Set Intersection is Self-Distributive

Theorem
Set intersection is distributive over itself:


 * $\forall A, B, C: \left({A \cap B}\right) \cap \left({A \cap C}\right) = A \cap B \cap C = \left({A \cap C}\right) \cap \left({B \cap C}\right)$

where $A, B, C$ are sets.

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

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