# Left inverse property implies two-sided inverses exist

From Groupprops

## Statement

Suppose is a loop with neutral element . Suppose is a left inverse property loop, i.e., there is a bijection such that for every , we have:

Then, is the unique two-sided inverse of (in a weak sense) for all :

Note that it is *not* necessary that the loop be a right-inverse property loop, so it is not necessary that be a right inverse for in the strong sense.

## Related facts

- Equality of left and right inverses in monoid
- Two-sided inverse is unique if it exists in monoid
- Equivalence of definitions of inverse property loop
- Equivalence of definitions of gyrogroup

## Proof

**Given**: A left-inverse property loop with left inverse map .

**To prove**: , where is the neutral element.

**Proof**: Putting in the left inverse property condition, we obtain that .

Next, putting , we obtain that:

Writing the on the right as and using cancellation, we obtain that:

This completes the proof.