Proposition 10

Let E, B and Z be topological spaces, $p: E\rightarrow B$ an étale mapping and $f: Z\rightarrow B$ a continuous mapping.

Let $z\in Z$ and $x\in E$ be points such that $f(z) =p(x)$. There exists a neighbourhood W of $z$ in Z and a continuous lifting $g$ of $f$ to E, defined on W, such that $g(z) =x$.

There exists an open neighbourhood V of $p(x)$ in B and a continuous section $s$ of $p$ over V such that $s(p(x)) =x$ (prop. 9). The set

$W =\overset{-1}{f}(V)$ is an open neighbourhood of $z$ and the mapping $g=s\circ (f|W)$ is a continuous lifting of $f$ to E, defined on W and such that $g(z) =x$.