# Decreasing an edge capacity that crosses the MIN-CUT; Can the Max Flow remain unchanged?

It's possible that by increasing an capacity in some edge that belongs to cross edge of an MIN-CUT, the max flow remain unchanged because there might be multiple min-cut. However, if I decrease the capacity; whether the MIN-CUT unique or not; it seems to me that the max flow will decrease the same amount. However, I can't prove that this holds for all cases (I mean not only integer capacity, but also non-integer, even irrationals)

Suppose $e$ is the edge whose capacity is reduced by $\delta$. The capacity of all cuts such that $e$ is one of their cross edges is decreased by $\delta$. The capacity of all other cuts is unchanged. If $e$ is a cross edge of a source-sink minimum cut, the capacity of the minimum cut decreases by $\delta$.