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 Th41:
  (A |^ (k + 1)) |^ (m, n) c= ((A |^ k) |^ (m, n)) ^^ (A |^ (m, n) )
proof
  let x be object;
  assume x in (A |^ (k + 1)) |^ (m, n);
  then consider mn such that
A1: m <= mn & mn <= n and
A2: x in (A |^ (k + 1)) |^ mn by Th19;
  A |^ mn c= A |^ (m, n) by A1,Th20;
  then
A3: ((A |^ k) |^ (m, n)) ^^ (A |^ mn) c= ((A |^ k) |^ (m, n)) ^^ (A |^ (m,
  n)) by FLANG_1:17;
  x in A |^ ((k + 1) * mn) by A2,FLANG_1:34;
  then x in A |^ (k * mn + mn);
  then x in (A |^ (k * mn)) ^^ (A |^ mn) by FLANG_1:33;
  then
A4: x in ((A |^ k) |^ mn) ^^ (A |^ mn) by FLANG_1:34;
  (A |^ k) |^ mn c= (A |^ k) |^ (m, n) by A1,Th20;
  then ((A |^ k) |^ mn) ^^ (A |^ mn) c= ((A |^ k) |^ (m, n)) ^^ (A |^ mn) by
FLANG_1:17;
  then x in ((A |^ k) |^ (m, n)) ^^ (A |^ mn) by A4;
  hence thesis by A3;
end;
