  A projective scheme $X$ is called `quadratic' if $X$ is scheme-theoretically
cut out by homogeneous equations of degree 2. Furthermore, we say $X$ satisfies
`property $\textbf{N}_{2,p}$' if it is quadratic and the quadratic ideal has
only linear syzygies up to first $p$-th steps. In the present paper, we compare
the linear syzygies of the inner projections with those of $X$ and obtain a
theorem on `embedded linear syzygies' as one of our main results. This is the
natural projection-analogue of `restricting linear syzygies' in the linear
section case, \cite{EGHP1}. As an immediate corollary, we show that the inner
projections of $X$ satisfy property $\textbf{N}_{2,p-1}$ for any reduced scheme
$X$ with property $\textbf{N}_{2,p}$.
  Moreover, we also obtain the neccessary lower bound $(\codim X)\cdot p
-\frac{p(p-1)}{2}$, which is sharp, on the number of quadrics vanishing on $X$
in order to satisfy $\textbf{N}_{2,p}$ and show that the arithmetic depths of
inner projections are equal to that of the quadratic scheme $X$. These results
admit an interesting `syzygetic' rigidity theorem on property
$\textbf{N}_{2,p}$ which leads the classifications of extremal and next to
extremal cases.
  For these results we develope the elimination mapping cone theorem for
infinitely generated graded modules and improve the partial elimination ideal
theory initiated by M. Green. This new method allows us to treat a wider class
of projective schemes which can not be covered by the Koszul cohomology
techniques, because these are not projectively normal in general.
