Corollary 3

Let B and $B'$ be topological spaces, let T be a simply connected and locally connected topological space. Let $f: B'\times T\rightarrow B$ be a continuous mapping and let E be a covering of B. Given a point $t$ of T and a continuous lifting $g_t: B'\times \{t\} \rightarrow E$ of $f|B'\times \{t\}$ to E, there exists a unique continuous lifting $g$ of $f$ to E extending $g_t$.

In view of prop. 3 of I, p. 9, it is a matter of proving that every continuous section of $f^*(E)$ over $B'\times \{t\}$ extends uniquely to a continuous section of $f^*(E)$. Now this follows from cor. 2 applied to the coverings $(B'\times T$, Id$_{B'\times T})$ and $f^*E$ of $B'\times T$.