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
  B c= A* implies A |^.. k ^^ B c= A |^.. k & B ^^ (A |^.. k) c= A |^.. k
proof
  assume
A1: B c= A*;
  then B ^^ (A |^.. k) c= A* ^^ (A |^.. k) by FLANG_1:17;
  then
A2: B ^^ (A |^.. k) c= A |^.. k ^^ (A*) by Th32;
  A |^.. k ^^ B c= A |^.. k ^^ (A*) by A1,FLANG_1:17;
  hence thesis by A2,Th17;
end;
