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 a 6, 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 = e = l where 

 I is any symbol oiK[wn]. 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 e. 



Then {loaob)e= \ lo(aob) fcby associative law, 

 = leo{aob)e by hypothesis, 

 = leo(aeobe) by hypothesis, 

 = leoacobe by associative law, 

 = (/oa)co6c. 

 Similarly in every other case when a, ^, e are replaced 

 by/oa, lob, loe respectively. But by definition 

 i(^oNa.) = noaDNa and so on. Therefore by strict 

 induction the formulas hold for every a, b, e 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 otK[wa] the definition gives 



(h o k) I = 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 f or any two unequal ele- 

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



33. Corollary 2. The higher rule is associative. 



