Here is what I was taught about coming up with my own results in math school:
A lemma builds up to a theorem, which implies corollaries. A theorem you are not proud of is a proposition.
However, I was never taught how I should refer, reference, cite or reprint other people's result.
- For example, suppose author $A$ has written a paper which contains theorem $A$ (for instance, De Moivre's Theorem). When referencing (and reprinting) his or her theorem $A$, should you refer to it as a theorem? But suppose you are using theorem $A$ to build your own theorem, then such a theorem could better be served as a lemma.
Should this theorem be referred to as "Lemma (theorem $A$ from author $A$)", or "Theorem (theorem $A$ from author $A$)"?
- Consider another scenario, an author $A$ has a famous so called "lemma $A$" (for instance, Borel Lemma). For your application, such a lemma could better serve the role of a theorem, for which you derive corollaries from. Should you cite his or her lemma as "lemma" or a "theorem"?
Should this lemma be referred to as, "Theorem (lemma $A$ from author $A$)", or "Lemma (Lemma $A$ from author $A$)"?
Similarly, how should you reference other people's propositions, corollaries, and or "remarks", "observations", "claims"? Is there a convention which is to be followed?