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 Th38:
  ((A`)^Foi)` = A^Fob
proof
  for x being object holds x in ((A`)^Foi)` iff x in A^Fob
  proof
    let x be object;
    thus x in ((A`)^Foi)` implies x in A^Fob
    proof
      assume
A1:   x in ((A`)^Foi)`;
      then reconsider y=x as Element of FMT;
A2:   not y in (A`)^Foi by A1,XBOOLE_0:def 5;
      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 not W c= A` by A2;
        then consider z being object such that
A3:     z in W and
A4:     not z in A`;
        z in A by A3,A4,XBOOLE_0:def 5;
        hence thesis by A3,XBOOLE_0:3;
      end;
      hence thesis;
    end;
    assume
A5: x in A^Fob;
    then reconsider y=x as Element of FMT;
    for W being Subset of FMT st W in U_FMT y holds not W c= A`
    proof
      let W be Subset of FMT;
      assume W in U_FMT y;
      then W meets A by A5,Th20;
      then ex z being object st z in W & z in A by XBOOLE_0:3;
      hence thesis by XBOOLE_0:def 5;
    end;
    then not y in (A`)^Foi by Th21;
    hence thesis by XBOOLE_0:def 5;
  end;
  hence thesis by TARSKI:2;
end;
