Identity of Monoid is Cancellable

Theorem
The identity of a monoid is cancellable.

Proof
Let $$\left({S, \circ}\right)$$ be a monoid whose identity is $$e$$.

Let $$x, y \in S$$ such that $$x \circ e = y \circ e$$

Then, by the definition of the identity:

$$x = x \circ e = y \circ e = y$$

... thus $$x = y$$ and the result is proved.