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 Th5:
  V is non empty & A is_without_nonatomicND_wrt V implies
  for n being Nat st 1 <= n & n < size &
  val.(n+1) in dom(LocalOverlapSeq(A,loc,val,d1,size).n) holds
  dom(LocalOverlapSeq(A,loc,val,d1,size).(n+1)) =
  { loc/.(n+1) } \/ dom(LocalOverlapSeq(A,loc,val,d1,size).n)
  proof
    assume that
A1: V is non empty and
A2: A is_without_nonatomicND_wrt V;
    let n be Nat;
    assume that
A3: 1 <= n and
A4: n < size;
    set F = LocalOverlapSeq(A,loc,val,d1,size);
    set X1 = F.n;
    set X2 = F.(n+1);
    assume
A5: val.(n+1) in dom X1;
A6: len F = size by Def4;
    set v = loc/.(n+1);
    set D = denaming(V,A,val.(n+1));
A7: dom D = {d where d is NonatomicND of V,A: val.(n+1) in dom d}
    by NOMIN_1:def 18;
    reconsider X1 as NonatomicND of V,A by A3,A4,A6,Def6;
    X1 in dom D by A5,A7;
    then reconsider d2 = D.X1 as TypeSCNominativeData of V,A
    by PARTFUN1:4,NOMIN_1:39;
    X2 = local_overlapping(V,A,X1,d2,v) by A3,A4,A6,Def4;
    hence thesis by A1,A2,NOMIN_4:4;
  end;
