MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 521 



*31'21. Proposition. Assuming (II. -VI. )HpR, the point-order ABC implies the 

 point-order CBA. In symbols, 



Proof. Cf. *1'63 and *20'51. 



*31'22. Proposition. Assuming (II. -VII, IX.-XI.) HpR, if A, B, are in the 

 point-order ABC, then A, B, and C are distinct. In symbols, 



I- .-. (II.-VIL, IX. -XI.) Hp R . a : R,, n i (ABCO . D . A ^ B . A ^ C . B ^ C 



Proof. There are (cf. *20'51) interpoints u, v, w on a common objective real a, 

 such that ~R- m '(uvwt) and [ABC] persp E( [M K( v Ki w m ], where (cf. *20'23 and *25'32) 

 u m, v Rt , w Rt are the dominant points of u, v, iv. But (cf. *1'62 and *22'33) u Rl , v Rt , w }it 

 are distinct points. Hence (cf. *20'32) A, B, C are distinct points. 



*31'23. Proposition. Assuming (IL XV.)HpR, the point-order ABC is incon- 

 sistent with the point-order BCA. In symbols, 



|- .'. (II. -XV.) Hp R . : R im ; (ABa) . D . - R ; (BCAO 



Proof. Since this proof is long, the paragraphs will be numbered for reference 

 by (i), (ii), &c., prefixed. 



(i) If u, v, w are interpoints on the objective real a, and u', v', w' interpoints on the 

 objective real a', and a and a' are cogredient, and [ihu v ntWui] persp R( [u'Rt v 'ntw'ni], then 

 (cf. *20'23'41 and *25'32) u, v, iv and u', v', iv' must agree in interpoint order, and 

 (cf. *1'64) each set of interpoints has only one (if any) interpoint order (counting uvw 

 and ivvu as the same order), and (cf. *27'43) a and a' are distinct. 



(ii) From *20 - 51, R 11U ; (ABC<) implies (a) that A, B, C are on an object real d; 

 (ft) that there are interpoints u, v, w on an objective real a, cogredient with d ; 

 (y) that l^'(uvwt) ; (8) that [ABC] persp Rj [u Et v Rt w Rt ~] ; and (e) (cf. *27'43) that a and 

 d are distinct. 



(iii) Assume that R pn : (BCA^) also holds. Then, in addition to the entities of (ii), 

 there exist interpoints u', 1/, iv' on an objective real a', satisfying all the conditions of 

 (ii) without changes, except that (ii, y) becomes R in ; (i/wV). This assumption (iii) 

 will now be proved to be absurd. 



(iv) From XIIIHpR(c/: *22'61) and *26'11'22, d, a, a' are copunctual and a and 

 a' are either cogredient or identical. 



(v) Hence (cf. *30'3) either (Case I.) [u Rt v Rt w Rt ~] persp B( [u' Rt v' Rt w' Rt '] or (Case II. ), 

 a and a' are identical. 



(vi). Case I. We have [_u Rt v Rt w Rt ']perBp Rt [u' Rt v' Rt U'' R t~]. Hence (cf. *20'41) the 

 interpoint orders of u, v, w and u', v', v/ must agree. Hence, from (ii, y), R in ; (u'v'iv't) 

 holds. But (cf. *1'64) this is inconsistent with ~R in -(v'iv'u't). Hence Case I. cannot 

 hold. 



VOL. CCV. A. 3 X 



