Ltac cases_parallel A B C D := elim (parallel_dec A B C D);intros.Definition Det3 (x1 x2 x3 y1 y2 y3 z1 z2 z3 : F) : F :=
  x1 * (y2 * z3) - x1 * (y3 * z2) - x2 * (y1 * z3) + x2 * (y3 * z1) +
  x3 * (y1 * z2) - x3 * (y2 * z1).

Lemma free_points_ratio_elimination_6_non_zero_denom: forall O U V A B C D: Point, parallel A B C D -> C<>D -> ~ Col O U V -> ~ Col O A C -> Col A C D -> S O U C * S O V D - S O V C * S O U D <> 0.