Let $\mathcal{P}$ be a polyhedron in $\mathbb{R}^n$. I want to show that the face $\mathcal{F}'$ of a face $\mathcal{F}$ of $\mathcal{P}$ is still a face of $\mathcal{P}$.
There are various proofs of this fact for polytopes. For example in Ziegler's "Lectures on Polytopes" proposition $2.3$. I want to show that the same holds for polyhedra. My idea is that I can take a large enough polytope $\mathcal{V}$ that intersects both $\mathcal{F}$ and $\mathcal{F}'$ in some way. At this point I find that $\mathcal{V} \cap \mathcal{F}'$ is a face of $\mathcal{V} \cap \mathcal{P}$, since the latter is a polytope (by the so-called main theorem in Ziegeler's textbook). From this I would want to conclude that $\mathcal{F}'$ is a face of $\mathcal{P}$. Maybe this is a typical problem: if the proof is pointed out somewhere reference would be fine for me as an answer.
EDIT about definitions:
A polyhedron is the intersection of a finite number of closed halfspaces. This is a convex set.
A polytope is the convex hull of e finite number of points.
A face $\mathcal{F}$ of a convex set $\mathcal{P}$ is any set defined by $\mathcal{F}=\mathcal{P}\cap\{ \mathbf{x}: \mathbf{a}\cdot \mathbf{x} - c = 0\},$ where $\mathbf{a}\cdot \mathbf{x} - c \ge 0$ is an inequality valid for all $\mathbf{x} \in \mathcal{P}$