Corollary

Let $s = (s_1, \ldots, s_p)$ and $s' = (s'_1, \ldots, s'_q)$ be two sequences of elements of $S$ such that $w = s_1 \ldots s_p$ and $w' = s'_1 \ldots s'_q$. If the sequence $(s_1, \ldots, s_p, s'_1, \ldots, s'_q)$ is a reduced decomposition of $ww'$, then $s$ is a reduced decomposition of $w$ and $s'$ is one of $w'$.

By hypothesis, $l(w) \leq p$, $l(w') \leq q$ and $l(ww') = p + q$. By (1), we must have $l(w) = p$ and $l(w') = q$, hence the corollary.