reserve X for non empty set;

theorem Th8:
  for ET being non empty strict FMT_Space_Str st ET is U_FMT_filter_base
  holds gen_filter(ET) is U_FMT_filter
  proof
    let ET be non empty strict FMT_Space_Str such that
A1: ET is U_FMT_filter_base;
    for x be Element of gen_filter(ET)
    holds U_FMT x is Filter of the carrier of gen_filter(ET)
    proof
      let x be Element of gen_filter(ET);
      reconsider x0=x as Element of ET;
      U_FMT x0 is filter_base of the carrier of ET by A1;
      then <.U_FMT x0.] is Filter of the carrier of ET by CARDFIL2:25;
      hence thesis by Def7;
    end;
    hence thesis;
  end;
