Left Identity while exists Left Inverse for All is Identity

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


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

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

Then $e_L$ is also a right identity, that is, is an identity.

Proof
From Left Inverse for All is Right Inverse we have that $x \circ x_L = e_L$.

Then:

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

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

Also see

 * Right Identity while exists Right Inverse for All is Identity