496 DR. A. N. WHITEHEAD ON 



General Deductions Concerning Dimensions. 



A large chapter of interesting propositions concerning the entities defined above in 

 *3 can be compiled. The following are chosen as being directly wanted in the 

 subsequent investigations : 



*4. On Common ^-subregions. 



*4'21. Proposition. If v is a subclass of u, then cm/?; is a subclass of cm/ it. In 



symbols, 



h : v c u . D . cm/i; c cm/ u 



Proof. Of. *3'12. 



*4'25. Proposition. A class u is itself a subclass of cm/M. In symbols 



I" . u c cm/ u 

 Proof. Cf. *3'12. 

 *4 - 27. Proposition. If u is a class with the property (f>, then u is identical with 



cm/ if. In symbols, 



1" : <f> ! u . D . u = cm/ u 



Proof. Cf. *3'12. 



*4'28. Proposition. If there exist two classes, both with the property <, which 

 possess no common member, then cm/ A is itself the null class (A). In symbols, 



l~ : (g; u, v) . u n v = A . < ! u . <f) ! v . D . cm/ A = A 



Proof. Note that cm/ A is the common part of all (^-classes. 



Corollary. If x and y exist such that they are distinct, and the two unit classes 

 with them as members respectively each have the property <, then cm/ A is A. 



Note that when *4 - 28 is appealed to, it will be this corollary which is directly used. 



*4'31. Proposition. The common c^-subregion for the common ^-subregion for u is 

 the common <-subregion for u. In symbols, 



I" . cm/cm/w = cm/w 



Proof. For cm/M is contained in every <-class containing u. Hence (of. *3'12) 

 cm/cm/M is contained in cm/w. Also (cf. *4'25'2l) cm/w is contained in 

 cm/ cm/ u. 



*4'32. Proposition. If u and v are (^-equivalent, and w is any class, then the 

 common ^-subregion for the logical sum of u and w is identical with the common 

 <-subregion for the logical sum of v and w. In symbols, 



I" : cm/tt = cm/v . => . cm/ (it u w) = cm/(v u w) 



Proof. For (cf. *4 P 21) cm/v is contained in cm/(i?uw), and hence (hypothesis) 

 cm/w is contained in cm/(vuw), and hence (c/ *4'25) MUW is contained in 

 cm/(vui0), and hence (c/ *4'21) cm/(wuw) is contained in cm/ cm/ (v u w), and 



