Proposition 3

Let G be a topological group, let B be a paracompact topological space and let $(E, p)$ be a principal fibre space of group G and base $B\times \mathbf{I}$. Put $E_1=\overset{-1}{p}(B\times \{1\})$ and let $p_1: E_0\rightarrow B$ be the mapping pr$_1\circ p|E_1$. Then, $(E, p)$ and $(E_1\times \mathbf{I}, p_1\times$ Id$_{\mathbf{I}})$ are isomorphic principal fibre spaces of group G and base $B\times \mathbf{I}$.

Let F be the topological space G endowed with the left operation of the group $G\times G$ given by $(g, g')\cdot f=g'f g^{-1}$. Let $(M, q)$ be the locally trivial fibre space of base $B\times \mathbf{I}$ and fibre type F associated with the principal fibre space $E\times_{B\times\mathbf{I}}(E_1\times \mathbf{I})$ of group $G\times G$. Put $M_1=\overset{-1}{q}(B\times \{1\})$ and $q_1=$ pr$_1\circ p|M_1$; the B-space $(M_1, q_1)$ is identified with the locally trivial fibre space of fibre type F associated with $E_1\times_BE_1$. By lemma 4, where one takes for principal fibre spaces E and $E'$ both equal to $E_1$, the B-space $M_1$ has a section. Since the $B\times \mathbf{I}$-spaces $(M, q)$ and $(M_1\times \mathbf{I}, q_1\times$Id$_{\mathbf{I}})$ are isomorphic (IV, p. 440, prop. 2), the $B\times \mathbf{I}$-space $(M, q)$ has a section, which implies that the principal fibre spaces of group G, E and $E_1\times \mathbf{I}$, are isomorphic.