Intersection Distributes over Intersection

Theorem
Set intersection is distributive over itself:


 * $\forall A, B, C: \paren {A \cap B} \cap \paren {A \cap C} = A \cap B \cap C = \paren {A \cap C} \cap \paren {B \cap C}$

where $A, B, C$ are sets.

Proof
We have:
 * Intersection is Associative
 * Intersection is Commutative
 * Intersection is Idempotent

The result follows from Associative Commutative Idempotent Operation is Distributive over Itself.