# Positive Integer is Well-Defined

## Theorem

"Positive" as applied to an integer is well-defined.

## 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:

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

By definition:

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

So:

 $\displaystyle b$ $<$ $\displaystyle a$ $\displaystyle \leadsto \ \$ $\displaystyle \exists p \in \N: a$ $=$ $\displaystyle b + p$ $\displaystyle \leadsto \ \$ $\displaystyle b + p + d$ $=$ $\displaystyle b + c$ $\displaystyle \leadsto \ \$ $\displaystyle p + d$ $=$ $\displaystyle c$ $\displaystyle \leadsto \ \$ $\displaystyle d$ $<$ $\displaystyle c$

$\blacksquare$