Two-sided inverse is unique if it exists in monoid
Suppose is a monoid with binary operation and identity element (neutral element) . Suppose has a two-sided inverse , i.e., . Then, is the only two-sided inverse for , i.e., if is an element such that , then .
Fact (1) says that every left inverse must equal every right inverse. Thus, if we have two two-sided inverses, we can treat one of them as a left inverse and the other as a right inverse, forcing both of them to be equal.