theorem
  for D9 being Subset of D, S being FinSequenceSet of D9
  holds S is FinSequenceSet of D
proof
  let D9 be Subset of D, S be FinSequenceSet of D9;
  S is FinSequenceSet of D
  proof
    let a;
    assume a in S;
    then reconsider p = a as FinSequence of D9 by Def3;
    rng p c= D9 by FINSEQ_1:def 4;
    then rng p c= D by XBOOLE_1:1;
    hence thesis by FINSEQ_1:def 4;
  end;
  hence thesis;
end;
