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 /\ B) |^ (m, n) c= (A |^ (m, n)) /\ (B |^ (m, n))
proof
  let x be object;
  assume x in (A /\ B) |^ (m, n);
  then consider mn such that
A1: m <= mn & mn <= n and
A2: x in (A /\ B) |^ mn by Th19;
A3: (A /\ B) |^ mn c= (A |^ mn) /\ (B |^ mn) by FLANG_1:39;
  then x in B |^ mn by A2,XBOOLE_0:def 4;
  then
A4: x in B |^ (m, n) by A1,Th19;
  x in A |^ mn by A2,A3,XBOOLE_0:def 4;
  then x in A |^ (m, n) by A1,Th19;
  hence thesis by A4,XBOOLE_0:def 4;
end;
