Product is Right Identity Therefore Right Cancellable

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

Let $e_R \in S$ be a right identity of $S$.

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

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

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

Then:

The result follows by definition of right cancellable.

Also see

 * Product is Left Identity Therefore Left Cancellable