Corollary 2

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

The proof is analogous to that of Corollary 1.