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CHAPTER II [:] The Real Number System [...] 2. ORDER. "end quote
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O8:       THE MULTIPLICATIVE INVERSE OF THE POSITIVE GREATER IS LESS

For all   a,b   such that   ab0   we have   .
   
   
   
  ab0  
  by O5 :   positive · positive = positive
   
   
   
  (ab)0  
  by O7  
   
   
   
  a(ab)b(ab)  
  by O4   where   c = d = (ab)  
   
   
   
  a(ab)(ab)b  
  by F9  
   
   
   
  (a·a)ba(b·b)  
  by the associativity of multiplication  
   
   
   
  (1)ba(1)  
  by the property of the multiplicative inverse  
   
   
   
  ba  
  by the property of the neutral element for multiplication  

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