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 implies L1 WHERE O1 c= L2 WHERE O2
  proof assume
A1: O1 c= O2 & L1 c= L2;
    let z be object; assume
A2: z in L1 WHERE O1; then
    reconsider z as Element of X;
A3: z.O1 <> {} & z in L1 by A2,Th3;
    z.O1 c= z.O2 by A1,Th1; then
    z.O2 <> {} by A3;
    hence thesis by A1,A3,Th3;
  end;
