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

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

From Real Numbers under Addition form Group:
 * $\forall a, b \in \R: a + \paren {-b} \in \R$

Therefore real number subtraction is closed.