%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO221+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 06:31:18 EDT 2022 % Result : Theorem 81.26s 81.51s % Output : Proof 83.01s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.12 % Problem : GEO221+3 : TPTP v8.1.0. Released v4.0.0. % 0.00/0.12 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.11/0.33 % Computer : n003.cluster.edu % 0.11/0.33 % Model : x86_64 x86_64 % 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.11/0.33 % Memory : 8042.1875MB % 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64 % 0.11/0.33 % CPULimit : 300 % 0.11/0.33 % WCLimit : 600 % 0.11/0.33 % DateTime : Sat Jun 18 18:37:13 EDT 2022 % 0.11/0.33 % CPUTime : % 81.26/81.51 % SZS status Theorem % 81.26/81.51 (* PROOF-FOUND *) % 81.26/81.51 (* BEGIN-PROOF *) % 81.26/81.51 % SZS output start Proof % 81.26/81.51 1. (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) ### Axiom % 81.26/81.51 2. (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### Axiom % 81.26/81.51 3. ((distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) => (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### Imply 1 2 % 81.26/81.51 4. (All Y, ((distinct_lines (orthogonal_through_point T_0 T_1) Y) => (convergent_lines (orthogonal_through_point T_0 T_1) Y))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) ### All 3 % 81.26/81.51 5. (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### All 4 % 81.26/81.51 6. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) ### Axiom % 81.26/81.51 7. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) ### Axiom % 81.26/81.51 8. (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) ### Axiom % 81.26/81.51 9. (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1)) ### Axiom % 81.26/81.51 10. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) ### Axiom % 81.26/81.51 11. (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) ### Axiom % 81.26/81.51 12. (-. (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) ### NotNot 11 % 81.26/81.51 13. (-. ((-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) ### NotOr 12 % 81.26/81.51 14. (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2)) ### Axiom % 81.26/81.51 15. (-. (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) ### NotNot 14 % 81.26/81.51 16. (-. ((-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) ### NotOr 15 % 81.26/81.51 17. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) ### Axiom % 81.26/81.51 18. (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### Axiom % 81.26/81.51 19. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) ### Axiom % 81.26/81.51 20. (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0) ### All 19 % 81.26/81.51 21. (convergent_lines (orthogonal_through_point T_0 T_2) T_0) (-. (convergent_lines (orthogonal_through_point T_0 T_2) T_0)) ### Axiom % 81.26/81.51 22. (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0))) => (-. (convergent_lines (orthogonal_through_point T_0 T_2) T_0))) (convergent_lines (orthogonal_through_point T_0 T_2) T_0) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) ### DisjTree 18 20 21 % 81.26/81.51 23. (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines (orthogonal_through_point T_0 T_2) N)))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (convergent_lines (orthogonal_through_point T_0 T_2) T_0) ### All 22 % 81.26/81.51 24. (unorthogonal_lines (orthogonal_through_point T_0 T_2) T_0) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) T_0)) ### Axiom % 81.26/81.51 25. (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (unorthogonal_lines (orthogonal_through_point T_0 T_2) T_0) ### All 24 % 81.26/81.51 26. ((convergent_lines (orthogonal_through_point T_0 T_2) T_0) \/ (unorthogonal_lines (orthogonal_through_point T_0 T_2) T_0)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines (orthogonal_through_point T_0 T_2) N)))) ### Or 23 25 % 81.26/81.51 27. (All M, ((convergent_lines (orthogonal_through_point T_0 T_2) M) \/ (unorthogonal_lines (orthogonal_through_point T_0 T_2) M))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines (orthogonal_through_point T_0 T_2) N)))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) ### All 26 % 81.26/81.51 28. ((((-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))))) => ((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines (orthogonal_through_point T_0 T_2) N)))) (All M, ((convergent_lines (orthogonal_through_point T_0 T_2) M) \/ (unorthogonal_lines (orthogonal_through_point T_0 T_2) M))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) ### DisjTree 13 16 17 27 % 81.65/81.90 29. (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_2) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) N)))) => ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All M, ((convergent_lines (orthogonal_through_point T_0 T_2) M) \/ (unorthogonal_lines (orthogonal_through_point T_0 T_2) M))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines (orthogonal_through_point T_0 T_2) N)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) ### All 28 % 81.65/81.90 30. (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_2) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_2) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines (orthogonal_through_point T_0 T_2) N)))) (All M, ((convergent_lines (orthogonal_through_point T_0 T_2) M) \/ (unorthogonal_lines (orthogonal_through_point T_0 T_2) M))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) ### All 29 % 81.65/81.90 31. (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 T_0 T_2) (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All M, ((convergent_lines (orthogonal_through_point T_0 T_2) M) \/ (unorthogonal_lines (orthogonal_through_point T_0 T_2) M))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines (orthogonal_through_point T_0 T_2) N)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) ### All 30 % 81.65/81.90 32. (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines (orthogonal_through_point T_0 T_2) N)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_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 31 % 81.65/81.90 33. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) ### Axiom % 81.65/81.90 34. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) ### Axiom % 81.65/81.90 35. (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2)) ### Axiom % 81.65/81.90 36. ((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) ### And 35 % 81.65/81.90 37. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) ### Axiom % 81.65/81.90 38. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) ### Axiom % 81.65/81.90 39. ((convergent_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))) ### And 20 % 81.65/81.90 40. (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) T_0)) (unorthogonal_lines (orthogonal_through_point T_0 T_2) T_0) ### Axiom % 81.65/81.90 41. ((convergent_lines (orthogonal_through_point T_0 T_2) T_0) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) T_0)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) T_0)) ### And 40 % 81.65/81.90 42. (((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) => (((convergent_lines (orthogonal_through_point T_0 T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) \/ ((convergent_lines (orthogonal_through_point T_0 T_2) T_0) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) T_0)))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) T_0)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### DisjTree 37 38 39 41 % 81.65/81.90 43. (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) => (((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 T_0 T_2) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) N))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) T_0)) ### All 42 % 81.65/81.90 44. (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) => (((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 T_0 T_2) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) N))))) ### All 43 % 81.75/81.95 45. (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) ### All 44 % 81.75/81.95 46. (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, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### All 45 % 81.75/81.95 47. ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) 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))))))) ### And 46 % 81.75/81.95 48. (((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) \/ ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))))) (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, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) ### DisjTree 33 34 36 47 % 81.75/81.95 49. (All N, (((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point T_0 T_2) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) 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 T_0 T_2) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) 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 48 % 81.75/81.95 50. ((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_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 L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point T_0 T_2) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) N)) \/ ((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))))) ### And 49 % 81.75/81.95 51. (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) ### Axiom % 81.75/81.95 52. ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) ### And 51 % 81.75/81.95 53. (((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) \/ ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All N, (((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point T_0 T_2) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) N)) \/ ((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) 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 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 (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines (orthogonal_through_point T_0 T_2) N)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) ### DisjTree 10 32 50 52 % 81.75/81.96 54. