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Write $N(w)$ for the inversion set of a Weyl-group element. Part b shows that $s_{\alpha }$ permutes $\Pi \setminus \{\alpha \}$. It follows that right multiplication by $s_{\alpha }$ changes the size of the inversion set byIndeed, all roots other than $\alpha$ are merely relabelled, while $w{s}_{\alpha }(\alpha )=-w(\alpha )$. Part c gives exactly the same recursion for the Coxeter length. Both quantities vanish at the identity, so induction along any word in the simple reflections gives

By the assumed transitivity on fundamental systems, some $w_0\in W$ sends $\Delta$ to $-\Delta$. It therefore sends the entire positive system of a root system $\Pi$ to $-\Pi$. Part d then givesFor every $w\in W$, its inversion set is contained in $\Pi$, so $\ell (w)\le |\Pi |$ and $w_0$ has maximal length.

If $u$ also has maximal length, then $N(u)=\Pi$, so $u(\Pi )=-\Pi$. Hence $w_0^{-1}u$ preserves $\Pi$ and has no inversions. Part d makes its Coxeter length zero, so it is the identity. Thus $u=w_0$, proving that the longest element of a finite Coxeter group is unique and has length $|\Phi |/2$.