O3:    (the sum of the greater) > (the sum of the less)

"If  a>b  and  [Graphics:../Images/index_gr_108.gif]  then  a+c > b+d.  In fact, the hypotheses mean [that]  a-b  [is in  R+]  and  c-d  [is either in  R+ or is zero]."  (see ita-2-2-8-pg20)

[Graphics:../Images/index_gr_109.gif]
[Graphics:../Images/index_gr_110.gif]

If  c-d  is zero, then we have the desired result because  a-b  is in  R+.  If  c-d  is  in  R+,  then we have the desired result because so is  a-b.

ita


Converted [ in part ] by Mathematica      December 4, 2007