}(J,\subseteq):J \in \Bbb I\}$
\sn
\item "{$(v)$}"  $\Bbb I$ is $\lambda$-complete \ub{iff} every 
$(L,<) \in {\Cal L}$ is $\lambda$-directed.
\endroster
\enddemo
\bigskip

\demo{Proof}  Easy.
\enddemo
\bigskip

\definition{\stag{3.9} Definition}  For a 
forcing notion $\Bbb Q$, satisfying the
$\kappa$-c.c., a $\Bbb Q$-name $\underset\tilde {}\to L$ of a 
quasi order with Dom$(\underset\tilde {}\to L) \in \bold V$ 
for notational simplicity given (i.e., is not just a $\Bbb Q$-name); 
let $L^* = \text{ ap}_\kappa(\underset\tilde {}\to L) = \text{
ap}_\kappa(\underset\tilde {}\to L,\Bbb Q)$ be the following quasi order

$$
\text{Dom}(L^*) = \{a:a \subseteq \text{ Dom}(\underset\tilde {}\to L)
\text{ and } |a| < \kappa\}
$$

$$
a \le^* b \text{ iff } \Vdash_{\Bbb Q} ``(\forall y \in a)(\exists x \in b)
[\underset\tilde {}\to L \models y < x]"
$$
\mn
(this is a quasi-order only, e.g. maybe $a \le^* b \le^* a$ but
$a \ne b$).
\enddefinition
\bigskip

\proclaim{\stag{3.10} Claim}  1) For a forcing notion $\Bbb Q$ satisfying the 
$\kappa$-c.c. and a 
$\Bbb Q$-name $\underset\tilde {}\to L$ of a $\lambda$-directed
quasi order (with ${\text{\rm Dom\/}}(\underset\tilde {}\to L) 
\in \bold V$ given,
not just a $\Bbb Q$-name, for simplicity) such
that $\lambda \ge \kappa$ we have:
\mr
\widestnumber\item{$(iii)$}
\item "{$(i)$}"  ${\text{\rm ap\/}}_\kappa(\underset\tilde {}\to L)$ 
is $\lambda$-directed
quasi order (in $\bold V$ and also in $\bold V^{\Bbb Q}$)
\sn
\item "{$(ii)$}"  $|{\text{\rm ap\/}}_\kappa(\underset\tilde {}\to L)| \le
|{\text{\rm Dom\/}}(\underset\tilde {}\to L)|^{< \kappa}$ 
\sn
\item "{$(iii)$}"  $\Vdash_{\Bbb Q} 
``{\text{\rm id\/}}_{{\underset\tilde {}\to L}[G]}
\le_{\text{RK}} { \text{\rm id\/}}_{\text{ap}_\kappa({\underset\tilde
{}\to L})}"$
\ermn
2) For a forcing notion $\Bbb Q$ satisfying the $\mu$-c.c., the local
$\underset\tilde {}\to \kappa$-c.c. and a $\Bbb Q$-name $\underset\tilde {}\to L$
of a $\lambda$-directed quasi order such that $\lambda \ge \kappa$ we
can find ${\Cal L} \in \bold V$ such that
\mr
\widestnumber\item{$(iii)$}
\item "{$(i)$}"  ${\Cal L}$ is a family of $< \mu \, \lambda$-directed quasi
orders (in $\bold V$ and also in $\bold V^{\Bbb Q}$)
\sn
\item "{$(ii)$}"  for each $L \in {\Cal L}$ for some $\theta$ we have
$|L| \le \theta^{< \kappa}$ and $\nVdash 
``|{\text{\rm Dom\/}}(\underset\tilde {}\to L)| \ne \mu,
\underset\tilde {}\to \kappa = \kappa"$
\sn
\item "{$(iii)$}"  $\Vdash_{\Bbb Q} 
``{\text{\rm id\/}}_{\underset\tilde {}\to L[G]}
\le_{\text{RK}} { \text{\rm id\/}}_L"$ for some $L \in {\Cal L}$.
\ermn
3) In (2), of course, we can replace $\Bbb I$ by a $\lambda$-complete
$\le_{\text{RK}}$-upper bound.
\endproclaim
\bigskip

\demo{Proof}  1) We leave (i), (ii) to the reader.  We check (iii).  Let
$G \subseteq \Bbb Q$ be generic over $\bold V$, and in $\bold V[G]$ 
we define a function $h$
from ap$_\kappa(\underset\tilde {}\to L)$ 
to Dom$(\underset\tilde {}\to L[G])$ as follows:
$h(a)$ will be an element of Dom$(\underset\tilde {}\to L[G])$ such that

$$
(\forall x \in a)[\underset\tilde {}\to L[G] \models ``x \le h(a)"].
$$
\mn
It exists by the ``$\lambda$-directed", ``$\lambda \ge \kappa$" assumptions.
We can now easily verify the condition in \scite{3.5}(4). \nl
2) Let $\{p_i:i < i^*\}$ be a maximal antichain of $\Bbb Q$ such that:
$p_i \Vdash ``\text{Dom}(\underset\tilde {}\to L)$ has cardinality $\mu_i,
\underset\tilde {}\to \kappa 
= \kappa_i"$ and $\Bbb Q_{\ge p_i}$ satisfies the
$\kappa_i$-c.c and \wilog \, $p_i \Vdash ``\text{Dom}(\underset\tilde {}\to
L) = \mu_i"$.  
Let $\Bbb Q_i = \Bbb Q_{\ge p_i}$ and ${\underset\tilde {}\to L_i}$ be
$\underset\tilde {}\to L$ restricted to $\Bbb Q_i$ and lastly let
${\Cal L} = \{\text{ap}_{\kappa_i}({\underset\tilde {}\to L_i},\Bbb
Q_i):i < i^*\}$, by part (1) we have $p_i \Vdash_{\Bbb Q}
``\text{id}_{\underset\tilde {}\to L[{\underset\tilde {}\to G_{\Bbb
Q}}]} \le_{\text{RK}}$; ap$_{\kappa_i}
({\underset\tilde {}\to L_i},{\Bbb Q}_i)"$ by $\{p_i:i < i^*\}$ is a maixmal antichain of $\Bbb
Q$ hence $\Vdash_{\Bbb Q} ``\text{id}_{\underset\tilde {}\to
L[{\underset\tilde {}\to G_{\Bbb Q}}]} \le_{\text{RK}}
\{\text{ap}_{\kappa_i}({\underset\tilde {}\to L_i},\Bbb Q_i):i <
i^*\}$. \nl
A base $|\text{ap}_{\kappa_i}({\underset\tilde {}\to L_i},\Bbb Q_i)|
\le |\text{Dom}({\underset\tilde {}\to L_i})|^{< \kappa_i} = \mu^{<
\kappa_i}_i$ and ap$_{\kappa_i}({\underset\tilde {}\to L_i},\Bbb Q_i)$
is $\lambda$-directed.  Together 
we are done as necessarily $i^* < \mu$. \nl
3) Easy by \scite{3.13}(3),(4) below.  \hfill$\square_{\scite{3.10}}$\margincite{3.10}
\enddemo
\bigskip

\demo{\stag{3.11} Conclusion}  1) 
Suppose $\Bbb Q$ is a forcing notion satisfying
the local $\kappa$-c.c., ${\underset\tilde {}\to {\Bbb I}_1}$ 
a $\Bbb Q$-name of 
a family of $\lambda$-complete filters and $\lambda \ge \kappa$. 
\underbar{Then} there is, (in $\bold V$), a family $\Bbb I_2$ of 
$\lambda$-complete filters such that:
\medskip
\roster
\widestnumber\item{$(iii)$}
\item "{$(i)$}"  $\Vdash_{\Bbb Q} ``{\underset\tilde {}\to {\Bbb I}_1} 
\le_{\text{RK}} \Bbb I_2"$
\sn
\item "{$(ii)$}"  if $\Bbb Q$ satisfies the $\mu$-c.c. then
$|\Bbb I_2| = |{\underset\tilde {}\to {\Bbb I}_1}| + \mu$ i.e.
$\Vdash_{\Bbb Q} ``|\Bbb I_2| = |\Bbb I_1|^V + \mu"$
\sn
\item "{$(iii)$}"  $\sup\{|\text{Dom}(J)|:J \in \Bbb I_2\} = 
\sup\{(2^\mu)^{< \kappa}:\text{some } q \in \Bbb Q$ \newline

$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad$ forces that some 
$J \in \Bbb I_1$ 
\newline

$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad$ has domain of 
power $\mu\}$.
\ermn
2) If in $\bold V^{\Bbb Q}$ the set $\Bbb I_1$ 
has the form $\{\text{id}_{(L,<)}:(L,<) \in {\Cal L}_1\}$; 
i.e., is a family of quasi order ideals, \underbar{then} in (iii) we can have
\mr
\item "{$(iii)'$}"   $\sup\{|\text{Dom}(J)|:J \in \Bbb I_2\} = 
\sup\{\mu^{< \kappa}:\text{some } q \in \Bbb Q$ \newline

$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad$ 
force some $(L,<) \in {\Cal L}$ has power $\mu\}$
\sn
\item "{$(iv)$}"  $\Bbb I_2$ is a family of quasi order ideals.
\endroster
\enddemo
\medskip

\demo{Proof}  Easy.
\enddemo
\bigskip

\remark{\stag{3.12} Remark}  The aim of \scite{3.10}, \scite{3.11} is the
following:  we will consider iterations $\langle \Bbb P_i,
{\underset\tilde {}\to {\Bbb Q}_i}:i < \alpha \rangle$ where
$\Vdash_{{\Bbb P}_i} ``{\underset\tilde {}\to {\Bbb Q}_i}$ satisfies 
UP$({\underset\tilde {}\to {\Bbb I}_i})"$, but 
${\underset\tilde {}\to {\Bbb I}_i}$ may 
not be a subset of the ground model $\bold V$.  
Now \scite{3.11} gives us a good
$\le_{\text{RK}}$-bound $\Bbb I'_i$ of $\Bbb I_i$ in $\bold V$, and we can 
prove (under suitable assumptions) that $\Bbb P_\alpha$ will satisfy the 
UP$(\dsize \bigcup_{i < \alpha} \Bbb I'_i)$.
\endremark
\bigskip

\definition{\stag{3.13} Definition}  1) For a family ${\Cal L}$ of
directed quasi
order let the $(< \kappa)$-closure $c \ell_\kappa({\Cal L})$ be

$$
{\Cal L} \cup \biggl\{ \dsize \prod_{i < \alpha} L_i:L_i \in {\Cal L}
\text{ for } i < \alpha \text{ and } \alpha < \kappa \biggr\}
$$
\mn
(the partial order on $\dsize \prod_{i < \alpha} L_i$ is natural). \newline
2) ${\Cal L}$ is $(< \kappa)$-closed if for any $\alpha < \kappa$ and $L_i
\in {\Cal L}$ for $i < \alpha$ \underbar{there is} $L \in {\Cal L}$ such
that $i < \alpha \Rightarrow L_i \le_{\text{RK}} L$.
\enddefinition
\bigskip

\proclaim{\stag{3.14} Claim}  1) If ${\Cal L}$ is $\lambda$-complete,
\underbar{then} $c \ell_\kappa({\Cal L})$ is $\lambda$-complete. \newline
2) $L_j \le_{\text{RK}} \dsize \prod_{i < \alpha} L_i$ for $j < \alpha$.
\newline
3) If $\kappa$ is regular, \ub{then} 
$c \ell_\kappa({\Cal L})$ is $\kappa$-directed
under $\le_{\text{RK}}$ and is $(< \kappa)$-closed.
\endproclaim
\bigskip

\demo{Proof}  Use \scite{3.5}(4).
\enddemo
\newpage

\head {\S4 UP reintroduced} \endhead  \resetall \sectno=4
 % \resetall 
\bn
The reader may concentrate on the case $\bold S = \{\aleph_1\}$.
\demo{\stag{4.0} Convention}  $\Bbb I$ will be a set of quasi order ideals,
i.e. $\Bbb I = \Bbb I_{\Cal L}$.
\enddemo
\bigskip

\definition{\stag{4.1} Definition}  Fix $\Bbb I,\bold S,\bold W$ assuming
\medskip
\roster
\item "{$(*)(a)$}"  $\Bbb I$ a family of $\aleph_2$-complete quasi-order
ideals
\sn
\item "{$(b)$}"  $\bold S$ is a set of regular cardinals to which $\aleph_1$
belongs
\sn
\item "{$(c)$}"  $\bold W \subseteq \omega_1$ is stationary.
\endroster
\medskip

\noindent
1) We say $\bar N$ is an $\Bbb I$-tagged tree of models (for $\chi$ or for
$(\chi,x)$) \ub{if} there is an $\Bbb I$-tagged tree $(T,\bold I)$ such that
$\bar N = \langle N_\eta:\eta \in (T,\bold I) \rangle$ satisfies 
the following:
\medskip
\roster
\item "{$(a)$}"  for $\eta \in T$ we have $N_\eta \prec ({\Cal H}(\chi),\in,
<^*_\chi)$ is a countable model
\sn
\item "{$(b)$}"  $\Bbb I \in N_{\langle \rangle}$ and $x \in
N_{\langle \rangle}$ (if $x$ is present, we can use $\Bbb I$ as a predicate)
\sn
\item "{$(c)$}"  $\eta \triangleleft \nu \in T$ implies $N_\eta \prec
N_\nu$
\sn
\item "{$(d)$}"  for $\eta \in T$ we have $\eta \in N_\eta$ and $\bold I_\eta
\in N_\eta \cap \Bbb I$.
\endroster
\medskip

\noindent
Whenever we have such an $\Bbb I$-tagged tree $\bar N$ of models, we write
$N_\eta = \dsize \bigcup_{k < \omega} N_{\eta \restriction k}$ for each
$\eta \in \text{ lim}(T)$.  
 \newline
1A)  $\bar N = \langle N_\eta:\eta \in (T,\bold I) \rangle$ is a tagged 
tree of models if this occurs for some $\aleph_2$-complete $\Bbb I$.  
If $x$ is not mentioned, we assume it contains all necessary information, 
in particular $\Bbb I,\bold S,\bold W$. \newline
1B) In part (1) we say ``weak $\Bbb I$-tagged tree of models" 
\ub{if} we replace
clause (d) by
\mr
\item "{$(d)^-(i)$}"  for $\eta \in T \cap \text{ Dom}(\bold I)$ 
we have $\bold I_\eta \in N_\eta \cap \Bbb I$
\sn
\item "{$(ii)$}"  if $\eta \in \text{ Split}(T,\Bbb I)$ then for some
one-to-one function $f \in N_\eta$ with domain $\supseteq \text{
Suc}_T(\eta)$ and range $\subseteq \text{ Dom}(\bold I_\eta)$ we have 
$\nu \in \text{ Suc}_T(\eta) \Rightarrow f(\nu) \in N_\nu$.
\ermn
We say weaker if we omit clause $(d)^-(ii)$.  In all the definitions
below we can use this version (i.e., adding weak/weaker and replacing
$(d)$ by $(d)^-/(d)^-(i)$
\nl
2) We say $\bar N$ is truely $\Bbb I$-suitable (tagged tree of models)
\ub{if} clauses $(a)-(d)$ and:
\medskip
\roster
\item "{$(e)$}"  $\Bbb I \in N_{\langle \rangle}$ and if $\eta \in T$ and
$I \in \Bbb I \cap N_\eta$ \ub{then} the set
\endroster

$$
\{\nu \in T^{[\eta]}:\nu \in \text{ split}(T) \text{ and } \bold I_\nu =
I \in N_\eta\}
$$
\medskip

\noindent
contains a front of $T^{[\eta]}$. 
So ``$\bar N$ is truly $\Bbb I$-suitable tree (of models)" does not imply
``$\bar N$ is an $\Bbb I$-tagged tree of models" as possibly $\Bbb I
\notin N_{<>}$.  \newline
3) We say $\bar N$ is $\Bbb I$-suitable (a tagged tree of models)
\ub{if} clauses $(a)-(d)$ and:
\medskip
\roster
\item "{$(e)^-$}"  if $\eta \in T$ and $I \in \Bbb I \cap N$, \ub{then} 
the set
$$
\{\nu \in T^{[\nu]}:\nu \in \text{ split}(T) \text{ and } I
\le_{\text{RK}} \bold I_\nu \in N\}
$$
contains a front of $T^{[\eta]}$.
\ermn
4) We say $\bar N$ is $\lambda$-strictly $(\Bbb I,\bold W)$-suitable \ub{if}
$\bar N$ is $\Bbb I$-suitable and in addition
\mr
\item "{$(b)^+$}"  $\Bbb I \in N_{\langle \rangle},\bold W \in N$
and $x \in N_{\langle \rangle}$
\sn
\item "{$(f)$}"  one of the following cases holds:
{\roster
\itemitem{ $(i)$ }   $\lambda = \aleph_2$ and for some $\delta \in
\bold W$, for all $\eta \in T$ we have: $N_\eta \cap \omega_1 = \delta$
\sn
\itemitem{ $(ii)$ }   $\lambda = \text{ cf}(\lambda) > \aleph_1,
N_{<>} \cap \omega_1 \in \bold W$ and $\eta
\triangleleft \nu \in T \Rightarrow N_\eta <_\lambda N_\nu$ (i.e.,
$N_\eta \cap \lambda$ is an initial segment of $N_\nu \cap \lambda$ \nl
(note: if $\lambda = \mu^+$ this means $N_\eta \cap \mu = N_\nu \cap \mu$
so no contradiction with the case $\lambda = \aleph_2$)
\sn
\itemitem{ $(iii)$ }  $\lambda = \aleph_1$ and we demand nothing. 
\endroster}
If we omit $\lambda$, we mean $\lambda = \aleph_2$.  So $\lambda = 
\text{ cf}(\lambda) > \aleph_1$ implies $\eta \in T \Rightarrow N_\eta 
\cap \omega_1 = N_{<>} \cap \omega_1 \in \bold W$.  If we omit ``strictly" we
demand only $(b)^+$.
\endroster
\medskip
\noindent
5) We say $\bar N$ is $\bold S$-strictly $(\Bbb I,\bold W)$-suitable, \ub{if}
in addition to clauses (a) - (d),(e)$^-$ we have:
\medskip
\roster
\item "{$(b)^+$}"  $\Bbb I \in N_{\langle \rangle},\bold W \in
N_{\langle \rangle},\bold S \in N_{\langle \rangle}$ and $x \in
N_{\langle \rangle}$
\sn
\item "{$(g)$}"  for all $\nu \in T$ and $\lambda \in \bold S \cap N_\delta$
there is $\delta_\lambda,\nu < \lambda$ such that \nl
$(\forall \eta \in T)[\nu \trianglelefteq \eta \in T
\Rightarrow \sup(N_\eta \cap \lambda) = \delta_\lambda]$.
\ermn
6) We say $\bar N$ is $\lambda^+$-uniformly $(\Bbb I,\bold
W)$-suitable \ub{if} it is $(\Bbb I,\bold W)$-suitable and
\mr
\item "{$(b)^+$}"  $\Bbb I \in N_{\langle \rangle},\bold W \in 
N_{\langle \rangle},\lambda \in N_{\langle \rangle}$ and $x \in
N_{\langle \rangle}$
\sn
\item "{$(f)'$}"  for some $a \in [\lambda]^{\aleph_0}$ for every
$\eta \in \text{ lim}(T)$ we have
$$
N_\eta \cap \lambda = a
$$
and
$$
\text{if } \lambda^+ = \aleph_2 \text{ then } a \in \bold W.
$$
\ermn
6A) We say $\bar N$ is $\bold S$-uniformly $(\Bbb I,\bold W)$-suitable
\ub{if} it is $(\Bbb I,\bold W)$-suitable and
\mr
\item "{$(b)^+$}"  $\Bbb I \in N_{\langle \rangle},\bold W \in
N_{\langle \rangle},\bold S \in N_{\langle \rangle}$ and $x \in
N_{\langle \rangle}$
\sn
\item "{$(g)'$}"  for every $\eta \in T,\lambda \in \bold S \cap N_\eta$
for some $a \in [\lambda]^{\aleph_0}$ for every $\nu$ satisfying $\eta 
\triangleleft \nu \in \text{ lim}(T)$ we have
$$
N_\nu \cap \lambda = a.
$$
\ermn
7) In 4), 5) if we add "truely" if (e)$^-$ is replaced by (e). \newline
8) If ${\underset\tilde {}\to {\bold S}}$ is a $\Bbb P$-name then 
in the clauses
above we mean $\bold S^* = \{\lambda:\,\,\Vdash_{\Bbb P} ``\lambda \notin 
{\underset\tilde {}\to {\bold S}}"\}$. \nl
9) $\Bbb I^{[\lambda]} = 
\{I \in \Bbb I:I \text{ is } \lambda \text{-complete}\}$.
\enddefinition
\bigskip

\definition{\stag{4.2} Definition}  1) We say $N \prec ({\Cal H}(\chi),\in,
<^*_\chi)$ is ${\underset\tilde {}\to {\bold S}}$-strictly or
$\lambda$-strictly $(\Bbb I,\bold W)$-suitable model if there is an
${\underset\tilde {}\to {\bold S}}$-strictly or a $\lambda$-strictly $(\Bbb I,
\bold W)$-suitable $\bar N$ such that $N = N_{\langle \rangle}$, (see 
\scite{4.1}(5),(9), and see \scite{4.1}(8), it applies).  
We can add ``truely". \nl
2) We say $N$ is $(\Bbb I,\bold W)$-suitable if it is strictly
$(\Bbb I,\bold W)$-suitable, that is $\aleph_2$-strictly $(\Bbb
I,\bold W)$-suitable.
\enddefinition
\bigskip

\definition{\stag{4.3} Definition}   In Definitions \scite{4.1},
\scite{4.2} we may omit $\bold W$ when it is 
$\omega_1$, and may omit $\bold S$ when $\bold S = \{\aleph_2\}$,
for \scite{4.1}(5) and \scite{4.1}(6),
\scite{4.1}(6A).  We may replace $\bold S$ by $*$ if $\bold S = U$Reg.
Let $\eta \in (T,\bold I)$ means $\eta \in T$ and we write $T$ 
when $\bold I$ is clear.
\enddefinition
\bigskip

\proclaim{\stag{4.8} Claim}  Assume
\mr
\item "{$(i)$}"  ${\underset\tilde {}\to {\bold S}},
\Bbb P,\Bbb I,\bold W,x \in
{\Cal H}(\chi)$ and $\bold S^*$ is as in \scite{4.1}(8) \nl
(i.e., $\bold S^* = \{\theta:\theta = \aleph_1 \text{ or } \aleph_1 \le
\theta = \text{ cf}(\theta) \le |\Bbb P|$ and $\nVdash_{\Bbb P} ``\theta \in
{\underset\tilde {}\to {\bold S}}"\} \cup \{\aleph_1\}$, \nl
e.g., ${\underset\tilde {}\to {\bold S}} = \{\aleph_1\}$), and
\sn
\item "{$(ii)$}"  $\Bbb I$ 
is a $\aleph_2$-complete or for each $I \in \Bbb I,
\kappa \in \bold S,I$ is $\kappa$-indecomposable.
\ermn
\ub{Then} there is an $\bold S$-strictly, truely $(\Bbb I,\bold W)$-suitable 
tree $\bar N$ with $x \in N_{\langle \rangle}$.
\endproclaim
\bigskip

\demo{Proof}  We will construct this tree in three steps: first we find a
suitable tree, then we thin it out to be a $\bold S^*$-uniformly suitable 
tree, then we blow up the models to make it $\bold S^*$-strict.  For 
notational simplicity let $\bold S^* = \{\aleph_1\}$ 
(the reader can check the others).
\mn
\underbar{First Step}:  An easy bookkeeping argument (to ensure 
\scite{4.1}(1)(e)) yields a truly
$(\Bbb I \cup \{J^{\text{bd}}_{\omega_1}\})$-suitable tree
$\langle N_\eta:\eta \in (T,\bold I) \rangle$ satisfying $\nu
\triangleleft \eta \Rightarrow \sup(N_\nu \cap \omega_1) < \sup(N_\eta
\cap \omega_1)$ such that $N_\eta \prec ({\Cal H}(\chi),\in,<^*_\chi)$
and $x \in N_{<>}$; so for $\eta \in \text{ lim}
(T)$ we let $N_\eta = \dsize \bigcup_{\ell < \omega} N_{\eta \restriction
\ell}$.  

Moreover we can we get that 
for all $\eta \in \text{ lim}(T)$, for each $I \in (\Bbb I 
\cap N_\eta) \cup \{J^{\text{bd}}_{\omega_1}\}$, there are 
infinitely many $k$ such 
that $\eta \restriction k \in \text{ split}(T,\bold I)$ 
and $\bold I_{\eta \restriction k} = I$ and Suc$_T(\eta \restriction k) =
\{\eta \char 94 \langle x \rangle:x \in \text{ Dom}(I)\}$.
\medskip

\noindent
\underbar{Second Step}:  Define $H:T \rightarrow \omega_1$ by $H(\eta) = \sup
(N_\eta \cap \omega_1) < \omega_1$.  Apply \scite{2.13} to get a subtree
$T'$ and a limit ordinal $\delta \in \bold W \subseteq \omega_1$ such that 
clauses (a) - (d) of \scite{2.13} hold for $\delta$.  By clause (d) of \scite{2.13}, for 
all $\eta \in T',N_\eta \cap \omega_1 \subseteq \delta$.  Let $\delta_0 < 
\delta_1 < \ldots \dsize \bigcup_n \delta_n = \delta$, and let

$$
\align
T'' = \biggl\{ \eta \in T':&\text{ for each } 
\forall k < \ell g(\eta),\text{ if Suc}_T(\eta
\restriction k) = \{\eta \restriction k \char 94 \langle \alpha \rangle:
\alpha < \omega_1\} \\
  &\text{ so Suc}_{T'}(\eta \restriction k) = \{\eta \restriction k \char 94
\langle \alpha \rangle:\alpha < \delta\}) \\
  &\text{ then } \eta(k) = \delta_k \biggr\}.
\endalign
$$
\medskip

\noindent
Clearly $T''$ will be $\aleph_1$-uniformly suitable; i.e. \newline
$\eta \in \text{ lim}(T) \Rightarrow N_{\eta,\ell} \cap \omega_1 = \delta$.
\mn
\underbar{Third Step}:  For $\eta \in T_2$, 
let $N'_\eta =$ the Skolem Hull of
$N_\eta \cup \delta$ in $({\Cal H}(\chi),\in,<^*_\chi)$.  
So $N'_\eta \cap \omega_1 \supseteq \delta$.
Conversely, let $\nu \in \text{ lim}(T_2),\eta \triangleleft \nu$, then
$N_\eta \cup \delta \subseteq N_\nu$, so $N'_\eta \subseteq N_\nu$ hence
$N'_\eta \cap \omega \subseteq \delta$.  So $N'_\eta \cap \omega_1 = \delta$,
i.e. $\langle N'_\eta:\eta \in T \rangle$ is an $\aleph_1$-strict, $(\Bbb I,
{\underset\tilde {}\to {\bold S}},\bold W)$-tree of models (see Definition
\scite{4.1}(4)).

We claim that this tree is still truly suitable.  
Indeed, assume $\eta \in T_2,\nu
\in \text{ lim}(T_2),\eta \trianglelefteq \nu$ and $I \in \Bbb I 
\cap N'_\eta$.
Then for some $\alpha < \delta,I$ is in the Skolem hull of $N_\eta \cup
\alpha$.  Let $k < \omega$ be such that $\alpha \in N_{\nu \restriction k}
\cap \omega_1$ and $k \le \ell g(\eta)$.  
Then since $\langle N_\eta:\eta \in T_2
\rangle$ was suitable, there is $\ell \ge k$ such that 
$\bold I_{\nu \restriction \ell} = I$.  So $\langle N'_\eta:\eta \in T_2
\rangle$ is also $(\Bbb I,{\underset\tilde {}\to {\bold S}},
\bold W)$-suitable. \hfill$\square_{\scite{4.8}}$\margincite{4.8}
\enddemo
\bigskip

\demo{\stag{4.4} Fact}  Assume $\Bbb I' \le_{\text{RK}} \Bbb I$, where 
$\Bbb I,\Bbb I'$ are families of ideals. \newline
1) If $\langle N_\eta:\eta \in (T,\bold I) \rangle$ is a $\Bbb I$-suitable 
tree and $\Bbb I' \in N_{\langle \rangle}$, \ub{then}
$\langle N_\eta:\eta \in (T,\bold I) \rangle$ is also $\Bbb I'$-suitable. \nl
2) If $\langle N_\eta:\eta \in (T,\bold I) \rangle$ is $\Bbb I$-suitable and 
$\Bbb I' \in N_{\langle \rangle}$, \underbar{then} there is a tree $(T',
\bold I')$ satisfying the following for some $T'',f$:
\medskip
\roster
\item "{$(a)$}"  $T'' \subseteq T$ and $f$ is an isomorphism from
$T''$ onto $T'$, (i.e., is one to one onto preserving length and
$\triangleleft$ and its negation) and $\eta\in T'' \Rightarrow \bold
I''_\eta \le_{\text{RK}} \bold I_{f(\eta)},\bold I''_\eta \ne \bold I'_\eta$
\sn
\item "{$(b)$}"  $\langle N_\eta:\eta \in (T',\bold I') \rangle$ is truely
$\Bbb I'$-suitable when we let $I''_\eta = \bold I'_{f(\eta)}$
\sn
\item "{$(c)$}"  split$(T'',\bold I'') = 
T'' \cap \text{ split}(T,\bold I)$ if $\Bbb I' \equiv_{\text{RK}} \Bbb I$.
\sn
\ermn
3) We can weaken the hypothesis to $\Bbb I' \le_{\text{RK}} 
\Bbb I \cup \{\text{the trivial ideal}\}$.  The same holds in 
similar situations. \newline
4) In Definition \scite{4.3}, if $\bold S = \{\theta:\aleph_1 \le \theta =
\text{ cf}(\theta) \le \lambda\},\lambda = \text{ cf}(\lambda)$, 
\underbar{then} clause (f) of \scite{4.1}(4) (i.e., $\lambda^+$-strictly) and
clause (g) of \scite{4.1}(5) (i.e. the demand concerning $\bold S$,
i.e., $\bold S$-strictly) are equivalent.  Similarly \scite{4.1}(6),
\scite{4.1}(6A) are equivalent. \nl
5) In part (2), $\langle N_\eta:\eta \in (T'',\bold I) \rangle$ is a
weak $\Bbb I'$-tagged tree, truely $\Bbb I'$-suitable; moreover it is
enough to assume $\langle N_\eta:\eta \in (T,\bold I) \rangle$ is a
weak $\Bbb I$-suitable tree (see \scite{5.2}). \nl
6) If $\langle N_\eta:\eta \in (T,\bold I) \rangle$ is a weak $\Bbb
I$-tagged tree \ub{then} for some tree $T'$ and tree isomorphic $f$ from
$T'$ onto $T$ letting $\bold I' = \langle \bold I_{f(\eta)}:\eta \in
T' \rangle$ we have $\langle N_{f(\eta)}:\eta \in (T',\bold I')
\rangle$ is a $\Bbb I$-tagged tree.  All relevant properties are
preserved.  [Check, see \scite{5.2}.]
\enddemo
\bigskip

\demo{Proof}  1) Should be clear, as $\le_{\text{RK}}$ is transitive (as a
relation among ideals and also among families of ideals). \newline
2) For every $\eta \in Y =: \{\eta \in T:(\exists I' \in \Bbb I')
(I' \le \bold I_\eta)$
and $\eta \in \text{ split}(T,\bold I)\}$ pick an ideal $\bold I''_\eta
\in \Bbb I \cap N_\eta,\bold I''_\eta \le_{\text{RK}} \bold I_\eta$
such that:
for every $\nu \in T$, for every 
$I' \in \Bbb I \cap N_\nu$ the set $\{\eta \in
T^{[\nu]}:I' = \bold I''_\eta \text{ and } \nu \triangleleft \eta$ and
$\eta \in Y\}$ contains a front of $T^{[\nu]}$.  This can
be done using a bookkeeping argument.

Now define $T'$ as follows.  We choose by induction on $n$, a function
$f_n$ with domain $\subseteq T \cap {}^n\text{Ord}$, such that $\eta
\in \text{ Dom}(f_\eta) \Rightarrow f(\eta) \in N_\eta \cap
{}^n\text{Ord}$.  Let $f_0$ be the identity on $\{<>\}$.  Assume $f_n$
has been defined and $\eta \in \text{ Dom}(f_n)$, and we shall define
$f_{n+1} \restriction \text{ Suc}_T(\eta)$.  If 
$\eta \in T \backslash Y$, then Dom$(f_{n+1}) \cap \text{ Suc}_T(\eta) \subseteq
\text{ Suc}_T(\eta)$ is a singleton $\{\nu_\eta\}$ and let
$f_{n+1}(\nu_\eta) = f_n(\eta) \char 94 \langle (\nu_\eta(n) \rangle$.  
If $\eta \in T \cap Y$, then $\bold I'_\eta$ is already defined and it 
belongs to $N_\eta$.  Let $g_\eta$
be a witness for $\bold I'_\eta \le_{\text{RK}} \bold I_\eta$ and
stipulate Dom$(\bold I_\eta) \supseteq \{x:\eta \char 94 <x> \in
\text{ Suc}_T(\eta)\}$.  Now $g_\eta$
introduces an equivalence relation on Dom$(\bold I_\eta)$.  Let $A_\eta$ be a
selector set for this equivalence relation; i.e. $g_\eta \restriction A_\eta$
is 1-1 and has the same range as $g_\eta$.  Note that we can choose $g_\eta$
in $N_\eta$ as $\bold I_\eta,\bold I'_\eta \in N_\eta$ 
(whereas Suc$_T(\eta)$
does not necessarily belong to $N_\eta$) and then choose $A_\eta$ and
let $A'_\eta =: \{x \in A_\eta:(\exists y \in \text{
Suc}_T(\eta))[g_\eta(x)= g_\eta(y)]\}$ (so possibly
$A_\eta \notin N_\eta$).  Now for $x \in A'_n$ so $\eta \char 94
<x> \in \text{ Suc}_T(\eta)$ we let $f_{n+1}(\eta \char 94 <x>) =
f_n(\eta) \char 94 \langle g_n(x) \rangle$ so Dom$(f_{n+1}) \cap
\text{ Suc}_T(\eta) = \{\eta \char 94 <x>:x \in A'_\eta\}$.  Lastly, let $T' =
\cup\{\text{Rang}(f_n):n < \omega\},T'' = \cup\{\text{Dom}(f_n):n <
\omega\}$ and for $\eta \in \text{ Dom}(f_n),n < \omega$ let $\bold
I'_{f_n(\eta)} = \bold I''_\eta$ and $f = \cup\{f_n:n < \omega\}$.
Now check. \nl
3), 4)  Left to the reader.   \hfill$\square_{\scite{4.4}}$\margincite{4.4}
\enddemo
\bigskip

\definition{\stag{4.5A} Definition}  1) Let $\chi > \aleph_0,
\Bbb I \in {\Cal H}(\chi)$ a set of ideals and $\bold S \in {\Cal H}(\chi)$
a set of regular cardinals (or just limit ordinals) such that $\aleph_1 \in
\bold S$.  For $N$ a countable elementary submodel of ${\frak B} =
({\Cal H}(\chi),\in)$ (or ${\frak B}$ an expansion of $({\Cal
H}(\chi),\in)$ with countable
vocabulary) such that $\Bbb I,\bold S \in N$ we define Dp$(N) = 
\text{ Dp}_{\Bbb I}(N) = \text{ Dp}_{\Bbb I}(N,{\frak B}) = 
\text{ Dp}_{\Bbb I}(N,\bold S,{\frak B}) \in \text{ Ord } \cup \{\infty\}$,
by defining when Dp$(N) \ge \alpha$ for an ordinal $\alpha$, by induction
on $\alpha$:

$$
\align
\text{Dp}(N) \ge \alpha \text{ \ub{iff} } &N \text{ is as above and for
every } I \in \Bbb I \cap N \\
  &\text{ and for every } \beta < \alpha \text{ and } X \in I 
\text{ there is } M \text{ satisfying}: \\
  &(i) \quad \text{ Dp}(M) \ge \beta ( \text{hence } M \prec {\frak B}
\text{ is countable})  \\
  &(ii) \quad N \prec M \\
  &(iii) \quad \text{ sup}(M \cap \omega_1) = \text{ sup}(N \cap \omega_1)
\text{ moreover} \\
  &\qquad \quad \theta \in \bold S \cap N \Rightarrow \text{ sup}(N \cap
\theta) = \text{ sup}(M \cap \theta) \\
  &(iv) \quad M \cap \text{ Dom}(I) \backslash X \ne \emptyset.
\endalign
$$
\mn
2) We 
define Dp$'(N) = \text{ Dp}'_{\Bbb I}(N)$ by defining: $\text{Dp}'(N) \ge
\alpha$ iff $N$ is as above and for every $J \in \Bbb I \cap N$ and $\beta <
\alpha$ for some $I \in N \cap \Bbb I$ we have $J \le_{\text{RK}} I$ 
and for every $X \in I$ there is $M$ satisfying (i)-(iv) above.
\enddefinition
\bigskip

\proclaim{\stag{4.5B} Claim}  1)  In Definition \scite{4.5A}:
\mr
\item "{$(a)$}"  ${\text{\rm Dp\/}}(N) \in { \text{\rm Ord\/}} 
\cup \{\infty\}$ if well defined
\sn
\item "{$(b)$}"  if ${\text{\rm Dp\/}}(N) = 
\infty,I \in \Bbb I \cap N$ then we can find 
$Y \in I^+$ (i.e., $Y \subseteq { \text{\rm Dom\/}}(I),
Y \notin I$) and $\bar N =
\langle N_t:t \in Y \rangle$ such that:
{\roster
\itemitem{ $(i)$ }  ${\text{\rm Dp\/}}(N_t) = \infty$
\sn
\itemitem{ $(ii)$ }  $N \prec N_t$
\sn
\itemitem{ $(iii)$ }  ${\text{\rm sup\/}}(N \cap \omega_1) = 
{ \text{\rm sup\/}}(N_t \cap
\omega_1)$ moreover $\theta \in \bold S \cap N \Rightarrow \sup
(N \cap \theta) = \sup(N_t \cap \theta)$.
\endroster}
\ermn
2) If $\Bbb I_1 \le_{\text{RK}} \Bbb I_2$ and $\{\Bbb I_1,\Bbb I_2\} \in N$
and Dp$_{{\Bbb I}_2}(N) \ge \alpha$ then Dp$_{{\Bbb I}_1}(N) \ge \alpha$.
\nl
3) Dp$_{\Bbb I}(N) = \text{ Dp}'_{\Bbb I}(N)$.
\endproclaim
\bigskip

\demo{Proof}  Straightforward.
\enddemo
\bigskip

\proclaim{\stag{4.5C} Claim}  1) Let $N \prec ({\Cal H}(\chi),\in)$ be
countable, $\Bbb I \in N, N \cap \omega_1 \in {\Cal W}$.  \ub{Then}
\mr
\item "{$(a)$}"   $N$ is strictly 
$(\Bbb I,{\Cal W})$-suitable \ub{iff} ${\text{\rm Dp\/}}_{\Bbb I}(N) = \infty$
\sn
\item "{$(b)$}"  $N$ is $(\Bbb I,{\Cal W})$-suitable \ub{iff}
${\text{\rm Dp\/}}'_{\Bbb I}(N) = \infty$.
\ermn
2) Similarly with $\bold S$.
\endproclaim
\bigskip

\demo{Proof}  Easy.
\enddemo
\bigskip

\definition{\stag{4.6} Definition}  1) The forcing notion $\Bbb P$ satisfies
UP$^\ell_{\underset\tilde {}\to \lambda}
(\Bbb I,{\underset\tilde {}\to {\bold S}},\bold W)$ 
(note if $\ell \ne 2$ we may omit $\underset\tilde {}\to \lambda$)
(adopting the
conventions of \scite{4.1}(8); $\underset\tilde {}\to \lambda$ is a
purely decidable $\Bbb P$-name of a $\bold V$-cardinal) when: 
$\ell \in \{0,1,2\}$ and if $\chi$ is 
large enough and $\bar N$ is ${\underset\tilde {}\to {\bold S}}$-strictly 
$(\Bbb I,\bold W)$-suitable and $p \in N_{\langle \rangle} \cap \Bbb P$ and 
$P \in N_{\langle \rangle}$, of course, \ub{then} 
there is $q \in \Bbb P$ such that $p \le_{\text{pr}} q \in \Bbb P$ and:
\mr
\item "{$(a)$}"  if $\ell = 0$ then $q \Vdash ``N_{\langle \rangle} 
[{\underset\tilde {}\to G_{\Bbb P}}] \cap \omega_1 = N_{\langle \rangle} 
\cap \omega_1$ and sup$(N_{<>}[{\underset\tilde {}\to G_{\Bbb P}}] 
\cap \theta) = \text{ sup}(N_{<>} \cap \theta)$ 
if $\theta \in \underset\tilde {}\to S"$
\sn
\item "{$(b)$}"   if $\ell = 1$ then $q \Vdash$ ``for some $\eta \in
\text{ lim}(T)$ we have $N_\eta
[{\underset\tilde {}\to G_{\Bbb P}}] \cap \omega_1 = 
N_{\langle \rangle} \cap \omega_1$ and for every $ \theta \in
{\underset\tilde {}\to {\bold S}}$ we have
sup$(N_\eta[{\underset\tilde {}\to G_{\Bbb P}}] \cap \theta) =
\sup(N_\eta \cap \theta)$ where $N_\eta = \cup\{N_{\eta \restriction
\ell}:\ell < \omega\}$ and $\eta$ is not necessarily from $\bold V$"
\sn
\item "{$(c)$}"  if $\ell = 2$ then $q \Vdash ``N_{\langle \rangle} 
[{\underset\tilde {}\to G_{\Bbb P}}]$ is $({\underset\tilde {}\to {\bold S}}
\backslash {\underset\tilde {}\to \lambda'})$-strictly 
$(\Bbb I^{[\underset\tilde {}\to \lambda]},\bold W)$-suitable and \nl
sup$(N_{<>}[{\underset\tilde {}\to G_{\Bbb P}}] 
\cap \theta) = \sup(N_{<>} \cap
\theta)$ for every $\theta \in \underset\tilde {}\to S$ (in particular
$\aleph^{\bold V}_1$)" where
${\underset\tilde {}\to \lambda'} = 
\aleph_2^{{\bold V}[{\underset\tilde {}\to G_{\Bbb P}}]}$.
\ermn
2) If we omit $\ell$ we mean $\ell=0$. \nl
If $\bold W = \omega_1$ we may omit it.  We write $*$ instead of
${\underset\tilde {}\to {\bold S}}$ if \newline
${\underset\tilde {}\to {\bold S}} =
\{\lambda:\bold V[{\underset\tilde {}\to G_{\Bbb P}}] \models \lambda \in 
\text{ UReg}^{\bold V[G_{\Bbb P}]}\}$.  
If we omit ${\underset\tilde {}\to {\bold S}}$ we mean
$\{\aleph^{\bold V}_1\}$. \nl
3)  The forcing notion $\Bbb P$ 
satisfies UP$^4_{\underset\tilde {}\to \kappa,
\underset\tilde {}\to \lambda}(\Bbb I,{\underset\tilde {}\to {\bold S}},
\bold W)$ when $\underset\tilde {}\to \kappa,\underset\tilde {}\to \lambda$ 
are $(\Bbb P,\le_{\text{pr}})$-names of regular $\bold V$-cardinals and for 
some $x$ we have: \underbar{if} $\chi$ is large enough and 
$\bar N = \langle N_\eta:\eta \in (T,\bold I) \rangle$ is an
${\underset\tilde {}\to {\bold S}}$-strictly $(\Bbb I,\bold W)$-suitable 
tree of models for $(\chi,x)$ and $p \in \Bbb P \cap N_{\langle \rangle}$ and
$\langle \underset\tilde {}\to \kappa,\underset\tilde {}\to
\lambda,\Bbb P,\Bbb I \rangle 
\in N_{\langle \rangle}$, \underbar{then} for some 
$q,\underset\tilde {}\to T$ we have:
\mr
\item "{$(a)$}"  $p \le_{\text{pr}} q \in \Bbb P$
\sn
\item "{$(b)$}"  $q$ is $(\bar N \restriction \underset\tilde {}\to T,
\underset\tilde {}\to \kappa,\underset\tilde {}\to \lambda,\Bbb P)$-semi$_4$ 
generic (see below). 
\ermn
3A) If $\underset\tilde {}\to \kappa = \infty$ we can replace
$\underset\tilde {}\to T$ by $\underset\tilde {}\to \eta$ such that
$\underset\tilde {}\to T = \{\underset\tilde {}\to \eta \restriction n:n <
\omega\}$ (see below) so $\Vdash ``\underset\tilde {}\to \eta \in
\text{ lim}(T)"$ and then in clause (b) write 
$N_{\underset\tilde {}\to \eta}$ instead of $\bar N \restriction
\underset\tilde {}\to T$.  We then may omit $\kappa$. 
We may $\underset\tilde {}\to \lambda$ if $\underset\tilde {}\to
\lambda = \underset\tilde {}\to \kappa(\Bbb P)$, 
see Definition \scite{1.28}(5).
\nl
3B) We say that $q$ is $(\bar N \restriction \underset\tilde {}\to T,
\underset\tilde {}\to \kappa,\underset\tilde {}\to \lambda,\Bbb P)$-semi$_4$ 
generic  where $\bar N,\underset\tilde {}\to T,\underset\tilde {}\to \kappa,
\underset\tilde {}\to \lambda$ is as above \ub{if}:

$$
\align
q \Vdash_{\Bbb P} ``&(i) \quad \underset\tilde {}\to T \text{ is a subtree of }
T \text{ (so } \underset\tilde {}\to T \subseteq T \text{ is closed under} \\
  &\qquad \text{initial segments } <> \in \underset\tilde {}\to T,\eta \in
\underset\tilde {}\to T \Rightarrow \text{ Suc}_{\underset\tilde {}\to T}
(\eta) \ne \emptyset) \\
  &(ii) \quad N_\eta[{\underset\tilde {}\to G_{\Bbb P}}] \cap \omega_1 = 
N_{\langle \rangle} \cap \omega_1 \text{ for } \eta \in
\underset\tilde {}\to T \\
  &(iii) \quad \bar N[{\underset\tilde {}\to G_{\Bbb P}}] \restriction
\underset\tilde {}\to T \text{ has }
(\underset\tilde {}\to \lambda,\underset\tilde {}\to \lambda) \text{-covering
which means: if} \\
  &\qquad \qquad \underset\tilde {}\to \eta \text{ is an } \omega
\text{-branch of } \underset\tilde {}\to T \text{ and }
y \in N_{\underset\tilde {}\to \eta}[{\underset\tilde {}\to G_{\Bbb P}}] \\
  &\qquad \qquad \text{ is a set of } < 
\underset\tilde {}\to \lambda[{\underset\tilde {}\to G_{\Bbb P}}] \text{ ordinals} \\
  &\qquad \qquad 
\text{(if } \underset\tilde {}\to \lambda[{\underset\tilde {}\to
G_{\Bbb P}}] 
\text{ is not a cardinal, this means } \le |\underset\tilde {}\to \lambda
[{\underset\tilde {}\to G_{\Bbb P}}]|) \\
  &\qquad \qquad 
\text{\ub{then} for some } A \in \bold V \cap \dbcu_{\ell < \omega} 
N_{{\underset\tilde {}\to \eta} \restriction \ell} \text{ we have} \\
  &\qquad \qquad |A|^{\bold V} < 
\underset\tilde {}\to \lambda [{\underset\tilde {}\to G_{\Bbb P}}]
\text{ and } y \subseteq A \\
  &(iv) \quad \langle N_\eta[{\underset\tilde {}\to G_{\Bbb P}}]:
\eta \in (\underset\tilde {}\to T,\bold I) \rangle
\text{ is a strictly} \\
  &\qquad \qquad \Bbb I^{[\underset\tilde {}\to \kappa[
{\underset\tilde {}\to G_{\Bbb P}}]]} \text{-suitable tree of models}".
\endalign
$$
\mn
4)  We define UP$^3_{\underset\tilde {}\to \kappa,
\underset\tilde {}\to \lambda}(\Bbb I,{\underset\tilde {}\to {\bold S}},
\bold W)$ similarly replacing clause (b) by
the weaker
\mr
\item "{$(b)^-$}"  $q$ is $(N_{\underset\tilde {}\to \eta},
\underset\tilde {}\to \kappa,\underset\tilde {}\to \lambda,\Bbb P)$-semi$_3$
-generic, (see below).
\ermn
4A) If $\underset\tilde {}\to \kappa = \infty$ we can replace
$\underset\tilde {}\to T$ by $\underset\tilde {}\to \eta$ such that
$\underset\tilde {}\to T = \{\underset\tilde {}\to \eta \restriction n:n <
\omega\}$ so $\Vdash_{\Bbb P} 
``\underset\tilde {}\to \eta \in \text{ lim}(T)"$
and replace $\bar N \restriction \underset\tilde {}\to T$ by
$N_{\underset\tilde {}\to \eta}$.  We may omit $\underset\tilde {}\to
\lambda$ if it is $\infty$ (but see \scite{4.7}(0)).
We then may omit $\underset\tilde {}\to \kappa$ if it is $\infty$, too. \nl
4B) $q$ is $(N,\underset\tilde {}\to \kappa,
\underset\tilde {}\to \lambda,\Bbb P)$-semi$_3$-generic is defined
as in (4) only replacing clause (iii) in $\boxtimes$ of (3B) by
\mr
\item "{$(iii)^-$}"  $\bar N[{\underset\tilde {}\to G_{\Bbb P}}]
\restriction \underset\tilde {}\to T$ has $(\underset\tilde {}\to \lambda,
1)$-covering which means for every 
$y \in \bold V \cap N_{\underset\tilde {}\to \eta}
[{\underset\tilde {}\to G_{\Bbb P}}]$
for some $A \in \bold V \cap 
N_{\underset\tilde {}\to \eta}$ we have
$|A|^V < \underset\tilde {}\to \lambda
[{\underset\tilde {}\to G_{\Bbb P}}]$ and $y \in A$, recalling
$N_{\underset\tilde {}\to \eta}[{\underset\tilde {}\to G_{\Bbb P}}] =
\cup\{N_{\underset\tilde {}\to \eta \restriction
\ell}[{\underset\tilde {}\to G_{\Bbb P}}]:\ell < \omega\}"$.
\ermn
5) We allow to use ${\underset\tilde {}\to {\Bbb I}}$, a $\Bbb P$-name of an 
element of $\bold V$ as above \ub{if}:
\mr
\item "{$(a)$}"  it is decidable purely
\sn
\item "{$(b)$}"  if $q \in \Bbb P$ 
decides ${\underset\tilde {}\to {\Bbb I}} =
\Bbb I$ then $\Bbb P_{\ge q}$ satisfies 
UP$^\ell(\Bbb I,{\underset\tilde {}\to {\bold S}},\bold W)$.
\ermn
6) We say that $\Bbb P$ satisfies the UP$^5_{\underset\tilde {}\to \kappa,
\underset\tilde {}\to \lambda}(\Bbb I,{\underset\tilde {}\to {\bold S}},
\bold W)$ \ub{iff}
\mr
\item "{$(a)$}"  $\underset\tilde {}\to \kappa$ and 
$\underset\tilde {}\to \lambda$ are $\Bbb P$-names of $\bold
V$-cardinals such that $\Bbb P$
satisfies the $\underset\tilde {}\to \kappa$-c.c. purely locally
\sn
\item "{$(b)$}"  ${\underset\tilde {}\to {\Bbb I}}$ is a $\Bbb P$-name 
of a set which belongs to $\bold V$, it is a set of ideals and
${\underset\tilde {}\to {\Bbb I}}$ is decidable purely
\sn
\item "{$(c)$}"  is $\kappa$-complete if 
$\nVdash_{\Bbb P} ``\neg(\underset\tilde {}\to \kappa = \kappa \and I \in
{\underset\tilde {}\to {\Bbb I}})"$,
\sn
\item "{$(d)$}"  if $p \in \Bbb P$ forces ${\underset\tilde {}\to {\Bbb I}} =
\Bbb I,\Bbb I \subseteq \Bbb I',
\underset\tilde {}\to \lambda = \lambda,\underset\tilde {}\to \kappa =
\kappa$ and $\Bbb I' \backslash \Bbb I$ is a set of $\lambda$-complete ideals
\ub{then} for some $x$ 
{\roster
\itemitem{ $\boxtimes$ }  if $\bar N = \langle N_\eta:\eta \in (T,\bold I)
\rangle$ is strictly $\Bbb I'$-suitable, $x \in N_{<>}$, then for some
$q,\underset\tilde {}\to T$ we have: $p \le_{\text{pr}} q \in \Bbb P$
and $q$ is $(\bar N \restriction \underset\tilde {}\to T,\underset\tilde
{}\to \kappa,\underset\tilde {}\to \lambda,\Bbb P)$-semi$_5$-generic,
(see below).
\endroster}
\ermn
7) We define $\Bbb P$ satisfies UP$^5_{\underset\tilde {}\to \kappa}
({\underset\tilde {}\to {\Bbb I}},{\underset\tilde {}\to {\bold
S}},\bold W)$ as in part (6) but restrict
ourselves to $\Bbb I' = \Bbb I$.  [others?] \nl
8) Assume $\bar N = \langle N_\eta:\eta \in (T,\bold I) \rangle$ and
$\Bbb P \in N_{<>}$ satisfies UP$^\ell_{\underset\tilde {}\to \kappa}
({\underset\tilde {}\to {\Bbb I}},\bold W)$ and $\langle \Bbb P,
{\underset\tilde {}\to {\Bbb I}},\bold W,\ldots \rangle \in N_{<>}$.  We say
$q$ is $(\bar N,\underset\tilde {}\to \kappa,\Bbb P)$-semi$_5$-generic (for
$\bar N$ when not understood from the context) if:

$$
\align
q \Vdash ``&\text{for some } \underset\tilde {}\to T \text{ we have }
(T,{\underset\tilde {}\to {\Bbb I}}) \le^{\underset\tilde {}\to \kappa}
({\underset\tilde {}\to T'},\bold I), \text{ see \scite{2.4}, 
clause (f) and} \\
  &\eta \in {\underset\tilde {}\to T'} \Rightarrow N_\eta
[{\underset\tilde {}\to G_{\Bbb P}}] \cap \omega_1 
= N_{<>} \cap \omega_1 \text{ and} \\
  &\eta \in T' \and \mu \in {\underset\tilde {}\to {\bold S}}
\Rightarrow \sup(N_{\underset\tilde {}\to \eta}[G_{\Bbb P}] \cap \mu) = \sup
(N_{\underset\tilde {}\to \eta} \cap \mu)".
\endalign
$$
\mn
9) We write UP$^3({\underset\tilde {}\to {\Bbb I}},\bold S,\bold W)$ for
UP$^3_{\underset\tilde {}\to \kappa}
({\underset\tilde {}\to {\Bbb I}},\bold S, \bold W)$ where
$\underset\tilde {}\to \kappa$ is $\underset\tilde {}\to \kappa(\Bbb
P)$ see \scite{1.28}(5), \scite{1.29}(1).
\enddefinition
\bigskip

