Elements of Cross-Relation Equivalence Class

Theorem
Let $\left[\!\left[{\left({x, y}\right)}\right]\!\right]_\boxtimes$ be the $\boxtimes$-equivalence class of $\left({x, y}\right)$, where $\left({x, y}\right) \in S_1 \times S_2$.

Then:

$\forall x, y \in S_1, a, b \in S_2:$


 * $(1): \quad \left[\!\left[{\left({x \circ a, a}\right)}\right]\!\right]_\boxtimes = \left[\!\left[{\left({y \circ b, b}\right)}\right]\!\right]_\boxtimes \iff x = y$


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