reserve FT for non empty RelStr;
reserve A for Subset of FT;
reserve T for non empty TopStruct;
reserve FMT for non empty FMT_Space_Str;
reserve x, y for Element of FMT;
reserve A, B, W, V for Subset of FMT;

theorem Th35:
  for FMT being non empty FMT_Space_Str holds FMT is Fo_filled iff
  for A being Subset of FMT holds A c= A^Fob
proof
  let FMT be non empty FMT_Space_Str;
A1: (for A being Subset of FMT holds A c= A^Fob) implies FMT is Fo_filled
  proof
    assume
A2: for A being Subset of FMT holds A c= A^Fob;
    assume not FMT is Fo_filled;
    then consider y being Element of FMT, V being Subset of FMT such that
A3: V in U_FMT y and
A4: not y in V;
A5: V misses {y}
    proof
      assume V meets {y};
      then ex z being object st z in V & z in {y} by XBOOLE_0:3;
      hence contradiction by A4,TARSKI:def 1;
    end;
    not {y} c= {y}^Fob
    proof
A6:   y in {y} by TARSKI:def 1;
      assume {y} c= {y}^Fob;
      hence contradiction by A3,A5,A6,Th20;
    end;
    hence contradiction by A2;
  end;
  FMT is Fo_filled implies for A being Subset of FMT holds A c= A^Fob
  proof
    assume
A7: FMT is Fo_filled;
    let A being Subset of FMT;
    let x be object;
    assume
A8: x in A;
    then reconsider y=x as Element of FMT;
    for W being Subset of FMT st W in U_FMT y holds W meets A
    proof
      let W be Subset of FMT;
      assume W in U_FMT y;
      then y in W by A7;
      hence thesis by A8,XBOOLE_0:3;
    end;
    hence thesis;
  end;
  hence thesis by A1;
end;
