  We first prove that, infinitely many pairs of trivial knot diagrams that are
transformed into each other by applying Reidemeister moves I and III are NOT
transformed into each other by a sequence of the Reidemeister moves I that
increase the number of crossings, followed by a sequence of Reidemeister moves
III, followed by a sequence of the Reidemeister moves I that decrease the
number of crossings. To create a simple sequence between link diagrams that are
transformed into each other by applying finitely many Reidemeister moves I and
III, we prove that the link diagrams are always transformed into each other by
applying an I-generalized ordered sequence.
