(** This file contains the proof that Dedekind completeness implies Hilbert's line completeness. *)
Section Line_extension.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.

(* Specifications: *)
Lemma line_extension_symmetry : forall {Tm : Tarski_neutral_dimensionless}
(f : @Tpoint Tn -> @Tpoint Tm) P Q, line_extension f P Q -> line_extension f Q P.
Lemma line_extension_stability : forall {Tm: Tarski_neutral_dimensionless}
(f : @Tpoint Tn -> @Tpoint Tm) P Q R,
Col P Q R -> P <> R -> line_extension f P Q -> line_extension f P R.
Lemma line_extension_reverse_bet : forall {Tm: Tarski_neutral_dimensionless}
{Tm2: Tarski_neutral_dimensionless_with_decidable_point_equality Tm}
(f : @Tpoint Tn -> @Tpoint Tm) P Q, line_extension f P Q ->
forall A B C, Col P Q A -> Col P Q B -> Col P Q C -> Bet (f A) (f B) (f C) -> Bet A B C.
Lemma pres_bet_line__col : forall {Tm: Tarski_neutral_dimensionless}
f (P Q : @Tpoint Tn), P <> Q -> pres_bet_line f P Q ->
forall A B C, Col P Q A -> Col P Q B -> Col P Q C -> Col (f A) (f B) (f C).