theorem Th22:
  V is non empty & A is_without_nonatomicND_wrt V implies
  for n being Nat st 1 <= n & n < len prg &
  PrgLocalOverlapSeq(A,loc,d1,prg,pos).n in dom(prg.(n+1)) holds
  dom(PrgLocalOverlapSeq(A,loc,d1,prg,pos).n) c=
  dom(PrgLocalOverlapSeq(A,loc,d1,prg,pos).(n+1))
  proof
    set F = PrgLocalOverlapSeq(A,loc,d1,prg,pos);
    assume
A1: V is non empty & A is_without_nonatomicND_wrt V;
    let n be Nat;
    assume 1 <= n & n < len prg & F.n in dom(prg.(n+1));
    then dom(F.(n+1)) = { loc/.(pos.(n+1)) } \/ dom(F.n) by A1,Th21;
    hence thesis by XBOOLE_1:7;
  end;
