theorem Th19:
  Y in Frechet_Ideal(X) iff card Y in card X
proof
  Y in Frechet_Ideal(X) iff Y` in Frechet_Filter(X) by SETFAM_1:def 7;
  then Y in Frechet_Ideal(X) iff card Y`` in card X by Th18;
  hence thesis;
end;
