Series Law for Extremal Length/Rho is Well Defined

Proof
To see that $\rho$ is a well-defined metric, we need to check that it transforms correctly when changing local coordinates.

Let $z = \map z t$ and $w = \map w t$ be charts on the Riemann surface $X$.

Let $\map {\rho_1^z} t$ and $\map {\rho_1^w} t$ be the coefficient functions when $\rho_1$ is expressed in the local coordinates $z$ and $w$, respectively.

We use the analogous notation for $\rho_2$ and $\rho$.

Since $\rho_j$ is a metric for $j \in \set {1, 2}$, we have:
 * $\map {\rho_j^w} t = \map {\rho_j^z} t \cdot \size {\dfrac {\d z} {\d w} }$

where $\dfrac {\d z} {\d w}$ denotes, the derivative of the coordinate change $z \circ w^{-1}$.

Thus we have:

This means that $\rho$ transforms correctly and is a metric, as desired.