theorem
  L WHEREge(O,1) = L WHERE O
  proof
    thus L WHEREge(O,1) c= L WHERE O by Th6;
    let z be object; assume
A1: z in L WHERE O; then
    reconsider z as Element of X;
A2: z.O <> {} & z in L by A1,Th3; then
    succ 0 c= card(z.O) by ORDINAL1:21,ORDINAL3:8;
    hence thesis by A2;
  end;
