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The first step in a proof by contraposition is to assume that the conclusion of the conditional statement "If is even, then is odd" is false; namely, assume that is even.
Then, by the definition of an even integer,
Substituting for we find that
This tells us that is odd, and therefore not even. This is the negation of the premise of the theorem.
Because the negation of the conclusion of the conditional statement implies that the hypothesis is false, the original conditional statement is true.
Our proof by contraposition succeeded; we have proved
the theorem " If is even, then is odd".