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;
