Series Law for Extremal Length/Rho is Well Defined

To see that $$\rho$$ is a well-defined metric, we need to check that it transforms correctly when changing local coordinates. Let $$z=z(t)$$ and $$w=w(t)$$ be charts on the Riemann surface $$X$$. Let $$\rho_1^z(t)$$ and $$\rho_1^w(t)$$ 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 ($$j\in\{1,2\}$$), we have
 * $$ \rho_j^w(t) = \rho_j^z(t) \cdot \left| \frac{dz}{dw}\right|$$.

(Here $$\frac{dz}{dw}$$ denotes, as usual, the derivative of the coordinate change $$z\circ w^{-1}$$.)

Thus we have
 * $$ \rho^w(t) = \max(\rho_1^w(t),\rho_2^w(t)) = \max(\rho_1^z(t),\rho_2^z(t))\cdot \left| \frac{dz}{dw}\right| = \rho^z(t)\cdot \left| \frac{dz}{dw}\right|.$$

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