(** *   Derivability of the reciprocal function        *)

Lemma derivable_pt_recip_interv
(f g:R->R) (lb ub x : R)
(lb_lt_ub:lb < ub) (x_encad:f lb < x < f ub)
(f_eq_g:forall x : R, f lb <= x -> x <= f ub -> comp f g x = id x)
(g_wf:forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub)
(f_incr:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y)
(f_derivable:forall a : R, lb <= a <= ub -> derivable_pt f a)
: derive_pt f (g x)
(derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub
x_encad g_wf f_derivable)
<> 0 ->
derivable_pt g x.