Left Inverse for All is Right Inverse

Theorem
Let $\left({S, \circ}\right)$ be a semigroup with a left identity $e_L$. such that:


 * $\forall x \in S: \exists x_L: x_L \circ x = e_L$

... i.e. every element of $S$ has a left inverse with respect to the left identity.

Then:


 * $x \circ x_L = e_L$, that is, $x_L$ is also a right inverse with respect to the left identity;
 * $e_L$ is also a right identity, that is, is an identity.

A similar complementary result appertains for right inverses with respect to a right identity.

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

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

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

Hence the result.