Proposition 7

Let $w$ be in $W$. There exists a subset $S_w$ of $S$ such that $\{s_1, \ldots, s_q\} = S_w$ for any reduced decomposition $(s_1, \ldots, s_q)$ of $w$.

Denote by $M$ the monoid consisting of the subsets of $S$ with the composition law $(A, B) \mapsto A \cup B$; the identity element of $M$ is $\varnothing$. Put $f(s) = \{s\}$ for $s \in S$. We are going to apply Prop. 5 of no. 5 to $M$ and $f$. We have $a(s, s') = \{s, s'\}$ for $s, s'$ in $S$ if $m(s, s')$ is finite, hence there exists a map $g : w \mapsto S_w$ from $W$ to $M$ such that $g(w) = f(s_1) \cup \cdots \cup f(s_q)$, that is $S_w = \{s_1, \ldots, s_q\}$ for any $w \in W$ and any reduced decomposition $(s_1, \ldots, s_q)$ of $w$.