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

Theorem
Then:
 * $\forall c, d \in C: \left({c, c}\right) \boxtimes \left({d, d}\right)$

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 Equivalence Class of Equal Elements of Cross-Relation.