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 $z = z \left({t}\right)$ and $w = w \left({t}\right)$ be charts on the Riemann surface $X$.

Let $\rho_1^z \left({t}\right)$ and $\rho_1^w \left({t}\right)$ 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 $j \in \left\{ {1, 2}\right\}$), we have:
 * $ \rho_j^w \left({t}\right) = \rho_j^z \left({t}\right) \cdot \left\vert{ \dfrac{\mathrm d z}{\mathrm d w} }\right\vert$

where $\dfrac {\mathrm d z} {\mathrm d w}$ 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.