I'm trying to prove that for a flow network, maximum flow f, and each minimum cut S that f(S)=c(S).  Max flow min cut promises that there exists a cut where f(S)=c(S); I want to prove that given multiple minimal s-t cuts Si, that c(S1)=f(S1)=c(S2)=f(S2)...,  meaning all outgoing edges of all minimal cuts are fully saturated.   It seems very intuitive to me but I can't seem to get it.  I want to say that the flow for any s-t cut is equal to the flow going into t.  How can I prove that?  I'm guessing that I need to rely on the fact that for each vertex the incoming flow equals the outgoing flow but I don't really see how.
All minimum cuts in flow network
graph theory;network flow