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 field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.

So $\struct {\Q, +, \times}$ is a field, and therefore a fortiori $+$ is well-defined and closed on $\Q$.

$\blacksquare$