= Solution
The <deletion condition for involutory generators> says that whenever a word
$$
s_1s_2\cdots s_m,qquad s_i\in S,
$$
is not reduced, there are indices $i<j$ such that deleting $s_i$ and $s_j$ leaves a word representing the same group element.
The <exchange condition for a Coxeter group> says that if $w=s_1\cdots s_m$ is reduced and $s\in S$ satisfies $\ell_S(sw)<\ell_S(w)$, then
$$
sw=s_1\cdots\widehat{s_i}\cdots s_m
$$
for some $i$. There is an equivalent right-handed form for $\ell_S(ws)<\ell_S(w)$.
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