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

theorem
  for f,g being nonnegative-yielding real-valued FinSequence holds
    (Sum f)*(Sum g) >= Sum(f(#)g)
  proof
    let f,g being nonnegative-yielding real-valued FinSequence;
    now let i be Nat;
      (Sum f)*g.i = ((Sum f)(#)g).i by VALUED_1:6;
      hence (f(#)g).i  <= ((Sum f)*g).i by FS;
    end; then
    Sum (f(#)g) <= Sum((Sum f)*g) by NYS;
    hence thesis by RVSUM_1:87;
  end;
