Proof:

Taking $\sigma \in Z_n(X)$ in (33.8), the term $L_{n-1}(\partial \sigma)$ drops out and we immediately see that the cycles $f_{\sharp}(\sigma)$ and $g_{\sharp}(\sigma)$ differ by a boundary. The proof is complete.

We see that equation (33.7) is the algebraic analogue of homotopy of continuous maps. As this phenomenon would recur often, we give a formal definition and a name for it.