  Following Haken and Casson-Gordon, it was shown in [Sc] that given a reducing
sphere or boundary-reducing disk E in a Heegaard split manifold M, the Heegaard
surface T can be isotoped so that it intersects E in a single circle. Here we
show that when this is achieved by two different positionings of T, one can be
moved to the other by a sequence of
  1) isotopies of T rel E
  2) pushing a stabilizing pair of T through E and
  3) eyegelass twists of T.
  The last move is inspired by one of Powell's proposed generators for the
Goeritz group.

Geometric Topology