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**emakarov** Proof by contradiction works like this. You need to prove P. Instead, you consider (not P). Then you prove that this (not P) implies a contradiction. Therefore, the original P is proved to be true.

If you need to show that the difference of two odds is even, you fix two *arbitrary* odd numbers x and y (so far the proof is like for every method) and assume that x - y is odd. From this assumption you derive contradiction, such as 0 = 1. Therefore, x - y being odd is false, i.e., x - y is even.