\definition{\stag{4.6A} Definition}[?]  We call $\Bbb I$ to be a name if
it is a name of an old family of ideals purely decidable.
\enddefinition
\bigskip

\proclaim{\stag{4.7} Claim}  0) In Definition \scite{4.6}(3)-(6), if
$\underset\tilde {}\to \lambda \ge \underset\tilde {}\to \kappa(\Bbb
P)$, \ub{then} the demand concerning $\underset\tilde {}\to \lambda$
(i.e., clause (iii) of \scite{4.6}(3B) holds trivially (as increasing
$p$ purely, $p \Vdash ``\underset\tilde {}\to \lambda = \lambda"$ and
$\Bbb P_{\ge p}$ satisfies the $\lambda$-c.c). \nl
1) If $\Bbb Q$ satisfies 
UP$^4_{\underset\tilde {}\to \kappa,\underset\tilde {}\to \lambda}(\Bbb I,
{\underset\tilde {}\to {\bold S}},\bold W)$, \ub{then} it satisfies
${\text{\rm UP\/}}^3_{\underset\tilde {}\to \kappa,
\underset\tilde {}\to \lambda}(\Bbb I,
{\underset\tilde {}\to {\bold S}},\bold W)$.  If $\Bbb Q$ satisfies
${\text{\rm UP\/}}^3_{\underset\tilde {}\to \kappa,
\underset\tilde {}\to \lambda}(\Bbb I,
{\underset\tilde {}\to {\bold S}},\bold W)$, \ub{then} $\Bbb Q$
satisfies ${\text{\rm UP\/}}^1
(\Bbb I,{\underset\tilde {}\to {\bold S}},\bold W)$ and
${\text{\rm UP\/}}^2_{\underset\tilde {}\to \kappa}(\Bbb I,\bold W)$.
If $\Bbb Q$ satisfies ${\text{\rm UP\/}}^2_{\underset\tilde {}\to \kappa}(\Bbb
I,{\underset\tilde {}\to {\bold S}},\bold W)$ or ${\text{\rm UP\/}}^1
(\Bbb I,\bold S,\bold W)$ then it satisfies ${\text{\rm UP\/}}^0
(\Bbb I,\bold S,\bold W)$.  
If $\ell \in \{3,4\}$ and $\Bbb Q$ satisfies 
${\text{\rm UP\/}}^\ell_{\underset\tilde {}\to \kappa,
\underset\tilde {}\to \lambda}
(\Bbb I,{\underset\tilde {}\to {\Bbb I}},\bold W)$ and
$\underset\tilde {}\to \kappa_1 \ge \underset\tilde {}\to \kappa,
{\underset\tilde {}\to \lambda_1} \ge \underset\tilde {}\to \lambda,[\ell =
4 \Rightarrow {\underset\tilde {}\to \lambda_1} =
\underset\tilde {}\to \lambda]$, \ub{then} $\Bbb Q$ satisfies
${\text{\rm UP\/}}^\ell_{{\underset\tilde {}\to \kappa_1},
{\underset\tilde {}\to \lambda_1}}(\Bbb I,
{\underset\tilde {}\to {\bold S}},\bold W)$. \nl
1A) If $\Bbb Q$ satisfies ${\text{\rm UP\/}}^\ell(\Bbb I,
{\underset\tilde {}\to {\bold S}},\bold W)$, \ub{then} it satisfies
${\text{\rm UP\/}}^0(\Bbb I,{\underset\tilde {}\to {\bold S}},\bold W)$
which implies ``$\Bbb Q$ has pure $\aleph_1$-decidability", see
Definition \scite{1.25}(2). \newline
2) The forcing notion $\Bbb Q$ satisfies ${\text{\rm UP\/}}^\ell(\Bbb I,
{\underset\tilde {}\to {\bold S}},\bold W)$ \ub{iff} its completion 
(i.e., $\Bbb Q^1$, or equivalently its completion
to a complete Boolean algebra) satisfies it assuming $\le_{\text{pr}} 
= \le$. \nl
3) If $\Bbb Q$ satisfies ${\text{\rm UP\/}}(\Bbb I,*,\bold W)$, 
(i.e., see \scite{4.6}(2)) and $\Bbb I$ is
$\mu^+$-complete (e.g., $\Bbb I = \emptyset$), \underbar{then} any ``new"
countable set of ordinals $< \mu$ is included in an ``old" countable set of
ordinals; i.e., one from $\bold V$. \newline
4) $\Bbb Q$ satisfies ${\text{\rm UP\/}}(\emptyset,*)$ 
\ub{iff} $\Bbb Q$ is purely proper (see Definition \scite{1.26A}(1)). \nl
5) $\Bbb Q$ satisfies ${\text{\rm UP\/}}(\emptyset,\{\aleph_1\})$ \ub{iff} 
$\Bbb Q$ is purely semiproper (see Definition \scite{1.26A}(2))
\footnote{that is: if $\Bbb Q \in N
\prec ({\Cal H}(\chi),\in),N$ is countable, $p \in \Bbb Q \cap N$,
\ub{then} for some $q$ we have $p \le_{pr} q$ and $q \Vdash_{\Bbb Q}
``\underset\tilde {}\to \tau \in N"$ for every $\Bbb Q$-name
$\underset\tilde {}\to \tau \in N$ of a countable ordinal}. 
\newline
6) If $\Bbb Q$ satisfies ${\text{\rm UP\/}}
(\Bbb I,{\underset\tilde {}\to {\bold S}},\bold W)$
and $\Bbb I \subseteq \Bbb I_1,{\underset\tilde {}\to {\bold S}_1} \subseteq
{\underset\tilde {}\to {\bold S}}$ and $\bold W_1 \subseteq \bold W$ 
\ub{then} $\Bbb Q$ satisfies ${\text{\rm UP\/}}^\ell
(\Bbb I_1,{\underset\tilde {}\to {\bold S}_1},
\bold W_1)$. \newline
7) In Definition \scite{4.1}, if $\Bbb P$ satisfies the $\kappa$-c.c. (e.g.
$\kappa = |\Bbb P|^+$) \underbar{then}:
\mr
\item "{$(a)$}"   we can replace
${\underset\tilde {}\to {\bold S}}$ by any set
${\underset\tilde {}\to {\bold S}'}$ of uncountable regular cardinals
of $\bold V$,
such that $\Vdash_{\Bbb P} ``{\underset\tilde {}\to {\bold S}} \cap \kappa =
{\underset\tilde {}\to {\bold S}'} \cap \kappa"$. 
\ermn
8) In Definition \scite{4.6} (in all the variants), \ub{if} we 
demand ``for $\chi$ large enough, for
some $x \in {\Cal H}(\chi)$, for every $\bar N$ such that $x \in
N_{\langle \rangle}$ and $\ldots$" we get an equivalent
definition. \nl
9) In Definition \scite{4.6} we can use weak $\Bbb I$-tagged trees,
i.e. we get with this an equivalent definition.
\endproclaim
\bigskip

\demo{Proof}  1), 2)  Trivial. \newline
3)  Straightforward. \newline  
4) Use \scite{4.8} below. \newline
5),6)  If $\Bbb I = \emptyset$, then every 
$N \prec ({\Cal H}(\chi),\in,<^*_\chi)$ is a $\Bbb I$-model. \newline
7) Easy. \newline
8) Check the Definition. \nl
9)  As in \cite{Sh:f}. \nl
10) The ``weak" version allows more trees of models so apparently is a
stronger condition, but by \scite{4.5B}(4) it is equivalent.
${{}}$ \hfill$\square_{\scite{4.7}}$\margincite{4.7}
\enddemo
\bigskip

\demo{\stag{4.9} Conclusion}  If $\Bbb P$ satisfies UP$^\ell(\Bbb I,
{\underset\tilde {}\to {\bold S}},\bold W)$ and 
${\underset\tilde {}\to {\bold S}}$ is as in \scite{4.1}$(*)(b)$ (or
${\underset\tilde {}\to {\bold S}} = \{\aleph_1\})$ (recall that this 
notation
implies $\Bbb I$ is $\aleph_2$-complete, $\aleph_1 \in
{\underset\tilde {}\to {\bold S}}$ and $\bold W \subseteq \omega_1$ 
stationary)
\underbar{then} $\Vdash_P ``\bold W$ is stationary".  Moreover, if $\bold W'
\subseteq \bold W$ is stationary then also $\Vdash_P ``\bold W'$ is a
stationary subset of $\omega_1"$.
\enddemo
\bigskip

\demo{Proof}  The ``moreover" fact is by \scite{4.7}(7) (i.e., monotonicity
in $\bold W$).

Assume that $p \Vdash ``\underset\tilde {}\to C$ is a club of $\omega_1$ and
$\underset\tilde {}\to C \cap \bold W = \emptyset"$.  By \scite{4.8} we can
find an $\aleph_1$-strictly $(\Bbb I,{\underset\tilde {}\to {\bold S}},
\bold W)$-suitable tree of models
$\langle N_\eta:\eta \in (T,\bold I)\rangle$ with $\underset\tilde {}\to C,p
\in N_{\langle \rangle}$.  Let $\delta = N_{<>} \cap \omega_1$, so $\delta \in
\bold W$.  By UP$^\ell(\Bbb I,{\underset\tilde {}\to {\bold S}},\bold W)$ we 
can find
a condition $q$ as in Definition \scite{4.1} in particular 
$p \le_{\text{pr}} q$.
Clearly $q \Vdash ``N_{\langle \rangle}[G] \cap \omega_1 = \delta"$ and,
trivially $p \Vdash_P ``\underset\tilde {}\to C$ is unbounded in
$N_{\langle \rangle}[G] \cap \omega_1"$ hence $p \Vdash ``N_{\langle \rangle}
[G] \cap \omega_1 \in \underset\tilde {}\to C"$.  So $q \Vdash_Q``\delta \in 
\underset\tilde {}\to C \cap \bold W"$, contradiction.
\hfill$\square_{\scite{4.9}}$\margincite{4.9}
\enddemo
\bigskip

\remark{\stag{4.10} Remark}  Usually we assume $\Bbb I,\bold S$ satisfies
\scite{4.1}(*)(a) + (c), ${\underset\tilde {}\to {\bold S}} = \{\aleph_1\}$
is the main case.
\endremark
\bigskip

\remark{\stag{4.11} Remark}  1) From the proof of \scite{4.8} we can 
conclude that in \scite{4.1}; we can replace ``$\bold S$-strictly 
$(\Bbb I,\bold W)$-suitable, $N_\eta \cap \omega_1 = \delta \in \bold W"$ by
$``(\Bbb I,\bold S,\bold W)$-suitable", and then the condition $q$ will be
$N_{\langle \rangle}$-semi-generic. \nl
2) As at present ${\underset\tilde {}\to {\bold S}} = \{\aleph_1\}$ seem to
suffice, we shall use only it for notational simplicity.
\endremark
\newpage

\head {\S5 An iteration theorem for UP} \endhead  \resetall \sectno=5
 % \resetall 
\bigskip

\proclaim{\stag{5.1} Claim}  1) If $\bar N = \langle N_\eta:\eta \in (T,
\bold I) \rangle$ is a tagged tree of models for $(\chi,\langle x,
\Bbb P \rangle),\Bbb P$ a forcing notion and $\Bbb P \in N_{\langle \rangle}$, \ub{then}
$\Vdash_{\Bbb P} ``\langle N_\eta[{\underset\tilde {}\to G_{\Bbb P}}]:
\eta \in (T,\bold I) \rangle$ is a tagged tree
of models for $(\chi,\langle x,\Bbb P,G \rangle)"$. \newline
2) If in addition $\Bbb P$ satisfies the $\kappa$-c.c. and 
$\Bbb I \in N_{\langle \rangle}$ is $\kappa$-closed (see Definition 
\scite{3.13}(2)) and $\bar N$ is $\Bbb I$-suitable, \ub{then}
\mr
\item "{$(*)$}" $\Vdash_{\Bbb P} ``\langle N_\eta[{\underset\tilde
{}\to G_{\Bbb P}}]:\eta \in (T,\bold I) \rangle \text{ is } \Bbb I \text{-suitable}"$.
\ermn
3) If in part (2) assume in addition that 
$\underset\tilde {}\to \kappa,{\underset\sim {}\to {\Bbb I}}$ 
to be $\Bbb P$-names of objects from $\bold V$ such that
${\underset\tilde {}\to {\Bbb I}}$ is purely decidable and 
$\Bbb P$ satisfies the local $\underset\tilde {}\to \kappa$-c.c. purely,  
\ub{then} for every $p \in N_{<>} \cap \Bbb P$ 
there is $q,p \le_{\text{pr}} q \in N_{<>} \cap \Bbb P$ 
forcing $(*)$ above.  If ${\underset\tilde {}\to {\Bbb I}}$ is purely
decidable and $\Bbb P$ is locally $\underset\tilde {}\to \kappa$-c.c. 
purely then we
can find $q$ satisfying $p \le_{\text{pr}} q \in N_{<>} \cap \Bbb P$ and forcing $(*)$.
\endproclaim
\bigskip

\demo{Proof}  1) Straight. \newline
2) So $\Bbb P$ satisfies the $\kappa$-c.c. and let
$G \subseteq \Bbb P$ be generic over $\bold V$.   Now from Definition 
\scite{4.1}(1), clearly $\langle N_\eta[G]:\eta \in (T,\bold I) \rangle$
satisfies clauses (a) - (d), so it is enough to check clause (e)$^-$ of
Definition \scite{4.1}(3).  So let $I \in \Bbb I \cap
N_\eta[G]$ where $\eta \in T$.  Hence there is $\underset\tilde {}\to I \in
N_\eta$ such that $\underset\tilde {}\to I$ is a $\Bbb P$-name and
$\underset\tilde {}\to I[G] = I$.  Let $\Bbb I' = \{J \in \Bbb I:
\text{for some } p \in \Bbb P$ we have $p \Vdash_{\Bbb P} ``\underset\tilde {}\to I = J"\}$.
So $\Bbb I'$ belongs to $\bold V$ and 
is a subset of $\Bbb I$ of cardinality $< \kappa$ and
$\Bbb I' \in N_\eta$ hence there is $I^* \in \Bbb I$ such that
$(\forall J)(J \in \Bbb I' \Rightarrow J \le_{RK} I^*)$, so \wilog \,
$I^* \in N_\eta$, hence as $N$ is $\Bbb I$-suitable clearly 
$\{\nu:\eta \triangleleft \nu \in T \text{ and } 
I^* \le_{RK} \bold I_\nu\}$ contains a front of $T[\eta]$.  
Hence in $\bold V[G]$, the set $\{\nu \in T:I \le_{\text{RK}} \bold I_\nu\}$
contains a front of $T^{[\eta]}$ as required.  \nl
3) Left to the reader.  \hfill$\square_{\scite{5.1}}$\margincite{5.1}
\enddemo
\bn
The point of the following claim is to get more from some UP$^\ell$ 
than seems on the surface; our aim is to help iterating.

\proclaim{\stag{5.2} Claim}  1) Assume 
$\ell \in \{3,4\}$ and the forcing notion
$\Bbb Q$ satisfies ${\text{\rm UP\/}}^\ell_{\underset\tilde {}\to \kappa,
\underset\tilde {}\to \lambda}({\underset\tilde {}\to {\Bbb I}},\bold W)$ and
$\underset\tilde {}\to \kappa,\underset\tilde {}\to \lambda,
{\underset\tilde {}\to {\Bbb I}},{\underset\tilde {}\to {\Bbb I}^+}$ are
$\Bbb Q$-names with pure decidability and 
$\Vdash ``{\underset\tilde {}\to {\Bbb I}^+} \backslash
{\underset\tilde {}\to {\Bbb I}}$ is 
$\underset\tilde {}\to \lambda$-complete".
\ub{Then} the forcing notion $\Bbb Q$ satisfies the 
${\text{\rm UP\/}}^\ell_{\underset\tilde {}\to \kappa,
\underset\tilde {}\to \lambda}
({\underset\tilde {}\to {\Bbb I}^+},\bold W)$. \nl 
2)  Suppose 
\mr
\item "{$(a)$}"  $\Bbb I_0,\Bbb I_1,\Bbb I_2,\Bbb I_3$ are sets of quasi-order
ideals, $\Bbb I_1 \subseteq \Bbb I_0 \subseteq \Bbb I_2,\Bbb I_3 = \Bbb I_1
\cup (\Bbb I_2 \backslash \Bbb I_0),\Bbb I_2 \backslash \Bbb I_0 =
\Bbb I_3 \backslash I_1$ is
$\kappa$-closed, $\kappa \le \lambda$ and $\Bbb I_2
\backslash \Bbb I_0$ is $\lambda$-complete
\sn
\item "{$(b)$}"  $\bar N = \langle N_\eta:\eta \in (T^*,\bold I^*) \rangle$
is a strict truely $\Bbb I_2$-suitable tree of models (for $\chi$ and $x =
\langle \Bbb Q,\Bbb I_0,\Bbb I_1,\Bbb I_2,\kappa,\lambda \rangle$
\sn
\item "{$(c)$}"  $p \in N_{<>},
\ell \in \{3,4\}$ and $\Bbb Q_{\ge p}$ satisfies the $\lambda$-c.c.
\sn
\item "{$(d)$}"  $\varphi(-)$ is a property with 
$\bar N,G_{\Bbb Q}$ as parameters (and possibly others)
\sn
\item "{$(e)$}"  for any $T'$ a subtree of $T^*$ such that $\langle N_\eta:
\eta \in (T',\bold I) \rangle$ is 
a truely $\Bbb I_0$-suitable tree of models 
there are $q = q_{T'},\underset\tilde {}\to T = 
\underset\tilde {}\to T(T')$ such that
{\roster
\itemitem{ $(i)$ }  $p \le_{\text{pr}} q \in \Bbb Q$
\sn
\itemitem{ $(ii)$ }  $(T',\bold I^*) \le (\underset\tilde {}\to T,\bold I^*)$
\sn
\itemitem{ $(iii)$ }  $q \Vdash_{\Bbb Q}$ ``$\langle N_\eta
[{\underset\tilde {}\to G_{\Bbb Q}}]:\eta \in (\underset\tilde {}\to T,\bold I)
\rangle$ is a truely $\Bbb I_1$-suitable tree of models and for every $\eta \in 
\underset\tilde {}\to T$ we have
\sn
\itemitem{ ${{}}$ }  $(\alpha) \quad 
N_\eta[{\underset\tilde {}\to G_{\Bbb Q}}] \cap \omega_1 = N_{<>} \cap \omega_1$ and
\sn
\itemitem{ ${{}}$ }  $(\beta) \quad \varphi[\eta]$
\sn
\itemitem{ ${{}}$ }  $(\gamma) \quad$ if $\ell = 3,y \in
N_\eta[{\underset\tilde {}\to G_{\Bbb Q}}]$ is a member of $\bold V$ then \nl

$\qquad \quad \{\nu:\eta
\triangleleft \nu \in \underset\tilde {}\to T$, and for some $A \in N_\nu,A$
a set of cardinality $< \lambda$ \nl

$\qquad \quad$ we have $y \in A\}$ contains a front of
$T^{[\eta]}$
\sn
\itemitem{ ${{}}$ }  $(\delta) \quad$ if $\ell = 4$ and $y \in N_\eta
[{\underset\tilde {}\to G_{\Bbb Q}}]$ is a set of $< \lambda$ members
of $\bold V$ then
\nl

$\qquad \quad \{\nu:\eta \triangleleft \nu \in \underset\tilde {}\to T$ 
and for some $A \in N_\eta$ a set of cardinality $< \lambda$ \nl

$\qquad \quad$ we have $y \subseteq A\}$
contains a front of $\underset\tilde {}\to T^{[\eta]}$.
\endroster}
\ermn
\ub{Then} there are $q,\underset\tilde {}\to T$ such that:
\mr
\widestnumber\item{$(iii)$}
\item "{$\circledast(i)$}"  $p \le_{\text{pr}} q \in \Bbb Q$
\sn
\item "{$(ii)$}"  $(T^*,\bold I^*) \le (\underset\tilde {}\to T,\bold I^*)$
\sn
\item "{$(iii)$}"  $q \Vdash_{\Bbb Q} ``\langle 
N_\eta[{\underset\tilde {}\to G_{\Bbb Q}}]:\eta \in (\underset\tilde {}\to T,
\bold I)\rangle$ is a truely $\Bbb I_3$-suitable tree of models and
for every $\eta \in \underset\tilde {}\to T$ we have
\sn
\item "{${{}}$}"  $(\alpha) \quad \quad N_\eta
[{\underset\tilde {}\to G_{\Bbb Q}}] \cap \omega_1 
= N_{<>} \cap \omega$
\sn
\item "{${{}}$}"   $(\beta) \quad \quad \models \varphi(\eta)$
\sn
\item "{${{}}$}"  $(\gamma),(\delta) \quad$  as in clause (iii)
of (e) above.
\ermn
3) In part (2), if $\Bbb Q$ satisfies the $\lambda$-c.c., 
\ub{then} we can omit $(\gamma),(\delta)$ in their two
appearances as they follow. \nl
4) In part (2), we can replace ``truely $\Bbb I_\ell$-suitable" by
``weakly $\Bbb I_\ell$-suitable".
\endproclaim
\bigskip

\remark{\stag{5.2a} Remark}  In part (2) clause (e) we can restrict $T'$ to
those needed.
\endremark
\bigskip

\demo{Proof}  1) As in the definition of UP$^\ell_{\underset\tilde
{}\to \kappa,\underset\tilde {}\to \lambda}(\Bbb I^+,\bold W)$ let
$\bar N = \langle N_\eta:\eta \in (T,\Bbb I) \rangle$ be a
strict $\Bbb I^+$-suitable tree of models for $\chi$ and $x \equiv
\langle \Bbb Q,
{\underset\tilde {}\to {\Bbb I}},{\underset\tilde {}\to {\Bbb I}^+},
\bold W,\underset\tilde {}\to \kappa,
\underset\tilde {}\to \lambda \rangle$ and $p \in N_{<>}$.  
We can find $p' \in N_{<>},p \le_{\text{pr}} p' \in \Bbb Q$ which forces 
$\underset\tilde {}\to \kappa,
\underset\tilde {}\to \lambda,{\underset\tilde {}\to {\Bbb I}},
{\underset\tilde {}\to {\Bbb I}^+}$ to be say $\kappa,\lambda,\Bbb I,\Bbb I^+$
respectively.
Now we can apply part (2) of the claim
with $\bar N,\Bbb I,\Bbb I^{[\kappa]},\Bbb I^+,\Bbb I^{[\kappa]} \cup
(\Bbb I^+ \backslash \Bbb I),x=x$ here standing for $\bar N,\Bbb I_0,\Bbb I_1,
\Bbb I_2,\Bbb I_3,\varphi$ there. \nl
2) Let ${\Cal T} = \{T:T$ is a subtree of $T^*$ such that $\bar N \restriction
T$ is a truely $\Bbb I_0$-suitable tree of models$\}$ or just 
${\Cal T} = \{T:T$ a subtree of
$T^*$ such that ($<> \in T,\eta \in T \Rightarrow \emptyset \ne
\text{ Suc}_T(\eta) \subseteq \text{ Suc}_{T^*}(\eta),T$ closed under initial
segments and): if $\eta \in \text{ split}(T^*,\bold I^*) \and \bold I_\eta
\in \Bbb I_0$ then Suc$_T(\eta) \in \bold I^+_\eta$ and if $\eta \notin
\text{ split}(T^*,\bold I^*) \vee \bold I_\eta \notin \Bbb I_0$ then
$|\text{Suc}_T(\eta)| = 1\}$. \nl
For any $T \in {\Cal T}$ by assumption (e) there are $q_T,
{\underset\tilde {}\to T^{\Bbb Q}}[T]$ as required there.
\sn
We shall show that some such $q_T$ is as required.  We define a $\Bbb Q$-name
${\underset\tilde {}\to T^\oplus}$ as follows:

for $G_{\Bbb Q} \subseteq \Bbb Q$ generic over $\bold V$ we let

$$
\align
T^\otimes = {\underset\tilde {}\to T^\otimes}[G_{\Bbb Q}] = \{\eta \in T^*:&N_\eta
[G] \cap \omega_1 = N_{<>} \cap \omega_1 \\
  &\text{and } \ell \le \ell g(\eta) \Rightarrow \varphi(\eta \restriction
\ell)\}.
\endalign
$$
\mn
Clearly
\mr
\item "{$(*)_1$}"  $q_T 
\Vdash_{\Bbb Q} ``{\underset\tilde {}\to T^{\Bbb Q}}[T]
\subseteq T^\otimes"$ for every $T \in {\Cal T}$.
\ermn
Working in $\bold V[G_{\Bbb Q}]$ we define a depth function Dp function from
$T^\otimes$ to Ord $\cup \{\infty\}$ by defining for any ordinal $\alpha$
when $Dp(\eta) \ge \alpha$ as follows:
\mr
\item "{$\boxtimes$}"  Dp$(\eta) \ge \alpha$ \ub{iff} the following
conditions hold:
{\roster
\itemitem{ $(\alpha)$ }  $\eta \in T^\otimes$
\sn
\itemitem{ $(\beta)$ }  for every $\beta < \alpha$ there is $\nu \in
\text{ Suc}_{T^\otimes}(\eta)$ such that $Dp(\nu) \ge \beta$
\sn
\itemitem{ $(\gamma)$ }  if $\beta < \alpha$ and $\ell \le \ell g(\eta)$
and $I \in N_\eta[G] \cap \Bbb I_3$ \ub{then} for some $\nu$ with 
Dp$(\nu) \ge \beta$ we have $\eta \trianglelefteq \nu \in T^\otimes \and 
\bold I = \bold I_\nu \in \Bbb I_3$ and \nl
$\{ \nu \in \text{ Suc}_{T^\otimes}(\nu):Dp(\nu) \ge \beta\} \ne \emptyset 
\text{ mod } \bold I_\nu$
\sn
\itemitem{ $(\delta)$ }  if $\beta < \alpha$ and $\underset\tilde {}\to y \in
N_\eta$ is a $\Bbb Q$-name of a member of $\bold V$ when $\ell=3$ and is a
set of cardinality $< \lambda$ when $\ell = 4$ \ub{then} for some 
$\eta',\eta \trianglelefteq \eta'$ with Dp$(\eta') \ge \beta$ and $A \in
N_{\eta'}$ of cardinality $< \lambda$ we have $[\ell = 3 \Rightarrow
\underset\tilde {}\to y[{\underset\tilde {}\to G_{\Bbb Q}}] \in A]$ and $[\ell=4
\Rightarrow \underset\tilde {}\to y[G_{\Bbb Q}] \subseteq A]$.
\endroster}
\ermn
Clearly it is enough to show in $\bold V$ that for some $T \in {\Cal T}$ we have
$q_T \Vdash ``\text{Dp}(<>) = \infty"$.  Note
\mr
\item "{$(*)_2$}"  if $\eta \triangleleft \nu \in T^\otimes$ then
Dp$(\eta) \ge \text{ Dp}(\nu)$.
\ermn
Clearly in $\bold V^{\Bbb Q}$ we have
\mr
\item "{$(*)_3$}"  if $\eta \in 
T^\otimes$, Dp$(\eta) < \infty$ and $\bold I_\eta
\in \Bbb I_3$ then $\{\nu \in \text{ Suc}_{T^\otimes}(\eta):\text{Dp}(\eta) <
\text{ Dp}(\nu)\} = \emptyset \text{ mod } \bold I_\eta$
\sn
\item "{$(*)_4$}"  if $\eta \in T^\otimes$, Dp$(\eta) < \infty$ 
and $\bold I_\eta \notin \Bbb I_3,\nu \in \text{ Suc}_{T^\otimes}
(\eta)$ then Dp$(\eta) \ge \text{ Dp}(\nu)$.
\ermn
For each $\eta \in T^\otimes$ such that $\bold I_\eta \in \Bbb I_2
\backslash \Bbb I_0$ define the set $A_\eta$ as follows:
\sn
First, ${\Cal A}_\eta$ is the minimal family of sets satisfying
\mr
\widestnumber\item{$(iii)$}
\item "{$(i)$}"   if $\ell =3$ and $\underset\tilde {}\to y \in N_\eta$ is a
$\Bbb Q$-name of a member of $\bold V$ \ub{then} the set
$A^3_{\eta,\underset\tilde {}\to y} =:
\{\rho \in \text{ Suc}_{T^*}(\eta):
\text{ for some set } A \in N_\rho 
\subseteq \bold V$ of cardinality $< \kappa,
\underset\tilde {}\to y \in A\}$ belongs to ${\Cal A}_\eta$
\sn
\item "{$(ii)$}"  if $\ell =4$, parallely (using
$A^4_{\eta,\underset\tilde {}\to y}$), i.e., if $y \in N_\eta$ is a
$\Bbb Q$-name of a family of $< \kappa$ members of $\bold V$ then the
set $A^4_{\eta,\underset\tilde {}\to y} =: \{\rho \in \text{
Suc}_{T^*}(\eta)$: for some set $A \in N_\rho \subseteq \bold V$ of
cardinality $, \kappa$ we have $\underset\tilde {}\to y \subseteq A\}$
belongs to ${\Cal A}_\eta$
\sn
\item "{$(iii)$}"  if $\ell \le \ell g(\eta),\beta = \text{ Dp}(\eta
\restriction \ell)$ then the set $A^*_{\eta,\ell} = 
\{\rho \in \text{ Suc}_{T^*}(\eta):\text{Dp}(\rho)
\ge \beta\}$ belongs to ${\Cal A}_\eta$.
\ermn
Let ${\Cal A}'_\eta = \{A \in {\Cal A}_\eta:A = \emptyset$ mod
$\bold I_\eta\}$, note that ${\Cal A}'_\eta$ is a 
countable family of members of $\bold I_\eta$ (more exactly the ideal
$\bold I^{{\bold V}^{\Bbb Q}}_\eta$, which $\bold I_\eta$ generates in
$\bold V^{\Bbb Q}$), and
so actually we have defined a $\Bbb Q$-name ${\underset\tilde {}\to
A_\eta} = \cup\{A:A \in {\Cal A}'_\eta\}$.
\nl
Now we can define in $\bold V$ a sequence $\langle B^*_\eta:\eta \in T^* \rangle$
such that
\mr
\item "{$(*)_5$}"  $B_\eta = \emptyset \text{ mod } \bold I_\eta$ for
$\eta \in T^*$
\sn
\item "{$(*)_6$}" $(i) \quad$ if Suc$_{T^*}(\eta) = \emptyset \text{ mod }
\bold I_\eta$ or $\bold I_\eta \notin \Bbb I_2 \backslash \Bbb I_0$ then
$B^*_\eta = \emptyset$
\sn
\item "{${{}}$}" $(ii) \quad$ if Suc$_{T^*}(\eta) \ne \emptyset \text{ mod }
\bold I_\eta$ and $\bold I_\eta \in \Bbb I_2 \backslash \Bbb I_0$ 
\ub{then} \nl

$\qquad \quad 
p \Vdash_{\Bbb Q} ``{\underset\tilde {}\to A_\eta} \subseteq B^*_\eta$ if
$\eta \in {\underset\tilde {}\to T^\otimes}"$. 
\ermn
This is possible as $\Bbb Q_{\ge p}$ satisfies the $\lambda$-c.c. and each
$I \in \Bbb I_2 \backslash \Bbb I_0$ is $\lambda$-complete.
\sn
Now we define

$$
T_0 = \{\eta \in T^*:\text{ for every } \ell < \ell g(\eta) \text{ we have }
\eta \restriction (\ell +1) \notin B^*_{\eta \restriction \ell}\}.
$$
\mn
Clearly we can find $T_1 \in {\Cal T}$ such that $T_1 \subseteq T_0$
(in particular by $(*)_6$(i)).
So if $q_{T_1} \Vdash_{\Bbb Q} ``\text{Dp}(<>) = 
\infty"$ then we are done so toward
contradiction we assume that this fails.  Hence, there is a subset
$G_{\Bbb Q}$ of $\Bbb Q$ generic over $\bold V$ 
such that $q_{T_1} \in G_{\Bbb Q}$ and $\bold V[G_{\Bbb Q}] \models
``\text{Dp}(<>) < \infty"$.  
As $q_{T_1} \in G_{\Bbb Q}$ clearly $\underset\tilde {}\to T
(T)[G] \subseteq T^\otimes[G_{\Bbb Q}]$. \nl
By the 
choice of $G_{\Bbb Q}$ we have $Dp(<>) < \infty$ hence by $(*)_2$ we have
$\eta \in \underset\tilde {}\to T(T_1)[G_{\Bbb Q}] \Rightarrow \eta \in 
{\underset\tilde {}\to T^\otimes}[G_{\Bbb Q}] \Rightarrow Dp(\eta) < \infty$.

Now we shall prove by induction on $\alpha \in \text{ Ord}$ that $\eta \in
\underset\tilde {}\to T(T_1)[G_{\Bbb Q}] \Rightarrow \text{ Dp}(\eta) \ge
\alpha$.  For
$\alpha = 0,\alpha$ limit we have no problem, so let $\alpha = \beta +1$,
assume toward contradiction Dp$(\eta) = \beta$ for some $\eta \in
\underset\tilde {}\to T(T_1)[G_{\Bbb Q}]$, hence by $(*)_2$ 
and the induction hypothesis we have
\mr
\item "{$\boxtimes_2$}"  $\eta 
\trianglelefteq \nu \in \underset\tilde {}\to T
(T_1)[G_{\Bbb Q}] \Rightarrow \text{ Dp}(\nu) = \beta$
\sn
\item "{$\boxtimes_3$}"  if $I \in (\Bbb I_2 \backslash \Bbb I_0) \cap
N_\eta[G_{\Bbb Q}]$ then for some $\rho \in (\underset\tilde {}\to
T(T_1)[G_{\Bbb Q}],
\eta \triangleleft \rho$ (in fact ``many $\rho$'s) and $J \in \Bbb I_3 \cap
N_\rho$ we have $I = \bold I_\rho$ \nl
[why?  by clause (e)(iii) of the assumption, i.e. choice of
$q_{T_1},\underset\tilde {}\to T(T_1)$.]
\sn
\item "{$\boxtimes_4$}"  if $I \in \Bbb I_3 \cap N_\eta[G_{\Bbb Q}]$ then there is
$\nu$ satisfying $\eta \triangleleft \nu \in \underset\tilde {}\to T(T_1)
[G_{\Bbb Q}]$ such that $I = \bold I_\nu$ \nl
[why? if $I \in \Bbb I_2 \backslash \Bbb I_1$ by $\boxtimes_3$, if 
$I \in \Bbb I_0$ use the
choice of $\underset\tilde {}\to T(T_1)$.]
\ermn
Now
\mr
\item "{$\boxtimes_5$}"  if $I \in N_\eta \cap \Bbb I_3$
\ub{then} for some $\nu$ satisfying
$\eta \triangleleft \nu \in \underset\tilde {}\to T
(T_1)[G_{\Bbb Q}]$ we have $\bold I_\ell = \bold I_\nu$ and \nl
$\{\rho \in \text{ Suc}_{T^\otimes}(\nu):\text{Dp}
(\rho) \ge \beta\} \ne \emptyset \text{ mod } \bold I_\nu$.
\ermn
[Why true?  We can choose $\nu$ such that $I = \bold I_\nu$ and 
$\eta \triangleleft \nu \in \underset\tilde {}\to T
(T)[G_{\Bbb Q}]$ and Suc$_{{\underset\tilde {}\to T}(T_1)
[G_{\Bbb Q}]}(\nu) \ne \emptyset$
mod $I_\nu$ and choose $\rho' \in 
\text{ Suc}_{{\underset\tilde {}\to T}(T_1)[G_{\Bbb Q}]}(\nu) \subseteq
{\underset\tilde {}\to T^\otimes}[G_{\Bbb Q}]$.  First assume $\bold
I_\nu \in \Bbb I_3 \backslash \Bbb I_0$. \nl
Now easily 

$$
{\underset\tilde {}\to A^*_{\nu,\ell g(\eta)}}[G_{\Bbb Q}] 
= \emptyset \text{ mod } \bold I_\nu
\Rightarrow {\underset\tilde {}\to A^*_{\nu,\ell g(\eta)}}
[G_{\Bbb Q}] \subseteq B^*_\nu
\Rightarrow \rho' \notin {\underset\tilde {}\to A_\nu}[G_{\Bbb Q}]
$$
\mn
by the definition of ${\underset\tilde {}\to A^*_{\nu,\ell
g(\eta)}}[G_{\Bbb Q}]$ and $\beta$ we get

$$
\rho' \in \{\rho \in \text{ Suc}_{T^\otimes[G_{\Bbb Q}]}(\nu):
\text{Dp}(\rho) \ge \text{ Dp}(\nu)(= \beta)\}
$$
\mn
easy contradiction.
\sn
Next assume $\bold I \in \Bbb I_1$.  Now
Suc$_{\underset\tilde {}\to T(T_1)[G_{\Bbb Q}]}(\nu)$ is a set witnessing the
requirement.]  \nl
Now for $\eta \in \underset\tilde {}\to T(G_1)
[G_{\Bbb Q}]$ we check the definition
of Dp$(\eta) \ge \beta + 1$: clause $(\alpha)$ holds as
$\underset\tilde {}\to T(T_1)[G_{\Bbb Q}] \subseteq \underset\tilde {}\to T^\otimes
[G_{\Bbb Q}]$, clause $(\beta)$ holds by the induction hypothesis, 
clause $(\gamma)$ holds by
$\boxtimes_3 + \boxtimes_4 + \boxtimes_5$ and clause $(\delta)$ by the choice
of $q_{T_1},\underset\tilde {}\to T(T_1)$.  So Dp$(\eta) \ge \beta +1$
contradiction. \nl
3),4)  Similar to the proof of part (2). \hfill$\square_{\scite{5.2}}$\margincite{5.2}
\enddemo
\bn
We now can deduce more implications between the UP$^\ell$-s.
\demo{\stag{5.3} Conclusion}  Assume that 
$\underset\tilde {}\to \kappa,\underset\tilde
{}\to \lambda$ are purely decided $\Bbb Q$-names and
$\Bbb I^{[\underset\tilde {}\to \kappa]}$ is 
$(< \underset\tilde {}\to \lambda)$-closed, (which just means: if
$r \Vdash ``\underset\tilde {}\to \kappa = \kappa,\underset\tilde
{}\to \lambda = \lambda$ and
${\underset\tilde {}\to {\Bbb I}} = \Bbb I"$ where $r \in \Bbb P$ then
$\Bbb I^{[\kappa]}$ is $\lambda$-closed) and $\Bbb Q$ satisfies the local
$\underset\tilde {}\to \lambda$-c.c. purely. \nl
1) If $\Bbb Q$ satisfies UP$^1_{\underset\tilde {}\to
\kappa,\underset\tilde {}\to \lambda}(\Bbb I,\bold W)$ 
\underbar{then} $\Bbb Q$ satisfies UP$^2_{\underset\tilde {}\to
\kappa}({\underset\tilde {}\to {\Bbb I}},
\bold W)$ and UP$^4_{\underset\tilde {}\to \kappa,\underset\tilde
{}\to \lambda}({\underset\tilde {}\to {\Bbb I}},\bold W)$,
UP$^3_{\underset\tilde {}\to \kappa,\underset\tilde {}\to
\lambda}({\underset\tilde {}\to {\Bbb I}},\bold W)$. \nl
2) UP$^1({\underset\tilde {}\to {\Bbb I}},\bold W)$ implies
UP$^5_{\underset\tilde {}\to \kappa,\underset\tilde {}\to
\lambda}({\underset\tilde {}\to {\Bbb I}},\bold W)$ (for $\Bbb Q$)
\ub{if} $\underset\tilde {}\to \kappa = \underset\tilde {}\to \lambda$
is a $\Bbb Q$-name decidably pure, $\Bbb Q$ satisfies the local
$\underset\tilde {}\to \kappa$-c.c. \nl
3) 4-5 fill!!!
\enddemo
\bigskip
 
\demo{Proof}  Why?  By Definition \scite{4.6} and Lemma \scite{5.2}.
\hfill$\square_{\scite{5.3}}$\margincite{5.3}
\enddemo
\bigskip

