reserve fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  X,Y for set,
  x,y for object;

theorem Th40:
  A is limit_ordinal implies A*^B is limit_ordinal
proof
A1: now
    deffunc F(Ordinal) = $1 *^ B;
    assume that
A2: A <> {} and
A3: A is limit_ordinal;
    consider fi such that
A4: dom fi = A & for C st C in A holds fi.C = F(C) from ORDINAL2:sch 3;
A5: A*^B = union sup fi by A2,A3,A4,ORDINAL2:37
      .= union sup rng fi;
    for C st C in A*^B holds succ C in A*^B
    proof
      let C;
      assume
A6:   C in A*^B;
      then consider X such that
A7:   C in X and
A8:   X in sup rng fi by A5,TARSKI:def 4;
      reconsider X as Ordinal by A8;
      consider D such that
A9:   D in rng fi and
A10:  X c= D by A8,ORDINAL2:21;
      consider x being object such that
A11:  x in dom fi and
A12:  D = fi.x by A9,FUNCT_1:def 3;
      succ C c= D by A7,A10,ORDINAL1:21;
      then
A13:  succ C in succ D by ORDINAL1:22;
      reconsider x as Ordinal by A11;
A14:  succ x in dom fi by A3,A4,A11,ORDINAL1:28;
      then fi.succ x = (succ x)*^B by A4
        .= x*^B+^B by ORDINAL2:36;
      then x*^B+^B in rng fi by A14,FUNCT_1:def 3;
      then
A15:  x*^B+^B in sup rng fi by ORDINAL2:19;
A16:  x*^B+^succ {} = succ(x*^B+^{}) by ORDINAL2:28;
      B<>{} by A6,ORDINAL2:38;
      then {} in B by Th8;
      then
A17:  succ {} c= B by ORDINAL1:21;
A18:  x*^B+^{} = x*^B by ORDINAL2:27;
      x*^B = fi.x by A4,A11;
      then succ D in sup rng fi by A12,A17,A16,A18,A15,ORDINAL1:12,ORDINAL2:33;
      hence thesis by A5,A13,TARSKI:def 4;
    end;
    hence thesis by ORDINAL1:28;
  end;
  assume A is limit_ordinal;
  hence thesis by A1,ORDINAL2:35;
end;
