MATHEMATICAL CONCEPTS OF THE MATEKIAL WOKLD. 497 



hence (cf. *4'31) cm^(u U w) is contained in cm^'(v u w). Then interchanging u and v, 

 and combining the two results, the proposition follows. 



The following propositions are not cited subsequently, so their verbal enunciations 

 are omitted : 



*4'41. |" . cm^'(cm^'?< n cm^'v) = cm^'u n cm^v 



*4'42. |" . n 'cm^'p = cm,/ n 'om^'p 



*4'43. h . cm^'(cm^M u cm^'v) = cm^'(u u v) 



*4'44. |" . cm,,,' u 'cm^'p = cm,/ u 'p 



*5. On ^-Primes. 



*5'23. Proposition. If u is a ^-prime, and v is a subclass of u, and is not the null 

 class, then v is a <-prime. In symbols, 



t" : u e rm . v e cls'u . \v . D . we 



Proof. For if w be any subclass (not the null class) of v, then (cf. *3'13) 

 cm^(tt-w) is not cm,/w. But (u-w) can be written {(v-w) u (u-v)}, and u can be 

 written {vu(u-v)}. Hence cm^' { (v - iv] u (u - v) } is not cm,,,' {v u (u-v)}. Hence 

 (cf. *4'32) cm^'(v-w) is not cm^'v. Hence (cf. *3'13) v is a ^-prime. 



Note. This theorem, together with *4'31'32, is the foundation of the whole theory. 

 It is remarkable that it requires no axiom concerning <f>. The companion theorem 

 (cf. *12 - 42), with ax,j, substituted for prm^, requires axioms respecting <f>. 



*5'231. Proposition. Necessary and sufficient conditions, that a class u may be 

 </>-prime, are : (1) u is not the null class, and (2) if x be any member of u, then 

 cm^'(M-t'a;) is not cm^u. In symbols, 



I" .'. g;!w : xeu . D Z . cm,/ (M-I' a;) ^ cm^'u : = . ?ieprm^ 



Proof Cf. *3'13 and *4'21. 



*5'233. Proposition. If cm,/ A = A, then every unit class is a <-prime. In 



symbols, 



I" : cm^'A = A . D . 1 c prm^, 



Proof Cf. *3'13 and *4'25. 



*5'235. Proposition. If x and y are distinct, and the unit classes i'x and i'y have 

 the property <j), then the class, which is the couple composed of x and y, is a ^>-prime. 

 In symbols, 



|" : x ? y . (j)\t.'x . <j>h'y . D . i'x u i'y eprm^, 



Proof. Cf. *3'13 and *4'25'27. 



*6. On (j>- Dimensions and <j>-Axial Classes. 



*6'23. Proposition. The (^-dimension of u, if there is such an entity, is a cardinal 

 number not zero. In symbols, 



|- : (ExYdim/tt) . D . dim,' u eNc-i'O 

 p roo f. _Cf. *3'21. 



VOL. CCV. A. 3 8 



