(** TT Ak; 0<=k<=N *)
Fixpoint prod_f_R0 (f:nat -> R) (N:nat) : R :=
  match N with
    | O => f O
    | S p => prod_f_R0 f p * f (S p)
  end.Notation prod_f_SO := (fun An N => prod_f_R0 (fun n => An (S n)) N).

Lemma RfactN_fact2N_factk : forall N k:nat, (k <= 2 * N)%nat -> Rsqr (INR (fact N)) <= INR (fact (2 * N - k)) * INR (fact k).