# Invertible implies cancellative in monoid

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## 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 with binary operation , and identity element (also called *neutral element*) . has a left inverse (i.e. an element )

*To prove*: is left-cancellative: whenever are such that , then

*Proof*: We start with:

Left-multiply both sides by :

Use associativity:

We now use that is the identity element, to conclude that .