What is the most appropriate way to cite others' results as a Lemma?

Question

Recently I am writing something up for possible publication. I am struggling with this question:

What is the most appropriate way to cite others' results as a Lemma?

For example, this is a theorem of Arhangelskii's:

Theorem 3.7 Suppose that $X$ is a regular space with a strong rank 1-digonal. Then any bounded subset $Y$ of $X$ is a Moore space.

I want to cite his theorem, however I don't need so much. (Note that if $X$ is a pseudocompact, then it is bounded.) I think if I cite the original theorem, it seems wordy, since I will need to explain not only pseudocompactness but also bounded set. Indeed, I want to write a Lemma as follows:

Lemma [I will note where it is from]: Suppose that $X$ is a regular pseudocompact space with a strong rank 1-diagonal. Then $X$ is a Moore space.

Could I directly do this?

Thanks for your help.

Answer 1

Either:

1. the fact that "$X$ is pseudocompact implies $X$ is bounded in the Arhangelskii sense" is well-known, or can be found in the literature, or
2. it is not.

In the first case you should instead state your Lemma without attribution to Arhangelskii and write

Proof. Since a pseudocompact space is bounded in the sense of Arhangelskii [citation needed], the conclusion is a easy corollary of Theorem 3.7 of [Arhangelskii's paper].

In the second case you absolutely must give the definition of boundedness and give the proof that pseudocompactness implies it.