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 Th39:
  A^Fodelta = (A^Fob) /\ ((A`)^Fob)
proof
  for x being Element of FMT holds x in A^Fodelta iff x in (A^Fob) /\ ((A`
  )^Fob)
  proof
    let x be Element of FMT;
    thus x in A^Fodelta implies x in (A^Fob) /\ ((A`)^Fob)
    proof
      assume
A1:   x in A^Fodelta;
      then for W being Subset of FMT st W in U_FMT x holds W meets A` by Th19;
      then
A2:   x in((A`)^Fob);
      for W being Subset of FMT st W in U_FMT x holds W meets A by A1,Th19;
      then x in (A^Fob);
      hence thesis by A2,XBOOLE_0:def 4;
    end;
    assume x in ((A^Fob) /\ ((A`)^Fob));
    then x in A^Fob & x in (A`)^Fob by XBOOLE_0:def 4;
    then
    for W being Subset of FMT st W in U_FMT x holds W meets A & W meets A`
    by Th20;
    hence thesis;
  end;
  hence thesis by SUBSET_1:3;
end;
