Right Operation is Distributive over Idempotent Operation

Theorem
Let $\struct {S, \circ, \rightarrow}$ be an algebraic structure where:
 * $\rightarrow$ is the right operation
 * $\circ$ is any arbitrary binary operation.

Then:
 * $\rightarrow$ is distributive over $\circ$


 * $\circ$ is idempotent.

Proof
From Right Operation is Left Distributive over All Operations:
 * $\forall a, b, c \in S: a \rightarrow \paren {b \circ c} = \paren {a \rightarrow b} \circ \paren {a \rightarrow c}$

for all binary operations $\circ$.

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

Necessary Condition
Let $\circ$ be idempotent.

Then:

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

Sufficient Condition
Let $\rightarrow$ be right distributive over $\circ$.

Let $c \in S$ be arbitrary.

Then:

Hence $\circ$ is idempotent.