# Left-invertible elements of monoid form submonoid

## Statement

Let be a monoid with neutral element . Then, the set of left-invertible elements of form a submonoid of .

## Related facts

Inverse map is involutive: Specifically the reversal law for inverses.

## Proof

**Given**: A monoid with neutral element

**To prove**: is left-invertible, and further, if are left-invertible elements of , so is

**Proof**: Clearly, is left-invertible, since .

Suppose are left-invertible and are left inverses for . Then consider:

Thus, is a left inverse for , so is left-invertible, completing the proof.