proposition: Lemma lower_dim' : PA <> PB /\ PB <> PC /\ PA <> PC /\ ~ Col_H PA PB PC.
proof: Proof. assert (HNCol : ~ Col PA PB PC) by (apply lower_dim). assert_diffs. apply ncols_coincide in HNCol. repeat split; auto. Qed.
imports: Morphisms GeoCoq.Axioms.hilbert_axioms GeoCoq.Axioms.parallel_postulates GeoCoq.Meta_theory.Parallel_postulates.tarski_playfair GeoCoq.Meta_theory.Parallel_postulates.SPP_ID GeoCoq.Meta_theory.Dimension_axioms.upper_dim_3 GeoCoq.Meta_theory.Parallel_postulates.parallel_postulates GeoCoq.Utils.triples