##
*Order04 - Larger Positives Multiplied*

If

*a* > *b* > 0

( meaning that *a* > *b* and *b* > 0 )

and

*c* ≥ *d* > 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.