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

theorem
  A^delta = A^deltai \/ A^deltao
proof
  for x being object holds x in A^delta iff x in A^deltai \/ A^deltao
  proof
    let x be object;
    thus x in A^delta implies x in A^deltai \/ A^deltao
    proof
      assume x in A^delta;
      then x in [#](the carrier of FT) /\ (A^delta) by XBOOLE_1:28;
      then x in (A \/ A`) /\ (A^delta) by SUBSET_1:10;
      hence thesis by XBOOLE_1:23;
    end;
    assume x in A^deltai \/ A^deltao;
    then x in (A \/ A`) /\ (A^delta) by XBOOLE_1:23;
    then x in [#](the carrier of FT) /\ (A^delta) by SUBSET_1:10;
    hence thesis by XBOOLE_1:28;
  end;
  hence thesis by TARSKI:2;
end;
