(* set (X_C := S O U C). set (X_A := S O U A). set (X_B := S O U B). set (X_D := S O U D). set (Y_C := S O V C). set (Y_A := S O V A). set (Y_B := S O V B). set (Y_D := S O V D). *)

Lemma free_points_ratio_elimination_7_non_zero_denom :
forall O U V A B C D: Point,
parallel A B C D ->
C<>D ->
~ Col O U V ->
~ Col U A C ->
Col A C D ->
(S O U V * (S O U D - S O U C) - S O U C * S O V D + S O U D * S O V C) <> 0.