502 DE. A. N. WHITEHEAD ON 



Proof. cnV is contained in O* (cf. *4'21'27 and *10'1 and *11"21). If x and y 

 are identical, cf. *12'11. If* is distinct from y, then c/ *3'21 and *12-ll'21-23. ^ 



*12'52. Proposition. Assuming (X - TT) Hj9 (/., if u has three members only, and is a 

 self-(/>-concurrence, and its (/.-dimension number is three, then u is <-axial and a 

 subclass of O^. In symbols, 



h.'. (\-ir)Hjpc . ID : e3 n conc^ . dim^, = 3 . => . u e ax^, n els' O^ 



Proof. I? v is contained in u and possesses two members, then (cf. *3'31'32) v is 

 f-axial, and (cf. *12'23) is not (/.-equivalent to u. Hence (cf. *5'231) M is 0-prime, 

 and hence (cf. *3'22) is (/.-axial, and also (cf. *11'21) is a subclass of O+. 



*12-53. Proposition. Assuming (X - TT) Hp (/>, if x, y, and z are three distinct 

 entities forming a (/.-axial class, then the common subclass of cm/('xUt'y) and 

 cm^'(i'a; u i'z) is the unit class L'X. In symbols, 



|-/. (\-ir)H2>^ .3: *'*U t'7/u t'2e3 nax^, .D. cm^'^'aju i c y) ncm^(i'aju t'z) = i'a; 

 Proof Cf. *4-21'27 and *10'2 and *12'23-42. 



*13. On ^-Maximal Classes and Self -^-Concurrences. 



*13'11. Proposition. Assuming (X - TT) lip </>, if p is a ^-maximal class and a 

 subclass of O^, and q is a subclass of p, not the null class, then q is a (/.-maximal 

 class. In symbols, 



|~ . . (X. - TT) Hp <j) .mpe mx^, n els' O^, . 7 e cls'j) .^Ip.^.qe mx^, 



Proof. The class q must (cf. *10'3 and *12'21) be of one, or two, or three 

 ^-dimensions. If the ^-dimension number of q is one or two, then (cf. *10'2 and 

 *12'lt'51) q is a ^-maximal class. If the ^-dimension number of q is three, then 

 (cf. *12'23) q is (^-equivalent to p. Hence if v be a subclass of q, which is ^-prime 

 and ^-equivalent to q, it is ^-equivalent to p, and hence (cf. *3'23) it is <^-axial. 

 Hence (cf. *3'23 and *12'37) q is ^.-maximal. 



*13'31. Proposition. Assuming (X ir)Hp^, if u is a self-(^-concurrence and a 

 subclass of O^, then u is a ^-maximal class. In symbols, 



I" . '. (X - TT) lip <f) . D : u e cone,,, n els' O^ . D . u e mx^, 



Proof. There exists (cf. *12'37) a subclass (v) of u, which is ^-prime and 

 ^-equivalent to u. If v is a unit class, then (cf. *10'2 and *1211) v and u are 

 identical, and u is of one (/(-dimension and ^-maximal. If v is a couple, then 

 (cf. *3'31-32) v is ^-axial, and (cf. *3'22'23) u is of two ^-dimensions and is 

 (/.-maximal. If v is composed of three members, then u is of three (/.-dimensions, and 

 neither of the previous cases can hold. Hence again u is (/.-maximal. 



