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 Th43:
  for p being FinSequence of D for f being Function of Seg i,Seg j
  st (j = 0 implies i = 0) & j <= len p holds p*f is FinSequence of D
proof
  let p be FinSequence of D;
  let f be Function of Seg i,Seg j such that
A1: ( j = 0 implies i = 0)& j <= len p;
  set q = p*f;
  rng p c= D & rng q c= rng p by FINSEQ_1:def 4,RELAT_1:26;
  then
A2: rng q c= D;
  q is FinSequence by A1,Th36;
  hence thesis by A2,FINSEQ_1:def 4;
end;
