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:
 * $a - b := a + \paren {-b}$

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

From Rational Numbers under Addition form Infinite Abelian Group, $\struct {\Q, +}$ forms a group.

Thus:
 * $\forall a, b \in \Q: a + \paren {-b} \in \Q$

Therefore rational number subtraction is closed.