Common nonseparating disks in a Heegaard splitting
= Common nonseparating disks in a Heegaard splitting
Two nonseparating compressing disks on opposite sides with the same boundary form an embedded sphere. Cutting each <handlebody> along its disk leaves it connected, so the sphere is nonseparating in the glued <three-manifold>. Capping the two resulting sphere boundaries gives a splitting of one lower genus of the remaining summand; the removed summand is $S^1\times S^2$.