  Suppose M is a compact orientable irreducible 3-manifold with Heegaard
splitting surfaces P and Q. Then either Q is isotopic to a possibly stabilized
copy of P or the Hempel distance of the splitting P is no greater than twice
the genus of Q.
  More generally, if P and Q are bicompressible but weakly incompressible
connected closed separating surfaces in M then either a) P and Q can be
well-separated or b) P and Q are isotopic or c) the Hempel distance of P is no
greater than twice the genus of Q.
