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.