Invertible Element of Associative Structure is Cancellable

Theorem
Let $\struct {S, \circ}$ be an algebraic structure where $\circ$ is associative.

Let $\struct {S, \circ}$ have an identity element $e_S$.

An element of $\struct {S, \circ}$ 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 $x \circ a = y \circ a \implies x = y$.