reserve E, x, y, X for set;
reserve A, B, C for Subset of E^omega;
reserve a, a1, a2, b for Element of E^omega;
reserve i, k, l, m, n for Nat;

theorem Th4:
  A |^.. n = {} iff n > 0 & A = {}
proof
  thus A |^.. n = {} implies n > 0 & A = {}
  proof
A1: A <> {} implies A |^.. n <> {}
    proof
      assume
A2:   A <> {};
      now
        let m;
        consider m such that
A3:     n <= m;
        A |^ m <> {} by A2,FLANG_1:27;
        then ex x being object st x in A |^ m by XBOOLE_0:def 1;
        hence thesis by A3,Th2;
      end;
      hence thesis;
    end;
    assume that
A4: A |^.. n = {} and
A5: n <= 0 or A <> {};
    n <= 0 implies <%>E in A |^.. n
    proof
      assume n <= 0;
      then
A6:   n = 0;
      A |^ 0 c= A |^.. 0 by Th3;
      then {<%>E} c= A |^.. 0 by FLANG_1:24;
      hence thesis by A6,ZFMISC_1:31;
    end;
    hence contradiction by A4,A5,A1;
  end;
  assume that
A7: n > 0 and
A8: A = {};
  now
    let x be object;
    assume x in A |^.. n;
    then ex m st n <= m & x in A |^ m by Th2;
    hence contradiction by A7,A8,FLANG_1:27;
  end;
  hence thesis by XBOOLE_0:def 1;
end;
