Let $\gamma:[a,b]\rightarrow\mathcal{H}$ be a curve that is continuous and piecewise $\mathcal{C}^1$, with $f(t) = -B(\gamma(a),\gamma(t))$ and $n(t) = \sqrt{B(\gamma'(t),\gamma'(t))}$ satisfying $f'(t) \leq \sqrt{f(t)^2-1}\,n(t)$.
Deduce that
$$-B(\gamma(a),\gamma(b)) \leq \operatorname{ch}(\ell(\gamma)).$$