Right Operation is Distributive over Idempotent Operation

Theorem
Let $\left({S, \circ, \rightarrow}\right)$ be an algebraic structure where $\rightarrow$ is the right operation and $\circ$ is any arbitrary binary operation.

Then $\rightarrow$ is distributive over $\circ$ iff $\circ$ is idempotent.

Proof
From Right Operation is Left Distributive over All Operations we have that:
 * $\forall a, b, c \in S: a \rightarrow \left({b \circ c}\right) = \left({a \rightarrow b}\right) \circ \left({a \rightarrow c}\right)$

for all binary operations $\circ$.

It remains to show that $\rightarrow$ is right distributive over $\circ$ iff $\circ$ is idempotent.

Necessary Condition
Suppose $\circ$ is idempotent.

Then by definition of the right operation:

Thus $\rightarrow$ is right distributive over $\circ$.

Sufficient Condition
Suppose $\rightarrow$ is right distributive over $\circ$.

Let $c \in S$ be arbitrary.

Then:

Hence $\circ$ is idempotent.