PROPERTIES OF CLASSES OF FUNCTIONS 41 



39a. Propositions. — 



(I) mi ^ W2 : 3 : ^1^_,„/"''"' . 1?_,„.f»'^-' . K,„„ . zd . Z^„,. 

 (2)* A . 2)2^"' : 3 : (3 m , 3 i5_„,) " (m . p . 3 . mp = 0). 

 (3) A.m. 3 . A^'<*'-\ 



(4)t A. W'. m .3. 9[)f^'**-'. 



(5) 9JJ^^ . 3 . 9J?/' . 93?/^ 



(6)t 9Dr^'' . m"^'" . 3 . m'w'T' ■ {W^v')!."' . fiorwo/^ 



m, . mz > m. :3 : ^l^„^ **"■"= . i^„,.„„^»'"^'"' ■ i?^„„.3 • ^ • -S^;-.™, 



(8) j^ „ 



(9) ?Mi . 7rt2 • ;li^^*'"i'"=' . Ws > 7no .Z) . ju** *">"='. 



(10) i^ . A . g??^'^'^^' : 3 : mi . 3 . 3 (mi . m[) i Mf^'*"'"'''- 

 Proof.— 



(o) ^^^'-'. = .^5(3^(19? :3 :3?«,9A'j„p,,„,.3.^(mp,-Mp2) ^e.4Mop,). 

 By (4) and § 39.8 and W' there are available a n and a mo such that 



(&) •K'^wi . 3 . Alp ^ 1 . Mop ^ 1 ; 



(c) -fi^pmi . -Kppimj • :3 • -4/ip ^ J. 



M and w. are available as the n^ and ??i| whose existence is affirmed 

 in (10). 



(II) "^ .A. m'-'"''''''' . : 3 : . wi : 3 : 3 m-. 9 m ^ ?«2 ■ 3 ■ P„n«/°''-"- 

 Proof.— 



(a) M'^'--' . = . ju 9 (3 Mo 9 e : 3 : 3 m^ 9 A'^^^- . 3 . .4mp ^ eAuop). 



By (10), (4) and 'jSl'''^' there are available m, mo and w^a such that 



(b) A^,„,,„3 .3. ^4mp ^ 1, 



Kpm^ . 3 . .4mop ^ 1- 



*That is, either there exists a stage m such that ^-„, is an existent class or SR-'^i consists 

 of a single function and that the function. 



t§ 34.11. 



{B3; B3 ; B3 are relative to W, A'; ^V, A"; 5p = «).V^^?", jv = A'" respectively. Cf. 

 §21a. 



§ By § lla4, §22.1, §32. .5. The ivi, /Co in the conclusion are of course relative to Afi_„. 



