
theorem Th31:
  for A being non empty Cantor-normal-form Ordinal-Sequence, a being object
  st a in dom A holds omega -exponent last A c= omega -exponent(A.a)
proof
  let A be non empty Cantor-normal-form Ordinal-Sequence, a be object;
  assume A1: a in dom A;
  consider A0 being Cantor-normal-form Ordinal-Sequence,
    a0 being Cantor-component Ordinal such that
    A2: A = A0 ^ <% a0 %> by Th29;
  per cases by A1, A2, AFINSQ_1:20;
  suppose A3: a in dom A0;
    0 in 1 by CARD_1:49, TARSKI:def 1;
    then 0 in dom <% a0 %> by AFINSQ_1:33;
    then len A0 + 0 in dom A by A2, AFINSQ_1:23;
    then omega -exponent(A.len A0) in omega -exponent(A.a)
      by A3, ORDINAL5:def 11;
    then omega -exponent a0 in omega -exponent(A.a) by A2, AFINSQ_1:36;
    then omega -exponent last A in omega -exponent(A.a) by A2, AFINSQ_1:92;
    hence thesis by ORDINAL1:def 2;
  end;
  suppose ex n being Nat st n in dom <% a0 %> & a = len A0 + n;
    then consider n being Nat such that
      A4: n in dom <% a0 %> & a = len A0 + n;
    n in Segm 1 by A4, AFINSQ_1:33;
    then n = 0 by CARD_1:49, TARSKI:def 1;
    then A.a = a0 by A2, A4, AFINSQ_1:36
      .= last A by A2, AFINSQ_1:92;
    hence thesis;
  end;
end;
