Corollary 10.8 -- the resistance triangle inequality

Let $c$ be a network on a finite vertex set: a symmetric nonnegative conductance function with total conductance $c(x)=\sum_yc(x,y)>0$ at every vertex, whose associated walk $P(x,y)=c(x,y)/c(x)$ is irreducible. For distinct vertices, the effective resistance $R(a\leftrightarrow z)$ is defined through the voltage $W(x)=\mathbb P_x\{\tau_a<\tau_z\}$ and the current $\|I\|=\sum_yc(a,y)[W(a)-W(y)]$ as $R(a\leftrightarrow z)=\|I\|^{-1}$.

The theorem (Corollary 10.8 of Levin–Peres–Wilmer) asserts that effective resistance satisfies the triangle inequality: for pairwise distinct vertices $a,b,z$,

Together with symmetry and positivity this makes $R$ a genuine metric on the vertices of a connected network — the resistance metric. In the book it is a direct consequence of the commute time identity, which converts the claim into the sub-additivity of round-trip times through the intermediate vertex $b$.