Construction of Inverse Completion/Equivalence Relation

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

Proof
From Semigroup is Subsemigroup of Itself, $\struct {S, \circ}$ is a subsemigroup of $\struct {S, \circ}$.

Also from Semigroup is Subsemigroup of Itself, $\struct {C, \circ {\restriction_C} }$ is a subsemigroup of $\struct {C, \circ {\restriction_C} }$.

The result follows from Cross-Relation is Equivalence Relation.