\definition{\stag{5.4} Definition}  1) We say $\bar \Bbb Q = \langle
\Bbb P_i,{\underset\tilde {}\to {\Bbb Q}_i},
{\underset\tilde {}\to {\Bbb I}_i},
{\underset\tilde {}\to \kappa_i},{\underset\tilde {}\to {\bold S}_i}:
i < \alpha \rangle$ is UP$^{4,e}(\bold W,W)$-suitable iteration \ub{if}:
\medskip
\roster
\item "{$(a)$}"  $\langle \Bbb P_i,
{\underset\tilde {}\to {\Bbb Q}_j}:i < \alpha,j <
\alpha \rangle$ is an $\aleph_1-\text{Sp}_e(W)$-iteration
\footnote{the reader can fix $W$ as the class of strongly inaccessible
cardinals}
\sn
\item "{$(b)$}"  ${\underset\tilde {}\to {\Bbb I}_i}$ is a 
$\Bbb P_i$-name of 
a set of quasi order ideals with domain a 
cardinal in $\bold V^{{\Bbb P}_i}$ for 
notational simplicity \ub{or} even just a $\Bbb P_{i+1}$-name of such objects
(i.e., $\Vdash_{{\Bbb P}_{i+1}} ``{\underset\tilde {}\to {\Bbb I}} \in
\bold V^{{\Bbb P}_i}"$) such that in $\bold V^{{\Bbb P}_i},
{\underset\tilde {}\to {\Bbb I}_i}/
{\underset\tilde {}\to G_{{\Bbb P}_i}}$ which is a 
${\underset\tilde {}\to {\Bbb Q}_i}$-name, is purely decidable
\sn
\item "{$(c)$}"  $\bold W \subseteq \omega_1$ is stationary
\sn
\item "{$(d)$}"  for each $i < \alpha$, we have: ${\underset\tilde {}\to 
\kappa_i}$ is a $\Bbb P_i$-name of a regular uncountable cardinal of
$\bold V^{{\Bbb P}_i}$, purely decidable
\sn
\item "{$(e)$}"  $\Vdash_{{\Bbb P}_i} 
``{\underset\tilde {}\to {\Bbb Q}_i}$ satisfies
UP$^4_{{\underset\tilde {}\to \kappa_i},{\underset\tilde {}\to \kappa_{i+1}}}
({\underset\tilde {}\to {\Bbb I}_i},
{\underset\tilde {}\to {\bold S}_i},\bold W)$ and 
${\underset\tilde {}\to {\Bbb I}_i}$ is
${\underset\tilde {}\to \kappa_i}$-complete set of partial order ideals
(from $\bold V$) and $({\underset\tilde {}\to {\Bbb Q}_i},
\le_{\text{vpr}})$ is $\aleph_1$-complete (see \scite{1.1})"
\sn
\item "{$(f)$}"  or $i<j$ we have $\Vdash_{P_j} ``
{\underset\tilde {}\to \kappa_i} \le {\underset\tilde {}\to \kappa_j}"$
\sn
\item "{$(g)$}"  $\Bbb P_{i +1}$ satisfies the ${\underset\tilde {}\to
\kappa_i}$-c.c., or just for every $p \in \Bbb P_{i+1}$ 
there are $\kappa',q$ such that 
{\roster
\itemitem{ $(\alpha)$ }   $p \le_{\text{pr}} q \in \Bbb P_{i+1}$
\sn
\itemitem{ $(\beta)$ }   $q \Vdash_{({\Bbb P}_{i+1},
\le_{\text{pr}})} ``{\underset\tilde {}\to \kappa_{i+1}} = \kappa'"$
\sn
\itemitem{ $(\gamma)$ }   $\kappa \le \kappa' \in$ UReg and ${\underset\tilde {}\to 
\zeta_\varepsilon}$ is a $\Bbb P_i$-name of an ordinal for $\varepsilon <
\varepsilon^* < \kappa'$ and $p',q \restriction i \le p' \in \Bbb P_i$ 
\ub{then} there is
$q',p' \le_{\text{pr}} q' \in \Bbb P_i$ and set $a$ of $< \kappa'$
ordinals such that $q' \Vdash_P ``{\underset\tilde {}\to \zeta_\varepsilon} \in a$ for
$\varepsilon < \varepsilon^*"$ 
\endroster}
\item "{$(h)$}"  [?] for any $i < \alpha$ for some $n < \omega$, if $i < j <
\alpha$ and $p \in \Bbb P_{j+1}$ and $\kappa^\otimes_{\underset\tilde {}\to \kappa_j}(p,
\Bbb P_{j+1}) = \kappa$ and ${\underset\tilde {}\to \zeta_\varepsilon}$ is a
$(\Bbb P_i)_{\ge p \restriction i}$-name of an ordinal for $\varepsilon <
\varepsilon^* < \kappa$ \ub{then} 
for some $q$ and $a$ we have: $p \restriction i
\le_{\text{pr}} q \in \Bbb P_i$ and $q \Vdash_{{\Bbb P}_i} 
``{\underset\tilde {}\to \zeta_\varepsilon} \in a$ for
$\varepsilon < \varepsilon^*"$ where $a$ is a set of $< \kappa$ ordinals.
\ermn
2) We may write UP$^{4,e}$ instead UP$^{4,e}(\omega_1$, the class of
strongly inaccessibles).  If we omit ${\underset\tilde {}\to {\bold S}_i}$ we mean
$\{\aleph_1\}$.  \nl
If we omit ${\underset\tilde {}\to {\Bbb I}_i}$, we mean ``some
${\underset\tilde {}\to {\Bbb I}}$ as required" (note that the requirements
on ${\underset\tilde {}\to {\Bbb I}_i}$ are actually on each member so the
family of candidates to being ${\underset\tilde {}\to {\Bbb I}}$ is closed
under union).  If we omit $e$ we mean $e=6$.  We may omit $W$ if it is the
class of strongly inaccessible cardinals.
\nl
If we omit ${\underset\tilde {}\to \kappa_i}$ we mean some such 
$\Bbb P_{i+1}$-name.  (Can we eliminate names?  
Well if we use the iteration as in \cite[Ch.X,\S1]{Sh:f},
(RCS) no, but if we waive associativity as done here, we can). \nl
3) We defined UP$^{3,e}(\bold W,W)$-suitable iterations similarly but
replace clause (e) by:
\mr
\item "{$(e)_a$}"  as above replacing 
UP$^4_{{\underset\tilde {}\to \kappa_i},
{\underset\tilde {}\to \kappa_{i+1}}}$
by UP$^3_{{\underset\tilde {}\to \kappa_i},
{\underset\tilde {}\to \kappa_{i+1}}}$.
\ermn
4) We define a UP$^\ell(\Bbb I,\bold W,W)$-iterations as above but with
${\underset\tilde {}\to {\Bbb I}_i} = 
{\underset\tilde {}\to {\Bbb I}^{[{\underset\tilde {}\to \kappa_i}]}}$
[Saharon like \S6, straight in successor, in limit work it out, Question:
${\underset\tilde {}\to \kappa_i}$ pure name?]]
\enddefinition
\bn
We can also deal with strong preservation
\definition{\stag{5.4B} Definition}  We say that $\bar{\Bbb Q} =
\langle \Bbb P_i,{\underset\tilde {}\to {\Bbb Q}_i},
{\underset\tilde {}\to \kappa_j}:
i < \alpha,j \le \alpha \rangle$ is a weak UP$^4(\bold W,W)$-iteration if
\mr
\item "{$(a)$}"  $\langle \Bbb P_i,
{\underset\tilde {}\to {\Bbb Q}_i}:i < \alpha \rangle$
is a $\aleph_1-\text{Sp}_6(W)$-iteration, 
${\underset\tilde {}\to \kappa_i}$ is
$(\Bbb P_i,\le^{{\Bbb P}_i}_{\text{pr}})$-name such that $j < i \Rightarrow
{\underset\tilde {}\to \kappa_j} \le {\underset\tilde {}\to \kappa_i}$
\sn
\item "{$(b)$}"  if $i < j \le \alpha$ and $i$ is non-limit, $p \in
\Bbb P_i$ and $p \Vdash_{({\Bbb P}_i,\le_{\text{pr}})} 
``{\underset\tilde {}\to \kappa_i} =
\kappa_i"$ then $p \Vdash_{{\Bbb P}_i} ``\Bbb P_j/G_{\Bbb P_i}$ satisfies
UP$^4_{{\underset\tilde {}\to \kappa_i},{\underset\tilde {}\to \kappa_j}}
({\underset\tilde {}\to {\Bbb I}},\bold W)$ for some $\Bbb P_i$-name
${\underset\tilde {}\to {\Bbb I}}$ of ${\underset\tilde {}\to 
\kappa_i}$-complete ideals".
\endroster
\enddefinition
\bigskip

\proclaim{\stag{5.5} Lemma}  Assume that $\bold W \subseteq \omega_1$ is 
stationary and $\bar{\Bbb Q} = 
\langle \Bbb P_i,{\underset\tilde {}\to {\Bbb Q}_i},
{\underset\tilde {}\to {\Bbb I}_i},{\underset\tilde {}\to \kappa_i}:
i < \alpha \rangle$ is a UP$^4(\bold W,W)$-suitable iteration, and 
$\Bbb P_\alpha = { \text{\rm Sp\/}}(W)-{\text{\rm Lim\/}}
(\bar{\Bbb Q})$ be the limit. \nl
For $j \le \alpha$ we define ${\underset\tilde {}\to \kappa^*_j}$, a
$(\Bbb P_j,\le_{\text{pr}})$-name of a member of 
${\text{\rm UReg\/}}^{\bold V}:
{\underset\tilde {}\to \kappa^*_j} =$ \nl
$\text{Min}\{\kappa \in { \text{\rm UReg\/}}^{\bold V}:
i < j \Rightarrow {\underset\tilde {}\to \kappa_i} \le \kappa \text{ and }
\kappa \ge \aleph_2 \text{ and } \kappa \ge j\}$.
\sn
For simplicity we can assume
\mr
\item "{$\boxtimes$}"  there are $\langle \kappa'_i:i < \alpha
\rangle,\Bbb P_i$
satisfies the $\kappa'_i$-c.c. and for each $i,i + \omega \le \alpha$ for
some $\Vdash_{(\Bbb P_i,\le_{\text{pr}})} ``{\underset\tilde {}\to \kappa_i} \le
\kappa'_i",\Vdash_{(\Bbb P_{i+1},\le_{\text{pr}})} ``\kappa'_i \le
{\underset\tilde {}\to \kappa'_{i+1}}"$ (and $\kappa^*_\beta = \beta
\Rightarrow \Bbb P_\beta = \dbcu_{i < \beta}\Bbb P_i$ (i.e. $\kappa^*_\beta$ is
strongly inaccessible $> |\Bbb P_i|$ for $i < \beta$ \ub{or} change support?)
\ermn
1) For each $\beta \le \alpha,\Bbb P_\beta$ satisfies 
UP$^5_{{\underset\tilde {}\to \kappa^*_\beta}}(\Bbb I'_\beta,\bold W)$ 
for some $\kappa^*$-complete $\Bbb I'_\beta \in \bold V$. \newline
1A) If $\gamma \le \beta \le \alpha,p \Vdash_{{\Bbb P}_\gamma}
``{\underset\tilde {}\to \kappa^*_\gamma} = \kappa_\gamma",p \in
G_{{\Bbb P}_\gamma}$ \ub{then} in $\bold V[G_{{\Bbb P}_\gamma}]$ 
the forcing notions $\Bbb P_\beta/
G_{{\Bbb P}_\gamma}$ satisfies 
${\text{\rm UP\/}}^5_{{\underset\tilde {}\to \kappa_\gamma},
{\underset\tilde {}\to \kappa_\beta}}
({\underset\tilde {}\to {\Bbb I}'_{\beta,
\delta}},\bold W)$ for some $\kappa_j$-complete
${\underset\tilde {}\to {\Bbb I}_{\beta,\gamma}}
\in \bold V^{{\Bbb P}_\gamma}$ (so this justifies the weak in
\scite{5.4B}). 
\nl
2) In fact each $I \in \Bbb I'_\beta$ has domain of cardinality 
$\le (\underset{\gamma < \beta} \to \sup\{ \lambda^\kappa:\,\, 
\nVdash_{{\Bbb P}_\gamma} ``\neg(\exists I \in \dbcu_{i<j}
{\underset\tilde {}\to {\Bbb I}_i})(\lambda = 
|{\text{\rm Dom\/}}(I)|)$ and 
${\underset\tilde {}\to \kappa_i} > \kappa$ for some
$i \le \gamma"\}$ and
$|\Bbb I'_\beta| \le 
\dsize \sum_{\gamma < \beta} (\aleph_0 + |\Bbb P_\gamma| +
\text{ sup}\{\lambda^\kappa:\,\, \nVdash_{{\Bbb P}_\gamma} 
``\neg(|\dbcu_{i \le \gamma}{\underset\tilde {}\to {\Bbb I}_i}| 
\le \lambda$ and $\kappa_i > \kappa$ for some $i \le \gamma)"\}$.
Similarly for ${\underset\tilde {}\to {\Bbb I}_{\gamma,\beta}}$. \nl
3) Similarly for weak ${\text{\rm UP\/}}^5(\bold W,W)$-iterations.
\endproclaim
\bigskip

\remark{\stag{5.5A} Remark}  1)  We can also get 
the preservation version of this Lemma. \nl
2) The reader can concentrate on the case that
${\underset\tilde {}\to \kappa'_\ell}$'s are objects and not names.
\endremark
\bigskip

\demo{Proof}  1) We prove this by induction on $\beta$, so \wilog \,
$\beta = \alpha$.  For each $\gamma < \alpha$ let ${\Cal J}_\gamma =:
\{q \in \Bbb P_{\gamma +1}:q \text{ forces a value to } 
{\underset\tilde {}\to \kappa_\gamma}$, called 
$\kappa_{\gamma,q}$ and $q$ forces 
${\underset\tilde {}\to {\Bbb I}_\gamma}$ to be equal to a $\Bbb P_\gamma$-name 
called ${\underset\tilde {}\to {\Bbb I}_{\gamma,q}}$ and
$q \restriction \gamma$ forces that 
$|{\underset\tilde {}\to {\Bbb I}_\gamma}|$ is 
$\le \mu_{\gamma,q}$ but no $q'$ such that $q \restriction \gamma 
\le_{\text{pr}} q' \in \Bbb P_\gamma$ forces a smaller bound$\}$.
Let $\mu_\gamma = \underset{q \in {\Cal J}_\gamma} \to \sup \mu_{\gamma,q}$. 
\newline
Let $q \Vdash_{{\Bbb P}_\gamma} ``{\underset\tilde {}\to {\Bbb I}_\gamma} = \{
{\underset\tilde {}\to I_{\gamma,\zeta}}:\zeta < 
{\underset\tilde {}\to \zeta_{\gamma,q}} \le \mu_{\gamma,q}\}"$ for
$q \in {\Cal J}_\gamma$ and let
${\Cal J}_{\gamma,\zeta} = \{q \in {\Cal J}_\gamma:\mu_{\gamma,q} > \zeta
\text{ and } q \Vdash ``\text{Dom}({\underset\tilde {}\to I_{\gamma,\zeta}})
\text{ is } \le \lambda_{\gamma,q,\zeta}"$ 
and no $q'$ such that $q \restriction \gamma \le_{\text{pr}} q' \in
\Bbb P_\gamma$ forces a smaller bound$\}$ and let
${\underset\tilde {}\to I_{\gamma,\zeta}} 
\text{ be id}_{{\underset\tilde {}\to L_{\gamma,q,\zeta}}}$, 
so ${\underset\tilde {}\to L_{\gamma,q,\zeta}}$ is a
$\Bbb P_\gamma$-name of a ${\underset\tilde {}\to \kappa_{\gamma,q}}$-directed 
quasi order on some $\lambda' \le \lambda_{\gamma,q,\zeta}$ 
(but $\Vdash_{{\Bbb P}_\gamma}$ ``if
$|{\underset\tilde {}\to {\Bbb I}_\gamma}| \le \zeta < \mu_\gamma$ then let
$L_{\gamma,\zeta}$ be trivial").  We can assume $L_{\gamma,q,\zeta}$ is a
quasi order on $\lambda_{\gamma,q,\zeta}$ (putting every $\beta \in
\lambda_{\gamma,q,\zeta} \backslash \text{ Dom}(I_{\gamma,q,\zeta})$ at the
bottom.

For $q \in {\Cal J}_\gamma$ let 
$L^*_{\gamma,q,\zeta}$ be ap$_{\kappa_{\gamma,q,q}}(
{\underset\tilde {}\to L_{\gamma,\zeta}})$ for the forcing notion \nl
$\Bbb P_\gamma^{[q]} = \{p \in \Bbb P_\gamma:q \restriction \gamma 
\le^{{\Bbb P}_\gamma}_{pr} p\}$ from Definition \scite{3.9}, so it is
defined in $\bold V$.  So by Claim \scite{3.10}
\medskip
\roster
\widestnumber\item{$(iii)$}
\item "{$(i)$}"   $L^*_{\gamma,q,\zeta}$ is $\kappa_{\gamma,q}$-directed
partial order on $[\lambda_{\gamma,q,\zeta}]^{< \kappa_{\gamma,q}}$
\sn
\item "{$(ii)$}"  $|L^*_{\gamma,q,\zeta}| \le (\lambda_{\gamma,q,\zeta})
^{< \kappa_{\gamma,q}}$
\sn
\item "{$(iii)$}"  $q \restriction \gamma \Vdash_{{\Bbb P}_\gamma} 
``{\underset\tilde {}\to I_{\gamma,\zeta}} = 
\text{ id}_{\underset\tilde {}\to L_{\gamma,q,\zeta}} 
\le_{\text{RK}} \text{ id}_{L^*_{\gamma,q,\zeta}}"$.
\endroster
\medskip

\noindent
Let $\kappa_\beta = \sup\{\kappa_{\gamma,q}:\gamma < \beta \text{ and } q \in
{\Cal J}_\gamma\}$. 
\sn
Let $\Bbb I^*_\beta$ be the $(< \kappa_\beta)$-closure of $\{
\text{id}_{L^*_{\gamma,q,\zeta}}:\gamma < \beta,q \in {\Cal J}_\gamma,
\zeta < \mu_{\gamma,q}\}$ (see Definition \scite{3.13}(1)).
\smallskip

Let $\bar N = \langle N_\eta:\eta \in (T^*,\bold I) \rangle$ be a
strict truely
$(\Bbb I^*_\alpha,\bold W)$-suitable tree of models for $(\chi,x),x$ coding 
enough information (so $\bar{\Bbb Q},
\Bbb I^*_\alpha,\bold W,W \in N_{\langle \rangle}$); why
truely? see \sciteu{4.x}.

Let ${\Cal T}_{\bar N}$ be the set of quadruples 
$(\gamma,q,\underset\tilde {}\to \nu,\underset\tilde {}\to T)$ such that:
\medskip
\roster
\item "{$\bigotimes_1$}"  $\gamma \le \alpha,q \in \Bbb P_\gamma,
\underset\tilde {}\to T$ is a $\Bbb P_\gamma$-name of a subtree of $T^*$, 
\newline
$q \Vdash_{({\Bbb P}_\gamma,\le_{\text{pr}})} 
``{\underset\tilde {}\to \kappa^*_\gamma} = \kappa_\gamma"$ and \nl
$q \Vdash_{{\Bbb P}_\gamma} ``\langle N_\eta[{\underset\tilde {}\to
G_{{\Bbb P}_\gamma}}]:
\eta \in (\underset\tilde {}\to T,\bold I \restriction 
\underset\tilde {}\to T) \rangle$ \nl

$\qquad \qquad$ is strictly 
$((\Bbb I^*_\alpha)^{[\kappa_\gamma]},\bold W)$-suitable tree, \nl

$\qquad \qquad N_{\langle \rangle}
[{\underset\tilde {}\to G_{{\Bbb P}_\gamma}}] \cap \omega_1 = N_{\langle \rangle}
\cap \omega_1$ and $\underset\tilde {}\to \nu \in \underset\tilde {}\to T$ 
\newline

$\qquad \qquad$ and $\gamma,\kappa \in N_{\underset\tilde {}\to \nu}
[{\underset\tilde {}\to G_{{\Bbb P}_\gamma}}]$ \newline

$\qquad \qquad$ and $\bar N[{\underset\tilde {}\to G_{{\Bbb P}_\gamma}}]$ has
$(\kappa)$-covering".
\endroster
\medskip
\noindent
Now ${\Cal T}'_{\bar N}$ is defined similarly as the set of quadruples
$(\underset\tilde {}\to \gamma,q,\underset\tilde {}\to \nu,
\underset\tilde {}\to T)$ such that:
$\underset\tilde {}\to \gamma$ is a simple $(\bar Q,W)$-named 
$[0,\alpha)$-ordinal, $q \in P_{\underset\tilde {}\to
\gamma},\underset\tilde {}\to \nu$ a $\Bbb P_{\underset\tilde {}\to \gamma}$-name and 
$\underset\tilde {}\to \gamma \in
N_{\underset\tilde {}\to \nu}[G_{{\Bbb P}_{\underset\tilde {}\to \gamma}}]$.  
(I.e. if $\zeta < \beta,G_{{\Bbb P}_\zeta} \subseteq \Bbb P_\zeta$ is generic over 
$\bold V$ and $\zeta = \underset\tilde {}\to \gamma
[G_{{\Bbb P}_\zeta}]$ then $r \in q \Rightarrow
{\underset\tilde {}\to \zeta_r}[G_{{\Bbb P}_\zeta}] < \zeta$, i.e. is well defined
$< \zeta$ \ub{or} is forced ($\Vdash_{{\Bbb P}_\alpha/G_{{\Bbb P}_\zeta}}$) to be not 
well defined), and $q \Vdash_{{\Bbb P}_{\underset\tilde {}\to \gamma}} 
``\underset\tilde {}\to \nu \in \text{ lim}(T)"$.
\medskip

We consider the statements, for $\gamma \le \beta < \alpha$
\medskip
\roster
\item "{$\boxtimes_{\gamma,\beta}$}"  for any 
$(\gamma,q,\underset\tilde {}\to \eta,\underset\tilde {}\to T) \in 
{\Cal T}_{\bar N}$ and $\underset\tilde {}\to \rho$ a $\Bbb P_\gamma$-name 
such that \newline
$q \Vdash_{P_\gamma} ``\underset\tilde {}\to \eta \triangleleft 
\underset\tilde {}\to \rho \in \underset\tilde {}\to T$ and $\gamma \in
N_{\underset\tilde {}\to \rho}[{\underset\tilde {}\to G_\gamma}]"$ \nl
and ${\underset\tilde {}\to p'}$ a $\Bbb P_\gamma$-name such that 
$q \Vdash_{{\Bbb P}_\gamma} ``{\underset\tilde {}\to p'}[G_{{\Bbb P}_\gamma}] \in
N_{\underset\tilde {}\to \rho}[{\underset\tilde {}\to G_{{\Bbb P}_\gamma}}] 
\cap \Bbb P_\beta/G_{{\Bbb P}_\gamma}$ and
$({\underset\tilde {}\to p'}[G_{{\Bbb P}_\gamma}]) \restriction \gamma \le_{pr} q"$
and ${\underset\tilde {}\to p'}[G_{{\Bbb P}_\gamma}]$ forces
$(\Vdash_{(P_\beta,\le_{\text{pr}})})$ a value $\kappa^*_\gamma$ to
${\underset\tilde {}\to \kappa_\gamma}$ (usually redundant) 
\ub{there is} $(\beta,q',
\underset\tilde {}\to \rho,{\underset\tilde {}\to T'})
\in {\Cal T}_{\bar N}$ such that 
${\underset\tilde {}\to p'} \le_{pr} q'$ \nl
(i.e., $p \Vdash_{{\Bbb P}_\gamma} ``{\underset\tilde {}\to p'}
[{\underset\tilde {}\to G_{{\Bbb P}_\gamma}}] \le_{pr} q"$) and 
$q' \restriction \gamma = q$ and $q' \Vdash_{{\Bbb P}_\beta} 
``\underset\tilde {}\to \rho \in {\underset\tilde {}\to T'} \subseteq
{\underset\tilde {}\to T}$".
\ermn
For simple $(\bar{\Bbb Q},W)$-names $[0,\alpha)$-ordinals
$\underset\tilde {}\to \gamma \le \underset\tilde {}\to \beta$ we define
\mr
\item "{$\boxtimes_{\underset\tilde {}\to \gamma,\underset\tilde {}\to
\beta}$}"
similarly $(\forall \beta < \beta^*)\forall \gamma \le
\beta(\square_{\gamma,\beta})$ and $\underset\tilde {}\to
\gamma,\underset\tilde {}\to \beta$.
\endroster
\bn
\ub{Observation}:  If $\forall < \beta^*,\forall \gamma \le
\beta(\boxtimes_{\gamma,\beta})$
and $\underset\tilde {}\to \gamma,\underset\tilde {}\to \beta$ are simple
$\bar Q$-named $[0,\beta)$-ordinals $\Vdash \underset\tilde {}\to \gamma
\le \underset\tilde {}\to \beta < \beta^*$ then
$\boxtimes_{{\underset\tilde {}\to \gamma^*},
{\underset\tilde {}\to \beta^*}}$
(defined naturally).
\enddemo
\bigskip

\demo{Proof}  By induction on the depth of $\underset\tilde {}\to \beta$
(see \scite{si.7}, fact A).

We prove by induction on $\beta \le \alpha$ that
\mr
\item "{$(a)$}"  $\Bbb P_\beta$ has pure 
${\underset\tilde {}\to \kappa_\beta}$-covering; i.e. if
$\underset\tilde {}\to \tau$ is a $\Bbb P_\beta$-name of an ordinal
$< {\underset\tilde {}\to \kappa_\beta}$ and $p \in \Bbb P_\beta$
\ub{then} for some $q$ and $a$ we have: $p \le_{\text{pr}} q \in \Bbb P_\beta,
a \in \bold V$ is a set of ordinals and $q \Vdash ``|a| <
{\underset\tilde {}\to \kappa_\beta} \and \underset\tilde {}\to \tau \le a"$
(even over $\Bbb P_\gamma$)
\sn
\item "{$(b)$}"  $\Bbb P_\beta$ has pure $(\aleph_1,\aleph_1)$-decidability
\sn
\item "{$(c)$}"  for every $\gamma \le \beta$ we have 
$\boxtimes_{\gamma,\beta}$ (but for \scite{5.5}(3) we have to
restrict ourselves to non-limit $\gamma$).
\ermn
Note that for $\gamma = \beta$ the statement in clause (c)
is trivial hence we shall consider only $\gamma < \beta$.
\bigskip

\noindent
\underbar{Case 1}:  $\beta = 0$.

Trivial.
\bigskip

\noindent
\underbar{Case 2}:  $\beta$ a successor ordinal.

Clauses (a), (b) follows easily from clause (c) so let us 
concentrate on clause (c).
As trivially $\boxtimes_{\gamma_0,\gamma_1} \and \boxtimes_{\gamma_1,\gamma_2}
\Rightarrow \boxtimes_{\gamma_0,\gamma_2}$, clearly without loss of
generality \nl
$\beta = \gamma + 1$.

Let $G_{{\Bbb P}_\gamma}$ be such that $q \in G_{{\Bbb P}_\gamma} 
\subseteq \Bbb P_\gamma$ and
$G_{{\Bbb P}_\gamma}$ generic over $\bold V$.

Let $T' = \{\nu:\rho \char 94 \nu \in \underset\tilde {}\to T[G_{{\Bbb P}_\gamma}]\},
\bar N' = \langle N'_\nu:\nu \in
(T',\bold I') \rangle$ where $N'_\nu = N_{\rho \char 94 \nu}[G_{{\Bbb P}_\gamma}],
\bold I'_\nu = \bold I^*_{\rho \char 94 \nu}$. \newline
By \scite{5.2} applied to $\bar N'$ we can find ${\underset\tilde {}\to p'},
{\underset\tilde {}\to T''}$ as required.
\bigskip

\noindent
\underbar{Case 3}:  $\beta$ a limit ordinal. \nl
[Saharon note: it would be if
$\underset\tilde {}\to \kappa$ is a $(\Bbb P_\beta,\le_{\text{pr}^+})$-name,
$\gamma < \beta,G_\gamma \subseteq \Bbb P_\gamma$ generic we should define
$\underset\tilde {}\to \kappa/G_\gamma$ anyhow we can use real names but
then ${\underset\tilde {}\to \kappa^*_\gamma}$ is just a
$(\Bbb P_\gamma,\le_{\text{pr}})$-name.  But if 
${\underset\tilde {}\to \kappa_\beta}$ are real cardinals no problem.  But see
clause (g) of definition of UP".]
\enddemo
\bigskip

\demo{Proof of Clause (a)}  If we use the c.c. version: easier, hardest 
case is $\kappa^*_\beta = \beta$, so $\beta$ strongly inaccessible. \nl
Note that we have to prove the weak version.  If the property fails for
$\beta,p,\underset\tilde {}\to \tau$ (so $p \Vdash_{P_\beta}
``\underset\tilde {}\to \tau$ an ordinal", etc.), \ub{then} by the
induction hypothesis
$\beta$ is minimal so definable in $({\Cal H}(\chi),\in)$ from 
$\bar{\Bbb Q}$ hence necessarily
$\beta \in N_{<>}$ and \wilog \, $p,\underset\tilde {}\to \tau \in N_{<>}$.
Also \wilog \, $p \Vdash_{({\Bbb P}_\beta,\le_{\text{pr}})}
``{\underset\tilde {}\to \kappa^*_\beta} = \kappa^*_\beta"$ for some
$\kappa^*_\beta \in$ UReg V.

We shall now choose $p_1$ as in the proof of \scite{1.27}.  Let
$\langle {\underset\tilde {}\to \zeta_\varepsilon}:\varepsilon < j \rangle$
be a witness for $p$ (see \scite{1.13} clause (F), in particular
$(a)(iii)$ of (F)), so \wilog \, $j \le \omega$.  For each 
$\varepsilon < j$ and $\xi < \beta$, let
${\underset\tilde {}\to a_{\varepsilon,\xi}}$ be a
$\Bbb P_{\xi +1}$-name of a set of $< \kappa^*_\beta$ ordinals and let
${\underset\tilde {}\to r_{\varepsilon,\xi}}$ be
a $\Bbb P_{\xi +1}$-name of a member of 
$\Bbb P_\beta/G_{\xi +1}$ with domain $\subseteq [\xi +1,\beta)$ 
such that if
$G_{\xi +1} \subseteq \Bbb P_{\xi +1}$ is generic over $\bold V$ and
${\underset\tilde {}\to \zeta_\varepsilon}[G_{\xi +1} \cap
\Bbb P_\xi] = \xi$, then $r = {\underset\tilde {}\to r_\varepsilon}
[{\underset\tilde {}\to G_{\xi +1}}]$ satisfies: 
\mr
\item "{$(a)$}"  if possible 
$p \le_{\text{pr}}(p \restriction (\xi +1)) \cup r \in \Bbb P_\beta$ and
$(p \restriction (\xi +1)) \cup r \Vdash_{{\Bbb P}_\beta} 
``\underset\tilde {}\to \tau \in 
a"$ where $a = {\underset\tilde {}\to a_{\varepsilon,\zeta}}[G_{\xi +1}] \in
\bold V[G_{\xi +1}]$ is a set of $< \kappa^*_\beta$ ordinals and
${\underset\tilde {} \to {\bold t}_{\varepsilon,\zeta}}[G_{\xi +1}] =$ true
\sn
\item "{$(b)$}"  if not possible $r = 
{\underset\tilde {}\to r_{\varepsilon,\xi}}[G_{\xi +1}]$ is the empty function
and ${\underset\tilde {}\to a_{\varepsilon,\xi}}
[G_{\xi +1}]  = \emptyset$ and ${\underset\tilde {}\to {\bold
t}_{\varepsilon,\xi}}[G_{\xi +1}] =$ false.
\nl
Also we can demand that:
\sn
\item "{$(c)$}"  ${\underset\tilde {}\to \zeta_{\varepsilon_1}}[G_\xi] = \xi 
= {\underset\tilde {}\to \zeta_{\varepsilon_2}}[G_\xi]$ then
${\underset\tilde {}\to r_{\varepsilon_1,\xi}}[G_{\xi +1} \cap \Bbb P_\xi] =
{\underset\tilde {}\to r_{\varepsilon_2,\xi}}[G_\xi]$. 
\ermn
Let ${\underset\tilde {}\to r_\varepsilon}[G_\beta] = r$
iff for some $\xi$ 
we have ${\underset\tilde {}\to \zeta_\varepsilon}[G_\beta \cap \Bbb P_\xi] = \xi$
and $r = {\underset\tilde {}\to r_{\varepsilon,\xi}}[G_\beta]$, similarly we
define ${\underset\tilde {}\to a_\varepsilon}$.  Let $p_1 = p \cup \bigcup
\{{\underset\tilde {}\to r_\varepsilon}:\varepsilon < j\}$.  Clearly
$p \le_{\text{pr}} p_1 \in \Bbb P_\beta$.
\mn
Next define $p_2$ as in \scite{1.27}: (recall (g) + (h) of Definition
\scite{5.3}).  I.e. for each $\varepsilon < j,\xi < \beta,G_\xi \subseteq
\Bbb P_\xi$ generic over $\bold V,p \restriction \xi \in G_\xi,
{\underset\tilde {}\to r_\varepsilon}[G_\xi] = \xi$ there are
${\underset\tilde {}\to r'_{\varepsilon,\xi}},a'_{\varepsilon,\xi} \in
\bold V[G_\beta]$ such that 
${\underset\tilde {}\to {\hat{\Bbb Q}}_\xi}[G_\xi] \models ``p_1
\restriction \{\xi\} \le_{\text{pr}} r^1_{\varepsilon,\xi}"$ and
$r'_{\varepsilon,\xi} \Vdash_{\underset\tilde {}\to {\hat{\Bbb Q}}_\xi}
``{\underset\tilde {}\to a_{\varepsilon,\xi}} \subseteq 
a'_{\varepsilon,\xi}"$.
So really \wilog \, we have $\Bbb P_\xi$-names
${\underset\tilde {}\to r'_{\varepsilon,\xi}},
{\underset\tilde {}\to a'_{\varepsilon,\xi}}$ such that
${\underset\tilde {}\to \zeta_{\varepsilon_1}}[G_\xi] = \xi =
{\underset\tilde {}\to \zeta_{\varepsilon_2}}[G_\xi]$ implies
${\underset\tilde {}\to r'_{\varepsilon_1,\xi}}[G_\xi] =
r'_{\varepsilon_2,\xi}[G_\xi]$ and
${\underset\tilde {}\to a'_{\varepsilon_1,\xi}}[G_\xi] =
{\underset\tilde {}\to a'_{\varepsilon_1,\xi}}[G_\xi]$.  We define
${\underset\tilde {}\to r'_\varepsilon},
{\underset\tilde {}\to a'_\varepsilon}$ by:
${\underset\tilde {}\to r'_\varepsilon}[G_\beta] = r$ iff for some $\xi <
\beta$ we have ${\underset\tilde {}\to \zeta_\varepsilon}[G_\beta \cap
\Bbb P_\xi] = \xi$ and 
$r = {\underset\tilde {}\to r_{\varepsilon,\xi}}[G_\beta
\cap \Bbb P_\xi]$ and similarly ${\underset\tilde {}\to
a'_\varepsilon}$.  
Now let $p_2 = p_1 \cup
\{{\underset\tilde {}\to r'_\varepsilon}:\varepsilon < j\}$ so $p_1 
\le_{\text{pr}} p_2$.  We can finish as in the proof of \scite{1.27} [fill]!!!
\bn
\ub{Clause (b)}:  

As in the proof of clause (a) \wilog \, we have $\beta,p,\underset\tilde {}\to
\zeta \in N_{<>}$. We define also $p_1$ as in the proof of clause (a) 
trying to force a countable
bound for $\underset\tilde {}\to \zeta$.  Let $\langle
{\underset\tilde {}\to \zeta^*_\varepsilon}:\varepsilon < \omega \rangle$ be
a witness for $p \in \Bbb P_\beta$ (see Definition \scite{1.13},
clause (F) in particular (a)(iii) of (F) and \wilog \, $\langle
{\underset\tilde {}\to \zeta^*_\varepsilon}:n < \omega \rangle$ belong
to $N_{<>}$).  We now choose
by induction on $n$ a quadruple $({\underset\tilde {}\to \gamma_n},
{\underset\tilde {}\to q_n},{\underset\tilde {}\to \nu_n},
{\underset\tilde {}\to t_n})$ such that
\mr
\widestnumber\item{$(iii)$}
\item "{$(i)$}"  ${\underset\tilde {}\to \gamma_n}$ is a simple
$(\bar{\Bbb Q} \restriction \beta,W)$-named $[0,\beta)$-ordinal
\sn
\item "{$(ii)$}"  ${\underset\tilde {}\to \gamma_0} = 0,
{\underset\tilde {}\to \gamma_n} < {\underset\tilde {}\to \gamma_{n+1}}$
\sn
\item "{$(iii)$}"  $\zeta_{\underset\tilde {}\to \zeta^*_n} + 1 \le
{\underset\tilde {}\to \gamma_{n+1}}$
\sn
\item "{$(iv)$}"  $({\underset\tilde {}\to \gamma_n},
{\underset\tilde {}\to q_n},{\underset\tilde {}\to \nu_n},
{\underset\tilde {}\to T_n}) \in
{\Cal T}'_{\bar N}$
\sn
\item "{$(v)$}"  ${\underset\tilde {}\to q_n} =
{\underset\tilde {}\to q_{n+1}} \restriction {\underset\tilde {}\to \gamma_n}$
\sn
\item "{$(vi)$}"  $p \restriction {\underset\tilde {}\to \gamma_n}
\le_{\text{pr}} {\underset\tilde {}\to q_n}$.
\ermn
No problem to carry it by the observation above.
\mn
Let $p_2 = \dbcu_{n < \omega} {\underset\tilde {}\to q_n} \cup p$, clearly
$p \le_{\text{pr}} p_2,{\underset\tilde {}\to q_n} \le p_2$, and so it
is enough to prove $p_2 \Vdash ``\underset\tilde {}\to \zeta < N_{<>} \cap
\omega_1"$.  So let $p_2 \in G_\beta$ with $G_\beta$ a subset of $\Bbb P_\beta$ generic
over $\bold V$.  So there is $p^+_2 \in G_\beta$ satisfying $p \le p_2
\le p^+_2$ such that 
$p^+_2 \Vdash ``\underset\tilde {}\to \zeta = \gamma^* < \omega_1"$ and so
$p \le p^+_2$ so \wilog \, $p \le p^+_2$ above $\xi_0 < \ldots <
\xi_{m-1}$, using \scite{1.15}.
There is $n$ such that $[{\underset\tilde {}\to \gamma_n}[G_\beta],
\dbcu_{\ell < \omega} {\underset\tilde {}\to \gamma_\ell}[G_\beta])$ is
disjoint to $\{\xi_0,\dotsc,\xi_{m-1}\}$, hence for some $\varepsilon <
\omega$, letting $\xi = {\underset\tilde {}\to \zeta_\varepsilon}[G_\beta]$
defining ${\underset\tilde {}\to r^0_{\varepsilon,\xi}}[G_\beta \cap
\Bbb P_{\xi +1}]$ we get ${\underset\tilde {}\to {\bold t}^0_{\varepsilon,\xi}} =$
truth (see? \scite{1.27}) and $\xi < {\underset\tilde {}\to \gamma_n}
[G_\beta]$.

Now consider $N' = N_{<>}[G_\beta \cap \Bbb P_{\xi +1}]$ we know $N' \cap \omega_1
= N_{<>} \cap \omega_2$ (by clause (iv) above), and in it we have $p
\restriction (\xi +1) \cup {\underset\tilde {}\to r^0_{\varepsilon,\xi}}
[G_\beta \cap p_{\xi +1}]$ forces a bound to $\underset\tilde {}\to \tau$,
but the condition is $\le_{\text{pr}} p_1 \le_{\text{pr}} p_2$, and it
belongs to $N'$, so the value is $< N_{<>} \cap \omega_1$ and
we are done.
\enddemo
\bn
\ub{Clause (c)}:  

By \scite{5.2} it suffices to prove
\mr
\item "{$\bigotimes_2$}"  there are $r,\underset\tilde {}\to \eta$ such that:
\nl
$\underset\tilde {}\to \eta$ is a $\Bbb P_\beta$-name, $r \in \Bbb P_\beta,
r \restriction \gamma = q \restriction \gamma,p' \le_{pr} q$ and
$r \Vdash_{{\Bbb P}_\beta} ``\underset\tilde {}\to \eta \in \text{ lim}
(\underset\tilde {}\to T)$ and
$N_{{\underset\tilde {}\to \eta} \restriction \ell}
[{\underset\tilde {}\to G_{{\Bbb P}_\beta}}] \cap \omega_1 = N_{\langle \rangle}
\cap \omega_1$ and for every $y \in \bold V \cap \dbcu_{\ell < \omega}
N_{{\underset\tilde {}\to \eta} \restriction \ell}[G_{{\Bbb P}_\beta}]$ for some
$A \in \bold V \cap \dbcu_{\ell < \omega} N_{{\underset\tilde {}\to \eta}
\restriction \ell}$ we have $|A|^{\bold V} < {\underset\tilde {}\to \kappa^*_\beta}
[G_{{\Bbb P}_\beta}]$ and $y \in A"$,
\ermn

We shall choose by induction on $n < \omega,{\underset\tilde {}\to \gamma_n},
q_n,{\underset\tilde {}\to \rho_n},{\underset\tilde {}\to T_n},
{\underset\tilde {}\to k_n},{\underset\tilde {}\to p_n}$ such that:
\medskip
\roster
\item "{$(a)$}"  $({\underset\tilde {}\to \gamma_n},q_n,
{\underset\tilde {}\to \rho_n},{\underset\tilde {}\to T_n}) \in 
{\Cal T}'_{\bar N}$ \newline
(so ${\underset\tilde {}\to \gamma_n}$ is a simple $\bar{\Bbb Q}$-named ordinal)
\sn
\item "{$(b)$}"  ${\underset\tilde {}\to k_n}$ is a
$\Bbb P_{\underset\tilde {}\to \gamma_n}$-name of a natural number
\sn
\item "{$(c)$}"  ${\underset\tilde {}\to \rho_n}$ is a 
$\Bbb P_{\underset\tilde {}\to \gamma_n}$-name
\sn
\item "{$(d)$}"  $q_n \Vdash_{{\Bbb P}_{\underset\tilde {}\to \gamma_n}}
``{\underset\tilde {}\to \rho_n} \in {\underset\tilde {}\to T_n} \cap
{}^{\underset\tilde {}\to k_n}\text{Ord}"$
\sn
\item "{$(e)$}"  ${\underset\tilde {}\to \gamma_0} = \gamma$ and
$\Vdash_{\bar{\Bbb Q}}``{\underset\tilde {}\to \gamma_n} < 
{\underset\tilde {}\to \gamma_{n+1}} < \beta$ and
${\underset\tilde {}\to \gamma_{n+1}} \text{ non-limt}"$ \newline
i.e., if $\zeta < \beta$ and $G_{{\Bbb P}_\gamma}
\subseteq \Bbb P_\zeta$ is generic over $\bold V$ and $\zeta = {\underset\tilde {}\to 
\gamma_n}[G_{{\Bbb P}_\zeta}]$ then \nl
$r \in q_n \Rightarrow {\underset\tilde {}\to \zeta_n}[G_\zeta] < \zeta$ 
(i.e., is well defined $< \zeta$ or is forced to be not well defined),
\sn
\item "{$(f)$}"  $q_{n+1} \restriction {\underset\tilde {}\to \gamma_n} =
q_n$
\sn
\item "{$(g)$}"  $q_{n+1} 
\Vdash_{{\Bbb P}_{\underset\tilde {}\to \gamma_{n+1}}} 
``{\underset\tilde {}\to \rho_n} \triangleleft
{\underset\tilde {}\to \rho_{n+1}}$, so 
${\underset\tilde {}\to k_n} < {\underset\tilde {}\to k_{n+1}}$ and
${\underset\tilde {}\to T_{n+1}} \subseteq {\underset\tilde {}\to T_n}"$
\sn
\item "{$(h)$}"  ${\underset\tilde {}\to p_n}$ is a
$\Bbb P_{\underset\tilde {}\to \gamma_n}$-name, $p_0 = p,
{\underset\tilde {}\to p_n} \restriction {\underset\tilde {}\to \gamma_n}
\le_{\text{pr}} q_n$ and \newline
$q_n \Vdash_{{\Bbb P}_{\underset\tilde {}\to \gamma_n}} ``
{\underset\tilde {}\to p_n} \in N_{\underset\tilde {}\to \rho_n}
[G_{{\Bbb P}_{\underset\tilde {}\to \gamma_n}}] \cap \Bbb P_\beta$ and 
$p_n \restriction {\underset\tilde {}\to \gamma_n} \in G_{{\Bbb P}_
{\underset\tilde {}\to \gamma_n}}"$
\sn
\item "{$(i)$}"  $q_n \Vdash_{{\Bbb P}_{\underset\tilde {}\to \gamma_n}}
``{\underset\tilde {}\to p_n} \le^{{\Bbb P}_\beta}_{\text{pr}}
{\underset\tilde {}\to p_{n+1}} \in
N_{{\underset\tilde {}\to \rho_{n+1}}}[G_{{\Bbb P}_{\gamma_n}}] 
\cap \Bbb P_\beta"$
\sn
\item "{$(j)$}"  letting $\langle {\underset\tilde {}\to \tau_{\nu,\ell}}:
\ell < \omega \rangle$ list the $\Bbb P_\beta$-names of ordinals from $N_\nu$: 
for $m,\ell \le n$ we have:
\endroster
\medskip
$$
\align
q_n \Vdash_{{\Bbb P}_{\underset\tilde {}\to \gamma_n}} ``{\underset\tilde {}\to 
p_{n+1}} \text{ force that: } &a) \,\,\,\text{if } 
{\underset\tilde {}\to \tau_{{\underset\tilde {}\to \rho_n} 
\restriction m,\ell}} 
\text{ is a countable ordinal}, m \le {\underset\tilde {}\to k_n} \\
  &\quad \text{ then it is smaller than some }  
{\underset\tilde {}\to \tau'_{{\underset\tilde {}\to \rho_n} \restriction
m,\ell}} \in N_{\underset\tilde {}\to \rho_{n+1}}[
G_{{\Bbb P}_{\underset\tilde {}\to \gamma_{n+1}}}], \\
  &\quad \text{ a } 
P_{\underset\tilde {}\to \gamma_n} \text{-name of a countable ordinal} \\
  &b) \,\,\, \text{for some } 
A \in \bold V \cap N_{{\underset\tilde {}\to \eta_{n+1}} 
\restriction {\underset\tilde {}\to k_{n+1}}}, \\
  &\quad \, |A|^{\bold V} < \kappa^*_\beta \text{ and }
{\underset\tilde {}\to \tau_{{\underset\tilde {}\to \rho_n} \restriction
m,\ell}} \in A".
\endalign
$$
\medskip

