Comparison Principle for Extremal Length

Theorem
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.