reserve phi,fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  f,g for Function,
  X for set,
  x,y,z for object;
reserve f1,f2 for Ordinal-Sequence;

theorem Th21:
  A <> {} & A is limit_ordinal implies for fi st dom fi = A & for
  B st B in A holds fi.B = exp(C,B) holds exp(C,A) is_limes_of fi
proof
  assume that
A1: A <> {} and
A2: A is limit_ordinal;
  consider psi such that
A3: dom psi = A and
A4: for B st B in A holds psi.B = exp(C,B) and
A5: ex D st D is_limes_of psi by A1,A2,Lm8;
  let fi such that
A6: dom fi = A and
A7: for B st B in A holds fi.B = exp(C,B);
  now
    let x be object;
    assume
A8: x in A;
    then reconsider B = x as Ordinal;
    thus fi.x = exp(C,B) by A7,A8
      .= psi.x by A4,A8;
  end;
  then fi = psi by A6,A3,FUNCT_1:2;
  then consider D such that
A9: D is_limes_of fi by A5;
  D = lim fi by A9,ORDINAL2:def 10
    .= exp(C,A) by A1,A2,A6,A7,ORDINAL2:45;
  hence thesis by A9;
end;
