reserve X for non empty set;

theorem Th5:
  for ET being non empty strict FMT_Space_Str st ET is Fo_filled &
  for x being Element of ET holds U_FMT x is non empty holds
  ET is U_FMT_with_point
  proof
    let ET be non empty strict FMT_Space_Str such that
A1: ET is Fo_filled and
A2: for x be Element of ET holds U_FMT x is non empty;
    for x be Element of ET, V be Element of U_FMT x  holds x in V
    proof
      let x be Element of ET,V be Element of U_FMT x;
      U_FMT x is non empty by A2;
      then V in U_FMT x;
      hence thesis by A1;
    end;
    hence thesis;
  end;
