Operations with Identities which Distribute over each other are Idempotent

Theorem
Let $S$ be a set.

Let $\odot$ and $\otimes$ be operations on $S$ which have identity elements $e$ and $u$ respectively.

Let $\odot$ and $\otimes$ be distributive over each other.

Then $\odot$ and $\otimes$ are both idempotent.

Proof
In the same way:

Then we have:

and:

Hence the result by definition of idempotent operation.