Theorem & Lemma In Short (arXiv Version) and Extended (Peer-Reviewed Version) Papers I'm writing a short paper from my master thesis to put on arXiv that includes a Definition and a Lemma/Proof.

If I want to publish another extended version for a journal or conference later, can I include the Lemma and Definition directly in the new paper or do I have to cite my previous paper only (without writing them)?
(They are the beauty of my work and I'd like to include them into the extended version as well)
Essentially, what you seem to be doing here is to write an extended version of a paper that previously appeared in a different venue (here, arXiv). In this case, it is customary to include all of the relevant definitions, theorem, lemmas, and the like in the extended version. 

The reason is that the longer paper - in a sense - replaces the shorter version. Thus, you would include them all. Nevertheless, you must state in your longer paper that it is an extended version of an earlier version of the paper (i.e., the short paper).

Note that you can just cite parts of your arXiv paper without including all of its results. It is probably not wise to pursue this approach, though. An arXiv paper is unreviewed, so the reviewers of your journal paper are more likely to question the correctness of the theorem in it. Since also most publishers do not consider arXiv papers to be a prior publication, there is no copyright or dual publication problem with just including theorems and proofs of your shorter paper.