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Mathematical proof

[File:Oxyrhynchus papyrus with Euclid's Elements.jpg|right|thumb|250px|One of the oldest surviving fragments of Euclid's *Elements*, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5.]] In mathematics, a **proof** is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing *all* possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.

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