(** *   Derivability of the reciprocal function        *)

Lemma derivable_pt_recip_interv_decr (f g:R->R) (lb ub x : R)
(lb_lt_ub:lb < ub)
(x_encad:f ub < x < f lb)
(f_eq_g:forall x : R, f ub <= x -> x <= f lb -> comp f g x = id x)
(g_wf:forall x : R, f ub <= x -> x <= f lb -> lb <= g x <= ub)
(f_decr:forall x y : R, lb <= x -> x < y -> y <= ub -> f y < f x)
(f_derivable:forall a : R, lb <= a <= ub -> derivable_pt f a)
: derive_pt f (g x)
(derivable_pt_recip_interv_prelim1_decr f g lb ub x lb_lt_ub
x_encad g_wf f_derivable)
<> 0 ->
derivable_pt g x.