Flows are maximal if there is no augmenting path

Statement

Lemma

Given a flow network a flow is maximal if there is no path from to in the residual network .

Proof

This follows from the proof of the Max-flow min-cut Theorem.

In particularly Claim 2 in the proof of the main result gives us that value of is the capacity of some st-cut. With the corollary of Claim 1 being that for these to be equal they must be maximal/minimal in their own respects.