reserve a,b,c,v,v1,x,y for object;
reserve V,A for set;
reserve d for TypeSCNominativeData of V,A;

theorem
  for d1 being NonatomicND of V,A,
      z being Element of V holds
  V is non empty & v in dom d1 & d = denaming(V,A,v).d1
  implies local_overlapping(V,A,d1,d,z).z = d1.v
  proof
    let d1 be NonatomicND of V,A,
        z be Element of V;
    assume
A1: V is non empty;
    set Dj = denaming(V,A,v);
    assume that
A2: v in dom d1 and
A3: d = Dj.d1;
    dom Dj = {d where d is NonatomicND of V,A: v in dom d}
    by NOMIN_1:def 18;
    then
A4: d1 in dom Dj by A2;
    thus local_overlapping(V,A,d1,d,z).z = Dj.d1 by A1,A3,Th10
    .= denaming(v,d1) by A4,NOMIN_1:def 18
    .= d1.v by A2,NOMIN_1:def 12;
  end;
