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

650

650 ~~PROLEGOMENA TO CARDINAL ARITHMETIC [ATI

[PART II

*96-47. I-:. 1?E C~s-, 1. y,z ER*'x.yRw.zRw.y~z.):w=-R'niaxli'JR'x:
y = maxIj'JR x * z { (IR'x) 1 RI 6RmaxR JR~x. v.
z=maxl>{JR'x. y = t(IR'x) 1 RJ'R'rnax U'JRx
[*96'441-462]
We still have to prove
4 -1? E Cis -+ 1.!JR'X. a! I.R'X. (dY, Z, w). Y, z e R*'x. ytw. ziw. y zf z,
or, what comes to the same thing because of *96'441,
1? E Cis -p+ 1. a! JR'X. a! IR'X.. (HY, Z, W). Y E JR'X. Z 6 IR'X. yf.? ZliW.
This is effected in the following propositions.
*96'472. F:BeCls —+1.a!JIz'x.a!JiR'x.).(ay).yEJR'x.RY'yEIR'X
Dem.
I- *901. F:.XEJR'X.R"'JR'xCJR'x. ):xR~y.). yEJR'x:
[*96-104]: I.Rx =A (1)
F. (1). Transp.*96-452.) F: Hp.:).af! R"fJp'x -JR X.
[*71'401] )(HY, Z).y eJIZ'X -Z=B'Y -Z'CJR'X.
[*13'195] ~(HY).YECJR'X R'cy,e JIjx.
[*96'102] ) (HY).YEGJR'X.RyE IB'Rx: ) F. Prop
*96'473. Fe Cis -+ 1. a!JR'x. a! IR'x. ).E! maxpi Jfl'x. E! R'rnaxlR'JR'x
[*96-461 '472]
*96-474. F: R e Cis -> 1. w = R? maxR"J.R~x.)
E! t(IR,'x) 1 RI}'w. E! maxpRJR'x. t(JR'x) 1 R}'w = max.R'J 'x
Dern.

F. *71-361.[*14-21]
[*93.11]
[(1).*96-45]
F. (2). *93'11.
[Hp.*96'102]
[*96'1.*91 '5-2]
[*34'1]
[*96-342]
[*96'45 ]

) F:Hp.:). (max.R'JR'x) Rw.
D. E! maxR'JR'w.
).max.R JR Ca- J 'x.
J (Jiitx) 1 RI 'w = maxRJlj(x
)F:Hp. D max"JRlx(..Je RJRX.
3] D. R'max 'Jl 'x,, cJR'x.
D. w 6 C~
D. wR~0w. wuR* I1w.
D (az). wI?0w. wR~z. zRw.
D (HZ). Z C I1"x. zRw.
D.E! f(I.'x) 1RI'w

(1)
(2)
(3)
(4)

F.(2). (3). (4). )I-. Prop