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
  O1 c= O2 & L1 c= L2 & O c= O3 implies L1 WHEREgt(O,O2) c= L2 WHEREgt(O3,O1)
  proof
    assume A1: O1 c= O2 & L1 c= L2 & O c= O3;
    let z be object; assume z in L1 WHEREgt(O,O2); then
    consider x such that
A2: z = x & card(x.O2) in card(x.O) & x in L1;
    x.O1 c= x.O2 & x.O c= x.O3 by A1,Th1; then
    card(x.O1) in card(x.O) & card(x.O) c= card(x.O3)
    by A2,CARD_1:11,ORDINAL1:12;
    hence z in L2 WHEREgt(O3,O1) by A2,A1;
  end;
