Left Cancellable Elements of Semigroup form Subsemigroup

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

Let $C_\lambda$ be the set of left cancellable elements of $\left ({S, \circ}\right)$.

Then $\left({C_\lambda, \circ}\right)$ is a subsemigroup of $\left ({S, \circ}\right)$.

Proof
Let $C_\lambda$ be the set of left cancellable elements of $\left ({S, \circ}\right)$:


 * $C_\lambda = \left\{{x \in S: \forall a, b \in S: x \circ a = x \circ b \implies a = b}\right\}$

Let $x, y \in C_\lambda$.

Then:

Thus $\left({C_\lambda, \circ}\right)$ is closed.

Therefore by the Subsemigroup Closure Test $\left({C_\lambda, \circ}\right)$ is a subsemigroup of $\left ({S, \circ}\right)$.