Field property proof

Question

I have tried quite a lot of "turn and twists" but I just can`t get around it. Here it is :

if x > y then x > z > y

for all $x,y$ in the partially ordered field $K$ exists such $z \in K$. For the real numbers f. e. that is perfectly clear but here I seem to believe that I don`t have enough info over $K$. Any thoughts, hints or suggestions on how to proceed?

Answer 1

Put $z = \frac{x+y}{2}$ (the field can't have characteristic $2$ except in the trivial case when $x > y$ is always false). Then $x > z > y$.