ScalingStacks

Proof. Consider $\gamma$ and $\gamma^{\prime}$ two injective composable paths such that $\gamma\circ\gamma^{\prime}$ is injective. We have $m_{c}^{\pm}([\gamma\gamma^{\prime}])=m_{c}^{\pm}([\gamma])+m_{c}^{\pm}([\gamma^{\prime}])$.

Consider now $\gamma$ a minimal path. We have $m_{c}^{\pm}([\gamma])=m_{c}^{\mp}([\gamma^{-1}])$, hence $m_{c}([\gamma])+m_{c}([\gamma^{-1}])=0=m_{c}([\gamma^{-1}\circ\gamma])$. The lemma follows. ∎