488 DR. A. N. WHITEHEAD ON 



Proof. If a; is a member of R ; ( ;;;t), y exists such that a? is a member of 

 R ; (2/;;;<) ; also (cf. *P61) x and y are distinct. Hence (cf. *1'21) P exists such that 

 it is a member of R ; (?/???), and a; is a member of it. Hence (cf. *1'41) P is a 

 member of E : (???<), and hence (cf. *l'2l) y is a member of R : (x ;;;t). Hence x is a 

 member of E'( ;*"*). 



*171. Proposition. Assuming aHpR, every interpoint possesses at least two 

 members. In symbols, 



(-.-. HpR. => : A e intpnt R( . D . Nc'A^ 2 



Proof. Cf. *1-13-21-22'51. 



*172. Proposition. Assuming (intpnt, a, /3, y, S) Up R, then on every objective 

 real there exist at least three interpoints. In symbols, 



|- .'. (intpnt, a, ft, y, 8) Rpll . D : a eR ; ( ;;;;<) . D . Nc'R ! (a???) ^ 3 



Proof. Cf. *l-2r31'62-65. 



*173. Proposition. Assuming (intpnt, a, /?, y, 8) Hp R, then, if there are any 

 objective reals, the interpoints are not all on any one objective real. In symbols, 



. I".', (intpnt, , J3, y, 8) Hp R . a : a !R ; (;;;;i) . a . 3! {intpnt H , - R ; (a???*)} 

 Proof. Cf. *1 -4271 72. 



(iii) CONCEPT IV. 



*2. This concept bifurcates into two alternate forms, namely IVA. and IVB. 

 Concept IVs. is related to IVA. just as Concept II. is related to the classical concept. 

 Thus Concept IVA. is dualistic, and Concept IVs. is the monistic variant of it. Both 

 concepts can initially be considered together as Concept IV. In Concept IV. the 

 essential relation (R) is pentadic, one of the terms being an instant of time. ~R-(abcdt) 

 can be read as a intersects b, c and d, in the order bed at the instant t. The class of 

 those entities, appearing among the first four terms in any instance of the relation 

 holding, is called the class (0) of " linear objective reals." The remaining class of 

 objective reals, required for Concept IVA., is called the class of " particles." 



The geometrical points of this concept are simply the interpoints of R, as defined 

 above (cf. *l). During the consideration of this concept they will be called points. 

 The further definitions, beyond those of *1, required for a concise statement of the 

 geometrical axioms are almost exactly those of Concept I., with the R jn of this 

 Concept IV. written for the R of Concept I., and modified by the mention oft, as in 

 Concept III. This mention of t can be managed in a similar (though not identical) 

 way to that in Concept III. by writing 



*2'0l. R < -(ABC = R ln =ABC Df 



