%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO223+3 : TPTP v8.1.0. Released v4.0.0. % Transfm : none % Format : tptp:raw % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % Computer : n003.cluster.edu % Model : x86_64 x86_64 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz % Memory : 8042.1875MB % OS : Linux 3.10.0-693.el7.x86_64 % CPULimit : 300s % WCLimit : 600s % DateTime : Sat Jul 16 06:31:19 EDT 2022 % Result : Theorem 33.39s 33.55s % Output : Proof 35.33s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.03/0.12 % Problem : GEO223+3 : TPTP v8.1.0. Released v4.0.0. % 0.03/0.12 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.12/0.33 % Computer : n003.cluster.edu % 0.12/0.33 % Model : x86_64 x86_64 % 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.33 % Memory : 8042.1875MB % 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.33 % CPULimit : 300 % 0.12/0.33 % WCLimit : 600 % 0.12/0.33 % DateTime : Sat Jun 18 18:33:13 EDT 2022 % 0.12/0.33 % CPUTime : % 33.39/33.55 % SZS status Theorem % 33.39/33.55 (* PROOF-FOUND *) % 33.39/33.55 (* BEGIN-PROOF *) % 33.39/33.55 % SZS output start Proof % 33.39/33.55 1. (distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### Axiom % 33.39/33.55 2. (distinct_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)) (-. (distinct_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) ### Axiom % 33.39/33.55 3. (All X, (-. (distinct_lines X X))) (distinct_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)) ### All 2 % 33.39/33.55 4. (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (-. (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### Axiom % 33.39/33.55 5. (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### Axiom % 33.39/33.55 6. (apart_point_and_line T_1 (parallel_through_point T_0 T_1)) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) ### Axiom % 33.39/33.55 7. (All Y, (-. (apart_point_and_line T_1 (parallel_through_point Y T_1)))) (apart_point_and_line T_1 (parallel_through_point T_0 T_1)) ### All 6 % 33.39/33.55 8. (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (apart_point_and_line T_1 (parallel_through_point T_0 T_1)) ### All 7 % 33.39/33.55 9. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### Axiom % 33.39/33.55 10. (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### Axiom % 33.39/33.55 11. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### Axiom % 33.39/33.55 12. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### Axiom % 33.39/33.55 13. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### Axiom % 33.39/33.55 14. (convergent_lines (parallel_through_point T_0 T_1) T_0) (-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) ### Axiom % 33.39/33.55 15. (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (parallel_through_point T_0 T_1) T_0) ### All 14 % 33.39/33.55 16. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) ### Axiom % 33.39/33.55 17. (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) ### All 16 % 33.39/33.55 18. (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### Axiom % 33.39/33.55 19. (-. (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### NotNot 18 % 33.39/33.55 20. (-. ((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### NotOr 19 % 33.39/33.55 21. (-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) (convergent_lines (parallel_through_point T_0 T_1) T_0) ### Axiom % 33.39/33.55 22. (-. (-. (convergent_lines (parallel_through_point T_0 T_1) T_0))) (-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) ### NotNot 21 % 33.39/33.55 23. (-. ((-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) T_0)))) (-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) ### NotOr 22 % 33.39/33.55 24. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) ### Axiom % 33.39/33.55 25. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) ### Axiom % 33.39/33.55 26. ((((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) T_0)))) => ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### DisjTree 20 23 24 25 % 33.39/33.55 27. (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) ### All 26 % 33.39/33.55 28. (convergent_lines (orthogonal_through_point T_0 T_1) T_0) (-. (convergent_lines (orthogonal_through_point T_0 T_1) T_0)) ### Axiom % 33.39/33.55 29. (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0))) => (-. (convergent_lines (orthogonal_through_point T_0 T_1) T_0))) (convergent_lines (orthogonal_through_point T_0 T_1) T_0) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) ### DisjTree 17 27 28 % 33.39/33.56 30. (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (orthogonal_through_point T_0 T_1) N)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (convergent_lines (orthogonal_through_point T_0 T_1) T_0) ### All 29 % 33.39/33.56 31. (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (convergent_lines (orthogonal_through_point T_0 T_1) T_0) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) ### All 30 % 33.39/33.56 32. (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0) ### Axiom % 33.39/33.56 33. ((convergent_lines (orthogonal_through_point T_0 T_1) T_0) \/ (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) ### Or 31 32 % 33.39/33.56 34. (All M, ((convergent_lines (orthogonal_through_point T_0 T_1) M) \/ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) ### All 33 % 33.39/33.56 35. (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) ### All 34 % 33.39/33.56 36. (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) ### All 35 % 33.39/33.60 37. (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) ### All 36 % 33.39/33.60 38. ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) T_0) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### DisjTree 13 15 37 % 33.39/33.60 39. (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) ### All 38 % 33.39/33.60 40. (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) ### All 39 % 33.39/33.60 41. (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) ### All 40 % 33.39/33.60 42. (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)) ### Axiom % 33.39/33.60 43. ((convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) ### And 42 % 33.39/33.60 44. (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### Axiom % 33.39/33.61 45. ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### And 44 % 33.39/33.61 46. (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### DisjTree 12 41 43 45 % 33.39/33.61 47. (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### All 46 % 33.39/33.61 48. (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### All 47 % 33.39/33.61 49. (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) ### All 48 % 33.39/33.61 50. (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) ### All 49 % 33.48/33.65 51. (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) ### All 50 % 33.48/33.65 52. (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)) ### Axiom % 33.48/33.65 53. (-. (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) ### NotNot 52 % 33.48/33.65 54. (-. ((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) ### NotOr 53 % 33.48/33.65 55. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### Axiom % 33.48/33.65 56. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### Axiom % 33.48/33.65 57. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### Axiom % 33.48/33.65 58. (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### Axiom % 33.48/33.65 59. (-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### Axiom % 33.48/33.65 60. ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) => ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) \/ (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) (-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### DisjTree 57 58 59 % 33.48/33.65 61. (All Z, ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) => ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z)))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### All 60 % 33.48/33.65 62. (All X, (-. (convergent_lines X X))) (-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All Z, ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) => ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z)))) ### All 61 % 33.48/33.65 63. (All Y, (All Z, ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All X, (-. (convergent_lines X X))) ### All 62 % 33.48/33.65 64. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### All 63 % 33.48/33.65 65. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) ### Axiom % 33.48/33.65 66. (distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (-. (distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### Axiom % 33.48/33.65 67. (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) ### Axiom % 33.48/33.65 68. (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (apart_point_and_line T_1 (parallel_through_point T_0 T_1)) ### Axiom % 33.48/33.65 69. (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### Axiom % 33.48/33.67 70. ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 (parallel_through_point T_0 T_1)) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### DisjTree 66 67 68 69 % 33.48/33.67 71. (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### All 70 % 33.48/33.67 72. (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)) ### Axiom % 33.48/33.67 73. ((convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) => ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) \/ (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) ### DisjTree 65 71 72 % 33.48/33.67 74. (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) => ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z)))) (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) ### All 73 % 33.48/33.67 75. (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) ### All 74 % 33.48/33.67 76. (-. (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) ### NotNot 75 % 33.48/33.67 77. (-. ((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) ### NotOr 76 % 33.48/33.67 78. (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### Axiom % 33.48/33.67 79. (-. (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### NotNot 78 % 33.48/33.67 80. (-. ((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### NotOr 79 % 33.48/33.67 81. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### Axiom % 33.53/33.70 82. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) ### Axiom % 33.53/33.70 83. (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### Axiom % 33.53/33.70 84. (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### Axiom % 33.53/33.70 85. ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) ### DisjTree 82 83 84 % 33.53/33.70 86. (All Z, ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### All 85 % 33.53/33.70 87. (All Y, (All Z, ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) ### All 86 % 33.53/33.70 88. ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### DisjTree 81 75 87 % 33.53/33.70 89. (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### All 88 % 33.53/33.70 90. (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### All 89 % 33.53/33.74 91. (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### Axiom % 33.53/33.74 92. ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) \/ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) ### Or 90 91 % 33.53/33.74 93. (All M, ((convergent_lines (parallel_through_point T_0 T_1) M) \/ (unorthogonal_lines (parallel_through_point T_0 T_1) M))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### All 92 % 33.53/33.74 94. (All X, (-. (convergent_lines X X))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All M, ((convergent_lines (parallel_through_point T_0 T_1) M) \/ (unorthogonal_lines (parallel_through_point T_0 T_1) M))) ### All 93 % 33.53/33.74 95. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All M, ((convergent_lines (parallel_through_point T_0 T_1) M) \/ (unorthogonal_lines (parallel_through_point T_0 T_1) M))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All X, (-. (convergent_lines X X))) ### All 94 % 33.53/33.74 96. (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### Axiom % 33.53/33.74 97. (-. (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### NotNot 96 % 33.61/33.76 98. (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### Axiom % 33.61/33.76 99. (-. (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### NotNot 98 % 33.61/33.76 100. (-. ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### NotAnd 97 99 % 33.61/33.76 101. (All M, (-. ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### All 100 % 33.61/33.76 102. (distinct_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (-. (distinct_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### Axiom % 33.61/33.76 103. (All X, (-. (distinct_lines X X))) (distinct_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### All 102 % 33.61/33.76 104. