unfold disjoint.
intro.
destruct H0 as [P []].
assert(HP:= betH_trans0 A P B C H0 H).
spliter.
apply between_only_one' in H1.
spliter.
contradiction.
Qed.

Lemma lower_dim_e : exists A, exists B, exists C, ~ (Bet A B C \/ Bet B C A \/ Bet C A B).