Product of Semigroup Element with Left Inverse is Idempotent

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

Let $$x \in S$$ such that $$\exists x_L: x_L \circ x = e_L$$, i.e. $$x$$ has a left inverse with respect to the left identity.

Then:


 * $$\left({x \circ x_L}\right) \circ \left({x \circ x_L}\right) = x \circ x_L$$

... that is, $$x \circ x_L$$ is idempotent.

Proof
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