MATHEMATICAL CONCEPTS otf THE MATERIAL WORLD. 437 



*1'53. yHpR is the statement that T&-(abcdt) and R-(acdbt) are inconsistent. In 



symbols, 



y Hp R . = :(, b, c, d, t) : U-(abcdt) . D . ^ E'(acdbt) Df 



*1'54. 8 Hp R is the statement that R ; (6ccfa) implies that b and d are distinct. 



In symbols, 



8 Hp R . = : (a, b, c, d, t) : H'(abcdt) .o.b^d Df 



*1'61. Proposition. Assuming (a, ft, y, 8) Hp R, then R,-(abcdt) implies that a, b, 

 c, d are all distinct. In symbols, 



t" . '. (a, ft, y, 8) Hp R . D : ~R-(abcdt) .^.a^b.a^c.a^d.b^c.b^d.c^d 



*T62. Proposition. Assuming (a, ft, y, 8) Up R, then R in ; (BCD<) implies that 

 B, C, D are all distinct. In symbols, 



\- . : (a, ft, y, 8) Up R . D : R in ; (BCD) .D.B^C.B^D.C^D 



Proof. By definition (cf. *l'3l) R In : (BCD() implies that a, x, y, z exist such that 

 B, C, D are members of R ; (a???), a; is a member of.B, y of C, z of D, and K,'(axyzt). 

 Hence (cf. *1'61) a, x, y, z are distinct. Now if any two of B, C, D are identical, 

 e.g., B and C, then x and y are both members of B. Hence (cf. *1 -12-21), since a is 

 distinct from x and y, x can be substituted for y in Hi'(axyzt). Hence ~R ; (axxzt), 

 which contradicts *1'61. 



*1'63. Proposition. Assuming ftEpR, then R in : (BCDO implies R in ; (DCBf). In 



symbols, 



I- /. ft Up R . D : R in '(BCD<) . 3 . R in '(DCB<) 



Proof. Cf. *1-31'52. 



*l-64. Proposition. Assuming (a, ft, y, 8) Hp R, then R In ; (BCD) and B ln ; (CDB) 

 are inconsistent. In symbols, 



I- .-. (, ft, y, 8) Hp R . D : B In '(BCD) . D . - R in ; (CDBO 



Proof. R in ; (BCD) implies (cf. *1'31) that a, x, y, z exist such that a is a common 

 member of B, C, D, x is a member of B, y of C, z of D, and ~R'-(axyzt), and B, C, D 

 are members of R : (???). Hence (cf. *1'61) a, x, y, z are all distinct. Similarly 

 also if R in ; (CDBi), then a', *', y f , z' exist with similar properties, viz., x' a member 

 of B, &c., except that T\,-(a'y'z'x't). Hence (cf. *r23'62) a and a' are identical. 

 Thus R : (axyzt) and ^(ai/z'x't). But (cf. *1'21) x can be substituted for x', y for y f , 

 and z for z'. Hence R-(axyzt) and ~R'(ayzxt). But this contradicts yHpR. 



*1'65. Proposition. Assuming (intpnt, a, ft, y, 8)HpR, the classes R ; (;-") and 

 R : ( ;;;;) are identical. In symbols, 



|- : (intpnt, a, ft, y, 8) Hp R . D . R ! ( ; *) = R : ( ; ; ; ;) 



