I am posting the question and the answer to the question below. I already have the right answer as the algorithm is given, please help me prove the correctness of the algorithm formally using induction. Thank you!

24. Let G=(V,E) be a directed graph, with source sV, sink tV, and nonnegative edge capacities {ce}. Give a polynomial-time algorithm to decide whether G has a unique minimum st cut (i.e., an s - t of capacity strictly less than that of all other st cuts). Solution: The polynomial time algorithm is shown below: - Compute a minimum s-t cut C, suppose its volume is C. - Increase eis c capacity by 1 and compute minimum cut in new graph. - Let new minimum cut be Cj and volume be Cj. - If C equals Cj then Cj is also a minimum cut in original graph and C=Cj. - So, it is not a unique minimum cut. - If in original graph, a different minimum cut C2 exists then there will be some ej that belongs to C and not in C. - The volume of C is not changed through increasing edge's capacity. - The graph has unique minimum cut if C