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Local Open Scope R_scope.
(** * Preliminaries lemmas *)
(* end hide *)
(** Intermediate Value Theorem on an Interval (Proof mainly taken from Reals.Rsqrt_def) and its corollary *)

Lemma IVT_interv_prelim0 : forall (x y:R) (P:R->bool) (N:nat),
x < y ->
x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= y.