  Let R be a semi-local regular domain containing an infinite perfect field k,
and let K be the field of fractions of R. Let G be a reductive semi-simple
simply connected R-group scheme such that each of its R-indecomposable factors
is isotropic. We prove that for any Noetherian affine scheme A over k, the
kernel of the map of etale cohomology sets H^1(A\times_k R,G)-> H^1(A\times_ k
K,G), induced by the inclusion of R into K, is trivial. If R is the semi-local
ring of several points on a k-smooth scheme, then it suffices to require that k
is infinite and keep the same assumption concerning G.
