Proposition 5

Let $C$ be a non-empty subset of $E$. Assume that there exists a subset $\mathfrak{H}'$ of $\mathfrak{H}$ with the following properties:

a) For any $H \in \mathfrak{H}'$, there exists an open half-space $D_H$ bounded by $H$ such that $C = \bigcap_{H \in \mathfrak{H}'} D_H$.

b) *The set C does not meet any hyperplane belonging to $\mathfrak{H} - \mathfrak{H}'$. Under these conditions, C is a chamber defined by $\mathfrak{H}$ in E, and $D_H = D_H(C)$ for all $H \in \mathfrak{H}$.

Properties a) and b) show that C is a convex subset of U; hence there is a chamber $C'$ with $C \subset C'$. Since $C \subset D_H$, we have $D_H = D_H(C)$ for all H in $\mathfrak{H}'$, hence $C = D_{\mathfrak{H}'}(C) \supset D_{\mathfrak{H}}(C)$ since $\mathfrak{H}' \subset \mathfrak{H}$; we have $D_{\mathfrak{H}}(C) = C'$ by (6), hence $C \supset C'$. Finally therefore, $C = C'$.