(** 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_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.