Proposition 27

Let $R_1$ be a subset of R, $Q_1$ the subgroup of Q(R) generated by $R_1$, and $W_1$ the subgroup of W(R) generated by the $s_\alpha$ ($\alpha \in R_1$). If $p \in P(R)$ and $w \in W_1$, then $p - w(p) \in Q_1$.

If $w = s_\alpha$ with $\alpha \in R_1$, then

If $w = s_{\alpha_1} s_{\alpha_2} \ldots s_{\alpha_r}$, with $\alpha_1, \ldots, \alpha_r \in R_1$, it is still true that $p - w(p) \in Q_1$, as we see by induction on r.

The group A(R) leaves P(R) and Q(R) invariant, and hence acts on the quotient P(R)/Q(R). By Prop. 27, the group W(R) acts trivially on P(R)/Q(R). Passing to the quotient, we see that the quotient group A(R)/W(R) (cf. Prop. 16, no. 5) acts canonically on P(R)/Q(R).