Lemma/Proposition/Theorem, which one should we pick?

This is something that confuses me.
I have read a few mathematical texts and they often seem to use Lemma/Proposition/Theorem if they have a particular statement.

Now, which one to use? A lemma can be something you need to prove a more important theorem, but then what about Fatou’s Lemma?

When to pick Proposition or Theorem?

Answer

There seem to be two issues here. One is why certain well-known results are called Lemmas, such as Zorn’s, Yoneda’s, Nakayama’s, and so on. I don’t know the answer to this; presumably it is a mixture of what was written in some original source and the results of the transmission of that original source through the mathematical tradition. (As one interesting example of how labels can be changed in the course of transmission, there is a result in the theory of automorphic forms and Galois representations, very well known to experts, universally referred to as “Ribet’s Lemma”; however, in the original paper it is labelled as a proposition!)

The second issue is how contemporary writers label the results in their papers. My experience is that typically the major results of the paper are called theorems, the lesser results are called propositions (these are typically ingredients in the proofs of the theorems which are also stand-alone statements that may be of independent interest), and the small technical results are called lemmas. This probably varies quite a bit from writer to writer (and perhaps also from field to field?).

Attribution
Source : Link , Question Author : JT_NL , Answer Author : Matt E

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