reserve a,b,i,j,k,l,m,n for Nat;

theorem FS:
  for f,g be nonnegative-yielding real-valued FinSequence holds
    (f(#)g).n <= (Sum f)*g.n
  proof
    let f,g being nonnegative-yielding real-valued FinSequence;
    per cases;
    suppose n in dom (f(#)g); then
      (f(#)g).n = f.n * g.n by VALUED_1:def 4;
      hence thesis by SN,XREAL_1:64;
    end;
    suppose
      not n in dom (f(#)g);
      hence thesis by FUNCT_1:def 2;
    end;
  end;
