Given a Heegaard splitting and
an incompressible surface S and a Heegaard splitting of an irreducible manifold, I
shall use a generalization of Haken’s lemma proved by Kobayashi in order to define a
pair of simple closed curves on the splitting surface such that each bounds a disc in
one of the handlebodies of the splitting. By modifying the proof of Kobayashi’s
lemma, I shall show that the sequence of boundary compressions used to isotope S
places a bound on the distance between these two simple closed curves in the
complex of curves. This will then place a bound on the distance of the Heegaard
splitting.