Theorem 37.1:

$H_n(X, A) = Z_n(X, A)/B_n(X, A)$.

We now consider the short exact sequence of complexes induced by the inclusion $i$ and $p$ denoting the projection onto the quotient:

$$\begin{CD} 0@> >> S(A) @> i_{\sharp} >> S(X) @> p >> S(X, A) @> >> 0. \end{CD}$$

Equation (37.1) states that $p$ is a chain map and exactness of this sequence is an easy exercise. Theorem (29.6) now gives