\noindent
The induction is straight (later we shall show that 
$\dsize \bigcup_{n < \omega} q_n$ and $\underset\tilde {}\to \eta = 
\dsize \bigcup_{n < \omega}
{\underset\tilde {}\to \rho_n}$ are as required in $\bigotimes_2$) 
by clauses (a) + (b) proved above.
\bn
\centerline {$* \qquad * \qquad *$}
\bn
Because we need and have $(*)_1$ or $(*)_2 + (*)_3$ below:
\medskip
\roster
\item "{$(*)_1$}"  Assume $\le_{pr},\le_{vpr}$ are equal to $\le$ \nl
(i.e., $\Vdash_{{\Bbb P}_\beta} 
``\le^{\underset\tilde {}\to {\Bbb Q}_\beta}_{vpr}$ is
$\le^{\underset\tilde {}\to {\Bbb Q}_\beta}"$ for each $\beta < \alpha$),
if $p \in \Bbb P_\beta,\gamma < \beta,\underset\tilde {}\to \tau$
a $\Bbb P_\beta$-name of an ordinal \ub{then} there are $p',
{\underset\tilde {}\to \tau'}$ such that:
{\roster
\itemitem{ (i) }  ${\underset\tilde {}\to \tau'}$ 
is a $\Bbb P_\gamma$-name of an ordinal
\sn
\itemitem{ (ii) }  $p \le_{\text{pr}} 
p' \in \Bbb P_\beta$ and $p \restriction \gamma =
p' \restriction \gamma$
\sn
\itemitem{ (iii) }  $p' \Vdash_{{\Bbb P}_\beta} ``
\underset\tilde {}\to \tau = {\underset\tilde {}\to \tau'}"$. \nl
[why?  straight by \scitet{1.16}].
\endroster}
\sn
[Saharon: maybe below we are stuck with $\underset\tilde {}\to \zeta \in
[\gamma,\beta)$, but this suffices - need to change?]
\sn
\item "{$(*)_2$}"  \ub{old proof of clause (b)}:  if 
$p \in \Bbb P_\beta,\gamma < \beta,
\underset\tilde {}\to \tau$ is a $\Bbb P_\beta$-name of a countable ordinal,
\ub{then} there are $p',\tau'$ such that
{\roster
\itemitem{ (i) }  $\tau'$ is a $\Bbb P_\gamma$-name of a countable ordinal
\sn
\itemitem{ (ii) }  $p \le_{pr} p' \in \Bbb P_\beta$ and $p \restriction \gamma =
p' \restriction \gamma$
\sn
\itemitem{ (iii) }  $p' \Vdash_{{\Bbb P}_\beta} ``\underset\tilde {}\to \tau \le
\tau'"$, 
\endroster}
\sn
[why $(*)_2$?  let $\underset\tilde {}\to \zeta$ be the following simple
$\bar{\Bbb Q}$-named $[\gamma,\beta)$-ordinal: \nl
for $G_\zeta \subseteq \Bbb P_\zeta$ is generic over $\bold V$ for 
$\zeta \in [\gamma,\beta)$ we let
$\underset\tilde {}\to \zeta[G_\zeta] = \zeta$ if 
{\roster
\itemitem{ $(a)$ }  $p \restriction \zeta \notin G_\zeta$ \ub{or}: for some 
$p' \in \Bbb P_\beta$ we have $p' \restriction \zeta = p \restriction \zeta$ 
and \nl
$\Bbb P \models p \le_{\text{pr}} p'$ and 
$p' \Vdash_{{\Bbb P}_\beta/G_\zeta} ``\underset\tilde {}\to \tau < \tau^*"$ 
for some countable ordinal $\tau^*$ 
\sn
\itemitem{ $(b)$ }  for no $\xi \in [\gamma,\zeta)$ does clause (a) hold for
$\xi,G_\zeta \cap \Bbb P_\xi$. 
\endroster}
Now if for some $\gamma \in [\alpha,\beta)$ we have 
$p \Vdash_{{\Bbb P}_\alpha} ``\underset\tilde {}\to \zeta = \gamma"$ we are
done.  Also $\Vdash_{{\Bbb P}_\alpha} ``\underset\tilde {}\to \zeta
[{\underset\tilde {}\to G_{{\Bbb P}_\alpha}}]$ is well defined" as if $p \in
G_\alpha \subseteq \Bbb P_\alpha$ and $G_\alpha$ is generic over
$\bold V$, then for
some $q \in G_\alpha$ and countable ordinal $\tau^*$ we have
$q \Vdash ``\underset\tilde {}\to \tau = \tau^*$.  By the definition of
$\aleph_1-\text{Sp}_e(W)$-iteration for some $\zeta \in
[\gamma,\beta)$ we have
$\xi \in [\zeta,\beta) \Rightarrow [p \restriction \{\xi\}
\le^{\underset\tilde {}\to {\Bbb Q}_\xi}_{pr} q \restriction \{\xi\}$ or
$e = 4 \and p \restriction \{\xi\}$ not defined[?]]. \nl
Define $p'$ by: $p' \restriction \underset\tilde {}\to \zeta = 
p \restriction \underset\tilde {}\to \zeta$, and for
$\xi \in [\underset\tilde {}\to \zeta,\beta)$ we let 
$p' \restriction \{\xi\}$ be $q \restriction \{\xi\}$ if: 
$p \restriction \{\xi\} \le^{\underset\tilde {}\to {\Bbb Q}_\xi}_{pr}
q \restriction \{\xi\}$ or $e = 4 \and p \restriction \{\xi\}$ not defined.
We shall show that $p'$ is as required, hence
really $\Vdash_{{\Bbb P}_\alpha} ``\underset\tilde {}\to \zeta \in [\gamma,\beta)$ is
well defined".
So there is a $\Bbb P_{\underset\tilde {}\to \zeta}$-name of
${\underset\tilde {}\to p'}$ as appearing in the definition of
$\underset\tilde {}\to \zeta$ and it is, essentially, a member of
$\Bbb P_\beta$.
Now as we have finite apure support, the proof of
``$\underset\tilde {}\to \zeta[{\underset\tilde {}\to G_{{\Bbb P}_\alpha}}]$ is well
defined" gives $\Vdash_{{\Bbb P}_\alpha} ``\underset\tilde {}\to \zeta$ is not a
limit ordinal $> \alpha$".  Lastly $\Vdash_{{\Bbb P}_\alpha} ``\underset\tilde {}\to 
\zeta$ is not a successor ordinal $> \gamma"$ is proved by the property of
each ${\underset\tilde {}\to {\Bbb Q}_\xi}$.]
\sn
\item "{$(*)_3$}"  \ub{old proof of clause (a)}:  if 
$p \in \Bbb P_\beta,\gamma < \beta,\underset\tilde {}\to 
\tau$ a $\Bbb P_\beta$-name of a set of $< \kappa$ ordinals, \ub{then} there are
$p',{\underset\tilde {}\to \tau'}$ such that:
\sn
{\roster
\itemitem { (i) }  ${\underset\tilde {}\to \tau'}$ is a $\Bbb P_\gamma$-name
of a set of $< \kappa$ ordinals
\sn
\itemitem{ (ii) }  $p \le_{pr} p' \in \Bbb P_\beta$
\sn
\itemitem{ (iii) }  $p' \Vdash_{{\Bbb P}_\beta} ``\underset\tilde {}\to \tau
\subseteq {\underset\tilde {}\to \tau'}$.
\endroster}
\sn
[Why?  Similar to the proof of $(*)_2$; note that it is automatic if the
${\underset\tilde {}\to \kappa_i}$'s increase fast enough].
\ermn
Finishing the induction we let
$\underset\tilde {}\to \eta = \dsize \bigcup_{n < \omega}
{\underset\tilde {}\to \rho_n}$ and we define $q_\omega \restriction
{\underset\tilde {}\to \gamma_n} = q_n$, \nl
$q_\omega \restriction [\dsize \bigcup_{n < \omega} 
{\underset\tilde {}\to \gamma_n},\beta)$ is defined as $\le_{vpr}$-upper 
bound of $\langle {\underset\tilde {}\to p_m} \restriction
[\dsize \bigcup_{n < \omega} \gamma_n,\beta):m < \omega \rangle$.
\medskip

More formally, let $\gamma^*,\beta$ and $G_{\gamma^*} \subseteq
\Bbb P_{\gamma^*}$ be such that: $G_{\gamma^*}$ is generic over $\bold V,\gamma^* =
\dsize \bigcup_{n < \omega} \gamma^*_n,\gamma^*_n = 
{\underset\tilde {}\to \gamma_n}[G_{\gamma^*}]$, let $p'_n =
{\underset\tilde {}\to p_n}[G_{\gamma^*} \cap \Bbb P_{\gamma^*_n}]$, let
$p'_n \restriction [\gamma^*,\alpha) = \{{\underset\tilde {}\to r^n_\zeta}:
\zeta < \zeta^*_n\}$ where ${\underset\tilde {}\to r^n_\zeta}$ is a simple
$[\gamma^*,\alpha)$-named atomic condition. \newline
Now we define ${\underset\tilde {}\to s^n_\zeta}$, a simple
$[\gamma^*,\alpha)$-named atomic condition as follows:
\medskip
\roster
\item "{$(a)$}"  $\zeta_{\underset\tilde {}\to s^n_\zeta} =
\zeta_{\underset\tilde {}\to r^n_\zeta}$
\sn
\item "{$(b)$}"  if $\zeta \in [\gamma^*,\alpha),G_{\gamma^*} \subseteq
G_\gamma \subseteq \Bbb P_\gamma,G_\gamma$ generic over $V,
\zeta_{\underset\tilde {}\to r^n_\zeta}[G_\zeta] = \zeta$ then
${\underset\tilde {}\to s^n_\zeta}[G_\zeta]$ is the $<^{*V[G_\zeta]}$-first
elements of ${\underset\tilde {}\to {\Bbb Q}_\zeta}[G_\zeta]$ which satisfies the
following:
{\roster
\itemitem{ $(*)(\alpha)$ }  
$({\underset\tilde {}\to p_n} \restriction \{\zeta\}) \le_{vpr} s$
\sn
\itemitem{ $(\beta)$ }  if 
$\langle \emptyset_{{\underset\tilde {}\to {\Bbb Q}_\zeta}[G_\zeta]} \rangle
\char 94 \langle ({\underset\tilde {}\to p_m} \restriction \{\xi\})
[G_\zeta]:m \in (n,\omega) \rangle$ has a $\le_{vpr}$-upper bound then
${\underset\tilde {}\to s^n_\zeta}[G_\zeta]$ is such upper bound.
\endroster}
\ermn
Now actually such $\le_{\text{vpr}}$-upper 
bound actually exists, and $q_\omega$
is as required.  \hfill$\square_{\scite{5.5}}$\margincite{5.5}
\bigskip

\proclaim{\stag{5.6} Claim}  Suppose 
$\bold W \subseteq \omega_1$ is stationary
and $\bar{\Bbb Q} = \langle \Bbb P_i,
{\underset\tilde {}\to {\Bbb Q}_i},\Bbb I_i,
{\underset\tilde {}\to \kappa_i}:i < \alpha \rangle$ is 
a ${\text{\rm UP\/}}^\ell(\bold W,W)$-suitable iteration 
(where $\ell \in \{4,5\}$), and 
$\Bbb P_\alpha = { \text{\rm Sp\/}}_6(W)-{\text{\rm Lim\/}}(\bar{\Bbb Q})$.

Each of the following is a sufficient condition for ``$\dsize \bigcup_{\beta <
\alpha} \Bbb P_\beta$ is a dense subset of \nl
$\Bbb P_\alpha$":
\medskip
\roster
\item "{$(A)_\alpha$}"  $\beta \in W$ is 
\footnote{remember $W$ is a parameter in the definition of $\kappa-Sp_e(W)$-
iteration} strongly inaccessible and
$\dsize \bigwedge_{\beta < \alpha}$ density $(\Bbb P_\beta) < \alpha$
\sn
\item "{$(B)_\alpha$}"  for every $i,
\le^{\underset\tilde {}\to {\Bbb Q}_i}_{\text{vpr}}$
is equality and $\bold V \models ``\text{cf}(\alpha) = \aleph_1"$ or at least
for some $\beta < \alpha$ we have 
$\Vdash_{{\Bbb P}_\beta} ``\text{cf}(\alpha) = \aleph_1"$.
\endroster
\endproclaim
\bigskip

\demo{Proof} \newline
\underbar{Case 1}:  $(A)_\alpha$ \newline
Straight by the definition of $\kappa-\text{Sp}_6(W)$-iteration 
(see \scite{1.13}).
\mn
\underbar{Case 2}:  $(B)_\alpha$ \newline
Follows by \scite{5.5}.  \hfill$\square_{\scite{5.6}}$\margincite{5.6}
\enddemo
\bigskip

\demo{\stag{5.7} Conclusion}  Assume $\bold W \subseteq 
\omega_1$ is stationary
and $\bar{\Bbb Q} = \langle \Bbb P_i,{\underset\tilde {}\to {\Bbb Q}_i},
{\underset\tilde {}\to {\Bbb I}_i},{\underset\tilde {}\to \kappa_i}:
i < \alpha \rangle$ is a
UP$^4(\bold W,W)$-iteration with Sp$_e(W)$-limit $\Bbb P_\alpha$.

If $\{\beta < \alpha:\bar{\Bbb Q} 
\restriction \beta$ satisfies $(A)_\beta \vee
(B)_\beta$ from \scite{5.6}$\}$ is a stationary subset of $\alpha$ and
$\beta < \alpha \Rightarrow \Bbb P_\beta$ satisfies the
cf$(\alpha)$-c.c. (e.g.
has cardinality or at least density $< \text{ cf}(\alpha)$), \ub{then}
$\Bbb P_\alpha$ satisfies the cf$(\alpha)$-c.c.
\enddemo
\bigskip

\demo{Proof}  Straight.

We may like to iterate up to e.g. the first inaccessible (we may below
weaken $|\Bbb P_\beta| < \alpha$ to 
$\Bbb P_\beta$ satisfies the $\alpha$-c.c. if
$\Bbb P_\alpha = \dbcu_{\beta < \alpha} \Bbb P_\beta$).
\enddemo
\bigskip

\proclaim{\stag{5.8} Claim}  [See \scite{si6.11}]??  Assume
\medskip
\roster
\item "{$(a)$}"  $\bold W \subseteq \omega_1$ is stationary
and $\bar{\Bbb Q} = \langle \Bbb P_i,{\underset\tilde {}\to {\Bbb Q}_i},
{\underset\tilde {}\to {\Bbb I}_i},{\underset\tilde {}\to \kappa_i}:
i < \alpha \rangle$ is a ${\text{\rm UP\/}}^4(\bold W,W)$-iteration 
$(y \in \{ a,b\})$ with ${\text{\rm Sp\/}}_6(W)$-limit $\Bbb P_\alpha$
\sn
\item "{$(b)$}"  $\Vdash_{{\Bbb P}_\beta}$ ``if in 
${\underset\tilde {}\to {\Bbb Q}_\beta},
\emptyset_{\underset\tilde {}\to {\Bbb Q}_\beta} \ne p
\le_{\text{vpr}} q$, then
$p = q$ and $\emptyset^+_{\underset\tilde {}\to {\Bbb Q}_\beta} \in
\Bbb Q_\beta$ is
$\ne \emptyset_{\underset\tilde {}\to {\Bbb Q}_\beta}$ but
$\emptyset_{\underset\tilde {}\to {\Bbb Q}_\beta} \ne p \in 
{\underset\tilde {}\to {\Bbb Q}_\beta} \Rightarrow
\emptyset^+_{\underset\tilde {}\to {\Bbb Q}_\beta} \le_{\text{pr}} p')"$ [?]
\sn
\item "{$(c)$}"  $S \subseteq \{\delta < \alpha:{\text{\rm cf\/}}(\delta) =
\aleph_1\}$ is stationary; for a club $E$ of $\kappa,\delta \in E \cap S \and
\delta \le \beta$ implies $\Vdash_{{\Bbb P}_\beta} ``(\{r \in
{\underset\tilde {}\to {\Bbb Q}_\beta}:
\emptyset_{\underset\tilde {}\to {\Bbb Q}_\beta} \le_{\text{vpr}} r\},
\le_{\text{vpr}})$ is $\delta^+$-directed (question directed above
$p,\emptyset <_{\text{vpr}} p$
\sn
\item "{$(d)$}"  $\alpha \notin W$ is strongly inaccessible and:
$\beta < \alpha \Rightarrow \Bbb P_\beta < \alpha$.
\ermn
\underbar{Then}:
\mr
\item "{$(\alpha)$}"  forcing with $\Bbb P_\alpha$ 
does not collapse $\alpha$
\sn
\item "{$(\beta)$}"  any function from any $\alpha(*) < \alpha$ to
ordinals in $\bold V^{{\Bbb P}_\alpha}$ belongs to some $\bold
V^{{\Bbb P}_\beta}$.
\endroster
\endproclaim
\bigskip

\demo{Proof}  Clearly clause $(\alpha)$ follows from clause $(\beta)$, so
we shall prove just clause $(\beta)$.  If $W \cap \alpha$ is stationary,
then by \scite{5.7} we are done, so assume not and let $E$ be a club of
$\alpha$ disjoint to $W$, \wilog \, $\beta < \delta \in E \Rightarrow$
density$(\Bbb P_\beta) < \delta$.
Suppose $p \in \Bbb P_\alpha$ and $p \Vdash_{{\Bbb P}_\alpha} ``\underset\tilde {}\to f:
\alpha(*) \rightarrow \text{ Ord}$ is not in any 
$\bold V^{{\Bbb P}_\beta}$ for $\beta < \alpha"$ where $\alpha(*) < \alpha$. \newline
We choose by induction on $\zeta < \alpha$ the tuple 
$(p_\zeta,\alpha_\zeta,\gamma_\zeta,\beta_\zeta,q_\zeta)$ such that:
\medskip
\roster
\item "{$(a)$}"  $\beta_\zeta < \alpha$ is increasing continuous in $\zeta$
and $\beta_{\zeta +1} > \text{ Min}(E \backslash B_\zeta)$
\sn
\item "{$(b)$}"  [?] $p_\zeta \in \Bbb P_{\beta_\zeta}$ is such that 
$\beta(*) \le \beta < \alpha_\zeta \Rightarrow p_\zeta \restriction \beta 
\Vdash_{{\Bbb P}_\beta} ``p_\zeta \restriction \{\beta\} \ne 
\emptyset_{\underset\tilde {}\to {\Bbb Q}_\beta}$ or
${\underset\tilde {}\to {\Bbb Q}_\beta} = 
\{\emptyset_{\underset\tilde {}\to \beta}\}"$
\sn
\item "{$(c)$}"  for $\xi < \zeta,p_\zeta \restriction \beta_\xi = p_\xi$
\sn
\item "{$(d)$}"  $p_\zeta \le q_\zeta \in \Bbb P_\alpha$
\sn
\item "{$(e)$}"  $q_\zeta \Vdash_{{\Bbb P}_\alpha} ``\underset\tilde {}\to f
(\alpha_\zeta) = \gamma_\zeta"$ but there is no $q' \in \Bbb P_{\beta_\zeta}$
compatible with $p_\zeta$ which forces this and $\alpha_\zeta < \alpha(*)$,
of course
\sn
\item "{$(f)$}"  if cf$(\zeta) = \aleph_1$, then for some 
$\beta'_\zeta < \beta_\zeta$, \newline
$\gamma \in [\beta'_\zeta,\beta_\zeta) \Rightarrow q_\zeta \restriction \gamma
\Vdash_{{\Bbb P}_\gamma} ``\emptyset_{\underset\tilde {}\to {\Bbb Q}_\gamma} 
\le_{\text{vpr}} = q_\zeta \restriction \{\gamma\}"$
\sn
\item "{$(g)$}"  $q_\zeta \in \Bbb P_{\beta_{\zeta +1}}$ and for every $\beta \in
[\beta_\zeta,\alpha)$ we have
$$
\align
p_{\zeta +1} \restriction \beta \Vdash_{{\Bbb P}_\beta} ``&\text{if }
\emptyset_{\underset\tilde {}\to {\Bbb Q}_\beta} <_{\text{vpr}} q_\zeta \restriction
\{\beta\} \text{ in } {\underset\tilde {}\to {\hat{\Bbb Q}}_\beta} \text{ then} \\
  &p_{\zeta +1} \restriction \{\beta\} = q_\zeta \restriction \{\beta\}; \\
  &\text{if not [?] then } r \in {\underset\tilde {}\to {\Bbb Q}_\beta} \text{ such
that } \emptyset_{\underset\tilde {}\to {\Bbb Q}_\beta} <_{\text{vpr}} r \text{ and }
\emptyset_{\underset\tilde {}\to {\Bbb Q}_\beta} \text{ if there is none}"
\endalign
$$
\noindent
(see clause (b) in the assumptions of \scite{5.8}).
\endroster
\medskip

\noindent
Having carried the definition, for some stationary $W' \subseteq S
\subseteq \{\delta < \alpha:\text{cf}(\delta) = \aleph_1\}$ and 
$\gamma^* < \alpha$ and $\beta'$ we have:
$\zeta \in W' \Rightarrow \gamma_\zeta = \gamma^* \and \beta'_\zeta \le
\beta' < \zeta$. \nl
As $|\Bbb P_{\gamma^*}| < \alpha = \text{ cf}(\alpha)$, \wilog \,
$\zeta \in W' \Rightarrow q_\zeta \restriction
\gamma_\zeta = q^*$.  Now choose $\xi < \zeta$ in $W'$ then $q_\zeta,q_\xi$
are compatible and an easy contradiction to clause (c) (with $q_\xi$ here
playing the role of $q'$ there). \hfill$\square_{\scite{5.8}}$\margincite{5.8}
\enddemo
\bn
Now we can refine \scite{1.16A} to the iteration theorem of this section.
\proclaim{\stag{5.9} Claim}  1) Suppose $\bold W \subseteq \omega_1$ 
be stationary, $F$ is a function, \ub{then} for every ordinal 
$\alpha$ there is ${\text{\rm UP\/}}^4(\bold W)$-iteration 
$\bar{\Bbb Q} = \langle \Bbb P_i,{\underset\tilde {}\to {\Bbb Q}_i},
\kappa_i:i < \alpha^\dag \rangle$, such that:
\mr
\item "{$(a)$}"  for every $i$ we have ${\underset\tilde {}\to {\Bbb Q}_i} = 
F(\bar{\Bbb Q} \restriction i)$ and $\kappa_i = \kappa_{cc}(\Bbb P_i *
\Bbb Q_i)$
\sn
\item "{$(b)$}"  $\alpha^\dag \le \alpha$
\sn
\item "{$(c)$}"  either $\alpha^\dag = \alpha$ or the following fails:
{\roster
\itemitem{ $(*)$ }  $F(\bar{\Bbb Q})$ is an (${\text{\rm
Sp\/}}_e(W)-{\text{\rm Lim\/}}(\bar{\Bbb Q}))$-name of a 
forcing notion forced to satisfy ${\text{\rm UP\/}}^4(\Bbb I,\bold W)$ 
for some $\Bbb I$
$\kappa$-complete set of ideals, where $\kappa$ is minimal such that
${\text{\rm Sp\/}}_e(W)-{\text{\rm Lim\/}}(\bar{\Bbb Q})$ satisfies 
the $\kappa$-c.c
\endroster}
\item "{$(d)$}"  ${\text{\rm Sp\/}}_e(W)-{\text{\rm Lim\/}}
(\bar{\Bbb Q})$ does not collapse $\aleph_1$ and
preserve stationary subsets of $\bold W$ (in fact it satisfies
${\text{\rm UP\/}}^1(\bold W)$.
\endroster
\endproclaim
\bigskip

\demo{Proof}  Straight.
\enddemo
\newpage

%begin part 2!!!!!! just inserted

\head {\S6 Preservation of UP$^0$} \endhead  \resetall \sectno=6
 % \resetall 
\bn
Here we present alternatives to \S5, i.e. to UP$^4(\bold W)$-iterations. 
In UP$^6$-iteration $\langle \Bbb P_j,{\underset\tilde {}\to {\Bbb
Q}_i},
\kappa_j:
k \le \alpha,i < \alpha \rangle$ the demands are weak but the
$\kappa_{i+j}$ may be large and it is quite similar to 
semi-properness (but for
fewer models).  In UP$^6$-iteration we carry with us trees.
\nl
Recall [?]
\definition{\stag{si.1} Definition}  1) We say $q$ is 
$(N,\kappa,\Bbb Q)$-semi-generic \ub{if}
$q$ is $(N,\Bbb Q)$-semi-generic and 
$q \Vdash ``\text{if } y \in
N[\underset\tilde {}\to G] \cap \bold V$ 
then for some $A \in N,|A|^{\bold V} < \kappa$ and $y \in A"$. \nl
2) Similarly with $\underset\tilde {}\to \kappa$ instead of $\kappa$.
\enddefinition
\bigskip

\remark{Remark}  In 
\scite{si.1}(1) \wilog \, $y \in N[\underset\tilde {}\to G]
\cap \text{Ord}$.
\endremark
\bigskip

\proclaim{\stag{si.2} Lemma}  Suppose
\mr
\item "{$(A)$}"  $\Bbb Q$ is a forcing notion
\sn
\item "{$(B)$}"  $\Bbb I \in N$ is a family of ideals, $\Bbb I$ is 
$\kappa$-closed, $\lambda \ge \aleph_2$ (and it is
natural but not needed to assume that $\Bbb I$ is $\lambda$-complete)
\sn
\item "{$(C)$}"  $N$ is $\lambda$-strictly $(\Bbb I,\bold W)$-suitable
for $(\chi,x)$ as witnessed by \newline
$\bar N = \langle N_\eta:\eta \in (T,\bold I)\rangle,
\Bbb Q \in N$ (so $N \prec ({\Cal H}(\chi),\in,<^*_\chi)$ is countable,
$N = N_{<>}$ and, of
course, $x$ codes $\langle \Bbb Q,\Bbb I,\bold W \rangle)$, see Definition
\scite{4.1})
\sn
\item "{$(D)$}"  $q$ is $(N,\kappa,\Bbb Q)$-semi-generic
\sn
\item "{$(E)$}"  at least one of the following holds:
{\roster
\itemitem{ $(\alpha)$ } $|{\text{\rm MAC\/}}(\Bbb Q)| < \lambda$ 
\sn
\itemitem{ $(\beta)$ }  $q$ is $(\bar N,\kappa,\Bbb Q)$-semi-generic, 
i.e. $(N_\eta,\kappa,\Bbb Q)$-semi-generic for every 
$\underset\tilde {}\to \eta \in T$ 
\sn
\itemitem{ $(\gamma)$ } $q \Vdash ``\underset\tilde {}\to T \subseteq
T$ is a subtree and every $\eta \in \underset\tilde {}\to T$ for some $\nu,
\eta \trianglelefteq \nu \in { \text{\rm split\/}}
(\underset\tilde {}\to T,\bold I),
\bold I_\eta \le_{RK} \bold I_\nu$ and $\eta \in 
\underset\tilde {}\to T \Rightarrow 
N_\eta[{\underset\tilde {}\to G_{\Bbb Q}}] \cap \omega_1 = 
N_\eta \cap \omega_1"$.
\endroster}
\ermn
\ub{Then} 
$q \models ``N[{\underset\tilde {}\to G_{\Bbb Q}}]$ is $\lambda$-strictly
$(\Bbb I,\bold W)$-suitable for $(\chi,\langle x,
{\underset\tilde {}\to G_{\Bbb Q}} \rangle)"$ (in fact, if \nl
$\langle N_\eta:\eta \in (T,\Bbb I) \rangle$ was a witness for 
$N$ \ub{then} $(\langle N_\eta
[\underset\tilde {}\to G]:\eta \in (T,\bold I) \rangle$ is a witness for
$N[\underset\tilde {}\to G]$ being $(\Bbb I,\bold W)$-suitable for
$(\chi,x)$).
\endproclaim
\bigskip

\demo{Proof}  Let $G \subseteq \Bbb Q$ be generic over $\bold V$ such that $q \in G$.

In $\bold V^{\Bbb Q}$, i.e., in $\bold V[G]$,  
clearly $N_\eta[G] \prec ({\Cal H}(\chi)^{\bold V^{\Bbb Q}},
\in,<^*_\chi)$ (see
\cite[III,2.11,p.104]{Sh:f}, $N_\eta[G]$ is countable (trivially) and
$N_{\eta \restriction k}[G] \prec N_\eta[G]$ for $k \le \ell g(\eta)$.

As $q$ is $(N,\kappa,{\Bbb Q})$-semi-generic and $N
= N_{\langle \rangle}$, clearly
$q$ is $(N_{\langle \rangle},\Bbb Q)$-semi-generic.  First assume
$|\text{MAC}(\Bbb Q)| < \lambda$: now $\Bbb Q \in N$ hence MAC$(\Bbb
Q) \in N$ hence $|\text{MAC}(\Bbb Q)| \in N$, and as $N_{\langle
\rangle} cap \lambda <_\lambda
N_\eta \cap \lambda$ clearly $N_\eta \cap |\text{MAC}(\Bbb Q)| =
N_{\langle \rangle} \cap |\text{MAC}(\Bbb Q)|$ hence also $N_\eta \cap
\text{ MAC}(\Bbb Q) = N_{\langle \rangle} \cap \text{ MAC}(\Bbb Q)$ hence
every $\Bbb Q$-name $\underset\tilde {}\to \tau \in N_\eta$ of an ordinal belongs
to $N_{\langle \rangle}$ (essentially).  Hence $q$ is
$(N_\eta,\kappa,\Bbb Q)$-semi-generic, so $q \Vdash_{\Bbb Q} ``N_\eta[G] \cap \omega_1 =
N_\eta \cap \omega_1 = N \cap \omega_1"$, (even $N_{\eta \restriction \ell}
[G] <_\lambda N_\eta[G]$).  So $q$ is
$(N_\eta,\kappa,\Bbb Q)$-semi-generic for any $\eta\in T$ and even $\eta 
\in (\text{lim }T)^{\bold V}$ or $\eta \in 
(\text{lim }T)^{\bold V^{\Bbb Q}}$.

This almost shows that $\langle N_\eta[G]:\eta \in (T,\bold I) \rangle$ is 
a witness to $N[G]$ being $\lambda$-strictly $(\Bbb I,\bold
W)$-suitable for $(\chi,x)$.

The missing point is clause $(e)^-$ of Definition \scite{4.1}, that is,
that there may be $\eta \in T$, and $J \in \Bbb I \cap
N_\eta[G]$ such that $J \notin N_\eta$.  So there is a $\Bbb Q$-name
$\underset\tilde {}\to \tau \in N_\eta$ satisfying 
$J = \underset\tilde {}\to \tau[G]$; choose
a $\underset\tilde {}\to \tau$ like that with min$\{|Y|:Y \in N_\eta$ and
$\underset\tilde {}\to \tau[G] \in Y\}$ minimal, and let the set $Y$
be $\{J_i:i < \alpha\}$ (without loss of generality $Y \subseteq \Bbb I$), so
without loss of generality $\langle J_i:i < \alpha \rangle$ belongs to
$N_\eta$: by the minimality of $|Y| = |\alpha|$ and $q$ being 
$(N_\eta,\kappa,\Bbb Q)$-semi-generic we have $\alpha < \kappa$.

So as $\{J_i:i < \alpha\} \in N_\eta \cap \Bbb I$ and $\Bbb I$ is 
$\kappa$-closed there is $J \in N_\eta \cap \Bbb I$ such that 
$\dsize \bigwedge_{i < \alpha} J_i \le_{\text{RK}} J$
hence the set $\{\nu:\eta \triangleleft \nu \in \text{ split}(T)$ and
$J \le_{\text{RK}} \bold I_\nu\}$ is a front of $T^{[\eta]}$.  So 
$\langle N_\eta[G]:\eta \in (T,\Bbb I) \rangle$ is a
witness for ``$N[G]$ is $\lambda$-strictly $\Bbb I$-suitable" for 
$(\lambda,\chi,\langle x,G \rangle)$ (see Definition \scite{4.1}) so by 
\scite{4.4} we know that $N[G]$ is $\Bbb I$-suitable. \nl
Now if the second phrase of (E) holds, the proof is similar and if the third
holds, we can prove that \wilog \, 
the second holds.  \hfill$\square_{\scite{si.2}}$\margincite{si.2}
\enddemo
\bigskip

\noindent
The proof suggests some definitions, but we first consider:
\definition{\stag{si.3} Definition}  1) For a forcing notion $\Bbb Q$, family 
$\Bbb I$ of ideals and cardinal and $\Bbb Q$-names
$\underset\tilde {}\to \lambda$ of a cardinal $\kappa$ and stationary 
$\bold W \subseteq \omega_1$, we say $\Bbb Q$ satisfies 
UP$^6(\Bbb I,\kappa,\underset\tilde {}\to \lambda,\bold W)$ or
UP$_{\kappa,\underset\tilde {}\to \lambda}(\Bbb I,\bold W)$ \ub{if}:
\mr
\item "{$(*)$}"  for every $\chi$ regular large enough, $p \in \Bbb Q$ 
and $N$ a strictly $(\Bbb I^{[\kappa]},\bold W)$-suitable model for 
$\chi$ satisfying $\{p,\Bbb Q,\kappa,\underset\tilde {}\to \lambda\} \in N$, 
\ub{there is} $q$ satisfying $p \le_{\text{pr}} q \in \Bbb Q$ such that 
$q$ is $(N,\Bbb Q)$-semi-generic
and $q \Vdash_{\Bbb Q} ``N[{\underset\tilde {}\to G_{\Bbb Q}}]$ is strictly
$\Bbb I^{[{\underset\tilde {}\to \lambda}]}$-suitable model for 
$\chi,\lambda$.
\ermn
1A)  We say ``$q$ is $(N,{\underset\tilde {}\to {\Bbb I}},\Bbb Q)$-semi$_6$ generic" if
$q$ is $(N,\Bbb Q)$-semi genric and $q \Vdash ``N[{\underset\tilde
{}\to G_{\Bbb Q}}]$
is strictly ${\underset\tilde {}\to {\Bbb I}}$-suitable". \nl
2) In part (1), if we omit $\lambda$, we mean $\lambda = \text{ Max}\{\lambda:\Bbb I
\text{ is } \lambda$-complete$\}$, so $\lambda$ is regular $> \aleph_1$. \nl
3) For 
$\Bbb Q,\Bbb I,\bold W,\kappa,\underset\tilde {}\to \lambda$ as in part
(1) and $\underset\tilde {}\to \theta$ a $\Bbb Q$-name of a cardinal; 
we say that
$\Bbb Q$ satisfies UP$^6_{\kappa,\underset\tilde {}\to
\lambda,\underset\tilde {}\to \theta}
(\Bbb I,\bold W)$ \ub{if}:
\mr
\item "{$(**)$}"  if $\Bbb I^+$ is a set of partial orders ideal extending
$\Bbb I,\chi$ large enough, $p \in \Bbb Q \cap N$ and $N$ a strictly
$(\Bbb I,\bold W)$-suitable model for $\chi,\{p,\Bbb Q,\kappa,
\underset\tilde {}\to \lambda,\underset\tilde {}\to \theta\} \in N$, \ub{then}
there is $q$ satisfying $p \le_{pr} q \in \Bbb Q$ such that $q$ is $(N,
\Bbb I^{[\underset\tilde {}\to \lambda]} \cup (\Bbb I^+ \backslash \Bbb I)
^{[\underset\tilde {}\to \theta]},\Bbb Q)$-semi$_6$-generic 
(see (1A) above).
\ermn
4) We say $\Bbb Q$ satisfies UP$^5_{\kappa,{\underset\tilde {}\to \lambda}}
(\Bbb I,\bold W)$ \ub{if}: 
for any $(\Bbb I^{[\kappa]},\bold W)$-suitable tree
$\langle N_\eta:\eta \in (T,\Bbb I) \rangle$ of models and $p \in N_{<>}$
there are $q,\underset\tilde {}\to T$ such that $p \le_{\text{pr}} q
\in \Bbb Q,q \Vdash ``\underset\tilde {}\to T \subseteq T$ is a subtree and
$\langle N_\eta[G_P]:\eta \in \underset\tilde {}\to T \rangle$ is a
$(\Bbb I^{[\lambda]},\bold W)$-suitable tree of  models and
$N_\eta[G_{\Bbb Q}] \cap \omega_1 = N_{<>} \cap \omega_1$.  [Saharon
but \sciteu{4.x}!]
\enddefinition
\bn
Some variants of this Definition are equivalent by the following claim.
\proclaim{\stag{si.4} Claim}  1) If $\kappa$ is regular uncountable and
$\Bbb Q$ satisfies the $\kappa$-c.c. \ub{then}:
$q \in \Bbb Q$ is $(N,\kappa,\Bbb Q)$-semi-generic \ub{iff} $q \in
\Bbb Q$ is $(N,\Bbb Q)$-semi-generic. \nl
2) If $N$ is 
$(\Bbb I,\bold W)$-suitable for $(\chi,x)$ (see Definition \scite{4.2}),
$\Bbb I$ is $\lambda$-complete, $\lambda$ is regular and 
$(\forall \alpha < \lambda)(|\alpha|^{\aleph_0} < \lambda)$,
\ub{then} there is a $\lambda$-strictly $(\Bbb I,\bold W)$-suitable $N'$
for $(\chi,x)$ such that $N \prec N'$ and $N' \cap \omega_1 = N \cap
\omega_1$.  \nl
3)  If $q$ is $(N,\kappa,\Bbb Q)$-semi generic, 
$|{\text{\rm MAC\/}}(\Bbb Q)| < \lambda,
\Bbb I \in N$ is $\kappa$-closed $\lambda$-complete and $N$ is
$\Bbb I$-suitable \ub{then} $q$ is $(N,\Bbb I^{[\lambda]},\Bbb Q)$-semi$_6$ 
generic. \nl
4) If $|{\text{\rm MAC\/}}(\Bbb Q)| < \lambda$ 
and $\Bbb Q$ is ${\text{\rm UP\/}}^0(\Bbb I^{[\kappa]},
\bold W)$ \ub{then} $\Bbb Q$ is 
${\text{\rm UP\/}}^6_{\kappa,\lambda}(\Bbb I,\bold W)$.
Similarly for $\lambda \, 
\Bbb Q$-name such that for a dense set of $p$ we have
$p \Vdash \underset\tilde {}\to \lambda = \lambda$ and 
${\text{\rm MAC\/}}(\Bbb Q_{\ge p}) <
\lambda$.
\endproclaim
\bigskip

\demo{Proof}  Straight. \nl
1) Reflect. \nl
2) Use the partition theorem \scite{2.12}. \nl
3) By \scite{si.2} using possibility (A). \nl
4) By \scite{si.2}, too.  \hfill$\square_{\scite{si.4}}$\margincite{si.4}
\enddemo
\bigskip

\proclaim{\stag{si.4a} Claim}  1) If $\Bbb Q$ satisfies 
${\text{\rm UP\/}}^6_{\kappa,{\underset\tilde {}\to \lambda}}
(\Bbb I,\bold W)$ and $\bold W_1 \subseteq
\bold W,\kappa_1 \le \kappa$ and 
$\Vdash_{\Bbb Q} ``\underset\tilde {}\to \lambda \le
{\underset\tilde {}\to \lambda_1}"$, \ub{then} $\Bbb Q$ satisfies
${\text{\rm UP\/}}^6_{\kappa_1,
{\underset\tilde {}\to \lambda_1}}(\Bbb I,\bold W_1)$. \nl
2) If $\Bbb Q_0$ is a forcing notion satisfying 
${\text{\rm UP\/}}^6_{\kappa_0,
{\underset\tilde {}\to \kappa_1}}(\Bbb I,\bold W)$ and
${\underset\tilde {}\to {\Bbb Q}_1}$ is a $\Bbb Q$-name 
of a forcing notion satisfying
${\text{\rm UP\/}}^6_{{\underset\tilde {}\to \kappa_1},
{\underset\tilde {}\to \kappa_2}}
(\Bbb I,\bold W_1)$, \ub{then} $\Bbb Q_0 * {\underset\tilde {}\to
{\Bbb Q}_1}$ is a
forcing notion satisfying 
${\text{\rm UP\/}}^6_{\kappa_0,{\underset\tilde {}\to \kappa_2}}
(\Bbb I,\bold W_1)$. \nl
3) If $\Bbb Q$ satisfies 
${\text{\rm UP\/}}^6_{\kappa,\underset\tilde {}\to \lambda,
\underset\tilde {}\to \theta}(\Bbb I,\bold W_1)$ and $\Vdash_{\Bbb Q}
\underset\tilde {}\to \theta \le \theta$ and $\Bbb I \subseteq \Bbb I^+$
an $\Bbb I^+ \backslash \Bbb I$ is $\theta$-complete \ub{then} 
$\Bbb Q$ satisfies
${\text{\rm UP\/}}^6_{\kappa,\underset\tilde {}\to \lambda}
(\Bbb I',\bold W_1)$.
\endproclaim
\bigskip

\definition{\stag{si.5} Definition}  1) We say that $\bar{\Bbb Q} = \langle
\Bbb P_j,{\underset\tilde {}\to {\Bbb Q}_i},
{\underset\tilde {}\to \kappa_j}:j \le \alpha$ 
and $i < \alpha \rangle$ 
is a UP$^{6,e}(\Bbb I,\bold W,W)$-suitable iteration (with $e=6$ if not
mentioned explicitly) \ub{if}:
\medskip
\roster
\item "{$(a)$}"  $\langle 
\Bbb P_j,{\underset\tilde {}\to {\Bbb Q}_i}:j < \alpha,i <
\alpha \rangle$ is an $\aleph_1-\text{Sp}_e(W)$-iteration
\sn
\item "{$(b)$}"  $\Bbb I$ is a set of partial 
order ideals such that $\Bbb I^{[\kappa]}$ is $\kappa$-closed for any
regular $\kappa,\aleph_2 \le \kappa \le |\Bbb P_\alpha|^+$
\sn
\item "{$(c)$}"  $\bold W \subseteq \omega_1$ is stationary
\sn
\item "{$(d)$}"  for each $i < \alpha$ we have: 
${\underset\tilde {}\to \kappa_i}$ is a $\Bbb P_i$-name of a regular cardinal
$> \aleph_1$ in $\bold V$ [purity?],
\sn
\item "{$(e)$}"  $(i) \quad$  for $i < j$ we have $\Vdash_{{\Bbb P}_j} 
``{\underset\tilde {}\to \kappa_i} \le
{\underset\tilde {}\to \kappa_j}"$
\sn
\item "{${{}}$}"  $(ii) \quad$ if $\delta \le \alpha$ is a limit ordinal,
then $\Bbb P_\delta$ satisfies the local 
${\underset\tilde {}\to \kappa_\delta}$-c.c. purely \nl

$\qquad$ [and (?) if
$\nVdash {\underset\tilde {}\to \kappa_\delta} \ne \kappa$ then some
$\kappa$-complete ideal on \nl

$\qquad \kappa$ belongs to $\Bbb I]$
\sn
\item "{$(f)$}"  $\Vdash_{{\Bbb P}_i} 
``{\underset\tilde {}\to {\Bbb Q}_i}$ satisfies
UP$^6_{{\underset\tilde {}\to \kappa_i},{\underset\tilde {}\to \kappa_{i+1}}}
({\underset\tilde {}\to {\Bbb I}_i},\bold W)$ and
$({\underset\tilde {}\to {\Bbb Q}_i},\le_{\text{vpr}})$ 
is $\aleph_1$-complete".
\ermn
2) We say $\langle \Bbb P_j,{\underset\tilde {}\to {\Bbb Q}_i},
{\underset\tilde {}\to \kappa_j}:j \le \alpha,i < \alpha \rangle$ is a
UP$^4(\Bbb I,\bold W)$-iteration \ub{if}:
\mr
\item "{$(a)-(e)$}"  as above
\sn
\item "{$(f)$}"  $\Bbb Q_i$ 
satisfies UP$^5_{{\underset\tilde {}\to \kappa_i},
{\underset\tilde {}\to \kappa_{i+1}}}(\Bbb I,\bold W)$.
\endroster
\enddefinition
\bigskip

\definition{\stag{si.6} Definition}  1) We 
say that $\langle \Bbb P_j,{\underset\tilde {}\to {\Bbb Q}_i},
{\underset\tilde {}\to \kappa_j}:j \le \alpha$ and $i < \alpha \rangle$ is a 
weak UP$^{6,e}(\Bbb I,\bold W)$-iteration \ub{if} 
(when $e=6$ we may omit it):
\mr
\item "{$(a)$}"  $\langle 
\Bbb P_i,{\underset\tilde {}\to {\Bbb Q}_i}:i < \alpha \rangle$ is 
an $\aleph_1$-SP$_e(W)$-iteration (and, of course, $\Bbb P_\alpha =
\text{ Sp}_e(W)$-Lim$_\kappa(\bar{\Bbb Q}))$
\sn
\item "{$(b)$}"  $\Bbb I$ is a set of partial order ideals
\sn
\item "{$(c)$}"  $\bold W \subseteq \omega_1$ stationary
\sn
\item "{$(d)$}"  ${\underset\tilde {}\to \kappa_j}$ is a $\Bbb P_j$-name of a
member of RCar$^V \backslash \omega_2$, increasing with $j$
\sn
\item "{$(e)$}"  for $i < j \le \alpha,i$ nonlimit we have
$$
\Vdash_{P_i} ``\Bbb P_j/\Bbb P_i 
\text{ satisfies the \, UP}_{{\underset\tilde {}\to \kappa_i},
{\underset\tilde {}\to \kappa_j}}(\Bbb I,\bold W)"
$$
\item "{$(f)$}"  $\Vdash_{{\Bbb P}_i} ``({\underset\tilde {}\to {\Bbb Q}_i},
\le_{\text{vpr}})$ is $\aleph_1$-complete".
\ermn
2) We say $\bar{\Bbb Q} = \langle \Bbb P_j,{\underset\tilde {}\to
{\Bbb Q}_i},
{\underset\tilde {}\to \kappa_j}:j \le \alpha,i < \alpha \rangle$ is a
UP$^5(\Bbb I,\bold W)$-iteration \ub{if}:
\mr
\item "{$(a)-(d),(f)$}"  as above
\sn
\item "{$(g)$}"  if $i < j \le \alpha,i$ non limit we have \nl
$\Vdash_{{\Bbb P}_i} 
``\Bbb P_j/{\underset\tilde {}\to G_{{\Bbb P}_i}}$ satisfies
UP$^5_{{\underset\tilde {}\to \kappa_i},{\underset\tilde {}\to \kappa_j}}
(\Bbb I,\bold W)$.
\endroster
\enddefinition
\bigskip

\proclaim{\stag{si.7} Claim}  1) If $\langle \Bbb P_j,
{\underset\tilde {}\to {\Bbb Q}_i},
{\underset\tilde {}\to \kappa_j}:j \le \alpha$ and $i < \alpha \rangle$ is
a ${\text{\rm UP\/}}^6(\Bbb I,\bold W,W)$-iteration, \ub{then} it is a weak
${\text{\rm UP\/}}^6(\Bbb I,\bold W,W)$-iteration, moreover, in clause
(e) also limit $i$ is O.K. \nl
2) Assume $\bar{\Bbb Q} = \langle \Bbb P_j,{\underset\tilde {}\to
{\Bbb Q}_i},
{\underset\tilde {}\to \kappa_j}:j \le \alpha,i < \alpha \rangle$ is
an $\aleph_1-{\text{\rm Sp\/}}_e(W)$-iteration
\mr
\item "{$(a)$}"  if $\alpha$ is a limit ordinal and $\beta < \alpha
\Rightarrow \bar{\Bbb Q} \restriction \beta = \langle \Bbb P_j,
{\underset\tilde {}\to {\Bbb Q}_i},{\underset\tilde {}\to \kappa_j}:
j \le \beta,i < \beta \rangle$ is a weak 
${\text{\rm UP\/}}^6(\Bbb I,\bold W)$-iteration and
${\underset\tilde {}\to \kappa_\alpha} = \text{ sup}\{\kappa_j:j < \alpha\}$,
\ub{then} $\bar{\Bbb Q}$ is a weak 
${\text{\rm UP\/}}^6(\Bbb I,\bold W)$-iteration
\sn
\item "{$(b)$}"  if $\alpha = \beta +1,\bar{\Bbb Q} \restriction \beta$ is a
weak ${\text{\rm UP\/}}^6(\Bbb I,\bold W)$-iteration and in $\bold V^{{\Bbb
P}_\beta},\Bbb Q_\beta$ satisfies
${\text{\rm UP\/}}^6_{{\underset\tilde {}\to \kappa_\beta},
{\underset\tilde {}\to \kappa_{\beta +1}}}(\Bbb I,\bold W)$ \ub{then}
$\bar{\Bbb Q}$ is a weak ${\text{\rm UP\/}}^6(\Bbb I,\bold W)$-iteration. \nl
[Saharon - compare with \scite{si.8} and \scite{1.27A}(2)]
\endroster
\endproclaim
\bigskip

\demo{Proof}  Let $\gamma \le \beta \le \alpha$ and we need
\mr
\item "{$\boxtimes_{\gamma,\beta}$}"  Assume $G_\gamma \subseteq \Bbb P_\gamma$
is generic over $\bold V,\kappa_i = {\underset\tilde {}\to \kappa_i}[G_i]$ and
let $N \in \bold V[G_\gamma],N$ is strictly 
$\Bbb I^{[\kappa]}$-suitable, $N \cap \omega_1 \in \bold W$ (so
$N \prec ({\Cal H}(\chi),\in)$ is countable) and $p \in
\Bbb P_\beta/G_\gamma$ and $p \in \Bbb P_\beta \cap N$ and 
$\bar{\Bbb Q},\beta,\gamma \in N$. 
\ub{Then} we can find $q$ satisfying 
$p \le_{\text{pr}} q \in \Bbb P_\beta/G_\gamma$ and
$q \Vdash_{{\Bbb P}_\beta/G_\gamma} ``N[{\underset\tilde {}\to G_\beta}] \cap
\omega_1 = N \cap \omega_1$ and $N[{\underset\tilde {}\to G_\beta}]$ is
$\Bbb I^{[{\underset\tilde {}\to \kappa_\beta}]}$-suitable".\nl
Without loss of generality [??] $p$ forces a value to 
${\underset\tilde {}\to \kappa_\beta}$ moreover
$\kappa_\beta = \kappa_{\underset\tilde {}\to \kappa_\beta}(p,\Bbb P_\beta)$].
\ermn
Naturally, we prove this by induction on $\beta$ (for all $\gamma$).  The
case $\gamma = \beta$ holds trivially so assume $\gamma < \beta$.
If $\beta = 0$, we have nothing to prove.  If $\beta$ is a successor 
ordinal say $\gamma_1 +1$  so $\gamma \le \gamma_1$, now we use first 
$\boxtimes_{\gamma,\gamma_1}$ and then
the demand on ${\underset\tilde {}\to Q_{\gamma_1}}$ in definition
\scite{si.6}, in clause (f).
\enddemo
\bn
So from now on we shall assume that $\beta$ is a limit ordinal.  As in the
proof of \scite{5.5} we can note \nl
\ub{Fact A}:  If ${\underset\tilde {}\to \gamma_1} \le
{\underset\tilde {}\to \gamma_2}$ are simple $\bar{\Bbb Q}$-named 
$[0,\beta)$-ordinals then $\boxtimes_{{\underset\tilde {}\to \gamma_1},
{\underset\tilde {}\to \gamma_2}}$ holds.
\bn
\ub{Proof of the fact}:  Here we use ``$e = 6$" rather than ``$e=4$".  On
$\Bbb P_{\underset\tilde {}\to \zeta}$ see Definition \scite{1.13}(F)(g).  We prove it
by induction on the depth of ${\underset\tilde {}\to \zeta_2}$, see
Definition \scite{1.6}(5).  So we are given
$G_{\underset\tilde {}\to \zeta_2} \subseteq 
\Bbb P_{\underset\tilde {}\to \zeta_1}$ generic over $\bold V$ and in particular let
$\zeta_1 = {\underset\tilde {}\to \zeta_2}
[G_{\underset\tilde {}\to \zeta_1}]$, (so it is simpler to say that
$G_{\zeta_1} \subseteq \Bbb P_{\zeta_1}$ is generic over $\bold V,
{\underset\tilde {}\to \zeta_1}[G_{\zeta_1}] = \zeta_1$).  Let
$\kappa_1 = {\underset\tilde {}\to \kappa_{\zeta_1}}
[G_{\underset\tilde {}\to \zeta_1}]$ and we are also given $N$ which is
strictly $\Bbb I^{[\kappa]}$-suitable, $p \in
\Bbb P_{\underset\tilde {}\to \zeta_2}/G_{\underset\tilde {}\to \zeta_1},p \in N,
\{\bar{\Bbb Q},{\underset\tilde {}\to \zeta_1},G_{\underset\tilde {}\to \zeta_1}\}
\in N,{\underset\tilde {}\to \zeta_2} \in N$.  We have to find $q \in
\Bbb P_{\underset\tilde {}\to \zeta_2}/G_{\underset\tilde {}\to \zeta_1}$ such that
$p \le_{\text{pr}} q,q$ is $(N,\Bbb P_{\underset\tilde {}\to \zeta_2}/
\Bbb P_{\underset\tilde {}\to \zeta_1})$-generic and $q \Vdash N
[G_{{\Bbb P}_{\underset\tilde {}\to \zeta_2}/G_{\underset\tilde {}\to \zeta_1}}]$
is strictly $\Bbb I^{[\kappa_2]}$-suitable.

