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 plane_separation_2D : forall A B X Y, ~ ColH A X Y -> ~ ColH B X Y -> cut' A B X Y \/ same_side' A B X Y.