Comparison Principle for Extremal Length

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

If every element of $$\Gamma_1$$ contains some element of $$\Gamma_2$$, then the extremal lengths of $$\Gamma_1$$ and $$\Gamma_2$$ are related by
 * $$ \lambda(\Gamma_1) \geq \lambda(\Gamma_2).$$

More precisely, for every conformal metric $$\rho$$ as in the definition of extremal length, we have
 * $$L(\Gamma_1,\rho) \geq L(\Gamma_2,\rho).$$

Proof
We have

$$ $$ $$

This proves the second claim. The second claim implies the first by definition.