Corollary 1

Let $B'$ be a topological space, let $A'$ be a subspace of $B'$, and let $f': A'\rightarrow$ X be a continuous mapping. Let $v: B\rightarrow B'$ be a continuous mapping such that $v(A)\subset A'$ and $f$ = $f'\circ (v|A)$. Let $w: X\cup_fB\rightarrow X\cup_{f'}B'$ be the unique continuous mapping such that $w\circ \alpha_f=\alpha_{f'}$ and $w\circ \beta_f=\beta_{f'}\circ v$. If $v$ defines a homotopy equivalence of the pair $(B,A)$ onto the pair $(B',A')$, then $w$ defines a homotopy equivalence of the pair $(X\cup_fB, \alpha_f(X))$ onto the pair $(X\cup_{f'}B', \alpha_{f'}(X))$.

Let $v': B'\rightarrow B$ be a continuous mapping, let $\tau : B\times \mathbf{I}\rightarrow B$ be a homotopy fixed on A joining Id$_B$ to $v'\circ v$, and let $\tau ': B'\times \mathbf{I}\rightarrow B'$ be a homotopy fixed on $A'$ joining Id$_{B'}$ to $v\circ v'$. For $a'\in A'$ and $a=v'(a')$, we have the relations $a'=v(a)$ and $\beta_f(v'(a')) =\beta_f(a) =\alpha_f(f(a)) =\alpha_f(f'(a'))$. By Prop. 8, a), there exists a unique mapping $w': X\cup_{f'}B'\rightarrow$ $X\cup_fB$ such that $w'\circ \alpha_{f'}=\alpha_{f'}$ and $w'\circ \beta_{f'}=\beta_f\circ v'$. By Prop. 8, b), there exists a unique homotopy $\eta : (X\cup_fB)\times \mathbf{I}\rightarrow X\cup_{f'}B'$ such that $\eta (\alpha_f(x), t) =\alpha_{f'}(x)$ and $\eta (\beta_f(b), t) =\beta_{f'}(\tau (b, t))$ for $x\in X,b\in B$ and $t\in \mathbf{I}$. Analogously there exists a unique homotopy $\eta ': (X\cup_{f'}B')\times \mathbf{I}\rightarrow$ $X\cup_fB$ such that $\eta '(\alpha_{f'}(x), t) =\alpha_f(x)$ and $\eta '(\beta_{f'}(b), t) =\beta_f(\tau '(b, t))$ for $x\in X,b\in B'$ and $t\in \mathbf{I}$. The mapping from $(X\cup_fB)\times \mathbf{I}$ into $X\cup_fB$ given by $(x, t)\mapsto \eta '(\eta (x, t), t)$ is then a homotopy fixed on $\alpha_f(X)$ joining the identity mapping of $X\cup_fB$ to the mapping $w'\circ w$. Analogously, the mapping from $(X\cup_{f'}B')\times \mathbf{I}$ into $X\cup_{f'}B'$ given by $(x, t)\mapsto \eta (\eta '(x, t), t)$ is a homotopy fixed on $\alpha_{f'}(X)$ joining the identity mapping of $X\cup_{f'}B'$ to the mapping $w\circ w'$. The corollary follows.