PROPERTIES OF CLASSES OF FUNCTIONS 33 



tions of the sj-stem ((6"*')) we have, denoting by n an arbitrary stage 

 and by ns the corresponding bipartite index, 



n . e : 3 : 3 7h,„ j (to ^ m„, . p) . ^ • E E ^^"■'■" ^i-' ^ ^^ 

 n . e/2 : 3 : 3 7H „ ,,o , (to ^ to„ „. . p) . 3 . E 45?t Z ^^""' ^ e/2- 



h s 



By condition la, § 17, and the hypothesis A'^=, there exists a stage no 

 which is complete and such that 



2J.3. E^^r ^ 1/2. 



Use this no as n in above and then any m exceeding both ?n„,e and too 

 such that "ip'""^ is effective* as the m^ desired in the theorem. 



/A\ A To mjLA.BiWlil _ Bo'J' • "nlfoimly binmded from I 



Proof. — By condition la, § 17, on developmental systems 



e : D : 3 m, 9 (to ^ toI . p) . ^ . A(E 5;,'^ - 1) ^ e. 



By (3) 



e .711, : =) : 3 »h > toI 9 p . d . E -'^^r'" ^ «• 



For the stage vii we have 



1 - 2e ^ E ^i"" ^ 1 + 2e. 



Take e = 1/4 and the function E^r' i^ t'^^ function desired. 



I 



Proposition (5) relative to a general class '^ is readily proved. 



/r\ gnBl'K*) , q Bull! . unifoimly bounded from ^ _j ^ J^J^^ 



Through (1 ... 5) we are able to prove the following funda- 

 mental propositions relative to a class 'ip with a development A con- 

 ditioned as indicated. 



(6) ^ A^^ . m"-^"'- . a . 9)r. 



,r^s aC; mjiAKiA-, OYJ/'Ki,. ^ LCD A \KaKn, ■ Biim ^ 



(8) A^^ . gir?^'^-^*'''^'^ . =1 . 9J?^^'"-^. 



(10) A^' . SQJ^'^-''"^ . 3 . SD^^^^-^'"-*. 



* The necessity of choosing a non-vacuous stage should be noticed. In case a stage is 

 vacuous any functions at all are effective as the developmental functions of that stage. 



I ^unifurmly bounded lr„m " . = . ^ , (3 e j Allp S 6 (p)). 



