reserve FT for non empty RelStr;
reserve x, y, z for Element of FT;
reserve A for Subset of FT;

theorem
  FT is symmetric iff for A holds A^b = A^f
proof
  thus FT is symmetric implies for A holds A^b = A^f
  proof
    assume
A1: FT is symmetric;
    let A be Subset of FT;
    thus A^b c= A^f
    proof
      let x be object;
      assume
A2:   x in A^b;
      then reconsider y = x as Element of FT;
      U_FT y meets A by A2,Th8;
      then consider z be object such that
A3:   z in U_FT y /\ A by XBOOLE_0:4;
      reconsider z as Element of FT by A3;
      z in U_FT y by A3,XBOOLE_0:def 4;
      then
A4:   y in U_FT z by A1;
      z in A by A3,XBOOLE_0:def 4;
      hence thesis by A4;
    end;
    let x be object;
    assume
A5: x in A^f;
    then reconsider y = x as Element of FT;
    consider z such that
A6: z in A and
A7: y in U_FT z by A5,Th11;
    z in U_FT y by A1,A7;
    then U_FT y meets A by A6,XBOOLE_0:3;
    hence thesis;
  end;
  assume
A8: for A being Subset of FT holds A^b = A^f;
  let x, y be Element of FT;
  assume y in U_FT x;
  then U_FT x meets {y} by ZFMISC_1:48;
  then x in {y}^b;
  then x in {y}^f by A8;
  then ex z st z in {y} & x in U_FT z by Th11;
  hence thesis by TARSKI:def 1;
end;
