Ltac out_exists A B X := try match goal with |H : A <> B |- _
                                 => assert(HH:=between_out A B H);auto; destruct HH as [X]
                              end.Ltac not_on_line A B X :=  try match goal with |H : A <> B |- _
                                 => assert(HH:=ncolH_exists A B H);auto; destruct HH as [X]
                              end.

Lemma upper_dim : forall A B C P Q, P <> Q -> A <> B -> A <> C -> B <> C -> 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).