Cross-Relation is Congruence Relation

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

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

We now need to show that:

First we note that:

Then:

So $\boxtimes$ is a congruence relation on $\struct {S \times C, \oplus}$.