Boundary submanifolds of a boundaryless manifold have half-slice charts

At an interior point, [L1] and [L3] give an ordinary slice chart. At a boundary point, choose boundary coordinates $x$ on $S$ and ordinary ambient coordinates. By [L2], extend the coordinate embedding to a smooth map $\tilde{F}$ on an open subset of ${\mathbb{R}}^k$. Its derivative at the boundary point equals the injective differential from [L1], so after shrinking an invertible $k\times k$ minor stays nonzero and $\tilde{F}$ has constant rank $k$.

Apply [L3] to $\tilde{F}$. In the resulting source coordinates $u=\alpha (x)$ and target coordinates $(u,w)$, it has the form $u\mapsto (u,0)$. The source change $\alpha$ need not preserve the face, but it can be absorbed into the target chart: postcompose that chart with the local diffeomorphism $(u,w)⟼({\alpha }^{-1}(u),w)\mathrm{.}$ In the new target coordinates, $\tilde{F}(x)=(x,0)$ in the original boundary coordinates.

Restricting $x$ back to the source half-space now gives $S=\{x^{k+1}=\cdots =x^n=0,x^k\ge 0\}$ near the boundary point, while step 1.1 gives the ordinary slice at interior points.