Corollary 4

*For any subset $X$ of $S$ and any $w$ in $W_X$, the length of $w$ with respect to the generating set $X$ of $W_X$ is equal to $l_S(w)$.

Let $(s_1, \ldots, s_q)$ be a reduced decomposition of $w$ considered as an element of $W$. We have $w = s_1 \ldots s_q$ and $s_j \in X$ for $1 \leq j \leq q$ (Cor. 1); moreover, $w$ cannot be a product of $q' < q$ elements of $X \subset S$ by definition of $q = l_S(w)$.*