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I want to prove a theorem using the result of a well-known theorem. Should I write the well-known theorem as a lemma since it aids the proof of my theorem or as a theorem?

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    $\begingroup$ What kind of article are you writing, and who are the intended readers? In a paper for a research journal you would generally just put in a reference to the well-known theorem: "by a theorem of Ploni (footnote or bibliography reference), X is a Y". In an expository article, you might write out the full statement of the theorem, and maybe even include a proof if it's short. But in any case, if it's generally called a theorem, don't call it a lemma. $\endgroup$ – Robert Israel Apr 19 at 1:23
  • $\begingroup$ @RobertIsrael The theorem is well-known in mathematics (it is in Wikipedia) but I am writing it in an engineering paper. So the reader is probably not familiar with it. $\endgroup$ – KRL Apr 19 at 1:28
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It depends on how well-known it is. If it's genuinely well-known, then I would just say "by Theorem XYZ," then the implication of the theorem as it is relevant to your proof, with any citations for the theorem as necessary. The less well-known it is, at least insofar as your audience is concerned, the more you should probably include, whether it be the statement of the theorem, or even the theorem and its proof as well (assuming it's not a hundred pages long or something).

As for what to call it - generally if it's referred to as a theorem, I'd just call it a theorem to avoid complications.

In short, though, it simply depends and you'll have to use your best judgment. I doubt you could really go wrong by stating the theorem at least, for clarity's sake if nothing else, but for really well-known theorems (e.g. Fermat's Last Theorem) that wouldn't even be necessary for the average mathematically-inclined person.

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