reserve k,n,m for Nat,
  A,B,C for Ordinal,
  X for set,
  x,y,z for object;
reserve f,g,h,fx for Function,
  K,M,N for Cardinal,
  phi,psi for
  Ordinal-Sequence;
reserve a,b for Aleph;
reserve a,b for Aleph;

theorem
  a is limit_cardinal & b in cf a implies exp(a,b) = Sum (b-powerfunc_of
  a)
proof
  assume that
A1: a is limit_cardinal and
A2: b in cf a;
  deffunc f(Ordinal) = Funcs(b,$1);
  consider L being Sequence such that
A3: dom L = a & for A st A in a holds L.A = f(A) from ORDINAL2:sch 2;
A4: now
    let x be object;
A5: card Union (b-powerfunc_of a) c= Sum (b-powerfunc_of a) by CARD_3:40;
    assume
A6: x in a;
    then reconsider x9 = x as Ordinal;
    set m = card x9;
A7: m in a by A6,CARD_1:8,ORDINAL1:12;
    then m in dom (b-powerfunc_of a) by Def2;
    then
A8: (b-powerfunc_of a).(card x9) in rng (b-powerfunc_of a) by FUNCT_1:def 3;
    x9,m are_equipotent by CARD_1:def 2;
    then
A9: card Funcs(b,x9) = card Funcs(b,card x9) by FUNCT_5:60;
    L.x = Funcs(b,x9) & (Card L).x = card (L.x) by A3,A6,CARD_3:def 2;
    then
A10: (Card L).x = exp(m,b) by A9,CARD_2:def 3;
    (b-powerfunc_of a).(card x9) = exp(card x9,b) by A7,Def2;
    then card exp(card x9,b) c= card Union (b-powerfunc_of a) by A8,CARD_1:11
,ZFMISC_1:74;
    then
A11: card exp(card x9,b) c= Sum (b-powerfunc_of a) by A5;
    thus (Card L).x c= (a --> Sum (b-powerfunc_of a)).x by A6,A10,A11,
FUNCOP_1:7;
  end;
  dom (a --> Sum (b-powerfunc_of a)) = a & dom Card L = dom L by CARD_3:def 2
;
  then
A12: Sum Card L c= Sum (a --> Sum (b-powerfunc_of a)) by A3,A4,CARD_3:30;
  a c= Sum (b-powerfunc_of a)
  proof
    let x be Ordinal;
     reconsider xx=x as set;
    assume
A13: x in a;
    reconsider x9 = x as Ordinal;
    set m = card x9;
    m in a by A13,CARD_1:8,ORDINAL1:12;
    then
A14: nextcard m c= a by CARD_3:90;
    nextcard m <> a by A1;
    then
A15: nextcard m in a by A14,CARD_1:3;
    then nextcard m in dom (b-powerfunc_of a) by Def2;
    then
A16: (b-powerfunc_of a).(nextcard m) in rng (b-powerfunc_of a) by FUNCT_1:def 3
;
    (b-powerfunc_of a).(nextcard m) = exp(nextcard m,b) by A15,Def2;
    then
A17: card exp(nextcard m,b) c= card Union (b-powerfunc_of a) by A16,CARD_1:11
,ZFMISC_1:74;
A18: nextcard m c= exp(nextcard m,b) by Th19;
    card xx = card card xx;
    then
A19: x9 in nextcard x9 & nextcard card xx = nextcard xx by CARD_1:18,CARD_3:88;
    card exp(nextcard m,b) = exp(nextcard m,b) & card Union (b
    -powerfunc_of a) c= Sum (b-powerfunc_of a) by CARD_3:40;
    then exp(nextcard m,b) c= Sum (b-powerfunc_of a) by A17;
    then nextcard card xx c= Sum (b-powerfunc_of a) by A18;
    hence thesis by A19;
  end;
  then
A20: a*`Sum (b-powerfunc_of a) = Sum (b-powerfunc_of a) by Th17;
  Funcs(b,a) c= Union L
  proof
    let x be object;
    assume x in Funcs(b,a);
    then consider f such that
A21: x = f and
A22: dom f = b and
A23: rng f c= a by FUNCT_2:def 2;
    reconsider f as Sequence by A22,ORDINAL1:def 7;
    reconsider f as Ordinal-Sequence by A23,ORDINAL2:def 4;
    rng f c= sup f by ORDINAL2:49;
    then
A24: x in Funcs(b,sup f) by A21,A22,FUNCT_2:def 2;
    sup f in a by A2,A22,A23,Th26;
    then
A25: L.sup f in rng L by A3,FUNCT_1:def 3;
    L.sup f = Funcs(b,sup f) by A2,A3,A22,A23,Th26;
    hence thesis by A24,A25,TARSKI:def 4;
  end;
  then
A26: card Funcs(b,a) c= card Union L by CARD_1:11;
  card Union L c= Sum Card L by CARD_3:39;
  then card Funcs(b,a) c= Sum Card L by A26;
  then
A27: exp(a,b) c= Sum Card L by CARD_2:def 3;
A28: Sum (b-powerfunc_of a) c= exp(a,b) by Th32;
  Sum (a --> Sum (b-powerfunc_of a)) = a*`Sum (b-powerfunc_of a) by CARD_2:65;
  then exp(a,b) c= a*`Sum (b-powerfunc_of a) by A27,A12;
  hence thesis by A28,A20;
end;
