I'm reading Jeffrey Lee's Manifolds and Differential Geometry section on manifolds with boundary.
In page 50, given an $n$-dimensional manifold $M$, he builds a smooth atlas for $\partial M$ in the following way:
Take $p \in \partial M$, then there is a chart $(U,\mathcal{x})$ such that $\mathcal{x}(p) \in \partial \mathbb{R}^n_{\lambda \geq c}$, so taking the restriction $\mathcal{x}\mid_{U \cap \partial M}$ and composing it with any linear isomorphism $F: \partial \mathbb{R}^n_{\lambda \geq c} \to \mathbb{R}^{n-1}$ we obtain a chart $(U\cap \partial M, F \circ \mathcal{x})$ for $\partial M$. The author says that with this construction we obtain smooth overlap maps but I don't see why:
Given 2 charts $(U, \mathcal{x}), (V, \mathcal{y})$ such that $p \in U\cap V$, we get the overlap map $$(F \circ \mathcal{x})\circ(F \circ \mathcal{y})^{-1} = F \circ (\mathcal{x} \circ \mathcal{y}^{-1})\circ F^{-1} $$
I know this is always a linear isomorphism, but why given any $F$ this map is always smooth?