The strong Haken theorem via sphere complexes

We give a short proof of Scharlemann's Strong Haken Theorem for closed 3-manifolds (and manifolds with spherical boundary). As an application, we also show that given a decomposing sphere R for a 3-manifold M that splits off an S² S¹ summand, any Heegaard splitting of M restricts to the standard Heegaard splitting on the summand.