Cartesian Product of Unions

Theorem

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

Corollary

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

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