Equivalence Classes of Cross-Relation on Natural Numbers

Theorem
Let $\left({\N, +}\right)$ be the semigroup of natural numbers under addition.

Let $\left({\N \times \N, \oplus}\right)$ be the (external) direct product of $\left({\N, +}\right)$ with itself, where $\oplus$ is the operation on $\N \times \N$ induced by $+$ on $\N$.

Let $\boxtimes$ be the cross-relation defined on $\N \times \N$ by:
 * $\left({x_1, y_1}\right) \boxtimes \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1$

Then $\left[\!\left[{\left({x, y}\right)}\right]\!\right]_\boxtimes$ is the equivalence class of $\left({x, y}\right)$ under $\boxtimes$, where:
 * $\left[\!\left[{\left({x_1, y_1}\right)}\right]\!\right]_\boxtimes = \left[\!\left[{\left({x_2, y_2}\right)}\right]\!\right]_\boxtimes \iff \left({x_1, y_1}\right) \boxtimes \left({x_2, y_2}\right)$

The equivalence class $\left[\!\left[{\left({x, y}\right)}\right]\!\right]_\boxtimes$ is more often denoted more simply as $\left[\!\left[{x, y}\right]\!\right]$.

Proof
We have that $\left({\N, +}\right)$ is a commutative semigroup from Natural Number Addition is Commutative.

We also have the result Cross-Relation on Natural Numbers is Equivalence Relation.

The result follows from the definition of equivalence class.