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 Th40:
  A? c= A |^.. k iff k = 0 or <%>E in A
proof
  thus A? c= A |^.. k implies k = 0 or <%>E in A
  proof
A1: <%>E in A? by FLANG_2:78;
    assume A? c= A |^.. k;
    hence thesis by A1,Th10;
  end;
  assume k = 0 or <%>E in A;
  then A |^.. k = A* by Th11;
  hence thesis by FLANG_2:86;
end;
