Product of Semigroup Element with Right Inverse is Idempotent

Theorem
Let $\struct {S, \circ}$ be a semigroup with a right identity $e_R$.

Let $x \in S$ such that $\exists x_R: x \circ x_R = e_R$, that is, $x$ has a right inverse with respect to the right identity.

Then:


 * $\paren {x_R \circ x} \circ \paren {x_R \circ x} = x_R \circ x$

That is, $x_R \circ x$ is idempotent.

Also see

 * Product of Semigroup Element with Left Inverse is Idempotent