Natural Numbers are Non-Negative Integers/Warning

Natural Numbers are Non-Negative Integers: Warning
From a strictly purist point of view it is inaccurate to say that the natural numbers are the non-negative integers, as an integer is technically an element of an equivalence class composed of pairs of elements of $\N$, constructed as detailed in Construction of Inverse Completion.

However, because an Inverse Completion is Unique, it follows that the natural numbers can be considered to be a substructure of the integers from the Inverse Completion Theorem.

With that caveat in mind, the theorem holds.