Positive Integer is Well-Defined

Theorem
"Positive" as applied to an integer is a well-defined operator:

Proof
Let us define $$\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$$ as in the formal definition of integers.

That is, $$\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $$\boxminus$$.

$$\boxminus$$ is the congruence relation defined on $$\N \times \N$$ by $$\left({x_1, y_1}\right) \boxminus \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1$$.

In order to streamline the notation, we will use $$\left[\!\left[{a, b}\right]\!\right]$$ to mean $$\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$$, as suggested.

Thus, what we are trying to prove is:


 * $$\left[\!\left[{a, b}\right]\!\right] = \left[\!\left[{c, d}\right]\!\right] \land b < a \Longrightarrow d < c$$

By definition, $$\left[\!\left[{a, b}\right]\!\right] = \left[\!\left[{c, d}\right]\!\right] \iff a + d = b + c$$.

So:

$$ $$ $$ $$ $$