reserve X,Y,z,s for set, L,L1,L2,A,B for List of X, x for Element of X,
  O,O1,O2,O3 for Operation of X, a,b,y for Element of X, n,m for Nat;

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;
