Proposition 6

Let $(P_s)_{s \in S}$ be a family of subsets of $W$ satisfying (C) and the following conditions:

(A') $1 \in P_s$ for all $s \in S$. (B') The sets $P_s$ and $sP_s$ are disjoint for all $s \in S$.

Then, $(W, S)$ is a Coxeter system and $P_s$ consists of the elements $w$ of $W$ such that $l(sw) > l(w)$.

Let $s \in S$ and $w \in W$. There are two possibilities:

a) $w \notin P_s$. Let $(s_1, \ldots, s_q)$ be a reduced decomposition of $w$ and

for $1 \leq j \leq q$; also put $w_0 = 1$. Since $w_0 \in P_s$ by (A') and since $w = w_q$ is not in $P_s$, there exists an integer $j$ with $1 \leq j \leq q$ such that $w_{j-1} \in P_s$ and $w_j = w_{j-1} s_j$ does not belong to $P_s$. By (C), We have thus proved the formula which implies that $sw = s_1 \ldots s_{j-1} s_{j+1} \ldots s_q$ and $l(sw) < l(w)$.

b) $w \in P_s$: put $w' = sw$, so that $w' \notin P_s$ by (B'). By a), we then have $l(sw') < l(w')$, that is $l(w) < l(sw)$.

Comparison of a) and b) proves that $P_s$ consists of those $w \in W$ such that $l(sw) > l(w)$. The exchange condition follows from this as we have seen in a), hence $(W, S)$ is a Coxeter system (Th. 1 of no. 6).