Corollary 2

Let B and $B'$ be topological spaces, let G be a topological group, and let E be a B-principal fibre space with group G. 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 $B'$-principal fibre spaces $f_0^*E$ and $f_1^*E$ are isomorphic.