READING:     II     1     F9                    Back to reading:     II     1     F8
begin quote ita"
CHAPTER II [:] The Real Number System [...] 1. THE FIELD PROPERTIES. "end quote
Please get the book and read along.    
topic index     contents

F9:       ( a · b )    =    a· b

For any   NONZERO   real numbers   a   and   b,    ( a · b ) = a· b
  1  
  =  
  (a·b) ·   (a·b)  
  existence of multiplication, neutral elements, and multiplicative inverses  
   
  =  
  1 · 1  
  existence and property of the neutral element for multiplication  
   
  =  
  1 · ( b · b )  
  multiplicative inverse of   b  
   
  =  
  (1 · b) · b  
  associativity  
   
  =  
  ( (a·a) · b ) · b  
  neutral element and multiplicative 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  
  1  
  =  
  x · (a · b
Therefore, by F4, we have the desired result because
the first line and the previous line are of this form.

on to F10