I am referring to the proof given in Stein's textbook on complex analysis. He assumes the disk is centered at the origin and given a point $z$ in this disk, he considers the polygonal line $\gamma_z$ that joins $0$ to $z$, first by moving horizontally from $0$ to Re($z$), then vertically from Re($z$) to z. It consists of (at most) 2 segments. Then, he defines $$F(z) = \int_{\gamma_z} f(w)dw.$$
The point I can't understand is the following: "The choice of $\gamma_z$ gives an unambiguous definition of the function $F (z)$. I understood the remainder of the proof. Thanks for the feedback!
Alternatively, one might define $$F(z) := \int_{\eta_z} f(w) \,dw ,$$ where $\eta_z$ is any curve in the disc $\Bbb D$ from $0$ to $z$, but in order for this to be well-defined, we'd need to check that the integral doesn't depend on the choice of $\eta_z$. (It turns out that this is the case, since $\Bbb D$ is simply connected and $f$ is holomorphic, the Cauchy Integral Formula implies that the definitions actually coincide.) Specifying for each $z \in \Bbb D$ a unique path means that there are no choices to be made in our definition, which means that $F(z)$ is automatically well-defined.