(** Lemma 14.40 states that we have an ordered field. *)

(* Specifications: *)
Lemma col_pos_or_neg : forall O E X,
O <> E -> O <> X -> Col O E X -> Ps O E X \/ Ng O E X.
Lemma ltP_neg : forall O E E' A, LtP O E E' A O -> Ng O E A.
Lemma ps_le : forall O E E' X,
~ Col O E E' -> Bet O X E \/ Bet O E X -> LeP O E E' O X.
Lemma lt_diff_ps : forall O E E' X Y XMY,
Col O E X -> Col O E Y -> LtP O E E' Y X -> Diff O E E' X Y XMY -> Ps O E XMY.
Lemma col_2_le_or_ge : forall O E E' A B,
~ Col O E E' -> Col O E A -> Col O E B -> LeP O E E' A B \/ LeP O E E' B A.
Lemma compatibility_of_sum_with_order : forall O E E' A B C APC BPC,
LeP O E E' A B -> Sum O E E' A C APC -> Sum O E E' B C BPC ->
LeP O E E' APC BPC.
Lemma compatibility_of_prod_with_order : forall O E E' A B AB,
LeP O E E' O A -> LeP O E E' O B -> Prod O E E' A B AB ->
LeP O E E' O AB.
Lemma pos_inv_pos : forall O E E' A IA,
O <> A -> LeP O E E' O A -> Prod O E E' IA A E -> LeP O E E' O IA.