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.