theorem
  for d being NonatomicND of V,A holds
    v in V & not d in A & not naming(V,A,v,f.d) in A & d in dom f implies
  dom (SC_assignment(f,v).d) = dom d \/ {v}
  proof
    set a = SC_assignment(f,v);
    let d be NonatomicND of V,A;
    assume that
A1: v in V & not d in A & not naming(V,A,v,f.d) in A and
A2: d in dom f;
A3: dom a = dom f by Def7;
    reconsider d1 = d as TypeSCNominativeData of V,A;
    reconsider d2 = f.d1 as TypeSCNominativeData of V,A
    by A2,PARTFUN1:4,NOMIN_1:39;
    dom local_overlapping(V,A,d,d2,v) = {v} \/ dom d by A1,Th14;
    hence thesis by A2,A3,Def7;
  end;
