ScalingStacks

Proof. The first statement is clear. Let us now prove the second statement. That statement is clear if $\gamma((0,1))\cap(X_{exc}\cup\{x\})=\emptyset$.

The left side of the equality is additive under compositions of paths, and so is the right side by Lemma 7.1.22 below.

Assume now $\gamma^{-1}(X_{exc}\cup\{x\})$ is finite. The path $\gamma$ is a (finite) composition of paths mapping $(0,1)$ into the complement of $X_{exc}\cup\{x\}$, hence the statement holds for $\gamma$.

Consider now the general case. The proof of Lemma 7.1.16 for $E=X_{exc}\cup\{x\}$ produces a path $\gamma^{\prime}$ homotopic to $\gamma$ such that $\gamma^{\prime-1}(E)$ is finite and such that $|I_{c}^{+}(\gamma)|-|I_{c}^{-}(\gamma)|=|I_{c}^{+}(\gamma^{\prime})|-|I_{c}^{-}(\gamma^{\prime})|$. Since the statement holds for $\gamma^{\prime}$, it follows that it holds for $\gamma$. ∎