Lemma 1

Let Y be a topological space and let $\varphi :\mathbf{I}\times \mathbf{I}\rightarrow Y$ be a continuous mapping. For $t\in \mathbf{I}$, put $c_0(t) =\varphi (t,0),c_1(t) =\varphi (t,1)$, $d_0(t) =\varphi (0, t)$ and $d_1(t) =\varphi (1, t)$. The paths $c_0*d_1$ and $d_0*c_1$ are strictly homotopic.

Let $c$ denote the path in $\mathbf{I}\times \mathbf{I}$ obtained by juxtaposing the paths $t\mapsto (t,0)$ and $t\mapsto (1, t)$; let $d$ denote the path in $\mathbf{I}\times \mathbf{I}$ obtained by juxtaposing the paths $t\mapsto (0, t)$ and $t\mapsto (t,1)$. The paths $c$ and $d$ have the same origin $(0,0)$ and the same end $(1,1)$. The mapping $(t, s)\mapsto$ $(1-s)c(t)+sd(t)$ of $\mathbf{I}\times \mathbf{I}$ into $\mathbf{I}\times \mathbf{I}$ is a strict homotopy joining $c$ to $d$. One has $c_0*d_1=\varphi \circ c$ and $d_0*c_1=\varphi \circ d$; these two paths are therefore strictly homotopic.