For $C$ a simple closed contour in the counterclockwise direction and $C_1$, $C_2$, $C_3$, $C_4$ are subsets in $C$ all in the counterclockwise direction, use the Cauchy-Goursat theorem to prove that:
If $f(z)$ is holomorphic on $C$, $C_1$, $C_2$, $C_3$, $C_4$ and throughout the multiple connected domain consisting of all points inside $C$ and exterior to each $C_k$, $k =1, 2, 3, 4$ then $$\int_C f(z) dz=\int_{C_1} f(z) dz+\int_{C_2} f(z) dz+\int_{C_3} f(z) dz+\int_{C_4} f(z) dz$$
So far I know that if a function $f$ is holomorphic at all points interior and on a simple closed contour $C$, then $$\int_C f(z) dz=0$$ I do not know how to go about doing this graphical proof.