Boundary reducible handle additions on simple 3-manifolds

Let M be a simple manifold, and F be a component of .M of genus two. For a slope . on F, we denote by M(.) the manifold obtained by attaching a 2-handle to M along a regular neighborhood of . on F. In this paper, we shall prove that there is at most one separating slope . on F such that M(.) is .-reducible.