Left Inverse for All is Right Inverse

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

Also see

 * Right Inverse for All is Left Inverse


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