Construction of Inverse Completion/Congruence Relation

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

Proof
From Construction of Inverse Completion/Equivalence Relation we have that $\boxtimes$ is an equivalence relation.

We now need to show that:

So:

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