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(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
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(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
Require Export Qpositive_order.
Definition Qpositive_sub (w w' : Qpositive) :=
let (p, q) := Qpositive_i w in
let (p', q') := Qpositive_i w' in
Qpositive_c (p * q' - p' * q) (q * q') (p * q' + p' * q + q * q').

(* Specifications: *)
Theorem lt_mult_conv : forall x y z : nat, S x * y < S x * z -> y < z.
intros x y z.