Let $A=\Delta^m A_1 A_2...A_k$ be the Garside's normal form of a braid. Then its canonical length is $k$. I need to prove that for any $A,B \in B_n$ we have ${\rm len}(AB) \leq{\rm len}(A)+{\rm len}(B)$ where ${\rm len}$ is its canonical length.
If $A=\Delta^m A_1 A_2...A_k$ and $B=\Delta^s B_1 B_2...B_r$, I can see how multiplying them may reduce the canonical factors but I am not sure how to go about it. Any hints or a nudge would be highly appreciated.
For definitions and concepts, one may check https://eprint.iacr.org/2018/1142.pdf