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In [mathematics] , a theorem is a [statement] which has been [proved] on the basis of previously established statements, such as other theorems, and previously accepted statements, such as [axiom] s. The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different [deductive system] s can yield other interpretations, depending on the meanings of the derivation rules. Theorems have two components, called the [hypotheses] and the conclusions. The proof of a mathematical theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions. The concept of a theorem is therefore fundamentally [deductive] , in contrast to the notion of a scientific [theory] , which is [empirical] .

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