Section Completeness.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
(** If there exists n such that (f A) X = n times (f A) (f B), then X belongs to the image of the line extension f *)
(** If there exists n such that (f A) X = (f A) (f B) divided by 2^n, then X belongs to the image of the line extension f *)
(** Given a line extension f to a line l in an Archimedean space, the image of f is dense in l *)

Lemma extension_image_density : forall {Tm: Tarski_neutral_dimensionless}
{TmEQD : 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.