516 DR. A. N. WHITEHEAD ON 



*27'22. Proposition. Assuming (II.-XIII.) HpR, a punctual line possesses either 

 more than one non-cogredient point or no such point. In symbols, 



I-.-. (IL-XIIL)HpR . D : me ling, . g!{ro- } . D . Nc'(m- OO B ) > 1 



Proof. Cf. *20'22 and *26'24 and *27'21. 



*27'23. Proposition. Assuming (II.-VI., IX.-XIII.)HpR, a punctual line, 

 possessing a non-cogredient point, possesses one and only one cogredient point. In 



symbols, 



.D :melin R( . a!(m-oo K ) .^.mn oo R( el 



Proof. Cf. *26'23 and *27'21. 



*27-31. Proposition. Assuming (II.-VL, IX.-XII.)HpR, if m is a punctual line 

 possessing a non-cogredient point, and A and B are two distinct points lying in it, 

 then m is the punctual associate of A n B. In symbols, 



I- /. (TI.-VL, IX.-XIL) HpK . D : 



melin B , . g;!(m- oo ){( ) . A, Be m . A ^ B . D . meass K( '(A n B) 



Proof. Cf. *14'21 and *27'11'12. 



*27'41. Proposition. Assuming (II. -XI.) Hp 11, if a is any objective real, then 

 there exist two planes p and q such that is the sole member of p n q. In symbols, 



|- .'. (II. -XL) Hp R . r> : a e O R( . D . (ftp, q) . p, q e ple R( . p n q = t'a 



Proof. Cf. *12'21 and *14'12'14 and *22'41 and *25'31. 



*27'42. Proposition. Assuming (II.-XIII.) HpR, if a be an objective real, then 

 ass K( 'i'a is a punctual line with a non-cogredient point, and conversely, if m is a 

 punctual line with a non-cogredient point, there exists an objective real a such that 

 m = assju't'a. In symbols, 



I" .'. (II.-XIII.) HpR . D : melinjj; . g!(m- CO K( ) . = . (get) . a e. O Rt . m = ass fi ,Ya 



Proof. Cf. *27-ll'13-21-41. 



Note. This proposition, *27'42, establishes the connection between the objective 

 reals and the punctual lines. 



*27'43. Proposition. Assuming (II.-XIII.)HpR, if a and c are cogredient, they 

 are distinct. In symbols, 



I" . '. (II.-XIII. ) Hp R . D : c cogrd Ht ' a . D . a ^ c 



Proof. Cf. *20'31-41 and *27'42. 



*27'51. Proposition. Assuming (IL-XIV.)HjpR, if A is any non-cogredient point 



