Proof of Theorem AIU

To prove uniqueness, a standard technique is to suppose the existence of two objects (Proof Technique U). So let $\vect{-u}_1$ and $\vect{-u}_2$ be two additive inverses for $\vect{u}$. Then \begin{align*} \vect{-u}_1&=\vect{-u}_1+\zerovector&&\knowl{./knowls/property.Z.knowl}{\text{Property Z}}\\ &=\vect{-u}_1+(\vect{u}+\vect{-u}_2)&&\knowl{./knowls/property.AI.knowl}{\text{Property AI}}\\ &=(\vect{-u}_1+\vect{u})+\vect{-u}_2&&\knowl{./knowls/property.AA.knowl}{\text{Property AA}}\\ &=\zerovector+\vect{-u}_2&&\knowl{./knowls/property.AI.knowl}{\text{Property AI}}\\ &=\vect{-u}_2&&\knowl{./knowls/property.Z.knowl}{\text{Property Z}} \end{align*}

So the two additive inverses are really the same.