FRIZELL: FOUNDATIONS OF ARITHMETIC. 387 



But eohoc = eo{boc) by definition. 

 Therefore e'oboc = e'o(boc) and so on. 

 Hence the theorem by strict induction. 



20. Corollary. The rule is also commutative. For 

 if eoa = aoe, then by the associative law eoaoe = 

 eo{aoe) = eo(eoa) . But eoe' = e'oe by § 19, there- 

 fore eoe'' = e"oe, and so on. And if /^oa = ao6for 

 a certain a and every h then ho{eoa) = bo{aoe) = 

 i>oaoe = aoboe = eo{aob) = eoaob. 



Hence the proposition follows by strict induction. 



21. Proposition VII . The formula of § 19 is also a 

 sufficient condition of the e-set constituting an abelian 

 semigroup for its generating C-rule. Proof. If aob 

 belongs to the given e-set, then eoaob — eo {aob) 

 also belongs to it. Thus the first group property is es- 

 tablished. We have proved the associative law and by 

 definition no two numbers of the set are equal. There- 

 fore a' o 6 is not =aob when a' is not = a. 



22. Postulates. Using capital letters M, N, . . . . 

 to denote elements of the lowest class K[tfo], and 

 italic letters a, /^ . . . for those of the class K[t^n] 

 where a shall precede />, we postulate for every MB = 

 BM 2i set MB o Na for every Na, then using letters 

 a, b, . . . . where a shall precede b to denote the re- 

 sulting symbols we postulate inductively new sets boa 

 where a = Na and b denotes successively the higher 

 ''polynomial" or composite symbols. Every b-set is to 

 be simply ordered and possess the first group property 

 for every boa. 



23. Proposition VIII. Every b-set is well ordered, 

 infinite and forms a K-class for the rule denoted by o . 

 For the b-set consists of the combinations b, b o a , 

 b o a li', b o a u", boa u'", .... and the reasoning 

 of Prop. I holds good. 



24. Proposition IX. The b-sets belonging to a 