If the depth of ${\underset\tilde {}\to \zeta_2}$ is 0, then
$\Vdash ``{\underset\tilde {}\to \zeta_2} = \zeta_2"$
and we can use $\boxtimes_{\zeta_1,\zeta_2}$.  So assume the depth of
${\underset\tilde {}\to \zeta_2}$ is $> 0$, and so for some $\gamma^*$ and
a sequence $\langle {\underset\tilde {}\to \zeta_{2,\varepsilon}}:
\varepsilon < \varepsilon^* \rangle$ of simple $\bar{\Bbb Q}$-named
$[\text{Max}\{\gamma^*,\gamma\},\beta)$-ordinals and $\Bbb P_{\gamma^*}$-name
$\underset\tilde {}\to \varepsilon$ we have
$\Vdash_{\bar{\Bbb Q}} ``{\underset\tilde {}\to \zeta_2} =
{\underset\tilde {}\to \zeta_{2,{\underset\tilde {}\to \varepsilon}}}"$.
So \wilog \, $\{\gamma^*,\langle
{\underset\tilde {}\to \zeta_{2,\varepsilon}}:
\varepsilon < \varepsilon^* \rangle,\underset\tilde {}\to \varepsilon\} \in
N$.  Let ${\underset\tilde {}\to \zeta_1} = \text{ Max}\{\gamma,\gamma^*,
{\underset\tilde {}\to \zeta_1}\},
\Vdash_{\Bbb Q} ``{\underset\tilde {}\to \zeta_1} \le
{\underset\tilde {}\to \zeta'_1} \le \zeta_2"$.

Now clearly ${\underset\tilde {}\to \zeta'_1}$ is a simple $\bar{\Bbb Q}$-named
$[0,\beta)$-ordinal, $\Vdash_{\bar{\Bbb Q}} ``{\underset\tilde {}\to \zeta_1} \le
{\underset\tilde {}\to \zeta'_1} \le {\underset\tilde {}\to \zeta'_2}"$
and $\boxtimes_{{\underset\tilde {}\to \zeta_1},
{\underset\tilde {}\to \zeta'_1}} \and
\boxtimes_{{\underset\tilde {}\to \zeta'_1},
{\underset\tilde {}\to \zeta_2}} \Rightarrow
\boxtimes_{{\underset\tilde {}\to \zeta_1},{\underset\tilde {}\to \zeta_2}}$
and $\boxtimes_{{\underset\tilde {}\to \zeta_1},
{\underset\tilde {}\to \zeta_2}}$ easily holds (by the cases proved above)
so it is enough to prove
$\boxtimes_{{\underset\tilde {}\to \zeta'_1},
{\underset\tilde {}\to \zeta_2}}$.  This just means that \wilog \,
$\zeta_1 \le \gamma^*$ and even ${\underset\tilde {}\to \zeta_1} =
\gamma^*$.  Now $\underset\tilde {}\to \varepsilon
[G_{\underset\tilde {}\to \zeta_1}] \in N$ so we use the induction hypothesis
to get the desired $q$.  \hfill$\square_{\text{fact}}$
\bn
\ub{Fact B}:  If $\underset\tilde {}\to \xi$ is a simple $\bar{\Bbb Q}$-named
$[0,\beta)$-ordinal, $p \in \Bbb P_\beta$ and $\underset\tilde {}\to \tau$ is
a $\Bbb P_\beta$-name of a countable ordinal, \ub{then} there are
$\underset\tilde {}\to \varepsilon$ and $q$ such that:
\mr
\item "{$(*)(i)$}"  $\Bbb P_\beta \models p_{\text{pr}} q$
\sn
\item "{$(ii)$}"  $q \restriction \underset\tilde {}\to \xi = p
\restriction \underset\tilde {}\to \xi$
\sn
\item "{$(iii)$}"  $\underset\tilde {}\to \varepsilon$ is a
$\Bbb P_{\underset\tilde {}\to \xi}$-name of a countable ordinal
\sn
\item "{$(iv)$}"  $q \Vdash_{{\Bbb P}_\beta} ``\underset\tilde {}\to \tau <
\underset\tilde {}\to \varepsilon"$.
\endroster
\bigskip

\demo{Proof}  Let $\langle {\underset\tilde {}\to \zeta_n}:n < \omega
\rangle$ be a witness for $p$, so each ${\underset\tilde {}\to \zeta_n}$
is a simple $\bar{\Bbb Q}$-named $[0,\beta)$-ordinal.  For each $n$ we define a
$P_{\underset\tilde {}\to \zeta_n}$-name of $\bold t_n$ of a truth value
and ${\underset\tilde {}\to r_n}$ of a member of $\Bbb P_\beta,
{\underset\tilde {}\to r_n} = {\underset\tilde {}\to r_n} \restriction
[{\underset\tilde {}\to \zeta_n},\beta)$, as follows: if $G^n \subseteq
\Bbb P_{\underset\tilde {}\to \zeta_n}$ is generic over $\bold V$ and $\neg
({\underset\tilde {}\to \zeta_n}[G^n] \ge \underset\tilde {}\to \xi
[G^n])$, and there are $q \in \Bbb P_\beta/G^n$ and $\varepsilon < \omega_1,
\Bbb P_\beta \Vdash ``p \le_{\text{pr}} q"$ and $q \Vdash_{{\Bbb P}_\beta/G^n}
``\underset\tilde {}\to \tau < \varepsilon"$, \ub{then}
${\underset\tilde {}\to {\bold t}_n}[G^n] =$ truth and then
${\underset\tilde {}\to r_n}[G^n]$ is $q \restriction
[{\underset\tilde {}\to \zeta_n},\beta)$ for some such $q$ otherwise
${\underset\tilde {}\to {\bold t}_n}[G^n] =$ false,
$\underset\tilde {}\to r[G^n] = \emptyset$.  Let $r'_n$ be the following
$\Bbb P_{\underset\tilde {}\to \zeta_n}$-name: \nl
for $G^n \subseteq \Bbb P_{\underset\tilde {}\to \zeta_n}$ \ub{if}:
\mr
\item "{$(a)$}"  ${\underset\tilde {}\to {\bold t}_n}[G^n] =$ truth and
\sn
\item "{$(b)$}"  for no $m < \omega$, for some $r \in G^n$ forces a value
to ${\underset\tilde {}\to \zeta_m}$, say $\zeta_m,\zeta_m < \zeta_n[G^n]
\vee (\zeta_m = {\underset\tilde {}\to \zeta_n}[G^n] \and m < n)$
\ermn
\ub{then}
${\underset\tilde {}\to \zeta'_n}[G^n]$ is ${\underset\tilde {}\to r_n}$,
otherwise ${\underset\tilde {}\to \zeta'_n}[G^n] = \emptyset$.  Let
$p^* = p \cup \dbcu_n {\underset\tilde {}\to r'_n}$, easily $q \in
\Bbb P_\beta$ and $p \le_{\text{pr}} p^*$.
\nl
Let ${\underset\tilde {}\to \xi_n} = \text{ Min}\{\underset\tilde {}\to \xi
+1,{\underset\tilde {}\to \zeta_0} +1,\dotsc,
{\underset\tilde {}\to \zeta_{n-1}} +1\}$, ${\underset\tilde {}\to \xi_n}$
is a simple $\bar{\Bbb Q}$-named $[0,\beta)$-ordinal, 
${\underset\tilde {}\to \xi_0} = \underset\tilde {}\to \xi,
{\underset\tilde {}\to \zeta_n} < {\underset\tilde {}\to \xi_{n+1}},
{\underset\tilde {}\to \xi_n} \le {\underset\tilde {}\to \xi_{n+1}}$.  Let
$N$ be a strictly $(\Bbb I,\bold W)$-suitable model for $\chi$ such that
$\{\bar{\Bbb Q},p,\langle {\underset\tilde {}\to \zeta_n},
{\underset\tilde {}\to \xi_n}:n < \omega \rangle,\langle
{\underset\tilde {}\to r_n},{\underset\tilde {}\to r'_n}:n < \omega\}$
belongs to $N$, it exists by \scite{4.8}.  Now we choose $q_n$ by induction
on $n < \omega$ such that

$$
q_n \in \Bbb P_{\underset\tilde {}\to \zeta_n} \text{ is }
(N,{\Bbb I}^{[{\underset\tilde {}\to \kappa_{\underset\tilde {}\to \zeta_n}}]},
P_{\underset\tilde {}\to \zeta_n})-\text{semi}_6 \text{ generic}
$$

$$
p^* \restriction \zeta_n,\beta \le_{\text{pr}} q_n
$$

$$
q_{n+1} \restriction {\underset\tilde {}\to \zeta_n} = q_n.
$$
\mn
This is possible by Fact A.  Now (see \scite{1.27}) $q^* =: \dbcu_{n < \omega}
q_n \cup p^*$ belongs to $\Bbb P_\beta,p^* \le_{\text{pr}} q^*$.  It is enough
to show that $q^* \Vdash \underset\tilde {}\to \tau \in N \cap \omega_1$,
assume not so there is $r,q^* \le r \in \Bbb P_\beta$ such that $r$ forces a value
$\varepsilon^* \in \omega_1 \backslash (N \cap \omega_1)$ to
$\underset\tilde {}\to \tau$.

By \scite{1.15}(1) \wilog \, $q^* \le r$ above $\{\Upsilon_1,\dotsc,
\Upsilon_m\}$ for some $m < \omega$ and $\Upsilon_\ell < \beta$.  There are
$r',k,\xi_k$ such that: $r \le r',\xi_k < \omega,r' \restriction [\xi_k,
\beta) = r \restriction [\xi_k,\beta),r'$ forces
${\underset\tilde {}\to \xi_k}$ is $\xi_k$ and for each $\ell = 1,2,\dotsc,m$
we have $\Upsilon_\ell < \xi_k$ \ub{or} 

$$
\neg(\exists r')(\exists \Upsilon < \beta)(\exists k')[r \restriction
\Upsilon \le r' \in \Bbb P_\Upsilon \and r' \Vdash {\underset\tilde {}\to \xi_{k'}}
= \Upsilon \and \Upsilon_\ell < \Upsilon].
$$
\mn
Clearly $r'$ forces that $(\forall n)[{\underset\tilde {}\to \zeta_n} \ge
\xi_k \rightarrow {\underset\tilde {}\to {\bold t}_n}$ is truth] and we
easily finish.  \hfill$\square_{\text{Fact B}}$.

Now we do not just have to find $q$ satisfying $\Bbb P_\beta/G_\gamma \models 
p \le_{\text{pr}} q$ and $q$ is $(N,\Bbb P_\beta/G_\gamma)$-semi-generic, but 
we need more in the $(N,\Bbb I^{[{\underset\tilde {}\to \kappa_\beta}]}
\restriction \Bbb P_\beta/G_\gamma)$-semi$_6$ generic.  
Now for $G_\beta \subseteq \Bbb P_\beta$ generic over $\bold V$, in
$\bold V[G_\beta]$
for every countable elementary submodel $M$ of $({\Cal H}(\chi)^{\bold V},G_\beta
\in,<^*),\langle \bar{\Bbb Q},\gamma,\beta,\Bbb I \rangle \in M$, we 
have (in \scite{4.5B}) defined Dp$(M)$, an ordinal or $\infty$ and $I_M$ 
such that
\mr
\item "{$(A)$}"  Dp$(M) = \infty$ iff $M[G_\beta] = \{
\underset\tilde {}\to \tau[G_\beta]:\underset\tilde {}\to \tau \in M
\text{ a } \Bbb P_\beta \text{-name}\}$ includes $M$, 
has the same countable ordinals, is $\prec({\Cal H}(\chi)^{\bold V[G_\beta]},\in)$ 
and $M[G_\beta]$ is strictly
$\Bbb I^{[{\underset\tilde {}\to \kappa_\beta}[G_\beta]]}$-suitable
\sn
\item "{$(B)$}"  if Dp$(M) = \alpha < \infty$ then $I_M$ is a member of 
$\Bbb I \cap M$ which
is ${\underset\tilde {}\to \kappa_\beta}[G_\beta]$-complete, and
$$
\align
Y_M = \{t \in \text{ Dom}(I):&\text{there is } N \text{ as above},
M \subseteq N,M \cap \omega_1 = N \cap \omega_1 \\
  &\text{and } t \in N \text{ and Dp}(N) \ge \alpha\} = 
\emptyset \text{ mod } I.
\endalign
$$
\ermn
So we have $\Bbb P_\beta$-names ${\underset\tilde {}\to {\text{Dp}}},
{\underset\tilde {}\to I_M}$. \nl
Consider 

$$
\align
{\frak K} = \bigl\{ (\zeta,G_\zeta,N):&\gamma \le \zeta < \beta,\zeta
\text{ nonlimit}, G_\zeta \subseteq \Bbb P_\zeta \\
  &\text{generic over } \bold V, \text{ in } \bold V[G_\zeta],N[G_\zeta] \text{ is } \\
  &\Bbb I^{[{\underset\tilde {}\to \kappa_\zeta}[G_\zeta]]} \text{-suitable}
\bigr\}.
\endalign
$$
\mn
We now define by induction $({\underset\tilde {}\to \zeta_n},q_n,
p_n,{\underset\tilde {}\to N_n})$ such that
\mr
\item "{$(a)$}"  ${\underset\tilde {}\to \zeta_n}$ is a simple $\bar{\Bbb Q}$-named
$[\gamma,\beta)$-ordinal, (as $e=6$, it is full)
\sn
\item "{$(b)$}"  ${\underset\tilde {}\to N_n}$ is a 
$\Bbb P_{\underset\tilde {}\to \zeta_n}$-name,
$q_n \in \Bbb P_{\underset\tilde {}\to \zeta_n},p_n$ is a
$\Bbb P_{\underset\tilde {}\to \zeta_m}$-name of a member of \nl
${\underset\tilde {}\to N_n}[G_{\underset\tilde {}\to \zeta_n}] \cap
(\Bbb P_\beta/{\underset\tilde {}\to G_{\zeta_n}})$
\sn
\item "{$(c)$}"  ${\underset\tilde {}\to \zeta_0} = \gamma,N_0 = N$
\sn
\item "{$(d)$}"  if $G^n \subseteq \Bbb P_{\underset\tilde {}\to \zeta_n}$ is
generic over $\bold V,q_n \in G^n,G_\gamma \subseteq G^n,
{\underset\tilde {}\to \zeta_n}[G^n] = \zeta_n$ (so essentially
$G_n$ is just a generic
subset of $G_{\zeta_n}$ over $\Bbb V$ such that ${\underset\tilde {}\to \zeta_n}
[G^n] = \zeta_n$), \ub{then} ${\underset\tilde {}\to N_n}[G^n]$ is a
countable elementary submodel of $({\Cal H}(\chi)^{\bold V[G^n]},\in)$ to which
$\bar{\Bbb Q},\gamma,\beta,\Bbb I$ belongs and is strictly
$\Bbb I^{[{\underset\tilde {}\to \kappa_{\zeta_n}}[G^n]]}$-suitable
\sn
\item "{$(e)$}"  ${\underset\tilde {}\to N_n} \subseteq N_{n+1},
{\underset\tilde {}\to N_n} \cap \omega_1 = 
{\underset\tilde {}\to N_{n+1}} \cap \omega_1,q_{n+1} \restriction
{\underset\tilde {}\to \zeta_n} = q_n,p_n \restriction
{\underset\tilde {}\to \zeta_n} \le q_n,p_n \le_{\text{pr}} p_{n+1}$
\sn
\item "{$(f)$}"  for $G^n,\zeta_n,N_n$ as in (d) and $I \in \Bbb I \cap
N_n$ there is $k > n$ such that: if $G^k,\zeta_k,N_k$ are as in (d),
$G^n \subseteq G^k$, \ub{then} there is $t \in \text{ Dom}(I) \cap N_k
\backslash {\underset\tilde {}\to Y_{({\underset\tilde {}\to N_n}[G^n])
[{\underset\tilde {}\to G_\beta}]}}$ (i.e. forced to be there) \nl
(recall $p$ forces a value to ${\underset\tilde {}\to \kappa_\beta})$
sn
\item "{$(g)$}"  for $G^n_n,\zeta_n,N_n$ as in clause (d) above and
$\underset\tilde {}\to \gamma \in N_n$ a $\Bbb P_\beta$-name of a countable
ordinal there is $k > n$ such that: if $G^k,\zeta_k,N_k$ are as in clause
(d), $G^n \subseteq G^k$ then $p_k$ forces $(\Vdash_{{\Bbb P}_\beta/G^k})$ to
be $< N \cap \omega_1$.
\ermn
No problem to carry the definition.  Now
\mr
\item "{$(*)_a$}"  $q = \dbcu_{n < \omega} q_n \in \Bbb P_\beta$ \nl
Here we use the $q_{n+1} \restriction {\underset\tilde {}\to \zeta_n} = q_n$
below $\dbcu_{n < \omega} {\underset\tilde {}\to \zeta_n}$ and
``$({\underset\tilde {}\to {\Bbb Q}_i},\le_{\text{vpr}})$ is $\aleph_1$-complete"
above $\dbcu_{n < \omega} {\underset\tilde {}\to \zeta_n}$
\sn
\item "{$(*)_b$}"  for $G^n,N_n,\zeta_n$ as in clause (d), $q$ is
$(N_n,\Bbb P_\beta/G^n)$-semi generic and above $p \restriction \zeta_n$ \nl
[why?  note clause (g)]
\sn
\item "{$(*)_c$}"  if $q \in G_\beta \subseteq \Bbb P_\beta,G_\beta$ is generic
over $\bold V$ extending $G_\gamma$ \ub{then} in $\bold V[G_\beta]$, 
Dp$({\underset\tilde {}\to N_n}[G_\beta])$ is
well defined
\sn
\item "{$(*)_d$}"  Dp$((N_0[G_\beta]))$ is $\infty$ \nl
(use clause (B) in the demands Dp and clause (f) above).  We use
$I^{[\kappa]}$ is $\kappa$-closed for the relevant $\alpha$'s.
\ermn
So we are done.  \hfill$\square_{\scite{si.7}}$\margincite{si.7}
\enddemo
\bigskip

\proclaim{\stag{si.8} Claim}  1) Assume 
that $\bar{\Bbb Q} = \langle \Bbb P_j,
{\underset\tilde {}\to {\Bbb Q}_i},
{\underset\tilde {}\to \kappa_j}:j \le \alpha,i < \alpha \rangle$ satisfies:
\mr
\item "{$(\alpha)$}"  $\alpha$ is a limit ordinal
\sn
\item "{$(\beta)$}"  if $\beta < \alpha$ then $\langle \Bbb P_j,
{\underset\tilde {}\to Q_i},{\underset\tilde {}\to \kappa_j}:j \le \beta,
i < \alpha \rangle$ is a weak 
${\text{\rm UP\/}}^6(\Bbb I,\bold W,W)$-iteration
\sn
\item "{$(\gamma)$}"  $\Bbb P_\alpha$ is the 
$\aleph_1-{\text{\rm Sp\/}}_6(W)$-limit of
$\langle \Bbb P_i,{\underset\tilde {}\to {\Bbb Q}_i}:i < \alpha \rangle$
\sn
\item "{$(\delta)$}"  ${\underset\tilde {}\to \kappa_\alpha}$, a
$\Bbb P_\alpha$-name is 
$\sup\{{\underset\tilde {}\to \kappa_i}:i < \alpha\}$.
\ermn
\ub{Then} $\bar{\Bbb Q}$ is a weak 
${\text{\rm UP\/}}^6(\Bbb I,\bold W,W)$-iteration. \nl
2) Assume that
\mr
\item "{$(\alpha)$}"  $\langle 
\Bbb P_j,\Bbb Q_i,{\underset\tilde {}\to \kappa_i}:
j \le \alpha,i < \alpha \rangle$ is a weak ${\text{\rm UP\/}}^6
(\Bbb I,\bold W,W)$-iteration
\sn
\item "{$(\beta)$}"  in 
$\bold V^{\Bbb P},\Bbb Q$ is a forcing notion satisfying
${\text{\rm UP\/}}^6\Bbb I,\kappa_\alpha,\underset\tilde {}\to \kappa,\bold W)$, where
$\kappa_\alpha$ is the interpretation of ${\underset\tilde {}\to
\kappa_\alpha}$) and let
${\underset\tilde {}\to {\Bbb Q}},\underset\tilde {}\to \kappa$ be
$\Bbb P_\alpha$-names
of those objects.
\ermn
\ub{Then} there is a ${\text{\rm UP\/}}^6(\Bbb I,\bold W,W)$-iteration
$\langle P_i,{\underset\tilde {}\to {\Bbb Q}_j},
{\underset\tilde {}\to \kappa_i}:
i \le \alpha +1,j < \alpha \rangle$ 
with ${\underset\tilde {}\to {\Bbb Q}_\alpha}
= {\underset\tilde {}\to {\Bbb Q}},{\underset\tilde {}\to \kappa_{\alpha +1}} =
\underset\tilde {}\to \kappa$.
\endproclaim
\bigskip

\demo{Proof}  1) By the proof of \scite{si.7}. \nl
2) Straightforward.
\enddemo
\bigskip

\proclaim{\stag{si6.9} Claim}  Assume
\mr
\item "{$(a)$}"  $\bar{\Bbb Q} = \langle \Bbb P_j,
{\underset\tilde {}\to {\Bbb Q}_i},
{\underset\tilde {}\to \kappa_j}:j \le \alpha^*,i < \kappa \rangle$ is a
weak ${\text{\rm UP\/}}^6(\Bbb I,\bold W,W)$-iteration
\sn
\item "{$(b)$}"  $\gamma < \beta \le \alpha^*$
\sn
\item "{$(c)$}"  $G_\gamma \subseteq \Bbb P_\gamma$ is generic over $V$
\sn
\item "{$(d)$}"  $N$ is a strictly $(\Bbb I,\bold W)$-suitable model $N$
for $(\chi,\langle \bar{\Bbb Q},\gamma,\beta \rangle)$ in 
$\bold V[G_\gamma]$
\sn
\item "{$(e)$}" $p \in N \cap (\Bbb P_\beta/G_\gamma)$.
\ermn
\ub{Then} there is $q$ such that:
\mr
\item "{$(\alpha)$}"  $p \le_{\text{pr}} q \in \Bbb P_\beta/G_\gamma$
\sn
\item "{$(\beta)$}"  $p \restriction \gamma = q \restriction \gamma$
\sn
\item "{$(\gamma)$}"  $q$ is $(N,\Bbb P_\beta/G_\gamma)$-semi generic
\sn
\item "{$(\delta)$}"  $q$ has 
a witness listing $\{\underset\tilde {}\to \zeta
\in N:\underset\tilde {}\to \zeta$ a simple $\bar{\Bbb Q}$-named
$[\gamma,\beta)$-ordinal$\}$.
\endroster
\endproclaim
\bigskip

\demo{Proof}  Like \scite{si.7}, forgetting to take care of the
$I \in \Bbb I \cap {\underset\tilde {}\to N_n}$ so we can use
$\underset\tilde {}\to N = 
N[{\underset\tilde {}\to G_{{\Bbb P}_{\underset\tilde {}\to
\zeta_n}}}]$. \nl
${{}}$  \hfill$\square_{\scite{si6.9}}$\margincite{si6.9}
\enddemo
\bigskip

\proclaim{\stag{si6.10} Claim}  Suppose $F_f,F_c$ are functions (possibly
classes), $\bold W \subseteq \omega_1$ is stationary, $\Bbb I$ is a class
of ($\aleph_2$-complete) quasi order ideals, $W$ a class of strongly
inaccessible cardinals.

\ub{Then} for every ordinal $\alpha$ there is a unique $\bar{\Bbb Q} =
\langle \Bbb P_j,
{\underset\tilde {}\to {\Bbb Q}_i},{\underset\tilde {}\to \kappa_j}:j \le 
\alpha^\dag,i < \alpha \rangle$ such that:
\mr
\item "{$(a)$}"  $\bar{\Bbb Q}$ is a ${\text{\rm UP\/}}^6
(\Bbb I,\bold W,W \cap (\alpha^\dag
+1))$-iteration
\sn
\item "{$(b)$}"  for every $i < \alpha^\dag$ we have 
${\underset\tilde {}\to {\Bbb Q}_i} = F_f(\bar{\Bbb Q} \restriction i),
{\underset\tilde {}\to \kappa_i} = F_c(\bar{\Bbb Q} \restriction i)$
\sn
\item "{$(c)$}"   $\alpha^\dag \le c$
\sn
\item "{$(d)$}"  for limit $\beta \le \alpha^\dag$ we have
${\underset\tilde {}\to \kappa_\beta} = \sup
\{{\underset\tilde {}\to \kappa_\gamma}:\gamma < \beta\}$
\sn
\item "{$(e)$}"  if $\alpha^\dag < \alpha$ then the following is impossible
{\roster
\itemitem{ $(\alpha)$ }   $F_f(\bar{\Bbb Q} \restriction i)$ is a
$\Bbb P_{\alpha^\dag}$-name of a forcing notion
\sn
\itemitem{ $(\beta)$ }  $F_c(\bar{\Bbb Q} \restriction i)$ is a
$\Bbb P_{\alpha^\dag}$-name of a $F_f(\bar{\Bbb Q} \restriction i)$-name of a
$\bold V$-cardinal $\ge \aleph_1$
\sn
\itemitem{ $(\gamma)$ }  $\Vdash_{{\Bbb P}_{\alpha^\dag}} 
``F_f(\bar{\Bbb Q} \restriction i)$ is 
${\text{\rm UP\/}}^6_{{\underset\tilde {}\to \kappa_{\alpha^\dag}},
{\underset\tilde {}\to \kappa}}(\Bbb I,\bold W)"$.
\endroster}
\endroster
\endproclaim
\bigskip

\demo{Proof}  Straight.
\enddemo
\bn
Next we give sufficient conditions for $\dbcu_{i < \delta} \Bbb P_i$ being a
dense subset of $\Bbb P_j$
\proclaim{\stag{si6.11} Claim}  Assume that $\langle \Bbb P_j,
{\underset\tilde {}\to {\Bbb Q}_i},{\underset\tilde {}\to \kappa_j}:
j \le \alpha^*,i < \alpha^* \rangle$ is a 
${\text{\rm UP\/}}^6(\Bbb I,\bold W,W)$-iteration. \nl
1) Assume
\mr
\item "{$(*)_1$}"  $(\forall i < \alpha^*)[\Vdash_{{\Bbb P}_i}
``\le^{{\Bbb Q}_i}_{\text{vpr}}$ is equality"].
\ermn
If ${\text{\rm cf\/}}(\delta) = \aleph_1 \and \delta \le \alpha^*$ \ub{then}
$\dbcu_{i < \delta} \Bbb P_i$ is a dense subset of $\Bbb P_\delta$ even under
$\le_{\text{pr}}$. \nl
2) Assume
\mr
\item "{$(*)_2$}"  $(\forall i < \alpha^*)(\Vdash_{{\Bbb P}_i}
``(\{r \in {\underset\tilde {}\to {\Bbb
Q}_i}:\emptyset_{\underset\tilde {}\to {\Bbb Q}_i}
\le_{\text{vpr}} r\},\le^{\underset\tilde {}\to {\Bbb Q}_i}_{\text{vpr}})$ is
directed).
\ermn
If $\delta \le \alpha^*,{ \text{\rm cf\/}}(\delta) = \aleph_1$ \ub{then}
\mr
\item "{$(a)$}"  $\dbcu_{i < \delta} {\Cal D}_{\delta,i}$ is a dense 
subset of $\Bbb P_\delta$ even under $\le_{\text{pr}}$ where
${\Cal D}_{\delta,i} = \{p \in \Bbb P_\delta:\Bbb P_\delta \models ``p \restriction
i \le_{\text{vpr}} p"\}$
\sn
\item "{$(b)$}"  if $i < \delta,\{p_0,p_1\} \subseteq {\Cal D}_{\delta,i}$
and $p_0 \restriction i = p_j \restriction i$ then $p_0,p_1$ has an upper
bound $p$ even in ${\Cal D}_{\delta,i},p \restriction i = p_0 \restriction
i = p_1 \restriction i$.
\ermn
3) If $\delta \in W \and \delta \le \alpha^* \and (\forall i < \delta)$
[${\text{\rm density\/}}(\Bbb P_i) 
< \delta$] \ub{then} $\Bbb P_\delta = \dbcu_{i < \delta} P_i$.
\endproclaim
\bigskip

\demo{Proof}  1), 2)  Let $p \in \Bbb P_\delta$, choose $\chi$ large enough.
There is a strictly $(\Bbb I,\bold W)$-suitable countable model
$N \prec ({\Cal H}(\chi),\in)$ to which $\{\bar{\Bbb Q},\delta,p\}$ belongs.
Applying \scite{si.7} for $\gamma = 0,\beta = \delta$
(i.e. $\boxtimes_{\gamma,\beta}$ from the proof) we can find $q \in
\Bbb P_\delta,p \le_{\text{pr}} q$ which is 
$(N,\Bbb I^{[{\underset\tilde {}\to \kappa_\delta}]},\Bbb P_\delta)$-semi$_6$ 
generic
and $q$ has a witness $\subseteq \{\underset\tilde {}\to \zeta \in N:
\underset\tilde {}\to \zeta$ a simple $\bar{\Bbb Q}$-named $[0,\delta)$-ordinal$\}$.
As cf$(\delta) = \aleph_1$ there is an increasing continuous $\bar \beta =
\langle \beta_\varepsilon:\varepsilon < \omega_1 \rangle$ with limit
$\delta$, \wilog \, $\bar \beta \in N$ so $q \Vdash ``
N[{\underset\tilde {}\to G_{{\Bbb P}_\delta}}] \cap \delta = \cup
\{\beta_\varepsilon:\varepsilon \in N
[{\underset\tilde {}\to G_{{\Bbb P}_\delta}}] \cap \omega_1\}"$ but
$q \Vdash \cup \{\beta_\varepsilon:\varepsilon \in 
N[{\underset\tilde {}\to G_{{\Bbb P}_\delta}}] \cap \omega_1\} = \cup
\{\beta_\varepsilon:\varepsilon \in N \cap \omega_1\} =
\beta_{N \cap \omega_1}$, so clearly $\Bbb P_\delta \models q \restriction
\beta_{N \cap \omega_1} \le_{\text{vpr}}$.  For (1) it follows that
$q \in P_{N \cap \delta}$ and we are done. \nl
For (2) just reflect. \nl
3) Straight.
\enddemo
\bigskip

\demo{\stag{si6.12} Conclusion}  Let $\bar{\Bbb Q} = \langle \Bbb P_j,
{\underset\tilde {}\to {\Bbb Q}_i},{\underset\tilde {}\to \kappa_j}:j \le \alpha^*,
i < \alpha^* \rangle$ be a UP$^6(\Bbb I,\bold W,W)$-iteration and
$\kappa = \text{ cf}(\kappa) \le \alpha^*$ and $(\forall i < \delta)$
(density$(\Bbb P_i) < \kappa)$. \nl
1) If $\{\theta < \kappa:\theta = \text{ cf}(\theta) \in W\}$ is
stationary.  
\ub{Then} $\Bbb P_\kappa$ satisfies the $\kappa$-c.c. (in fact a strong
version and even under $\le^{{\Bbb P}_\kappa}_{\text{vpr}}$. \nl
2) If $(*)_1$ from \scite{si6.11}(1), i.e. $\Vdash_{{\Bbb P}_i} 
``\le^{\underset\tilde {}\to {\Bbb Q}_i}_{\text{pr}}$ is equality" 
for $i < \kappa$ \ub{then} 
$\Bbb P_\kappa$ satisfies the $\kappa$-c.c. (in fact a strong version)
even for $\le^{{\Bbb P}_\kappa}_{\text{vpr}})(\theta \le_{\text{vpr}})$ is
$(2^{|\delta|})^+$-complete. \nl
3) Assume $\kappa \notin W$ and $S \subseteq \{\delta < \kappa:
\text{cf}(\delta) = \aleph_1\}$ is stationary and $i \ge \delta \and \delta
\in S \Rightarrow \Vdash_{{\Bbb P}_i} 
``(\{r:\emptyset_{\underset\tilde {}\to {\Bbb Q}_i}
\le_{\text{vpr}} r\}$.  \ub{Then} forcing with $\Bbb P_\kappa$ does not add a
function in ${}^\omega\text{Ord}$ not in $\dbcu_{\beta < \kappa}
\bold V^{{\Bbb P}_\beta}$, even any function in ${}^{\alpha(*)}$Ord $\backslash
\dbcu_{\beta < \kappa} \bold V^{{\Bbb P}_\beta},\alpha(*) < \kappa$.
\enddemo
\bigskip

\demo{Proof}  1) Let $S = \{\theta < \kappa:\theta = \text{ cf}(\theta) 
\in W\}$ 
and let $\langle \Bbb P_\theta:\theta \in S \rangle$ be a sequence of
members of 
$\Bbb P_\kappa$.  So for each $\theta \in S$ for some $i(\theta) < 
\theta$ 
we have $p_\theta \restriction \theta \in \Bbb P_{i(\theta)}$ and we can
find a pressing down function $h$ on $S$ such that $h(\theta_1) =
h(\theta_2) \Rightarrow p_{\theta_1} \restriction \theta_1 = p_{\theta_2}
\restriction \theta_2$.  Clearly there is a club $E$ or $\kappa$ such that
$\theta_i \in \theta_2 \cap s \and \theta_2 \in S \cap S \cap E \Rightarrow
p_{\theta_1} \in P_{\theta_2}$.

Lastly, if $\theta_1,\theta_2 \in E \cap S$ and $f(\theta_1) = f(\theta_2)$
then $p_{\theta_1} \cup p_{\theta_2}$ is a common upper bound of 
$p_{\theta_1},p_{\theta_2}$ (even a $\le_{\text{vpr}}$ one). \nl
2) Similar using $S = \{\delta < \kappa:\text{cf}(\delta) = \aleph_1$ and
$\alpha < \delta \Rightarrow$ density$(\Bbb P_\alpha) < \delta\}$. \nl
3) If $W \cap \kappa$ is stationary in $\kappa$ use part (2), so let $E$ be
a club of $\kappa$ disjoint to $W$.  Assume toward contradiction that
$p \Vdash_{{\Bbb P}_\kappa} ``\text{the function } \underset\tilde {}\to \tau:
\omega_1 \rightarrow {}^\kappa\text{Ord}$ is not in $\dbcu_{\beta < \kappa}
\bold V^{{\Bbb P}_\beta}"$, let $S$ be as in part (2).  We choose by induction on
$j < \kappa,(p_j,\alpha_j)$ and if $\alpha _j \in \text{ Salso}(q_j,
\varepsilon_j,\gamma_j,\beta_j)$ such that:
\mr
\widestnumber\item{$(iii)$}
\item "{$(i)$}"  $\alpha_j \in E$ is increasing continuous
\sn
\item "{$(ii)$}"  $p_j \in \Bbb P_\kappa,p_0 = p$
\sn
\item "{$(iii)$}"  $i < j \Rightarrow p_j  \restriction \alpha_i = p_i$
\sn
\item "{$(iv)$}"  $i < j \and \beta \in [\alpha_i,\alpha_j) \Rightarrow
\Vdash_{{\Bbb P}_\beta} ``\emptyset \le_{\text{vpr}} p_j \restriction \{\beta\}$
in ${\underset\tilde {}\to {\hat{\Bbb Q}}_\beta}"$
\sn
\item "{$(v)$}"  if $\alpha_i \in S,j < \alpha_i$ \ub{then} for every
$p_i \le q \in {\Cal D}_{\alpha_i,j}$ there is $\varepsilon_i(q) < \omega_i$
such that: \ub{if} there are $\varepsilon,r$ such that $p_{i+1} \cup 
\le_{\text{vpr}} r,r \restriction \alpha_i = q,\varepsilon < \omega_1,r$
forcing a value to $\underset\tilde {}\to \tau(\varepsilon)$ but for no
$r,q \le r' \in \Bbb P_{\alpha_i}$, does $P_{i+1} \cup r'$ forces a value to
$\underset\tilde {}\to \tau(\varepsilon)$ then $\varepsilon = \varepsilon_i
(q)$ satisfies this
\sn
\item "{$(vi)$}"  $\alpha_i \in S$ then $p_{i+1} \le q_i,q_i \Vdash
``\underset\tilde {}\to \tau(\varepsilon_i) = \gamma_i",\varepsilon_i <
\alpha(*),\beta_i < \alpha_i,q_i \restriction \beta_i \le_{\text{vpr}} q_i
\restriction \alpha_i$
\sn
\item "{$(vii)$}"  there is no $q,q_i \restriction \alpha_i \le q \in
\Bbb P_{\alpha_i}$ such that $p_{i+1} \cup (q_i \restriction \alpha_i)$ forces a
value to $\underset\tilde {}\to \tau(\varepsilon_i)$.
\ermn
For any $j$ we choose $(p_j,\alpha_j)$. \nl
For $j=0$ let $p_0=p$ and by \scite{si.8} for some $\alpha_0 < \kappa,p_0
\in \Bbb P_{\alpha_0}$.  For $j=i+1$, first choose $p_j$ to satisfy clause (v)
and then $\alpha_j$ such that $p_j,q \in \Bbb P_{\alpha_j}$.  Lastly for $j$
limit let $p_j = \dbcu_{i < j} p_i,\alpha_j = \dbcu_{i < j} \alpha_i$ and
check.
The contradiction is easy.

Let $G_{\alpha_i} \subseteq \Bbb P_{\alpha_i}$ be generic over $\bold V$ such that
$p_i \in G_{\alpha_i}$.  Clearly for some $\varepsilon < \alpha(*)$ for no
$q \in \Bbb P_{\alpha_i}$ do we have $q \cup p_{i+1}$ forces a value to
$\underset\tilde {}\to \tau(\varepsilon)$ as otherwise
$p_{\alpha_{i+1}} \Vdash_{{\Bbb P}_\kappa/G_{\alpha_i}} \underset\tilde {}\to \tau
\in \bold V[G_{\alpha_i}]$.  Choose $\varepsilon_i < \alpha(*)$ as above, choose
$q'_i \in G_{\alpha_i}$ which forces this choose $q_i \in \Bbb P_\kappa$ above
$q'_i \cup p_{i+1}$ which forces a value to $\underset\tilde {}\to \tau
(\varepsilon_i)$ and \wilog \, there is $\gamma _i < \alpha_i$ such that
$q_i \restriction \gamma_i  \le_{\text{vpr}} q_i \restriction \alpha_i$.
Lastly let $\alpha_{i+1}$ be such that $\beta \in [\alpha_{i+1},\kappa)
\Rightarrow \Vdash_{P_\beta} ``\emptyset_{\underset\tilde {}\to {\Bbb Q}_\beta}
\le_{\text{vpr}} p_{i+1} \restriction \{\beta\}$. \nl
[Saharon: the role of $W$].
\enddemo
\newpage

\head {\S6B On UP$^2$-iteration} \endhead  \resetall \sectno=6
 % \resetall 
\bigskip

\proclaim{\stag{6.7}(?)Lemma}  Assume that $\bold W \subseteq \omega_1$ is 
stationary and $\bar{\Bbb Q} = \langle \Bbb P_i,{\underset\tilde
{}\to {\Bbb Q}_i},
{\underset\tilde {}\to {\Bbb I}_i},{\underset\tilde {}\to \kappa_i}:
i < \alpha \rangle$ is a ${\text{\rm UP\/}}^{2,e}(\bold W,W)$-suitable 
iteration, and $\Bbb P_\alpha 
= { \text{\rm Sp\/}}_e(W)-{\text{\rm Lim\/}}_\kappa
(\bar{\Bbb Q})$ be the limit and 
$\underset\tilde {}\to \kappa(\beta) = \sup
\{{\underset\tilde {}\to \kappa_\gamma}:\gamma < \beta\}$ (this is a
$\Bbb P_\beta$-name of a $V$-cardinal) and

$$
\kappa^-(\beta) = {\text{ \rm Sup\/}}
\{\kappa:\,\,\nVdash_{{\Bbb P}_\gamma} ``\kappa \ne
\underset\tilde {}\to \kappa(\beta)" \text{ for some } \gamma < \beta\}.
$$
\mn
1) For each $\beta \le \alpha,\Bbb P_\beta$ satisfies 
${\text{\rm UP\/}}^{0,e}_{\underset\tilde {}\to \kappa
(\beta)}(\Bbb I'_\beta,\bold W)$ for 
some $\aleph_2$-complete $\Bbb I'_\beta \in \bold V$. \newline
2) In fact, $\Bbb I_\beta$ is $\kappa^\beta$-complete where
$\kappa^\beta = \text{ min}\{\kappa:\text{for some } \gamma < \beta$ we
have \nl
$\nVdash_{{\Bbb P}_{\gamma +1}} ``{\underset\tilde {}\to \kappa_\gamma} \ne
\kappa"\}$, and each $I \in \Bbb I_\beta$ has domain of cardinality \nl
$\le (\underset{\gamma < \beta} \to \sup\{ \lambda < \kappa_{\delta +1}:\,
\nVdash_{{\Bbb P}_\gamma} ``\neg(\exists I \in 
{\underset\tilde {}\to {\Bbb I}_\gamma})(\lambda = |\text{Dom}(I)|)"\})$ and
\nl
$|\Bbb I'_\beta| 
\le \dsize \sum_{\gamma < \beta} (\aleph_0 + |\Bbb P_\gamma| +
\text{ min}\{\lambda:\Vdash_{{\Bbb P}_\gamma} 
``|{\underset\tilde {}\to {\Bbb I}_\gamma}| \le \lambda)^{< \kappa}\}$. \nl
3) Similarly for ${\text{\rm UP\/}}^{0,e}_a$ 
and weak ${\text{\rm UP\/}}^{0,e}_y(\bold W)$-iterations.
\endproclaim
\bigskip

\remark{\stag{6.7A} Remark}  We can also get the preservation version of this
Lemma.
\endremark
\bigskip

\demo{Proof}  For each $\gamma < \alpha$ let ${\Cal J}_\gamma =:
\{q \in \Bbb P_{\gamma +1}:q \text{ forces a value to } 
{\underset\tilde {}\to \kappa_\gamma}$, called 
$\kappa_{\gamma,q}$ and $q$ forces
${\underset\tilde {}\to {\Bbb I}_\gamma}$ to be equal to a $\Bbb P_\gamma$-name
${\underset\tilde {}\to {\Bbb I}_{\gamma,q}}$ and
$q \restriction \gamma$ forces a value to
$|\Bbb I_\gamma|$ says $\mu_{\gamma,q}$ is this is purely decidable, if not,
just an upper bound to it$\}$; let ${\Cal J}'_\gamma \subseteq
{\Cal J}_\gamma$ a maximal antichain.  Let $\mu_\gamma =
\underset{q \in {\Cal J}'_\gamma} \to \sup \mu_{\gamma,q}$. \newline
Let $q \Vdash_{{\Bbb P}_\gamma} ``{\underset\tilde {}\to {\Bbb I}_\gamma} = \{
{\underset\tilde {}\to I_{\gamma,\zeta}}:\zeta < \mu_{\gamma,q}\}"$ for
$q \in {\Cal J}_\gamma$ and let
${\Cal J}_{\gamma,\zeta} = \{q \in {\Cal J}_\gamma:\mu_{\gamma,q} > \zeta
\text{ and } q \Vdash \text{ Dom}({\underset\tilde {}\to I_{\gamma,\zeta}})
\text{ is } \lambda_{\gamma,q,\zeta}\}$ and let
${\underset\tilde {}\to I_{\gamma,\zeta}} \text{ be id}_{L_{\gamma,\zeta}}$, 
so $L_{\gamma,q,\zeta}$ is a
$\Bbb P_\gamma$-name of a ${\underset\tilde {}\to \kappa_{\gamma,q}}$-directed 
partial order on $\lambda_{\gamma,q,\zeta}$ (but $\Vdash_{{\Bbb P}_\gamma}$ ``if
$|{\underset\tilde {}\to {\Bbb I}_\gamma}| \le \zeta < \mu_\gamma$ then let
$L_{\gamma,\zeta}$ be trivial").

For $q \in {\Cal J}_\gamma$ let 
$L^*_{\gamma,q,\zeta}$ be ap$_{\kappa_{\gamma,q}}(
{\underset\tilde {}\to L_{\gamma,\zeta}})$ for the forcing notions \nl
$\Bbb P_\gamma^{[q]} = \{p \in \Bbb P_\gamma:q \restriction \gamma 
\le^{{\Bbb P}_\gamma}_{pr} p\}$ from Definition \scite{3.9}.  So by Claim \scite{3.10}
\medskip
\roster
\widestnumber\item{$(iii)$}
\item "{$(i)$}"   $L^*_{\gamma,q,\zeta}$ is $\kappa_{\gamma,q}$-directed
partial order on $[\lambda_{\gamma,q,\zeta}]^{< \kappa_{\gamma,q}}$
\sn
\item "{$(ii)$}"  $|L^*_{\gamma,q,\zeta}| \le (\lambda_{\gamma,q,\zeta})
^{< \kappa_{\gamma,q}}$
\sn
\item "{$(iii)$}"  $q \restriction \gamma \Vdash_{{\Bbb P}_\gamma} 
``{\underset\tilde {}\to I_{\gamma,\zeta}} = 
\text{ id}_{\underset\tilde {}\to L_{\gamma,\zeta}} \le_{\text{RK}} \text{ id}
_{L^*_{\gamma,q,\zeta}}"$.
\endroster
\medskip

\noindent
Let $\kappa_\beta = \sup\{\kappa_{\gamma,q}:\gamma < \beta,q \in
{\Cal J}_\gamma\}$. 
\sn
Let $\Bbb I^*_\beta$ be the $(< \kappa_\beta)$-closure of $\{
\text{id}_{L^*_{\gamma,q,\zeta}}:\gamma < \beta,q \in {\Cal J}_\gamma,
\zeta < \mu_{\gamma,q}\}$ (see Definition \scite{3.13}(1)).
\smallskip

Let $N$ be $(\Bbb I^*_\alpha,\bold W)$-suitable model for $(\chi,\lambda),x$
code enough informtion so for some $\bar N,N = N_{\langle \rangle}$ and 
$\bar N = \langle N_\eta:\eta \in (T,\bold I) \rangle$ be a strict 
$(\Bbb I^*_\alpha,\bold W)$-suitable tree of models for $(\chi,x),x$ 
coding enough information
(so $\bar{\Bbb Q},\Bbb I^*_\alpha,{\underset\tilde {}\to {\bold S}},\bold W \in
N_{\langle \rangle}$).

