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

SECTION B] ON THE ARITHMETICAL PRODUCT OF TWO CARDINALS

117

*113-251. I. yx 3cyNc'yx Nc' [*113-25.*100-3]
*113-26. -:Pa,v ENC. H!smq"jt. a!smS"v. ).: X,a xv=sm, x sm"'v
Dent.
F. *37-29. Transp. F: Hp.). D!,at!.
[*102-64] ) (~,fy,3)qL=Nc(a)'ey. v=Nc(/)'( (1)
F. *102~88. ) F: = Nc (t)'ry. ii = Nc (P3)'(3. ~j! sm,"C. ~j! srn5~"v.).
sm 1.k= Nc (,q)',y. smSjv = Nc ( g)%.!! Nc (,q)6y. a[! Nc e
[*113221] D. sm,rn" x, sm'"v = Nc'(y X 8) (2)
F. *37-29. Transp. *113-221.D
F-: u= Nc (a)'~y. v = Nc (3C. t~~~~! s,~..~xvN~c ) (3)
F. (2). (3). D F Nc (a)C'y. v = Nc (i3)'3. E! smq",a. st!iSrnv.).
4 xe v = sim, tk X, smL"V (4)
F. (4). *1111-35-45. (1). D F. Prop
*113-261. F: P, v e NC D., v = PI(') xe Xv0) = ' u(O) xo Xv(0) = etc.
Here "etc." inclutles all ascending derivatives of At. We shall only prove
the result for t(') and.pm), since it is proved in just the same way for the
other cases. a(') x,v 0) or p/i) x' v(,o) or etc. will serve equally well; i.e. it is
not necessary to take the same derivative of lk as of v.
Dent.
F. *1O426-2645). )
F: Hp. g!,a. a! v. V1'' ~rr. () MC ~i~U'.~ 2v'
[*113-26] D:,U X, v= pA" X, p(') (1)
F. *104-264. *113-204. D
F-:, (5 i,. 2[ v). D. p x, v = A. it(') x, 0() = A ~ (2)
F.(1). (2). DF.Prop
As appears in the above proof; if p' and ri are any derivatives of p. and v,
the above proposition holds provided we have
2! P. a! V. D.2[! pA P!.
Thus it holds for all ascemding derivatives, buit not always for descending
derivatives.
*113-27. F.Ftx, v = v x,
Dern.
F. *1132141.)
F: e L x, Pv..(~ja, fl). = N,,c'. v = N,,c'. sm (3 x a).
[*113-2].ev x,k: )F. Prop
Note that this proposition is not confined to the case in which Ftk and v are
cardinals. When either or both are not cardinals,
Pt x V=A =Av x. P.