
theorem Th84:
  for A, B being finite Ordinal-Sequence st A^B is Cantor-normal-form
  holds (Sum^ A) (+) (Sum^ B) = (Sum^ A) +^ (Sum^ B)
proof
  defpred P[Nat] means for A, B being finite Ordinal-Sequence
    st len A = $1 & A^B is Cantor-normal-form
    holds (Sum^ A) (+) (Sum^ B) = (Sum^ A) +^ (Sum^ B);
  A1: P[0]
  proof
    let A, B being finite Ordinal-Sequence;
    assume len A = 0 & A^B is Cantor-normal-form;
    then A is empty;
    then A2: Sum^ A = 0 by ORDINAL5:52;
    hence (Sum^ A) (+) (Sum^ B) = Sum^ B by Th82
      .= (Sum^ A) +^ (Sum^ B) by A2, ORDINAL2:30;
  end;
  A3: for n being Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
    assume A4: P[n];
    let A, B being finite Ordinal-Sequence;
    assume A5: len A = n+1 & A^B is Cantor-normal-form;
    then A6: A <> {} & A is Cantor-normal-form by ORDINAL5:66;
    then consider a0 being Cantor-component Ordinal,
      A0 being Cantor-normal-form Ordinal-Sequence such that
      A7: A = <% a0 %> ^ A0 by ORDINAL5:67;
    A8: <% a0 %> ^ (A0 ^ B) is Cantor-normal-form by A5, A7, AFINSQ_1:27;
    then A9: A0 ^ B is Cantor-normal-form by ORDINAL5:66;
    n+1 = len <% a0 %> + len A0 by A5, A7, AFINSQ_1:17
      .= 1 + len A0 by AFINSQ_1:34;
    then A10: (Sum^ A0) (+) (Sum^ B) = (Sum^ A0) +^ (Sum^ B) by A4, A9;
    consider b being Ordinal, m being Nat such that
      A11: 0 in Segm m & a0 = m *^ exp(omega,b) by ORDINAL5:def 9;
    reconsider m as non zero Nat by A11;
    0 in m & m in omega by A11, ORDINAL1:def 12;
    then A12: omega -exponent a0 = b by A11, ORDINAL5:58;
    A13: omega -exponent Sum^ A0 c= b
    proof
      per cases;
      suppose A14: 0 in Sum^ A0;
        Sum^ A0 in exp(omega, omega -exponent a0) by A6, A7, Th43;
        hence thesis by A12, A14, Th23, ORDINAL1:def 2;
      end;
      suppose not 0 in Sum^ A0;
        then omega -exponent Sum^ A0 = 0 by ORDINAL5:def 10;
        hence thesis;
      end;
    end;
    A15: omega -exponent (Sum^ A0 (+) Sum^ B) c= b
    proof
      A16: Sum^ A0 (+) Sum^ B = Sum^ (A0^B) by A10, Th24;
      per cases;
      suppose A17: 0 in Sum^ (A0^B);
        Sum^ (A0^B) in exp(omega, omega -exponent a0) by A8, Th43;
        hence thesis by A12, A16, A17, Th23, ORDINAL1:def 2;
      end;
      suppose not 0 in Sum^ (A0^B);
        then omega -exponent Sum^ (A0^B) = 0 by ORDINAL5:def 10;
        hence thesis by A16;
      end;
    end;
    thus (Sum^ A) (+) (Sum^ B) = (a0 +^ Sum^ A0) (+) Sum^ B by A7, ORDINAL5:55
      .= (a0 (+) Sum^ A0) (+) Sum^ B by A11, A13, Th83
      .= a0 (+) (Sum^ A0 (+) Sum^ B) by Th81
      .= a0 +^ (Sum^ A0 (+) Sum^ B) by A11, A15, Th83
      .= a0 +^ Sum^ A0 +^ Sum^ B by A10, ORDINAL3:30
      .= Sum^ A +^ Sum^ B by A7, ORDINAL5:55;
  end;
  A18: for n being Nat holds P[n] from NAT_1:sch 2(A1,A3);
  let A, B being finite Ordinal-Sequence;
  assume A19: A^B is Cantor-normal-form;
  len A is Nat;
  hence thesis by A18, A19;
end;
