Recall that an implication \(P \imp Q\) is logically equivalent to its contrapositive \(\neg Q \imp \neg P\text{.}\) There are plenty of examples of statements which are hard to prove directly, but whose contrapositive can easily be proved directly. This is all that proof by contrapositive does. It gives a direct proof of the contrapositive of the implication. This is enough because the contrapositive is logically equivalent to the original implication.
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