I would like to prove (for purposes of illustration mainly) that the symmetric group $S_n$ with the set $S$ of adjacent transpositions $(i, i+1)$ is a Coxeter group by proving that it satisfies the following exchange condition:
(E) Let $w\in S_n$ and $s\in S$ be such that $\ell(sw) \leq \ell(w)$. For any reduced decomposition $(s_1, \dots, s_q)$ of $w$, there exists an integer $j \in \{1, \dots, q\}$ and $$ ss_1 \dots s_{j-1} = s_1\dots s_{j-1}s_j. $$
Here, $\ell(w)$ denotes the minimal number $q$ of elements in an expression $w = s_1 ... s_q$ with $s_i\in S$.
Does anyone have a suggestion how to go about that?!