Corollary 2

Let B and $B'$ be topological spaces, and let E be a locally trivial fibre space over B. Let $f_0$ and $f_1$ be continuous mappings of $B'$ into B. Suppose that the space $B'$ is paracompact. If the mappings $f_0$ and $f_1$ are homotopic, the locally trivial fibre spaces over $B'$ $f_0^*E$ and $f_1^*E$ are isomorphic.

Let $\sigma : B'\times \mathbf{I}\rightarrow B$ be a homotopy joining $f_0$ to $f_1$ and let us denote by $E'$ the space over $B'\times \mathbf{I}$ $\sigma^*E$; this is a locally trivial fibre space. Let us denote by $i_0$ and $i_1$ the mappings of $B'$ into $B'\times \mathbf{I}$ given by $x\mapsto (x,0)$ and $x\mapsto (x,1)$. By Cor. 1, the fibre spaces over $B'$ $i^*_0E'$ and $i^*_1E'$ are isomorphic. Since $\sigma \circ i_0=f_0$, the space over $B'$ $i^*_0E'$ is identified with $f_0^*E$; analogously, the space over $B'$ $i^*_1E'$ is identified with $f_1^*E$. Consequently, the fibre spaces over $B'$ $f_0^*E$ and $f_1^*E$ are isomorphic, as was to be proved.