Equivalence Class of Equal Elements of Cross-Relation

Theorem
Then:
 * $\forall c, d \in S_1 \cap S_2: \tuple {c, c} \boxtimes \tuple {d, d}$

Proof
Note that in order for $\tuple {c, c}$ and $\tuple {d, d}$ 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.