# Inverse Completion of Natural Numbers

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## Theorem

There exists an inverse completion of the natural numbers under addition.

## Proof

The set of natural numbers under addition can be denoted $\left ({\N, +}\right)$.

From Natural Numbers under Addition form Commutative Monoid, the algebraic structure $\left ({\N, +}\right)$ is a commutative monoid.

Therefore by definition of commutative monoid, $\left ({\N, +}\right)$ is a commutative semigroup.

From Natural Number Addition is Cancellable, all of the elements of $\left ({\N, +}\right)$ are cancellable.

The result follows from the Inverse Completion Theorem.

$\blacksquare$