Integer Subtraction is Closed

Theorem

The set of integers is closed under subtraction:

$\forall a, b \in \Z: a - b \in \Z$

Proof

From the definition of subtraction:

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

where $-b$ is the inverse for integer addition.

From Integers under Addition form Abelian Group, the algebraic structure $\struct {\Z, +}$ is a group.

Thus:

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

Therefore integer subtraction is closed.

$\blacksquare$