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

theorem Th10:
  v in V implies
  for d1,d2 being TypeSCNominativeData of V,A
   for L being Function st L = local_overlapping(V,A,d1,d2,v)
  holds L.v = d2
  proof
    assume
A1: v in V;
    let d1,d2 be TypeSCNominativeData of V,A;
    let L be Function such that
A2: L = local_overlapping(V,A,d1,d2,v);
A4: naming(V,A,v,d2) = v.-->d2 by A1,NOMIN_1:def 13;
A6: v in {v} by TARSKI:def 1;
A7: (v.-->d2).v = d2 by FUNCOP_1:72;
    per cases;
    suppose not d1 in A & not naming(V,A,v,d2) in A;
      then consider f1,f2 being Function such that
      f1 = d1 and
A8:   f2 = naming(V,A,v,d2) and
A9:   L = f2 \/ (f1|(dom(f1)\dom(f2))) by A2,NOMIN_1:def 16;
      thus L.v = f2.v by A4,A6,A8,A9,GRFUNC_1:15
      .= d2 by A8,A4,A6,FUNCOP_1:7;
    end;
    suppose d1 in A or naming(V,A,v,d2) in A;
      hence thesis by A4,A7,A2,NOMIN_1:def 16;
    end;
  end;
