PROPERTIES OF CLASSES OF FUNCTIONS 35 



Theorem I. In the case of two systems: 



(21; ^V; A'^'; W"); (31; %V'; A"""; m"'"), 

 where 



p'^la'k:-, p"^ la"k':, 



the composite systems : 



(3( ; ^;^ ; A'" ; m'm") ; (51 ; %^ ; A'" ; m'^")^), 

 are such that 



From Theorem I and § 32.9 we secure the following theorem: 

 Theorem II. The genus of all systems: 



(51; %^; A'-'; 500, 

 where 



is closed under the operations A, C: 



A : ^-extension of classes SJf ; 



C: simultaneous com,position of classes ^ and developments A and 

 ^composition of classes 9}J ; 

 and combinations of these operations. 



The subgenus of all systems obtained by the operations A, C and their 

 combinations is the genus of all systems: 



(51; ^iv A^>; W"), 

 P, = LC{DA)AiK,)K,{K,^K,,^). 



This genus is closed under the operations A, C and their combinations. 

 To this genus belong the systems: 



(31; iV; A'; W'); (51; %V'"; A""; 2«""); (5(; r''; ^'^■, ^l'"')- 



Here the properties occurring in parentheses are implied by the 

 other properties of the composite property in which they occur in 

 view of A^> and § 32.6, 9, 10; § 27.11; § 20al. 



34. Propositions. — 



(1) i^.pi.A'=""'.M''"'-^.Mp, = 0, 



viz., if the development A of 'i^ has the property* W as to pi a particular 



* A^'st-P) .=. A 3 (m . 3 . 3 m„> m j p-"o). Cf. § 22. 



