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
  exp(nextcard a,b) = exp(a,b) *` nextcard a
proof
  now
    per cases by CARD_1:4;
    suppose
A1:   a in b;
      then a c= b by CARD_1:3;
      then
A2:   exp(a,b) = exp(2,b) by Th30;
      nextcard a c= b by A1,CARD_3:90;
      then exp(nextcard a,b) = exp(2,b) & nextcard a in exp(2,b) by Th14,Th30,
ORDINAL1:12;
      hence thesis by A2,Th17;
    end;
    suppose
A3:   b c= a;
      deffunc f(Ordinal) = Funcs(b,$1);
      consider phi being Sequence such that
A4:   dom phi = nextcard a & for A st A in nextcard a holds phi.A = f
      (A) from ORDINAL2:sch 2;
A5:   cf nextcard a = nextcard a by Th22;
A6:   b in nextcard a by A3,CARD_3:91;
      Funcs(b,nextcard a) c= Union phi
      proof
        let x be object;
        assume x in Funcs(b,nextcard a);
        then consider f be Function such that
A7:     x = f and
A8:     dom f = b and
A9:     rng f c= nextcard a by FUNCT_2:def 2;
        reconsider f as Sequence by A8,ORDINAL1:def 7;
        reconsider f as Ordinal-Sequence by A9,ORDINAL2:def 4;
        sup f in nextcard a by A6,A5,A8,A9,Th26;
        then
A10:    phi.sup f in rng phi by A4,FUNCT_1:def 3;
        rng f c= sup f by ORDINAL2:49;
        then
A11:    f in Funcs(b,sup f) by A8,FUNCT_2:def 2;
        phi.sup f = Funcs(b,sup f) by A6,A5,A4,A8,A9,Th26;
        hence thesis by A7,A11,A10,TARSKI:def 4;
      end;
      then
A12:  card Funcs(b,nextcard a) c= card Union phi by CARD_1:11;
      card Funcs(b,nextcard a) = exp(nextcard a,b) & card Union phi c=
      Sum Card phi by CARD_2:def 3,CARD_3:39;
      then
A13:  exp(nextcard a,b) c= Sum Card phi by A12;
      a in nextcard a by CARD_1:18;
      then
A14:  exp(nextcard a,b) *` exp(nextcard a,b) = exp(nextcard a,b) & exp(a,
      b) c= exp (nextcard a,b) by CARD_2:92,CARD_4:15;
      exp(nextcard a,1) = nextcard a & 1 in b by Lm1,Th15,CARD_2:27;
      then nextcard a c= exp(nextcard a,b) by CARD_2:92;
      then
A15:  exp(a,b)*`nextcard a c= exp(nextcard a,b) by A14,CARD_2:90;
A16:  now
        let x be object;
        reconsider xx=x as set by TARSKI:1;
A17:    card card xx = card xx & card b = card b;
        assume
A18:    x in nextcard a;
        then reconsider x9 = x as Ordinal;
A19:    phi.x9 = Funcs(b,x9) by A4,A18;
        card xx in nextcard a by A18,CARD_1:8,ORDINAL1:12;
        then card xx c= a by CARD_3:91;
        then Funcs(b,card xx) c= Funcs(b,a) by FUNCT_5:56;
        then
A20:    card Funcs(b,card xx) c= card Funcs(b,a) by CARD_1:11;
A21:    card Funcs(b,a) = exp(a,b) by CARD_2:def 3;
        (nextcard a --> exp(a,b)).x = exp(a,b) & Card phi.x = card (phi.x
        ) by A4,A18,CARD_3:def 2,FUNCOP_1:7;
        hence Card phi.x c= (nextcard a --> exp(a,b)).x by A19,A17,A20,A21,
FUNCT_5:61;
      end;
      dom Card phi = dom phi & dom (nextcard a --> exp(a,b)) = nextcard a
      by CARD_3:def 2;
      then
A22:  Sum Card phi c= Sum (nextcard a --> exp(a,b)) by A4,A16,CARD_3:30;
      Sum (nextcard a --> exp(a,b)) = (nextcard a)*`exp(a,b) by CARD_2:65;
      then exp(nextcard a,b) c= exp(a,b)*`nextcard a by A13,A22;
      hence thesis by A15;
    end;
  end;
  hence thesis;
end;
