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

636 SERIES [PART V
*211-361. F: PeSer.ID'P1= 'PP. ). sect'P=D'P,
Dem.
F. *201P63. -: Hp. ). DP, CD'PR.
[*93-103]. B'P=A (1)
F. (1). *211-36 F: Hp.:)~. sect"P - D"P, = A (2)
F.(2).*211-15. )F. Prop
*211-371. F:. P E trans n connex (a). a E U'rriaxp v U'seqp: ). D'PE C tI'seqp
[*211P32]
*211-372. F:. P E trans A connex (a). a e Gi'maxpv l'seqp: ). D'Pe = P'%'CP
Dem.
F. *211P371.)F:. Hp.) a,D'PeP. ). E!seqp'a.
[*206-3.*211115]. a = P'seqp'a.
[*206-18] a.aeP"'P (1)
F.(1). *2113.) F-.Prop
*211-38. I-:. P eSer. ) (a). ae I'maxp v U'seqp.. D'P = P"c'P
Dem.
F.*211-11. F:.D'PIe=P"G'P.E-:(13):(a[x).P"/38=P'x.xeG'P (1)
F. *206174. *205111.)
-: Pe Ser. a! maxp',8. ). seqp/3 = CCP n 5^ (P",8 = P'x) (2)
F.(1). (2).) F:. Pc Ser. D'Pe = P""C,'P.): a! maxp'. ). D! seqp'lB:
[*3341]:) f e U'maxp v UI'seqp (3)
F.(3).a*211-372. F. Prop
The following propositions are concerned with D'(Pe A I), i.e. with those
sections of P which have no maximum. If P is compact (i.e. if P2 = P),
D'(Pe A I) = D'Pe. If P is also a Dedekindian series, D'(Pe A I)= P"C'tP.
This is the mark of Dedekindian continuity, since it states that, if P"Ca has
no maximum, there is an x for which P"a = P'C, and this x is the upper
limit of P"a; while conversely, if x is any term of C'P, P'w has no
maximum, so that the series is compact.
*211-4. F. D'(P6 A I) C - U'maxp
Demn.
F. *211-12. ) F: a e D'(P6 A I)..a - PCCa = A.
[*205-111]. maxp'a = A:) F. Prop