MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 501 



The following proposition should be compared with *12'12 : 



*12'37. Proposition. Assuming (X - IT) Hp <f>, if u is a subclass of 0^, and is not the 

 null class, there exists a subclass of u which is <-prime and ^-equivalent to u. In 



symbols, 



h . . (X - TT) Hp (f> . D : u e cls'O^ . g ! M . D . g ! (prm^, n els' u n equiv,,,' u) 



Proof. From *12'12'21, u is either of one, or of two, or of three ^-dimensions. 

 If u is of one (^-dimension, the conclusion follows from *5'233 and *10'2 and *11'12. 

 If u is of two (^-dimensions, then (of. *5'235 and *10'2) any two members of it form a 

 <-prime class, and (cf. *12'21) the ^-dimension number of this class is not greater 

 than two, and hence (cf. *6'25) it is two, and hence (cf. *12'23) this subclass is 

 (^-equivalent to u. If u is of three ^-dimensions, it must contain at least one subclass 

 v consisting of two members, and, as before, v must be </>-prime. If v is of three 

 ^-dimensions, then (cf. *12'23) u arid v are ^-equivalent. Iff is of two ^-dimensions, 

 then there is a member of u, x say, which is not a member of cm^'v. Then, either 

 v u L'X is (^-prime and (cf. *12 - 23) ^-equivalent to u, or the class composed ofx and 

 some one (not necessarily any one) of the members off is <-prime and ^-equivalent to u. 

 The following proposition should be compared to *3'13 and *12'33 : 

 *12'41. Proposition. Assuming (X- 77) ~Rp<f>, if u is <-axial, and v is a subclass of 

 u, and both v and (u-v) are not the null class, then dim^r is less than dim^K In 

 symbols, 



|~ . '. (X - TT) Hp ( . D : u e ax^, . v e els ' u . g ! v . 3 ! (u - r) . D . dim^' v < dim,/ u 



Proof. Cf. *3'13 and *4'21 and *11'21 and *12-21'23. 

 The following proposition should be compared to *5'23 : 



*12'42. Proposition. Assuming (X ir)Hp$, then, if u is a subclass of O (/) and is 

 (^-axial, any subclass of u, not the null class, is <^-axial. In symbols, 



1" . '. (X - TT) Up <f> . D : u f. ax^, n els ' O^, . v e els' u . g ! v . D . v e ax^, 



Proof. From *6'26 we have Nc'tt = dim^'w ; from *5'23 and *6'25 we have 

 Nc'v = dim^'n Hence (cf. *12'41), if v is not identical with v, we have 



Nc' v = dim^,' v < Nc' u ......... (1) 



Firstly, assume that v omits one member of u only. Then Nc'-j'+l = Nc'?*. 

 Hence, from (l), Nc'w = dini^'w, and hence (cf. *3'21) v is ^-axial. 



Secondly, if v omits two members of u, then it is a unit class, and (cf. *12'11) is 



It is convenient to conclude this section (*12) with three theorems which are 

 fundamental to the theory of (/>-points and of (^-planes. 



*12'51. Proposition. Assuming (X - TT) H_p <, if u is of two ^-dimensions, and x 

 and y are members of cm^'w, then the class composed of x and y is ^>-axial and is a 

 subclass of O.J,. In symbols, 



I" .. (X-7r)Hp( . D : dim^'M = 2 . x, yfcm+u . D. L'X u t'/eax^ n cls'O^, 



