Cross-Relation is Equivalence Relation

Theorem
Then $\boxtimes$ is an equivalence relation on $\left({S_1 \times S_2, \oplus}\right)$.

Reflexivity

 * $\forall \left({x_1, y_1}\right) \in S_1 \times S_2: x_1 \circ y_1 = x_1 \circ y_1 \implies \left({x_1, y_1}\right) \boxtimes \left({x_1, y_1}\right)$

So $\boxtimes$ is a reflexive relation.

Symmetry
So $\boxtimes$ is a symmetric relation.

Transitivity
So $\boxtimes$ is a transitive relation.

All the criteria are therefore seen to hold for $\boxtimes$ to be an equivalence relation.