Corollary 3

Let B be a connected and locally connected topological space, let $(E, p)$ and $(E', p')$ be coverings of B. For every point $b$ of B, the mapping $f\mapsto f_b$ of $\mathscr{C}_B(E'; E)$ into $\mathscr{C}(E'_b; E_b)$ is injective.

Let $b$ be a point of B. Let $f$ and $g$ be B-morphisms from $E'$ into E such that $f_b=g_b$. The set of points $a$ of B such that $f_a=g_a$ is open and closed in B (Corollary 2) and contains $b$. It is therefore equal to B, and $f$ is equal to $g$.