
theorem Th65:
  for A being Cantor-normal-form Ordinal-Sequence, a being object st a in dom A
  holds A.a = (omega -leading_coeff A).a *^ exp(omega, (omega -exponent A).a)
proof
  let A be Cantor-normal-form Ordinal-Sequence, a be object;
  assume A1: a in dom A;
  hence A.a = (omega -leading_coeff(A.a)) *^ exp(omega, omega -exponent(A.a))
      by Th64
    .= (omega -leading_coeff A).a *^ exp(omega, omega -exponent(A.a))
      by A1, Def3
    .= (omega -leading_coeff A).a *^ exp(omega, (omega -exponent A).a)
      by A1, Def1;
end;
