Invertible Element of Associative Structure is Cancellable

Theorem
Let $$\left({S, \circ}\right)$$ be an 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 $$x \circ a = y \circ a \implies x = y$$.