reserve D for non empty set;
reserve f1,f2,f3,f4,f5,f6,f7,f8,f9,f10 for BinominativeFunction of D;
reserve p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11 for PartialPredicate of D;
reserve q1,q2,q3,q4,q5,q6,q7,q8,q9,q10 for total PartialPredicate of D;
reserve n,m,N for Nat;
reserve fD for PFuncs(D,D)-valued FinSequence;
reserve fB for PFuncs(D,BOOLEAN)-valued FinSequence;
reserve V,A for set;
reserve val for Function;
reserve loc for V-valued Function;
reserve d1 for NonatomicND of V,A;
reserve p for SCPartialNominativePredicate of V,A;
reserve d,v for object;
reserve size for non zero Nat;
reserve inp,pos for FinSequence;
reserve prg for non empty FPrg(ND(V,A))-valued FinSequence;

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;
