Let $P$ and $V$ be, respectively, a bounded full-dimensional polytope and an $m$-dimensional affine subspace of $\mathbb R^n$, and assume $P \cap V$ is non-empty.
Q1. Could you provide me with a reference where the following (or even better a generalization of it) is explicitly stated and proved?
Lemma. There exists a face $F$ of $P$ such that $\dim(F) \le n-m$ and $F \cap V \ne \emptyset$.
This is certainly well-known for everyone familiar with polyhedral theory (I'm stressing it due to the tone of certain comments below), but I wouldn't say that it is trivial (whatever this may mean), and I need it as an intermediate step in the proof of a certain result in zero-sum theory. Yet, I would like not to include it in my paper, since otherwise I feel that, in the interest of my two readers, I should also introduce some lexicon and so on. That's why I'm looking for a reference.
In particular, I gave a look at A. Brøndsted's An Introduction to Convex Polytopes and G. M. Ziegler's Lectures on Polytopes, but I don't seem to find it. Thank you in advance for any help.
Q2 (added later). Is there any "non-trivial" generalization of the lemma to a larger class of convex sets than the one of polytopes or to something different from an affine subspace?