Require Export GeoCoq.Axioms.continuity_axioms.
Require Export GeoCoq.Meta_theory.Parallel_postulates.parallel_postulates.
Section Szmielew.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
Definition hyperbolic_plane_postulate := forall A1 A2 P,
~ Col A1 A2 P -> exists B1 B2 C1 C2,
Par A1 A2 B1 B2 /\ Col P B1 B2 /\
Par A1 A2 C1 C2 /\ Col P C1 C2 /\
~ Col C1 B1 B2.

Theorem szmielew_s_theorem :
aristotle_s_axiom ->
(forall P : Prop,
(playfair_s_postulate -> P) ->
(hyperbolic_plane_postulate -> ~ P)->
(P <-> playfair_s_postulate)).