The questions come from a paragraph of page 271 of Rudin's "Principles of Mathematical Analysis", as the image below shows.

I completely do not understand what this paragraph is discussing. First, why "$\displaystyle \Phi$ is the same as the 2-chain $\displaystyle \Phi\circ\sigma_1+\Phi\circ\sigma_2$"? which is equivalent to $\displaystyle \int_\Phi \omega=\int_{\Phi\circ\sigma_1}\omega+\int_{\Phi\c irc\sigma_2}\omega$ for all 2-forms $\displaystyle \omega$. Second, why does the equalities at the end of this paragraph hold? Better use definitions and theorems of this book.
This paragraph may contain materials of above paragraphs, as is shown below.

please help me with these questions, thanks!