(* 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 continuity_pt_recip_prelim : forall (f g:R->R) (lb ub : R) (Pr1:lb < ub), (forall x y, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x, lb <= x <= ub -> (comp g f) x = id x) -> (forall a, lb <= a <= ub -> continuity_pt f a) -> forall b, f lb < b < f ub -> continuity_pt g b.