reserve D for non empty set;
reserve m,n,N for Nat;
reserve size for non zero Nat;
reserve f1,f2,f3,f4,f5,f6 for BinominativeFunction of D;
reserve p1,p2,p3,p4,p5,p6,p7 for PartialPredicate of D;
reserve d,v for object;
reserve V,A for set;
reserve z for Element of V;
reserve val for Function;
reserve loc for V-valued Function;
reserve d1 for NonatomicND of V,A;
reserve T for TypeSCNominativeData of V,A;

theorem Th2:
  V is non empty & v in dom d1 implies
  local_overlapping(V,A,d1,denaming(V,A,v).d1,z).z = d1.v
  proof
    assume
A1: V is non empty;
    set Dj = denaming(V,A,v);
    assume that
A2: v in dom d1;
    dom Dj = {d where d is NonatomicND of V,A: v in dom d}
    by NOMIN_1:def 18;
    then
A3: d1 in dom Dj by A2;
    then Dj.d1 is TypeSCNominativeData of V,A by PARTFUN1:4,NOMIN_1:39;
    hence local_overlapping(V,A,d1,Dj.d1,z).z = Dj.d1 by A1,NOMIN_2:10
    .= denaming(v,d1) by A3,NOMIN_1:def 18
    .= d1.v by A2,NOMIN_1:def 12;
  end;
