Assume $x,y \in F$ and F is an ordered field, if $x<0$ and $y<0$, then $xy>0$.

Question

I was thinking since $x<0$ and $y<0$, $-x>0$ and -y<0 by adding the additive inverse. Then we multiply -x and -y to get $(-x)(-y)>0$ since $-x>0$ and $-y>0$, but I feel like this is wrong.

Is it $x=-x$ and $y=-y$ since $x<0$ and $y<0$? If so, why? Can someone help me, please?

Answer 1

By your work $$(-x)(-y)+(-x)y>(-x)y,$$ which gives $$(-x)y<0$$ and from here $$(-x)y+xy<xy,$$ which gives $$xy>0.$$ We used that $0x=0,$ which I hope you can prove.