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 WHEREge(O,n+1) = L WHEREgt(O,n)
  proof
    thus L WHEREge(O,n+1) c= L WHEREgt(O,n)
    proof
      let z be object; assume z in L WHEREge(O,n+1); then
      consider x such that
A1:   z = x & n+1 c= card(x.O) & x in L;
      Segm(n+1) = succ Segm n by NAT_1:38; then
      n in card(x.O) by A1,ORDINAL1:21;
      hence thesis by A1;
    end;
    let z be object; assume z in L WHEREgt(O,n); then
    consider x such that
A2: z = x & n in card(x.O) & x in L;
    Segm(n+1) = succ Segm n by NAT_1:38; then
    n+1 c= card(x.O) by A2,ORDINAL1:21;
    hence thesis by A2;
  end;
