(* begin hide *)
Ltac case_le H :=
   let t := type of H in
   let h' := fresh in
      match t with ?x <= ?y => case (total_order_T x y);
         [intros h'; case h'; clear h' |
          intros h'; clear -H h'; exfalso; lra ] end.

Lemma derivable_pt_recip_interv_prelim1_decr : forall (f g:R->R) (lb ub x : R), lb < ub -> f ub < x < f lb -> (forall x : R, f ub <= x -> x <= f lb -> lb <= g x <= ub) -> (forall a : R, lb <= a <= ub -> derivable_pt f a) -> derivable_pt f (g x).