End Line_extension.
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 *)

(* Specifications: *)
Lemma extension_grad : 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 -> Grad (f A) (f B) X ->
exists C, Col P Q C /\ Grad A B C /\ f C = X.