0PAG Proof. Let ζ′\zeta^{\prime} and ζ′′\zeta^{\prime\prime} be homotopy classes of admissible paths such that ζ=ζ′∘ζ′′\zeta=\zeta^{\prime}\circ\zeta^{\prime\prime}. We have i(ζ,ζ1∘ζ2)≤i(ζ′,ζ1)+i(ζ′′,ζ2).i(\zeta,\zeta_{1}\circ\zeta_{2})\leq i(\zeta^{\prime},\zeta_{1})+i(\zeta^{\prime\prime},\zeta_{2}). Let γ\gamma be a minimal path in ζ\zeta and let t∈(0,1)t\in(0,1). We have mζ(0)+(ζ2)=i(γ|[0,t],ζ2)m_{\zeta(0)^{+}}(\zeta_{2})=i(\gamma_{|[0,t]},\zeta_{2}) for tt small enough. Since i([γ[t,1],ζ1)≤i(ζ,ζ1)CLOSEi([\gamma_{[t,1]},\zeta_{1})\leq i(\zeta,\zeta_{1}), it follows that i(ζ,ζ1∘ζ2)≤i(ζ,ζ1)+mζ(0)+(ζ2).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 ZZ by ZoppZ^{\mathrm{opp}}. ∎