Cartesian Product of Intersections

Theorem

 * $\left({S_1 \cap S_2}\right) \times \left({T_1 \cap T_2}\right) = \left({S_1 \times T_1}\right) \cap \left({S_2 \times T_2}\right)$

Corollary 1

 * $A \times \left({B \cap C}\right) = \left ({A \times B}\right) \cap \left ({A \times C}\right)$

Corollary 2

 * $\left({A \times B}\right) \cap \left({B \times A}\right) = \left ({A \cap B}\right) \times \left ({A \cap B}\right)$

Proof of Corollary 1
Put $S_1 = S_2 = A, T_1 = B, T_2 = C$:

Proof of Corollary 2
Put $S_1 = A, S_2 = B, T_1 = B, T_2 = A$: