reserve D for non empty set;
reserve f1,f2,f3,f4 for BinominativeFunction of D;
reserve p,q,r,t,w for PartialPredicate of D;
reserve d,v,v1 for object;
reserve V,A for set;
reserve z for Element of V;
reserve d1 for NonatomicND of V,A;
reserve f for SCBinominativeFunction of V,A;
reserve T for TypeSCNominativeData of V,A;

theorem Th3:
  A is_without_nonatomicND_wrt V & v in V & v <> v1 & v1 in dom d1
  implies local_overlapping(V,A,d1,T,v).v1 = d1.v1
  proof
    assume that
A1: A is_without_nonatomicND_wrt V and
A2: v in V & v <> v1;
    not d1 in A & not naming(V,A,v,T) in A by A1,NOMIN_4:3;
    hence thesis by A2,NOMIN_2:12;
  end;
