
theorem Th51:
  for A being Cantor-normal-form Ordinal-Sequence
  holds 0 in rng(omega -exponent A) iff A <> {} & omega -exponent last A = 0
proof
  let A be Cantor-normal-form Ordinal-Sequence;
  hereby
    assume 0 in rng(omega -exponent A);
    then consider x being object such that
      A1: x in dom(omega -exponent A) & (omega -exponent A).x = 0
      by FUNCT_1:def 3;
    thus A2: A <> {} by A1;
    A3: x in dom A by A1, Def1;
    then omega -exponent last A c= omega -exponent(A.x) by A2, Th31;
    then omega -exponent last A c= 0 by A1, A3, Def1;
    hence omega -exponent last A = 0;
  end;
  assume A4: A <> {} & omega -exponent last A = 0;
  then consider A0 being Cantor-normal-form Ordinal-Sequence,
    a0 being Cantor-component Ordinal such that
    A5: A = A0 ^ <% a0 %> by Th29;
  0 in 1 by CARD_1:49, TARSKI:def 1;
  then 0 in dom <% a0 %> by AFINSQ_1:33;
  then A6: len A0 + 0 in dom A by A5, AFINSQ_1:23;
  then A7: len A0 in dom(omega -exponent A) by Def1;
  0 = omega -exponent a0 by A4, A5, AFINSQ_1:92
    .= omega -exponent(A.len A0) by A5, AFINSQ_1:36
    .= (omega -exponent A).len A0 by A6, Def1;
  hence thesis by A7, FUNCT_1:3;
end;
