O2:    Transitivity    ( a>b and b>c  implies  a>c )

"If  a>b  and  b>c  then  a>c.  For we are given [that  a-b  and  b-c  are both  R+];  it therefore follows [by the first part of the order property] that  a-c = (a-b) + (b-c)  is in  R+."  (see ita-2-2-7-pg20)

We can see this result at a glance by remembering the first field property consequence that any number of numbers have a unique sum, and that parentheses may therefore be omitted or inserted at will.  Then  b  and its additive inverse will sum to zero leaving  a-c  in the sum of four numbers.

ita


Converted [ in part ] by Mathematica      December 4, 2007