  Suppose T is a Heegaard splitting surface for a compact orientable 3-manifold
M, and S is a reducing sphere for M. In 1968 Haken showed that there is then
also a reducing sphere S* for the Heegaard splitting. That is, S* is a reducing
sphere for M and the surfaces T and S* intersect in a single circle. In 1987
Casson and Gordon extended the result to boundary-reducing disks in M and noted
that in both cases S* is obtained from S by a sequence of operations called
1-surgeries. Here we show that in fact one may take S* = S.
