theorem
  L WHEREgt(O,0) = L WHERE O
  proof
    thus L WHEREgt(O,0) c= L WHERE O by Th8;
    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
    0 in card(z.O) by ORDINAL3:8;
    hence thesis by A2;
  end;
