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
  A c= B implies exp(A,C) c= exp(B,C)
proof
  defpred P[Ordinal] means exp(A,$1) c= exp(B,$1);
  assume
A1: A c= B;
A2: for C st P[C] holds P[succ C]
  proof
    let C;
A3: exp(B,succ C) = B*^exp(B,C) by ORDINAL2:44;
    exp(A,succ C) = A*^exp(A,C) by ORDINAL2:44;
    hence thesis by A1,A3,ORDINAL3:20;
  end;
A4: for C st C <> 0 & C is limit_ordinal & for D st D in C holds P[D] holds
  P[C]
  proof
    deffunc F(Ordinal) = exp(A,$1);
    let C;
    assume that
A5: C <> 0 and
A6: C is limit_ordinal and
A7: for D st D in C holds exp(A,D) c= exp(B,D);
    consider f1 such that
A8: dom f1 = C & for D st D in C holds f1.D = F(D) from ORDINAL2:sch
    3;
    deffunc F(Ordinal) = exp(B,$1);
    consider f2 such that
A9: dom f2 = C & for D st D in C holds f2.D = F(D) from ORDINAL2:sch
    3;
A10: now
      let D;
      assume
A11:  D in dom f1;
      then
A12:  f1.D = exp(A,D) by A8;
      f2.D = exp(B,D) by A8,A9,A11;
      hence f1.D c= f2.D by A7,A8,A11,A12;
    end;
A13: exp(A,C) is_limes_of f1 by A5,A6,A8,Th21;
    exp(B,C) is_limes_of f2 by A5,A6,A9,Th21;
    hence thesis by A8,A9,A13,A10,Th6;
  end;
  exp(A,{}) = 1 by ORDINAL2:43;
  then
A14: P[0] by ORDINAL2:43;
  for C holds P[C] from ORDINAL2:sch 1(A14,A2,A4);
  hence thesis;
end;
