Is it legal to include another person's result in my research paper?
Theorem (Name of the author): Statement
Proof. The proof of this appeared in .
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Isaac Newton said, "If I have seen further, it is by standing on the shoulders of giants." Research would grind to a halt if each scientist had to start from "scratch."
So the question becomes, "How should I use the work of others?"
The answer: By citing their published work. Since the number of times a publication has been cited is often used to measure its impact, this is both fair and ethical.
Not only is this completely "legal", but -- up to issues of formatting and style that are up to you to decide -- it is a practice that occurs in the vast majority of contemporary mathematical papers.
If you are unsure about how to word things, consulting your advisor (I sure hope you have one, since mathematics is a difficult profession to break into unaided, intellectual issues aside) and consulting many published math papers are both good ideas.
Let me address the style issue a bit: this occurs in your choice to either set off Prof. A's result as a theorem in its own right versus just citing it when needed in the proof of a result in your paper. There are good reasons to do both. I'll start you off with one sample paper to look at and try to get some feel for when each is done. Then look at a few dozen others. Anyway, it's really up to you. One tip: the reader should not be confused, even for a second, as to whether the result she is reading is due to you or is a recalled result of someone else. So please choose a style/formatting that makes this immediately clear.