24 pitcher: interrelations of 



the composite developmental sj'stem Sj' " is provabh' a developmental 

 system as to (A'A", WW). 



In the composite development we see that 



(PV ) -^-P -P ; {])])) '".^.p : p' . 



21a. Propositions. — Using the previous notation and the notations 

 K'i; K"i, Ki (i = 1, 2, 12) for properties of classes of functions on 

 1^'; '^" ; f respectively, we have the propositions: 



22. Reduction of developments A. — The reduction of a class of ele- 

 ments '^ with a development A leads to a reduced development A^ 

 of 'iPjj. We use the index rre^ and the bipartite index ?n^?^ relative to 

 the development A^. These indices have the same range of values 

 as the corresponding indices m and ?nl of the development A. Then 

 the development A^ of ^g is completely defined by : 



Thus 



In case a reduce^ development A^ possesses a property' P, then the 

 development A is said to possess the property P on '^^ in notation :§ 



* I. G. A., § 83, p. 147, as re\-ised. 



t Cf. § 19.3. 



I In I. G. A., § 7263 1, Sj similar propositions are given in a more general setting. 



§ Cf. § Ua for similar notations as to properties of functions and classes of functions 



