(* 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 derive_pt_recip_interv_decr : forall (f g:R->R) (lb ub x:R) (lb_lt_ub:lb < ub) (x_encad:f ub < x < f lb) (f_decr:forall x y : R, lb <= x -> x < y -> y <= ub -> f y < f x) (g_wf:forall x : R, f ub <= x -> x <= f lb -> lb <= g x <= ub) (Prf:forall a : R, lb <= a <= ub -> derivable_pt f a) (f_eq_g:forall x, f ub <= x -> x <= f lb -> (comp f g) x = id x) (Df_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1_decr f g lb ub x lb_lt_ub x_encad g_wf Prf) <> 0), derive_pt g x (derivable_pt_recip_interv_decr f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_decr Prf Df_neq) = 1 / (derive_pt f (g x) (Prf (g x) (derive_pt_recip_interv_prelim1_1_decr f g lb ub x lb_lt_ub x_encad f_decr g_wf f_eq_g))).