"If 
then 
, so that 
."
"Since 
, we have 
."
"Hence %3E0)
."
This is true by the second rule for ordered fields.
"And therefore +xy%3E0+xy=xy)
."
Thus we have 
.
"By (a), (b), and Proposition 1.16(c), )=(-x)(z-y)%3E0)
."
I find that using (b) makes this proof unnecessarily complicated. Instead, here is another proof.
First, since 
, by (a) we have 
. Furthermore, since , we have 
(see above for proof). By 1.16(c), )=(-x)(z-y))
. Since both elements of the right-hand side of the equation
exceed zero, the second rule for ordered fields mandates that the product of these elements also exceed zero. Thus, )%3E0)
.
"...so that %3C0)
."
Because )%3E0)
, we have
"Hence 
."
therefore, by the first rule for ordered fields, 
. By (A4), 
, and also,
Thus, 
.
"If , then ."
This is because of (a).
"Hence ^{2}%3E0)
."
This is because of the second rule for ordered fields.
"If 
and 
, then 
."
If 
, we have
If 
, consider (c). Take 
as the 
used in (c), 
as the 
used in (c), and our current as (c)'s 
. Then, by (c), 
. We have
and (M2) gives 
. Thus, if 
, we have 
.
Therefore, if and , we have .
"But =1%3E0 "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
."
"Hence %3E0 "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
."
We know 
, and 
. Now assume that 
. Then, taking our 
as the 
in (b), 
as (b)'s , and 
as (b)'s 
, we must have
This, however, is exactly contrary to what was just proved above. Therefore our assumption must be false, and 
.
"If we multiply both sides of the inequality 
by the positive quantity (1/y) "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
..."
As we have just proved, both and 
exceed zero. Therefore, by the second rule for ordered fields, their product also exceeds zero.
"...we obtain 
."
Take \cdot%20\left(\frac{1}{y}\right) "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
as in (b), our as (b)'s , and our as (b)'s . Then %20\left(%20\frac{1}{y}\right)%20\cdot%20x%20%3C%20\left(\frac{1}{x}\right)%20\left(\frac{1}{y}\right)%20\cdot%20y "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
. Now consider:
Similarly, %20\left(\frac{1}{y}\right)%20\cdot%20y%20=%20\left(\frac{1}{x}\right) "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
. Then 
when 
.
