Ltac Col := auto 3 with col.Ltac line A B l Hdiff := let H:=fresh in (assert(H:= line_existence A B Hdiff); destruct H as [l]; spliter).Ltac ColHR :=
 let tpoint := constr:(Point) in
 let col := constr:(ColH) in
   Col_refl tpoint col.Ltac two_points l X Y := assert(_H := two_points_on_line l); destruct _H as [X _H]; destruct _H as [Y _H]; spliter.

(* Specifications: *)
Lemma ncolH_exists : forall A B, A <> B -> exists C, ~ColH A B C.
Lemma ncolH_distincts : forall A B C, ~ColH A B C -> A <> B /\ B <> C /\ A <> C.
Lemma betH_expand : forall A B C, BetH A B C -> BetH A B C /\ A <> B /\ B <> C /\ A <> C /\ ColH A B C.
Lemma inter_uniquenessH : forall A B A' B' X Y, A' <> B' -> ~ColH A B A' -> ColH A B X -> ColH A B Y -> ColH A' B' X -> ColH A' B' Y -> X = Y.
Lemma inter_incid_uniquenessH : forall P X Y l m, ~IncidL P l -> IncidL P m -> IncidL X l -> IncidL Y l -> IncidL X m -> IncidL Y m -> X = Y.
Lemma between_only_one' : forall A B C, BetH A B C -> ~ BetH B C A /\ ~ BetH B A C.
Lemma betH_colH : forall A B C, BetH A B C -> ColH A B C /\ A <> B /\ B <> C /\ A <> C.
Lemma cut_comm : forall A B l, cut l A B -> cut l B A.
Lemma line_on_plane' : forall A B C p, A <> B -> IncidP A p -> IncidP B p -> ColH A B C -> IncidP C p.
Lemma inner_pasch_aux : forall A B C P Q, ~ ColH B C P -> Bet A P C -> Bet B Q C -> exists X, Bet P X B /\ Bet Q X A.
Lemma betH_trans1 : forall A B C D, BetH A B C -> BetH B C D -> BetH A C D.
Lemma betH_trans2 : forall A B C D, BetH A B C -> BetH B C D -> BetH A B D.
Lemma betH_trans : forall A B C D, BetH A B C -> BetH B C D -> BetH A B D /\ BetH A C D.
Lemma not_cut3 : forall A B C l p, IncidP A p -> IncidP B p -> IncidP C p -> IncidLP l p -> ~ IncidL A l -> ~ ColH A B C -> ~ cut l A B -> ~ cut l A C -> ~ cut l B C.
Lemma betH_trans0 : forall A B C D, BetH A B C -> BetH A C D -> BetH B C D /\ BetH A B D.
Lemma betH_outH2__betH: forall A B C A' C', BetH A B C -> outH B C C' -> outH B A A' -> BetH A' B C'.
Lemma cong_existence' : forall A B l M, A <> B -> IncidL M l -> exists A' B', IncidL A' l /\ IncidL B' l /\ BetH A' M B' /\ CongH M A' A B /\ CongH M B' A B.