So I've been trying for a while to write an article on 3 applications of some theorem for a small journal. But after finding my second example I realized that I didn't need the theorem to prove it. Now, the two proofs are linked by another theorem which I used in both for some constructions, but I use it just so I could generalize them from a particular case which I proved using other method. Now the question is do these two proofs still count as an application the second theorem?



Can it be an application of a theorem if I only use it to generalize?
It's really hard to answer this question when you phrase things so vaguely. I take your question to mean:


* There is a theorem, let's call it Theorem A.
* You thought you had three new results that are applications of Theorem A.
* You realized you could prove those results via a generalization of Theorem B instead.


If that's right, then from a mathematical-writing perspective, how about: "Three applications of (a generalization of) Theorem B"


I have certainly seen text like this in plenty of papers. Alternatively, you could ditch the language about "Applications of" and just focus your paper on the three examples, no matter how you prove whatever you are trying to prove about them.


