Left Self-Distributive Operation with Right Identity is Idempotent

Theorem
Let $\struct {S, \circ}$ be an algebraic structure.

Let $\circ$ be left self-distributive.

Let $\struct {S, \circ}$ have a right identity.

Then $\circ$ is an idempotent operation.

Proof
Let the right identity of $\struct {S, \circ}$ be $e_R$.

We have:

The result follows by definition of idempotent operation.