  Let F be an arbitrary local field. Consider the standard embedding of GL(n,F)
into GL(n+1,F) and the two-sided action of GL(n,F) \times GL(n,F) on GL(n+1,F).
  In this paper we show that any GL(n,F) \times GL(n,F)-invariant distribution
on GL(n+1,F) is invariant with respect to transposition.
  We show that this implies that the pair (GL(n+1,F),GL(n,F)) is a Gelfand
pair. Namely, for any irreducible admissible representation $(\pi,E)$ of
(GL(n+1,F), $$dimHom_{GL(n,F)}(E,\cc) \leq 1.$$
  For the proof in the archimedean case we develop several new tools to study
invariant distributions on smooth manifolds.
