Construction of Inverse Completion/Equivalence Relation

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

Proof
From Semigroup is Subsemigroup of Itself, $\left({S, \circ}\right)$ is a subsemigroup of $\left({S, \circ}\right)$.

Also from Semigroup is Subsemigroup of Itself, $\left({C, \circ_{\restriction_C}}\right)$ is a subsemigroup of $\left({C, \circ_{\restriction_C}}\right)$.

The result follows from Cross-Relation is Equivalence Relation.