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
  D = Seg i implies for p being FinSequence of D9 for q being
  FinSequence of D st i <= len p holds p*q is FinSequence of D9
proof
  assume
A1: D = Seg i;
  let p be FinSequence of D9;
  let q be FinSequence of D;
  assume i <= len p;
  then reconsider pq = p*q as FinSequence by A1,Th28;
  rng pq c= rng p & rng p c= D9 by FINSEQ_1:def 4,RELAT_1:26;
  then rng pq c= D9;
  hence thesis by FINSEQ_1:def 4;
end;
