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 Th87:
  A+ |^ k c= A |^.. k
proof
  per cases;
  suppose
    k > 0;
    then (A |^.. 1) |^ k c= A |^.. (1 * k) by Th19;
    hence thesis by Th50;
  end;
  suppose
A1: k = 0;
    A |^.. 0 = A* by Th11;
    then
A2: <%>E in A |^.. 0 by FLANG_1:48;
    A+ |^ k = {<%>E} by A1,FLANG_1:24;
    hence thesis by A1,A2,ZFMISC_1:31;
  end;
end;
