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.
