Corollary 1

Let X and Y be topological spaces and let $x$ be a point of X. Let $f_0$ and $f_1$ be continuous mappings of X into Y. If there exists a pointed homotopy at $x$ joining $f_0$ to $f_1$, one has $\pi_1(f_0, x) =$ $\pi_1(f_1, x)$.

Let $\sigma$ be a pointed homotopy at $x$ joining $f_0$ to $f_1$. With the notation of Prop. $3,\delta$ is the class of a constant path with image $f_0(x) =$ $f_1(x)$. The assertion follows.