Associative Commutative Idempotent Operation is Self-Distributive

Theorem
Let $\left({S, \circ}\right)$ be an algebraic structure, such that:
 * $(1): \quad \circ$ is associative
 * $(2): \quad \circ$ is commutative
 * $(3): \quad \circ$ is idempotent.

Then $\circ$ is distributive over itself.

That is:
 * $\forall a, b, c \in S: \left({a \circ b}\right) \circ \left({a \circ c}\right) = a \circ b \circ c = \left({a \circ c}\right) \circ \left({b \circ c}\right)$