Deformation Theorem without a point

Question

So I essentially have to prove the deformation theorem, but in this case the domain $D$ contains a point $z_0$ where $f$ may or may not be holomorphic. One path is $\gamma_1$ a circle of radius $r$ around $z_0$ and another path $\gamma_2$ which is the path in which $D$ is bounded by. I need to show that $\int_{\gamma_1}f(z)dz=\int_{\gamma_2}f(z)dz$ and the hint is to use Cauchy's Theorem. Any hints will be appreciated!