TITLE: mixing theorem with definition (definition with proof)
QUESTION [5 upvotes]: I often find myself writing a definition which requires a proof. You are defining a term and, contextually, need to prove that the definition makes sense. 
How can you express that? What about a definition with a proof?
Sometime one can write the definition and then the theorem. But often happens that many definition which should stay together need to be split 
because a theorem is required in between.
A tentative example:
Definition (rational numbers)
Let $\sim$ be the equivalence relation on $\mathbb Z^*\times \mathbb Z$ given 
by
$$
  (q,p) \sim (q',p') \iff pq' = p'q.
$$
We define $\mathbb Q= (\mathbb Z^*\times \mathbb Z)/\sim$. 
On $\mathbb Q$ we define addition and multiplication as follows 
$$
   [(q,p)] + [(q',p')] = [(qq',pq'+p'q)] \\
   [(q,p)] \cdot[(q',p')] = [(qq',pp')]
$$
With these operations and choosing 
$0_\mathbb Q=[(1,0)]$, and $1_\mathbb Q=[(1,1)]$ 
turns out that $\mathbb Q$ is a field.
Proof.
We are going to prove that $\sim$ is indeed an equivalence relation,
that addition and multiplication are well defined and that the resulting 
set is a field. [...]

REPLY [2 votes]: I disagree with the implication that it's always necessary, when trying to write something to be understood, that every sentence has to be a logical consequence of previous sentences or assumed background knowledge.
If you're careful about it, and you say this is what you're doing, I think it can be pedagogical to state a definition first and then show that it makes sense. Especially if the verification is routine and you don't want the definition to be buried among a mass of trivialities. I can give examples of this from my own writing.