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

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

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