PROPERTIES OF CLASSES OF FUNCTIONS 37 



Theorem I. If a class W of functions of a system (51; ^l.V; A''''; W) 

 has the properties Di, A', K[ then 



<^>f . '>ffl"LCD, . -, . CJiJl'^")^ = [all ^A-„'m,M").B„-l.",p')] 



= [all ^•K.'i)i'(i'!").s„-i"(y)j. 

 (6) W has the properties Kl, K[^, K[^^; 



(c) 9}?i atid 9)?^ each has the properties L, Di, A', K\.,, ivj^, K[„^. 



Theorem la furnishes, under special hypotheses, a functional 

 characterization of (^Jf'SO?")* where 9}J" is subject to conditions less 

 exacting than the LCD used in the definitions of K\^, ii^j,^, K\„j^.* 



In proposition (2) under the hypothesis A^' we secure A',^, K^.,jf, 

 as a result of D\, A, Ki. This is to be compared with Theorem II 

 of § 82 of the Introduction to General Analysis.! 



Theorem II. In the case of two syste77is : 



(21; %V; A'''"; WV'"); (31; %V'; A"'\ m"'"), 

 where 



P' ^ DiA'K[; P" = DiA"K[', 



the composite systems: 



(31; %^; A'"; m'W); (31; ^; A'"; (WW) J; (31; %^; A'"; m'W)^) 



are such that 



This theorem follows from Theorem I and § 21al, 3, 7; § 20a2. 

 Theorem III. The genus of all systems: 



(2l;^;A'-';g[yn 

 where 



P^^ D,AK,{K,)K,^{K,.J, 



is closed under the five operations A, B, Co, Ci, d: 

 A : linear extension of classes 90? ; 

 B : *-extension of classes M ; 



C: simultaneous composition of classes *i|.^ and developments A and 

 Co : multiplication of classes Wl ; 

 Ci'. linear composition of classes W; 

 C2: ^-composition of classes 3)1, 

 and the combinations of these operations. 



*Cf. I. G. A., §65(iII;§67. 



t § 23al7. 



