Invertible implies cancellative in monoid

Statement
In a monoid (a set with an associative binary operation possessing a multiplicative identity element) the following are true:


 * Any left invertible element (element having a left inverse) is left cancellative.
 * Any right invertible element (element having a right inverse) is right cancellative.
 * Any invertible element is cancellative.

Proof
We'll give here the proof for left invertible and left cancellative. An analogous proof works for right invertible and right cancellative.

Given: A monoid $$M$$ with binary operation $$*$$, and identity element (also called neutral element) $$e$$. $$a \in M$$ has a left inverse $$b$$ (i.e. an element $$b * a = e$$)

To prove: $$a$$ is left-cancellative: whenever $$c,d \in M$$ are such that $$a * c = a * d$$, then $$c =d $$

Proof: We start with:

$$a * c = a * d$$

Left-multiply both sides by $$b$$:

$$b * (a * c) = b * (a * d)$$

Use associativity:

$$(b * a) * c = (b * a) * d$$

We now use that $$b * a = e$$ is the identity element, to conclude that $$c = d$$.