Relations between two versions of the rank theorem in Rudin and Lee This a follow-up question to a previous one of mine regarding Rudin's rank theorem:



@Jack Lee gives the following theorem in his Introduction to Smooth Manifolds:



As I understand, Theorem 4.12 should be a generalization of 9.32. But I don't completely understand how one can be translated into another. Here is my question:

How is (66) related to (4.1)?


One can see that $(H,V)$ (or $(H^{-1},V)$ according to different convention) in Theorem 9.32 plays the role of the chart $(U,\varphi)$ in Theorem 4.12. What really puzzles me is that how one can translate the R.H.S of (66) to that of (4.12).
 A: Rudin's theorem stops one step short of constructing the coordinates that I construct. To get my result from his, the first thing to do is to choose bases in $\mathbb R^n$ and $\mathbb R^m$ such that $A$ is given by 
$$
A\big(x^1,\dots,x^n\big) = \big(x^1,\dots,x^r,0,\dots,0\big).
$$
(Theorem B.20 in my Appendix B shows that this can always be done for a linear map of rank $r$.) Then you can choose the complementary subspace $Y_2$ to
be the set of points of the form $\big(0,\dots,0,x^{r+1},\dots,x^n\big)$,
and $P$ is just the coordinate projection 
$$\big(x^1,\dots,x^m\big)\mapsto 
\big(x^1,\dots,x^r,0,\dots,0\big).
$$
With these choices, Rudin's theorem shows that $F\circ H$
can be written as
$$
F\circ H(x) = \big( x^1,\dots, x^r,\varphi^{r+1}\big( x^1,\dots, x^r\big),\dots,
\varphi^m\big( x^1,\dots, x^r\big)\big),
$$
where $\big(\varphi^{r+1},\dots,\varphi^m\big)$ are the coordinate functions of the map $\varphi$.
(The construction up to this point corresponds to formula (4.6) in my proof of the rank theorem.)
To get the coordinates in my version of the theorem, define new coordinates on a neighborhood of $0$ in $\mathbb R^m$ by $y=\Psi(x)$, where
$$
\Psi\big(x^1,\dots, x^m\big) = \big( x^1,\dots, x^r,x^{r+1}-\varphi^{r+1}\big( x^1,\dots, x^r\big),\dots,
x^m-\varphi^m\big( x^1,\dots, x^r\big)\big).
$$
Then a straightforward computation shows that the composite function $\Psi\circ F \circ H$ is given by my formula 4.1.
