Corollary 1

Let A be a topological space, B, $B'$ A-spaces. Let E and $E'$ be locally trivial fibre spaces ( resp. coverings) with bases B and $B'$; let $p$ and $p'$ denote their projections. The $B\times_AB'$-space $E\times_AE'$ obtained from $p\times p'$ by passing to the subspaces is then a locally trivial fibre space ( resp. a covering). It is trivializable if E and $E'$ are.

Since the set $E\times_AE'$ is the inverse image of $B\times_AB'$ by the mapping $p\times p': E\times E'\rightarrow B\times B'$, the corollary follows from proposition 2 and remark 1 (I, p. 68).