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 Th17:
  ((A`)^b)` = A^i
proof
  for x being object holds x in ((A`)^b)` iff x in A^i
  proof
    let x be object;
    thus x in ((A`)^b)` implies x in A^i
    proof
      assume
A1:   x in ((A`)^b)`;
      then reconsider y=x as Element of FT;
      not y in (A`)^b by A1,XBOOLE_0:def 5;
      then U_FT y misses A`;
      then U_FT y /\ A` = {};
      then U_FT y \ A = {} by SUBSET_1:13;
      then U_FT y c= A by XBOOLE_1:37;
      hence thesis;
    end;
    assume
A2: x in A^i;
    then reconsider y=x as Element of FT;
    U_FT y c= A by A2,Th7;
    then U_FT y \ A = {} by XBOOLE_1:37;
    then U_FT y /\ A` = {} by SUBSET_1:13;
    then U_FT y misses A`;
    then not y in (A`)^b by Th8;
    hence thesis by SUBSET_1:29;
  end;
  hence thesis by TARSKI:2;
end;
