Proof or citation? I'm writing an article. I suppose that I'll submit it to a more or less decent journal (in English). I have doubts about the following: I have a lemma (with quite a trivial proof). I don't want to include this lemma with complete proof due to its triviality and because it has been already proved before in my Ph.D. thesis.
So I may either prove the lemma in the article (this certainly will not be very good) or simply include a citation of my dissertation instead of proof. But the thesis is in Ukrainian and it's not even available freely via the Internet. Which of the two alternatives would you choose?  
Do there exist other ways to do this well?  
 A: I would include the short proof, but also mention that the lemma appeared before.
A: If you use the word "trivial" in its usual meaning, then no proof is needed --
usually, "trivial" means "immediately obvious to any reader".
But since you ask, it seems that you use the word in some other meaning --
perhaps in the meaning of "elementary", but this is only a guess.
If this is so, then what I would do in this situation depends on the length
of the proof, both in terms of pages as well as in terms of percentage of
the total length of the paper, as well as on whether the lemma is needed
in the proof of some key result of your paper or just in some example, etc..
If the proof does not make the paper significantly longer, or if it is
not longer than one or two pages, I would certainly include it.
A: A possible solution is writing before your lemma a sentence like the following: "The following result is easy and probably already known, however we include a proof for the convenience of the reader."
A: One should distinguish between short, trivial and elementary proofs. Trivial proofs can be omitted with no harm. Omission of proof in the other two classes does seldom any good to the reader. In your shoes I would put full proof in the preprint on arXiv. It can be easily edited out in the published version.
A: On my opinion, the main criterion for a reference is that it must be AVAILABLE.
Either on Internet or in most university libraries. An unpublished thesis in Ukrainian
which is not available on Internet does not satisfy this criterion. Such a reference
is meaningless.
So the real options are: a) to leave the proof to the reader or b) to include it.
I do not understand why you call the second option "not very good". What's wrong with including a trivial lemma? A reader for whom it is trivial will just skip the proof.
PS. Why don't you scan your thesis and put in on Internet? Even in Ukrainian.
