Let G=(V,E) be a flow network with source s and sink t. Suppose that G has a modified capacity function c:VVR{}. Note that c can adopt negative values, which is different from our original definition. In such a network, a feasible flow need not exist. Prove that if there is a feasible flow f in G, then there is a maximal flow with a value equal to that of the minimal cut. (This problem is motivated by a question in class about whether capacities can be negative.) (One natural interpretation of a "negative" capacity is that this is a way to enforce a mandatory minimum flow. Suppose there is an edge between vertex x and y such that cyx=3. It means that the total flow from y to x must be at most 3; in other words, the total flow from x to y is at least 3. It is natural to consider a flow network with negative capacity; in plumbing, for example, you often want to ensure a minimal amount of flow through your pipes to stop them from bursting when it drops below freezing in the winter.)