READING:     II     1     F10                    Back to reading:     II     1     F9
begin quote ita"
CHAPTER II [:] The Real Number System [...] 1. THE FIELD PROPERTIES. "end quote
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F10:       -a   =   (-1) · a

For any real number   a,    -a = (-1)·a
  0  
  =  
  0 · a  
  F5: Zero times any real number equals zero.  
   
  =  
  a · (1+(-1))  
  commutativity and additive inverse  
   
  =  
  a + (-1)·a  
  distributivity, neutral element for multiplication, and commutativity  
  But  
   
   
  we also have  
  0  
  =  
  a + (-a)  
  existence of additive inverse  
  And  
   
   
  both the previous equation and the one before it are of the form  
  0  
  =  
  a + x  
  So, by F3, we have the desired result.  

Before going on to the next section,
Rosenlicht remarks that there is a field consisting of just the two elements,   0   and   1.
In my first reading, I looked at this simple field here.

on to section 2