(** This development is inspired by Theorem 32 of Hilbert's Foundations of Geometry (10th edition). It deduces completeness of a 2 or 3-dimensional space from completeness of lines. The original proof is due to Paul Bernays. *)
Section Completeness.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.

(* Specifications: *)
Lemma extension__line_extension : forall {Tm: Tarski_neutral_dimensionless}
(f : @Tpoint Tn -> @Tpoint Tm) P Q,
P <> Q -> extension f -> line_extension f P Q.
Lemma extension_reverse_bet : forall {Tm: Tarski_neutral_dimensionless}
{Tm2 : Tarski_neutral_dimensionless_with_decidable_point_equality Tm}
(f : @Tpoint Tn -> @Tpoint Tm),
extension f ->
forall A B C, Bet (f A) (f B) (f C) -> Bet A B C.
Lemma extension_reverse_col : forall {Tm: Tarski_neutral_dimensionless}
{Tm2 : Tarski_neutral_dimensionless_with_decidable_point_equality Tm}
(f : @Tpoint Tn -> @Tpoint Tm),
extension f ->
forall A B C, Col (f A) (f B) (f C) -> Col A B C.
Lemma line_completeness_aux : line_completeness ->
forall (Tm: Tarski_neutral_dimensionless)
(Tm2 : Tarski_neutral_dimensionless_with_decidable_point_equality Tm)
(f : @Tpoint Tn -> @Tpoint Tm),
@archimedes_axiom Tm ->
extension f ->
forall A P Q R, ~ Col P Q R -> Coplanar (f P) (f Q) (f R) A ->
exists B, Coplanar P Q R B /\ f B = A.
Lemma line_completeness__completeness_for_planes : line_completeness -> completeness_for_planes.
Lemma completeness_aux : forall {Tm: Tarski_neutral_dimensionless}
P Q R (f : @Tpoint Tn -> @Tpoint Tm) A,
(exists B, Coplanar P Q R B /\ f B = A) -> exists B, f B = A.
Lemma line_completeness__completeness_for_3d_spaces :
(exists P Q R S, ~ Coplanar P Q R S) ->
line_completeness -> completeness_for_3d_spaces.