Union Distributes over Union

Theorem
Set union is distributive over itself:


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

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.