 Conformal invariance of extremal length

For a conformal map $f$ and a conformal metric $\rho$ on its image, the pullback density is $\rho (f(z))|f^{\prime }(z)|$. The change of variables formula preserves its metric area, and the chain rule preserves the length of each corresponding curve. Applying this to $f$ and its inverse proves the title equality. This allows extremal length computations on rectangles, discs and annuli to be transported to other domains.