Several of the above theorems have interesting demonstrations when applied to the crazy vector space, $C$ (Example CVS). We are not proving anything new here, or learning anything we did not know already about $C$. It is just plain fun to see how these general theorems apply in a specific instance. For most of our examples, the applications are obvious or trivial, but not with $C$.
Suppose $\vect{u}\in C$. Then, as given by Theorem ZSSM, \begin{equation*} 0\vect{u}=0(x_1,\,x_2)=(0x_1+0-1,\,0x_2+0-1)=(-1,-1)=\zerovector \end{equation*} And as given by Theorem ZVSM, \begin{align*} \alpha\zerovector &=\alpha(-1,\,-1)=(\alpha(-1)+\alpha-1,\,\alpha(-1)+\alpha-1)\\ &=(-\alpha+\alpha-1,\,-\alpha+\alpha-1)=(-1,\,-1)=\zerovector \end{align*} Finally, as given by Theorem AISM, \begin{align*} (-1)\vect{u} &=(-1)(x_1,\,x_2)=((-1)x_1+(-1)-1,\,(-1)x_2+(-1)-1)\\ &=(-x_1-2,\,-x_2-2)=-\vect{u} \end{align*}