End Completeness.
(** In the following sections, we prove that our formalizations of Hilbert's axiom of completeness are always true in spaces with dimension respectively greater than 2 and 3. *)
Section Dimension.
Context `{Tn:Tarski_neutral_dimensionless}.

(* Specifications: *)
Lemma extension_to_plane__plane : forall {Tm: Tarski_neutral_dimensionless}
{Tm2 : Tarski_neutral_dimensionless_with_decidable_point_equality Tm}
{M : Tarski_2D Tm2}
(f : @Tpoint Tn -> @Tpoint Tm),
extension f ->
@upper_dim_axiom Tn.
Lemma nupper_dim__completeness_for_planes : ~ upper_dim_axiom -> completeness_for_planes.
Lemma extension_to_3d__upper_dim_3 : forall {Tm: Tarski_neutral_dimensionless}
{Tm2 : Tarski_neutral_dimensionless_with_decidable_point_equality Tm}
{M : Tarski_3D Tm2}
(f : @Tpoint Tn -> @Tpoint Tm),
extension f ->
@upper_dim_3_axiom Tn.
Lemma nupper_dim_3__completeness_for_3d_spaces : ~ upper_dim_3_axiom -> completeness_for_3d_spaces.
End Dimension.