Slice-chart restrictions form a smooth atlas

Let $\phi :U\to \phi (U)$ and $\psi :V\to \psi (V)$ be slice charts. By [F1], after projecting away the zero normal coordinates, the overlap transition on $S$ is $x\mapsto {\pi }_k\text{\hspace{0.17em}⁣}(\psi \circ {\phi }^{-1}(x,0))$, where ${\pi }_k$ denotes projection onto the first $k$ coordinates. Because $\psi \circ {\phi }^{-1}$ is a smooth map between open Euclidean sets and restriction to $(x,0)$ plus projection are smooth, the restricted transition maps are smooth.

The restricted charts cover $S$ because the slice charts do. Their images are open in ${\mathbb{R}}^k$: indeed $\phi (U)\cap ({\mathbb{R}}^k\times \{0\})$ equals $({\pi }_k(\phi (U)\cap ({\mathbb{R}}^k\times \{0\})),0)$, and the slice condition in [F1] says that every point of this set has an ambient product neighbourhood whose first-factor projection stays inside the image.

By [L1], each ambient chart map $\phi$ is a homeomorphism, so its restriction identifies $U\cap S$ with $\phi (U)\cap ({\mathbb{R}}^k\times \{0\})$. Therefore the restricted charts make $U\cap S$ open exactly when it is open in the subspace topology from [F2]. The atlas therefore generates precisely the subspace topology on $S$.

Steps 1.1-3.1 prove smooth compatibility and the topology claim.