I'm trying to understand the proof of Theorem 4.23 (Case 1) in Allen Hatcher's Algebraic Topology.
We have a map $f$, for which $f^{-1}(\Delta ^{n+1})$ is a finite union of convex polyhedra, on each of which $f$ is the restriction of a linear surjective map from $\mathbb{R}^{i}$ to $\mathbb{R^{n+1}}$.
And the implication I don't get is this: "For a $q \in \Delta^{n+1}$, $f^{-1}(q)$ is a finite union of convex polyhedra of dimension $\leq i - n -1 $, since $f^{-1} : (\Delta ^{n+1})$ is a finite union of convex polyhedra on each of which $f$ is the restriction of a linear surjection $\mathbb{R}^{i} \rightarrow \mathbb{R}^{n+1}$."
This is as follows. Let $f^{-1}(\Delta^{n+1})=\bigcup_{j=1}^k D_j$, where $D_j$ is a convex polyhedron in $\mathbb{R}^i$. And let $f|_{D_j}=f_j|_{D_j}$, where $f_j:\mathbb{R}^{i}\to \mathbb{R^{n+1}}$ is a linear surjection. Then $f^{-1}(q)=\bigcup_{j=1}^k (D_j\cap f_j^{-1}(q))$. Note that $f_j^{-1}(q)$ is a subspace of dimension $i-n-1$, and hence the intersection $(D_j\cap f_j^{-1}(q))$ is a convex polyhedron of dimension $\le i-n-1$. QED.