Peter Shor teaches this session of MIT's 18.200, Principles of Discrete Applied Mathematics, on the max-flow min-cut theorem. He sets up the maximum flow problem on a network and first proves, by a direct combinatorial argument, that any flow value is bounded above by the capacity of any cut, so max flow never exceeds min cut. He then formulates maximum flow as a linear program and takes its dual, showing that the dual program is exactly the minimum cut problem. Working through the duality argument on the board, Shor demonstrates why the two quantities must actually be equal, not just bounded, tying together combinatorial optimization and linear programming duality. The seventy-eight minute lecture assumes familiarity with linear programming from earlier in the course and builds toward a complete, rigorous proof of one of the central theorems in network optimization.