Rational Subtraction is Closed

Theorem
The set of rational numbers is closed under subtraction:
 * $\forall a, b \in \Q: a - b \in \Q$

Proof
From the definition of subtraction:
 * $a - b := a + \left({-b}\right)$

where $-b$ is the inverse for rational number addition.

As the algebraic structure $\left({\Q, +}\right)$ forms a group, it follows that:
 * $\forall a, b \in \Q: a + \left({-b}\right) \in \Q$

Therefore rational number subtraction is closed.