# Natural Number Addition is Closed

## Theorem

The operation of addition on the set of natural numbers $\N$ is closed:

$\forall x, y \in \N: x + y \in \N$

## Proof

Follows directly from Natural Numbers under Addition form Commutative Monoid.

A monoid by definition is a semigroup.

Again by definition, the operation in a semigroup is closed.

$\blacksquare$