Let ${\Cal T}_{\bar N}$ be the set of pairs $(\gamma,p)$ such that:
\medskip
\roster
\item "{$\bigotimes$}"  $\gamma \le \alpha,p \in \Bbb P_\gamma$, and for some
$\kappa$, \newline
$p \Vdash_{P_\gamma} ``N[{\underset\tilde {}\to G_{P_\gamma}}] \cap \omega_1 
= N_{\langle \rangle} \cap \omega_1$ and $\gamma \in N[G_{P_\gamma}]$".
\endroster
\medskip

\noindent
${\Cal T}'_{\bar N}$ is defined similarly as the set of pairs
$(\underset\tilde {}\to \gamma,p)$ such that:
$\underset\tilde {}\to \gamma$ is a simple $(\bar{\Bbb Q},W)$-named ordinal, $p \in
\Bbb P_{\underset\tilde {}\to \gamma}$.  (I.e. if $\zeta <
\beta,G_{{\Bbb P}_\gamma}
\subseteq \Bbb P_\zeta$ is generic over $\bold V$ and $\zeta = {\underset\tilde {}\to 
\gamma_n}[G_{{\Bbb P}_\zeta}]$ then $r \in q_n \Rightarrow
{\underset\tilde {}\to \zeta_n}[G_\zeta] < \zeta$, i.e. is well defined
$< \zeta$ \ub{or} is forced ($\Vdash_{{\Bbb P}_\alpha/G_\zeta}$) to be not 
well defined, and $p \Vdash_{{\Bbb P}_{\underset\tilde {}\to \gamma}} 
``\eta \in \text{ lim } T"$).
\medskip

We consider the statements, for $\gamma \le \beta < \alpha$ (or restrict
ourselves to $\gamma$ non-limit)
\medskip
\roster
\item "{$\boxtimes_{\gamma,\beta}$}"  for any 
$(\gamma,p) \in {\Cal T}_{\bar N}$ and $\rho$ such that \newline
$p \Vdash_{{\Bbb P}_\gamma} ``\rho \triangleleft \underset\tilde {}\to \eta"$
\newline
and ${\underset\tilde {}\to p'}$ a $\Bbb P_\gamma$-name such that 
$p \Vdash_{{\Bbb P}_\gamma} ``{\underset\tilde {}\to p'}[G_{{\Bbb P}_\gamma}] \in
N[{\underset\tilde {}\to G_{{\Bbb P}_\gamma}}] \cap P_\beta/G_{{\Bbb P}_\gamma}$ and
$({\underset\tilde {}\to p'}[G_{{\Bbb P}_\gamma}]) \restriction \gamma \le_{pr} p"$ 
\ub{there is} $(\beta,q) \in {\Cal T}$ such that 
${\underset\tilde {}\to p'} \le_{pr} q$ \nl
(i.e. $p \Vdash_{{\Bbb P}_\gamma} ``{\underset\tilde {}\to p'}
[{\underset\tilde {}\to G_{{\Bbb P}_\gamma}}] \le_{pr} q"$) and 
$q \restriction \gamma = p"$.
\endroster
\medskip

We prove by induction on $\beta \le \alpha$ that $(\forall \gamma \le \beta)
\, \boxtimes_{\gamma,\beta}$ (but for \scite{6.7}(3), we use
($\forall$ non-limit $\gamma \le \beta$) $\boxtimes_{\gamma,\beta}$), note 
that for $\gamma = \beta$ the statement is trivial hence we shall consider 
only $\gamma < \beta$.
\bigskip

\noindent
\underbar{Case 1}:  $\beta = 0$.

Trivial.
\bigskip

\noindent
\underbar{Case 2}:  $\beta$ a successor ordinal (for part (3), $\beta$
successor of non-limit ordinal).

As trivially $\boxtimes_{\gamma_0,\gamma_1} \and \boxtimes_{\gamma_1,\gamma_2}
\Rightarrow \boxtimes_{\gamma_0,\gamma_2}$, clearly without loss of
generality \nl
$\beta = \gamma + 1$.

Let $G_{{\Bbb P}_\gamma}$ be such that $p \in G_{{\Bbb P}_\gamma} 
\subseteq \Bbb P_\gamma$ and
$G_{{\Bbb P}_\gamma}$ generic over $V$.

Let $\bar N' = \langle N_\nu[G_{{\Bbb P}_\gamma}]:\eta \in (T',\bold I) \rangle$. \nl
In $\bold V[G_{{\Bbb P}_\gamma}]$ we apply \sciteu{6.2} for $\lambda = \aleph_2$ to
$\bar N'$ and find $T' \subseteq T$ such that $\bold V[G_{{\Bbb P}_\gamma}] \models 
``\langle N'_\eta[G_{{\Bbb P}_\gamma}]: \eta \in (T'',\bold I') \rangle$ is strict 
$(\Bbb I^*_\gamma)^{[\kappa(\gamma)]},\bold W)$-suitable".  So we can apply
clause (f) of \sciteu{6.4A}.
\bn
\ub{Discussion}:  This is a question whether there is an $\Bbb I$-tree of
model $\langle N_\eta:\eta \in (T,\Bbb I) \rangle$ such that:
\mr
\item "{{}}"  if $\Bbb I_\eta$ is $\lambda(\Bbb I_\eta)$-complete,
$\lambda(\bold I_\eta)$ regular, $\alpha < \lambda(\bold I_\eta)
\Rightarrow |\alpha|^{\aleph_0} < \lambda(\bold I_\eta)$, \ub{then}
$\nu \in \text{ Suc}_T(\eta) \Rightarrow N_\eta <_\lambda N_\nu$.
\ermn
This would make \scite{si.2}? more effective.
\bigskip

\noindent
\underbar{Case 3}:   $\beta$ is a limit ordinal.

We shall choose by induction on $n < \omega,{\underset\tilde {}\to \gamma_n},
q_n,{\underset\tilde {}\to p_n}$ such that:
\medskip
\roster
\item "{$(a)$}"  $({\underset\tilde {}\to \gamma_n},q_n) \in 
{\Cal T}'_{\bar N}$ \newline
(so ${\underset\tilde {}\to \gamma_n}$ is a simple $(\bar{\Bbb Q},W)$-named ordinal)
\sn
\item "{$(b)$}"  ${\underset\tilde {}\to \gamma_0} = \gamma$ and
$\Vdash_{\bar{\Bbb Q}}``{\underset\tilde {}\to \gamma_n} < 
{\underset\tilde {}\to \gamma_{n+1}} < \beta"$ \newline
i.e. if $\zeta < \beta$ and $G_{{\Bbb P}_\gamma}
\subseteq \Bbb P_\zeta$ is generic over $V$ and $\zeta = {\underset\tilde {}\to 
\gamma_n}[G_{{\Bbb P}_\zeta}]$ then \nl
$r \in q_n \Rightarrow {\underset\tilde {}\to \zeta_n}[G_\zeta] < \zeta$ 
(i.e. is well defined $< \zeta$ or is forced to be not well defined),
\sn
\item "{$(c)$}"  $q_{n+1} \restriction {\underset\tilde {}\to \gamma_n} =
q_n$
\sn
\item "{$(d)$}"   ${\underset\tilde {}\to p_n}$ is a
$\Bbb P_{\underset\tilde {}\to \gamma_n}$-name, $p_0 = p,
{\underset\tilde {}\to p_n} \restriction {\underset\tilde {}\to \gamma_n}
\le_{pr} q_n$ and \newline
$q_n \Vdash_{{\Bbb P}_{\underset\tilde {}\to \gamma_n}} ``
{\underset\tilde {}\to p_n} \in N_{\underset\tilde {}\to \rho_n}
[G_{{\Bbb P}_{\underset\tilde {}\to \gamma_n}}] \cap \Bbb P_\beta$ and 
$p_n \restriction {\underset\tilde {}\to \gamma_n} \in G_{{\Bbb P}_
{\underset\tilde {}\to \gamma_n}}"$
\sn
\item "{$(e)$}"  $q_n \Vdash_{{\Bbb P}_{\underset\tilde {}\to \gamma_n}}
``{\underset\tilde {}\to p_n} \le^{{\Bbb P}_\beta}_{pr}
{\underset\tilde {}\to p_{n+1}} \in N[G_{{\Bbb P}_{\gamma_n}}] \cap P_\beta"$
\sn
\item "{$(f)$}"  letting $\langle {\underset\tilde {}\to \tau_\ell}:
\ell < \omega \rangle$ list the $P_\beta$-names of ordinals from $N$: 
for $m,\ell \le n$ we have:
\endroster
\medskip
$$
\align
q_n \Vdash_{{\Bbb P}_{\underset\tilde {}\to \gamma_n}} ``{\underset\tilde {}\to 
p_{n+1}} \text{ force } (\Vdash_{{\Bbb P}_{\underset\tilde {}\to \gamma_{n+1}}})
\text{ that:} \, &\text{ if } 
{\underset\tilde {}\to \tau_\ell} 
\text{ is a countable ordinal, then it is smaller} \\
  &\quad \text{than some }  
{\underset\tilde {}\to \tau'_\ell} \in N
[G_{{\Bbb P}_{\underset\tilde {}\to \gamma_{n+1}}}], \\
  &\quad \text{a } 
\Bbb P_{\underset\tilde {}\to \gamma_n} \text{-name of a countable ordinal}".
\endalign
$$
\medskip

\noindent
The induction is straight and $\dsize \bigcup_{n < \omega} q_n$
are as required noting we need and have $(*)_1$ or $(*)_2$ below:
\mr
\item "{$(*)_1$}"  Assume $\le_{\text{pr}},\le_{\text{vpr}}$ 
are equal to $\le$ \nl
(i.e., $\Vdash_{{\Bbb P}_\beta} 
``\le^{\underset\tilde {}\to {\Bbb Q}_\beta}_{\text{vpr}}$ is
$\le^{\underset\tilde {}\to {\Bbb Q}_\beta}"$ for each $\beta < \alpha$),
if $p \in \Bbb P_\beta,\gamma < \beta,\underset\tilde {}\to \tau$
a $\Bbb P_\beta$-name of an ordinal \ub{then} there are $p',
{\underset\tilde {}\to \tau'}$ such that:
{\roster
\itemitem{ (i) }  ${\underset\tilde {}\to \tau'}$ 
is a $\Bbb P_\gamma$-name of an ordinal
\sn
\itemitem{ (ii) }  $p \le_{\text{pr}} 
p' \in \Bbb P_\beta$ and $p \restriction \gamma = p' \restriction \gamma$
\sn
\itemitem{ (iii) }  $p' \Vdash_{{\Bbb P}_\beta} ``
\underset\tilde {}\to \tau = {\underset\tilde {}\to \tau'}"$. \nl
[why?  straight by \scitet{1.16}].
\endroster}
\sn
\item "{$(*)_2$}"  if $p \in \Bbb P_\beta,\gamma < \beta,
\underset\tilde {}\to \tau$ is a $\Bbb P_\beta$-name of a countable ordinal,
\ub{then} there are $p',\tau'$ such that
{\roster
\itemitem{ (i) }  $\tau'$ is a $\Bbb P_\gamma$-name of a countable ordinal
\sn
\itemitem{ (ii) }  $p \le_{\text{pr}} 
p' \in \Bbb P_\beta$ and $p' \restriction \gamma = p \restriction \gamma$
\sn
\itemitem{ (iii) }  $p' 
\Vdash_{{\Bbb P}_\beta} ``\underset\tilde {}\to \tau \le \tau'\,"$ 
\endroster}
\sn
[why $(*)_2$?  let $\underset\tilde {}\to \zeta$ be the following simple
$(\bar{\Bbb Q},W)$-named $[\gamma,\beta)$-ordinal: \nl
$G_\zeta \subseteq \Bbb P_\zeta$ is generic over $\bold V$ for 
$\zeta \in [\gamma,\beta)$ we let
$\underset\tilde {}\to \zeta[G_\zeta] = \zeta$ if
{\roster
\itemitem{ $(a)$ }  $p \restriction \zeta \notin G_\zeta$ \ub{or}: \nl
for some $p' \in \Bbb P_\beta$ we have $p' \restriction \zeta
= p$ and $\Bbb P_\beta \models p \le_{\text{pr}} p'$ \nl
and $p' \Vdash_{{\Bbb P}_\beta/G_\zeta} ``\underset\tilde {}\to \tau < \tau^*"$ for 
some countable ordinal $\underset\tilde {}\to \tau$ 
\sn
\itemitem{ $(b)$ }  for no $\xi \in [\gamma,\zeta)$ does clause (a) hold for
$\xi,G_\zeta \cap \Bbb P_\xi$. 
\endroster}
\sn
Now if $p \Vdash_{{\Bbb P}_\alpha} ``\underset\tilde {}\to \zeta = \gamma"$ we are
done.  Also $\Vdash_{{\Bbb P}_\alpha} ``\underset\tilde {}\to \zeta
[{\underset\tilde {}\to G_{{\Bbb P}_\alpha}}]$ is well defined" as if $p \in
G_\alpha \subseteq \Bbb P_\alpha$ and $G_\alpha$ is generic over
$\bold V$, then for
some $q \in G_\alpha$ and countable ordinal $\tau^*$ we have
$q \Vdash ``\underset\tilde {}\to \tau = \tau^*$.  By the definition of
$\aleph_1-Sp_e(W)$-iteration for some $\zeta \in [\gamma,\beta)$ we have
$\xi \in [\zeta,\beta) \Rightarrow [p \restriction \{\xi\}
\le^{\underset\tilde {}\to {\Bbb Q}_\xi}_{\text{pr}} 
q \restriction \{\xi\}$ or
$e = 4 \and p \restriction \{\xi\}$ not defined]. \nl
Define $p'$ by: $p' \restriction \underset\tilde {}\to \zeta = p 
\restriction \underset\tilde {}\to \zeta$, and for
$\xi \in [\zeta,\beta)$ we let $p' \restriction \{\xi\}$ be $q \restriction
\{\xi\}$ if: $p \restriction \{\xi\} 
\le^{\underset\tilde {}\to {\Bbb Q}_\xi}_{\text{pr}}
q \restriction \{\xi\}$ or $e=4 \and p \restriction \{\xi\}$ not defined.
Now $p'$ is as required.
\sn
So there is a $\Bbb P_{\underset\tilde {}\to \zeta}$-name of
${\underset\tilde {}\to p'}$ as appearing in the definition of
$\underset\tilde {}\to \zeta$ and it is, essentially, a member of
$\Bbb P_\beta$.
Now as we have finite apure support, the proof of
``$\underset\tilde {}\to \zeta[{\underset\tilde {}\to G_{{\Bbb P}_\alpha}}]$ is well
defined" gives $\Vdash_{{\Bbb P}_\alpha} ``\underset\tilde {}\to \zeta$ is not a
limit ordinal $> \gamma$".  Lastly $\Vdash_{{\Bbb P}_\alpha} ``\underset\tilde {}\to 
\zeta$ is not a successor ordinal $> \gamma"$ is proved by the property of
each ${\underset\tilde {}\to {\Bbb Q}_\xi}$.]
\ermn
Finishing the induction we define $q_\omega \restriction
{\underset\tilde {}\to \gamma_n} = q_n,
q_\omega \restriction [\dsize \bigcup_{n < \omega} 
{\underset\tilde {}\to \gamma_n},\beta)$ is defined as $\le_{vpr}$-upper 
bound of $\langle {\underset\tilde {}\to p_m} \restriction
[\dsize \bigcup_{n < \omega} \gamma_n,\beta):m < \omega \rangle$.
\medskip

More formally, let $\gamma^*,\beta$ and $G_{\gamma^*} \subseteq
\Bbb P_{\gamma^*}$ be such that: $G_{\gamma^*}$ is generic over $\bold V,\gamma^* =
\dsize \bigcup_{n < \omega} \gamma^*_n,\gamma^*_n = 
{\underset\tilde {}\to \gamma_n}[G_{\gamma^*}]$, let $p'_n =
{\underset\tilde {}\to p_n}[G_{\gamma^*} \cap \Bbb P_{\gamma^*_n}]$, let
$p'_n \restriction [\gamma^*,\alpha) = \{{\underset\tilde {}\to r^n_\zeta}:
\zeta < \zeta^*_n\}$ where ${\underset\tilde {}\to r^n_\zeta}$ is a simple
$[\gamma^*,\alpha)$-named atomic condition. \newline
Now we define ${\underset\tilde {}\to s^n_\zeta}$, a simple
$(\bar{\Bbb Q},W)$-named $[\gamma^*,\alpha)$-ordinal atomic condition as follows:
\medskip
\roster
\item "{$(a)$}"  $\zeta_{\underset\tilde {}\to s^n_\zeta} =
\zeta_{\underset\tilde {}\to r^n_\zeta}$
\sn
\item "{$(b)$}"  if $\zeta \in [\gamma^*,\alpha),G_{\gamma^*} \subseteq
G_\gamma \subseteq \Bbb P_\gamma,G_\gamma$ generic over $\bold V,
\zeta_{\underset\tilde {}\to r^n_\zeta}[G_\zeta] = \zeta$ then
${\underset\tilde {}\to s^n_\zeta}[G_\zeta]$ is the $<^{*{\bold V}[G_\zeta]}$-first
elements of ${\underset\tilde {}\to {\Bbb Q}_\zeta}[G_\zeta]$ which satisfies the
following:
{\roster
\itemitem{ $(*)(a)$ }  $({\underset\tilde {}\to p_n} \restriction \{\zeta\})$
\sn
\itemitem{ $(b)$ }  if 
$\langle \emptyset_{{\underset\tilde {}\to {\Bbb Q}_\zeta}[G_\zeta]} \rangle
\char 94 \langle ({\underset\tilde {}\to p_m} \restriction \{\xi\})
[G_\zeta]:m \in (n,\omega) \rangle$ has a $\le_{vpr}$-upper bound then
${\underset\tilde {}\to s^n_\zeta}[G_\zeta]$ is such upper bound.
\endroster}
\ermn
Now actually such $\le_{vpr}$-upper bound actually exists, and $q_\omega$
is as required.  \hfill$\square_{\scite{6.7}}$\margincite{6.7}
\enddemo
\bn
Now we can refine \scite{1.16A} to our iteration theorem. \nl
\ub{Saharon} revise.
\bigskip

\proclaim{\stag{6.8} Claim}  1) Suppose $F$ is a function, 
$\bold W \subseteq \omega_1$ stationary, $\Bbb I$ a class of quasi order
ideals, \ub{then} for every ordinal $\alpha$ there is 
UP$^6(\Bbb I,\bold W)$-iteration $\bar{\Bbb Q} = \langle \Bbb P_j,
{\underset\tilde {}\to {\Bbb Q}_i},{\underset\tilde {}\to \kappa_j}:
j \le \alpha^\dag,i < \alpha^\dag \rangle$, such that:
\mr
\item "{$(a)$}"  for every $i < \alpha^\dag$ we have 
${\underset\tilde {}\to {\Bbb Q}_i} = F(\bar Q \restriction i)$,
\sn
\item "{$(b)$}"  $\alpha^\dag \le \alpha$
\sn
\item "{$(c)$}"  for $\delta \le \alpha^\dag,{\underset\tilde {}\to \kappa
_\delta}$ is as in clause (d) of Definition \sciteu{6.3}
\sn
\item "{$(d)$}"  either $\alpha^\dag = \alpha$ or $F(\bar{\Bbb Q})$ is not a
pair 
$({\underset\tilde {}\to {\Bbb Q}},\underset\tilde {}\to \kappa)$ such that:
$\underset\tilde {}\to \kappa$ is a $\Bbb P_{\alpha^\dag} *
{\underset\tilde {}\to {\Bbb Q}}$-name of a cardinal from $\bold V$ and
$\Vdash_{{\Bbb P}_{\alpha^\dag}} ``{\underset\tilde {}\to {\Bbb Q}}$ 
satisfies
UP$^6_{{\underset\tilde {}\to \kappa_{\alpha^\dag}},
\underset\tilde {}\to \kappa}(\Bbb I,\bold W)$.
\ermn
2) Suppose $\beta < \alpha,G_\beta \subseteq \Bbb P_\beta$ is generic over
$\bold V$, then in $\bold V[G_\beta],\bar{\Bbb Q}/G_\beta = \langle
\Bbb P_i/G_\beta:
{\underset\tilde {}\to {\Bbb Q}_i},{\underset\tilde {}\to \kappa_i}:
\beta \le i < \alpha \rangle$ is an
${\text{\rm UP\/}}^6
(\Bbb I^{[{\underset\tilde {}\to \kappa_\beta}[G_\beta]]},
\bold W)$-iteration. \nl
3) If $\bar{\Bbb Q}$ is an ${\text{\rm UP\/}}^6(\Bbb I,\bold W)$-iteration, 
$p \in { \text{\rm Sp\/}}_e(W)-{\text{\rm Lim\/}}(\bar{\Bbb Q}),
\Bbb P'_i = \{q \in \Bbb P_i:q \ge p \restriction i\},{\underset\tilde
{}\to {\Bbb Q}'_i} =
\{p \in {\underset\tilde {}\to {\Bbb Q}_i}:
p \ge p \restriction \{i\}\}$, \ub{then}
$\bar{\Bbb Q} = \langle \Bbb P'_i,\Bbb Q'_i:i < \ell g(\bar{\Bbb Q}) 
\rangle$ is (essentially) an
${\text{\rm UP\/}}^6(\Bbb I,\bold W)$-iteration.
\endproclaim
\newpage

\head {\S7 No New Reals, Replacements for Completeness} \endhead  \resetall \sectno=7
 % \resetall 
\bn
Now we turn to ``No New Reals", there are versions corresponding to
\cite[Ch.V,\S1-\S3]{Sh:f} ($\bold W$-complete), \cite[Ch.V,\S5-\S7]{Sh:f}
($\dsize \bigwedge_{\alpha < \omega_1} \alpha$-proper 
$+ {\Cal D}$-completeness) and better \cite[Ch.VIII, \S4]{Sh:f} 
(making the previous preserved) and in different directions
\cite[Ch.XVIII,\S2]{Sh:f} and \cite{Sh:656}.
\bn
We deal here with the first (here we are interested in the cases
$\le_{pr} = \le$)

\definition{\stag{stc.0} Definition}  For $p \in \Bbb Q$ let
$\Bbb Q^{\text{pr}}_p = 
\{q \in \Bbb Q:p \le_{\text{pr}} q\}$.  A point which may confuse is that the
pure extension notion used in 
Definition \scite{stc.1}, is not necessarily the
one used seriously in the iteration.  This is the reason for the case
$e=5$ in \S1.  [Saharon check: main question: do we really need the purity
in the iteration for Nm$'$.  For Nm it is not needed (as in
\cite[Ch.XI]{Sh:f}.]
\enddefinition
\bigskip

\definition{\stag{stc.1} Definition}  1) UP$^4_{\text{com}}(\Bbb I,\bold W)$ is
satisfied by the forcing notion $\Bbb Q$, \ub{iff}: 
for any $\langle N_\eta:\eta \in
(T,\bold I) \rangle$ a strict $(\Bbb I,\bold W)$-suitable tree of models 
for $(\chi,x),x$ coding enough information, we have $(*)_1 \Rightarrow (*)_2$
where:
\medskip
\roster
\item "{$(*)_1$}"  for every $\eta,\nu \in T$, of the same length we have
$(N_\eta,\Bbb Q) \cong (N_\nu,\Bbb Q)$ and letting $h_{\eta,\nu}$ be the isomorphism 
from $N_\eta$ onto $N_\nu$ we have $h_{\eta,\nu}(x) = x$ and
$\ell < \ell g(\eta) \Rightarrow h_{\eta \restriction \ell,\nu 
\restriction \ell} \subseteq h_{\eta,\nu}$; (for $\eta,\nu \in \text{ lim}
(T)$ let $h_{\nu,\eta} = \dbcu_{\ell < \omega}
h_{\nu \restriction \ell,\eta \restriction \ell})$
\sn
\item "{$(*)_2$}"  if $\eta^* \in \text{ lim}(T),p \in N_{\langle \rangle}
\cap G$ and $G_{\eta^*}$ is a $\le_{pr}$-directed subset of 
$N_{\eta^*} \cap \Bbb Q^{pr}_p = \dbcu_{\ell < \omega} N_{\eta^* \restriction
\ell} \cap \Bbb Q^{pr}$, not disjoint to any dense subset of $\Bbb Q^{pr}_p \cap
\dsize \bigcup_{m < \omega} N_{\eta^* \restriction m}$ definable in
$(\dbcu_{m < \omega} N_{\eta^* \restriction m},N_{\eta^* \restriction m},
\bold I_{\eta^* \restriction m})_{m < \omega}$, \underbar{then}
there is $q \in \Bbb Q$ such that $p \le_{pr} q$ and 
$q \Vdash_{\Bbb Q}$ ``there is $\nu \in \text{ lim}(T)$
(in $\bold V^{\Bbb Q}$) such that $\dsize \bigcup_{\ell < \omega}
h_{\eta^* \restriction \ell,\nu \restriction \ell} (G \cap N_{\nu \restriction
\ell})$ is a subset of ${\underset\tilde {}\to G_{\Bbb Q}}$".
\ermn
2) UP$^4_{\text{stc}}(\Bbb I,\bold W)$ is satisfied by the forcing
notion $\Bbb Q$
\ub{iff} for any $\bar N = \langle N_\eta:\eta \in (T,\bold I))$ a strict
$(\Bbb I,\bold W)$-suitable tree of models for $(\chi,x),x$ coding enough
information we have $(*)_1 \Rightarrow (*)_2$ where
\mr
\item "{$(*)_1$}"  for every $\eta,\nu \in T$, of the same length we have
$(N_\eta,\Bbb Q) \cong (N_\nu,\Bbb Q)$ and letting $h_{\eta,\nu}$ be the isomorphism 
from $N_\eta$ onto $N_\nu$ we have $h_{\eta,\nu}(x) = x$ and
$\ell < \ell g(\eta) \Rightarrow h_{\eta \restriction \ell,\nu 
\restriction \ell} \subseteq h_{\eta,\nu}$; (for $\eta,\nu \in \text{ lim}
(T)$ let $h_{\nu,\eta} = \dbcu_{\ell < \omega}
h_{\nu \restriction \ell,\eta \restriction \ell})$
\sn
\item "{$(*)_2$}"  if $\eta^* \in \text{ lim}(T);\rho \in N_{<>} \cap G$ then
in the following game $\Game = \Game_{{\Bbb Q},p,\bar N,\eta}$; player complete has
a winning strategy.  The plays last $\omega$-moves, in the $n$-th move (we
can incorporate the last into the game) where
$\bold t_{\eta \restriction n} = T * x(\text{Suc}(\eta^* \restriction n) \in
\bold I^+_\eta)$ a condition $p_n \in P \cap N_{\eta^*}$ is chosen such that
$p \le p_0,[n > 0 \Rightarrow p_{n-1} \le p_n]$.  In the $n$-th move, the
anti-completeness player chooses $q_n \in \Bbb Q \cap N_{\eta^*}$ such that
$n = 0 \Rightarrow p = q_n$ and $n > 0 \Rightarrow p_{n-1} 
\le_{\text{pr}} q_n$ and the completeness player chooses $p_n \in \Bbb Q \cap N$
such that $q_n \le p_n \in \Bbb Q \cap N_{\eta^*}$.  In the end the completeness
player wins iff one of the following occurs:
{\roster
\itemitem{ $(\alpha)$ }  for some $\Bbb Q$-name $\underset\tilde {}\to \tau \in
N_{\eta^*}$ of a countable ordinal, no $p_n$ forces a value
\sn
\itemitem{ $(\beta)$ }  not $(\alpha)$ but there is $q$ such that:
$p \le_{\text{pr}} q \in \Bbb Q$ and $q \Vdash_{\Bbb Q}$ ``there is $\eta \in
\text{ lim}(T)$ such that $n < \omega \Rightarrow
h_{{\underset\tilde {}\to \eta} \restriction n,\eta^* \restriction n}(p_n)
\in {\underset\tilde {}\to G_{\Bbb Q}}"$.
\endroster}
\ermn
3) We define ``$\Bbb Q$ satisfies UP$^1_{\text{com},\underset\tilde {}\to \kappa}
(\Bbb I,W)$ is defined as in (1) but we replace the conclusion of $(*)_2$
by: there is $q \in \Bbb Q$ such that $p \le_{\text{pr}} q$ and

$$
\align
q \Vdash_{\Bbb Q} ``&\text{there is } (T',\bold I) \text{ satisfying }
(T,\bold I) \le^{\underset\tilde {}\to \kappa} (T',\bold I) \\
  &\text{(see \scite{2.4}(d)) such that for every} \\
  &\nu \in \text{ lim}(T') \text{ we have } 
h_{\nu,\eta^*}(G^*_\eta) \subseteq {\underset\tilde {}\to G_{\Bbb Q}}".
\endalign
$$
\mn
4) Similarly UP$^1_{\text{stc},\underset\tilde {}\to \kappa}(\Bbb I,W)$.
\nl
5) We say UP$^4_{\text{stc}}(\Bbb I,\bold W)$ if letting ${\frak p} = 
\langle({\underset\tilde {}\to \tau_n},I_n,\eta^*(n):I_{\eta^* \restriction
n},(\bold t_{\eta^* \restriction n}),n < \omega \rangle$ be such that
$\{{\underset\tilde {}\to \tau_n}:n < \omega\}$ list the $\Bbb Q$-names of
ordinals in $N_{\eta^*},\{I_n:n < \omega\}$ lists $\Bbb I \cap
N_{\eta^*}$, the winning strategy on each stage depends just on
$\theta,N_{<>} \cap \omega_1$ and in ${\frak p}$ continuously.
\enddefinition
\bigskip

\remark{\stag{stc.2} Remark}  1) This property relates to the UP$(\Bbb I,
\bold W)$ just as $E$-complete relates to $E$-proper (see 
\cite[Ch.V,\S1]{Sh:f}). \newline
2) Who satisfies this condition?  See section 8, so $\bold W$-complete
forcing notions, Nm$'(D)$,Nm$(D)(D$ is $\aleph_2$-complete) 
Nm$^{(\prime)}(T,\bold I)$ (when $\bold I$ is $\aleph_2$-complete), 
and shooting a club through a stationary subset of
some $\lambda = \text{ cf}(\lambda) > \aleph_1$ consisting of ordinals of
cofinality $\aleph_0$ (and generally those satisfying the $\Bbb I$-condition
from \cite[Ch.XI]{Sh:f}).
\endremark
\bigskip

\proclaim{\stag{stc.3} Claim}  If $\Bbb Q$ satisfies 
${\text{\rm UP\/}}^1_{\text{com}}
(\Bbb I,\bold W)$ or ${\text{\rm UP\/}}^1_{\text{stc}}(\Bbb I,\bold W)$ and 
$\Bbb I$ is $(2^{\aleph_0})^+$-complete, and $\Bbb Q$ has
$(\omega_1,2)$-pure decidability, \ub{then} 
forcing by $\Bbb Q$ add no new real.
\endproclaim
\bigskip

\demo{Proof}  Immediate.
\enddemo
\bigskip

\proclaim{\stag{stc.4} Claim}  Suppose:
\mr
\item "{$(a)$}"  $\Bbb Q$ is a 
forcing notion satisfying the ${\text{\rm UP\/}}^1_{\text{com}}
(\Bbb I,\bold W)$ and the local $\underset\tilde {}\to \kappa$-c.c. where
$\underset\tilde {}\to \kappa$ is a purely decidable $\Bbb Q$-name 
\sn
\item "{$(b)$}"  $\bar N = \langle N_\eta:\eta \in (T,\bold I) \rangle$
is a strict $(\Bbb I,\bold W)$-suitable tree of models (for $\chi$ and 
$x = \langle \Bbb Q,\underset\tilde {}\to \kappa,\Bbb I,\bold W) \rangle)$ 
satisfying $(*)_1$ of Definition \scite{stc.1}
\sn
\item "{$(c)$}"  the family 
${\underset\tilde {}\to {\Bbb I}'} =: \{I \in \Bbb I:I
\text{ is } \underset\tilde {}\to \kappa \text{-complete}\}$ is 
$(< \underset\tilde {}\to \kappa)$-closed.
\ermn
\ub{Then} $\Bbb Q$ satisfies ${\text{\rm UP\/}}^1_{\text{stc},
\underset\tilde {}\to \kappa}
(\Bbb I,\bold W)$.
\endproclaim
\bigskip

\demo{Proof}  Let $(T^*,\bold I^*)$ and $\bar N = \langle N_\eta:\eta \in
(T^*,\bold I^*) \rangle,\langle h_{\eta,\nu}:\eta,\nu \in T^* \cap
{}^n\text{Ord for some } n \rangle$ be as in Definition \scite{stc.1}. \nl
Let $\eta^* \in \text{ lim}(T^*),p \in \Bbb Q \cap N_{<>}$ be given and we choose
as our strategy for proving UP$^1_{\text{stc}}(\Bbb I,\bold W)$ the same
strategy that exists as UP$^1_{\text{stc}}(\Bbb I,\bold W)$ and let
$\langle p_n:n < \omega \rangle$ be a play as in Definition \scite{stc.1} in
which the completeness player uses his winning strategy. \nl
Let ${\Cal T} = \{T:(T^*,\bold I^*) \le^* (T,\bold I^* \restriction T)\}$.
As we can replace $p$ by $p'$ if $p \le_{\text{pr}} p' \in \Bbb Q \cap N_{<>}$,
\wilog \, $p$ forces a value to $\kappa$.  So for every $T \in {\Cal T}$
there are $q$ and $\underset\tilde {}\to \eta$ such that

$$
\align
p \le_{\text{pr}} q \in \Bbb Q \text{ and } q \Vdash_Q 
``&\underset\tilde {}\to \eta
\in \text{ lim } \underset\tilde {}\to T \text{ and } h = \bigcup
h_{\underset\tilde {}\to \eta} \restriction n,\eta^* \restriction n \\
  &\text{satisfies } n < \omega \Rightarrow h(\beta) \in
{\underset\tilde {}\to G_{\Bbb Q}} \\
  &\text{(hence } N_{{\underset\tilde {}\to \eta} \restriction n} \cap
\omega_1 = N_{<>} \cap \omega_1)".
\endalign
$$
\mn
Remember that we can replace $\eta^*$ by any $\eta^{**} \in \text{ lim}(T)$.
Let 

$$
\align
{\underset\tilde {}\to T^*}[G_{\Bbb Q}] = \{\nu \in T^*:&G_{\Bbb Q} \cap N_\nu \supseteq
\{h_{\eta,\eta^* \restriction \ell g(\eta)}(r), \\
  &r \in \Bbb Q \cap N_{\eta^* \restriction (\ell g(\nu))} \text{ and }
r \le p_n \text{ for some } n\}
\endalign
$$
\mn
clearly it is a subtree.  We continue as in the proof of \scite{5.2}.
\hfill$\square_{\scite{stc.4}}$\margincite{stc.4}
\enddemo
\bigskip

\proclaim{\stag{stc.5} Claim}  Suppose:
\mr
\item "{$(a)$}"  $\bar{\Bbb Q} = \langle \Bbb P_i,{\underset\tilde
{}\to {\Bbb Q}_i},
{\underset\tilde {}\to {\Bbb I}_i},{\underset\tilde {}\to \kappa_i}:
i < \alpha \rangle$ is a ${\text{\rm UP\/}}^{1,e}(\bold
W,W)$-iteration with \nl 
${\text{\rm Sp\/}}_e(W)$-limit $\Bbb P_\alpha$
\sn
\item "{$(b)$}"  $\Vdash_{{\Bbb P}_i} ``{\underset\tilde {}\to {\Bbb Q}_i}$ 
satisfies ${\text{\rm UP\/}}^1_{\text{stc}}
({\underset\tilde {}\to {\Bbb I}_i},\bold W)$ 
\sn
\item "{$(c)$}"  ${\underset\tilde {}\to \kappa^-}(\beta) = {
\text{\rm Min\/}} 
\{{\underset\tilde {}\to \kappa_\gamma}:\gamma < \gamma\}$, \newline
$\kappa^+(\beta) = { \text{\rm Sup\/}}
\{ \kappa:\text{for some } \gamma < \beta,
\nVdash_{{\Bbb P}_\gamma} 
{\underset\tilde {}\to \kappa_\gamma} \ne \kappa\}$, \nl
$\kappa^-(\beta) = { \text{\rm Min\/}}
\{ \kappa:\text{for some } \gamma < \beta
\text{ we have } \nVdash_{{\Bbb P}_\gamma} 
{\underset\tilde {}\to \kappa_\gamma} \ne \kappa\}$.
\ermn
\underbar{Then} \,1) for each $\beta \le \alpha,\Bbb P_\beta$ satisfies
${\text{\rm UP\/}}^1_{\text{com}}({\Bbb I}'_\beta,\bold W)$ for some 
$\kappa^-(\beta)$-complete set ${\Bbb I}'_\beta$ of (partial order)
ideals. \newline 
2) In fact, ${\Bbb I}_\beta$ is $\kappa^\beta$-complete where \newline
$\kappa^\beta = \min\{\kappa:\text{for some } \gamma < \beta$ we
have $\nVdash_{{\Bbb P}_{\gamma +1}} ``{\underset\tilde {}\to \kappa_\gamma} \ne
\kappa"\}$, and each $I \in \Bbb I_\beta$ \newline
has domain of cardinality
$\le (\underset{\gamma < \beta} \to \sup\{ \lambda < \kappa_{\delta +1}:\,
\nVdash_{{\Bbb P}_\gamma} ``\neg(\exists I \in 
{\underset\tilde {}\to {\Bbb I}_\gamma})(\lambda = |{\text{\rm Dom\/}}
(I)|)"\})$
\newline
and $|{\Bbb I}'_\beta| \le \dsize \sum_{\gamma < \beta} 
(\aleph_0 + |\Bbb P_\gamma| + \min\{(\lambda:\,\Vdash_{{\Bbb P}_\gamma} 
``|{\underset\tilde {}\to {\Bbb I}_\gamma}| \le \lambda)^{< \kappa}\}$.
\endproclaim
\bigskip

\remark{\stag{stc.5A} Remark}  We can 
use this for iteration as in \scite{5.8},
the version with clauses (b), (d) or (d)$'$, $W \cap \alpha = \emptyset$.
To prove $\Bbb P_\alpha$ does not add reals, it is enough to prove
that for each
$\beta < \alpha$, forcing with $\Bbb P_\beta$ does not add reals.  By
$\{p \in \Bbb P_\beta:(\forall \gamma < \beta) \Vdash_{{\Bbb P}_\gamma}
``\emptyset_{\underset\tilde {}\to {\Bbb Q}_\gamma} < p \restriction 
\{\gamma\}"\}$ is $\le_{\text{vpr}}$-dense.  This should be useful in \cite{GoSh:511}.
\endremark
\bn
SAHARON:  1) Use less $\underset\tilde {}\to \kappa$. \nl
2) What requirements will resurrect $\le_{\text{vpr}}$?
\bigskip

\demo{Proof}  Similar to the one of \scite{5.5}. \newline

For each $\gamma < \alpha$ let ${\Cal J}_\gamma =:
\{q \in \Bbb P_{\gamma +1}:q \text{ forces a value to } 
{\underset\tilde {}\to \kappa_\gamma}$, called 
$\kappa_{\gamma,q}$ and $q$ forces
${\underset\tilde {}\to {\Bbb I}_\gamma}$ to be equal to a $\Bbb P_\gamma$-name 
${\underset\tilde {}\to {\Bbb I}_{\gamma,q}}$ and
$q \restriction \gamma$ forces a value to
$|\Bbb I_\gamma|$ says $\mu_{\gamma,q}\}$; let ${\Cal J}'_\gamma \subseteq
{\Cal J}_\gamma$ be a maximal antichain.  Let $\mu_\gamma =
\underset{q \in {\Cal J}'_\gamma} \to \sup \mu_{\gamma,q}$. \newline
Let $q \Vdash_{{\Bbb P}_\gamma} ``{\underset\tilde {}\to {\Bbb I}_\gamma} = \{
{\underset\tilde {}\to I_{\gamma,\zeta}}:\zeta < \mu_{\gamma,q}\}"$ for
$q \in {\Cal J}_\gamma$ and let
${\Cal J}_{\gamma,\zeta} = \{q \in {\Cal J}_\gamma:\mu_{\gamma,q} > \zeta
\text{ and } q \Vdash ``\text{ Dom}({\underset\tilde {}\to I_{\gamma,\zeta}})
\text{ is } \lambda_{\gamma,q,\zeta}"$ if this is purely decidable$\}$ and let
${\underset\tilde {}\to I_{\gamma,\zeta}} \text{ be id}_{L_{\gamma,\zeta}}$, 
so $L_{\gamma,q,\zeta}$ is a
$\Bbb P_\gamma$-name of a ${\underset\tilde {}\to \kappa_{\gamma,q}}$-directed 
partial order on $\lambda_{\gamma,q,\zeta}$ (but $\Vdash_{{\Bbb P}_\gamma}$ ``if
$|{\underset\tilde {}\to {\Bbb I}_\gamma}| \le \zeta < \mu_\gamma$ then let
$L_{\gamma,\zeta}$ be trivial").

For $q \in {\Cal J}_\gamma$ let 
$L^*_{\gamma,q,\zeta}$ be ap$_{\kappa_{\gamma,q}}(
{\underset\tilde {}\to L_{\gamma,\zeta}})$ for the forcing notions \nl
$\Bbb P_\gamma^{[q]} = \{p \in P_\gamma:q \restriction \gamma
\le^{{\Bbb P}_\gamma}_{pr} p\}$ from Definition \scite{3.9}.  So by Claim \scite{3.10}
\medskip
\roster
\widestnumber\item{$(iii)$}
\item "{$(i)$}"   $L^*_{\gamma,q,\zeta}$ is $\kappa_{\gamma,q}$-directed
partial order on $[\lambda_{\gamma,q,\zeta}]^{< \kappa_{\gamma,q}}$
\sn
\item "{$(ii)$}"  $|L^*_{\gamma,q,\zeta}| \le (\lambda_{\gamma,q,\zeta})
^{< \kappa_{\gamma,q}}$
\sn
\item "{$(iii)$}"  
$q \restriction \gamma \Vdash_{{\Bbb P}_\gamma} 
``{\underset\tilde {}\to I_{\gamma,\zeta}} = 
\text{ id}_{\underset\tilde {}\to L_{\gamma,\zeta}} \le_{\text{RK}} \text{ id}
_{L^*_{\gamma,q,\zeta}}"$.
\endroster
\medskip

\noindent
Note: $\kappa^-(\beta) = \text{ Min}\{\kappa_{\gamma,q}:\gamma < \beta,q \in
{\Cal J}_\gamma\}$. 
\sn
Let $\Bbb I^*_\beta$ be the $(< \kappa^-(\beta))$-closure of $\{
\text{id}_{L^*_{\gamma,q,\zeta}}:\gamma < \beta,q \in {\Cal J}_\gamma,
\zeta < \mu_{\gamma,q}\}$ (see Definition \scite{3.13}(1)).
\smallskip

Let $\bar N = \langle N_\eta:\eta \in (T^*,\bold I) \rangle$ be a strict 
$(\Bbb I^*_\alpha,\bold W)$-suitable tree of models for $(\chi,x),x$ coding 
enough information (so $\bar{\Bbb Q},\Bbb I^*_\alpha,\bold W,W \in 
N_{\langle \rangle}$).  
For any $\gamma < \alpha$ and $G_\gamma \subseteq \Bbb P_\gamma$
generic over $\bold V,
T \subseteq T^*$ and $\bold I^{[\kappa^-(\gamma)]}$-tree and $\nu \in T$ and
$\eta^* \in \text{ lim}(T)$ and $p \in N_\nu[G_\gamma] \cap (\Bbb P_\alpha/
G_\gamma)$ then let $\bar N_{\nu,T}[G] = \langle N_{\nu \char 94 \rho}[G]:
\nu \char 94 \rho \in T \rangle$ then we can find a winning strategy
{\bf St} for the completeness player in the game $\Game = 
\Game_{\bar N_{\nu,T}}[G],p,\eta^*,\Bbb P_\alpha/G_\gamma$ of \scite{stc.1}(2).
Without loss of generality if $\eta^*_1,\eta^*_2 \in \text{ lim}(T)$ the
isomorphism from $N_{\eta^*_1}[G]$ onto $N_{\eta^*_2}[G]$ commutes with the
winning strategies; so the choice of $\eta^*$ is not important.  Of course,
we have a name ${\underset\tilde {}\to {\bold{St}}} = 
\bold{St}_{\underset\tilde {}\to \nu,\underset\tilde {}\to T,
{\underset\tilde {}\to \eta^*}}$.  
\sn
Now fix $\eta^* \in \text{ lim}(T^*)$ and we define a strategy {\bf St} for
the game.  For each simple $(\bar{\Bbb Q},W)$-name of an $[0,\alpha)$-ordinal
$\underset\tilde {}\to \gamma \in N_{\eta^*}$, let $\langle
(\tau^{\underset\tilde {}\to \gamma}_n,{\underset\tilde {}\to I^\gamma_n},
\eta^*(n),\bold I_{\eta^* \restriction n},\bold t_{\eta^* \restriction m}):
n < \omega \rangle$ be as in \scite{stc.1}(2) for
${\underset\tilde {}\to {\Bbb Q}_{\underset\tilde {}\to \gamma}}$.
\sn
We define {\bf St} such that if $\bar p = \langle p_n:n < \omega \rangle$ is
a play in which the completeness player uses his winning strategy then this
holds for $\langle p_n(\underset\tilde {}\to \gamma):n < \omega \rangle$
for each $\gamma$, i.e.,
\mr
\item "{$(*)$}"  if $\gamma < \alpha,G_\gamma \subseteq \Bbb P_\gamma$ is
generic over $\bold V$ and $p \restriction \gamma \in G$ and
$\underset\tilde {}\to \gamma[G_\gamma] = \gamma$ and $T \in \bold V[G_\gamma]$ is
a subtree of $T^*,\Bbb I^{[\kappa^-(\gamma)]}$-large and $\eta^{**} \in
\text{ lim}(T)^{V[G_\gamma]}$ and $p'_\gamma = (h^*_{\eta^{**},\eta^*}(p_n))
(\gamma)[G_\gamma] \in {\underset\tilde {}\to {\Bbb Q}_\gamma}[G_\gamma]$, \ub{then}
in $\langle p'_n:n < \omega \rangle$ the completeness player uses his
winning strategy from above.
\ermn
So fix such $\bar p^* = \langle p^*_n:n < \omega \rangle$, we would like to
find $q$ as in Definition \scite{stc.1}(2).
\sn
Let ${\Cal T}_{\bar N}$ be the set of quadruples 
$(\gamma,q,\underset\tilde {}\to \nu,\underset\tilde {}\to T)$ such that:
\medskip
\roster
\item "{$\bigotimes_1$}"  $\gamma \le \alpha,q \in \Bbb P_\gamma,
\Bbb P_\gamma \models p \restriction \gamma \le_{\text{pr}} q$ and \newline
$q \Vdash_{{\Bbb P}_\gamma} ``(\alpha) \quad 
\underset\tilde {}\to \nu \in \underset\tilde {}\to T
\subseteq T^*$, where $\underset\tilde {}\to \nu,\underset\tilde {}\to T$
are $\Bbb P_\gamma$-names \nl

$\qquad (\beta) \quad 
N_{\langle \rangle}[{\underset\tilde {}\to G_{{\Bbb P}_\gamma}}]
\cap \omega_1 = N_{\langle \rangle} \cap \omega_1$ 

$\qquad (\gamma) \quad \gamma \in \dbcu_{\ell < \omega} 
N_{\underset\tilde {}\to \nu}[G_{{\Bbb P}_\gamma}]$ \nl

$\qquad (\delta) \quad \langle N_{\eta,\ell}
[{\underset\tilde {}\to G_{{\Bbb P}_\gamma}}]:\eta \in \underset\tilde {}\to T$ is a
strictly 
$(\Bbb I^*_\gamma)^{[{\underset\tilde {}\to \kappa}(\gamma)]}$-suitable 
tree, \nl

$\qquad (\varepsilon) \quad$ for every
$\underset\tilde {}\to \eta \in \text{ lim}(\underset\tilde {}\to T)$ we have
\nl

$\qquad \qquad 
\{h_{{\underset\tilde {}\to \eta},\eta^*}(p^*_n) \restriction \gamma:n <
\omega\}$ is a subset of ${\underset\tilde {}\to G_{{\Bbb P}_\gamma}}"$.
\endroster
\medskip

\noindent
Now ${\Cal T}'_{\bar N}$ is defined similarly as the set of quadruples
$(\underset\tilde {}\to \gamma,q,\underset\tilde {}\to \nu,
\underset\tilde {}\to T)$ such that: as in $\otimes_1$ but we have
$\underset\tilde {}\to \gamma$ is a simple $(\bar{\Bbb Q},W)$-named ordinal, $q \in
\Bbb P_{\underset\tilde {}\to \gamma}$ and in clause $(\gamma)$ \,
$\Vdash \underset\tilde {}\to \gamma
\in 
N_{\underset\tilde {}\to \nu}[G_{{\Bbb P}_{\underset\tilde {}\to \gamma}}]$.  
(I.e., if $\zeta < \beta,G_{{\Bbb P}_\gamma}
\subseteq \Bbb P_\zeta$ is generic over $\bold V$ and $\zeta = {\underset\tilde {}\to 
\gamma_n}[G_{{\Bbb P}_\zeta}]$ then $r \in q_n \Rightarrow
{\underset\tilde {}\to \zeta_n}[G_\zeta] < \zeta$, i.e., is well defined
$< \zeta$ \ub{or} is forced ($\Vdash_{{\Bbb P}_\alpha/G_\zeta}$) to be not 
well defined, and $p \Vdash_{{\Bbb P}_{\underset\tilde {}\to \gamma}} 
``\eta \in \text{ lim}(T)"$).
\medskip

We consider the statements, for $\gamma \le \beta < \alpha$
\medskip
\roster
\item "{$\boxtimes_{\gamma,\beta}$}"  \ub{for any} 
$(\gamma,p,\underset\tilde {}\to \eta,\underset\tilde {}\to T) \in 
{\Cal T}_{\bar N}$ and $\underset\tilde {}\to \rho$ such that \newline
$p \Vdash_{{\Bbb P}_\gamma} ``\underset\tilde {}\to \eta \triangleleft 
\underset\tilde {}\to \rho \in \underset\tilde {}\to T"$
\newline
and ${\underset\tilde {}\to p'}$ a $\Bbb P_\gamma$-name such that \nl
$p \Vdash_{{\Bbb P}_\gamma} ``{\underset\tilde {}\to p'}
[{\underset\tilde {}\to G_{{\Bbb P}_\gamma}}] \in
N_{\underset\tilde {}\to \rho}
[{\underset\tilde {}\to G_{{\Bbb P}_\gamma}}] \cap 
P_\beta/{\underset\tilde {}\to G_{{\Bbb P}_\gamma}}$ and \nl
$({\underset\tilde {}\to p'}[G_{P_\gamma}]) \restriction \gamma \le p"$ 
\ub{there is} $(\beta,q,\underset\tilde {}\to \nu,{\underset\tilde {}\to T'}) 
\in {\Cal T}$ such that \nl
${\underset\tilde {}\to p'} \le q$ 
(i.e., $p \Vdash_{{\Bbb P}_\gamma} ``{\underset\tilde {}\to p'}
[{\underset\tilde {}\to G_{{\Bbb P}_\gamma}}] \le q"$) and 
$q \restriction \gamma = p$ and ${\Cal T}' \subseteq {\Cal T}"$.
\endroster
\medskip

We prove by induction on $\beta \le \alpha$ that $(\forall \gamma \le \beta)
\, \boxtimes_{\gamma,\beta}$ (or, for strong preservation), that 
($\forall$ non-limit
$\gamma \le \beta) \boxtimes_{\gamma,\beta})$, note that for $\gamma = \beta$ 
the statement is trivial hence we shall consider only $\gamma < \beta$.
\bigskip

\noindent
\underbar{Case 1}:  $\beta = 0$.

