Proposition 8

Every hyperplane $H$ belonging to $\mathfrak{H}$ is the wall of at least one chamber.

By Prop. 7, (iii), there exists a point $a$ of $H$ that does not belong to any hyperplane $H' \neq H$ of $\mathfrak{H}$; by Prop. 6, there exists a chamber $C$ such that $a \in \overline{C}$; Prop. 4 then shows that $H$ is a wall of $C$.