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CHAPTER II [:] The Real Number System [...] 2. ORDER. "end quote
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For any real numbers   a, b, c:      ab,   and   bc   imply that   ac
  From previous experience, we might write immediately that  
  a-c = (a-b) + (b-c)  
We recognize in   a+b-b+c   that the b's cancel, and the quantities
in parentheses are in by the meaning of the greater than symbol,
and therefore by the order property the sum of the two parenthetical
terms is in .

However, one of the main points of this work is to derive everything from a few first principles - in order to clear up uncertainties the reader may have about the validity of various operations. Therefore we have the following:

  a-c = a + (-c)  
  definition of subtraction  
  (a+0) + (-c)  
  existence of the neutral element for addition  
  ( a + ((-b)+b) ) + (-c)  
  existence of the additive inverse for additon  
  ( (a + (-b)) + b ) + (-c)  
  associativity of addition  
  (a + (-b)) + ( b + (-c) )  
  associativity of addition  
  (a-b) + (b-c)  
  definition of subtraction  

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