(* 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_prelim0 (f g:R->R) (lb ub x:R) (Prf:derivable_pt f (g x)) (Prg:derivable_pt g x) : lb < ub -> lb < x < ub -> (forall x, lb < x < ub -> (comp f g) x = id x) -> derive_pt f (g x) Prf <> 0 -> derive_pt g x Prg = 1 / (derive_pt f (g x) Prf).