is, admissible after removing from $\Gamma_1$ a subfamily of modulus zero). We extend the definition of $\rho$ to the entire space $X$ by setting $\rho(x) = 0$ for all $x \in X \setminus Q$.

Let $\Gamma_0$ be the family of curves in $Q$ with a subcurve on which does not hold. Note that $\operatorname{mod}\Gamma_0 = 0$. We define the function $u$ as follows. Let $x \in Q$. If there exists a curve $\gamma \in \Gamma_1 \setminus \Gamma_0$ whose image contains $x$, then define $$\label{equ:u_definition}
    u(x) = \inf \int_{\gamma_x} \rho\,ds,$$ where the infimum is taken over all such curves $\gamma$ and over all subcurves $\gamma_x$ of $\gamma$ joining $\zeta_1$ and $x$. Otherwise, define $u(x)$ by $$u(x) = \liminf_{y \in E, y \rightarrow x} u(y),$$ where $E$ is the set of those $y \in Q$ such that $u(y)$ is defined by . Lemma 4.1 of [@Raj:16] shows that $u$ is well-defined in $Q$.

We recall Lemma 4.3 of [@Raj:16], which states that $\rho$ is a weak upper gradient of $u$. More precisely, $$\label{equ:upper_gradient}
    |u(x) - u(y)| \leq \int_{\gamma} \rho\,ds$$ for all curves $\gamma$ in $Q$ with $\gamma \notin \Gamma_0$. In particular, $u$ is absolutely continuous along any curve $\gamma \notin \Gamma_0$. We 