The proof of this is everywhere skipped and said to be corollary of Min-Cut-Max-Flow theorem ... It's usually something like:
Let S1 and S2 be minimum cuts of a flow network. Then S1∪S1 and S1∩S2 are also min cuts.
Can anyone tell me how exactly this is proved?
By the min-cut-max-flow theorem, for every max flow and every cut, that cut is minimum if and only if all arcs that cross it are saturated (this is the analog of complementary slackness, provable by observing that the total flow crossing the cut is the overall flow). Given min cuts S1 and S2, every arc that crosses S1 ∪ S2 crosses S1 or crosses S2, so every such arc is saturated. Ditto for S1 ∩ S2.