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines (orthogonal_through_point T_0 T_2) N)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (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 N, (((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point T_0 T_2) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) 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 T_0 T_1) (orthogonal_through_point T_0 T_1))) ### All 53 % 81.75/81.96 55. (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_2) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) M)) => (((convergent_lines (orthogonal_through_point T_0 T_2) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_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 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 (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines (orthogonal_through_point T_0 T_2) N)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) ### All 54 % 81.75/81.96 56. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines (orthogonal_through_point T_0 T_2) N)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (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 (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) ### All 55 % 81.75/81.96 57. (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_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 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 (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines (orthogonal_through_point T_0 T_2) N)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) ### All 56 % 81.75/81.96 58. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines (orthogonal_through_point T_0 T_2) N)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (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 (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) ### All 57 % 81.75/81.96 59. (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_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 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 (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) ### All 58 % 81.76/81.99 60. (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (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 (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) ### All 59 % 81.76/81.99 61. ((distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) => ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) \/ ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_1)) \/ (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_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 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 (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (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)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2))) (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) ### DisjTree 7 8 9 60 % 81.76/81.99 62. (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Z) => ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (orthogonal_through_point T_0 T_2) Z))))) (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2))) (-. (apart_point_and_line T_1 (orthogonal_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 L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (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 (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) ### All 61 % 81.76/81.99 63. (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) ### Axiom % 81.76/81.99 64. ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) \/ (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_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 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 (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2))) (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Z) => ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (orthogonal_through_point T_0 T_2) Z))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### DisjTree 6 62 63 % 81.76/81.99 65. (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Z) => ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (orthogonal_through_point T_0 T_2) Z))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2))) (-. (apart_point_and_line T_1 (orthogonal_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 L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (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 (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) ### All 64 % 81.81/82.03 66. (All X, (-. (convergent_lines X X))) (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, (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 (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2))) (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Z) => ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) \/ ((apart_point_and_line T_1 Z) \/ (convergent_lines (orthogonal_through_point T_0 T_2) Z))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z)))) ### All 65 % 81.81/82.03 67. (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 Z, ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2))) (-. (apart_point_and_line T_1 (orthogonal_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 L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (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 X, (-. (convergent_lines X X))) ### All 66 % 81.81/82.03 68. (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, (-. (convergent_lines X X))) (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, (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 (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z)))) ### All 67 % 81.81/82.03 69. (distinct_points T_1 T_2) (-. (distinct_points T_1 T_2)) ### Axiom % 81.81/82.03 70. (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) ### Axiom % 81.81/82.03 71. (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1)) ### Axiom % 81.81/82.03 72. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2))) ### Axiom % 81.81/82.03 73. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) ### Axiom % 81.81/82.03 74. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) ### Axiom % 81.81/82.03 75. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) ### Axiom % 81.81/82.03 76. (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) ### Axiom % 81.81/82.03 77. (-. (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) ### NotNot 76 % 81.81/82.03 78. (-. ((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) ### NotOr 77 % 81.81/82.03 79. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### Axiom % 81.81/82.03 80. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### Axiom % 81.81/82.03 81. (-. (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 % 81.81/82.03 82. ((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))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) ### And 81 % 81.81/82.03 83. (((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_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))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### DisjTree 79 80 52 82 % 81.84/82.07 84. (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) => (((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 (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 T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (convergent_lines (orthogonal_through_point T_0 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 83 % 81.84/82.07 85. (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_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) (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### All 84 % 81.84/82.07 86. (-. (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (convergent_lines (orthogonal_through_point T_0 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 M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) ### NotNot 85 % 81.84/82.07 87. (-. ((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_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 T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_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) (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### NotOr 86 % 81.84/82.07 88. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) 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))) ### Axiom % 81.84/82.07 89. (-. (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 % 81.84/82.07 90. (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### Axiom % 81.84/82.07 91. ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) 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 (orthogonal_through_point T_0 T_1) T_1)) \/ (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) (-. (convergent_lines (orthogonal_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) (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 T_0 T_1)) ### DisjTree 88 89 90 % 81.84/82.07 92. (All Z, ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) => ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z)))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) 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 (orthogonal_through_point T_0 T_1) T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### All 91 % 81.84/82.07 93. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### Axiom % 81.84/82.07 94. ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))))) => ((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) \/ (-. (unorthogonal_lines (orthogonal_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) (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 T_0 T_1)) (All Z, ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) => ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z)))) (unorthogonal_lines (orthogonal_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) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) ### DisjTree 78 87 92 93 % 81.84/82.10 95. (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)))) /\ ((-. (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 T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_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) (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All Z, ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) => ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z)))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) 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 (orthogonal_through_point T_0 T_1) T_1))) ### All 94 % 81.84/82.10 96. (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))))) (-. (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 T_0 T_1)) (unorthogonal_lines (orthogonal_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) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)))) /\ ((-. (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 T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))))) ### All 95 % 81.84/82.10 97. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)))) /\ ((-. (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 T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_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) (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_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) (orthogonal_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 96 % 81.84/82.10 98. (All X, (-. (convergent_lines X X))) (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 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) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)))) /\ ((-. (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 T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 97 % 81.84/82.10 99. (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_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)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_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) (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_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) (orthogonal_through_point T_0 T_1)) (All X, (-. (convergent_lines X X))) ### All 98 % 81.84/82.10 100. (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 X, (-. (convergent_lines X X))) (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 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) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_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 99 % 81.93/82.14 101. (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### Axiom % 81.93/82.14 102. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))) ### Axiom % 81.93/82.14 103. (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)))) => (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_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) (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)) (All X, (-. (convergent_lines X X))) (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))))))) ### DisjTree 100 101 102 % 81.93/82.14 104. (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) 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 X, (-. (convergent_lines X X))) (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 M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) ### All 103 % 81.93/82.14 105. (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_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) (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)) (All X, (-. (convergent_lines X X))) (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 104 % 81.93/82.14 106. (-. (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_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 X, (-. (convergent_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) ### NotNot 105 % 81.93/82.14 107. (-. (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 % 81.93/82.14 108. (-. (-. (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))) ### NotNot 107 % 81.93/82.14 109. (-. ((-. (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 M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_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) (orthogonal_through_point T_0 T_1))) (All X, (-. (convergent_lines X X))) (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))))))) ### NotAnd 106 108 % 81.93/82.17 110. (All M, (-. ((-. (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))))) (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 X, (-. (convergent_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) ### All 109 % 81.93/82.17 111. (-. (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_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) (orthogonal_through_point T_0 T_1))) (All X, (-. (convergent_lines X X))) (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, (-. ((-. (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))))) ### NotNot 110 % 81.93/82.17 112. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2))) ### Axiom % 81.93/82.17 113. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### Axiom % 81.93/82.17 114. (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2)) ### Axiom % 81.93/82.17 115. (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (-. (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))) ### Axiom % 81.93/82.17 116. (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) ### Axiom % 81.93/82.17 117. ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) ### Imply 115 116 % 81.93/82.17 118. (All Y, ((distinct_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) Y))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))) ### All 117 % 81.93/82.17 119. ((distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2)) \/ (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)))) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))) (All Y, ((distinct_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) Y))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### DisjTree 113 114 118 % 81.93/82.17 120. (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All Y, ((distinct_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) Y))) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))) ### All 119 % 81.99/82.21 121. (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))) (All Y, ((distinct_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) Y))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### All 120 % 81.99/82.21 122. (-. (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 % 81.99/82.21 123. ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) (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)))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All Y, ((distinct_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) Y))) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) ### DisjTree 112 121 122 % 81.99/82.21 124. (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))) (All Y, ((distinct_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) Y))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### All 123 % 81.99/82.21 125. (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All Y, ((distinct_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) Y))) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) ### All 124 % 81.99/82.21 126. (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2)) ### Axiom % 81.99/82.21 127. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))) ### Axiom % 81.99/82.21 128. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))) ### Axiom % 81.99/82.21 129. (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) ### Axiom % 81.99/82.21 130. ((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) ### And 129 % 81.99/82.21 131. (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))) => (((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 (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (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))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) ### DisjTree 127 128 82 130 % 81.99/82.21 132. (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))) => (((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 T_0 T_2) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (-. (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 T_0 T_2) (orthogonal_through_point T_0 T_1))) ### All 131 % 81.99/82.21 133. (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)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (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))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) ### All 132 % 81.99/82.24 134. (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 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (-. (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 T_0 T_2) (orthogonal_through_point T_0 T_1))) ### All 133 % 81.99/82.24 135. ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2)) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (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))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (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, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (All Y, ((distinct_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) Y))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) ### DisjTree 125 126 134 % 81.99/82.24 136. (All Z, ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All Y, ((distinct_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) Y))) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line 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))))))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (-. (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 T_0 T_2) (orthogonal_through_point T_0 T_1))) ### All 135 % 81.99/82.24 137. (All Y, (All Z, ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (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))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (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, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (All Y, ((distinct_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) Y))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) ### All 136 % 81.99/82.24 138. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All Y, ((distinct_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) Y))) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line 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))))))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (-. (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 T_0 T_2) (orthogonal_through_point T_0 T_1))) ### All 137 % 81.99/82.24 139. (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (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))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)) (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, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 138 % 82.07/82.28 140. (-. (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line 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))))))) (-. (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 T_0 T_2) (orthogonal_through_point T_0 T_1))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### NotNot 139 % 82.07/82.28 141. (-. ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_2))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_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 Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 M, (-. ((-. (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))))) (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 X, (-. (convergent_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) ### NotAnd 111 140 % 82.07/82.28 142. (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_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) (orthogonal_through_point T_0 T_1))) (All X, (-. (convergent_lines X X))) (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, (-. ((-. (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))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line 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))))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### All 141 % 82.07/82.28 143. (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_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 Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 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 X, (-. (convergent_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) ### All 142 % 82.07/82.31 144. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) ### Axiom % 82.07/82.31 145. (distinct_points T_1 T_2) (-. (distinct_points T_1 T_2)) ### Axiom % 82.07/82.31 146. (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (-. (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1))) ### Axiom % 82.07/82.31 147. (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) ### Axiom % 82.07/82.31 148. (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1)) ### Axiom % 82.07/82.31 149. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) ### Axiom % 82.07/82.31 150. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) ### Axiom % 82.07/82.31 151. (distinct_points T_2 T_1) (-. (distinct_points T_2 T_1)) ### Axiom % 82.07/82.31 152. (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (-. (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1))) ### Axiom % 82.07/82.31 153. (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) ### Axiom % 82.07/82.31 154. (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1)) ### Axiom % 82.07/82.31 155. (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) ### Axiom % 82.07/82.31 156. (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1)) ### Axiom % 82.07/82.31 157. (((distinct_points T_2 T_1) /\ (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1))) => ((apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) \/ ((apart_point_and_line T_2 (orthogonal_through_point T_0 T_1)) \/ ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) \/ (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (distinct_points T_2 T_1) ### DisjTree 151 152 153 154 155 156 % 82.07/82.31 158. (All V, (((distinct_points T_2 T_1) /\ (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) V)) => ((apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_2 T_1) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) ### All 157 % 82.07/82.31 159. (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (distinct_points T_2 T_1) ### All 158 % 82.07/82.31 160. (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) ### Axiom % 82.07/82.31 161. ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) => ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) \/ (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (distinct_points T_2 T_1) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) ### DisjTree 150 159 160 % 82.07/82.31 162. (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) => ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (distinct_points T_2 T_1) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) ### All 161 % 82.07/82.31 163. (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (distinct_points T_2 T_1) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) ### All 162 % 82.15/82.36 164. (All Y, (All U, (All V, (((distinct_points T_2 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (distinct_points T_2 T_1) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) ### All 163 % 82.15/82.36 165. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (distinct_points T_2 T_1) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) ### All 164 % 82.15/82.36 166. (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) ### Axiom % 82.15/82.36 167. ((apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) => ((distinct_points T_2 T_1) \/ (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) ### DisjTree 149 165 166 % 82.15/82.36 168. (All Z, ((apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))))) (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) ### All 167 % 82.15/82.36 169. (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) ### All 168 % 82.15/82.36 170. (-. (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)))) (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) ### NotNot 169 % 82.15/82.36 171. (-. ((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) ### NotOr 170 % 82.15/82.38 172. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1))) ### Axiom % 82.15/82.38 173. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1))) ### Axiom % 82.15/82.38 174. ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))))) => ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1))))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) ### DisjTree 171 78 172 173 % 82.15/82.38 175. (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)))) /\ ((-. (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 (orthogonal_through_point T_0 T_2) T_2) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) ### All 174 % 82.15/82.38 176. (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_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_2) T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) ### All 175 % 82.15/82.38 177. (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1)) ### Axiom % 82.15/82.38 178. (((distinct_points T_1 T_2) /\ (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1))) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) \/ ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_1)) \/ ((apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) \/ (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1)))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (distinct_points T_1 T_2) ### DisjTree 145 146 147 148 176 177 % 82.15/82.43 179. (All V, (((distinct_points T_1 T_2) /\ (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) V)) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) \/ (apart_point_and_line T_2 V)))))) (distinct_points T_1 T_2) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_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_2) T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) ### All 178 % 82.15/82.43 180. (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) ### Axiom % 82.15/82.43 181. ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) => ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) \/ (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (distinct_points T_1 T_2) (All V, (((distinct_points T_1 T_2) /\ (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) V)) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) \/ (apart_point_and_line T_2 V)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) ### DisjTree 144 179 180 % 82.15/82.43 182. (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) => ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (All V, (((distinct_points T_1 T_2) /\ (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) V)) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) \/ (apart_point_and_line T_2 V)))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_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_2) T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) ### All 181 % 82.25/82.44 183. (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (distinct_points T_1 T_2) (All V, (((distinct_points T_1 T_2) /\ (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) V)) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) \/ (apart_point_and_line T_2 V)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) ### All 182 % 82.25/82.44 184. (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_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_2) T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) ### All 183 % 82.25/82.44 185. (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (-. (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) ### Axiom % 82.25/82.44 186. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) ### Axiom % 82.25/82.44 187. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) ### Axiom % 82.25/82.44 188. (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2)) ### Axiom % 82.25/82.44 189. ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) ### And 188 % 82.25/82.44 190. (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) ### Axiom % 82.25/82.44 191. ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) ### And 190 % 82.25/82.44 192. (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) ### DisjTree 186 187 189 191 % 82.32/82.50 193. (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) ### All 192 % 82.32/82.50 194. (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) ### All 193 % 82.32/82.50 195. ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) ### Imply 185 194 % 82.32/82.50 196. (All Y, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) ### All 195 % 82.32/82.50 197. (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2)) ### Axiom % 82.32/82.50 198. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) ### Axiom % 82.32/82.50 199. (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2)))) => (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (All Y, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y))) ### DisjTree 196 197 198 % 82.32/82.50 200. (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N))) => (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) N)))) (All Y, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) ### All 199 % 82.32/82.53 201. (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N))) => (-. (convergent_lines M N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (All Y, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y))) ### All 200 % 82.32/82.53 202. (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) ### Axiom % 82.32/82.53 203. ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) => ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) \/ (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)))) (All Y, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y))) (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N))) => (-. (convergent_lines M N))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (distinct_points T_1 T_2) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) ### DisjTree 184 201 202 % 82.32/82.53 204. (All Z, ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2)) => ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Z) \/ (apart_point_and_line T_1 Z)))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_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_2) T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N))) => (-. (convergent_lines M N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y))) ### All 203 % 82.39/82.58 205. (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Y, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y))) (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N))) => (-. (convergent_lines M N))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (distinct_points T_1 T_2) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) ### All 204 % 82.39/82.58 206. (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_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_2) T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) ### All 205 % 82.39/82.58 207. (-. (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Y, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y))) (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) ### NotNot 206 % 82.39/82.60 208. (-. ((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_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_2) T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) ### NotOr 207 % 82.39/82.60 209. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1))) ### Axiom % 82.39/82.60 210. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1))) ### Axiom % 82.39/82.60 211. ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))))) => ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1))))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Y, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y))) (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) ### DisjTree 208 78 209 210 % 82.39/82.64 212. (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1)))) /\ ((-. (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 (orthogonal_through_point T_0 T_2) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) N))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_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_2) T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) ### All 211 % 82.39/82.64 213. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Y, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y))) (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) ### All 212 % 82.39/82.66 214. (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, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_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_2) T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All Y, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) ### All 213 % 82.39/82.66 215. (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Y, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L 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 214 % 82.39/82.66 216. (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, (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, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All Y, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) ### All 215 % 82.53/82.70 217. (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Y, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y) => (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) Y))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L 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 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 216 % 82.53/82.70 218. (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (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, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) ### All 217 % 82.53/82.70 219. (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) ### Axiom % 82.53/82.70 220. ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) \/ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L 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 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### Or 218 219 % 82.53/82.72 221. (All M, ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) M) \/ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) M))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (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, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) ### All 220 % 82.53/82.72 222. (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L 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 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### All 221 % 82.53/82.72 223. (-. (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (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, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) ### NotNot 222 % 82.56/82.77 224. (-. (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L 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 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)))) ### NotNot 223 % 82.56/82.77 225. (-. ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (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, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) ### NotOr 224 % 82.56/82.77 226. (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2)) ### Axiom % 82.56/82.77 227. (-. (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2)))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) ### NotNot 226 % 82.56/82.77 228. (-. ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) ### NotOr 227 % 82.56/82.77 229. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) ### Axiom % 82.56/82.77 230. