Set Union is Self-Distributive

Theorem
Set union is distributive over itself:


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

where $A, B, C$ are sets.

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

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