504 DK. A. N. WHITEHEAD ON 



t'aui'&uc'c and i'a u i'6 u iVZ are both of them three ^-dimensional and <-axial. 

 Hence (cf. *10'5 and *13'32) cHs a member of O/(i'a u I'b u i'c), and hence (cf. *3'42 

 and * H'll '13) d is a member of P. Thus P and Q are identical. 



*16. On <f>-Planes. 



*16'11. Proposition. Assuming (X-7r)Hp<, every ^-plane is (^-maximal, self-<- 

 concurrent, and a subclass of 0+. In symbols, 



i" : (X - TT) Hp <f) . z> . ple^, c mx,,, n conc^, n cls'O^ 



Proof. Cf. *12-21-23'51. 



*16'21. Proposition. Assuming (X - TT) Hp <, < -planes with more than one member 

 in common are identical. In symbols, 



tv. (\-7r)Hp<f> . D \p, Q-eple^ . Nc'(^ n </) > 1 . D . p = q 



Proof. Cf. *12'23 and *16'11. 



*1G'31. Proposition. Assuming (X - n-) H/> <, every self-<-coucurrence is either 

 0-copunctual or ^-coplanar. In symbols, 



\ . : (X - TT} Hp 4> ^ : u e conc^, . => . copnt^, ! u . V . cople^, ! u 



Proof. A proof is only required when u is of three ^-dimensions. Then , b, c 

 exist, such that they are three distinct members of u and are not a <-coplanar class. 

 Hence (cf. *12'52) they form a <-axial class of three members. Hence (cf. *3'42) 

 u is, in this case, <-copunctual. 



*16'32. Proposition. Assuming (\-p)H^>(, if p is a <-plane, and P and Q are 

 distinct (^-points, and p and P have common members, and also p and Q, then the 

 member (if any) common to P and Q is a member of p. In symbols, 



i- .-. (X -p) Hp< . D : p eple* . P, Q epnt, . P * Q . R\(p n P) . ^\(p n Q) . D . P n Q cp 



Proof. If P n Q is the null class, then P n Q is contained in p. If P n Q is not 

 null, let c be a member ; also let a and b be, respectively, members of p n P and of 

 p n Q, which exist by hypothesis, (i) If c is identical with a or b, then (cf. *14'2l) 

 P n Q is contained in p. Again (ii) if c is not identical with a or 6, then (cf. *14'11 

 and *16'll)a, b and c form a self-<-concurrence. Hence (cf. *16'3l) this class is either 

 <-copunctual or ^-coplanar. If the class is <-copunctual, then (cf. *14'21) P and Q 

 are identical. Hence it is ^-coplanar, and hence (cf. *16'2l) c is a member of p. 



*16'33. Proposition. Assuming (X-/3)H^<^, if P is a <-point and p and q are 

 distinct ^-planes, and P and p have common members, and so, also, have P and q, 

 then the member (if any) common to p and q is a member of P. In symbols, 



.p j q . an^ . a 

 Proof. The proof is in all respects similar to that of *16'32. 



