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'Maximum Flow' printed from https://nrich.maths.org/
If your 'cut' were to block the edges it cuts through there would
be no flow from $A$ to $B$ but if those edges take as much flow as
they can they only allow through the sum of the flows on each of
the 'cut' edges. Whatever cut you make you still have to check that
sufficient flow reaches that cut to allow through the flow given by
that sum.