
theorem Th67:
  for A, B being Cantor-normal-form Ordinal-Sequence
  st omega -exponent A = omega -exponent B &
    omega -leading_coeff A = omega -leading_coeff B
  holds A = B
proof
  let A, B be Cantor-normal-form Ordinal-Sequence;
  assume that A1: omega -exponent A = omega -exponent B and
  A2: omega -leading_coeff A = omega -leading_coeff B;
  A3: dom A = dom(omega -exponent A) by Def1
    .= dom B by A1, Def1;
  now
    let a be object;
    assume A4: a in dom A;
    hence A.a =
      (omega -leading_coeff A).a *^ exp(omega, (omega -exponent A).a) by Th65
      .= B.a by A1, A2, A3, A4, Th65;
  end;
  hence thesis by A3, FUNCT_1:2;
end;
