Exercise 2.4.7.7. Let $W$ be a Coxeter group, and let $\overrightarrow {v} = ( v_1, v_2, \ldots , v_ m )$ and $\overrightarrow {w} = ( w_1, \ldots , w_ n )$ be elements of $M(W)$. Show that, if $\overrightarrow {v}$ is a refinement of $\overrightarrow {w}$, then there is a unique sequence of integers $0 = j_0 < j_1 < \cdots < j_ m = n$ satisfying the condition specified in Notation 2.4.7.6.
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