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 Th69:
  A |^ k |^ (m, n) c= A*
proof
  let x be object;
  assume x in A |^ k |^ (m, n);
  then consider mn such that
  m <= mn and
  mn <= n and
A1: x in A |^ k |^ mn by Th19;
A2: A |^ (k * mn) c= A* by FLANG_1:42;
  x in A |^ (k * mn) by A1,FLANG_1:34;
  hence thesis by A2;
end;
