Rational Addition is Closed

Theorem
The operation of addition on the set of rational numbers $$\Q$$ is well-defined and closed:
 * $$\forall x, y \in \Q: x + y \in \Q$$

Proof
Follows directly from the definition of rational numbers as the quotient field of the integral domain $$\left({\Z, +, \times}\right)$$ of integers.

So $$\left({\Q, +, \times}\right)$$ is a field, and therefore a priori $$+$$ is well-defined and closed on $$\Q$$.