Fundamental Theorem of Complex Calculus

Hello everyone, I was having trouble with the proof of this and I was wondering if anyone could help me!

Let be analytic (where S is an open set), and such that if is a piecewise continuously differentiable path in S, then if f' is continuous on (which is always true anyway), and

then

PROOF: Write as , where each is a simple smooth path, with

Then now here I have a problem because it says this line is equal to

by the fundamental theorem of calculus for REAL functions. however is not real valued is it? so why can i use it here?