(** *   Derivability of the reciprocal function        *)

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.