# Understanding Difference Between Cauchy-Goursat and Related Theorem

I am reading Brown and Churchill's Introductory complex analysis book, which states the Cauchy-Goursat theorem as follows: If a function $$f$$ is analytic at all points interior to and on a simple closed contour $$C$$, then $$\int_{C} f(z) dz =0$$ I understand this result and its proof just fine. However, there is a second theorem which claims the following: If a function $$f$$ is analytic throughout a simply connected domain $$D$$, then $$\int_{C} f(z) dz =0$$ for every closed contour $$C$$ lying in $$D$$. So in this second theorem we don't $$C$$ to be simple due to the fact that $$D$$ is simply connected. However, if $$f$$ is analytic at all points interior to and on a closed contour $$C$$, isn't the interior of the closed contour $$C$$ a simply-connected domain by default?

To be more precise, if I rewrite the first theorem as: If a function $$f$$ is analytic at all points interior to and on a closed contour $$C$$, then $$\int_{C} f(z) dz =0$$ is this not true? I don't see why we require $$C$$ to be simple, because even if $$C$$ intersects itself, we can just treat our non-simple closed contour as a union of simple closed contours, and those integrals are all $$0$$.