Real Subtraction is Closed

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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 Abelian Group:

$\forall a, b \in \R: a + \paren {-b} \in \R$

Therefore real number subtraction is closed.

$\blacksquare$


Sources