Proof with Geometry.
intros.
apply co_side_elim_1...
Qed.

(* Specifications: *)
Theorem aux_co_side_3 :
forall A B P Q M : Point, B <> M -> inter_ll M A B P Q -> S B P Q <>0.
Theorem co_side_elim_3 : forall A B P Q M : Point,
B <> M -> inter_ll M A B P Q -> A ** M / B ** M = S A P Q / S B P Q.
Theorem aux_co_side_4 :
forall A B P Q M : Point, A <> M -> inter_ll M A B P Q -> S A P Q <>0.
Theorem co_side_elim_4 : forall A B P Q M : Point,
A <> M -> inter_ll M A B P Q -> B ** M / A ** M = S B P Q / S A P Q.
Lemma elim_length_ratio_inter_ll_1_spec :
forall A C U V P Q Y : Point,
inter_ll Y P Q U V -> S A U V <> 0 -> C <> Y -> parallel A Y C Y ->
A ** Y / C ** Y = S A U V / S C U V.
Lemma elim_length_ratio_inter_ll_2_spec :
forall A C U V P Q Y : Point,
inter_ll Y P Q U V -> S A U V = 0 -> C <> Y -> parallel A Y C Y -> A<>Y ->
A ** Y / C ** Y = S A P Q/ S C P Q.