proposition: Lemma Qclt_shift_div_r : forall q q' q'', 0 < q' -> (q / q' < q'' <-> q < q'' * q').
proof: Proof. intros q q' q'' Hq'. unfold Qclt,Qcdiv,Qcmult,Qcinv in *. repeat rewrite Q2Qc_correct in *. change (q * / q' < q'')%Q with (q / q' < q'')%Q. split. intro. rewrite <- (Qmult_div_r q q'). rewrite Qmult_comm. now apply Qmult_lt_compat_r. intro Goal. symmetry in Goal. revert Goal. now apply Qlt_not_eq. now apply Qlt_shift_div_r. Qed.
imports: Qfield Qreduction Qcanon