(** Definition of the collinearity predicate. We say that three points are collinear if they belongs to the same line. *)
Definition Col_H := fun A B C =>
exists l, IncidentL A l /\ IncidentL B l /\ IncidentL C l.
(** We show that the notion of collinearity we just defined is equivalent to the notion of collinearity of Tarski. *)

(* Specifications: *)
Lemma cols_coincide_1 : forall A B C, Col_H A B C -> Col A B C.
Lemma cols_coincide_2 : forall A B C, Col A B C -> Col_H A B C.
Lemma cols_coincide : forall A B C, Col A B C <-> Col_H A B C.
Lemma ncols_coincide : forall A B C, ~ Col A B C <-> ~ Col_H A B C.
(** There exists three non collinear points. *)
Lemma lower_dim' : PA <> PB /\ PB <> PC /\ PA <> PC /\ ~ Col_H PA PB PC.
Record Plane := Plan {M1; M2; M3; NCol : ~ Col_H M1 M2 M3}.