Product is Left Identity Therefore Left Cancellable

Theorem
Let $\struct {S, \circ}$ be a semigroup.

Let $e_L \in S$ be a left identity of $S$.

Let $a \in S$ such that:
 * $\exists b \in S: b \circ a = e_L$

Then $a$ is left cancellable in $\struct {S, \circ}$.

Proof
Let $x, y \in S$ be arbitrary.

Then:

The result follows by definition of left cancellable.

Also see

 * Product is Right Identity Therefore Right Cancellable