(* 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_prelim0 : forall (f g : R -> R) (lb ub x : R) (Prf : forall a : R, g lb <= a <= g ub -> derivable_pt f a), continuity_pt g x -> lb < ub -> lb < x < ub -> forall Prg_incr : g lb <= g x <= g ub, (forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0) -> derive_pt f (g x) (Prf (g x) Prg_incr) <> 0 -> derivable_pt g x.