Are these version of Cauchy’s Theorem (complex analysis) equivalent?

I am looking through different complex analysis texts and comparing their statement and proofs of Cauchy’s theorem. One version I found is:

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A second version of Cauchy’s Theorem is:
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Now I was wondering if the corollary of the second version is in fact equivalent to the first version. The biggest difference I see is that the first version makes it clear that we need f to be analytic inside the closed curve whereas the second version doesn’t say that. Is one of the four criterions equivalent to saying the interior of the curve is analytic?