Construction of Inverse Completion/Quotient Structure

Theorem
This cross-relation is a congruence relation on $S \times C$.

Let the quotient structure defined by $\boxtimes$ be:
 * $\struct {T', \oplus'} := \struct {\dfrac {S \times C} \boxtimes, \oplus_\boxtimes}$

where $\oplus_\boxtimes$ is the operation induced on $\dfrac {S \times C} \boxtimes$ by $\oplus$.

Proof
From the defined equivalence relation, we have that:
 * $\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 \circ y_2 = x_2 \circ y_1$

is a congruence relation on $\struct {S \times C, \oplus}$.

From the definition of the members of the equivalence classes:
 * $(1) \quad \forall x, y \in S, a, b \in C: \tuple {x \circ a, a} \boxtimes \tuple {y \circ b, b} \iff x = y$


 * $(2) \quad \forall x, y \in S, a, b \in C: \eqclass {\tuple {x \circ a, y \circ a} } \boxtimes = \eqclass {\tuple {x, y} } \boxtimes$

From the definition of the equivalence class of equal elements:
 * $(3) \quad \forall c, d \in C: \tuple {c, c} \boxtimes \tuple {d, d}$

where $\eqclass {\tuple {x, y} } \boxtimes$ is the equivalence class of $\tuple {x, y}$ under $\boxtimes$.

Hence we are justified in asserting the existence of the quotient structure:
 * $\struct {T', \oplus'} = \struct {\dfrac {S \times C} \boxtimes, \oplus_\boxtimes}$