Lemma 1

Let $s_1, \ldots, s_q \in S$ and let $w \in W$. We have

where $(i_1, \ldots, i_p)$ denotes the set of strictly increasing sequences of integers in the interval $[1, q]$.

We argue by induction on $q$, the case $q = 0$ being trivial. If $q \geqslant 1$, we have $C(s_1 \ldots s_q).C(w) \subset C(s_1).C(s_2 \ldots s_q).C(w)$. By the induction hypothesis, $C(s_2 \ldots s_q).C(w)$ is contained in the union of the $C(s_{j_1} \ldots s_{j_p} w)$, where

By (T3), the set $C(s_1).C(s_{j_1} \ldots s_{j_p} w)$ is contained in the union of the sets $C(s_1 s_{j_1} \ldots s_{j_p} w)$ and $C(s_{j_1} \ldots s_{j_p} w)$. This proves the lemma.