Right Inverse for All is Left Inverse

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 $x_R \circ x = e_R$, that is, $x_R$ is also a left inverse with respect to the right identity.

Proof
Let $y = x_R \circ x$. Then:

So $x_R \circ x = e_R$, and $x_R$ behaves as a left inverse as well as a right inverse with respect to the right identity.

Also see

 * Left Inverse for All is Right Inverse


 * Right Identity while exists Right Inverse for All is Identity
 * Left Identity while exists Left Inverse for All is Identity