reserve FT for non empty RelStr;
reserve A for Subset of FT;

theorem Th2:
  A^delta = (A^b) /\ ((A^i)`)
proof
  for x being object holds x in A^delta iff x in (A^b) /\ ((A^i)`)
  proof
    let x be object;
    thus x in A^delta implies x in (A^b) /\ ((A^i)`)
    proof
      assume
A1:   x in A^delta;
      then reconsider y=x as Element of FT;
      U_FT y meets A` by A1,FIN_TOPO:5;
      then y in ((A`)^b)``;
      then
A2:   y in ((A^i)`) by FIN_TOPO:17;
      U_FT y meets A by A1,FIN_TOPO:5;
      then y in (A^b);
      hence thesis by A2,XBOOLE_0:def 4;
    end;
    assume
A3: x in ((A^b) /\ ((A^i)`));
    then reconsider y=x as Element of FT;
    x in ((A^i)`) by A3,XBOOLE_0:def 4;
    then x in ((((A`)^b)`)`) by FIN_TOPO:17;
    then
A4: U_FT y meets A` by FIN_TOPO:8;
    x in (A^b) by A3,XBOOLE_0:def 4;
    then U_FT y meets A by FIN_TOPO:8;
    hence thesis by A4;
  end;
  hence thesis by TARSKI:2;
end;
