How to prove this lemma about Weyl group?

Question

Let $\mathscr{W}$ be the Weyl group of a root system $\Phi$ with basis $\Delta$. If $\sigma\in \mathscr{W}$, $\sigma = \sigma_{\alpha_1} .. \sigma_{\alpha_t}$ where $\alpha_1, ...,\alpha_t \in \Delta$, and $t$ is as small as possible, then $\sigma(\alpha_t)$ is a negative root.

Answer 1

This is the Corollary to Lemma $C$ in Humphreys text "Introduction on Lie algebras and Representation Theory" in section $10.2$. The proof involves some lemmas on simple roots, which are elementary, but a bit lengthy. In particular we need that for a simple root $\alpha$ the Weyl group element $\sigma_{\alpha}$ permutes the positive roots other than $\alpha$, and, writing $\sigma_i$ for $\sigma_{\alpha_i}$, if $\sigma_1\cdots \sigma_{t-1}(\alpha_t)$ is negative, then $ \sigma_1\cdots \sigma_t=\sigma_1\cdots \sigma_{s-1}\sigma_{s+1}\cdots \sigma_{t-1} $ for some index $1\le s<t$.