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;
