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

118

118 ~~~~CARDINAL ARITHMETIC PRII

[PART III

*113-3. F:. Mult ax. D:K e Nc',& n Cl'Nc'a. D. 1c,6Nc'a x, Nc'/3
Dent.
F. *112-24. *113-12.)D
F:. Mult ax. a! a.:): K E Nc'fl rn C1'Nc'a.. 16 sin 6'a 4. "
[*113-141L2-51 D. 111 E Nc'ca x. Nc'/3 (1)
F.-*1 1:3114-25. DF a = A. D.Nc'a x.Nc' = 0 (2)
F. *101 -14. D F:a A. Ke NC'/3A CI'Nc'a. D: K eCl't"A:
[*60-362] D: K * t * K, = A:
[*112-3301] D:Z'-,K=A (3)
F.- (2). (3). *54>102.)F:a=A.KceNc'I8rCCl'Nc'a.:).El'eNc'ax,,Nc',t (4)
F.(1). (4). D F. Prop
*113-31. F:.Multax.):L,aveNC.KcevAC1',t.).Y.'KteaxCv [*113'3]
*113-32. F:.Multax.):,-LLeNC./Ke vnClexcl',4. ).s'KCEpx~v
*113-33. F:.Multax.):iz,veNC.KccvACl',a.XE/tinCl'v.:).
Z.Nc'K =Y Nc'X =px, v [*113-31-27-23]
*113-34. F:.Multax.):-,WVENC.KEvnClexcl',u.Xe1LtiClexc1'v.:).
Nc's' = Nc'sX =pxa v [*113-32-27]
The above propositions give the connection of addition and -multiplication.
The following propositions are concerned with various forms of the distributive law.
*113-4 F. (/3vy) xa(/x a)vu (y xa)
Demn.
F. *1 131.)F. (,3 v y) x a= s'a 4. "(fi v y
[*40-31] = s'a 4."p3 v s'a 4.c"
[*113-1] -(3 xa) u(y xa). DF. Prop
*113-401. F:/3r y =A. )(s3 x a) (ey xa) =A [*113,19.Transp]
*113-41. F. Nc'(fl + 'y) x,, Nc'a = Nc'tj(fl + ey) x a} = Nc'{(fl x a) + (ey x a)}
= Nc'(/ x a) -i- Nc'( x a)
Dent.
F. *113-25. *110-3. )F.Nc'(/3~+y) x, Nc'a =Nc'{(/ +,y) x a}
Ncx a)+,y x a)}=Nc'(/8x a)-i-Nc',y x a) (1)
F. *1 13-4. (*II10'Ol).:) F. (fl+y) xa=( 4.A./t"flxa)vj(Ap 4."t ryxa) (2)
F. *1 13-13 *1 10-12.:)F.4.Ayl"t"fl8xasmflxa.Ap 4."t",yxasmyxa (3))
F.*113-401.*110-11. - ) F.- (4. A-y"tff'/x a) n (A4 "t"Y~y x a) =A (4)
F. (1).(5). FProp