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
  A^delta = (A^b) /\ ((A`)^b)
proof
  for x being object holds x in A^delta iff x in (A^b) /\ ((A`)^b)
  proof
    let x be object;
    thus x in A^delta implies x in (A^b) /\ ((A`)^b)
    proof
      assume
A1:   x in A^delta;
      then reconsider y=x as Element of FT;
      U_FT y meets A` by A1,Th5;
      then
A2:   x in (A`)^b;
      U_FT y meets A by A1,Th5;
      then x in A^b;
      hence thesis by A2,XBOOLE_0:def 4;
    end;
    assume
A3: x in (A^b) /\ ((A`)^b);
    then reconsider y=x as Element of FT;
    x in ((A`)^b) by A3,XBOOLE_0:def 4;
    then
A4: U_FT y meets A` by Th8;
    x in A^b by A3,XBOOLE_0:def 4;
    then U_FT y meets A by Th8;
    hence thesis by A4;
  end;
  hence thesis by TARSKI:2;
end;
