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.