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(** *   Value of the derivative of the reciprocal function        *)
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Lemma derive_pt_recip_interv_prelim1_0 : forall (f g:R->R) (lb ub x:R),
lb < ub ->
f lb < x < f ub ->
(forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) ->
(forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub) ->
(forall x, f lb <= x -> x <= f ub -> (comp f g) x = id x) ->
lb < g x < ub.