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A theorem is a statement which can be proven true within some logical framework. Proving theorems is a central activity of mathematics. Note that 'theorem' is distinct from 'theory'.

A theorem generally has a set-up - a number of conditions, which may be listed in the theorem or described beforehand. Then it has a conclusion - a mathematical statement which is true under the given set up. The proof, though necessary to the statement's classification as a theorem is not considered part of the theorem.

In general mathematics a statement must be interesting or important in some way to be called a theorem. Less important statements are called:

A mathematical statement which is believed to be true but has not been proven is known as a conjecture.

As noted above, a theorem requires some sort of logical framework, this will consist of a basic set of axioms (see axiomatic system), as well as a process of inference, which allows to derive new theorems from axioms and other theorems that have been derived earlier. In propositional logic, any proven statement is called a theorem.

See also