In [RS] we use Cerf theory to compare irreducible Heegaard splittings of the same irreducible non-Haken orientable 3-manifold. A critical part of the argument is the observation that any two Heegaard surfaces may be isotoped so that they intersect in a nonempty collection of simple closed curves, each of which is essential in both surfaces. Here we describe an analog to this theorem that applies to the Haken case. An eventual goal, not yet realized here, is a bound for the number of stabilizations needed to make two distinct Heegaard splittings equivalent. Such a bound is found, for the non-Haken case, in [RS].
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