reserve v for object;
reserve V,A for set;
reserve f for SCBinominativeFunction of V,A;

theorem
  A is_without_nonatomicND_wrt V implies
  for d being NonatomicND of V,A holds
    v in V & d in dom f implies
  dom (SC_assignment(f,v).d) = dom d \/ {v}
  proof
    assume
A1: A is_without_nonatomicND_wrt V;
    let d be NonatomicND of V,A;
    not d in A & not naming(V,A,v,f.d) in A by A1,Th3;
    hence thesis by NOMIN_2:33;
  end;
