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I'm think you misunderstood the question.

You are not forced to use all numbers. (And in fact, my explicit solution doesn't use all numbers.) There is no restriction that the union of both subsets must be equal to the full set. Note that this restriction would change the question dramatically. See also: https://news.ycombinator.com/item?id=10023098

Also note that it is allowed for both chosen subsets to overlap. In fact, the proof just says that two different (i.e. not entirely equal) subsets with the same sum exist.

However, once you have a pair of different overlapping subsets of the same sum, you can simply remove the intersection from both sets. Both sums decrease by the same amount. You then get a pair of two disjoint sets that have the same sum.

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