Theorem 2

(i) For any subset $X$ of $S$, the pair $(W_X, X)$ is a Coxeter system.

(ii) Let $(X_i)_{i \in I}$ be a family of subsets of $S$. If $X = \bigcap_{i \in I} X_i$, then $W_X = \bigcap_{i \in I} W_{X_i}$.

(iii) Let $X$ and $X'$ be two subsets of $S$. Then $W_X \subset W_{X'}$ (resp. $W_X = W_{X'}$) if and only if $X \subset X'$ (resp. $X = X'$).

Every element of $X$ is of order 2 and $X$ generates $W_X$. Let $x \in X$ and $w \in W_X$ with $l_X(xw) \leq l_X(w) = q$. By Cor. 4 of Prop. 7, we have

Let $x_1, \ldots, x_q$ be elements of $X$ such that $w = x_1 \ldots x_q$. Since $(W, S)$ satisfies the exchange condition (Th. 1 of no. 6), there exists an integer $j$ such that $1 \leq j \leq q$ and $xx_1 \ldots x_{j-1} = x_1 \ldots x_{j-1} x_j$. Thus, $(W_X, X)$ satisfies the exchange condition and is therefore a Coxeter system (Th. 1 of no. 6). This proves (i).

Assertions (ii) and (iii) follow immediately from Cor. 1 of Prop. 7.