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
  for FMT being non empty FMT_Space_Str, A,B being Subset of FMT st FMT
is Fo_filled holds (for x being Element of FMT holds {x} in U_FMT x ) implies (
  A /\ B)^Fob = ((A^Fob) /\ (B^Fob))
proof
  let FMT be non empty FMT_Space_Str;
  let A,B be Subset of FMT;
  assume
A1: FMT is Fo_filled;
  assume
A2: for x being Element of FMT holds {x} in U_FMT x;
A3: for C being Subset of FMT holds C^Fob c= C
  proof
    let C be Subset of FMT;
    for y being Element of FMT holds y in C^Fob implies y in C
    proof
      let y be Element of FMT;
      assume
A4:   y in C^Fob;
      {y} in U_FMT y by A2;
      then {y} meets C by A4,Th20;
      then ex z being object st z in {y} & z in C by XBOOLE_0:3;
      hence thesis by TARSKI:def 1;
    end;
    hence thesis;
  end;
A5: for C being Subset of FMT holds C^Fob = C
  by A1,A3,Th35;
  then (A /\ B)^Fob = (A /\ B) & (A^Fob) = A;
  hence thesis by A5;
end;
