34 pitcher: interrelations of 



(6) follows from (1, 2, 3, 4, 5). (7) follows from (6) and from 

 § 23al7. (8) is a corollary of (7). (9) follows from (7) and § 16a2; 

 § 20al ; § 27.11. (10) is a corollary of (9). We state (7) as the follow- 

 ing theorem: 



Theorem I. If a class 9.11 of functions on ^, icith a development A 

 having the property Cn, has the properties: 



L', '^', Ki] Ki, 



then the class 93? has also the properties: 



D; X,,*, 



and the class SW* has the properties: 



L; C; D; A; A; A'.; K,; K,,^; B,319JJ. 



Here under the condition that the development A has the property 

 Co we secure the properties D; iv",,^ as a result of the properties L; 

 A; Ki; Ko. This is to be compared with Theorem II of § 82 of The 

 Introduction to General Analysis* where it is shown that for a general 

 development A the property K^^* is imphed by the properties D; A; 

 Ki] Ki. It will be seen that the theorem which we give is appHcable 

 in many important cases. The ordinary development of the linear 

 interval (01), for instance, has the property C and therefore the prop- 

 erties Ci and Co. 



33. Relative to classes %' ; iV' with developments A'; A" and the 

 composite class ^ = ^^3'^" with the composite development! A'" we 

 have the following proposition: 



(1) ■ ■ 



This follows from § 32.6; § 12a2; § 21al, 2, 3, 4; and § 23al7. 



The following proposition is a corollary of (1) through § 16a2 

 and §20al; §27.11. 



A'"'' . A"""' . W''^'"'' . ^Ol"^-""-^'" . z) . m'yjl")i>^"'^''.K,-'^'2. 

 (2) 



aSl'^V) ^'"°^-^' "^1!^!'^12- ■ Bl^'St'W 



By (2) and § 21a9 we have the following theorem: 



* Cf. § 23al7. 

 t§21. 



