Consider www-math.mit.edu/~goemans/18997-CO/co-lec22.ps page 4 first paragraph below figure 4.
They claim that graph G' has a s'-t' flow with value d1+d2 due to the max flow min cut theorem because the min cut has a value of d1+d2.
I do indeed see that there is a cut with value d1+d2, namely the edges (s',s1)+(s',s2). But I do not understand why this is the smallest cut? Couldn't there be a cut which seperates both s1 and t1 and which seperates s2 from t2 with less capacity? If that would be the case, then the demand d1 for (s1, t1) and (d2 for (s2,t2) could not be satisfied since the capacity of the edges between those source/terminal pairs is insufficient to satisfy both flows. Please explain why it would hold that there is a flow from s' to t' with value d1+d2.