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### Axiom % 33.61/33.76 105. (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) ### Axiom % 33.61/33.76 106. (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### Axiom % 33.61/33.76 107. (-. (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### NotNot 106 % 33.61/33.76 108. (-. ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### NotOr 107 % 33.61/33.76 109. (-. ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### NotOr 99 % 33.61/33.76 110. (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) ### Axiom % 33.61/33.76 111. (-. (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) ### NotNot 110 % 33.61/33.76 112. (-. ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) ### NotOr 111 % 33.61/33.76 113. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) ### Axiom % 33.61/33.76 114. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) ### Axiom % 33.61/33.76 115. ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))))) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### DisjTree 109 112 113 114 % 33.61/33.76 116. (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) ### All 115 % 33.61/33.80 117. (-. (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) ### NotNot 116 % 33.61/33.80 118. (-. ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) ### NotOr 117 % 33.61/33.80 119. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### Axiom % 33.61/33.80 120. (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### Axiom % 33.61/33.80 121. (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) ### Axiom % 33.61/33.80 122. (-. (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) ### NotNot 121 % 33.61/33.80 123. (-. ((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) ### NotOr 122 % 33.61/33.80 124. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) ### Axiom % 33.61/33.80 125. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) ### Axiom % 33.61/33.80 126. ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) T_0)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)))) (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0) (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) ### DisjTree 123 23 124 125 % 33.61/33.80 127. (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) T_0)) (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0) (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0) ### All 126 % 33.61/33.80 128. (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0) (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) ### All 127 % 33.66/33.84 129. (-. (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) ### NotNot 128 % 33.66/33.84 130. (-. ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) ### NotOr 129 % 33.66/33.84 131. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0)) ### Axiom % 33.66/33.84 132. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0)) ### Axiom % 33.66/33.84 133. ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0)))) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### DisjTree 109 130 131 132 % 33.66/33.84 134. (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) ### All 133 % 33.66/33.84 135. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0)) ### Axiom % 33.66/33.84 136. (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0))) => (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0))) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### DisjTree 120 134 135 % 33.70/33.85 137. (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) ### All 136 % 33.70/33.85 138. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### Axiom % 33.70/33.85 139. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### Axiom % 33.70/33.85 140. (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### Axiom % 33.70/33.85 141. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### Axiom % 33.70/33.85 142. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### Axiom % 33.70/33.85 143. ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) ### And 17 % 33.70/33.85 144. (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### Axiom % 33.70/33.85 145. (-. (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### NotNot 144 % 33.70/33.85 146. (-. ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### NotOr 145 % 33.70/33.85 147. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) ### Axiom % 33.70/33.85 148. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) ### Axiom % 33.70/33.85 149. (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0) ### Axiom % 33.70/33.85 150. (-. (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) ### NotNot 149 % 33.70/33.85 151. (-. ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) ### NotOr 150 % 33.70/33.85 152. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0)) ### Axiom % 33.70/33.85 153. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0)) ### Axiom % 33.70/33.85 154. ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0)))) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### DisjTree 109 151 152 153 % 33.70/33.89 155. (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) ### All 154 % 33.70/33.89 156. ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) ### And 155 % 33.70/33.89 157. (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0) ### Axiom % 33.70/33.89 158. ((convergent_lines (orthogonal_through_point T_0 T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) ### And 157 % 33.70/33.89 159. (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0)) \/ ((convergent_lines (orthogonal_through_point T_0 T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) ### DisjTree 147 148 156 158 % 33.70/33.89 160. (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) ### All 159 % 33.70/33.89 161. (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) ### All 160 % 33.79/33.94 162. (-. (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)))) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) ### NotNot 161 % 33.79/33.94 163. (-. ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) ### NotOr 162 % 33.79/33.94 164. (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) ### Axiom % 33.79/33.94 165. (-. (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) ### NotNot 164 % 33.79/33.94 166. (-. ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) ### NotOr 165 % 33.79/33.94 167. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) (-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) ### Axiom % 33.79/33.94 168. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) ### Axiom % 33.79/33.94 169. ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### DisjTree 108 166 167 168 % 33.79/33.94 170. (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) ### All 169 % 33.79/33.94 171. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) ### Axiom % 33.79/33.94 172. ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### DisjTree 146 163 170 171 % 33.79/33.95 173. (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) ### All 172 % 33.79/33.95 174. ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) ### And 173 % 33.79/33.99 175. (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) \/ ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### DisjTree 141 142 143 174 % 33.79/33.99 176. (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) ### All 175 % 33.86/34.03 177. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) (-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) ### Axiom % 33.86/34.03 178. (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)))) => (-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### DisjTree 140 176 177 % 33.86/34.03 179. (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) ### All 178 % 33.86/34.05 180. ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) ### And 179 % 33.86/34.05 181. (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### DisjTree 138 139 180 143 % 33.93/34.08 182. (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) ### All 181 % 33.96/34.13 183. (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0)) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) ### All 182 % 33.96/34.13 184. (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) T_0))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) ### DisjTree 116 137 183 % 33.99/34.14 185. (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) ### All 184 % 33.99/34.18 186. ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) T_0) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) T_0))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### DisjTree 119 15 185 % 33.99/34.19 187. (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) ### All 186 % 34.07/34.23 188. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) ### Axiom % 34.07/34.23 189. ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### DisjTree 108 118 187 188 % 34.07/34.28 190. (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) ### All 189 % 34.07/34.28 191. (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### All 190 % 34.14/34.29 192. (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### Axiom % 34.14/34.29 193. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) ### Axiom % 34.14/34.29 194. (((-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) ### DisjTree 191 192 193 % 34.14/34.33 195. (All N, (((-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### All 194 % 34.14/34.34 196. (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) ### All 195 % 34.24/34.38 197. ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### DisjTree 104 105 196 % 34.24/34.39 198. (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) ### All 197 % 34.30/34.43 199. (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z)))) ### All 198 % 34.30/34.45 200. (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) ### All 199 % 34.30/34.50 201. ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((distinct_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (-. ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))))) ### DisjTree 101 103 200 % 34.30/34.50 202. (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((distinct_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (All M, (-. ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) ### All 201 % 34.41/34.54 203. (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (-. ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))))) (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((distinct_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z)))) ### All 202 % 34.41/34.57 204. (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((distinct_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (All M, (-. ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) ### All 203 % 34.41/34.58 205. (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((-. (unorthogonal_lines (parallel_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((distinct_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) ### All 204 % 34.48/34.60 206. (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) => ((distinct_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z)))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) ### All 205 % 34.48/34.63 207. (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) ### All 206 % 34.48/34.65 208. (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) ### All 207 % 34.55/34.69 209. (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) ### Axiom % 34.55/34.69 210. ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) \/ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) ### Or 208 209 % 34.55/34.69 211. (All M, ((convergent_lines (parallel_through_point T_0 T_1) M) \/ (unorthogonal_lines (parallel_through_point T_0 T_1) M))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) ### All 210 % 34.55/34.72 212. (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All M, ((convergent_lines (parallel_through_point T_0 T_1) M) \/ (unorthogonal_lines (parallel_through_point T_0 T_1) M))) ### All 211 % 34.55/34.74 213. ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All M, ((convergent_lines (parallel_through_point T_0 T_1) M) \/ (unorthogonal_lines (parallel_through_point T_0 T_1) M))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) ### DisjTree 77 80 95 212 % 34.65/34.78 214. (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All M, ((convergent_lines (parallel_through_point T_0 T_1) M) \/ (unorthogonal_lines (parallel_through_point T_0 T_1) M))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) ### All 213 % 34.65/34.79 215. (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) ### All 214 % 34.69/34.83 216. (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### Axiom % 34.69/34.83 217. ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### And 216 % 34.69/34.83 218. (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) \/ ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### DisjTree 64 215 43 217 % 34.69/34.85 219. (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) ### All 218 % 34.69/34.88 220. (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 219 % 34.78/34.90 221. (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### Axiom % 34.78/34.90 222. ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) \/ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) ### Or 220 221 % 34.78/34.93 223. (All M, ((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) \/ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 222 % 34.78/34.95 224. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) ### All 223 % 34.86/34.99 225. (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (orthogonal_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 224 % 34.86/34.99 226. (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) ### All 225 % 34.86/35.02 227. (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) ### All 226 % 34.86/35.04 228. (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) ### All 227 % 34.86/35.05 229. (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (parallel_through_point T_0 T_1) N)))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) ### All 228 % 34.96/35.08 230. (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) ### All 229 % 34.96/35.08 231. (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) ### All 230 % 34.96/35.09 232. (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) ### All 231 % 34.97/35.13 233. (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point (parallel_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N))))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) ### All 232 % 34.97/35.13 234. (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) ### All 233 % 35.01/35.15 235. (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) ### All 234 % 35.01/35.15 236. (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (-. (distinct_lines X X))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) ### All 235 % 35.01/35.17 237. ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) ### And 236 % 35.01/35.18 238. (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (-. (distinct_lines X X))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### DisjTree 55 56 43 237 % 35.01/35.18 239. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All N, (((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) ### All 238 % 35.10/35.22 240. (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (-. (distinct_lines X X))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### All 239 % 35.10/35.22 241. (-. (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) ### NotNot 240 % 35.10/35.23 242. (-. ((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (-. (distinct_lines X X))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### NotOr 241 % 35.10/35.23 243. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### Axiom % 35.10/35.23 244. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### Axiom % 35.10/35.23 245. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) ### Axiom % 35.10/35.23 246. ((((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))))) => ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### DisjTree 20 54 244 245 % 35.10/35.23 247. (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### All 246 % 35.10/35.26 248. (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### All 247 % 35.10/35.26 249. ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) ### DisjTree 54 242 243 248 % 35.10/35.27 250. (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (-. (distinct_lines X X))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### All 249 % 35.17/35.28 251. ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) ### And 250 % 35.17/35.28 252. (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) \/ ((convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### DisjTree 11 51 251 43 % 35.17/35.28 253. (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) ### All 252 % 35.17/35.32 254. (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### All 253 % 35.17/35.32 255. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N))))) ### All 254 % 35.17/35.33 256. (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All M, (All N, (((convergent_lines (parallel_through_point T_0 T_1) M) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) M)) => (((convergent_lines (parallel_through_point T_0 T_1) N) /\ (unorthogonal_lines (parallel_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### All 255 % 35.17/35.33 257. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) ### All 256 % 35.26/35.36 258. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Y) => ((convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### All 257 % 35.26/35.36 259. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 258 % 35.28/35.38 260. (All Y, (-. (apart_point_and_line T_1 (parallel_through_point Y T_1)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Z, ((distinct_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (parallel_through_point (parallel_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### All 259 % 35.28/35.38 261. (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) M))) /\ ((-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All Y, (-. (apart_point_and_line T_1 (parallel_through_point Y T_1)))) ### All 260 % 35.28/35.38 262. (All Y, (-. (apart_point_and_line T_1 (parallel_through_point Y T_1)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All M, (All N, ((((-. (convergent_lines (parallel_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (parallel_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (parallel_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) ### All 261 % 35.28/35.38 263. (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All Y, (-. (apart_point_and_line T_1 (parallel_through_point Y T_1)))) ### All 262 % 35.28/35.38 264. (All Y, (-. (apart_point_and_line T_1 (parallel_through_point Y T_1)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) ### All 263 % 35.28/35.38 265. (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (-. (convergent_lines (parallel_through_point (parallel_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 264 % 35.28/35.39 266. