I'm really struggling with the inductive proof of the associativity of free groups, given about halfway down page 6 of this pdf.
The bit I'm not getting is this:
Suppose now that bc involves a cancellation; so c = b-1v starts with b-1. Therefore, (ab)c = wc = w(b-1v) while a(bc) = a(v) = (wb-1)v. Since l(w) + l(v) < l(a) + l(c), the last two terms are equal by induction hypothesis. Hence, the associativity holds when l(b) = 1.
I don't understand what $l(a) + l(c)$ has to do with anything, and why the fact that $l(w) + l(v) < l(a) + l(c)$ means that "the last two terms are equal". I am familiar with proof by induction, but I don't get how it works in this case.
Any help would be greatly appreciated!