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

SECTION B] ON SECTIONS AND SEGMENTS 627
*211-727. F:. Pcconnex n Rl'J. aesect'P.):
E! limaxp'a.. E! liminp'(C'P - a)
*211-728. F:. P Econnex n R'J. Ja Esect'P:,- E! imaxp a. v.
NE!minp'(CGP-a):.limaxp'a =liminp(C'P -a)
The remaining propositions are mainly occupied with relation-arithmetic.
rihe most important of them is
*211-82. F::PeSer.QeD'P>. ):.
C'Q Esect/P. -: (aR). P= Q4.R. v. (ax). P= Q+*x:
u:(aR).P=Qt R.v.P=Qf+ B'P
That is, given any series contained in P, if something can be added to
make it into P, its field is a section of P, and vice versa.
*211-01. sect'P=&(a CCG'P. P Caa) Df
*2111. F:aesect'P.a.=aCC'P.P"aCa [(*211-01)]
*211-11. F: as e D'P,. =. (Hj3). a = P"/3 [*37-101]
*211-12. F:aeD'(PtAI:).=E.a=P"a
Dem.
F. *37 101.*50-1. F: a e D(PE I)..(a13). a = P'Cfl. a3 = 8.
[*13-195]. a = P"a: ) F. Prop
*211413. F:ae6sect'P.E.a=P*"C Ca a E DIfD't(P*)E AI.a e D'(P*)e
Dem.
F.*2111.*90-23. )F:aesect'P. =.a=P* (1)
F. *90-17. ) F ccP*6cj3 = * Cc
[*13-12] F: a= P*11 R. P* Cfa = a:
[*211-11] ) F: ae D'(P*)e ). a = P*ca (2)
F. *10-24. *211 11.F: a= P*"1a - D..ae D'(P*), (3)
F. (1). (2). (3). *211-12.) F. Prop
In virtue of the above proposition, the properties of sect'P can be deduced
from those of D'PE or D'(PE A I) by substituting P* for P.
*2114131. F: a E sect'P. ). PL1a ==PP0
Dem.
F. *211~13. F: Hp.). P~Ca = CP"
[*91-52] -P0"(C:a F. Prop
40-2