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) ### Axiom % 82.56/82.77 231. ((((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))))) => ((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))))) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L 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 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### DisjTree 225 228 229 230 % 82.56/82.78 232. (All N, ((((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)))) => ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (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, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### All 231 % 82.56/82.78 233. (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) M))) /\ ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L 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 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### All 232 % 82.63/82.80 234. (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (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, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### All 233 % 82.63/82.80 235. (-. (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)))) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L 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 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### NotNot 234 % 82.63/82.80 236. (-. (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (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, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)))) ### NotNot 235 % 82.63/82.84 237. (-. ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1))))) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L 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 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### NotOr 236 % 82.63/82.84 238. (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)) ### Axiom % 82.63/82.84 239. (-. (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2)))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) ### NotNot 238 % 82.63/82.84 240. (-. ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) ### NotOr 239 % 82.63/82.84 241. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) ### Axiom % 82.63/82.84 242. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) ### Axiom % 82.63/82.84 243. ((((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))))) => ((-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (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, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### DisjTree 237 240 241 242 % 82.63/82.84 244. (All N, ((((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1))) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_1)))) /\ ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) N)))) => ((-. (convergent_lines (orthogonal_through_point T_0 T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))))) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L 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 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### All 243 % 82.63/82.85 245. (All M, (All N, ((((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) M)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) M))) /\ ((-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) N)) \/ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (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, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### All 244 % 82.63/82.85 246. (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L 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 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### All 245 % 82.63/82.85 247. ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (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, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### And 246 % 82.70/82.89 248. (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L 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 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) ### All 247 % 82.70/82.89 249. (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (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, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) ### All 248 % 82.70/82.89 250. (((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) => (((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) \/ ((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_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) (orthogonal_through_point T_0 T_1))) (All X, (-. (convergent_lines X X))) (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 X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line 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))))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### DisjTree 75 143 249 36 % 82.70/82.91 251. (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) => (((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 T_0 T_2) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) N))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_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 Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 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 X, (-. (convergent_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_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 M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) ### All 250 % 82.70/82.91 252. (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_1) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) => (((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_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) (orthogonal_through_point T_0 T_1))) (All X, (-. (convergent_lines X X))) (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 X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line 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))))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### All 251 % 82.70/82.92 253. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_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 Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 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 X, (-. (convergent_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_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 M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) ### All 252 % 82.70/82.92 254. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) ### Axiom % 82.70/82.92 255. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) ### Axiom % 82.70/82.92 256. (((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) \/ ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L 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 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) ### DisjTree 254 255 36 247 % 82.70/82.96 257. (All N, (((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point T_0 T_2) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) 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 T_0 T_2) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (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, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### All 256 % 82.70/82.96 258. (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_2) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) M)) => (((convergent_lines (orthogonal_through_point T_0 T_2) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L 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 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) ### All 257 % 82.70/82.96 259. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (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, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_1) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### All 258 % 82.80/82.98 260. (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_2) T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L 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 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) ### All 259 % 82.80/82.98 261. (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (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, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) ### All 260 % 82.80/82.98 262. ((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L 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 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) ### And 261 % 82.80/82.99 263. (((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) \/ ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines (orthogonal_through_point T_0 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 X, (-. (convergent_lines X X))) (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 X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) ### DisjTree 74 253 262 52 % 82.80/82.99 264. (All N, (((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point T_0 T_2) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) 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 T_0 T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 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 X, (-. (convergent_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_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 M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) ### All 263 % 82.82/83.04 265. (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All X, (-. (convergent_lines X X))) (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 X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) => (((convergent_lines (orthogonal_through_point T_0 T_2) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) N)) \/ ((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))))) ### All 264 % 82.82/83.04 266. (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_2) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) M)) => (((convergent_lines (orthogonal_through_point T_0 T_2) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 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 X, (-. (convergent_lines X X))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_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 M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) ### All 265 % 82.82/83.04 267. (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All X, (-. (convergent_lines X X))) (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 X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (All M, (All N, (((convergent_lines (orthogonal_through_point T_0 T_2) M) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) M)) => (((convergent_lines (orthogonal_through_point T_0 T_2) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_2) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) ### All 266 % 82.87/83.05 268. