Cartesian Product Distributes over Union

Theorem
Cartesian product is distributive over union:


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

Proof
Take the result Cartesian Product of Unions:
 * $\left({S_1 \cup S_2}\right) \times \left({T_1 \cup T_2}\right) = \left({S_1 \times T_1}\right) \cup \left({S_2 \times T_2}\right) \cup \left({S_1 \times T_2}\right) \cup \left({S_2 \times T_1}\right)$

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

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

The other result is proved similarly.