(** 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 *)

(* Specifications: *)
Lemma extension_gradexp : forall {Tm: Tarski_neutral_dimensionless}
{Tm2 : Tarski_neutral_dimensionless_with_decidable_point_equality Tm}
P Q (f : @Tpoint Tn -> @Tpoint Tm),
line_extension f P Q ->
forall A B X, Col P Q A -> Col P Q B -> GradExp (f A) X (f B) ->
exists C, Col P Q C /\ GradExp A C B /\ f C = X.
Lemma extension_image_density_aux : 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, A <> B ->
Lt (f P) (f Q) A B -> Bet (f P) (f Q) A -> Bet (f P) A B ->
exists X, Col P Q X /\ Bet A (f X) B /\ f X <> A /\ f X <> B.
Lemma extension_image_density_aux2 : 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, A <> B -> Col (f P) (f Q) A -> Col (f P) (f Q) B ->
Lt (f P) (f Q) A B -> Bet (f P) A B ->
exists X, Col P Q X /\ Bet A (f X) B /\ f X <> A /\ f X <> B.