Refer to Definition 1.3, which states, an ordered field is a field F that is ordered set with the following additional properties:
If x>0 and y>0, then x+y>0.
If x>0 and y>0, then xy>0.
x<y if and only if y-x>0.
Prove that property (3) implies property (1). Be certain to use only properties that have been proved or already assumed to be true.
Suppose that $a > 0$ and $b > 0$. We want to show that $a + b > 0$.
Now take $y = 0$ and $x = -b$. Then since $0 - (-b) = b > 0$, it follows by property $(3)$ that $-b < 0$. But since $0 < a$, it follows by the transitivity of $<$ that $-b < a$. Hence, by taking $x = -b$ and $y = a$, it follows by property $(3)$ that $a - (-b) = a + b > 0$, as desired.