Real Subtraction is Closed

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

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

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

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

Therefore real number subtraction is closed.