What is the terminology for "lemma of lemma" Let's say I need to prove a main theorem, to prove which I need three lemmas. Thus in writing the structure is as follows:

Lemma 1
Proof
Lemma 2
Proof
Lemma 3
Proof
Theorem
Proof

But if when proving Lemma 1, I need to prove another two results ("lemmas for the proof of a lemma"). Is there any standard terminology for this situation? It seems to me that calling them Lemmas makes the structure of the article messier.

You can use Claim to set it on a lower footing than a lemma, but it's really not a big deal.  One lemma can build on other lemmas.
The biggest distinction you want to make is between the results you consider broad and important (theorems), narrow but important for the proof of a theorem (lemmas), and immediate consequences of theorems (corollaries).
