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CHAPTER II [:] The Real Number System [...] 2. ORDER. "end quote
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O3:       THE SUM OF THE GREATER IS GREATER.    ALSO:    THE SUM OF THE LESS IS LESS!!

For any real numbers   a, b, c, d:      ab,   and   cd   imply that   a+cb+d
  By the meaning  
  of  
  the symbol ,  
  we need to look at the quantity   (a+c)-(b+d)   and show that it is in .  
   
   
   
  Furthermore, the given inequalities lead us to look at   a-b   and   c-d.  
   
   
   
  If we know algebra well, this might lead us to jump directly to the last equals sign below. 
  But here are the steps:  
   
   
   
  (a+c)-(b+d)  
  =  
  (a+c) + ( -(b+d) )  
  by the definition of subtraction  
   
  =  
  (a+c) + ( (-1)(b+d) )  
  by F10  
   
  =  
  (a+c) + ( (-1)b + (-1)d )  
  by the distributivity of muliplication over addition  
   
  =  
  (a+c) + ( (-b) + (-d) )  
  by F10  
   
  =  
  ( (a+c) + (-b) ) + (-d)  
  by the associativity of addition  
   
  =  
  ( ( c+a ) + (-b) ) + (-d)  
  by the commutativity of addition  
   
  =  
  ( c + ( a+(-b) ) ) + (-d)  
  by the associativity of addition  
   
  =  
  ( c + ( a-b ) ) + (-d)  
  by the definition of subtraction  
   
  =  
  ( (a-b) + c ) + (-d)  
  by the commutativity of addition  
   
  =  
  (a-b) + ( c + (-d) )  
  by the associativity of addition  
   
  =  
  (a-b) + (c-d)  
  by the definition of subtraction  
  But:  
   
  (a-b)  
  by the given inequality,   ab  
  Furthermore:  
   
  cd  
  was given and implies that either   (c-d)   or   (c-d) = 0  

Consequently, the sum   (a-b) + (c-d)   is in     because of the order property.
Finally, by the chain of equal signs, we have the desired result:   (a+c)-(b+d)

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