This is a question about Spviak's Calculus on Manifolds, page 115.
Let $M$ be a $k$-dimension manifold in $\mathbb{R}^n$ and let $f:W\rightarrow \mathbb{R}^n$ be a coordinate system around $x=f(a)$, where $W$ is some open subset of $\mathbb{R}^k$. We know f'(a) has rank $k$, by definition), so the linear transformation $f_*:\mathbb{R}^k_a\rightarrow \mathbb{R}^n_x$ is 1-1, and the image $f(\mathbb{R}^k_a)$ is $k$-dimensional. If $g:V\rightarrow \mathbb{R}^n$ is another coordinate system, with $x=g(b)$. Spviak claims:
$g_*(\mathbb{R}^K_b) = f_*(f^{-1}\circ g)_*(\mathbb{R}^K_b) = f_*(\mathbb{R}^K_a)$
I realize the chain rule is being used, but I am having trouble following the implications. In particular:
1) I think the chain rule gives the first implication, but I don't see why we can take the derivative of $f^{-1}$, because it is a map between spaces of different dimensions (so the inverse function theorem doesn't apply).
2) The last equality implies that (f^{-1}\circ g)' is the identity. I do not see why this is true (but I do see that $(f^{-1}\circ g)(b)=a$, so the subscript works out).