Max-flow min-cut theorem

Fix a network of $m$ directed arcs on nodes $\{0,\dots,n-1\}$, capacities $u_k\in[0,\infty]$, and nodes $s\ne t$, with no self-loops and every capacity strictly positive. Writing feasibility of $f$ for the max-flow problem as: conservation $(Nf)_i=0$ for $i\notin\{s,t\}$, $(Nf)_t=-(Nf)_s$, and $0\le f_k$, $\mathrm{ofReal}(f_k)\le u_k$ per arc; and $\mathrm{val}(f)=(Nf)_s$; the theorem asserts three conjuncts. (i) For every feasible $f$ admitting no augmenting path (no step-list that is a node-distinct path $s\to t$ using forward steps with $\mathrm{ofReal}(f_k)<u_k$ and backward steps with $f_k>0$), $f$ is a maximum flow: $\mathrm{val}(f')\le\mathrm{val}(f)$ for all feasible $f'$. (ii) The extended-real supremum $\sup_f \mathrm{val}(f)$ equals the infimum, over all finite node sets $S$ with $s\in S,\ t\notin S$, of the cut capacity $\sum_k [\mathrm{tail}_k\in S,\ \mathrm{head}_k\notin S]\,u_k$, coerced from $[0,\infty]$ into the extended reals ($\infty\mapsto+\infty$); since $s\ne t$, $S=\{s\}$ is such a cut, so the infimum is over a nonempty family. (iii) If the supremum is not $+\infty$, it is attained: some feasible $f$ has $\mathrm{val}(f)$ equal to it. All three parts are asserted together under the same hypotheses.