  Let $X$ be a smooth complete curve, and let $Bun_n$ be the moduli stack of
rank $n$ vector bundles on $X$. Let $E$ be a local system on $X$.
  In a recent paper of E.Frenkel, K.Vilonen and the author, it was shown that
the vanishing of a certian functor $Av_E^d$ acting from the category $D(Bun_n)$
to itself, implies the geometric Langlands conjecture.
  In this paper we establish the required vanishing result.
  Our proof works for sheaves with char=0 coefficients, or with torsion
coefficients when the parameter $d$ is less than the characteristic.
