I am reading Hatcher, theorem 2.10.
( http://www.math.cornell.edu/~hatcher/AT/ATpage.html page 112 )
I mostly understand the proof, but am having trouble verifying that (except for the two), the terms with $i = j$ cancel. Why should the restriction of the homotopy to the faces $\sigma [v_0 , \ldots \hat v_i w_i \ldots w_n]$ and $\sigma [v_0 , \ldots v_i \hat w_i \ldots w_n]$ be equal? That doesn't seem to make sense - since these are definitely different subsets of $X \times I$. Or maybe they're not - I don't know. I'm finding the decomposition of $\Delta^n \times I$ into simplices difficult to visualize (too many lines for me in dimension three - are any tricks for seeing this more clearly? Or maybe a good picture...).