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**HallsofIvy** ?? Just **having** a positive cut and a negative cut doesn't necessarily give us anything! If we have a positive cut, x, and a negative cut, y, then **adding** the two, which is what you appear to be doing, does not necessarily give a negative cut. But if we have a positive cut, x, and negative cut, y, **and** |y|> x, then x+ y is a negative cut. As for writing it as -(x)(-y), I suspect that was because they define the sum of two **positive** cuts first and wanted to write the general case in terms of that.