In this page Cauchy Integral Theorem, the three Cauchy integral theorems are presented. The first is about simply connected domains and the second is a generalisation of the 1st that includes removable singularities. The third is the Cauchy Goursat theorem regarding triangles or rectangles. At one point (just before the 3rd theorem) it writes:
"Goursat's argument makes use of rectangular contour (many authors use triangles though), but the extension to an arbitrary simply-connected domain is relatively straight-forward"
How is that extension straight-forward? What is the proof of that? In other words how can I prove the 1st theorem for the 3rd (using the triangle version of Cauchy Goursat prefferably) and then how do I prove that the 1st theorem implies the 2nd? ($3\Rightarrow 1\Rightarrow 2$). It would be preferrable if someone linked a pdf file containg the proofs I am asking for. Also note that I am looking for elementary self contained answers that don't invole a lot of topological notions but basic topology. Thank you for your answers.