Theorem 1 (Minimal cut theorem), p. 400 — the maximal flow value equals the minimum value of a disconnecting set

Let $N$ be a network: a finite graph whose arcs are undirected (parallel arcs allowed), with a source $a$, a sink $b\neq a$ and a positive capacity $c(e)$ on every arc. A flow sends non-negative amounts along chains (self-avoiding paths) joining $a$ and $b$, so that the total amount through each arc is at most its capacity; its value is the total amount sent. A disconnecting set is a set of arcs meeting every chain joining $a$ and $b$, and its value is $v(D)=\sum_{e\in D}c(e)$.

Theorem (Ford–Fulkerson). The maximal flow value obtainable in $N$ is the minimum of $v(D)$ over all disconnecting sets $D$: there is a number $m$ such that

both the maximum and the minimum being attained.

This is the max-flow min-cut theorem in its original form, for undirected networks with flows decomposed along chains.

Formalization Note The statement says that $m$ is the greatest element of the set of flow values and the least element of the set of values of disconnecting sets, so both extrema are attained and no supremum of a possibly empty or unbounded set is used. Flows are functions on finite sets of arcs (a collection of chain flows listing a chain twice can be merged without changing the value or any arc load).