Change of variables for multiple integrals. Fact used in proof

Question

This is a part of the proof of The Change of Variables theorem in Spivak's "Calculus on Manifolds". I don't understand how that Remark in the end follows from (1). I tried to plug in $g^{-1}$, but that doesn't seem to lead me anywhere.

Answer 1

Actually, it seems like the problem is that the statement "This follows from (1) applied to $g^{-1}$" is a little inaccurate here. The remark can be obtained like this. Since $\{V_{\alpha}\}$ covers $g(A)$ and $g$ is 1-1, $\{g^{-1}(V_{\alpha})\}$(which are well-known to be open as image of inverse of 1-1 function) covers $A$. And we know that for $V\in \{V_{\alpha}\}$(and thus for $g^{-1}(V) \in \{g^{-1}(V_{\alpha})\}$):

$$\int_Vf=\int_{g^{-1}(V)} (f\circ g)|det {g'}|$$
which is nothing but statement (1) with $g(U)=V$ and $g^{-1}(V)=U$.