MATHEMATICAL CONCEPTS OF THE MATERIAL WOELD. 503 



*13'32. Proposition. Assuming (\ - TT) Up (j>, all subclasses of O^ which are 

 ^-maximal are self-<-concurrent, and conversely. In symbols, 



I" : (X - TT) Hp (f> . D . mx^, n els' O = conc^ n els' O 

 Proof. Cf. * 13-1 1-31. 



*14. On Points. 



*14'11. Assuming (X-/3)Hjp<, every ^-point is a self-(-concurrence and a subclass 

 of O^. In symbols, 



h : (X - p) Up (j> . => . pnt^, c confy n els' O^ 



Proof. Every <-point (cf. *ll*2l) is a subclass of O^. Again let x and y be two 

 distinct members of a <-point P. Then (cf. *3'42) a, b, c exist such that 

 i'rt u i'6 u I'c is <-axial and of three dimensions, and x and y are each ^-concurrent 

 with each of a, b, and c. Hence (cf. *12'53) at least one pair of a, b, and c exist 

 (say a and b) such that i' x u i' a u i' b and t' y u <,' a u <.' b are both three ^-dimensional and 

 self-(^-concurrences. Hence (cf. * 1 2 - 52) t' x u t' a u i' b and i' x u t' u t' b are both ^>-axial. 

 Hence (cf. *10'5 and *13'32) i'.x u i'y is </>-axial. Hence P is a self-(/>-concurrence. 



*14'12. Proposition. Assuming (A.-p)Hp<, every ^-point is ^-maximal. In 

 symbols, 



h : (X - 



Proof Cf. *13'32 and *14'11. 



*14'13. Proposition. Assuming (X'-p)Hp^, if P is a ^-point, then P is the 

 ^-concurrence of O^, with P. In symbols, 



I- .'. (X-p) % </, . D : P epnt, . D . P = (\'P 



Proof. Cf. *3'42 and *8'21'22 and *14'11. 



*14 - 14. Proposition. Assuming (X-TT) Hp ^>, if P is a ^>-point, it possesses at least 

 three members. In symbols, 



|-.\ (\-7r)Hpc/> . = : Pepnt* . D.Nc'P^ 3 



Proof. P possesses (cf. *3'42) every member of O^ which is ^-concurrent with 

 each of a certain <-axial set of three members. Hence (cf. *12'42) P possesses this 

 set of three members. 



*14'21. Proposition. Assuming (\-p)Hp<, ^-points with more than one member 

 in common are identical. In symbols, 



I- .'. (X-p) Hp . D : P, Q epnt^ . Nc'(P n Q) > 1 . o . P = Q 



Proof. Let a and 6 be two distinct members of P n Q. Then (cf. *3'42 and 

 *12'52 and *14'11) c and d exist, such that c is a member of P and d of Q, and 



