(* 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_lim_recip_interv : forall (f g:R->R) (lb ub x:R) (Prf:forall a : R, g lb <= a <= g ub -> derivable_pt f a) (Prg : continuity_pt g x), lb < ub -> lb < x < ub -> forall (Prg_incr:g lb <= g x <= g ub), (forall x, lb <= x <= ub -> (comp f g) x = id x) -> derive_pt f (g x) (Prf (g x) Prg_incr) <> 0 -> derivable_pt_lim g x (1 / derive_pt f (g x) (Prf (g x) Prg_incr)).