ScalingStacks

Proof. Let $\zeta^{\prime}$ and $\zeta^{\prime\prime}$ be homotopy classes of admissible paths such that $\zeta=\zeta^{\prime}\circ\zeta^{\prime\prime}$. We have

$$i(\zeta,\zeta_{1}\circ\zeta_{2})\leq i(\zeta^{\prime},\zeta_{1})+i(\zeta^{\prime\prime},\zeta_{2}).$$

Let $\gamma$ be a minimal path in $\zeta$ and let $t\in(0,1)$. We have $m_{\zeta(0)^{+}}(\zeta_{2})=i(\gamma_{|[0,t]},\zeta_{2})$ for $t$ small enough. Since $i([\gamma_{[t,1]},\zeta_{1})\leq i(\zeta,\zeta_{1})$, it follows that

$$i(\zeta,\zeta_{1}\circ\zeta_{2})\leq i(\zeta,\zeta_{1})+m_{\zeta(0)^{+}}(\zeta_{2}).$$

The second inequality follows from the first one by replacing $Z$ by $Z^{\mathrm{opp}}$. ∎