Trivial.
\bigskip

\noindent
\underbar{Case 2}:  $\beta$ a successor ordinal.

As trivially $\boxtimes_{\gamma_0,\gamma_1} \and \boxtimes_{\gamma_1,\gamma_2}
\Rightarrow \boxtimes_{\gamma_0,\gamma_2}$, clearly without loss of
generality \nl
$\beta = \gamma + 1$.

Let $G_{{\Bbb P}_\gamma}$ be such that $p \in G_{{\Bbb P}_\gamma} 
\subseteq \Bbb P_\gamma$ and
$G_{{\Bbb P}_\gamma}$ generic over $V$.

Let $T' = \{\nu:\rho \char 94 \nu \in \underset\tilde {}\to T
[G_{{\Bbb P}_\gamma}]\},\bar N' = \langle N'_\nu:\nu \in
(T',\bold I') \rangle$ where $N'_\nu = N_{\rho \char 94 \nu}[G_{{\Bbb P}_\gamma}]$,
\nl
$\bold I'_\nu = \bold I^*_{\rho \char 94 \nu}$. \newline
By \scite{stc.4} applied to $\bar N'$ we can find 
$\underset\tilde {}\to p',{\underset\tilde {}\to T''}$ as required.
\bigskip

\noindent
\underbar{Case 3}:   $\beta$ is a limit ordinal.

By \scite{stc.4} it suffices to prove $\bigotimes_2$ there are $q$ and
$\underset\tilde {}\to \eta$ such that: $\underset\tilde {}\to \eta$ is a
$\Bbb P_\beta$-name, $q \in P_\beta,q \restriction \gamma = p \restriction \gamma,
p \le q$ and $q \Vdash ``\underset\tilde {}\to \eta \in \text{ lim}
(\underset\tilde {}\to T)"$ and $\dbcu_{\ell < \omega}
N_{{\underset\tilde {}\to \eta} \restriction \ell}
[{\underset\tilde {}\to G_{{\Bbb P}_\beta}}] \cap \omega_1 = N_{\langle \rangle}
\cap \omega_1$ and $\{ h_{{\underset\tilde {}\to \eta},\eta^*}(r)
\restriction \beta:r \in G_{\eta^*}\} \subseteq 
{\underset\tilde {}\to G_{{\Bbb P}_\beta}}$.

We should choose by induction on $n < \omega,{\underset\tilde {}\to \gamma_n},
q_n,{\underset\tilde {}\to \rho_n},{\underset\tilde {}\to \eta_n},k_n$ 
such that:
\medskip
\roster
\item "{$(a)$}"  $({\underset\tilde {}\to \gamma_n},q_n,
{\underset\tilde {}\to \eta_n}) \in {\Cal T}'_{\bar N}$ \newline
(so ${\underset\tilde {}\to \gamma_n}$ is a $\bar{\Bbb Q}$-named ordinal)
\sn
\item "{$(b)$}"  ${\underset\tilde {}\to k_n}$ is a
$P_{\underset\tilde {}\to \gamma_n}$-name of a natural number
\sn
\item "{$(c)$}"  ${\underset\tilde {}\to \rho_n}$ is a $\Bbb P_{\gamma_n}$-name
(of a member of $T$)
\sn
\item "{$(d)$}"  $q_n \Vdash_{{\Bbb P}_{\underset\tilde {}\to \gamma_n}}
``{\underset\tilde {}\to \eta_n} \restriction {\underset\tilde {}\to k_n} =
{\underset\tilde {}\to \rho_n}"$
\sn
\item "{$(e)$}"  ${\underset\tilde {}\to \gamma_0} = \gamma$ and
$\Vdash_{\bar{\Bbb Q}}``{\underset\tilde {}\to \gamma_n} < 
{\underset\tilde {}\to \gamma_{n+1}} < \beta$ and 
${\underset\tilde {}\to \gamma_{n+1}}$ non-limit" \newline
i.e., if $\zeta < \beta$ and $G_{{\Bbb P}_\gamma}
\subseteq \Bbb P_\zeta$ is generic over $V$ and $\zeta = {\underset\tilde {}\to 
\gamma_n}[G_{{\Bbb P}_\zeta}]$ then \nl
$r \in q_n \Rightarrow {\underset\tilde {}\to \zeta_n}[G_\zeta] < \zeta$ 
(i.e. is well defined $< \zeta$ or is forced to be not well defined),
\sn
\item "{$(f)$}"  $q_{n+1} \restriction {\underset\tilde {}\to \gamma_n} =
q_n$
\sn
\item "{$(g)$}"  $q_{n+1} 
\Vdash_{{\Bbb P}_{\underset\tilde {}\to \gamma_{n+1}}} 
``{\underset\tilde {}\to \rho_n} \triangleleft
{\underset\tilde {}\to \rho_{n+1}}$, so 
${\underset\tilde {}\to k_n} < {\underset\tilde {}\to k_{n+1}}"$.
\ermn
Finishing the induction we let
$\underset\tilde {}\to \eta = \dsize \bigcup_{n < \omega}
{\underset\tilde {}\to \rho_n}$ and we define $q_\omega \restriction
{\underset\tilde {}\to \gamma_n} = q_n$ and \nl
$q_\omega \in \Bbb P
\dbcu_{n < \omega} {\underset\tilde {}\to \gamma_n}$. \nl
We shall check that $\bigotimes_2$ holds which is straight.
\enddemo
\bigskip

\demo{\stag{7.6} Discussion}  1) As in \S6 (not \S5)? \nl
2) The other $NNR$.

Like V and like XVIII. \nl
A. \,\, Like XVIII,\S2 - seem straight but check. \nl
B. \,\, Like V,\S6 - think. \nl
3) Explain the specific choice for \scite{stc.1}. \nl
4) Think Ch.VI,\S1, $\le = \le_{pr}$. \nl
\S3 not necessarily.
\enddemo 
\newpage

\head {\S8 Examples} \endhead  \resetall \sectno=8
 % \resetall 
\bigskip

\noindent
Namba [Nm] defines Nm$(J^{bd}_\lambda)$ (and also with $\omega$ ideals)
as examples of forcing notion preserving $\aleph_1$ but changing the 
cofinality of some $\lambda = \text{ cf}(\lambda)$ to $\aleph_0$. \newline
More \cite{RuSh:117}, \cite[X,XI,XV,XIV,\S5]{Sh:f}.
\bigskip

\definition{\stag{8.1} Definition}  1) For an ideal $I$ on a cardinal
$\lambda$, let the forcing notion Nm$(I)$ be

$$
\align
\text{Nm}(I) = \biggl\{ T:&T \subseteq {}^{\omega >}\lambda 
\text{ is non-empty, closed under initial segments and} \\
  &(\forall \eta \in T)(\exists \nu)[\eta \triangleleft \nu \in T \and
   (\exists^{I^+} \alpha < \lambda)(\eta \char 94 \langle \alpha \rangle \in
    T)] \biggr\}
\endalign
$$
\mn
where $(\exists^{I^+} \alpha < \lambda)\text{Pr}(\alpha) \text{ means }
\{\alpha < \lambda:\text{Pr}(\alpha)\} \in I^+ 
\text{ and } I^+ = \{A \subseteq \lambda:A \notin I\}$ ordered by 
inverse inclusion and let $<_{\text{pr}} = \le$ 
and $\le_{\text{vpr}}$ be the equality
$p \in \text{ Nm}(I)$ 
is normal if $\forall \eta \in p \Rightarrow |\text{Suc}
(\eta)| = 1 \vee \text{ Suc}_T(\eta) \ne \emptyset \text{ mod } I$. \nl
2) For an ideal $I$ and a cardinal $\lambda$, let the forcing notions
Nm$'(I)$ be

$$
\align
\text{Nm}'(I) = \biggl\{ T:\,&T \subseteq {}^{\omega >}\lambda 
\text{ is non-empty, closed under initial segments and for some} \\
  &\,n = n(T) < \omega \text{ we have}: \\
  &\,(i) \quad \ell \le n \Rightarrow |T \cap {}^\ell \lambda| = 1 \\
  &\,(ii) \quad \eta \in T \and \ell g(\eta) \ge n \Rightarrow
(\exists^{I^+} \alpha < \lambda)[\eta \char 94 \langle \alpha \rangle \in
T] \biggr\}
\endalign
$$
\mn
we call the $\eta \in T \cap {}^{n(T)}\lambda$ 
the trunk of $T$ and denote it by tr$(T)$) 
\sn
\underbar{ordered} by inverse inclusion 
and let $\le_{\text{pr}} = \le^*$ (see \S2)
and $\le_{\text{vpr}}$ be the equality.
\sn
3) Writing a filter $D$ means the dual ideal.
\enddefinition
\bigskip

\proclaim{\stag{8.2} Claim}  Let $I$ be a $\kappa$-complete ideal on
$\lambda,\lambda \ge \kappa \ge \aleph_2,I \in \Bbb I,\Bbb I$ is 
(restriction closed and) $\kappa$-complete. \newline
1) \text{\rm Nm}$(I)$ and 
\text{\rm Nm}$'(I)$ satisfies ${\text{\rm UP\/}}^1(\Bbb I)$ 
and ${\text{\rm UP\/}}^4_{\lambda^+}(\Bbb I)$
and ${\text{\rm UP\/}}^6_{\lambda^+}(\Bbb I)$ 
so does not collapse $\aleph_1$. \newline
) If $I$ is uniform, then $\Vdash_{\text{Nm}(I)} ``{\text{\rm cf\/}}
(\lambda) = 
\aleph_0"$ and 
$\Vdash_{\text{Nm}'(I)} ``{\text{\rm cf\/}}(\lambda) = \aleph_0"$, 
in fact if $[A \in I^+ \Rightarrow I \restriction A$ is 
$\lambda'$-decomposable] and $\lambda'$ is regular, 
\ub{then} the same holds for $\lambda'$. \newline
3) $|{\text{\rm Nm\/}}(I)|,|{\text{\rm Nm\/}}'(I)| \le 2^\lambda$. \newline
4) If in addition $2^{\aleph_0} < \kappa$, \underbar{then} forcing with
${\text{\rm Nm\/}}(I)$ does not add reals, moreover it satisfies the 
condition from \scite{stc.1}, ${\text{\rm UP\/}}^4_{\text{com}}(\Bbb I)$. \nl
5) If in addition $2^{\aleph_0} < \kappa$ \ub{then} forcing with
${\text{\rm Nm\/}}(I),{\text{\rm Nm\/}}'(I)$ 
does not add reals; moreover, they satisfy the condition
${\text{\rm UP\/}}^{4,+}_{\text{stc}}(\Bbb I)$.
\endproclaim
\bigskip

\demo{Proof}  1)  We will use the following fact about $\Bbb Q = \text{ Nm}(I)$ 
and $\Bbb Q = \text{ Nm}'(I)$:
\medskip
\roster
\item "{$(*)$}"  If $p \in \Bbb Q,\underset\tilde {}\to \alpha$ is a
$\Bbb Q$-name of 
an ordinal, \newline
then there is $q,p \le_{pr} q$ such that the set \newline
$\{\eta \in q:\text{for some } \beta \text{ we have } q^{[\eta]} \Vdash
``\underset\tilde {}\to \alpha = \beta"\}$ contains a front.
\endroster
\medskip

This fact follows easily from \scite{2.14} (let $H:p \rightarrow \{0,1\}$
(i.e., Dom$(H) = T^p$) be defined by $H(\eta) = 1$ iff $(\exists q)[p
^{[\eta]} \le_{pr} q \and q$ decides $\underset\tilde {}\to \alpha]$, define
$H(\eta) =$ \nl
$\text{lim}_{n < \omega}(H(\eta \restriction n))$ for
$\eta \in \text{ lim}(p)$, and find $q$ such that $H$ is constant on
lim$(q)$).  Let $Y = \{\eta \in T^q:H(\eta) = 1 \and (\forall \nu)(\nu
\triangleleft \eta \rightarrow H(\eta) = 0]\}$, so $Y$ is a front of $q$.
For $\eta \in Y$ let $q_\eta$ be such that $p^{[\eta]} \le_{\text{pr}} 
q_\eta$ and $q_\eta$ forces a value 
to $\underset\tilde {}\to \alpha$ let $r$ be such that
$T^r = \dbcu_{\eta \in Y} T^{q_\eta}$.  So clearly $r$ is as required, $Y$
such a front.

Now let $\langle N_\eta:\eta \in (T,\bold I) \rangle$ be a strictly 
$\Bbb I$-suitable tree of models for $\chi,x$ satisfying 
$\{p,I,\Bbb I\} \in N_{\langle \rangle}$ where $p \in \Bbb Q \cap N_{<>}$ is a
condition.  We can now find a condition $q,p \le_{pr} q$, a family $\langle
p_\eta:\eta \in p \rangle$ of conditions and a function $f:q \rightarrow T$
satisfying the following:
\medskip
\roster
\item  If $\eta \triangleleft \nu$ in $q$, then $f(\eta) \triangleleft
f(\nu)$.
\sn
\item  For all $\eta$ in $q$, Suc$_T(f(\eta)) \ne 0 \text{ mod } I$ and
$\bold I_\eta = I$.
\sn
\item  For all $\eta$ in $q$, Suc$_q(\eta) \subseteq \text{ Suc}_T(f(\eta))$.
\sn
\item  For all $\eta$ in $q,p_\eta \in N_{f(\eta)},\text{tr}(p_\eta) =
\eta,p^{[\eta]} \le_{pr} p_\eta$.
\sn
\item  For all $\eta$ in $q,p_\eta \le_{pr} q^{[\eta]}$.
\sn
\item  For all $\eta$ in $q$, all names $\underset\tilde {}\to \alpha$ in
$N_{f(\eta)}$, the set \newline
$\{\nu \in q:p_\nu \text{ decides } \alpha\}$ contains a front of
$p^{[\eta]}$.
\endroster
\medskip

We can do this as follows: by induction on $\eta \in p$ we define
$f(\eta),p_\eta$ and Suc$_q(\eta)$.  We can find $f(\eta)$ satisfying (2)
+ (3) because $T$ is $\Bbb I$-suitable and $I \in \Bbb I$ and $\Bbb I$ is
restriction closed.  We choose $p_\eta$ using a bookkeeping
argument to take care of a case of (6), using $(*)$.  Then we 
choose Suc$_q(\eta)$ such that (3) are satisfied.  

Lastly, let $q = \{\nu$: for some $\eta,p_\eta$ is well defined and $\nu
\trianglelefteq \eta\}$.  Clearly $p \le_{\text{pr}} q \in \Bbb Q$.

Now let $G$ be $\Bbb Q$-generic, $q \in G$.  Now $G$ defines a generic branch
$\eta$ through $q$.  This induces a branch $\nu$ through $T:\nu =
\dsize \bigcup_{n < \omega} f(\eta \restriction n)$.  Let
$\underset\tilde {}\to \alpha \in N_{\nu \restriction k}$, then there is
$\ell$ such that $p_{\eta \restriction \ell} \Vdash ``
\underset\tilde {}\to \alpha = \beta$ and $\beta \in N_{f(\eta \restriction
\ell)} \subseteq N_\nu"$. \nl
2) It is enough to prove the second version for any condition $p$, let
$\bold I^p_\eta$ be $I$ ``mapped" to Suc$_p(\eta)$. \nl
For any condition $p$, for each $\eta \in p$ such that Suc$_p(\eta) \ne
\emptyset$ mod $I$ let $h_\eta:\text{Suc}_p(\eta) \rightarrow \lambda'$ be
such that $(\forall \alpha < \lambda')[\{\nu \in \text{ Suc}_p(\eta):h_\eta
(\nu) < \alpha\} \in \bold I^p_\eta]$.  Now letting 
$\underset\tilde {}\to \eta \in {}^\omega \text{Ord},
\underset\tilde {}\to \eta \restriction \ell =
{}^n \text{Ord} \cap r$ for any $r \in {\underset\tilde {}\to G_{\Bbb Q}}$ large
enough, we let $\underset\tilde {}\to A = \{
h_{{\underset\tilde {}\to \eta} \restriction \ell}(\underset\tilde {}\to \eta
\restriction (\ell + 1)):\ell < \omega\}$. \nl
So easily $\Vdash_{\Bbb Q} ``\underset\tilde {}\to A \subseteq \lambda'$ is
unbounded. \nl
3) Trivial. \nl
4) Without loss of generality $\Bbb I$ is $\kappa$-complete (as we can
decrease it).  So by \scite{stc.3} it suffices to prove
UP$^1_{\text{com}}(\Bbb I,\bold W)$.  So assume $\langle N_\eta:\eta \in
(T,\bold I) \rangle,h_{\eta,\nu}$ (for $\eta,\nu \in T \cup \text{ lim}(T),
\ell g(\eta) = \ell g(\nu))$ are as in Definition \scite{stc.1}, $(*)_1$ and
$\eta^* \in \text{ lim}(T),G_{\eta^*}$ as in the assumption of $(*)_2$ there.
Now choose inductively on $n < \omega,p_n$ and $k_n$ such that:
$p_0 = p,k_0 = 0,p_n \in G_{\eta^*},p_n \in N_{\eta^* \restriction
k_{n+1}}$, and $p_n < p_{n+1},k_n < k_{n+1},\eta_n$ is the trunk of
$p_n,\eta_n \triangleleft \eta_{n+1}$, Suc$_{p_n}(\eta_n) \ne \emptyset
\text{ mod } I$ (as in proof of part (1)) and Suc$_T(\eta \restriction
k_{n+1}) = \text{ Suc}_{p_n}(\eta_n)$ and 
$p^{[\eta_n \char 94 \langle \eta^*(k_{n+1})]}_n \le_{pr} p_{n+1}$ and if
$\underset\tilde {}\to \tau \in N_{\eta^*}$ is a Nm$(I)$-name of a countable
ordinal then for some $n,p_n$ decides its value. \nl
5) The winning strategy of the completeness player is, given $q_n$, let
$\nu = tr(T)$ and let $n$ be minimal such that $q_n \in N_{\eta^* \restriction
n}$ and $I \restriction \text{ Suc}_{q_n}(\nu) = I_{\eta^* \restriction n}$
and let $p_n = (q_n)^{[\nu \restriction \eta^*(n)]}$. 
\hfill$\square_{\scite{8.2}}$\margincite{8.2}
\enddemo
\bigskip

\definition{\stag{8.3} Definition}  1) We can consider an $\Bbb I$-suitable 
tree of models \newline
$\bar N = \langle N_\eta:\eta \in (T^*,{\Bbb I}^*) \rangle$, and let

$$
\align
a) \qquad \Bbb Q_{\bar N} = \biggl\{ T \subseteq T^*:\,&T \text{ non-empty, closed
under initial segments} \\
  &\,\text{such that } \langle N_\eta:\eta \in (T,\bold I \restriction T)
\rangle \text{ is an} \\
  &\,{\Bbb I} \text{-suitable tree of models} \biggr\}
\endalign
$$

\noindent
ordered by inverse inclusion. \newline
2) We can consider for any tagged tree $(T^*,\bold I^*)$

$$
\align
\Bbb Q^0_{(T^*,\bold I^*)} = \biggl\{T \subseteq T^*:\,&T \text{ non-empty, closed
under initial segments} \\
  &\,\text{such that for some } n = n(T), \\
  &\,(i) \qquad \ell \le n \Rightarrow |T \cap {}^n\text{Ord}| = 1 \\
  &\,(ii) \qquad \text{if } \eta \in T \and \ell g(\eta) 
\ge n \and \text{ Suc}_{T^*}
(\eta) \ne \emptyset \text{ mod } \bold I_\eta \\
  &\qquad \qquad \,\,\text{ then Suc}_T(\eta) \ne 
\emptyset \text{ mod } \bold I_\eta \biggr\}
\endalign
$$
\mn
is ordered by inverse inclusion.

$$
\align
\Bbb Q^1_{(T,{\bold I}^*)} = \biggl\{ (T,\bold I):&(T^*,\bold I^*) \le
(T,\bold I), \text{ and for every } \eta \in \text{ lim}(T) \\
  &\text{ we have } (\forall k)(\exists^\infty n)[\eta \restriction n
\text{ is a splitting point of } (T,\bold I) \\
 &\text{ and } \bold I_{\eta \restriction k} \le_{\text{RK}}
\bold I_{\eta \restriction n} \biggr\}
\endalign
$$

\noindent
ordered by inverse inclusion.  [Saharon" [$\Bbb Q^0 \ne \Bbb Q^2$?]

$$
\Bbb Q^2_{(T,{\bold I}^*)} = \biggl\{ (T,\bold I):(T^*,\bold I^*)^{[\eta]} \le^*
(T,\bold I), \text{ for some } \eta \in T^* \biggr\}
$$

\noindent
ordered by inverse inclusion. 
\enddefinition
\bigskip

\proclaim{\stag{8.3A} Claim}  For the forcing notions defined in Definition
\scite{8.3} for $\Bbb I$ being $\aleph_2$-complete, of course, we have: if
$P \in \{\Bbb Q_{\bar N},\Bbb Q^0_{(T^*,\bold I^*)},\Bbb Q^1_{(T,\bold I^*)}\}$, \ub{then}
\medskip
\roster
\item "{$(a)$}"  $P$ satisfies ${\text{\rm UP\/}}^1(\Bbb I)$
\sn
\item "{$(b)$}"  if $I \in \Bbb I \Rightarrow |{\text{\rm Dom\/}}
(I)| < \lambda =
{ \text{\rm cf\/}}(\lambda)$, \ub{then} $|\Bbb P| \le 2^{< \lambda}$ and even
$\le 2^\mu$ for some $\mu < \lambda$
\sn
\item "{$(c)$}"  if for $\lambda$ regular \newline
$(\forall I \in \Bbb I)(\forall A \in (I)^+[I \restriction A$ is not
$\lambda$-indecomposable] \newline
\underbar{then} $\Vdash_{\Bbb P} ``{\text{\rm cf\/}}
(\lambda) = \aleph_0"$
\sn
\item "{$(d)$}"  if $\Bbb P = \Bbb Q^0_{(T^*,\bold I^*)}$ then 
$(\forall \eta \in \text{ lim}(T^*)) \, 
\exists^\infty n \dsize \bigwedge_{m \ge n} \forall A \in
(\bold I^*_{\eta \restriction n})^+[\bold I_\eta \restriction A$ is not
$\lambda$-indecomposable] 
\underbar{then} $\Vdash_{\Bbb P} ``{\text{\rm cf\/}}
(\lambda) = \aleph_0"$
\sn
\item "{$(e)$}"  if $\Bbb I$ is $(2^{\aleph_0})^+$-complete then forcing with
$\Bbb P$ add no new reals, moreover it satisfies 
${\text{\rm UP\/}}^{4,+}_{\text{stc}}(\Bbb I)$
and if $p \in \{{\Bbb Q}_{\bar N},\Bbb Q^0_{(T^*,\bold I^*)}\}$ \ub{then} it 
satisfies ${\text{\rm UP\/}}^4_{\text{com}}(\Bbb I)$.
\endroster
\endproclaim
\bigskip

\demo{Proof}  Left to the reader. \hfill$\square_{\scite{8.3A}}$\margincite{8.3A}
\enddemo
\bigskip

\definition{\stag{8.4} Definition}  Let $\lambda = \text{ cf}(\lambda) >
\aleph_1,S \subseteq \{\delta < \lambda:\text{cf}(\delta) = \aleph_0\}$ be
stationary and

$$
\align
\text{club}_S(S) = \biggl\{ h:\,&\text{for some non-limit } \alpha 
< \omega_1,\\
  &\,h \text{ is an increasing function from } \alpha \text{ to } S \biggl\}
\endalign
$$

\noindent
ordered by inverse inclusion, $\le_{\text{pr}} = \le,\le_{vpr}$ is
equality.
\enddefinition
\bigskip

\proclaim{\stag{8.5} Claim}  For $\lambda,S$ as in Definition \scite{8.4}
we have (for any $I,I$ is an $\aleph_2$-complete ideal on $\lambda$ extending
$J^{bd}_\lambda$). \newline
1) Club$(S)$ satisfies ${\text{\rm UP\/}}^1(\{I\})$ 
of cardinality $\le \lambda^{\aleph_0}$.
\newline
2) If $I$ is $(2^{\aleph_0})^+$-complete and $I \in \Bbb I$, \ub{then} 
Club$(S)$ satisfies ${\text{\rm UP\/}}^4_{\text{com}}(\Bbb I)$, hence 
${\text{\rm UP\/}}^{4,+}_{\text{stc}}(\Bbb I)$.
\endproclaim
\bigskip

\demo{Proof}  Left to the reader \ub{or} follows from \cite[XI,4.6]{Sh:f} by
\scite{8.7} below. \hfill$\square_{\scite{8.5}}$\margincite{8.5}
\enddemo
\bigskip

\proclaim{\stag{8.5A} Lemma}  Let $\bar W = \langle W_i:i < \omega_1 \rangle$ 
be a sequence of stationary subsets of $\{\alpha < \lambda:
{\text{\rm cf\/}}(\alpha)
= \omega\}$ where $\lambda = { \text{\rm cf\/}}(\lambda) > 
\aleph_0$ and let the
forcing notion $\Bbb P[\bar W]$ be defined by

$$
\align
\Bbb P[\bar W] =: \biggl\{ f:\,&f \text{ is an increasing and continuous function
from} \\
  &\,\alpha + 1 \text{ into } W_0 \text{ for some } \alpha < \omega_1, \\
  &\,\text{ such that for every } i \le \alpha \text{ we have } f(i) \in
W_i \biggr\}
\endalign
$$
\mn
(ordered by inclusion).  If $I \supseteq J^{bd}_\lambda$ be
$\aleph_2$-complete, \ub{then} $\Bbb P[\bar W]$ satisfies 
${\text{\rm UP\/}}^{4,+}(\Bbb I)$ for
any $\Bbb I$ such that $I \in \Bbb I$ and if $\Bbb I$ is
$(2^{\aleph_0})^+$-complete it also satisfies 
${\text{\rm UP\/}}^4_{\text{com}}(\Bbb I)$ 
hence ${\text{\rm UP\/}}^{4,+}_{\text{stc}}(\Bbb I)$.
\endproclaim
\bigskip

\demo{Proof}  Left to the reader or follows from \cite[Ch.XI,4.6A]{Sh:f} by
\scite{8.7} above.  \hfill$\square_{\scite{8.5A}}$\margincite{8.5A}
\enddemo
\bn
Concerning $\bold W$-completeness (see \cite[Ch.V]{Sh:f}):
\proclaim{\stag{8.6} Claim}  Assume $\bold W \subseteq \omega_1$ is 
stationary and $\Bbb Q$ is $\bold W$-complete forcing notion (i.e., 
if $\chi$ is large enough, $\Bbb Q \in N \prec ({\Cal H}(\chi),\in,
<^*_\chi),N$ countable, 
$p_n \in \Bbb Q \cap N$ 
is $\le_{\Bbb Q}$-increasing and $(\forall {\Cal I} \in N)
({\Cal I} \subseteq \Bbb Q$ is dense $\rightarrow \dsize \bigvee_n p_n \in
{\Cal I})$ \ub{then} $\{p_n:n < \omega\}$ has an upper bound in $\Bbb Q$).

\underbar{Then} $\Bbb Q$ satisfies 
${\text{\rm UP\/}}^{4,+}_{\text{com}}(\Bbb I,\bold W)$ (i.e.,
for any $\Bbb I$).
\endproclaim  
\bigskip

\demo{Proof}  Trivial (see Definition \sciteu{7.1}), for the $\omega$-branch
$\eta^*$ of $T$ there is $q,p \le q,q$ is an upper bound of $\Bbb Q^*$ hence is
$(\dsize \bigcup_{\ell < \omega} N_{\eta^* \restriction \ell},\Bbb Q)$-generic.
\hfill$\square_{\scite{8.6}}$\margincite{8.6}
\enddemo
\bn
Comparing to \cite{Sh:f} we have
\proclaim{\stag{8.7} Claim}  1) If $\Bbb Q$ satisfies the $\Bbb I$-condition
of \cite[Ch.XI,Def.2.6,2.7]{Sh:f} $\Bbb I$ is $(2^{\aleph_0})^+$-complete,
\ub{then} $\Bbb Q$ satisfies ${\text{\rm UP\/}}^4_{\text{com}}(\Bbb I)$. \nl
2) If $\Bbb Q$ satisfies condition 
${\text{\rm UP\/}}(\Bbb I,\bold W)$ of \cite[Ch.XV, Definition 2.7A]{Sh:f}, 
\ub{then} it satisfies ${\text{\rm UP\/}}^4
(\Bbb I,\bold W)$ here. \nl
3) If $\Bbb Q$ is a proper or just semi-proper forcing notion,
\ub{then} $\Bbb Q$
satisfies ${\text{\rm UP\/}}^6(\Bbb I)$ and all the UP$^\ell(\Bbb I)$.
\endproclaim
\bigskip

\demo{Proof}  Part (2) holds by \cite[XV,2.11]{Sh:f}. \nl
Part (1) follows by part (2) and \cite[Ch.XV,2.11]{Sh:f}. \nl
Part (3) is immediate by the definition (can use any fix branch).
\hfill$\square_{\scite{8.7}}$\margincite{8.7}
\enddemo
\newpage

\head {\S9 Reflection in $[\omega_2]^{\aleph_0}$} \endhead  \resetall \sectno=9
 % \resetall 
\bn
As an exercise 
we answer a question (and variants) of Jech \footnote{Done 10/97}.
\proclaim{\stag{11.1} Theorem}  Assume
\mr
\item "{$(A)$}"  $\kappa$ is large enough (supercompact or just Woodin).
\ermn
\ub{Then} \nl
1) there is a $\kappa$-c.c. semi-proper forcing notion $\Bbb P$ of cardinality
$\kappa$ such that in $\bold V^{\Bbb P}$
\mr
\item "{$(\alpha)$}"  $\aleph_2 = \aleph^V_1,\aleph_2 = \kappa$ and cardinal
$\ge \kappa$ are the same as in $\bold V$ and $2^{\aleph_0} = \aleph_2$
\sn
\item "{$(\beta)$}"  every stationary subset of $[\omega_2]^{\le \aleph_0}$
reflect (in a set of cardinality $\aleph_1$)
\sn
\item "{$(\gamma)$}"  $D_{\omega_1}$ is $\aleph_2$-saturated
\sn
\item "{$(\delta)$}"  there is a projectively stationary $S \subseteq
[\omega_2]^{\le \aleph_0}$, see below 
such that there is no sequence $\langle a_i:i < \omega_1 \rangle$ increasing
continuous, $a_i \in S,a_i \ne a_{i+1}$.
\ermn
2) Assume in addition
\mr
\item "{$\boxtimes$}"   $\{\lambda:\lambda \text{ measurable}\}$ is
not in the weakly compact ideal of $\kappa$. 
\ermn
We can add to (1) the statement (on $\bold V^{\Bbb P}$)
\mr
\item "{$(*)$}"  every stationary $S \subseteq S^2_0 = \{\delta < \omega_2:
\text{cf}(\delta) = \aleph_2\}$ contains a closed copy of $\omega_1$.
\ermn
3) We may strengthen clause $(\delta)$ of (1) to ``$S$ is 
$S^2_1$-projectively stationary".
\endproclaim 
\bigskip

\definition{\stag{11.1A} Definition}  1) We call 
$S \subseteq [\omega_2]^{\aleph_0}$ projectively stationary if: \nl
for every club $E$ of $[\omega_2]^{\aleph_0}$ and stationary
co-stationary $W \subseteq \omega_1$ we can find a sequence
$\langle a_i:i < \omega_1 \rangle$ increasing continuous, 
$a_i \in [\omega_2]^{\aleph_0},a_i \in E$
and $\{i \in W:a_i \notin S\}$ is not a stationary subset of $\omega_1$. \nl
2) We say $S \subseteq [\omega_2]^{\aleph_0}$ is $S^2_1$-projectively 
stationary for $W \subseteq \omega_1$ stationary co-stationary, for 
stationarily many $\delta \in S^2_1$ if we let 
$\delta = \dbcu_{i < \omega_1} a^\delta_i,a^\delta_i$
countable increasing continuous we have 
$\{i \in W:a^\delta_i \notin S\}$ non-stationary.
\enddefinition
\bigskip

\demo{Proof}  Like \cite[Ch.XVI,2.4]{Sh:f}'s proof.
\mn
1), 2)  We 
define a RCS iteration $\langle \Bbb P_i,{\underset\tilde {}\to {\Bbb Q}_j}:i \le
\kappa,j < \kappa \rangle$ such that:

$$
|\Bbb Q_j| \le (2^{\aleph_2})^{\bold V^{{\Bbb P}_j}}
$$
\mn
in $\bold V^{{\Bbb P}_j},\Bbb Q_j$ is the disjoint union of the following (so the choice by
which of them we force is generic), \ub{but} if $j$ is non-limit only (a)
is allowed
\mr
\item "{$(a)$}"  $\Bbb Q^0_j = \text{ Levy}(\aleph_1,2^{\aleph_1}) *$ Cohen
\sn
\item "{$(b)$}"  $\Bbb Q^1_j =$ sealing all semi-proper maximal antichanges of
$D_{\omega_1}$ provided that strong Chang conjecture holds in
$\bold V^{{\Bbb P}_j}$ (true if $j$ is measurable $> \aleph_0$)
\sn
\item "{$(c)$}"  if we like to have $(*)$ and in $\bold V^{{\Bbb P}_j}$, strong
Chang conjecture holds then allow: $\Bbb Q^2_{j,S}$ where 
$S \subseteq S^2_0$ stationary not containing a closed copy of $\omega_1$
and $\Bbb Q^2_{j,S}$ semi-proper where $\Bbb Q^2_{j,S}$ shoot an $\omega_1$-increasingly
continuous chain i.e.
$$
\align
\Bbb Q^2_{j,S} = \{(S,f):&S \subseteq S^2_0 \text{ stationary, Dom}(f)
\text{ is a successor} \\
  &\text{countable ordinal, } f \text{ is increasingly continuous into } S\}
\endalign
$$

$$
(S,f_1) \le (S,f_2) \Leftrightarrow S_1 = S_2 \wedge f_1 \subseteq f_2.
$$
\ermn
So $\Vdash_{{\Bbb P}_j} ``{\underset\tilde {}\to {\Bbb Q}_j}$ is semi-proper of cardinality
$\le (2^{\aleph_1})^{V^{P_j}}"$. \nl
So by \cite[Ch.XVI,\S2,2.4,2.5]{Sh:f}
\mr
\item "{$\bigotimes_1$}"  for $i < j \le \kappa,\Bbb P_j/\Bbb P_i$ is semi-proper,
so $\aleph_1$ is not collapsed
\sn
\item "{$\bigotimes_2$}"  $\Bbb P_\kappa$ collapses every 
$\theta \in (\aleph_1,\kappa)$, satisfies the $\kappa$-c.c. and has
cardinality $\kappa$
\sn
\item "{$\bigotimes_3$}"  $\Vdash_{{\Bbb P}_\kappa} 
``{\underset\tilde {}\to D_{\omega_1}}$ is $\aleph_2$-saturated".
\ermn
By preliminary forcing, \wilog \, there is 
$S_0 = \{\delta < \kappa:\delta \text{ strong limit, cf}(\delta) =
\aleph_0\}$, stationary in $\kappa$, reflecting only in inaccessibles.
Let $S_1 = \{\lambda < \kappa:\lambda
\text{ is measurable}\}$ so we know $S_1$ is stationary.  If we are proving
the version with $(*)$, note that $\lambda \in S_1 \Rightarrow$ in
$\bold V^{{\Bbb P}_\lambda}$, the strong Chang conjecture holds (\cite[Ch.XIII,1.9]{Sh:f})
hence $\Bbb Q^2_{\lambda,S}$ is semi-proper for every stationary $S \subseteq
\{\delta < \lambda:\bold V^{{\Bbb P}_\lambda} \models \text{ cf}(\delta) = \aleph_0\}$.
Also if $\boxtimes$ holds then

$$
\align
\Vdash_{{\Bbb P}_\kappa} ``&\underset\tilde {}\to S \subseteq \{\delta < \lambda:
\text{in } \bold V^{{\Bbb P}_\kappa},\text{cf}(\delta) = \aleph_0\} \text{ is
stationary } \Rightarrow \\
  &S'_1 = \{\lambda \in S_1:\underset\tilde {}\to S \restriction \lambda
\text{ is a } \Bbb P_\lambda \text{-name of a stationary} \\  
  &\text{subset of } \{\delta < \lambda:\text{cf}(\delta) = \aleph_0
\text{ in } \bold V^{{\Bbb P}_\lambda}\}\} \text{ is stationary}".
\endalign
$$
\mn
So
\mr
\item "{$\bigotimes_4$}"  if we are proving $(*)$, then in $\bold
V^{{\Bbb P}_\kappa}$ 
every stationary $S \subseteq \{\delta < \kappa:\text{cf}(\delta) = 
\aleph_0\}$ contains a close copy of $\omega_1$.
\ermn
Now we have to deal with the ``projectively stationary".  We can find
function $h$, Dom$(h) = S_0,h(\delta)$ is a $\Bbb P_\delta$-name of a stationary
co-stationary subset of $\omega_1$ (even $\Bbb P_\alpha$-name for some
$\alpha < \delta$) such that: every such name appears
stationarily often.  Let $\langle {\underset\tilde {}\to a^\delta_i}:i <
\omega_1 \rangle$ be a $\Bbb P_\delta$-name such that

$$
\align
\Vdash_{{\Bbb P}_\delta} ``&{\underset\tilde {}\to a^\delta_i} 
\subseteq \delta \text{ is countable unbounded
in } \delta \text{ increasingly continuous in } i \\
  &\text{and } \delta = \dbcu_{i < \omega_1}
{\underset\tilde {}\to a^\delta_i}".
\endalign
$$
\mn
Let

$$
{\underset\tilde {}\to {\Cal W}_\delta} = \{a^\delta_i:i \in h(\delta)\}
$$

$$
{\underset\tilde {}\to {\Cal W}_{< \alpha}} = \cup\{{\Cal W}_\delta:
\delta \in \alpha \cap S_0\}
$$

$$
\underset\tilde {}\to {\Cal W} = {\Cal W}_{< \kappa}.
$$
\mn
So 
\mr
\item "{$\bigotimes_5$}"  $\underset\tilde {}\to {\Cal W}$ is a
$\Bbb P_\kappa$-name of a subset of $[\kappa]^{\aleph_0}$ and $S_0$ is
stationary in $\bold V^{{\Bbb P}_\kappa}$ (as $\Bbb P_\kappa \models \kappa$-c.c.)
\sn
\item "{$\bigotimes_6$}"  $\Vdash_{{\Bbb P}_\kappa} ``\underset\tilde {}\to {\Cal W}$
is stationary.
\ermn
[why?  If 
$\Vdash_{{\Bbb P}_\kappa} ``\underset\tilde {}\to M$ is a model with countable
vocabulary and universe $\kappa$" \ub{then} $\underset\tilde {}\to E = \{
\delta < \lambda:\underset\tilde {}\to M \restriction \delta \text{ is a }
\Bbb P_\delta$-name and is an elementary submodel of $M\}$ is a $\Bbb P_\kappa$-name
of a club of $\kappa$ hence contains a club $E^*$ of $\kappa$ from
$\bold V$.  
So for a club $i < \omega_1,M \restriction a^\delta_i$ is an 
elementary submodel of $M$.  But for stationarily many $i < \omega_1,
a^\delta_i \in {\Cal W}_\delta \subseteq {\Cal W}$, so really ${\Cal W}$ is
stationary. If
$\underset\tilde {}\to W$ is a $\Bbb P_\kappa$-name of a stationary co-stationary
subset of $\omega_1$ then for some, even for stationarily many 
$\delta \in E^* \cap S_0$ we have $h(\delta) =
\underset\tilde {}\to W$ and so easily
\mr
\item "{$\bigotimes_7$}"  $\Vdash_{{\Bbb P}_\kappa} 
``\underset\tilde {}\to {\Cal W}$ is projectively stationary".
\ermn
Lastly, why would ${\Cal W}$ contain no increasing $\omega_1$-chains?  Assume
$p^* \Vdash ``\langle {\underset\tilde {}\to a_i}:i < \omega_1 \rangle$ is
increasing continously and ${\underset\tilde {}\to a_i} \in W"$.  
So \wilog \, for some $\delta^*$ either
\mr
\item "{$(\alpha)$}"  $p^* \Vdash ``\sup(a_i)$ is strictly increasing with
limit $\delta^*"$ or
\sn
\item "{$(\beta)$}"  $p^* \Vdash ``\sup {\underset\tilde {}\to a_i}$ is
constantly $\delta^*$ for $i \ge i^*,i^* < \omega_1$ so \nl
without loss of generality $i^*=0"$.
\ermn
\ub{Case A}:  The possibility $(\beta)$ holds. \nl
Necessarily $\delta^* \in S_0,p^* \Vdash_{\Bbb P} ``\{{\underset\tilde {}\to a_i}:
i < \omega_1\} \subseteq {\Cal W}_{\delta^*}"$ and as $\Vdash_{P_{\delta^*}}
``h(\delta^*)$ is co-stationary subset of $\omega_1"$ and $\Bbb P_\kappa/
\Bbb P_{\delta^*}$ is semi-proper hence preserve stationarity of subsets of
$\omega_1$ we are done. 
\mn
\ub{Case B}:  Possibility $(\alpha)$ holds and $\delta^*$ not 
strongly inaccessible.  So $S_0 \cap \delta^*$ 
is not stationary in $\delta^*$ hence ${\underset\tilde {}\to {\Cal W}} \cap
[\delta^*]^{\aleph_0} = \dbcu_{\delta \in \delta^* \cap S_0}{\Cal W}_\delta$
is not even stationary.
\mn
\ub{Case C}:  Possibility $(\alpha)$ holds and not case B, in 
$V^{P_{\delta^*}}$ strong Chang conjecture fails.

