Resistors in dual networks

Let $G$ be a finite plane multigraph and $G^{\mathrm{\prime }}$ its dual. Each edge $e$ of $G$ is interpreted as a resistor of resistance $R_e$, and the dual edge $e^{\mathrm{\prime }}$ is assigned the dual resistance $R_{e^{\mathrm{\prime }}}:=1\mathrm{/}{R}_e$. Then the equivalent resistance $r_e$ over $e$ and the equivalent resistance $r_{e^{\mathrm{\prime }}}$ over $e^{\mathrm{\prime }}$ satisfy $r_e\mathrm{/}{R}_e+r_{e^{\mathrm{\prime }}}\mathrm{/}{R}_{e^{\mathrm{\prime }}}=1$. We provide a graph theoretic proof of this relation by expressing the resistances in terms of sums of weights of spanning trees in $G$ and $G^{\mathrm{\prime }}$ respectively.