proposition: Lemma le_cong_le : forall A B C A' B' C', Bet A B C -> Bet A' B' C' -> Le A B A' B' -> Cong B C B' C' -> Le A C A' C'.
proof: Proof. intros. eapply l5_5_2. unfold Le in H1. ex_and H1 P. prolong A C T P B'. exists T. split. assumption. assert(Bet A B T). eapply between_exchange4. apply H. assumption. apply l2_11 with B P. apply H6. eapply between_exchange4. apply H1. assumption. assumption. apply cong_left_commutativity. eapply l2_11 with C B'. apply between_symmetry. eapply between_exchange3. apply H. assumption. eapply between_exchange3. apply H1. assumption. Cong. Cong. Qed.
imports: GeoCoq.Tarski_dev.Ch12_parallel_inter_dec