reserve E, x, y, X for set;
reserve A, B, C, D for Subset of E^omega;
reserve a, a1, a2, b, c, c1, c2, d, ab, bc for Element of E^omega;
reserve e for Element of E;
reserve i, j, k, l, n, n1, n2, m for Nat;

theorem
  A |^ n = {<%>E} iff n = 0 or A = {<%>E}
proof
  thus A |^ n = {<%>E} implies n = 0 or A = {<%>E}
  proof
    assume
A1: A |^ n = {<%>E};
    now
      assume n > 0;
      then consider m such that
A2:   m + 1 = n by NAT_1:6;
      A |^ n = (A |^ m) ^^ A by A2,Th23;
      hence A = {<%>E} by A1,Th14;
    end;
    hence thesis;
  end;
  assume n = 0 or A = {<%>E};
  hence thesis by Th24,Th28;
end;
