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(* Copyright 2006 Milad Niqui                                           *)
(* This file is distributed under the terms of the                      *)
(* GNU Lesser General Public License Version 2.1                        *)
(* A copy of the license can be found at                                *)
(*                  <http://www.gnu.org/licenses/lgpl-2.1.html>         *)
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Require Export Qsyntax.
Require Export Field_Theory_Q.
Definition Qmult_Qinv_l:= Qinv_defT.
Definition Qmult_resp_Qlt_pos_r:= Qlt_Qmult_cancel_r :  forall t x y : Q, Zero < t -> x * t < y * t -> x < y.
Definition Qmult_resp_Qlt_pos_l:= Qlt_Qmult_cancel_l: forall t x y : Q, Zero < t -> t * x < t * y -> x < y.
Hint Resolve Qinv_pos Qinv_resp_nonzero Qminus_Qeq Qeq_Qminus
Qlt_not_eq' Qinv_neg Qle_Qopp_pos Qlt_Qopp_pos
Qlt_Qopp_neg Qopp_Qone_Qlt_Qone.

Lemma Qdiv_num_denom_explicit: forall q1 q2 p : Q, q2 <> Zero -> p <> Zero -> q1 / q2 = (q1 * p)/ (q2 * p).