O4:    (the product of the positive greater) > (the product of the positive less)

"If  a>b>0  (meaning that  a>b  and  b>0) and  [Graphics:../Images/index_gr_111.gif],  then  ac > bd."  (see ita-2-2-9-pg20)

"For  a-b  [is in  R+]  and  c  [is in  R+],  so  ac-bc = (a-b)c  [is in  R+],  and similarly  c-d  [is either in  R+  or is zero] and  b  [is in  R+]  together imply that  bc-bd  [is either in  R+  or is zero]; it necessarily follows that  ac-bd = (ac-bc)+(bc-bd)  [is in  R+],  that is  ac>bd."  (see ita-2-2-9-pg20)

Because:  If  bc-bd=0,  then  ac-bd = ac-bc  which is in  R+,  and if  bc-bd  is in  R+,  then  ac-bd  is in  R+  because both  ac-bc  and  bc-bd  are in  R+.


Converted [ in part ] by Mathematica      December 4, 2007