
theorem Th32:
  for A being non empty Cantor-normal-form Ordinal-Sequence, a being object
  st a in dom A holds omega -exponent(A.a) c= omega -exponent(A.0)
proof
  let A be non empty Cantor-normal-form Ordinal-Sequence, a be object;
  assume A1: a in dom A;
  consider a0 being Cantor-component Ordinal,
    A0 being Cantor-normal-form Ordinal-Sequence such that
    A2: A = <% a0 %> ^ A0 by ORDINAL5:67;
  per cases by A1, A2, AFINSQ_1:20;
  suppose a in dom <% a0 %>;
    then a in Segm 1 by AFINSQ_1:33;
    hence thesis by CARD_1:49, TARSKI:def 1;
  end;
  suppose ex n being Nat st n in dom A0 & a = len <% a0 %> + n;
    then consider n being Nat such that
      A3: n in dom A0 & a = len <% a0 %> + n;
    reconsider n1 = a as Nat by A3;
    n1 = n+1 by A3, AFINSQ_1:34;
    then 0 in Segm n1 by NAT_1:44;
    then A4: 0 in n1;
    n1 in dom A by A2, A3, AFINSQ_1:23;
    hence thesis by A4, ORDINAL5:def 11, ORDINAL1:def 2;
  end;
end;
