= Solution
We induct on the <Coxeter length> $\ell(w)$. There is nothing to prove when $\ell(w)=0$. Otherwise choose a simple root $\alpha$ such that
$$
\ell(ws_\alpha)=\ell(w)-1;
$$
equivalently, $w\alpha$ is negative. Since $\lambda$ and $\mu=w\lambda$ lie in the <Closed dominant Weyl chamber>,
$$
0\leq\langle\lambda,\alpha^\vee\rangle
=\langle\mu,(w\alpha)^\vee\rangle\leq0.
$$
Therefore $\langle\lambda,\alpha^\vee\rangle=0$, and the simple reflection $s_\alpha$ fixes $\lambda$. Moreover,
$$
(ws_\alpha)\lambda=w\lambda=\mu.
$$
The induction hypothesis writes $ws_\alpha$ as a product of simple reflections that fix $\lambda$. Multiplying on the right by $s_\alpha$ gives the required expression for $w$. This is the <Weyl stabilizer of a dominant point> lemma.
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