49 DE. A. N. WHITEHEAD ON 



*2-ll. I-.'. (Ill, IV, V, VI, VII) Hp-R . D : t T . 3 . Nc<R t >(;;;) 5 2 

 Proof. Cf. *172. 



*2-21. I" .-. IV Hp R . 3 : R ( : (ABC) . o . BV(CBA) 

 Proof Cf. *1'63. 



*2-22. |- .-. (Ill, IV, V, VI) HpR . D : R ( ; (ABC) . D . - R, : (BCA) 

 Proof Cf. *1'G4. 



*2-23. |- .-. (Ill, IV, V, VI) Hj?R . D : BY (ABC) . D . A * C 

 Proof Cf.*V&2. 



*2'31. Proposition. Assuming (VII, IX)HpR, if A and B are two distinct points 

 at the time t, they possess one, and only one, common member. In symbols, 



i- .-. (VII, IX)HpR . D : A, BeR ( ; (;;;) . A ^ B . D . AnBel 



Proof Cf. *1-31'42 and (VII, IX)HpR. 



*2'32. Proposition. Assuming (VII, VIII, IX) Hp R, a line at any instant t (i.e., a 

 member of lin H( ) is the complete class of points (Intel-points) possessing some linear 

 objective real. In symbols, 



L :(VII,Vin,IX)HpR.D.lin lu =^[( a a).eR ; (;;;;0.p = A {AeR,'(;;;) . ae A}] 



Proof. Cf. VIII Hp R and *2'31. 



Propositions *2'31'32 effect the identification of the punctual line, as defined above, 

 and the class of points on some linear objective real. Thus a straight line considered 

 as an entity with parts is a punctual line, and considered as a simple unit is a linear 

 objective real. 



*2'33. Proposition. Assuming (VII, VIII, IX) HpR, if C and D are two points in 

 the punctual line R ( ; AB, then A is a point in the punctual line R ( ; CD. In symbols, 



t- .-. (VII, VIII, IX) Hp R . 3 : C, D e R ( : AB . C ^ D . D . A eR, ! CD 



Proof Cf. *2'32. 



*2-41. I- .-. (Ill -IX) Hp R . 3 : * e T . 3 . ( a A, B, C) . A , i( ; (ABC) 



Proof Cf. *17273.*2-32. 



*2'5. Proposition. Assuming (III-XIV) Hp R of Concept IV., then all the 

 axioms of Concept I. hold when the R t of Concept IV. is substituted for the R of 

 Concept I., and t is a member of T. 



Proof. Cf *2-ll-21-22-23'33-41 and (IX-XIV)HpR (of Concept IV.) and 

 (I-XII) Hp R of Concept I. 



It will be noticed that IHpR (of Concept IV.) is not required in the above 



