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 c= dom O implies L WHERE O = L
  proof
    assume
A1: L c= dom O;
    thus L WHERE O c= L by Th4;
    let z be object; assume
A2: z in L; then
    reconsider z as Element of X;
    z.O <> {} by A1,A2,RELAT_1:170;
    hence thesis by A2,Th3;
  end;
