Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

SECTION Al

SEQUENT POINTS

591

*206-551. I-: P eSer. ).seq p'a =seq p'p'Pp'cP"p(a riC'P).
maxp 'a = maxp'pPIpP"IIII(a n G'P)
Dem.
-~ -~ 4 -F. *206-13. )F.seqp'a = minp'p'P"I(a rn C'IP) (1)
F.- (1). *206-53.) F:Hp. ). seTp'a =z seqp'~p'PIIfp'P"(a nC'P) A C'P} (2)
F. (2). *200-54. ) F: Hp. A4! P.:). seq p'a =seqp'p'P"p'IP"I(a A C'P) (3)
F. *206-18. )F: P= A).seqpl'a= A. seqp'~p'Pc~p6P"(a A C'P)=A (4)
F. (3). (4). F F: Hp. ).seqp'a = seqp'~p'P'~~p'P"(a nC'P) (5)
F. *206-53. ) F: Hp.. maxp'a = precp'lp'P"1(a A C'P)
[*206K13-101.*200-54] = maxp'~p'P"p'P"P(a nI G'P) (6)
F. (5). (6). F. Prop
*206-56. F:. PeSer. ): (a).p'P"I(anCG'P),EU'lmaxp u UI'seqp.=(a). ae E GImaxp v U'seq p
Dem.
F.*10-1-11. )F: (a). aEG'maxpv UI'seqp.:).
(a). p'IP"6(o A G'P) E G'maxp v G'cseqp (1)
F. *10-1. F: (a) p'P"1(a A C'P) eG'maxp v G'seqp.).
pPtpP"(C/3C( A, C'P) e G'tmaxp v Ui'seqp.
[*206-551] ).86 /3EEI'maxp v P1seq p (2)
F. (1).(2). ) F. Prop
*206-57. F:.PcSer.:):(a).aeU'lminpvU'rprecp.n-.
(a). a e U'6maxp v U'1seqp [*206-35556]
This proposition is important, since it shows that when a serial relation
satisfies Dedekind's axiom., so does its converse. Thus if all classes which
have no maximum have an uipper limit, then all classes which have no
minimum have a lower limit, and vice versa.
*206-6. F: SEP -mor Q. ).pP"rII(a A GP) = SCCpQ"CS6a
Dem.
F.- *1 51 11.) F: Hp. ).pP"rI(a A G'P) = p'&"cQ("'S"(a A% D'rS)
[*72-341] = SetP'Q"'St(a r% D'S)
4- -J
[*71L613] = SC(p'Q"S81"a: F. Prop