(** Given a line extension f to a line l in an Archimedean space, the image of f is dense in l *)

(* Specifications: *)
Lemma extension_image_density : forall {Tm: Tarski_neutral_dimensionless}
{Tm2 : Tarski_neutral_dimensionless_with_decidable_point_equality Tm}
P Q (f : @Tpoint Tn -> @Tpoint Tm),
@archimedes_axiom Tm ->
line_extension f P Q ->
forall A B, Col (f P) (f Q) A -> Col (f P) (f Q) B -> A <> B ->
exists X, Col P Q X /\ Bet A (f X) B /\ f X <> A /\ f X <> B.
Lemma dedekind__completeness : dedekind_s_axiom -> line_completeness.
End Completeness.