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 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 X, (-. (convergent_lines X X))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_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 M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_2) L))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) ### All 267 % 82.87/83.05 269. (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))) => (-. (convergent_lines M N))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (All X, (-. (convergent_lines X X))) (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 X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) ### All 268 % 82.87/83.07 270. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 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 X, (-. (convergent_lines X X))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_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 L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) ### All 269 % 82.87/83.07 271. (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_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, (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 X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All Y, (All Z, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (distinct_lines Y Z))))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) ### All 270 % 82.87/83.07 272. (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 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 X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) ### All 271 % 82.87/83.09 273. ((distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) => (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_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, (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 X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) ### Imply 73 272 % 82.87/83.09 274. (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 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 X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) ### All 273 % 82.87/83.11 275. (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1)) ### Axiom % 82.87/83.11 276. ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_1)) \/ (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1)))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_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, (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 X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line 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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) ### DisjTree 72 274 275 % 82.87/83.11 277. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 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 X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) ### All 276 % 82.87/83.11 278. (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_points T_1 T_2) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_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, (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 X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) ### All 277 % 82.94/83.12 279. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 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 X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_2) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) ### All 278 % 82.94/83.12 280. (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1)) ### Axiom % 82.94/83.12 281. (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2)) ### Axiom % 82.94/83.12 282. (((distinct_points T_1 T_2) /\ (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) => ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_1)) \/ ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) \/ ((apart_point_and_line T_2 (orthogonal_through_point T_0 T_1)) \/ (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2)))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_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, (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 X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (distinct_points T_1 T_2) ### DisjTree 69 70 71 279 280 281 % 82.94/83.12 283. (All V, (((distinct_points T_1 T_2) /\ (distinct_lines (orthogonal_through_point T_0 T_1) V)) => ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_1)) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 (orthogonal_through_point T_0 T_1)) \/ (apart_point_and_line T_2 V)))))) (distinct_points T_1 T_2) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 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 X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) ### All 282 % 82.94/83.16 284. (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All U, (All V, (((distinct_points T_1 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_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, (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 X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (distinct_points T_1 T_2) ### All 283 % 82.94/83.16 285. (All Y, (All U, (All V, (((distinct_points T_1 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points T_1 T_2) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X 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))))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 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 X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) ### All 284 % 82.94/83.16 286. (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_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, (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 X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (distinct_points T_1 T_2) ### All 285 % 83.01/83.17 287. (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2)) ### Axiom % 83.01/83.17 288. ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_points T_1 T_2) \/ (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2)))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (apart_point_and_line T_1 (orthogonal_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 L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (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 X, (-. (convergent_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))))))) ### DisjTree 68 286 287 % 83.01/83.17 289. (All Z, ((apart_point_and_line T_1 (orthogonal_through_point T_0 T_2)) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z (orthogonal_through_point T_0 T_2))))) (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, (-. (convergent_lines X X))) (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, (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 (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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)))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z)))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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 L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### All 288 % 83.01/83.19 290. (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (apart_point_and_line T_1 (orthogonal_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 L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (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 X, (-. (convergent_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 289 % 83.01/83.19 291. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (convergent_lines X X))) (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, (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 (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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)))))) (-. 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(apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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 L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### All 290 % 83.01/83.19 292. (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z)))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (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)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (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 X, (-. (convergent_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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 291 % 83.01/83.19 293. (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (convergent_lines X X))) (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, (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 (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) => ((distinct_lines (orthogonal_through_point T_0 T_2) Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z)))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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)))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### All 292 % 83.01/83.19 294. (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (orthogonal_through_point T_0 T_1) Z))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (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)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (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 X, (-. (convergent_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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) ### All 293 % 83.01/83.19 295. (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (convergent_lines X X))) (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, (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 (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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)))))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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)))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) ### All 294 % 83.01/83.22 296. (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2)) ### Axiom % 83.01/83.22 297. ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) => ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) \/ (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) (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)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (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 X, (-. (convergent_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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### DisjTree 5 295 296 % 83.01/83.22 298. (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) => ((convergent_lines (orthogonal_through_point T_0 T_1) Z) \/ (convergent_lines (orthogonal_through_point T_0 T_2) Z)))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (convergent_lines X X))) (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, (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 (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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)))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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)))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 297 % 83.01/83.22 299. (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((convergent_lines (orthogonal_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) (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)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (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 X, (-. (convergent_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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### All 298 % 83.01/83.23 300. (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (convergent_lines X X))) (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, (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 (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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)))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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)))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 299 % 83.01/83.23 301. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_2))) (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)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (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 X, (-. (convergent_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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### All 300 % 83.01/83.23 302. (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (convergent_lines X X))) (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, (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 (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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 L, (-. (apart_point_and_line T_2 (orthogonal_through_point L T_2)))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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)))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 301 % 83.01/83.24 303. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, ((distinct_lines (orthogonal_through_point T_0 T_2) Y) => (convergent_lines (orthogonal_through_point T_0 T_2) Y))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_2 (orthogonal_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 L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (convergent_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (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 X, (-. (convergent_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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### All 302 % 83.01/83.24 304. (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (convergent_lines X X))) (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, (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 (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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)))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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)))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 303 % 83.01/83.24 305. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (-. (distinct_lines (orthogonal_through_point T_0 T_2) (orthogonal_through_point T_0 T_2))) (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, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_2 (orthogonal_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 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, (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 X, (-. (convergent_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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### All 304 % 83.01/83.24 306. (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (convergent_lines X X))) (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, (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, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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 L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 305 % 83.01/83.24 307. (-. (-. (distinct_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (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, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_2 (orthogonal_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 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, (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 X, (-. (convergent_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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) ### NotNot 306 % 83.01/83.24 308. (-. (equal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (convergent_lines X X))) (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, (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, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (-. (apart_point_and_line T_2 (orthogonal_through_point T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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 L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### Definition-Pseudo(equal_lines) 307 % 83.01/83.24 309. (incident_point_and_line T_2 (orthogonal_through_point T_0 T_1)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (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, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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 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, (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 X, (-. (convergent_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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (-. (equal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2))) ### Definition-Pseudo(incident_point_and_line) 308 % 83.01/83.25 310. (-. ((incident_point_and_line T_2 (orthogonal_through_point T_0 T_1)) => (equal_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_2)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (convergent_lines X X))) (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, (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, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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 L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### NotImply 309 % 83.01/83.25 311. (-. (All L, ((incident_point_and_line T_2 (orthogonal_through_point L T_1)) => (equal_lines (orthogonal_through_point L T_1) (orthogonal_through_point L T_2))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (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, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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 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, (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 X, (-. (convergent_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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) ### NotAllEx 310 % 83.01/83.25 312. (-. (All B, (All L, ((incident_point_and_line B (orthogonal_through_point L T_1)) => (equal_lines (orthogonal_through_point L T_1) (orthogonal_through_point L B)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (convergent_lines X X))) (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, (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, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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 L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### NotAllEx 311 % 83.01/83.25 313. (-. (All A, (All B, (All L, ((incident_point_and_line B (orthogonal_through_point L A)) => (equal_lines (orthogonal_through_point L A) (orthogonal_through_point L B))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (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, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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 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, (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 X, (-. (convergent_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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) ### NotAllEx 312 % 83.01/83.25 % SZS output end Proof % 83.01/83.25 (* END-PROOF *) %------------------------------------------------------------------------------