Associative Commutative Idempotent Operation is Distributive over Itself

Theorem
Let $\struct {S, \circ}$ 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:
 * $\forall a, b, c \in S: \paren {a \circ b} \circ \paren {a \circ c} = a \circ b \circ c = \paren {a \circ c} \circ \paren {b \circ c}$