Today we continue reading the flow networks. In particular, we look at the max flow min cut theorem. A cut of the flow network is a partition of its vertices into set S and T = V- S such that s belongs to S and t belongs to T. A minimum cut of the network is a cut whose capacity is the minimum over all cuts in the network. The max flow min cut theorem states the following:
1) f is a maximum flow in network.
2) there are no augmenting paths in the residual network.
3) the magnitude of the flow is the capacity of the cut for some cut (S,G)
This is proved by contradiction. The first implies the second. If there is an augmenting path in the residual network while there is a maximum flow in the network, the maximum flow can be exceeded with the augmentation. The second implies the third because if the residual neywork does not have any flows  the a cut between vertices s and t that do not have a path will have only forward flow and no reverse flow which implies it is the capacity.