reserve FT for non empty RelStr,
  A,B,C for Subset of FT;

theorem
  for X being non empty SubSpace of FT st FT is symmetric holds X is symmetric
proof
  let X be non empty SubSpace of FT;
  assume
A1: FT is symmetric;
  for x, y being Element of X holds y in U_FT x implies x in U_FT y
  proof
    let x, y be Element of X;
    assume
A2: y in U_FT x;
    the carrier of X c= the carrier of FT by Def2;
    then reconsider x2=x, y2=y as Element of FT;
A3: U_FT x=U_FT x2 /\ ([#]X) & U_FT y=U_FT y2 /\ ([#]X) by Def2;
    y2 in U_FT x2 implies x2 in U_FT y2 by A1;
    hence thesis by A2,A3,XBOOLE_0:def 4;
  end;
  hence thesis;
end;
