reserve E, x, y, X for set;
reserve A, B, C for Subset of E^omega;
reserve a, b for Element of E^omega;
reserve i, k, l, kl, m, n, mn for Nat;

theorem Th75:
  A? = (A |^ 0) \/ (A |^ 1)
proof
  now
    let x be object;
    x in A? iff ex k st (k = 0 or k = 1) & x in A |^ k
    proof
      thus x in A? implies ex k st (k = 0 or k = 1) & x in A |^ k
      proof
        assume x in A?;
        then consider k such that
A1:     k <= 1 and
A2:     x in A |^ k by Th73;
        k = 0 or k = 1 by A1,NAT_1:25;
        hence thesis by A2;
      end;
      given k such that
A3:   ( k = 0 or k = 1)& x in A |^ k;
      thus thesis by A3,Th73;
    end;
    hence x in A? iff x in A |^ 0 or x in A |^ 1;
  end;
  hence thesis by XBOOLE_0:def 3;
end;
