reserve i,j,k,l for natural Number;
reserve A for set, a,b,x,x1,x2,x3 for object;
reserve D,D9,E for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve d9,d19,d29,d39 for Element of D9;
reserve p,q,r for FinSequence;

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;
