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 WHEREle(O,n) = L WHERElt(O,n+1)
  proof
    thus L WHEREle(O,n) c= L WHERElt(O,n+1)
    proof
      let z be object; assume z in L WHEREle(O,n); then
      consider x such that
A1:   z = x & card(x.O) c= n & x in L;
      Segm(n+1) = succ Segm n by NAT_1:38; then
      card(x.O) in n+1 by A1,ORDINAL1:22;
      hence thesis by A1;
    end;
    let z be object; assume z in L WHERElt(O,n+1); then
    consider x such that
A2: z = x & card(x.O) in n+1 & x in L;
    Segm(n+1) = succ Segm n by NAT_1:38; then
    card(x.O) c= n by A2,ORDINAL1:22;
    hence thesis by A2;
  end;
