For example, the set {1,2,3} is in Partition, but f({1,2,3}) = {1,2,3}, which is not in 3-Partition (nor is {1,2,3,3}, if we choose to view it in that fashion). In general, the entire proof hinges on the idea that the number representing half of the overall sum is not in the initial set. This is not an assumption that seems warranted by the question or proved in the solution given.