Cartesian Product of Unions

Theorem

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

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Corollary

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

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

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