518 DK. A. N. WHITEHEAD ON 



the plane ,, there exists at least one non-cogredient point not lying in it. In 

 symbols, 



|- . '. (II.-XIII.) Hp R . D : p pple E < - t < OCR, . a . (3 A) . A e (pnt B( - COR,) -p 



Proof Cf. *12'21 and *16'32 and *22'41 and *26"24 and *28'11. 



*28'21. Proposition Assuming (II.-XIV.) HpR, if a: be any objective real and p 

 be any plane, then either the punctual associates of x and p have one and only one 

 common member, or x is a member of p. In symbols, 



h : : (II. -XI V.) Hp R . D . '. x e O K( . p e ple K( . ^ : ass R( Ya; n ass R( > c 1 . V . x ep 



Proof. _ Take (cf. *26'24 and*28'll)two non-cogredient points A and B upon x, and 

 a non-cogredient point C in ass R( > but not on x. If either A or B lie in uss^'p, 

 then cf. *16'32. If neither A nor B lie in ass, t /_p, then (cf. *16'32 and *27'5l) 

 cm,,/ {(B n 0) u (A n C)} is a plane possessing x, and its punctual associate possesses C. 

 Hence (cf. *22'51) there is a common member of p and this plane, y say, and x and y 

 are coplanar, hence (cf. *10'4 and *1G'11) the punctual associates of a; and y possess a 

 common point. Hence ass R ,'i'ic and ass Kt 'p possess a point in common, and then 

 cf. *16'32. 



*28'22. Proposition. Assuming (II. -XIV.) Hp II, if p and q are punctual planes, 

 and p is not identical with QO H( , and p n q is contained in QO H< , then p n q and 

 p n oo u( are identical. In symbols, 



I".-. (II. -XIV.) HpR .=>:p, tfepplejj, .p ^ co m . jongcoo R( . z> .pr\q =pn<x> Rt 



Proof If p and q are identical, then pcoo K( , but (cf. *28'll) this is impossible. 

 Hence p ^ q. If q = OO H( , then p n q = p n oo lt( . Assume q ^ OO R( . Then (cf. *20'6l) 

 there exist planes, p' and q' say, such that p is ass m 'p' and q = ass R ,'(/. Since 

 pru/coo K! , there is (cf. *26'24) no objective real common to p' and q'. Hence 

 (cf. *28 - 21) upon every objective real possessed by p' there is one and only one point 

 lying in </. Hence (cf. *26'23) p n q = p n oo K( . 



*28'31. Proposition. Assuming (II. -XIV.) Hp R, in every punctual line there lie 

 at least three points. In symbols, 



[ . : (II.-XIV.) Rp R . a : m e lin H( . 3 . Nc'm ^ 3 



Proof If m possesses non-cogredient points, then cf. *27'22'23. If m is contained 

 in oo Rj , then (cf *28'22) m is identical with pr\a RI , where p is a punctual plane. 

 But (cf *27'51'52 and *28'll) there are three distinct punctual lines contained in p, 

 meeting two by two in three non-cogredient points ; then cf. *27'23. 



*28'32. Proposition. Assuming (II.-XIV.) HpR, a punctual line is the common 

 meeting of two punctual planes, and conversely. In symbols, 



h . '. (II.-XIV.) Hp R . D : lin B( = m {(ftp, q) . p, q e pple B .p^q.m=pnq} 



