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In mathematics, a proof is a convincing demonstration (within the accepted standards of the field) that some mathematical statement is necessarily true.^{[1]}^{[2]} Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single exception. An unproven proposition that is believed to be true is known as a conjecture.
The statement that is proved is often called a theorem.^{[1]} Once a theorem is proved, it can be used as the basis to prove further statements. A theorem may also be referred to as a lemma, especially if it is intended for use as a stepping stone in the proof of another theorem.
Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasiempiricism in mathematics, and socalled folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.
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