MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 515 



*26'23. Proposition. Assuming (II. -VI, IX -XL, XIII. )HpR, there is one and 

 only one cogredient point lying in the punctual associate of an objective real. In 

 symbols, 



|- . '. (II. -VI, IX.-XI., XIII.) Hp R . s : a e OH, . 3 . > R( n ass K Ya 1 



Proof. Cf. *26'22. 



*26'24. Proposition. Assuming (II. -XL, XIII.) HpR, there are at least two 

 points, not cogredient points, lying in the punctual associate of an objective real. In 

 symbols, 



h .-. (II.-XL, XIII.)HpR.D: eO R( . D . Nc'{ass R( Va- <,} S 2 

 Proof. Cf. *25'31 and *26'23. 



*27. Ow Punctual L/ines. 



*27'11. Proposition. Assuming (II. -VI, IX.-XI.) Hj? R, if p and q are distinct 

 planes, and p n q possesses a member, then ass Kt 'p n ass Kt 'q is identical with 

 &ss Kt '(p n q). In symbols, 



|- .". (II.-VI, IX.-X 



p, geple R( .p ? q . R\(p n q) . z> . &ss Kt 'p n &ss Rt f q = ass R( '(p n q) 



Proof. Cf. *16-33 and *20'21. 



*27'12. Proposition. Assuming (II.-VI, IX.-XII)HpR, if p and q are distinct 

 planes, and a point, not a cogredient point, lies in the punctual associates both of p 

 and also of q, then p n q possesses one and only one member. In symbols, 



tV. (II.-VI, IX. -XII.) HpR . ^ : 



p, q e ple u , . p ^ q . 3 ! {(ass K( ^p n ass K( ' q) - OO E( } . D . p n q e 1 



Proof. Cf. *1 6 -21 and *22'51. 



*27'13. Proposition. Assuming (II-XIII.) HpR, if p and q are distinct planes, 

 then if p n q possesses one member, there are non-cogredient points lying in the 

 punctual associates both of p and of q ; and also conversely. In symbols, 



I- : : (II-XIII) Up R . D .'. p, q e ple R( . p * q . D : 



3 ! {(ass R( 'jp n ass K( 'g) - a> R <} . = . p n q e 1 

 Proof. Cf. *26'24 and *27'12. 



*27'21. Proposition. Assuming (II.-VI, IX.-XLT.) Hp R, if m is a punctual line 

 possessing a non-cogredient point, then there exists an objective real such that m is 

 its punctual associate. In symbols, 



I .'. (II -VI, IX.-XII.) HpR . D : 



melin R( . ft!(m- CO R( ) . D. (ga) . aeO R( . m = 



Proof. Cf. *27'11-12. 



3 u 2 



