# Flows are maximal if there is no augmenting path
Last edited: 2023-11-11
# Statement
Lemma
Given a flow network $(G, c, s, t)$ a flow $f$ is maximal if there is no path from $s$ to $t$ in the residual network $(G^f, c^f, s, t)$.
# 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 $f$ 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.