Definition
Mathematical theorem
A statement established from the axioms and hypotheses of a theory.
A theorem is a statement established by a proof from the axioms and stated hypotheses of a theory. Its usable content includes all assumptions and quantifiers, not just the conclusion.
Lemmas and corollaries
The labels “lemma,” “proposition,” and “corollary” primarily describe a result's role in an exposition. A lemma is organized to support another argument; a corollary is presented as a consequence of preceding results. These labels do not change the standard of proof or permit dropping hypotheses.
Using a result
To apply a theorem to particular data, first verify that the data satisfy its hypotheses. When a theorem asserts an object exists for each parameter, any additional claim that the choices vary continuously or smoothly requires justification in its own right.