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 Th6:
  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) c=
  dom(LocalOverlapSeq(A,loc,val,d1,size).(n+1))
  proof
    set F = LocalOverlapSeq(A,loc,val,d1,size);
    assume
A1: V is non empty & A is_without_nonatomicND_wrt V;
    let n be Nat;
    assume 1 <= n & n < size & val.(n+1) in dom(F.n);
    then dom(F.(n+1)) = {loc/.(n+1)} \/ dom(F.n) by A1,Th5;
    hence thesis by XBOOLE_1:7;
  end;
