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:
 * $\displaystyle \left({T', \oplus'}\right) := \left({\frac {S \times C} \boxtimes, \oplus_\boxtimes}\right)$

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

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

is a congruence relation on $\left({S \times C, \oplus}\right)$.

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


 * $(2) \quad \forall x, y \in S, a, b \in C: \left[\!\left[{\left({x \circ a, y \circ a}\right)}\right]\!\right]_\boxtimes = \left[\!\left[{\left({x, y}\right)}\right]\!\right]_\boxtimes$

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

where $\left[\!\left[{\left({x, y}\right)}\right]\!\right]_\boxtimes$ is the equivalence class of $\left({x, y}\right)$ under $\boxtimes$.

Hence we are justified in asserting the existence of the quotient structure:
 * $\displaystyle \left({T', \oplus'}\right) = \left({\frac {S \times C} \boxtimes, \oplus_\boxtimes}\right)$