Exercise 17, § 1

We retain the notation of Exerc. 16 and assume that $W$ is finite. Let $\mathfrak{H}$ be the set of walls of $A$. To any $H \in \mathfrak{H}$, we associate the half $H^+$ of $A$ determined by $H$ containing $C$. Show that the elements of $\mathfrak{H}$ can be numbered in such a way that the map $j \mapsto \bigcap_{i \leq j} H_i^+$ is strictly decreasing on the interval $[1, \operatorname{Card}(\mathfrak{H})]$. (Consider the family $\mathfrak{F}$ of intersections of the sets $H^+$ ordered by inclusion and consider a strictly decreasing sequence $(F_0, \ldots, F_q)$ of elements of $\mathfrak{F}$ of maximal length. For any $H \in \mathfrak{H}$, there exists an $i$ such that $H^+ \supset F_j$ for $j \geq i$ and $H^+ \not\supset F_{i-1}$: show that $F_i = H^+ \cap F_{i-1}$.)