reserve E, x, y, X for set;
reserve A, B, C for Subset of E^omega;
reserve a, a1, a2, b for Element of E^omega;
reserve i, k, l, m, n for Nat;

theorem Th44:
  (A /\ B) |^.. k c= (A |^.. k) /\ (B |^.. k)
proof
 let x be object;
    assume x in (A /\ B) |^.. k;
    then consider m such that
A1: k <= m and
A2: x in (A /\ B) |^ m by Th2;
A3: B |^ m c= B |^.. k by A1,Th3;
    (A /\ B) |^ m c= (A |^ m) /\ (B |^ m) by FLANG_1:39;
    then
A4: x in (A |^ m) /\ (B |^ m) by A2;
    A |^ m c= A |^.. k by A1,Th3;
    then (A |^ m) /\ (B |^ m) c= (A |^.. k) /\ (B |^.. k) by A3,XBOOLE_1:27;
    hence thesis by A4;
end;
