Cuts and Capacity

A cut divides the nodes into two sets $S$ and $T$, where the source is in $S$ and the sink is in $T$. The capacity of the cut is the sum of capacities of edges going from $S$ to $T$. Edges from $T$ to $S$ do not count toward cut capacity.

Only forward edges (from $S$ to $T$) matter. Every flow from source to sink must cross the cut. So the flow value is at most the cut capacity. This is true for every cut.