Positive Integer is Well-Defined

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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:

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

$\blacksquare$