Equivalence Class of Equal Elements of Cross-Relation

Theorem
Then:
 * $\forall c, d \in S_1 \cap S_2: \left({c, c}\right) \boxtimes \left({d, d}\right)$

Proof
Note that in order for $\left({c, c}\right)$ and $\left({d, d}\right)$ to be defined, $c$ and $d$ must be in both $S_1$ and $S_2$.

Hence the restriction given:
 * $\forall c, d \in S_1 \cap S_2$

Then:

Hence the result.