End HilbertContext_2.
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.
Hint Resolve betH_trans0 betH_trans1 betH2_out out2_out : bet.
Hint Resolve congH_sym congH_perm congH_perml cong_permr congH_refl
cong_pseudo_transitivity congH_perms: cong.
Ltac Cong := eauto 3 with cong.
Section HilbertContext_2D.
Context `{H2D : Hilbert_neutral_2D}.

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.