Inductive Point :=
 P0.Inductive Point :=
 P0.Definition Bet (A B C : Point) := False.Definition Cong (A B C D : Point) := True.

(* Specifications: *)
Lemma between_identity : forall A B, Bet A B A -> A=B.
Lemma cong_pseudo_reflexivity : forall A B, Cong A B B A.
Lemma cong_identity : forall A B C, Cong A B C C -> A = B.
Lemma cong_inner_transitivity : forall A B C D E F, Cong A B C D -> Cong A B E F -> Cong C D E F.
Lemma inner_pasch : forall A B C P Q, Bet A P C -> Bet B Q C -> exists x, Bet P x B /\ Bet Q x A.
Lemma five_segment : forall A A' B B' C C' D D', Cong A B A' B' -> Cong B C B' C' -> Cong A D A' D' -> Cong B D B' D' -> Bet A B C -> Bet A' B' C' -> A <> B -> Cong C D C' D'.
Lemma not_segment_construction : ~ (forall A B C D, exists E, Bet A B E /\ Cong B E C D).
Lemma lower_dim : exists A, exists B, exists C, ~ (Bet A B C \/ Bet B C A \/ Bet C A B).
Lemma upper_dim : forall A B C P Q , 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).
Lemma euclid : forall A B C, ~ (Bet A B C \/ Bet B C A \/ Bet C A B) -> exists CC, Cong A CC B CC /\ Cong A CC C CC.