Then ${\underset\tilde {}\to {\Bbb Q}_\delta}$ is Levy$(\aleph_1,2^{\aleph_1}) *$
Cohen (as in clauses (b) and (c) in $\bold V^{{\Bbb P}_\delta}$ strong Chang conjecture 
holds), so as
clearly in $\bold V^{{\Bbb P}_\delta},2^{\aleph_0} = \aleph_2$ (by the Cohen in (a), i.e.
$\Bbb Q^0_j$), then in $\bold V^{{\Bbb P}_{\delta^*}}$ for every club $E'$ of $[\delta^*]
^{\aleph_0}$, we can find some $\delta < \delta^*$ and $2^{\aleph_0}$
members of $E' \cap [\delta]^{\aleph_0} \backslash
{\underset\tilde {}\to W_{< \delta^*}}$.  So in $\bold V^{{\Bbb P}_{\delta^*}},
[\delta^*]^{\aleph_0} \backslash 
{\underset\tilde {}\to {\Cal W}_{< \delta^*}}$ is stationary
and ${\underset\tilde {}\to {\Bbb Q}_\delta}$ is proper so this
holds in $\bold V^{{\Bbb P}_{\delta^*+1}}$.  But 
$\Bbb P_\kappa/\Bbb P_{\delta^*+1}$ preserves
stationarity of subsets of $\omega_1$ hence in $\bold V^{{\Bbb P}_\kappa}
[\delta^*]^{\aleph_0} < {\Cal W} < \delta$ is stationary, so we are done.
\mn
\ub{Case D}:  Possibility $(\alpha)$ holds, not case B and in 
$\bold V^{{\Bbb P}_{\delta^*}}$ strong Chang conjecture holds.
\mn
Just note: in $\bold V^{{\Bbb P}_{\delta^*}}$, let $p \in \Bbb Q_{\delta^*}$, let 
$\chi$ large enough $N \prec ({\Cal H}(\chi),\in)$ is countable
to which $\bar{\Bbb Q},\delta^*,G_{{\Bbb P}_{\delta^*}},p,\langle {\Cal W}_\delta:
\delta \in S_0 \cap \delta^* \rangle$ belong, \ub{then} we can find 
(see \cite[Ch.XIII]{Sh:f}) $T \subseteq {}^{\omega >} \omega_2$ closed 
under initial segments $T \cap N = \emptyset$, 
satisfying $(\forall \eta \in T)(\exists^{\aleph_2}\alpha)
(\eta \char 94 \langle \alpha \rangle \in T)$ and $\langle N_\eta:\eta \in T
\rangle$ such that
\mr
\widestnumber\item{$(iii)$}
\item "{$(i)$}"  $N_{<>} = N$
\sn
\item "{$(ii)$}"  $N_\eta \prec ({\Cal H}(\chi),\in)$ is countable
\sn
\item "{$(iii)$}"  $\eta \in N_\eta,N_\eta \cap \omega_1 = N \cap \omega_1$
\sn
\item "{$(iv)$}"  $\nu \triangleleft \eta \Rightarrow N_\eta \subseteq N_\nu$
\sn
\item "{$(v)$}"  if ${\Cal I} = \{A_\zeta:\zeta < \zeta^*\}$ is a maximal
antichain of ${\Cal D}_{\omega_1}$ which is semi-proper 
and ${\Cal I} \in N_\eta$ \ub{then} for
some $k < \omega,\eta \triangleleft \nu \in T \and \ell g(\nu) \ge k
\Rightarrow N \cap \omega_1 \in \dbcu_{\zeta \in N_\nu} A_\zeta$.
\ermn
Let

$$
\align
E = \{\delta < \omega_2:&\text{ if } \eta \in {}^{\omega >}\delta
\text{ then } N_\eta \cap \omega_2 \\  
  &\text{is a bounded subset of } \delta\}.
\endalign
$$
\mn
Now if $p \in \Bbb Q^0_\delta$ we do as in Case C.  If $p \in \Bbb Q^1_{\delta^*}$, 
choose $\delta \in E$, cf$(\delta) = \aleph_0$, and
such that for every $\eta \in T \cap {}^{\omega >} \delta,\delta =
\text{ otp}\{\beta < \delta:\eta \char 94 \langle \beta \rangle \in T\}$ and
$\eta \char 94 \langle \alpha \rangle \in T \and \alpha < \delta \Rightarrow 
\text{ sup}(N_\alpha \cap \omega_2) < \delta\}$.  Now 
we can by cardinality  considerations
$(2^{\aleph_0} > \aleph_1)$ find $\eta \in \lim(T) \cap {}^\omega \delta$
such that letting $M = \dbcu_{\ell < \omega} N_{\eta \restriction \ell},
M \cap \omega_2 = M \cap \delta \notin {\Cal W}_{< \delta^*}$.  
So there is $q \in \Bbb Q_\delta$ which is $(M,\Bbb Q_\delta)$-generic, 
$p \le q$ (by the definition of $\Bbb Q^1_{\delta^*}$).  Now $q$  forces
${\underset\tilde {}\to a_{M \cap \omega_1}} = a_{M \cap \omega_1}$ to be
$M \cap \omega_2$ which is not in 
${\underset\tilde {}\to {\Cal W}_{< \delta^*}}$.
\sn
Lastly if $q \in \Bbb Q^2_{j,S}$ (in $\bold V^{{\Bbb P}_{\delta^*}}$) as $S$
does not reflect we can find $\delta \in E$ as above, $\delta \in S$, cf$(\delta) =
\aleph_0$ and choose $\eta,M$ as above. \nl
3) We may like to adapt the proof above.  \nl
We omit the choice of $\langle {\underset\tilde {}\to a^\delta_i}:i <
\omega_1 \rangle$, \ub{but} in ${\underset\tilde {}\to {\Bbb Q}_j}$ if $j \in S_0$
we also choose a $\Bbb P_j$-name of a countable unbounded subset of $\delta,
{\underset\tilde {}\to a_\delta}$ and let 
${\underset\tilde {}\to {\Cal W}_\delta} = \{{\underset\tilde {}\to a_\delta}
\}$ so $\Bbb Q_j$ is replaced by 
$\Bbb Q_j \times \{\underset\tilde {}\to a:\underset\tilde {}\to a$
a name as above$\}$.  Now $h_0$ has domain $S^* = \{\delta:\delta$
strongly inaccessible, in $\bold V^{{\Bbb P}_\delta}$, strong Chang conjecture holds$\},
h_0(\delta)$ a $\Bbb P_\delta$-name of a stationary co-stationary subset of
$\omega_1$ and we add to clauses (a), (b), (c) above also 
\smallskip

(d)   define in $\bold V^{{\Bbb P}_\delta}$:

$$
\align
\Bbb Q^3_\delta = \biggl\{ \langle M_i:i \le j \rangle:&\text{the ordinal }
j \text{ is countable and} \\
  &M_i \prec ({\Cal H}
((2^\delta)^+),\in) \text{ is countable increasing} \\
  &\text{continuous, and: \ub{if} } M_i \cap \omega_1 \in h_0(\delta)
\text{ \ub{then}} \\
  &M_i \in {\Cal W}_{< \delta} \text{ and if } 
M_i \cap \omega_1 \notin h(\delta) \text{ then } 
M_i \notin {\Cal W}_{< \delta} \biggr\}.
\endalign
$$
\mn
Now again we use $\langle N_\eta:\eta \in T \rangle$ and choosing $M$ it is
enough to show that
\mr
\item "{$\boxtimes_1$}"  for some $\eta \in \lim T,\dbcu_{i < \omega}
N_{\eta \restriction \ell} \cap \omega_2 \in {\Cal W}_{< \delta}$
\sn
\item "{$\boxtimes_2$}"  some $\eta \in \lim T,\dbcu_{\ell < \omega}
N_{\eta \restriction \ell} \cap \omega_2 \notin {\Cal W}_{< \delta}$.
\ermn
Now $\boxtimes_2$ is as before, $\boxtimes_1$ O.K. by the way
${\underset\tilde {}\to {\Cal W}_{< \delta}}$ is defined.
\hfill$\boxtimes_{\scite{11.1}}$
\enddemo
\newpage

\head {\S10 Mixing finitary norms and ideals} \endhead  \resetall \sectno=10
 % \resetall 
\bn
We may consider replacing families of ideals by families of creatures
see \cite{RoSh:470} on creatures: \nl
We hope it will gain something
\definition{\stag{0.A} Definition}:  1)  A $\lambda$-creature ${\frak c}$ 
consists of $(D^{\frak c},\le,\text{val}^{\frak c},\text{nor}^{\frak c},
\lambda^{\frak c})$, where: 
\mr
\item "{{}}"  $\lambda^{\frak c} = \lambda$
\sn
\item "{{}}"  $D^{\frak c}$ the domain,
\sn
\item "{{}}"  $\le$ a partial order on $D^{\frak c}$,
\sn
\item "{{}}"  val$^{\frak c}$ is a function from $D^{\frak c}$ to
${\Cal P}(\lambda) \backslash \{\emptyset\}$
\sn
\item "{{}}"  nor$^{\frak c}:D^{\frak c} \rightarrow \omega$ or to??
\ermn
2) It is called simple if nor$^{\frak c}$ is always $>0$ (without loss 
of generality constant, e.g. Rang(val$^{\frak c}$) = $I^+,I$ an ideal 
on $\lambda$).  A creature is a $\lambda$-creature for some $\lambda$. \nl
$\Bbb I$ will be a set of creatures. 
\enddefinition
\bigskip

\definition{\stag{0.B} Definition}  1) An $\Bbb I$-tree is 
$(T,\bold I,\bold d)$ such that: \newline
for some ordinal $\alpha,T \subseteq {}^{\omega >}\alpha$ closed 
under initial segments, $\ne \emptyset$ \newline
$\bold I$ is a partial function Dom$(\bold I) \subseteq T,\bold I_\eta \in
\Bbb I$, \newline
$\bold d$ has domain Dom$(\bold I),\bold d(\eta) \in D^{\bold I_\eta}$ \nl
val$^{\bold I_\eta}(\bold d(\eta)) = 
\{\alpha:\eta \char 94 \langle \alpha \rangle \in T\}$. \newline
2) Let $(T^*,\bold I^*,\bold d^*)$ be an $\Bbb I$-tree, such that
\medskip
\roster
\item "{$(*)$}"  $(\forall \eta \in \text{ lim } T^*)
[\text{lim } \sup {n < \omega} \text{ nor}^{\bold I^*_{\eta \restriction n}}
(\bold d^*_\eta) = \infty]$ \newline
and Dom$(\bold I_\eta) = T^*$.
\endroster
\medskip

\noindent
We define a forcing notion $\Bbb Q = \Bbb Q_{(T^*,\bold I^*,\bold d^*)}$:

$$
\align
\Bbb Q = \biggl\{(T,\bold I,\bold d):&T \subseteq T^*,\bold I = \bold I^*
\restriction T, \\
  &(T,\bold I,\bold d) \text{ an } \Bbb I \text{-tree}, \\
  &(\forall \eta \in T)(\bold d^*_\eta \le^{\bold I_\eta} \bold d_\eta) \\
  &\text{and } ((\forall \eta \in \text{ lim } T^*) \text{ lim sup nor}
^{{\bold I}^*_{\eta \restriction n}}(\bold d_\eta) = \infty) \biggr\}.
\endalign 
$$
\mn
Order: natural.
\sn
3) Let $(T^*,I^*,\bold d^*)$ be an $\Bbb I$-tree such that
\medskip
\roster
\item "{$(**)$}"  $(\forall \eta \in \text{ lim }T^*)(\forall n)
(\forall^* \ell)(\text{nor}^{I^*_{\eta \restriction n}}(\bold d^*_\eta) \ge
n)$ \newline
(i.e. lim inf $= \infty$).
\endroster
\medskip

\noindent
and define $\Bbb Q' = \Bbb Q'_{(T^*,{\bold I}^*,{\bold d}^*)}$ parallelly. \nl
4) For $p \in \Bbb Q$ (or $p \in \Bbb Q'$) we write $p = (T^p,\bold I^p,\bold d^p)$.
In this case for $\eta \in T^p$ we define $q = p^{[\eta]}$ by: \nl
$T^{[q]} = \{\nu \in T^p:\nu \trianglelefteq \eta$ or $\eta \trianglelefteq
\nu\},\bold I^q = \bold I^p \restriction T^q,\bold d^q = \bold d^p 
\restriction T^p$.  Clearly $p \in \Bbb Q \Rightarrow p \le q \in \Bbb Q$ and $p \in
\Bbb Q' \Rightarrow p \le q \in \Bbb Q'$.
\enddefinition
\bigskip

\proclaim{\stag{0.C} Claim}  Let $(T,\bold I^*,\bold d^*)$ and $\Bbb
Q,\Bbb Q'$ be as in
\scite{0.B}.  A sufficient condition for ``$\aleph_1$ not collapsed" is:
\mr
\item "{$(a)$}"  for $\Bbb Q$: $(**)$ below
\sn
\item "{$(b)$}"  for $\Bbb Q':(*) + (**)$ below where
\sn
{\roster
\itemitem{ $(*)$ }  $\Bbb I$ has $\aleph_1$-bigness: 
$$
\align
(\forall {\frak c} \in \Bbb I)(\forall x \in D^{\frak c})
\biggl[ \text{nor}^{\frak c}(x) > 0 \rightarrow &(\forall h \in
{}^{(\lambda^{\frak c})}\omega_1)(\exists y) \\
  &[x \le^{\frak c} y \and \text{ nor}^{\frak c}(y) \ge \text{ nor}^{\frak c}
(x)-1 \and \\
  &(h \restriction \text{ val}^{\frak c}(y) \text{ is constant}] \biggr]
\endalign
$$

\itemitem{ $(**)$ }  $\Bbb I$ is $(\aleph_1,\aleph_1)$-indecomposable where
$\Bbb I$ is $(\mu,\kappa)$-indecomposable means:
\mn
\block
$\boxtimes_{\Bbb I,\mu,\kappa} \qquad$ if ${\frak c} \in \Bbb I$ and $x \in 
D^{\frak c}$ satisfies nor$^{\frak c}(x) > 2$ and $A_\alpha \subseteq
\lambda^{\frak c}$ for $\alpha < \mu$ are such that $(\forall y)(x \le y \in 
D^{\frak c} \wedge \text{ val}^{\frak c}(y) \subseteq A_\alpha \rightarrow 
\text{ nor}^{\frak c}(y)+2 \le \text{ nor}^{\frak c}(x))$, \ub{then} we can
find $u \subseteq \lambda^{\frak c}$ of cardinality $< \mu$ such that for
every large enough $\alpha < \mu$ we have $u \nsubseteq A_\alpha$.
\endblock
\endroster}
\endroster
\endproclaim
\bigskip

\demo{Proof for $\Bbb Q$}  Lets use given $p = (T,\bold I,\bold d) \in
\Bbb Q$ and 
$\Bbb Q$-name $\underset\tilde {}\to \tau$ such that $\Vdash 
{\underset\tilde {}\to \tau}:\omega \rightarrow \omega_1$.  Now we 
choose by induction on $n,p^n,A_n$ such that:
\medskip
\roster
\item "{$(a)$}"  $p^n \le p^{n+1}$
\sn
\item "{$(b)$}"  $A_0,\dotsc,A_n$ are fronts of $p^n$ which means
$(\forall \eta \in \lim^{p^n})(\exists!n)(\eta \restriction n \in A_\ell)$
\sn
\item "{$(c)$}"  $A_\ell$ below $A_{\ell + 1}$ which means \nl 
$(\forall \eta \in A_{\ell +1})(\exists \nu \triangleleft \eta)
\nu \in A_\ell$ \newline
(so $A_n \subseteq T^{p^{n+1}},(\forall \eta \in T^{p^n} \backslash
T^{p^{n+1}})(\exists \nu \triangleleft \eta)(\nu \in A_n))$
\sn
\item "{$(d)$}"  $(\forall \nu \in A_n)(\forall \eta \in \text{ Suc}_{T^{p_n}}
(\nu))(p^{[\eta]}_n$ forces a value to $\underset\tilde {}\to \tau(n))$
\sn
\item "{$(e)$}"  $\eta \in A_n \Rightarrow \text{ nor}^{{\bold I}^*_\eta}
(\bold d^{p_n}_\eta) \ge n \and \bold d^{p_n}_\eta = \bold d^{p_{n+1}}_\eta$,
it follows that $\ell < n \and \eta \in A_\ell \Rightarrow \bold d^{p_\ell}
_\eta = \bold d^{p_n}_\eta \and \text{ Suc}_{T_{p_n}}(\eta) =
\text{ Suc}_{T^{p_\ell}}(\eta)$.
\ermn
So $p^*$ is defined by $T^{p^*} = \dbca_{n < \omega} T^{p_n},\bold I^{p^*} =
\bold I \restriction T^{p^*},\bold d^{p^*} = \dbcu_n\{\bold d^{p_n}
\restriction \{\eta \restriction \ell:\eta \in A_n$ and $\ell \le \ell g
(\eta)\}:n < \omega\}$ belong to $\Bbb Q$ and is an upper bound of $\{p_n:n <
\omega\}$. 

We define $h:T^{p^*} \rightarrow \omega_1$ as follows: if 
$\eta \in T^{p^*},\nu \triangleleft \eta \trianglelefteq \nu',\nu \in 
A_{\ell -1},\nu' \in A_\ell$ \newline
(if $\ell = 0$ omit $\nu'$, so just $\eta \trianglelefteq \nu'$), then 
$p^{*^{[\eta]}}$ forces value to $\underset\tilde {}\to \tau \restriction 
\ell$ call it $(\tau \restriction \ell)^{p^{*^{[\eta]}}}$ and let

$$
h(\eta) = \text{ Sup Rang}(\underset\tilde {}\to \tau \restriction 
\ell)^{p^{*^{[\eta]}}}.
$$
\medskip

\noindent
For notational simplicity $A_n = T^{p^*} \cap {}^n\text{Ord}$. \newline
We now define a game $\Game = \Game^\alpha_{T^{p^*}}$ for each $\alpha < 
\omega_1$:
\sn

A play of the game last $\omega$ moves, in the $(n-1)$-th move a member
$\eta_n$ of $A_n$ is chosen such that $m < n \Rightarrow \eta_m
\triangleleft \eta_n$, and fixing some $\eta_{-1} \in A_n$.
\sn
In the $n$-the move:
\mr
\item "{$(a)$}"  the anti-decidability player chooses a set $A_n \subseteq
\text{ Suc}_{T^{p^*}}(\eta_{n-1})$ such that
$$
B_n = \emptyset \vee (\text{nor}^{{\bold I}_\eta}(A_n) \le n-2,n \ge 2)
$$
{\roster
\itemitem{ $\boxtimes_1$ }  $B_n \ne \emptyset \text{ \ub{or} } n \ge 3$
and for no $d$, satisfying $\bold d^{p^*}(\eta) \le d \wedge
\text{ nor}^{{\bold I}^*_2}(d) \ge n-2$ do we have $\eta \char 94 \langle
\alpha \rangle \in A_n \Rightarrow \alpha \in \text{ val}^{{\bold I}^*_\eta}
(d)$
\endroster}
\item "{$(b)$}"  the decidability player chooses $\eta_n \in A_n$ such that
$\neg(\exists \nu \in B_n)(\nu \trianglelefteq \eta_n)$ and $n \ge 1
\Rightarrow (\eta_n \restriction (\ell g(\eta_{n-1})-1)) \le \alpha$.
\ermn
Without loss of generality $A_0 = \{<>\}$.

If for some $\alpha$ and decidability player has a winning strategy, we can
produce a condition as required.

If not, for every $\alpha < \omega_1$ the antidecidability player has a
winning strategy {\bf St}$_\alpha$.  For each $\eta \in T^{p^*}$ and $\alpha
< \omega_1$, we consider the play of the game in which the antidecidability
player has winning strategy {\bf St}$_\alpha$ and in some move $n$ the
decidability player chooses $\eta_n = \eta$.  Reflecting there is no freedom
left so there is at most one such play and $n$ and let the antidecidability
player choose set $B_{\eta,\alpha}$ as there (if no such game let
$B_{\eta,\alpha} = \emptyset$).
\bn
\ub{Case 1}:  For some $n$ and $\eta \in A_n$, we have: there is no
countable $u \subseteq \text{ Suc}_{T^{p^*}}(\eta)$ such that for every
large enough $\alpha < \omega_1,u \nsubseteq B_{\eta,\alpha}$ so by the
assumption $(**)$ we get a contradiction.
\bn
\ub{Case 2}:  Not Case 1.

We can choose by induction on $n$, a countable subset $u_n \subseteq A_n$
such that: $u_0 = \{<>\}$

$$
\align
\text{if } \eta \in A_n &\text{ then for some } \alpha_\eta < \omega_1
\text{ for every } \alpha \in [\alpha_\eta,\omega_1), \\
  &\text{ if in the play in which the antidecidability player uses }
{\bold {St}}_\alpha \\
  &\text{ and they arrive to } \eta, \text{ there is } \eta',\eta
\triangleleft \eta' \in u_n +1 \\
  &\text{ which is a legal response of the decidability player}.
\endalign
$$
mn
Now let

$$
\align
\alpha^* = &\sup\{h(\eta)+1:\text{ for some } \nu \in \dbcu_n u_n,
\eta \triangleleft \nu,\eta \in \text{ Dom}(h)\} + \\
  &\sup\{\alpha_\eta:\eta \in \dbcu_n u_n\}
\endalign
$$
\mn
and we can find a play of $\Game^{\alpha^*}$ as above where the
decidability player chooses $\eta$'s from $\dbcu_{n < \omega} u_n$.  We get
a contradiction.
\enddemo
\bigskip

\demo{Proof for $\Bbb Q'$}?  We should make changes: in $p^{n+1}$ we shrink
$p^{[\eta]}_n$ for each $\eta \in T^{p^n} \cap {}^n\text{Ord}$, to $q_\eta,
p^{[\eta]}_n \le_{pr} [\eta]$ and for each $\ell \le n$, if possible, 
$q_\eta$ forces a bound to $\underset\tilde {}\to \tau(\ell)$
and, of course, $p^{[\eta]}_{n+1} = q_\eta$ for each such $\eta$ and
$T^{p^{n+1}} \cap {}^{\eta \ge}\text{Ord } = 
T^{p^n} \cap {}^{n \ge}\text{Ord}$ and 
$\bold d^{p^{n+1}} \restriction {}^{n \ge}\text{Ord } = \bold d^{p^n}
\restriction {}^{n \ge}\text{Ord}$.  
So let $p^* = \dbca_n p_n$ be naturally defined, and we use 2-bigness to
prove enough times $q_\eta$ forces a bound.
\enddemo
\bn
Now we give details.  
\demo{Proof for $\Bbb Q$}  Given $p = (T,\bold I,\bold d)$, for notational
simplicity tr$(p) = <>$ and nor$^{{\bold d}^p_\eta}(\text{Suc}_{T^p}(\eta))
> 2$ and $\Bbb P$-name $\underset\tilde {}\to \tau$ such that 
$\Vdash {\underset\tilde {}\to \tau}:\omega \rightarrow \omega_1$ we 
choose by induction on $n,p^n$ such that:
\medskip
\roster
\item "{$(a)$}"  $p^n \le p^{n+1}$, and $p^n$ has trunk $n$
\sn
\item "{$(b)$}"  $A_0,\dotsc,A_n$ are fronts of $p^n$
\sn
\item "{$(c)$}"  $A_\ell$ below $A_{\ell + 1}$ which means \nl 
$(\forall \eta \in A_{\ell +1})(\exists \nu \triangleleft \eta)
\nu \in A_\ell$ \newline
(so $A_n \subseteq T^{p^{n+1}},(\forall \eta \in T^{p^n} \backslash
T^{p^{n+1}})(\exists \nu \triangleleft \eta)(\nu \in A_n))$
\sn
\item "{$(d)$}"  $A_0 = \{<>\}[\eta \in A_n \wedge \eta \trianglelefteq
\nu \in T^{p^n} \Rightarrow \text{ nor}^{{\bold I}_\nu}(\bold d^{p_n}_\nu)
> n+2]$
\sn
\item "{$(e)$}"  when $\eta \in A_n$ let $\ell = \ell_\eta \le n$ be
maximal such that there are $\alpha_m < \omega_1,m < \ell$ and $q$ satisfying
$\text{tr}q = \eta,p^{[\eta]}_n \le q \in P,q \Vdash ``\dsize \bigwedge_{m <
\ell} \underset\tilde {}\to \tau(m) < \alpha'_n$ and $\eta \trianglelefteq \nu
\in T^q \Rightarrow \text{ nor}^{{\bold d}^*_\nu}(\bold d^q_\nu) \ge n$ and
we demand: $p^{[\eta]}_{n+1}$ satisfies the demand on $q$ for some $\langle
\alpha_m:m < \ell_\eta \rangle$, note possible $\ell_\eta = \nu$ then we are
left with demand on norm.
\ermn
So $p^*,T^{p^*} = \dbca_{n < \omega} T^{p_n}$ is an upper bound of 
$\{p_n:n < \omega\}$.

Clearly $p^* \in \Bbb P$ and $n < \omega \Rightarrow p_n \le p^*$.  Let for
$\eta \in T^{p^*}$, let $n(\eta) = \text{ Max}\{n:\text{there is } \nu
\triangleleft \eta,\nu \in A_n\}$ and $\nu_\eta \triangleleft \eta$ be in
$A_n$ and $\beta_\eta < \omega_1$ be minimal such that $p^{[\nu_\eta]}_{n+1}$
forces $\tau(0),\dotsc,\tau(\ell_{\nu_\eta}-1) < \beta_\eta$.  Using games as
in the proof for $\Bbb Q'$ there is $p^+$ such that:
\mr
\item "{$(a)$}"  $p^* \le p^* \in \Bbb P$
\sn
\item "{$(b)$}"  $\rho \in T^{p^+} \Rightarrow \text{ nor}^{{\bold I}^*_\rho}
(\bold d^{p^+}_\rho) \ge \text{ nor}^{{\bold I}^*_\rho}(\bold d^{p^*}_\rho)-1$
\sn
\item "{$(c)$}"  $\beta^* = \sup\{\beta_\eta:\eta \in T^{p^+}\} < \omega_1$.
\ermn
We continue as in \cite[Ch.XIV,\S5]{Sh:f}.
\enddemo
\bn
\centerline {$* \qquad * \qquad *$}
\bn
\ub{Discussion}:  We can continue to do iteration.  \newline
But more urgent: can $\Bbb Q,\Bbb Q'$ like this do anything not already covered by
composition?

A natural thought is splitting or reaping numbers.  We can think of the
tree splitting in $T^*$ as a list of the reals.  BUT, what is the norm?
\bigskip

\centerline {$* \qquad * \qquad *$}
\bigskip

Not finished...check the better's theorem proof?
\bigskip

\noindent
\underbar{Assignment}: \cite[XIII,XVI]{Sh:f} and here put together, so does
the reflection $Pr_a(\lambda,f)$ works for ???
\bn
\ub{97/2/2 - Discussion}:

Saying a creature is $\mu$-complete meana that for pure extensions, increasing
chains of length $< \mu$ have pure upper bounds?  Probably pure means not
changing the norm; maybe the $\aleph_1$-indecomposable is enough. \nl
So the $\Bbb I$-th condition has a new meaning. 
\mn
\ub{Question}:  Does the theorem here hold?
\mn
\ub{Question}:  Does this new context have real applications?

The first result to be discussed is moving from 
${\underset\tilde {}\to {\Bbb I}}$ to one in the ground model. \nl
The second are \scite{5.2}, \sciteu{6.2} preservation of $N$ being suitable.
\newpage

\head{\S11 Variants of the iteration} \endhead  \resetall \sectno=11
 % \resetall 
\bigskip

As mentioned in \S1, we can consider $\kappa$-RS iteration and
variants of the Sp$_e$ iteration.
\bigskip

\definition{\stag{11.13} Definition/Claim}
Let $\kappa$ be a successor cardinal or
an infinite ordinal not a cardinal but an ordinal of power $|\kappa|,\kappa$ 
fix \footnote{For $\kappa$ inaccessible, see \sciteu{1.22}.}.  
We define and prove the following by induction on $\alpha$ (here 
$e = \{3,4,5,6\}$).  If $\kappa = \aleph_1$, we may omit it and this is 
the main case.  

We repeat \scite{1.13} with the following changes in the proof and
definition:
\mr
\item "{$(B)$}"  We say $\underset\tilde {}\to \zeta$ is a simple
$\bar{\Bbb Q}$-named$_e$ \, $[j,\beta)$-ordinal if
{\roster
\itemitem{ $(*)_1$ }  $\underset\tilde {}\to \zeta$ is a simple
$\bar{\Bbb Q}$-named$^1$ \, $[j,\beta)$-ordinal and may restrict ourselves
to $\kappa = \aleph_1 \Rightarrow e \in \{3,4\}$
\sn
\itemitem{ $(*)_2$ }  if $e \in \{5,6\}$ and $\kappa = \aleph_1$, \ub{then}
$\underset\tilde {}\to \zeta$ is a simple $\bar{\Bbb Q}$-named$^2$ \, 
$[j,\beta)$-ordinal.
\endroster}
\item "{$(F)(a)$}"  in (v) replace the remark in the end by: \nl
``if $e \in \{5,6\},\alpha \in w$ then this demand follows by \scite{1.6A}
\nl
and add:
{\roster
\itemitem{ $(vii)$ }  if $e=3,5$ then for some $n < \omega$ and simple
$\bar{\Bbb Q}$-named $[0,\ell g(\bar{\Bbb Q}))$-ordinals
${\underset\tilde {}\to \xi_1},\dotsc,{\underset\tilde {}\to \xi_n}$ 
we have, for every $\xi < \ell g(\bar{\Bbb Q}) 
\Vdash_{{\Bbb P}_\xi}$ ``if for $\ell = 1,\dotsc,n$ we have
${\underset\tilde {}\to \xi_\ell}
[{\underset\tilde {}\to G_{{\Bbb P}_\xi}}] \ne \xi$
(for example ${\underset\tilde {}\to \xi_\ell}[G_{{\Bbb P}_\xi}]$ 
not well defined) then 
$\emptyset_{\underset\tilde {}\to {\Bbb Q}_\xi} \le_{\text{pr}}
p \restriction \{\xi\}$ in ${\underset\tilde {}\to {\hat{\Bbb Q}}_\xi}"$ 
\endroster}
\item "{$(F)(e)(iii)$}"   inside change $p_2 \restriction \xi
\Vdash_{{\Bbb P}_\xi} ``\ldots"$ by \nl
$p_2 \restriction \xi \Vdash_{{\Bbb P}_\xi}$ ``if
$\xi \ne {\underset\tilde {}\to \xi_\ell}
[{\underset\tilde {}\to G_{{\Bbb P}_\xi}}]$ for \nl

$\qquad \qquad \ell = 1,\dotsc,n$ and \nl

$\qquad \qquad [e=4 \vee e=6 \Rightarrow 
\neg(\emptyset_{\underset\tilde {}\to {\Bbb Q}_\xi} \le_{vp} p^1 
\restriction \{\xi\})]$  \nl

$\qquad \qquad$ then: 
${\underset\tilde {}\to {\hat{\Bbb Q}}_\xi} 
\models p^1 \restriction
\{\xi\} \le_{pr} p^2 \restriction \{\xi\}"$.
\endroster
\enddefinition
\bigskip

\proclaim{\stag{11.14} Claim}  1) As in \scite{1.14} adding $\kappa \ne
\aleph_1 \vee e \in \{3,4\}$. \nl
2)  If $\bar{\Bbb Q}$ is an $\kappa-\text{Sp}_e(W)$-iteration, 
and for each $i$ the quasi-order $\le^{{\Bbb Q}_i}_{\text{pr}}$ is 
equality hence $\le^{{\Bbb Q}_i}_{\text{vpr}}$ is equality,
\ub{then} $\bar{\Bbb Q}$ is essentially a finite support iteration. 
\nl
[Saharon:  maybe restrict yourself above the constantly function
$\zeta \mapsto \emptyset_{{\Bbb Q}_\zeta}$, so we have to use $\kappa >
\ell g(\bar{\Bbb Q})$.]
\endproclaim
\bigskip

\proclaim{\stag{11.15} Claim}  1)  Add
\mr
\item "{$(d)$}"  if $e=3,5$ then $r$ is pure outside
$\{\xi_1,\dotsc,\xi_n\}$. 
\ermn
2) In the proof on ``$\xi^*_{\ell \times 1}$" ?? we say, i.e., simple$^1$
if $e \in \{3,4\}$ and simple$^2$ if $e \in \{5,6\}$.
\endproclaim
\bigskip

\citewarning{\noindent \llap{---$\!\!>$} MARTIN WARNS: Label 1.16 on next line is also used somewhere else (Perhaps should have used scite instead of stag?)}
\proclaim{\stag{1.16} Claim}
5) If $e \in \{3,4\}$ and \footnote{for the parallel for $e \in \{5,6\},
\kappa = \aleph_1$ we need pure decidability and restrict ourselves to
``above $p$" 
for purely dense sets of $p-s$} for each $\beta < \ell g(\bar{\Bbb Q}),
{\underset\tilde {}\to {\bold t}_\beta}$ is a $\Bbb P_\beta$-name of a truth
value, \ub{then} there is a simple $(\bar{\Bbb Q},W)$-named 
$[0,\alpha)$-ordinal
$\underset\tilde {}\to \zeta$ such that $\underset\tilde {}\to \zeta[G_\beta]
= \beta$ iff ${\underset\tilde {}\to {\bold t}_\beta}[G_\beta] =$ truth and
$\gamma < \beta \Rightarrow {\underset\tilde {}\to {\bold t}_\gamma}
[G_\beta] =$ false for any subset $G_\beta$ of $\Bbb P_\beta$ generic
over $\bold V$.
\endproclaim
\bigskip
 
We can deal with the parallel of hereditarily countable names.  This is not
used in later sections.
\definition{\stag{11.21} Definition}  We define for an 
$\kappa-\text{Sp}_e(W)$-iteration $\bar{\Bbb Q}$, and 
cardinal $\mu$ ($\mu$ regular), when is a $(\bar{\Bbb Q},W)$-name 
hereditarily $< \mu$, and in particular when a
$(\bar{\Bbb Q},W)$-named $[j,\alpha)$-ordinal is
hereditarily $< \mu$ and a $(\bar{\Bbb Q},W)$-named 
$[j,\alpha)$-atomic condition hereditarily $< \mu$, 
and which conditions of $\text{Sp}_e(W)$-Lim$_\kappa(\bar{\Bbb Q})$ 
are hereditarily $< \mu$.  For simplicity we are assuming that the set of 
members of $\Bbb Q_i$ is in $\bold V$.  This is done by induction on $\alpha = 
\ell g(\bar{\Bbb Q})$.
\mn
\underbar{First Case}:  $\alpha = 0$.

Trivial.
\mn
\underbar{Second Case}:  $\alpha > 0$.
\mr
\item "{$(A)$}"  A $\bar{\Bbb Q}$-named $[j,\alpha)$-ordinal 
$\underset\tilde {}\to \xi$ 
hereditarily $< \mu$ is a $(\bar{\Bbb Q},W)$-named
$[j,\alpha)$-ordinal which can be represented as follows: there is
$\langle (p_i,\xi_i):i < i^* \rangle,i^* < \mu$, each $\xi_i$ an ordinal in
$[j,\alpha),p_i \in \Bbb P_{\xi_i}$ is a member of $\Bbb P_{\xi_i}$ 
hereditarily $< \mu$ and for any $G \in \text{ Gen}^r(\bar{\Bbb Q}),
\underset\tilde {}\to \zeta[G]$ is $\zeta$ iff for some $i$ we have:
{\roster
\itemitem{ $(a)$ }  $p_i \in G,\zeta_i = \zeta$
\sn
\itemitem{ $(b)$ }  if $p_j \in G$ then $\zeta_i < \zeta_j \vee (\zeta_i =
\zeta_j \and i < j)$
\endroster}
\item "{$(B)$}"  A $(\bar{\Bbb Q},W)$-named $[j,\alpha)$-atomic condition 
$\underset\tilde {}\to q$ hereditarily $< \mu$, is a $(\bar{\Bbb Q},W)$-named
$[j,\alpha)$-atomic condition which can be represented as follows: there is
$\langle (p_i,\zeta_i,q_i):i < i^* \rangle,i^* < \mu,\zeta_i \in [j,\alpha),
p_i \in \Bbb P_{\zeta_i},q_i \in \bold V$, and for any $G \in \text{ Gen}^r(\bar{\Bbb Q}),
\underset\tilde {}\to q[G]$ is $q$ iff for some $i$ we have:
\sn
{\roster
\itemitem{ $(a)$ }  $p_i \in G,q = q_i$, and $p_i \Vdash_{{\Bbb P}_{\zeta_i}}
``q \in {\underset\tilde {}\to {\Bbb Q}_{\zeta_i}}"$
\sn
\itemitem{ $(b)$ }  if $p_j \in G$ then $\zeta_i < \zeta_j \vee (\zeta_i =
\zeta_j \and i < j)$
\endroster} 
\sn
\item "{$(C)$}"  A member $p$ of $\Bbb P_\alpha =
Sp_e(W)$-Lim$_\kappa(\bar{\Bbb Q})$ is 
hereditarily $< \mu$ if each member of 
$r$ is a $(\bar{\Bbb Q},W)$-named atomic
condition hereditarily $< \mu$
\sn
\item "{$(D)$}"  A $(\bar{\Bbb Q},W)$-name of a member of 
$\bold V$ hereditarily $< \kappa$
is defined as in clause $(B)$, similarly for member $x \in \bold
V^{{\Bbb P}_\alpha}$ such
that $y \in$ transitive closure of $x \Rightarrow |y| < \mu$.
\endroster
\enddefinition
\bigskip

\remark{\stag{11.22} Concluding Remarks}  1) We 
have not really dealt with the 
case $\kappa$ is inaccessible.  The point is that in this case, we do not 
know a priori the length of the list of the members of a condition 
(which are atomic conditions).  It is natural to work on it together 
with ``decidability on bound
on $\underset\tilde {}\to \alpha < \kappa$ by pure extensions",
see \scite{1.25} below. \newline
2) We can think of putting together \cite[Ch.XIV]{Sh:f} and \cite{Sh:587}.
\newline
3) We can ask:  Does ``Souslin forcing notions" help?
\endremark
\bigskip

\citewarning{\noindent \llap{---$\!\!>$} MARTIN WARNS: Label 11.27 on next line is also used somewhere else (Perhaps should have used scite instead of stag?)}
\proclaim{\stag{11.27} Claim}  $e \in \{4,5\}$ is O.K.
\endproclaim
\bigskip

\citewarning{\noindent \llap{---$\!\!>$} MARTIN WARNS: Label 11.27 on next line is also used somewhere else (Perhaps should have used scite instead of stag?)}
\proclaim{\stag{11.27} Claim}  In the proof of \scite{1.27}, in case 3
add:
\nl
(the point is that $e \in \{4,6\}$).  Instead $e \in \{4,6\}$ it is enough to
assume:
\mr
\item "{$\boxtimes_{\underset\tilde {}\to {\Bbb Q}_\beta}$}"  for every
$q',q'' \in {\underset\tilde {}\to {\Bbb Q}_{\beta_0}}$ we have \nl
$\emptyset_{{\Bbb Q}_{\beta_0}} \le_{\text{vpr}} q' \le q'' \Rightarrow q' 
\le_{\text{pr}} q''$.
\endroster
\endproclaim
\bigskip

\remark{\stag{11.27B} Remark}  1) Add: \nl
but for $e=4$ we could use appropriate $p_1 = p \cup \{
{\underset\tilde {}\to r_1}\},\underset\tilde {}\to r$ an atomic
$(\bar{\Bbb Q},W)$-named condition,
${\underset\tilde {}\to \zeta_{\underset\tilde {}\to r}} =
\underset\tilde {}\to \zeta$, see \scite{1.6}(5).  \nl
2) Holds for $e \in \{4,6\}$.
\endremark
\bigskip

\definition{\stag{n.2} Definition}  0) Let $\bar{\Bbb Q}$ be a
$\kappa_1$-Sp$_e(W)$-iteration of length $\alpha$.  Let 
$\underset\tilde {}\to \zeta$ denote a simple $\bar{\Bbb Q}$-named
$[0,\alpha)$-ordinal \ub{or} a simple $\bar{\Bbb Q}$-named$^2 
[0,\alpha)$-ordinals and $\Xi$ a countable set of such objects. \nl
1) For an atomic simple $\bar{\Bbb Q}$-named condition $r,r \restriction
\underset\tilde {}\to \zeta$ is defined by $r \restriction
\underset\tilde {}\to \zeta[G] = r^* \in \Bbb P_\zeta$ if
$\underset\tilde {}\to \zeta[G] \ge {\underset\tilde {}\to \zeta_r}[G],
r[G] = r^*$ and $\emptyset_{p_\zeta}$ otherwise. \nl
2) For $q \in \Bbb P_\alpha,q \restriction \underset\tilde {}\to \zeta =
\{r \restriction \zeta:r \in q\}$ and $q \restriction \Xi =
\dbcu_{{\underset\tilde {}\to \zeta} \in \Xi} q \restriction \zeta$. \nl
3) $\Bbb P_{\underset\tilde {}\to \zeta} 
= \{p \in \Bbb P_\alpha:p \restriction
\underset\tilde {}\to \zeta = p$, i.e., for every $G \subseteq \Bbb P_\alpha$
generic over $\bold V,p \restriction \underset\tilde {}\to \zeta[G] = p[G]\}$

$$
P_\Xi = \{p \in \Bbb P_\alpha:p \restriction \Xi = p\}
$$
\mn
both with the order inherited from $\Bbb P_\alpha$.
\enddefinition
\bigskip

\proclaim{\stag{n.3} Claim}  Let $\bar{\Bbb Q}$ be an
$\aleph_1-{\text{\rm Sp\/}}_e(W)$-iteration. \nl
1) If ${\underset\tilde {}\to \zeta_1}$ is a simple $\bar{\Bbb Q}$-named
$[\beta_1,\beta_2)$-ordinal, ${\underset\tilde {}\to \zeta_2}$ is a simple
$\bar{\Bbb Q}$-named$^1\,[\beta_1,\beta_2)$-ordinal, 
\ub{then} there is a simple
$\bar{\Bbb Q}$-named 
$[\beta_1,\beta_2)$-ordinal $\underset\tilde {}\to \zeta$
such that for $G \subseteq \Bbb P_\alpha$ is generic over $\bold V$:
\mr
\item "{$(a)$}"  if ${\underset\tilde {}\to \zeta_1}[G] =
{\underset\tilde {}\to \zeta_1}[G \cap \Bbb P_\xi] = \xi$ 
and ${\text{\rm Min\/}}\{\varepsilon$:
some $p \in G \cap \Bbb P_\varepsilon$ decided to be $\varepsilon$ or be
undefined$\} > \varepsilon$ \ub{then} $\underset\tilde {}\to \zeta[G] =
\underset\tilde {}\to \zeta[G \cap \Bbb P_\xi] = \xi$
\sn
\item "{$(b)$}"  otherwise undefined.
\ermn
2) Let 
$\underset\tilde {}\to \zeta$ be a simple $\bar{\Bbb Q}$-named$^1$ ordinal.
For $r$ an atomic $\bar{\Bbb Q}$-named condition $r \restriction
\underset\tilde {}\to \zeta$ is an atomic $\bar{\Bbb Q}$-named condition. \nl
3) For $q \in \Bbb P_\alpha$ we have 
$q \restriction \underset\tilde {}\to \zeta \in \Bbb P_\alpha$. \nl
4) For $q_1,q_2 \in \Bbb P_\alpha,q_1 \le q_2 \Rightarrow q_1 \restriction
\underset\tilde {}\to \zeta \le q_2 \restriction 
\underset\tilde {}\to \zeta$.
\nl
5) If $q 
\in \Bbb P_{\underset\tilde {}\to \zeta},p \in \Bbb P_\alpha,p \restriction
\zeta \le q$ \ub{then} $p \cup q \in \Bbb P_\alpha$ is a lub of $p$
and $q$. 
\nl
6) $\Bbb P_{\underset\tilde {}\to \zeta} \lessdot \Bbb P_\alpha$. \nl
7) If $G \subseteq \Bbb P_{\underset\tilde {}\to \alpha}$ is generic
over $\bold V,\xi = \underset\tilde {}\to \zeta[G]$ \ub{then} $G \cap
P_{\underset\tilde {}\to \zeta},G \cap \Bbb P_\xi$ are 
essentially the same. \nl
8) The parallel statements with $\Xi$ instead of $\underset\tilde {}\to 
\zeta$.
\endproclaim
\bigskip

\remark{Remark}  In fact 
by part (1), part (6) follows from the parts (2)-(5).
\endremark
\bigskip

\proclaim{\stag{n.4} Claim}  Let $e = 4(2)$.
Assume ${\underset\tilde {}\to \zeta_n}$ is a
simple $\bar{\Bbb Q}$-named for $n < \omega,
{\underset\tilde {}\to \zeta_n} < 
{\underset\tilde {}\to \zeta_{n+1}}$ and 
for every $G \subseteq \Bbb P_\alpha$ generic over $\bold V$, 
for some $n,\varphi(n,G \cap 
\Bbb P_{{\underset\tilde {}\to \zeta_n}[G]})$.  \ub{Then} for some simple
$\bar{\Bbb Q}$-name ordinal $\underset\tilde {}\to \xi$, we have

$$
\Vdash_{{\Bbb P}_\alpha} ``\text{for some } n,\underset\tilde {}\to \xi
[{\underset\tilde {}\to G_{{\Bbb P}_\alpha}}] = 
{\underset\tilde {}\to \zeta_n}
[{\underset\tilde {}\to G_{{\Bbb P}_\alpha}}]
\text{ and } \varphi(n,{\underset\tilde {}\to G_{{\Bbb P}_\alpha}} \cap
\Bbb P_{{\underset\tilde {}\to \zeta_n}[G_{{\Bbb P}_\alpha}]})".
$$
\endproclaim
\bigskip




 





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\newpage
    
REFERENCES.  
\bibliographystyle{lit-plain}
\bibliography{lista,listb,listx,listf,liste}

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311.bbl
\def\germ{\frak} \def\scr{\cal} \ifx\documentclass\undefinedcs
  \def\bf{\fam\bffam\tenbf}\def\rm{\fam0\tenrm}\fi % f**k-amstex!
  \def\defaultdefine#1#2{\expandafter\ifx\csname#1\endcsname\relax
  \expandafter\def\csname#1\endcsname{#2}\fi} \defaultdefine{Bbb}{\bf}
  \defaultdefine{frak}{\bf} \defaultdefine{mathfrak}{\frak}
  \defaultdefine{mathbb}{\bf} \defaultdefine{mathcal}{\cal}
  \defaultdefine{beth}{BETH}\defaultdefine{cal}{\bf} \def\bbfI{{\Bbb I}}
  \def\mbox{\hbox} \def\text{\hbox} \def\om{\omega} \def\Cal#1{{\bf #1}}
  \def\pcf{pcf} \defaultdefine{cf}{cf} \defaultdefine{reals}{{\Bbb R}}
  \defaultdefine{real}{{\Bbb R}} \def\restriction{{|}} \def\club{CLUB}
  \def\w{\omega} \def\exist{\exists} \def\se{{\germ se}} \def\bb{{\bf b}}
  \def\equivalence{\equiv} \let\lt< \let\gt> \def\implies{\Rightarrow}
\begin{thebibliography}{RuSh 117}
\makeatletter \renewcommand{\@biblabel}[1]{[#1]} \makeatother
\def\eprintfootnotetext{References of the form {\tt math.XX/$\cdots$}
 refer to the {\tt xxx.lanl.gov} archive}
\ifx\documentstyle\undefinedcontrolsequence
   \def\anyfootnote{\footnote{*}}
   \else\def\anyfootnote{\footnote}\fi
\def\eprintfn{\ifEprint\anyfootnote{\eprintfootnotetext}\fi\Eprintfalse }
\newif\ifEprint  \Eprinttrue

\bibitem[GiSh 191]{GiSh:191}Moti Gitik and Saharon Shelah.
\newblock {On the $\bbfI$-condition}.
\newblock {\em {Israel Journal of Mathematics}}, {\bf 48}:148--158, 1984.

\bibitem[GoSh 511]{GoSh:511}Martin Goldstern and Saharon Shelah.
\newblock {Forcing the failure of CH by adding a real, II}.
\newblock {\em {in preparation}}.

\bibitem[LrSh 746]{LrSh:746}Paul Larson and Saharon Shelah.
\newblock {Bounding by canonical functions, with CH}.
\newblock {\em {Journal of Mathematical Logic}}, {\bf submitted}.
\newblock math.LO/0011187.

\bibitem[RoSh 470]{RoSh:470}Andrzej Ros{\l}anowski and Saharon Shelah.
\newblock {Norms on possibilities I: forcing with trees and creatures}.
\newblock {\em {Memoirs of the American Mathematical Society}}, {\bf 141}(671),
  1999.
\newblock math.LO/9807172.

\bibitem[RuSh 117]{RuSh:117}Matatyahu Rubin and Saharon Shelah.
\newblock {Combinatorial problems on trees: partitions, $\Delta$-systems and
  large free subtrees}.
\newblock {\em {Annals of Pure and Applied Logic}}, {\bf 33}:43--81, 1987.

\bibitem[Sh 656]{Sh:656}Saharon Shelah.
\newblock {NNR Revisited}.
\newblock {\em {Journal of Symbolic Logic}}, {\bf submitted}.
\newblock math.LO/0003115.

\bibitem[Sh:e]{Sh:e}Saharon Shelah.
\newblock {\em {Non--structure theory}}, accepted.
\newblock {Oxford University Press}.

\bibitem[Sh:b]{Sh:b}Saharon Shelah.
\newblock {\em {Proper forcing}}, volume 940 of {\em {Lecture Notes in
  Mathematics}}.
\newblock {Springer-Verlag, Berlin-New York, xxix+496 pp}, 1982.

\bibitem[Sh 136]{Sh:136}Saharon Shelah.
\newblock {Constructions of many complicated uncountable structures and Boolean
  algebras}.
\newblock {\em {Israel Journal of Mathematics}}, {\bf 45}:100--146, 1983.

\bibitem[Sh:f]{Sh:f}Saharon Shelah.
\newblock {\em {Proper and improper forcing}}.
\newblock {Perspectives in Mathematical Logic}. {Springer}, 1998.

\bibitem[Sh 587]{Sh:587}Saharon Shelah.
\newblock {Not collapsing cardinals $\leq\kappa$ in $(<\kappa)$--support
  iterations}.
\newblock {\em {Israel Journal of Mathematics}}, {\bf 136}:29--115, 2003.
\newblock math.LO/9707225.

\end{thebibliography}
