Suppose you have two reduced expressions $s_1\dots s_n=t_1\dots t_n$ in a Coxeter group. If $r$ is the maximum index such that $t_1s_1\dots s_r$ is reduced, why does $t_1s_1\cdots s_r=s_1\dots s_{r+1}$? A paper I'm reading says it follows from the exchange rule.
I see that $t_1s_1\dots s_r=t_2\dots t_ns_n\dots s_{r+1}$, so by the deletion condition I should be able to deleter $2n-r+1$ terms from the right, but I don't see what that should be equal to $s_1\dots s_{r+1}$. In what way is the exchange condition being used here?
Since $t_1s_1\ldots s_{r+1}$ is not reduced, we have $$ \ell(t_1s_1\ldots s_{r+1})\leq r<\ell(s_1\ldots s_{r+1}). $$ The strong exchange condition then ensures $$ t_1s_1\ldots s_{r+1}=s_1s_2\ldots\hat s_i\ldots s_{r+1} $$ for some $i$ with $1\leq i\leq r+1$. If $i<r+1$ then we would have $$ \ell(t_1s_1\ldots s_r)=\ell(s_1\ldots\hat s_i\ldots s_r)<r, $$ contradicting the assumption that $t_1s_1\ldots s_r$ is reduced. Thus $i=r+1$, so $$ t_1s_1\ldots s_{r+1}=s_1s_2\ldots s_r. $$