Construction of Inverse Completion/Equivalence Relation/Equivalence Class of Equal Elements

Theorem
Then:
 * $\forall c, d \in C: \tuple {c, c} \boxtimes \tuple {d, d}$

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 Equivalence Class of Equal Elements of Cross-Relation.