(* Gabriel Braun April 2013 *)
Require Export GeoCoq.Tarski_dev.Ch13_4_cos.
Require Export GeoCoq.Tarski_dev.Annexes.project.
Section Pappus_Pascal.
Context `{T2D:Tarski_2D}.
Context `{TE:@Tarski_euclidean Tn TnEQD}.

(* Specifications: *)
Lemma l13_10_aux1 : forall O A B P Q la lb lp lq,
Col O A B -> Col O P Q -> Perp O P P A -> Perp O Q Q B ->
Q_Cong la -> Q_Cong lb -> Q_Cong lp -> Q_Cong lq -> la O A -> lb O B -> lp O P -> lq O Q ->
exists a, Q_CongA_Acute a /\ Lcos lp la a /\ Lcos lq lb a.
Lemma l13_10_aux2 : forall O A B la lla lb llb,
Col O A B  -> Q_Cong la -> Q_Cong lla -> Q_Cong lb -> Q_Cong llb -> la O A -> lla O A -> lb O B -> llb O B ->
A <> O -> B <> O -> exists a, Q_CongA_Acute a /\ Lcos lla la a /\ Lcos llb lb a.
Lemma l13_6_bis : forall lp l1 l2 a, Lcos lp l1 a -> Lcos lp l2 a -> EqL l1 l2.
Lemma lcos3_lcos2 : forall l1 l2 a b c d n, Eq_Lcos3 l1 a b n l2 c d n -> Eq_Lcos2 l1 a b l2 c d.
Lemma lcos2_lcos : forall l1 l2 a b c, Eq_Lcos2 l1 a c l2 b c -> Eq_Lcos l1 a l2 b.
Lemma lcos_per_anga : forall O A P la lp a, Lcos lp la a -> la O A -> lp O P -> Per A P O -> a A O P.
Lemma lcos_lcos_cop__col : forall la lb lp a b O A B P,
Lcos lp la a -> Lcos lp lb b -> la O A -> lb O B -> lp O P -> a A O P -> b B O P -> Coplanar O A B P ->
Col A B P.