theorem
  L WHEREgt(O,O1) c= L WHERE O
  proof
    let z be object; assume z in L WHEREgt(O,O1); then
    consider x such that
A1: z = x & card(x.O1) in card(x.O) & x in L;
    x.O <> {} by A1;
    hence thesis by A1,Th3;
  end;
