On an atom pair of $\Gamma$, the two couplings carry the same mass

Let $\mu\perp\nu$ be mutually singular Borel probability measures, $\Gamma$ the paired-route set of equation (14), and let $\pi,\pi'$ be transport plans with marginals $\mu,\nu$ both concentrated on $\Gamma$. Then for every paired route $(a,b)\in\Gamma$,

This is the remaining, countable, part of Proposition 3.6: the two couplings have already been shown to agree off the set of atom pairs $A\times B$, which is countable, so only the individual masses at such pairs remain to be matched.

Assuming $a<b$ with $F_\sigma(a^-)\ge F_\sigma(b)$, the definition of $\Gamma$ forces the mass of $\mu$ on $[a,b[$ to be transported into $]a,b]$, whence for either coupling

an expression depending only on the marginals (equation (17)). The generalized intermediate value theorem shows $F_\sigma(a')>h$ for every $a'\in\,]a,b[$, so the same identity holds with any $a'$ in place of $a$ for which $(a',b)\in\Gamma$; taking the supremum over such $a'$ gives $\pi(]a,b[\times\{b\})=\pi'(]a,b[\times\{b\})$, and subtracting yields the claim at $(a,b)$.