Construction of Inverse Completion/Natural Number Difference

Natural Number Difference in Inverse Completion
In the context of the natural numbers, the difference is defined as:


 * $n - m = p \iff m + p = n$

from which it can be seen that the above congruence can be understood as:


 * $\left({x_1, y_1}\right) \boxtimes \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1 \iff x_1 - y_1 = x_2 - y_2$

Thus this congruence defines an equivalence between pairs of elements which have the same difference.