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(** *   Value of the derivative of the reciprocal function        *)
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Lemma derive_pt_recip_interv_prelim1_1_decr : forall (f g:R->R) (lb ub x:R),
lb < ub ->
f ub < x < f lb ->
(forall x y : R, lb <= x -> x < y -> y <= ub -> f y < f x) ->
(forall x : R, f ub <= x -> x <= f lb -> lb <= g x <= ub) ->
(forall x, f ub <= x -> x <= f lb -> (comp f g) x = id x) ->
lb <= g x <= ub.