(** *   Derivability of the reciprocal function        *)

Lemma derivable_pt_recip_interv_prelim1_decr : forall (f g:R->R) (lb ub x : R),
lb < ub ->
f ub < x < f lb ->
(forall x : R, f ub <= x -> x <= f lb -> lb <= g x <= ub) ->
(forall a : R, lb <= a <= ub -> derivable_pt f a) ->
derivable_pt f (g x).