(** 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.

(* 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.
Lemma lower_dim' : PA <> PB /\ PB <> PC /\ PA <> PC /\ ~ Col_H PA PB PC.