In particular, assuming associativity and the existence of a left identity e (that is, ⁠ e ⋅ ⋅ f = f ⁠) and a left inverse f − − 1 for each element f (that is, ⁠ f − − 1 ⋅ ⋅ f = e ⁠), one can show that every left inverse is also a right inverse of the same element as follows. Indeed, one has