Invertible Element of Associative Structure is Cancellable

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

An element of $\left({S, \circ}\right)$ which is invertible is also cancellable.

Proof
Let $a \in S$ be invertible.

Suppose $a \circ x = a \circ y$.

Then:

A similar argument shows that $x \circ a = y \circ a \implies x = y$.