Exercise 1, § 1

a) Let $(W, S)$ be a Coxeter system and let $s_1, \ldots, s_r$ be elements of $S$. Put $w = s_1 \ldots s_r$. Show that if $l_S(w) < r$, there exist two integers $p$ and $q$ with $1 \leq p < q \leq r$ such that $w = s_1 \ldots s_{p-1} s_p + 1 \ldots s_{q-1} s_q + 1 \ldots s_r$. Show that there is a strictly increasing sequence of integers $j(1), \ldots, j(k)$ between 1 and $r$ such that $(s_{j(1)}, \ldots, s_{j(k)})$ is a reduced decomposition of $w$.

b) Let $(W, S)$ be a Coxeter system and $X, Y, Z$ three subsets of $S$. Show that

(show that every element $w \in W_Y . W_Z$ has a reduced decomposition such that $s_i \in Y$ and $t_j \in Z$ and use Cor. 1 of Prop. 7 of no. 8). Show that