How to use lemmas and proofs from another paper?

Question

Suppose that paper A has a lemma called Lemma A, along with its proof.

I want to use this lemma in another paper B. However, in the scope of my paper, I have to change it slightly, but without losing the general idea.

For instance, the original lemma might read as follows:

Lemma A: The intersection of two straight lines in the plane is either empty, a single point, or a straight line.

Suppose I need the following variant in paper B:

Lemma B: The intersection of two planes in 3-space is either empty, a single line, or a plane.

Also, the proofs for both lemmas are very similar. Therefore, I have two questions:

1. Can I use the same methodology and same terminology with similar words to prove my own lemmas?
2. If I can, is citing paper A in my lemma confusing? If I cannot, can I just specify the lemma in paper A and say that this lemma and proof can also be used etc.?

Answer 1

You should certainly cite paper A in any case.

One way this is commonly handled: state your Lemma B.1 and give the complete proof. At the beginning of the proof, write something like "This closely follows the proof of Lemma A.1 from [A]." Now your paper is self-contained and you have given appropriate credit. It is fine if your proof is similar in structure to theirs; in some ways this is better, because a reader who looks at both will more easily be able to see the similarities and differences. But do not simply copy and paste their proof and change the necessary words. Your proof should be your words, even if it is from their ideas.

Or, state your Lemma B.1, but instead of giving a complete proof, say "The proof is very similar to that of Lemma A.1 from [A]". This saves space but will be more annoying to the reader, who in order to check your result will have to find the paper [A] and read through the proof, adapting it to prove B.1 instead of A.1. (The referee may be similarly annoyed.)

Some people would omit the statement of Lemma B.1 altogether, and when they need to use it, would say "By a slight modification of the proof of Lemma A.1 from [A], we have blah blah blah...". This is even more annoying.

Worst of all is to just say "By Lemma A.1 from [A], we have blah blah blah" where Lemma A.1 claims something different from (and not obviously implying) the statement you want.

Answer 2

In your example, those are not equivalent mathematical assertions, and so they are not identical lemmas. What I have done in similar cases is to say something along the lines of:

Lemma X is closely based on Lemma Y in [cite], and follows a similar proof structure.

This way you give appropriate credit to the original source, while still making your new assertion as you need.

Answer 3

Although many people do it, it is bad style and confusing to cite a lemma and restate it in a way that is not equivalent to the original one. I would suggest to state the lemma you need, and, instead of proving the whole lemma, explain in the proof that your lemma is very similar to the lemma A.1 and that the proof can be reused making the changes ... .

If, though, you just steal an idea, it might be better to completely prove your lemma in your version.