Flow network - minimum capacity cuts proof Let's start out by reviewing max-flow min-cut, as well as the flow networks they operate on.
http://en.wikipedia.org/wiki/Flow_network
http://en.wikipedia.org/wiki/Max-flow_min-cut_theorem
Let $G = (V,E)$ be a flow network. Prove a minimum cut is also a minimum-capacity
cuts of $G$.
Thanks for any help, this problem has been throwing me for a loop! 
Note: NOT homework, extra problems from our book I'm working through for exam preparation.

Here is a way to do this. Consider a max flow $f$ and the residual network $E_f$. From the max flow min cut theorem it is easy to get that $E_f$ doesn't contain any edge that goes from $S$ to $T$. Neither does it contain any edges from $S'$ to $T'$. Then it also has no edges going from $S \cup S'$ to $T \cap T'$.
But if no edge from $E_f$ crosses the cut $(S \cup S',T \cap T')$, then
$$
    c(S \cup S', T \cap T') = f(S \cup S', T \cap T'),
$$
and using the min cut max flow theorem once again we see that cut $(S \cup S', T \cap T')$ is also minimal.