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (-. (convergent_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) ### All 265 % 35.28/35.39 267. (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All Z, ((convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 266 % 35.28/35.39 268. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((convergent_lines (parallel_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) ### All 267 % 35.28/35.39 269. (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All Y, (All Z, ((convergent_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 268 % 35.31/35.42 270. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All Y, (-. (convergent_lines (parallel_through_point Y T_1) Y))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) ### All 269 % 35.31/35.42 271. (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 270 % 35.31/35.42 272. ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) => ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) \/ (convergent_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### DisjTree 9 10 271 % 35.31/35.42 273. (All Z, ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) => ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (parallel_through_point T_0 T_1) Z)))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 272 % 35.31/35.42 274. (All Y, (All Z, ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Y) => ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z) \/ (convergent_lines Y Z))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### All 273 % 35.31/35.43 275. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 274 % 35.31/35.43 276. ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 (parallel_through_point T_0 T_1)) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (-. (distinct_lines X X))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) ### DisjTree 4 5 8 275 % 35.31/35.43 277. (All Z, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z))))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (-. (distinct_lines X X))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 276 % 35.31/35.43 278. ((distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((distinct_lines (parallel_through_point T_0 T_1) (parallel_through_point T_0 T_1)) \/ (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (parallel_through_point T_0 T_1)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All Z, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z))))) (All X, (-. (distinct_lines X X))) (distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### DisjTree 1 3 277 % 35.31/35.43 279. (All Z, ((distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((distinct_lines (parallel_through_point T_0 T_1) Z) \/ (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All X, (-. (distinct_lines X X))) (All Z, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z))))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 278 % 35.31/35.43 280. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line T_1 Y) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All X, (-. (distinct_lines X X))) (distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All Z, ((distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((distinct_lines (parallel_through_point T_0 T_1) Z) \/ (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) ### All 279 % 35.31/35.43 281. (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All Z, ((distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((distinct_lines (parallel_through_point T_0 T_1) Z) \/ (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 280 % 35.31/35.43 282. (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (All X, (-. (distinct_lines X X))) (distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All Z, ((distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((distinct_lines (parallel_through_point T_0 T_1) Z) \/ (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) ### All 281 % 35.31/35.43 283. (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All Z, ((distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((distinct_lines (parallel_through_point T_0 T_1) Z) \/ (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All X, (-. (distinct_lines X X))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 282 % 35.31/35.43 284. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (All X, (-. (distinct_lines X X))) (distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All Z, ((distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((distinct_lines (parallel_through_point T_0 T_1) Z) \/ (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) ### All 283 % 35.31/35.43 285. (All Y, (All Z, ((distinct_lines (parallel_through_point T_0 T_1) Y) => ((distinct_lines (parallel_through_point T_0 T_1) Z) \/ (distinct_lines Y Z))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All X, (-. (distinct_lines X X))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 284 % 35.31/35.43 286. (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (All X, (-. (distinct_lines X X))) (distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) ### All 285 % 35.33/35.44 287. (-. (-. (distinct_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### NotNot 286 % 35.33/35.44 288. (-. (equal_lines (parallel_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) ### Definition-Pseudo(equal_lines) 287 % 35.33/35.44 289. (-. (All L, (equal_lines (parallel_through_point L T_1) (orthogonal_through_point (orthogonal_through_point L T_1) T_1)))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### NotAllEx 288 % 35.33/35.44 290. (-. (All A, (All L, (equal_lines (parallel_through_point L A) (orthogonal_through_point (orthogonal_through_point L A) A))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (All Y, (-. (convergent_lines (parallel_through_point Y X) Y)))) (All X, (All Y, (-. (apart_point_and_line X (parallel_through_point Y X))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) ### NotAllEx 289 % 35.33/35.44 % SZS output end Proof % 35.33/35.44 (* END-PROOF *) %------------------------------------------------------------------------------