Parallel Law for Extremal Length

Proposition
Let $$X$$ be a Riemann surface. Let $$\Gamma_1$$, $$\Gamma_2$$ be families of rectifiable curves (or, more generally, families of disjoint unions of rectifiable curves) on $$X$$.

Suppose that $$\Gamma_1$$ and $$\Gamma_2$$ are disjoint, in the sense that there exist disjoint Borel subsets $$A_1,A_2\subseteq X$$ such that, for any $$\gamma_1\in \Gamma_1$$ and $$\gamma_2\in\Gamma_2$$, we have $$\gamma_1\subseteq A_1$$ and $$\gamma_2\subseteq A_2$$.

Suppose that $$\Gamma$$ is a third curve family, with the property that every element $$\Gamma_1$$ and every element of $$\Gamma_2$$ contains some element of $$\Gamma$$.

Then the extremal length of $$\Gamma$$ satisfies
 * $$\frac{1}{\lambda(\Gamma)} \geq \frac{1}{\lambda(\Gamma_1)} + \frac{1}{\lambda(\Gamma_2)}.$$

Proof
The assumption means that every element of $$\Gamma_1\cup \Gamma_2$$ contains some element of $$\Gamma$$. Hence

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