FRIZELL: FOUNDATIONS OF ARITHMETIC. 389 



30. Definition. The higher rule shall be defined 

 inductively for the remaining symbols of our abelian 

 semigroup on the lower rule according to the formulas 



{aob)c = acobc and a{boc) = aboac, 

 where the combinations ab, ac, be shall have been 

 already defined. 



31. Proposition XIII. The higher rule is distribu- 

 tive over the lower throughout the class of symbols 

 defined in § 22. Proof. First let a = b = c = l where 

 I is any symbol oi K[iva]. Then II is defined by § 28 

 and l{lol) =lloll = {lol)l by § 30. Now suppose 

 that the formulas of § 30 have been established for all 

 symbols of K [ ^ o ] up to and including a certain a and 

 for every b and c. 



Then {loaob)c ^ \ lo{aob) fcby associative law, 



= lco(aob)c by hypothesis, 

 = lco{acobc) by hypothesis, 

 = lcoacobc by associative law, 

 = (loa)cobc. 

 Similarly in every other case when a, h, c are replaced 

 by^oa, lob, loc respectively. But by definition 

 ^(^oNa) = /^oanNa and so on. Therefore by strict 

 induction the formulas hold for every a, b, c belonging 

 to the same K[^o]. Hence by Prop. V the theorem 

 is true in this case. If h, k, I denote different mem- 

 bers of K [ -w; D ] the definition gives 



{h o k)l = hi okl Bind h(kol) —hkohl. 

 Then the proof is completed inductively by the same 

 reasoning as above. 



32. Corollary 1. The higher rule is a C-rule and 

 by it unequals with equals give unequals. For since 

 our symbols form a well ordered set according to the 

 lower rule, we have a = b o x for any two unequal ele- 

 ments a , b and therefore ac=^bcoxc .". not = b c . 



33. Corollary 2. The higher rule is associative. 



