Corollary 2

Under the hypotheses of prop. 8, let $(E, p)$ and $(E', p')$ be coverings of $B\times T$ and let $k: E_t\rightarrow E'_t$ be a B-morphism. There exists a unique $(B\times T)$-morphism $\widetilde{k}: E\rightarrow E'$ which extends $k$. If $k$ is a B-isomorphism, $\widetilde{k}$ is a $(B\times T)$-isomorphism.

By proposition 8, one may suppose that there exist coverings F and $F'$ of B such that E and $E'$ are respectively the $(B\times T)$-spaces $F\times T$ and $F'\times T$. One then has $E_t= F\times \{t\},E'_t= F'\times \{t\}$ and the mapping $k$ is written $(x, t)\mapsto (k'(x), t)$, where $k'$ is a B-morphism of F into $F'$. The mapping $k'\times$Id$_T$ is a $B\times T$-morphism extending $k$; it is an isomorphism if $k$ is one.

Let $\widetilde{k}: E\rightarrow E'$ be a $(B\times T)$-morphism extending $k$. Let $x$ be a point of F. Denote by $q$ the projection of F and put $b=q(x)$. The mappings $u\mapsto \widetilde{k}(x, u)$ and $u\mapsto (k'(x), u)$ are liftings in $E'$ of the mapping $u\mapsto (b, u)$ of T into $B\times T$. They coincide at $t$. Since the space T is connected, they are equal. Thus, $\widetilde{k}$ is the $(B\times T)$-morphism $k'\times$ Id$_T: F\times T\rightarrow F'\times T$.