Exercise 4, § 3

Let $\top$ and $\perp$ be two internal laws on $E$, each with an identity element; if the left action of $E$ on itself derived from each of the laws is distributive with respect to the other, every element of $E$ is idempotent under both these laws (if $e$ is the identity element for $\top$, $u$ the identity element for $\perp$, prove first that $e \perp e = e$, noting that $e = e \perp (u \top e)$).