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.

(* Specifications: *)
Lemma aah__hpp : hypothesis_of_acute_saccheri_quadrilaterals -> hyperbolic_plane_postulate.
Theorem szmielew_s_theorem : aristotle_s_axiom -> (forall P : Prop, (playfair_s_postulate -> P) -> (hyperbolic_plane_postulate -> ~ P)-> (P <-> playfair_s_postulate)).