Corollary 1

Let X be a normal topological space, let $f$ be a continuous mapping of X into a topological space Z. Let A be a closed subspace of X, let U be a neighbourhood of A in X and let $\sigma : U\times$ $\mathbf{I}\rightarrow Z$ be a homotopy whose term is the mapping $f|U$. There exists a homotopy $\widetilde{\sigma}: X\times \mathbf{I}\rightarrow Z$ whose term is the mapping $f$ and which coincides with $\sigma$ in $A\times \mathbf{I}$.

Put $X'= X$ and let $r: X\times \mathbf{I}\rightarrow$ Cyl($i_U$) be the continuous mapping given by Prop. 1. With the notations introduced in the proof of this proposition, there exists a unique continuous mapping F: Cyl($i_U$)$\rightarrow Z$ such that $F(\alpha (x, t)) =\sigma (x, t)$ for $(x, t)\in U\times \mathbf{I}$ and $F(\beta (x)) =f(x)$ for $x\in$ X (III, p. 238, Prop. 4). The mapping $F\circ r: X\times \mathbf{I}\rightarrow$ Z is a homotopy ; its term maps a point $x\in$ X to $F(r(x,1)) = F(\beta (x)) =f(x)$. If $(x, t)\in A\times \mathbf{I}$, $F(r(x, t)) = F(\alpha (x, t)) =\sigma (x, t)$, therefore this homotopy coincides with $\sigma$ on $A\times \mathbf{I}$.