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

theorem Th12:
  for d1 being NonatomicND of V,A
    for d2 being TypeSCNominativeData of V,A
   st v in V & not v in dom d1 & not d1 in A & not naming(V,A,v,d2) in A
  holds dom local_overlapping(V,A,d1,d2,v) = {v} \/ dom d1
  proof
    let d1 be NonatomicND of V,A;
    let d2 be TypeSCNominativeData of V,A such that
A1: v in V and
A2: not v in dom d1 and
A3: not d1 in A & not naming(V,A,v,d2) in A;
    set n = naming(V,A,v,d2);
A4: n = v .--> d2 by A1,NOMIN_1:def 13;
A5: dom (d1|(dom(d1)\dom(n))) = dom d1
    proof
      thus dom(d1|(dom(d1)\dom(n))) c= dom d1 by RELAT_1:60;
      dom(d1) c= dom(d1)\dom(n) by A2,A4,ZFMISC_1:34;
      hence thesis by RELAT_1:62;
    end;
    local_overlapping(V,A,d1,d2,v) = n \/ (d1|(dom(d1)\dom(n)))
    by A3,NOMIN_1:64;
    hence thesis by A4,A5,XTUPLE_0:23;
  end;
