Proposition 5

A homotopism is a homologism; a morphism composed of homotopisms is a homotopism. A morphism homotopic to a homotopism is a homotopism.

Let $f : C \to C'$ and $f_1 : C' \to C''$ be homotopisms of complexes, $f' : C' \to C$ and $f'_1 : C'' \to C'$ morphisms reciprocal up to homotopy. We have

and analogously $\mathrm{H}(f) \circ \mathrm{H}(f') = 1_{\mathrm{H}(C')}$, so $\mathrm{H}(f)$ is bijective and $f$ is a homology isomorphism. On the other hand, $(f' \circ f'_1) \circ (f_1 \circ f)$ is homotopic to $f' \circ 1_C' \circ f$ (prop. 4), hence to $1_C$; analogously, $(f_1 \circ f) \circ (f' \circ f'_1)$ is homotopic to $1_{C''}$ and $f_1 \circ f$ is a homotopy equivalence. Finally, if $g : C \to C'$ is a morphism homotopic to $f$, $f' \circ g$ is homotopic to $f' \circ f$, hence to $1_C$; analogously, $g \circ f'$ is homotopic to $f \circ f'$, hence to $1_{C'}$ and $g$ is a homotopy equivalence.