Elements of Cross-Relation Equivalence Class

Theorem
Let $\eqclass {\tuple {x, y} } \boxtimes$ be the $\boxtimes$-equivalence class of $\tuple {x, y}$, where $\tuple {x, y} \in S_1 \times S_2$.

Then:

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


 * $(1): \quad \eqclass {\tuple {x \circ a, a} } \boxtimes = \eqclass {\tuple {y \circ b, b} } \boxtimes \iff x = y$


 * $(2): \quad \eqclass {\tuple {x \circ a, y \circ a} } \boxtimes = \eqclass {\tuple {x, y} } \boxtimes$