Lemma 5

The morphism of sheaves $\alpha$ is an isomorphism.

Suppose first that the principal fibre spaces E and $E'$ are both trivializable; choose sections $i: B\rightarrow E$ and $i': B\rightarrow E'$ of them. Then there exists a unique continuous mapping $\theta$ from M into $B\times G$ such that one has

for $b\in B,g\in G,g'\in G$ and $f\in$ F; it is an isomorphism of B-spaces. Let $\varphi$ be an isomorphism of principal fibre spaces from E onto $E'$; there exists a unique continuous mapping $\gamma : B\rightarrow G$ such that $\varphi (i(b)) =i'(b)\cdot \gamma (b)$ for $b\in B$. The image of $\varphi$ under the mapping $\alpha_B$ is the mapping $b\mapsto \theta^{-1}(b, \gamma (b))$ from B into M. This implies that $\alpha_B$ is a bijection.

Consequently, $\alpha_U$ is a bijection for every open set U of B such that the principal fibre spaces $E_U$ and $E'_U$ are trivializable. By corollary 2 of I, p. 55, this implies that $\alpha$ is an isomorphism of sheaves.