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
  (A |^ m) |^.. n c= (A |^.. n) |^ m
proof
  per cases;
  suppose
A1: m > 0;
    (A |^ m) |^.. n c= A |^.. (m * n) by Th35;
    hence thesis by A1,Th19;
  end;
  suppose
    m <= 0;
    then
A2: m = 0;
    then (A |^ m) |^.. n = {<%>E} |^.. n by FLANG_1:24
      .= {<%>E} by Th15
      .= (A |^.. n) |^ m by A2,FLANG_1:24;
    hence thesis;
  end;
end;
