Right Identity while exists Right Inverse for All is Identity

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


 * $\forall x \in S: \exists x_R: x \circ x_R = e_R$

That is, every element of $S$ has a right inverse with respect to the right identity.

Then $e_R$ is also a left identity, that is, is an identity.

Proof
Let $x \in S$ be any element of $S$.

From Right Inverse for All is Left Inverse we have that $x_R \circ x = e_R$.

Then:

So $e_R$ behaves as a left identity as well as a right identity.

That is, by definition, $e_R$ is an identity element.

Also see

 * Left Identity while exists Left Inverse for All is Identity