I understand that this is the difficult direction of Morera's proof, applied to disks, rather than triangles.
However, the trick of defining $$F(z)=\int_{\gamma(t)}f(w)\,dw, \text{ with } \gamma(t)=tz+(1-t)z_0$$
No longer works, as this function is no longer path invariant and therefore not well defined, although the particulars of this are not completely clear to me. Should I think of this definition of $F$ to be on cosets of $\mathbb{C}$ quotiented by paths in some way?
Any tips on how to approach this proof would be much appreciated.