# Cartesian Product of Intersections/Corollary 1

## Corollary to Cartesian Product of Intersections

$A \times \paren {B \cap C} = \paren {A \times B} \cap \paren {A \times C}$

## Proof

Take the result Cartesian Product of Intersections:

$\paren {S_1 \cap S_2} \times \paren {T_1 \cap T_2} = \paren {S_1 \times T_1} \cap \paren {S_2 \times T_2}$

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

 $\ds A \times \paren {B \cap C}$ $=$ $\ds \paren {A \cap A} \times \paren {B \cap C}$ Intersection is Idempotent $\ds$ $=$ $\ds \paren {A \times B} \cap \paren {A \times C}$ Cartesian Product of Intersections

$\blacksquare$