Theorem 1(Cauchy Theorem for star shaped domains)
Let $\Omega \subseteq \mathbb{C}$, be a star shaped domain. If $f$ is holomorphic on $\Omega$, then it possesses a primitive and $\int_{\gamma}f(z)dz=0$, for every closed curve $\gamma$ in $\Omega$.
Theorem 2(Cauchy Theorem homology version)
Let $\Omega \subseteq \mathbb{C}$ be an open set and let $f$ be holomorphic on $\Omega$. If $\Gamma$ is a cycle contained in $\Omega$, such that $\Gamma$ is 0-homologous, then $\int_{\Gamma}f(z)dz=0$.
I know that the second Theorem is the more general one. So there is an exercise that asks us to formally derive the Star shaped version of Cauchy's theorem by using the homology version. Since I am new to the homology version of Cauchy Theorem, I am not sure if what I did was correct.
My attempt: A cycle $\Gamma$ in $\Omega$ is called 0-homologous if $Ind_{\Gamma}(a)=0 \forall a \notin \Omega $. If $\Gamma=\gamma$, where $\gamma$ is some star shaped curve, then $Ind_{\Gamma}(a)=0 \forall a \notin \Omega$, and thus by the homology version we get $\int_{\gamma} f(z)dz=0$.
Question: Is there a better way of doing this? (I kind of have the feeling that my argumentation is somewhat handwavy). How do I argue for the existence of a primitive?