ScalingStacks

Proof. The first statement follows from Proposition 3.1.3. The second statement is a property of the multiplication of $T_{w}$’s.

For the third statement, let us assume $r<r^{\prime}$. We have $s_{i_{r+1}}\cdots s_{i_{r^{\prime}}}=s_{i_{r}}\cdots s_{i_{r^{\prime}-1}}$ reduced, hence $s_{i_{r}}s_{i_{r+1}}\cdots s_{i_{r^{\prime}}}$ is not reduced, a contradiction. ∎