Inverse Completion of Natural Numbers

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 (and therefore a commutative semigroup) all of whose elements are cancellable.

The result follows from the Inverse Completion Theorem.