Ok, I know this the statement "if A#B is homeomorphic to C#B, then A and C are homeomorphic" is not valid for n-manifolds A, B, and C if B is non-orientable.
Ex) T#P is homeomorphic to (P#P)#P, where T is the torus and P is the projective plane. However, T is not homeomorphic to P#P.
So, my question is: if we restrict B to being an orientable n-manifold, is the statement "if A#B is homeomorphic to C#B, then A and C are homeomorphic" valid for any n-manifolds A and C, be they orientable or non-orientable?