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 |^ (k + l)) |^ (m, n) c= ((A |^ k) |^ (m, n)) ^^ ((A |^ l) |^ (m, n ))
proof
  let x be object;
  assume x in (A |^ (k + l)) |^ (m, n);
  then consider mn such that
A1: m <= mn & mn <= n and
A2: x in (A |^ (k + l)) |^ mn by Th19;
  x in A |^ ((k + l) * mn) by A2,FLANG_1:34;
  then x in A |^ (k * mn + l * mn);
  then x in (A |^ (k * mn)) ^^ (A |^ (l * mn)) by FLANG_1:33;
  then consider a, b such that
A3: a in A |^ (k * mn) and
A4: b in A |^ (l * mn) and
A5: x = a ^ b by FLANG_1:def 1;
  b in (A |^ l) |^ mn by A4,FLANG_1:34;
  then
A6: b in (A |^ l) |^ (m, n) by A1,Th19;
  a in (A |^ k) |^ mn by A3,FLANG_1:34;
  then a in (A |^ k) |^ (m, n) by A1,Th19;
  hence thesis by A5,A6,FLANG_1:def 1;
end;
