(** 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 extension__line_extension : forall {Tm: Tarski_neutral_dimensionless}
(f : @Tpoint Tn -> @Tpoint Tm) P Q,
P <> Q -> extension f -> line_extension f P Q.