# Constant Rank Theorem for Manifolds with Boundary

I'm trying to answer problem 4-3 from Lee's Introduction to Smooth Manifolds, 2nd edition. The problem says:

Formulate and prove a version of the rank theorem for a map of constant rank whose domain is a smooth manifold with boundary.

Lee himself gave a hint in another question. Here is what I have so far:

I'm assuming I have a smooth map of constant rank $$F: \mathbb{H}^m\rightarrow\mathbb{R}^n$$. Let's say $$\mathrm{rank}(F)=k$$. Then I extend $$F$$ to a smooth map $$\tilde{F}:\mathbb{R}^m\rightarrow\mathbb{R}^n$$. I can shrink the domain to a small enough neighborhood of $$\mathbf{0}$$, $$U$$, such that there is a projection $$\pi:\mathbb{R}^n\rightarrow\mathbb{R}^k$$ with $$\pi\tilde{F}_{|U}$$ a submersion. The actual constant-rank theorem then says we have charts $$(A,\alpha)$$ and $$(B,\beta)$$ with $$\beta\circ\pi\tilde{F}\circ\alpha^{-1}:\alpha(A)\rightarrow\beta(B)$$

If I write the coordinates of $$\mathbb{H}^m$$ as $$(x,y)$$ and the coordinates of of $$\alpha(A)$$ as $$(a,t)$$, then this map looks like $$\beta\circ\pi\tilde{F}\circ\alpha^{-1}(a,t) = a$$

I can switch the order so the projection is the last map applied: $$\pi\circ(\beta\times\mathrm{id})\circ\tilde{F}\circ\alpha^{-1}\equiv\beta\circ\pi\tilde{F}\circ\alpha^{-1}$$

Which finally means $$(\beta\times\mathrm{id})\circ\tilde{F}\circ\alpha^{-1}(a,t)=(a,S(a,t))$$ Where $$S:\alpha(A)\rightarrow\mathbb{R}^{(n-k)}$$.

This is pretty similar to how the constant-rank theorem is proved, but now I'm stuck. The map above, restricted to $$\alpha(A\cap\mathbb{H}^m)$$, has rank $$k$$. But that does not mean $$S$$ is independent of $$y$$. That's because $$\alpha$$ is not necessarily a boundary chart on $$A\cap\mathbb{H}^m$$. I also don't see any way to make $$\alpha$$ a boundary chart (something similar is proved in Lee's book, but crucially uses that $$F$$ is an immersion).

Can someone give me a hint as to how to finish this? Ideally I'd get to a place where (with possibly different charts) $$\tilde{\beta}\circ F\circ\tilde{\alpha}^{-1}(a,t) = (a,0)$$ and $$\tilde{\alpha}$$ is a boundary chart.

Bonus question: where do I use Lee's assumption (from the hint) that $$\ker dF_p\not\subseteq T_p\partial\mathbb{H}^m$$?