Proposition 2

Let B be a paracompact topological space and let $(E, p)$ be a locally trivial fibre space over $B\times \mathbf{I}$. Put $E_1=\overset{-1}{p}(B\times \{1\})$ and denote by $p_1: E_1\rightarrow B$ the mapping pr$_1\circ p|E_1$. Then the $B\times \mathbf{I}$-spaces $(E, p)$ and $(E_1\times \mathbf{I}, p_1\times$ Id$_{\mathbf{I}})$ are isomorphic.

Let us first prove two lemmas.