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 Th28:
  D = Seg i implies for p being FinSequence for q being
  FinSequence of D st i <= len p holds p*q is FinSequence
proof
  assume
A1: D = Seg i;
  let p be FinSequence;
  let q be FinSequence of D;
  assume i <= len p;
  then Seg i c= Seg len p by FINSEQ_1:5;
  then
A2: Seg i c= dom p by FINSEQ_1:def 3;
  rng q c= Seg i by A1,FINSEQ_1:def 4;
  then dom (p*q) = dom q by A2,RELAT_1:27,XBOOLE_1:1;
  then dom (p*q) = Seg len q by FINSEQ_1:def 3;
  hence thesis by FINSEQ_1:def 2;
end;
