reserve v for object;
reserve V,A for set;
reserve f for SCBinominativeFunction of V,A;
reserve d for TypeSCNominativeData of V,A;
reserve d1 for NonatomicND of V,A;

theorem Th6:
  for d1 being NonatomicND of V,A holds
   v in V & A is_without_nonatomicND_wrt V implies
  local_overlapping(V,A,d1,d,v) in dom denaming(V,A,v)
  proof
    let d1 being NonatomicND of V,A;
    set L = local_overlapping(V,A,d1,d,v);
    set D = denaming(V,A,v);
A1: dom D = {d where d is NonatomicND of V,A: v in dom d} by NOMIN_1:def 18;
A2: L is NonatomicND of V,A by NOMIN_2:9;
    assume v in V & A is_without_nonatomicND_wrt V;
    then
A3: dom L = {v} \/ dom d1 by Th4;
    v in {v} by TARSKI:def 1;
    then v in dom L by A3,XBOOLE_0:def 3;
    hence thesis by A1,A2;
  end;
