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

SECTION C]

SIMILARITY OF CLASSES

487

F.*71L362. )F: Hp (4). D. y# R'x.
[*2292.*33-44] ). (['R - 'y- t'R'w) v tR'x= PR - c'y (5)
F. (4). (5).*737143. (2). DF: HP (4). ). (D'R - 'x) sin ('R - t'y) (6)
F. (3). (6). )i: Hp(2). D. (D'R - t'x) sin (UR - t'y) (7)
F.*51P21122. F F: D'R = a v t'x. EP'R =3 v tc'y. xc, e a. y E3.
). D'R - t'x = a. R-'y = 3 (8)
F. (7)-(8). F: Re I-+1. Hp (8). D. a smf3 (9)
F. (1). (9). F. Prop
The following propositions give the proof of the Schrdder-Bernstein
theorem, namely: If one class is similar to part of another, and the other
is similar to part of the one, then the two classes are similar. The proof
here given is due to Zermelo*. An explanation of the following proof is
given in connection with another proof in the summary of *95.
*73-8. F: U'R C)3./3 C D'R. tc = a (a C D'R./ - U'R C a. Rcia C a)..
D'Re Ic.p"K C D'R
Dem.
F. *22-42-43-44. )F:Hp..)D'R C D'R.3 - 'R C D'R (1)
F.*22-44. *37-25. D)F: Hp.)D. _IDIR C D'R (2)
F.(1).(2). DF: Hp. D. D'ReK (3)
F.(3). *40-12. D)F. Prop
*73-801.: Hp *73&8.:)./3- EIR C p'l
Here "Hp *73-8 " means " the hypothesis of *73-8."
Dem.
F. *2033. D F:. Hp. D: a E K. ) Q. / - (t'R C a:. F F. Prop
*73-802. F: Hp *73-8.. Rh'p'K C p'lc
Dern.
F.*20333.:.Hp.):ae K.a.R"a C a (1)
F. (1).*40-81.)F Prop
*73-81. F:Hp *7388. D).p'lEK Ic
Demn.
F.*73&80801802. )F: Hp. ).p'c CD'R./3-Et'R C p'tc. A"p'1KCp'1: DF. Prop
*73'811. F: Hp *73-8. R"p'lc C p'Kc - (/ - (1'R)
Dem.
F.- *37-16.DF. Rp'rti C C(1R
[*22-8] C - (- U'AR)
[*21-62-81-43] C - (3 - U 'R) (1)
F. (1). *73802. D F. Prop
* Math. Annalen, vol. Lxv. Heft 2, February 1908.