Cross-Relation is Congruence Relation

Theorem
The cross-relation $\boxtimes$ is a congruence relation on $\left({S_1 \times S_2, \oplus}\right)$.

Proof
From Cross-Relation is Equivalence Relation we have that $\boxtimes$ is an equivalence relation.

We now need to show that:

So:

So $\boxtimes$ is a congruence relation on $\left({S \times C, \oplus}\right)$.