486 OR A. N. WHITEHEAD ON 



*T23. Proposition. If P and Q are distinct members of R ! (a???), then a is the 

 sole member common to P and Q. In symbols, 



h : P, Q '(???) .P^Q.3.i'a = PnQ 



p roo f.Cf. *1'1 1-12-21. 



*1'31. Definition. The interpoints B, C, D will be said to be in the interpoint- 

 order BCD at the instant t with respect to the relation R, when there exist objective 

 reals a, x, y, z, such that (1) B, C, D are members of R ! (???); (2) a; is a member of 

 B, y of C, z of D; (3) U'(axyzt) holds. The symbol R in ; (BCD) stands for the 

 statement that the interpoints B, C, D are in the interpoint-order BCD at the instant 

 t with respect to the relation R. In symbols, 



R in -(BCD) . = . (fta, x, y,z) . B, C, DeR'(a???0 .xeB.yeC.2eD. ll^axyzt) Df 



*l-32. |-.intpnt K( = R in : (;;;0 



Proof. The class Rj,,^;;;^) is part (or all) of the class intpnt H( (cf. *l'3l). Again 

 (cf. *1'22), if B is a member of intpnt R( , objective reals a and x exist, such that x is a 

 member of R : (;;;<), and B is the iuterpoint possessing a and x. Hence there are 

 objective reals y and z, such that either ~R l (axyzt) or R,'(ayxzt) or ^(ayzxt). Hence 

 (cf. *1'31), there are interpoints C and D, such that either R in ; (BCD) or R ; (CBD) 

 or R' (CDBi). Hence B is a member of R jn : ( ; ; ; t ). 



The theory of interpoints has its chief interest when the following axiom is 

 satisfied. It is named intpnt Hp R. 



*1'41. Intpnt Hp R is the statement that if A be an interpoint at the instant t, 

 and a be any member of A., then A is a member o/"R : (a ???<). In symbols, 



Intpnt Hp R . = : Aeintpnt m . aeA . D, ( , A , t . AeR ; (a???) Df 



*T42. Proposition. Assuming intpnt HpR, then if A and B are distinct members 

 of intpnt H j, A and B have either no members in common or one only. In symbols, 



I" .'. intpnt Hp R . D : A, B e intpnt R( .A^B.D.AnBeOul. 



Proof Cf. * 1-23 -41. 



The interest of the relation of interpoint-order (R hl ) arises when the relation R 

 satisfies four axioms specifying the idea that ~R-(abcdt) expresses that a intersects b, 

 c, d in the order bed. These axioms (together with intpnt Hp R) will be employed 

 both in Concept IV. and in Concept V. They will be named aHpR, H|?R, 

 yHpR, SHpR. 



*1 - 51. allpR is the statement that a is not a member of R ; (a;;;). In symbols, 



HpR. = . (a, t) .a^e~R<(a;;;t) Df 



*1'52. ^HpR is the statement that R'(abcdt) implies ^'-(adcbt). In symbols, 



R . = : (a, b, c, d, t) : ~R''(abcdt) . o . R(adcbt) Df 



