(** 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}.

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.