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 Th37:
  ((A`)^Fob)` =A^Foi
proof
  for x being object holds x in ((A`)^Fob)` iff x in A^Foi
  proof
    let x be object;
    thus x in ((A`)^Fob)` implies x in A^Foi
    proof
      assume
A1:   x in ((A`)^Fob)`;
      then reconsider y=x as Element of FMT;
      not y in (A`)^Fob by A1,XBOOLE_0:def 5;
      then consider V be Subset of FMT such that
A2:   V in U_FMT y and
A3:   V misses A`;
      V /\ A` = {} by A3;
      then V \ A = {} by SUBSET_1:13;
      then V c= A by XBOOLE_1:37;
      hence thesis by A2;
    end;
    assume
A4: x in A^Foi;
    then reconsider y=x as Element of FMT;
    consider V be Subset of FMT such that
A5: V in U_FMT y and
A6: V c= A by A4,Th21;
    V \ A = {} by A6,XBOOLE_1:37;
    then V /\ A` = {} by SUBSET_1:13;
    then V misses A`;
    then not y in (A`)^Fob by A5,Th20;
    hence thesis by XBOOLE_0:def 5;
  end;
  hence thesis by TARSKI:2;
end;
