Miss Wrinch, On the exponentiation of well-ordered series 231 



[*176-19] D f- :. Hp . D : /? = i'PQ'X . D . G'PQ'X ■-= C'Q (2) 

 l-.(l).(2).DI-.Prop 



*-451. t-:.P,QeO.\CC*P«.X-'eOul .p\Tp^Ay\ = A. 



D . a'PQ'X =C'Q:D.E ! tl (P'?)'\ [**45-44] 



r46. f- :, P, Q e a . \ C 0'P« . \ ~ e u 1 . p'(Tp*Ay\ = A . 



D^ . E! limin (P^Yfi [*rll-21 8-451] 



r5. l-.Hpr42.D.maJ<2^a'PQ'X = A 



Pe?u. 

 [*-04] h. Hp.^ 6 a*PQ*\. 



D.'^fX,.fl€(Tp^AyX.2Q{Qm'f^) (1) 



[r04] h.Hp.yLie(rp*^yx. 



D.ai..^ = TpV.Q'QmV.ca'PQ'x (2) 



1- . r21G . (1) . (2) . D I- . Hp . ^ e a'PQ'X . 



D.'^z'.zQz'.z'ea'PQ'X (3) 



I- . (3) . D h . Prop 



r51. l-:.Hpr42:D:a*PQ'\=C'Q. 



V . a^ . ^ e c"Q - a'Q, . a'PQ'X = Q'^ [*-5] 



-r52. \-:.P,Qen.C'QC Q'Q, . D : ya ~ e . /^ C G'P^ . 



D^ . E ! limin (P«)V [*r46-51] 



r53. h : P, Q e n . C'Q C Q'Q, . E ! P'Cnv'P« . D . P<^' e Ded 



[**-52-l] 



r531. \-:P,Qea.C'QC a'Q, .~B'Cnv'P'i = A . 



D . P« e semi-Ded [**-52-l] 



r5401. h :. P, Qe n : D : p = B'Cnv'P^ .^.Rep. 



■D,.I)'R = i'B'P [*17ryl9] 



*'541. f- :. P, Q e O . D : E ! 5'Cnv'P« . = . E ! 5'P [r5401] 



*-55. h : . P, Q e n . (7'Q C a^Qi : D : P« e semi-Ded : 



P« e Ded . = . E ! B'P [**-541-53-531] 

 *-56. h : . P, Q e n . Nr'Q ^ « . D : P« e semi-Ded : 



P«eDed.= .E!7i'P [r55] 



