READING:     II     1     F8                    Back to reading:     II     1     F7
begin quote ita"
CHAPTER II [:] The Real Number System [...] 1. THE FIELD PROPERTIES. "end quote
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F8:       - ( a + b )    =    (-a) + (-b)

For any real numbers   a   and   b,    - ( a + b ) = (-a) + (-b)
  0  
  =  
  -(a+b)   +  (a+b)  
  existence of addition, neutral elements, and additive inverses  
   
  =  
  0 + 0  
  existence and property of the neutral element for addition  
   
  =  
  0 + ( b + (-b) )  
  additive inverse of   b  
   
  =  
  (0 + b) + (-b)  
  associativity  
   
  =  
  ( (a+(-a)) + b ) + (-b)  
  neutral element and additive inverse of   a  
   
  =  
  ( ((-a)+a) + b ) + (-b)  
  commutativity  
   
  =  
  ( (-a) + (a + b) ) + (-b)  
  associativity  
   
  =  
  ( (a + b) + (-a) ) + (-b)  
  commutativity  
   
  =  
  (a + b) + ((-a) + (-b))  
  associativity  
   
  =  
  ((-a) + (-b)) + (a + b)  
  commutativity  
  0  
  =  
  x + (a + b
Therefore, by F3, we have the desired result because
the first line and the previous line are of this form.

on to F9