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People will talk about axioms and rules of deduction, and say that a proof consists of (effectively) a web of statements, each of which is either an axiom, or the result of applying a rule of deduction/derivation from earlier statements.

Well, perhaps, in theory, yes.

In practice, that's not how it works. It's much murkier.

A proof is a mathematical statement which has been demonstrated to the satisfaction of enough mathematicians of repute normally using the standard methods of mathematical logic, deduction and induction.

Such a statement is then normally called a theorem.

Here is a list of pages on proofs:

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