
theorem Th82:
  for a being Ordinal holds a (+) 0 = a
proof
  let a be Ordinal;
  set E1 = omega -exponent CantorNF a, E2 = omega -exponent CantorNF 0;
  set L1 = omega -leading_coeff CantorNF a;
  set L2 = omega -leading_coeff CantorNF 0;
  consider C being Cantor-normal-form Ordinal-Sequence such that
    A1: a (+) 0 = Sum^ C & rng(omega -exponent C) = rng E1 \/ rng E2 and
    A2: for d being object st d in dom C holds
      (omega -exponent(C.d) in rng E1 \ rng E2 implies
        omega -leading_coeff(C.d) = L1.(E1".(omega -exponent(C.d)))) &
      (omega -exponent(C.d) in rng E2 \ rng E1 implies
        omega -leading_coeff(C.d) = L2.(E2".(omega -exponent(C.d)))) &
      (omega -exponent(C.d) in rng E1 /\ rng E2 implies
        omega -leading_coeff(C.d) = L1.(E1".(omega -exponent(C.d))) +
          L2.(E2".(omega -exponent(C.d)))) by Def5;
  A3: rng E2 is empty;
  then A4: rng(omega -exponent C) = rng E1 by A1;
  A5: dom C = card dom(omega -exponent C) by Def1
    .= card rng(omega -exponent C) by CARD_1:70
    .= card dom E1 by A4, CARD_1:70
    .= dom CantorNF a by Def1;
  for x being object st x in dom C holds C.x = (CantorNF a).x
  proof
    let x be object;
    A6: omega -exponent C = E1 by A4, Th34;
    assume A7: x in dom C;
    then A8: x in dom(omega -exponent C) by Def1;
    then (omega -exponent C).x in rng E1 by A4, FUNCT_1:3;
    then omega -exponent(C.x) in rng E1 \ rng E2 by A3, A7, Def1;
    then A9: omega -leading_coeff(C.x) = L1.(E1".(omega -exponent(C.x)))
        by A2, A7
      .= L1.(E1".((omega -exponent C).x)) by A7, Def1
      .= L1.x by A6, A8, FUNCT_1:34;
    A10: x in dom CantorNF a by A6, A8, Def1;
    thus C.x = L1.x *^ exp(omega, omega -exponent(C.x)) by A7, A9, Th64
      .= L1.x *^ exp(omega, E1.x) by A6, A7, Def1
      .= L1.x *^ exp(omega, omega -exponent((CantorNF a).x)) by A10, Def1
      .= (omega -leading_coeff((CantorNF a).x)) *^
        exp(omega, omega -exponent((CantorNF a).x)) by A10, Def3
      .= (CantorNF a).x by A10, Th64;
  end;
  then C = CantorNF a by A5, FUNCT_1:2;
  hence thesis by A1;
end;
