proposition: Lemma eq_o_refl : forall A B P, ~Col A B P -> eq_o A B P A B P.
proof: Proof. intros. unfold eq_o. repeat split; auto. intros. unfold Midpoint in H8. apply l7_3 in H6. subst M. spliter. assert(proj P A C C1). eapply proj_col_proj. apply H3. apply perp_not_eq_2 in H0. auto. eapply perp2__col. apply perp_comm. apply perp_sym. apply H2. apply perp_comm. Perp. assert(C=C1). eapply proj_uniqueness. apply H1. assumption. subst C1. left. assumption. Qed.
imports: GeoCoq.Tarski_dev.Ch12_parallel_inter_dec