What do you call a proof of a claim inside a lemma? And that lemma is inside a theorem.
As I already indicated in this thread you might want to revise your document structure. I don't think it is common to have nested statements (like lemma's inside theorems) at all.
The normal presentation for this would go something like:
Statement of Theorem
Comment that to prove the theorem we will use some simple lemmas
Statements and proofs of lemmas
Restatement of theorem, or just a statement that theorem X above now follows.
You should avoid a cascade of statements whose proofs depend on the following statements. Instead put the thing you prove first at the top, and perhaps precede with a comment such as 'we will use the following small result later', and then reference it when you do you use it.
What happens if you need to prove something inside a lemma?
Depending on how important it is and how large the proof is, you have three options IMO:
Make it a separate lemma which you prove separately, and refer to in the main proof.
Prove it "inline", e.g. in a new sentence or even in brackets, right after giving the statement.
Mention the statement and defer the proof to the end ("This concludes the proof of the statement, it just remains to show [... conclusion which you had already used ...] -- which we shall do now."
Or if you're feeling particular wicked just insert the statement:
The proof is left as an excercise to the reader.
In exams you might want to try the following variant:
The proof is left as an excerise for the grader.
Does the sublemma even merit a title? Why can't it just be part of the proof of the containing lemma? If this lemma requires another result to be proven before it, then why is it a lemma itself? Why not put the sublemma first as another lemma? You have been given many alternatives.
It certainly could, just as subroutines could be included in the computer program where they are called. A "lemma" is just a part of the main proof that is simpler to understand if it is done separately. The same could be true of a "sub-lemma".
If I am reading the original post correctly, a "proof of a claim inside a lemma", if it is written as a separate proof, would, indeed, be a "sub-lemma".
If the "sub-lemma" is a really small result, one will often just put "Claim: xxx" Then "Proof of Claim:" in the middle of the proof of the lemma. If it's a result that will be used later in the paper, it deserves its own "Lemma" status. I have never read a paper that refers to "sub-lemmas". But then again I'm sure they are out there.
No one is disputing that, Halls, merely the OP's motivation for doing it and his desire both for the numbering (see parallel thread) and the wish to nest 3 proofs inside each other, rather than have them run sequentially: this subresult appears to be so pivotal as to merit its own number (and indeed counter). At which point you might wish to ask if it needs to stand alone as a separate statement.
Separate names with a comma.