Suppose $f$ is analytic on the complex plane except at $z_1,z_2$, that $\gamma_1$ and $\gamma_2$ are simple closed curves with $z_1,z_2$ in their interiors and $\gamma_1$ and $\gamma_2$ are in the interior of $\gamma$. $\gamma$, $\gamma_1$, $\gamma_2$ are all traveling counterclockwise. Give an argument to show that $\int_{\gamma}f(z)dz = \int_{\gamma_1}f(z)dz + \int_{\gamma_2}f(z)dz$.
I assume that we will use the Cauchy Integral Theorem that if $f$ is analytic in a domain $D$, then any simple closed curve yields an integral of $0$ (very summarized), but we are unsure how to deal with the fact $f$ is not analytic at $z_1$ and $z_2$.