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

theorem
  (A |^ (m, n)) |^ (k, l) c= A |^ (m * k, n * l)
proof
  let x be object;
  assume x in (A |^ (m, n)) |^ (k, l);
  then consider kl such that
A1: k <= kl & kl <= l and
A2: x in (A |^ (m, n)) |^ kl by Th19;
  m * k <= m * kl & n * kl <= n * l by A1,NAT_1:4;
  then
A3: A |^ (m * kl, n * kl) c= A |^ (m * k, n * l) by Th23;
  x in A |^ (m * kl, n * kl) by A2,Th40;
  hence thesis by A3;
end;
