Every min-cut has no flow going backwards along it in a max-flow

Statement

Lemma

Given a Flow network with a max flow . Then for every min st-cut (, ) we have all edges in for and .

Proof

From the Max-flow min-cut Theorem we know that for any min st-cut we have As the flow across any st-cut is equal to the value of the flow itself we have So we have

Given for . This gives that for any and with and .

Which is the required result.

Note this also gives every min-cut is at full capacity in a max-flow.