Showing that $\mathbb{C}$ under the lexicographic order does not satisfy $x,y>0\implies xy>0$

Question

Identifying $\mathbb{C}$ with $\mathbb{R} \times \mathbb{R}$, we can consider the lexicographic order: we define $x_1 + iy_1 < x_2 +iy_2$ if either

A) $x_1 < x_2$ or B) $x_1 = x_2$ and $y_1 < y_2$.

Prove that this relation does not satisfy Axiom 8 of Apostles chapter.

where Axiom 8 states If $x > 0$ and $y > 0$, then $xy > 0$

I don't know how to go about this question, yeah I understand what we were defined on the complex line. But I don't know how to relate this to axiom 8. Please help out, thank you.

Answer 1

HINT: $0<i$; what about $i^2$?