500 DK. A. N. WHITEHEAD ON 



class, has a set of <-axes. (Note. A class which is (/.-axial and (/.-equivalent to a 

 class tt is said to be a set of $-axes ofu.) In symbols, 



|- /. (X-ir) Hp </> . r> : u ecls'O^, . R\u . D . g!(ax^ n equiv^w) 



Proof. Since there is at least one member of u, there is (cf. *12'll) a (/.-axial class 

 contained in . Hence (c/ *10'4) this class can be augmented so as to become a set 



of -axes of u. 



*12'13. Proposition. Assuming (X-7r)Hp(, if u and v are subclasses of 0^, and 

 v is not the null class, and cm^'-u is contained in cm/w, then there exist two sub- 

 classes of 0^, w and w', say, such that w is a set of (/.-axes of v, and wuw? is a set of 

 (/.-axes of u. In symbols, 



I".'. (X-7r) Hp(/> . D : M, -wccls'O^ . g!w . cm/vccm^w . => . 



(3*0, w') . weaXf n equiv^'v . wuw/e ax^ n equiv^'w. 



. Cf. *10'4 and *12'12. 



*12'21. Proposition. Assuming (X - TT) Hp (/>, if w and v are subclasses of O^, and 

 v is not the null class, and cm^'v is contained in cm^'u, then the (/.-dimension number 

 of v is less than, or equal to, the (/.-dimension number of u. In symbols, 



I:.'. (X-7r) Hp (/>.=>: u, -uecls'O^ . Q\v . cm+'v c cm^'u . => . dim,,,' v = dim^,' M. 



p roo y; From *6'26 and *12'13, w and w' exist (assuming wnu/ = A) such that 

 Nc'?0 = dim^'v and Nc'w + Nc'w' = dim^,'?<. Hence dim^,'v < dim^'w, unless w 1 is the 

 null class, or unless the numbers are not finite, in which cases dim/w = dim^'tt is 

 possible. 



*12'23. Proposition. Assuming (X-7r)Hp(/>, if u and v are subclasses of 0,,,, and v 

 is not the null class, and cm^'v is contained in cm^'w, then, if dim^'-y = dim^'n, 

 we have cm^'v = cm^'u, and conversely. In symbols, 



I" : : (X - TT) Hp <j> . D . '. v, u e cls'O^, . 3 ! v . cm,,,' v c cm^,' u . D : 



cm^'-w = cm^'tt . = . dim^'v = dim/u 



Proof. Assuming dim^,' v = dim^' u, and also assuming the notation of the proof of 

 *12'21, then w and w' are such that (l) w is (/.-equivalent to v and wuw' to w, 

 (2) Nc'w = dim^'v and Nc'ty + Nc'w/ = dim^'u. Hence, by hypothesis and (2), 

 Nc'w + NcV = Nc ( w. Also (cf. *10'3 and *12'2l) Nc'w + NcV = 3. Hence NcV = 0, 

 that is, W = A. Hence from (1), cm^'w = cm^'v. The converse is obvious. 



*12'33. Proposition. Assuming (X-Tr)Hp^., if u and v are subclasses of O^, and v 

 is not the null class, and cm^'v is contained in, but is not identical with, cm^'w, then 

 dim^'v is less than dim^w. In symbols, 



1" . '. (X - TT) Hp < . D : u, v f. cls'O^, . 3 ! v . cm^,' v c. cm,,,' u . 



cm 6 'v T cm/M . D . dim/!' < dim/ 

 Proof.-Cf. *12'21-23. 



