Product of Semigroup Element with Right Inverse is Idempotent

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

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

Then:


 * $$\left({x_R \circ x}\right) \circ \left({x_R \circ x}\right) = x_R \circ x$$

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

Proof
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