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

theorem Th16:
  ((A`)^i)` = A^b
proof
  for x being object holds x in ((A`)^i)` iff x in A^b
  proof
    let x be object;
    thus x in ((A`)^i)` implies x in A^b
    proof
      assume
A1:   x in ((A`)^i)`;
      then reconsider y=x as Element of FT;
      not y in (A`)^i by A1,XBOOLE_0:def 5;
      then not U_FT y c= A`;
      then consider z being object such that
A2:   z in U_FT y and
A3:   not z in A`;
      z in A by A2,A3,SUBSET_1:29;
      then U_FT y meets A by A2,XBOOLE_0:3;
      hence thesis;
    end;
    assume
A4: x in A^b;
    then reconsider y=x as Element of FT;
    U_FT y meets A by A4,Th8;
    then ex z being object st z in U_FT y & z in A by XBOOLE_0:3;
    then not U_FT y c= A` by XBOOLE_0:def 5;
    then not y in (A`)^i by Th7;
    hence thesis by SUBSET_1:29;
  end;
  hence thesis by TARSKI:2;
end;
