Definition upper_dim_axiom := forall A B C P Q : Tpoint,
  P <> Q -> Cong A P A Q -> Cong B P B Q -> Cong C P C Q ->
  (Bet A B C \/ Bet B C A \/ Bet C A B).Definition all_coplanar_axiom := forall A B C D, Coplanar A B C D.

(* Specifications: *)
Lemma upper_dim_implies_per2__col : upper_dim_axiom -> (forall A B C X, Per A X C -> X <> C -> Per B X C -> Col A B X).
Lemma upper_dim_implies_col_perp2__col : upper_dim_axiom -> (forall A B X Y P, Col A B P -> Perp A B X P -> Perp P A Y P -> Col Y X P).
Lemma upper_dim_implies_perp2__col : upper_dim_axiom -> (forall X Y Z A B, Perp X Y A B -> Perp X Z A B -> Col X Y Z).
Lemma upper_dim_implies_not_two_sides_one_side_aux : upper_dim_axiom -> (forall A B X Y PX, A <> B -> PX <> A -> Perp A B X PX -> Col A B PX -> ~ Col X A B -> ~ Col Y A B -> ~ TS A B X Y -> OS A B X Y).
Lemma upper_dim_implies_not_two_sides_one_side : upper_dim_axiom -> (forall A B X Y, ~ Col X A B -> ~ Col Y A B -> ~ TS A B X Y -> OS A B X Y).
Lemma upper_dim_implies_not_one_side_two_sides : upper_dim_axiom -> (forall A B X Y, ~ Col X A B -> ~ Col Y A B -> ~ OS A B X Y -> TS A B X Y).
Lemma upper_dim_implies_one_or_two_sides : upper_dim_axiom -> (forall A B X Y, ~ Col X A B -> ~ Col Y A B -> TS A B X Y \/ OS A B X Y).
Lemma upper_dim_implies_all_coplanar : upper_dim_axiom -> all_coplanar_axiom.
Lemma all_coplanar_implies_upper_dim : all_coplanar_axiom -> upper_dim_axiom.
Lemma all_coplanar_upper_dim : all_coplanar_axiom <-> upper_dim_axiom.
Lemma upper_dim_stab : ~ ~ upper_dim_axiom -> upper_dim_axiom.