= Solution
Let $C$ be the <fundamental chamber of a root system>. The chambers $wC$ and $ws_\alpha C$ are adjacent across the reflecting hyperplane orthogonal to $w(\alpha)$. The chamber $wC$ lies on the side on which $w(\alpha)$ is positive. If $w(\alpha)\in\Pi$, then $C$ lies on that same side, so crossing this wall moves one step farther from $C$; if $w(\alpha)\in-\Pi$, it moves one step nearer. The gallery distance from $C$ to $wC$ is the <Coxeter length> $\ell(w)$, and adjacent chamber distances differ by one. Consequently
$$
w(\alpha)\in\Pi
\quad\Longleftrightarrow\quad
\ell(ws_\alpha)=\ell(w)+1
\quad\Longleftrightarrow\quad
\ell(ws_\alpha)>\ell(w).
$$
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