Proposition 4

Let $p: E\rightarrow B$ be an étale and separated mapping which satisfies the path-lifting property. Let $a$ and $b$ be points of B and let $x$ be a point of E such that $p(x) =a$. Let $c_0$ and $c_1$ be two strictly homotopic paths joining $a$ to $b$ in B. The paths with origin $x$ in E which lift respectively $c_0$ and $c_1$ have the same term and are strictly homotopic.

Let $\sigma :\mathbf{I}\times \mathbf{I}\rightarrow$ B be a strict homotopy joining $c_0$ to $c_1$. For $s\in \mathbf{I}$, let $c'_s$ be the unique path with origin $x$ in E which lifts the path $t\mapsto \sigma (t, s)$. For $(t, s)\in \mathbf{I}\times \mathbf{I}$, put $\sigma '(t, s) =c'_s(t)$; we have $p\circ \sigma '=\sigma$. The mapping $\sigma '$ is constant on $\{0\} \times \mathbf{I}$; by construction, it is continuous on $\mathbf{I}\times \{s\}$ for every $s\in \mathbf{I}$. It is therefore continuous, by I, p. 37, cor. 1 of th. 1. In particular, the mapping from $\mathbf{I}$ into $E_b$ defined by $s\mapsto \sigma '(1, s)$ is a continuous lifting of the constant path with image $b$; it is therefore constant. This proves that the mapping $\sigma '$ is a strict homotopy joining $c'_0$ to $c'_1$.