I'm having trouble understanding a certain part of the proof for the homological version of the Cauchy theorem. I'm going through a proof that given in this link:

starting on page 9.
At the end of page 10, it states that g(w,z)= (f(w)-f(z))/w-z, since z is not in the trace of the path.

Looking at the definition of g(w,z) as described earlier, all it states for w is that it exists in Omega - so why does this mean that z is not equal to w?

I've been poring over this for quite some time to no avail, so any help would be much appreciated! Thanks