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CHAPTER II [:] The Real Number System [...] 2. ORDER. "end quote
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O1:       TRICHOTOMY

For any real numbers   a, b:      ab,   a=b,   or   ab
   
Apply the order property to the number   a-b.
Then we have
   
  1.  
  a-b    
   
   
   
   
  or 2.  
  0 = a-b  
   
  =  
  a + (-b)  
  by the definition of subtraction  
  So:  
  -b = -a  
  by the uniqueness of the additive inverse  
  And then  
  -(-a) = -(-b)  
   
  implies  
  a = b  
  by F6  
   
   
   
  or 3.  
  -(a-b)    
   
  But  
  -(a-b) = (-1)·(a-b)  
  by F10  
  =  
  (-1)( a + (-b) )  
  by the definition of subtraction  
  =  
  (-1)(a) + (-1)(-b)  
  by the distributive property of multiplication over addition  
  =  
  (-a) + ( -(-b) )  
  by F10  
  =  
  (-a) + b  
  by F6  
  =  
  b + (-a)  
  by the commutativity of addition  
  =  
  b-a  
  by the definition of subtraction  

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