The reducibility of surgered 3-manifolds and homology 3-spheres

Suppose K(r) is reducible, by Theorem 3 of Gordon and Luecke, K(r) has a lens space as a connected summand. We will show that if the order of the fundamental group of the lens space is r then K(r) is the connected sum of L(r,.) and an irreducible homology 3-sphere.