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

theorem
  v in V & v <> v1 implies
  for d1 being NonatomicND of V,A
  for d2 being TypeSCNominativeData of V,A
   st v1 in dom d1 & not d1 in A & not naming(V,A,v,d2) in A
  holds local_overlapping(V,A,d1,d2,v).v1 = d1.v1
  proof
    assume that
A1: v in V and
A2: v <> v1;
    let d1 be NonatomicND of V,A;
    let d2 be TypeSCNominativeData of V,A such that
A4: v1 in dom d1 and
A5: not d1 in A & not naming(V,A,v,d2) in A;
A7: naming(V,A,v,d2) = v.-->d2 by A1,NOMIN_1:def 13;
    consider f1,f2 being Function such that
A10: f1 = d1 and
A11: f2 = naming(V,A,v,d2) and
A12: local_overlapping(V,A,d1,d2,v) = f2 \/ (f1|(dom(f1)\dom(f2)))
     by A5,NOMIN_1:def 16;
    not v1 in {v} by A2,TARSKI:def 1;
    then v1 in dom(f1)\dom(f2) by A4,A7,A10,A11,XBOOLE_0:def 5;
    then
A13: v1 in dom(f1|(dom(f1)\dom(f2))) by RELAT_1:57;
    hence local_overlapping(V,A,d1,d2,v).v1 = (f1|(dom(f1)\dom(f2))).v1
    by A12,GRFUNC_1:15
    .= d1.v1 by A10,A13,FUNCT_1:47;
  end;
