
theorem Th66:
  for A being decreasing Ordinal-Sequence
  for B being natural-valued non-empty Ordinal-Sequence st dom A = dom B
  ex C being Cantor-normal-form Ordinal-Sequence
  st omega -exponent C = A & omega -leading_coeff C = B
proof
  let A be decreasing Ordinal-Sequence;
  let B be natural-valued non-empty Ordinal-Sequence;
  assume A1: dom A = dom B;
  deffunc F(Ordinal) = B.$1 *^ exp(omega, A.$1);
  consider C being Ordinal-Sequence such that
    A2: dom C = dom A & for a being Ordinal st a in dom A holds C.a = F(a)
    from ORDINAL2:sch 3;
  A3: now
    let a be Ordinal;
    assume A4: a in dom C;
    then A5: C.a = B.a *^ exp(omega, A.a) by A2;
    B.a <> {} by A1, A2, A4, FUNCT_1:def 9;
    hence C.a is Cantor-component by A5;
  end;
  now
    let a, b be Ordinal;
    assume A6: a in b & b in dom C;
    then A7: C.a = B.a *^ exp(omega, A.a) & C.b = B.b *^ exp(omega, A.b)
      by A2, ORDINAL1:10;
XA: rng B c= NAT by VALUED_0:def 6;
X0: b in dom B by A6,A1,A2; then
    b c= dom B by ORDINAL1:def 2; then
xy: B.b in rng B & B.a in rng B by A6, X0, FUNCT_1:3;
    B.a <> {} & B.b <> {} by A1, A2, A6, ORDINAL1:10, FUNCT_1:def 9;
    then 0 c< B.a & 0 c< B.b by XBOOLE_1:2, XBOOLE_0:def 8;
    then 0 in B.a & 0 in B.b by ORDINAL1:11;
    then omega -exponent(C.b) = A.b & omega -exponent(C.a) = A.a
      by A7, ORDINAL5:58, XA, xy;
    hence omega -exponent(C.b) in omega -exponent(C.a)
      by A2, A6, ORDINAL5:def 1;
  end;
  then reconsider C as Cantor-normal-form Ordinal-Sequence
    by A3, ORDINAL5:def 11;
  take C;
  A9: dom(omega -exponent C) = dom A by A2, Def1;
  now
    let a be object;
    assume a in dom(omega -exponent C);
    then A10: a in dom C by Def1;
    then A11: C.a = B.a *^ exp(omega, A.a) by A2;
    B.a <> {} by A1, A2, A10, FUNCT_1:def 9;
    then 0 c< B.a by XBOOLE_1:2, XBOOLE_0:def 8;
    then A12: 0 in B.a by ORDINAL1:11;
Sa: B.a in rng B by FUNCT_1:3,A10,A1,A2;
    rng B c= NAT by VALUED_0:def 6; then
    omega -exponent(C.a) = A.a by A11, A12, ORDINAL5:58, Sa;
    hence (omega -exponent C).a = A.a by A10, Def1;
  end;
  hence omega -exponent C = A by A9, FUNCT_1:2;
  A13: dom(omega -leading_coeff C) = dom B by A1, A2, Def3;
  now
    let a be object;
    assume a in dom(omega -leading_coeff C);
    then A14: a in dom C by Def3;
    then C.a = B.a *^ exp(omega, A.a) by A2;
    then omega -leading_coeff(C.a) = B.a by Th57, ORDINAL1:def 12;
    hence (omega -leading_coeff C).a = B.a by A14, Def3;
  end;
  hence omega -leading_coeff C = B by A13, FUNCT_1:2;
end;
