= Sphere surgery preserving nontrivial complementary pieces
Suppose disjoint separating <spheres> cut a <3-manifold> into pieces none of which is a <punctured three-sphere>. Compress one <sphere> along a <disk> disjoint from the others. The old <sphere> and the two new <spheres> enclose a three-times punctured <sphere>. At least one of the two pieces on the compressed side is not a punctured <sphere>; retain that new <sphere> and discard the other. The piece on its opposite side cannot become a punctured <sphere>, since capping its boundaries and applying the <Three-dimensional Schoenflies theorem> would make the original opposite piece a punctured <sphere> as well. Hence the number of <spheres> and the nontriviality condition can be preserved.
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