Proof of Theorem ZVU

To prove uniqueness, a standard technique is to suppose the existence of two objects (Proof Technique U). So let $\zerovector_1$ and $\zerovector_2$ be two zero vectors in $V$. Then \begin{align*} \zerovector_1 &=\zerovector_1+\zerovector_2 &&\knowl{./knowls/property.Z.knowl}{\text{Property Z}}\text{ for }\zerovector_2\\ &=\zerovector_2+\zerovector_1 &&\knowl{./knowls/property.C.knowl}{\text{Property C}}\\ &=\zerovector_2 &&\knowl{./knowls/property.Z.knowl}{\text{Property Z}}\text{ for }\zerovector_1 \end{align*}

This proves the uniqueness since the two zero vectors are really the same.