
theorem Th43:
  for A being finite Ordinal-Sequence, a being Ordinal
  st <% a %> ^ A is Cantor-normal-form
  holds Sum^ A in exp(omega, omega -exponent a)
proof
  let A be finite Ordinal-Sequence, a be Ordinal;
  assume <% a %> ^ A is Cantor-normal-form;
  then reconsider B = <% a %> ^ A as Cantor-normal-form Ordinal-Sequence;
  now
    let c be Ordinal;
    assume A1: c in dom A;
    then reconsider n = c as Nat;
    len <% a %> + n in dom B by A1, AFINSQ_1:23;
    then A2: n + 1 in dom B by AFINSQ_1:34;
    B.(len <% a %> + n) = A.n by A1, AFINSQ_1:def 3;
    then A3: A.n = B.(n+1) by AFINSQ_1:34;
    0 in Segm (n+1) by NAT_1:44;
    then omega -exponent(B.(n+1)) in omega -exponent(B.0)
      by A2, ORDINAL5:def 11;
    then exp(omega,omega -exponent(A.n)) in exp(omega,omega -exponent(B.0))
      by A3, ORDINAL4:24;
    then A4: exp(omega,omega -exponent(A.n)) in exp(omega,omega -exponent a)
      by AFINSQ_1:35;
    B.(n+1) is Cantor-component by A2, ORDINAL5:def 11;
    then consider b being Ordinal, m being Nat such that
      A5: 0 in Segm m & A.n = m *^ exp(omega,b) by A3, ORDINAL5:def 9;
    0 in m & m in omega by A5, ORDINAL1:def 12;
    then omega -exponent(A.n) = b by A5, ORDINAL5:58;
    hence A.c in exp(omega,omega -exponent a) by A4, A5, Th42;
  end;
  hence Sum^ A in exp(omega,omega -exponent a) by Th41;
end;
