Proof:

We shall only prove (29.7). It suffices to check these on singular simplices. So let $\sigma:\Delta_n\longrightarrow X$ be a singular $n$ simplex in $X$. Using (29.4) we get

$$(f_{\sharp}\circ \partial _n)\sigma = f_{\sharp}\Big( \sum_{i = 0... ...gma) = \partial _n(f_{\sharp}(\sigma)) = (\partial _n\circ f_{\sharp})\sigma.$$