theorem Th22:
  for F being being_ultrafilter Filter of X for Y holds Y in F iff
  not Y in dual F
proof
  let F be being_ultrafilter Filter of X;
  let Y;
  thus Y in F implies not Y in dual F
  proof
    assume Y in F;
    then not Y` in F by Th6;
    hence thesis by SETFAM_1:def 7;
  end;
  assume not Y in dual F;
  then not Y` in F by SETFAM_1:def 7;
  hence thesis by Def7;
end;
