A complex is a non-empty finite set $\mathcal{C}$ of polytopes of $\mathbb{R}^n$ such that for all $P, Q \in \mathcal{C}$, $P \cap Q$ is either empty or simultaneously a face of both $P$ and $Q$. A face of $\mathcal{C}$ is a subset $F \subset |\mathcal{C}|$ that is a face of one of the $P \in \mathcal{C}$.
Show that if $P$ is a polytope of $\mathbb{R}^n$ of dimension $k > 0$, the set of its faces of dimension $k-1$ forms a complex.