  Let $U$ be a regular connected affine semi-local scheme over a field $k$. Let
$G$ be a reductive group scheme over $U$. Assuming that $G$ has an appropriate
parabolic subgroup scheme, we prove the following statement. Given an affine
$k$-scheme $W$, a principal $G$-bundle over $W\times_kU$ is trivial if it is
trivial over the generic fiber of the projection $W\times_kU\to U$. We also
simplify the proof of the Grothendieck-Serre conjecture: let $U$ be a regular
connected affine semi-local scheme over a field $k$. Let $G$ be a reductive
group scheme over $U$. A principal $G$-bundle over $U$ is trivial if it is
trivial over the generic point of $U$. We generalize some other related results
from the simple simply-connected case to the case of arbitrary reductive group
schemes.
