I am reading Braid Groups (GTM 247) by Kassel Christian and Turaev Vladimir and am puzzled by a detail in the proof of a theorem:
I do not quite see the reason of the inequality sign in the second last row. Here the authors are basically using the inequality that $$|b'|\geq \rho(b_i(t),b_j(t)).$$ But it seems to me that by the definition of $|b'|$, it should be
$$|b'|\leq \rho(b_i'(t),b_j'(t)).$$ Am I right in saying that the proof of the theorem is fallacious?