Cross-Relation on Natural Numbers is Equivalence Relation

Theorem
Let $\left({\N, +}\right)$ be the semigroup or 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 $\N$.

The relation $\mathcal R$ defined on $\N \times \N$ by:
 * $\left({x_1, y_1}\right) \ \mathcal R \ \left({x_2, y_2}\right) \iff x_1 \circ y_2 = x_2 \circ y_1$

is an equivalence relation on $\left({\N \times \N, \oplus}\right)$.

Proof
$\mathcal R$ is an instance of a cross-relation.

The result therefore follows from Cross-Relation is Equivalence Relation.