12 1. ELEMENTAR Y EMBEDDING S Since C G jo(^i), w e may fi x a function he '• Xo V suc h tha t jo(hc)(do) = C, z = { d G X0 | h c (d) G /xi} G /i0 and d o G jo(^) « Le t S = {/ic(d ) | d G 2:} . W e hav e tha t XQ G V^ , s o |Xo| K\. Therefore , |*S | K\ an d s o T = f]S G /xi , sinc e /i i i s KI- complete. Sinc e T n - A 7 ^ 0 , i t suffice s t o sho w tha t .70(a ) £ C fo r eac h a G T . W e hav e essentiall y show n thi s already : th e constan t functio n f a from XQ t o {a } represent s .70(a ) i n th e /io-ultrapower , he represent s C , an d {d G XQ I fa{d) G /ic(^)} contain s z an d s o i s i n JJLQ. Thi s prove s Clai m 3 . Proof o f Clai m 1 : W e ar e give n function s / an d g i n MQ from jo(Xi) t o the ordinal s suc h tha t th e se t C = {bejo(X 1 )\f{b)=g(b)} is i n jo(fii). Fo r eac h a G Xll 7r(/)(a ) = f(jo(a)) an d 7r(p)(a ) = p(jo(a)) . We hav e t o sho w tha t {a G Xi I 7r(/)(a) = 7r( 5 )(a)} G /ii. Suppose t o th e contrar y tha t A = {aeX l \ n(f)(a) ± Tr(g)(a)} e fii. By Clai m 3 , ther e exist s a G A suc h tha t jo {a) £ C . Fixin g suc h a n a , w e have tha t n(f)(a) 7 ^ n(g)(a)1 whic h mean s tha t f(jo(a)) ^ g(jo(o)), whic h gives a contradictio n sinc e jo(a) G C. Thi s prove s Clai m 1 , an d th e proo f o f Claim 2 i s almos t identical . We no w hav e tha t n induce s a n orde r preservin g 1- 1 ma p 7f : (Ord^Xl^)Mo/jo(fJLi) - Ord Xl /in. We hav e t o sho w tha t n i s onto . Thi s amount s t o showin g tha t fo r eac h function / : X\ Ord ther e exist s a functio n g: jo(Xi) Ord suc h tha t g G MQ and n(g) = f mo d \i\. Th e difficult y i s that th e functio n / nee d no t be i n Mo . For eac h a G X\ , le t h a : Xo » Ord b e a functio n whic h represent s /(a) , i.e. , a functio n suc h tha t jo(h a )(do) = /(a) , wher e d o a s befor e i s th e element o f M o represente d b y th e identit y functio n o n XQ . Defin e a functio n G: Xo Ord Xl b y lettin g G(b) b e a functio n fro m X\ t o Or d suc h tha t G(b)(a) = h a (b) for al l a G X\. Le t g = jo(G)(do) . Applyin g jo, w e se e tha t fo r eac h d G jo(Xo), jo(G)(d) i s a functio n fro m jo(X\) t o th e ordinals , s o g i s suc h a functio n an d g G Mo. Finally , w e sho w tha t 7r(g) f. Fi x a E X\. B y the definitio n o f 7r , ^(«7)(«) = 9(h(a)) = Uo(G)(d 0 ))(Jo(a)) = jo(h a )(d0) = f(a),
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