Suppose I am writing a paper in which an important lemma has a proof which is either long and unenlightening or requires additional background of the reader (or both). Thus, to avoid disrupting the narrative flow, I want to postpone the proof of the lemma to a later section. So I have a lemma in section 3 (say) whose proof is postponed to section 7 (say). And in section 7, I restate the lemma before proving it.
Now I can think of three ways to assign numbers to the lemma:
- The statement in section 3 and the restatement in section 7 each get their own number. Thus we have Lemma 3.4 and Lemma 7.2 which are identical.
- The restatement in section 7 gets the same number as the original statement in section 3. Thus we have only Lemma 3.4, which is stated in section 3 and then restated with the same number and proven in section 7.
- The statement in section 3 gets the same number as the restatement in section 7. Thus we have only Lemma 7.2, which is stated and proven in section 7, and also stated with a forward reference in section 3.
Which of these options is preferable?
Edit: Note that the first statement of the lemma is in section 3. This is not the introduction (at least, I've never read a paper where section 3 was the introduction). The situation of stating one or more of the paper's main theorems in the introduction is similar, but not quite the same.