Since $u$ is defined by means of the curve family $\Gamma_1(Q)$, this provides the necessary link between $\Gamma_1(Q)$ and $\Gamma_2(Q)$. Roughly speaking, if there are few curves in $\Gamma_1(Q)$, as quantified by modulus, then these corresponding curves in $\Gamma_2(Q)$ must be short, which implies that the modulus of $\Gamma_2(Q)$ is large. The organization of the paper is the following. Section \[sec:preliminaries\] contains some basic notation and background, including an overview of the construction of the harmonic function $u$ described in the previous paragraph. In Section \[sec:level\_sets\], we prove several properties of the level sets of $u$ which are required for the proof of Theorem \[thm:main\]. This section expands on the material present in Section 4 of [@Raj:16]. Section \[sec:lower\_bound\] contains the main technical portion of our paper, the coarea inequality for $u$ described previously valid for all metric surfaces, as well as the proof of Theorem \[thm:main\]. Section \[sec:continuity\_u\] contains a final auxiliary result, namely that the harmonic function $u$ is continuous in general. The continuity of $u$ had previously been proved as Theorem 5.1. of [@Raj:16] using the reciprocality conditions and .

Preliminaries {#sec:preliminaries}
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In this section, we give a review of notation and auxiliary results from [@Raj:16] 