Order04 - Larger Positives Multiplied

If
a > b > 0
( meaning that   a > b   and   b > 0 )
and
cd > 0
then
ac > bd .
See ita-2-2-9-pg20.

We have
.

By the hypotheses, both (a-b) and c are in , while b is in and (c-d) is in or is equal to zero. Thus it follows from the order property that the first term on the right hand side above is in , while the second is either in or is equal to zero. It follows, then, that the right hand side is in , and we have the required result.