PROPERTIES OF CLASSES OF FUNCTIONS 27 



Propositions and theorems relative to systems (31; ^; A; SO?) where 'ip is 



unconditioned except by the development A. Theorems relative 



to the closure of certain genera of systems (21 ; "ij? ; A ; SO*?) . 



§§ 25-40. 



25. The following propositions are relative to a general class ^ 

 and to classes 9)2 and ))l of functions on "ip to 21: 



(1)* go?*''"' . 9I*'"-" : 3 : 9J?^' . ~ . gi^', 



(2) * 50?^''"' . 9t^''"" : 3 : a«^ . ~ . sjj^, 



viz., if SO*? is dominated by SCiJt and 9*? by 219)? then the statement that 

 9)J has the property Di; D, is equivalent to the statement that 91 

 has the property Di\ D, respectively. 



In addition we have the following proposition relative to any de- 

 velopment A of the class ^ : 



(3) m^'""' . 9^'"^ : D : 9JJ^' . ~ . 9J''\ 



26. We have the following simple propositions relative to a class 

 ip with a development A : 



(1) 6'^™'"'""'*'. 0- .3. 0^'-; 



(2) m"'- . z, . [all r°"''""' '«f "■=■"•; 



(3) m = [0] t . 3 . 9)r*-. 



In case the development of ^V is such that there exists a pair of 

 elements p[p!, such that for every stage m there exists a stage Wo ^ w* 

 such that the elements p[, p'^ are directly connected at stage mo, i. e., 



A' 9 (3 {p[, p'.^ 9 ?« . 3 . 3 (Wo > W . Zo ^ L,) 9 (p;"""'" . pT'')) 



then 



(5) <7'<7" : 3 : ^"^■''-"K 3 . v,'.. = ^.>" (p") • 



In case the elements p[, p'.-, are each ultimately developed, i. e., 



3 Wo 3 m > 7??o . 3 . /jj"" . P2'" 

 we have the proposition: 



(6) 9)r9M'.3.M;,. = M;':3:2)^''^''•• 

 * Cf. I. G. A., § 22a2, 2'. 



t [0] = class consisting of a single function whose every functional value is 0. 



