%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO191+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 : n023.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 06:30:52 EDT 2022 % Result : Theorem 5.29s 5.53s % Output : Proof 6.42s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.03/0.12 % Problem : GEO191+3 : TPTP v8.1.0. Released v4.0.0. % 0.03/0.12 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.12/0.33 % Computer : n023.cluster.edu % 0.12/0.33 % Model : x86_64 x86_64 % 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.33 % Memory : 8042.1875MB % 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.33 % CPULimit : 300 % 0.12/0.33 % WCLimit : 600 % 0.12/0.33 % DateTime : Sat Jun 18 06:26:04 EDT 2022 % 0.12/0.34 % CPUTime : % 5.29/5.53 % SZS status Theorem % 5.29/5.53 (* PROOF-FOUND *) % 5.29/5.53 (* BEGIN-PROOF *) % 5.29/5.53 % SZS output start Proof % 5.29/5.53 1. (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) ### Axiom % 5.29/5.53 2. (distinct_lines T_2 T_1) (-. (distinct_lines T_2 T_1)) ### Axiom % 5.29/5.53 3. (-. (convergent_lines T_2 T_1)) (convergent_lines T_2 T_1) ### Axiom % 5.29/5.53 4. ((distinct_lines T_2 T_1) => (convergent_lines T_2 T_1)) (-. (convergent_lines T_2 T_1)) (distinct_lines T_2 T_1) ### Imply 2 3 % 5.29/5.53 5. (All Y, ((distinct_lines T_2 Y) => (convergent_lines T_2 Y))) (distinct_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### All 4 % 5.29/5.53 6. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 5.29/5.53 7. (apart_point_and_line (intersection_point T_0 T_1) T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) ### Axiom % 5.29/5.53 8. ((convergent_lines T_0 T_1) => (-. (apart_point_and_line (intersection_point T_0 T_1) T_1))) (apart_point_and_line (intersection_point T_0 T_1) T_1) (convergent_lines T_0 T_1) ### Imply 6 7 % 5.29/5.53 9. (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) Y)))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### All 8 % 5.29/5.53 10. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_1) (convergent_lines T_0 T_1) ### All 9 % 5.29/5.53 11. ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 T_1) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_2 T_1)) (All Y, ((distinct_lines T_2 Y) => (convergent_lines T_2 Y))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### DisjTree 1 5 10 % 5.29/5.53 12. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Y, ((distinct_lines T_2 Y) => (convergent_lines T_2 Y))) (-. (convergent_lines T_2 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) ### All 11 % 5.29/5.53 13. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_2 T_1)) (All Y, ((distinct_lines T_2 Y) => (convergent_lines T_2 Y))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### All 12 % 5.29/5.53 14. (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 (intersection_point T_0 T_1) T_2) (All Y, ((distinct_lines T_2 Y) => (convergent_lines T_2 Y))) (-. (convergent_lines T_2 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) ### All 13 % 5.29/5.53 15. (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_2 T_1)) (apart_point_and_line (intersection_point T_0 T_1) 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 14 % 5.29/5.53 16. (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) ### Axiom % 5.29/5.53 17. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 5.29/5.53 18. (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) ### Axiom % 5.29/5.53 19. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1))) ### Axiom % 5.29/5.53 20. (distinct_lines T_1 T_0) (-. (distinct_lines T_1 T_0)) ### Axiom % 5.29/5.53 21. (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### Axiom % 5.29/5.53 22. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 5.29/5.53 23. (apart_point_and_line (intersection_point T_0 T_1) T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) ### Axiom % 5.29/5.53 24. ((convergent_lines T_0 T_1) => (-. (apart_point_and_line (intersection_point T_0 T_1) T_0))) (apart_point_and_line (intersection_point T_0 T_1) T_0) (convergent_lines T_0 T_1) ### Imply 22 23 % 5.29/5.53 25. (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### All 24 % 5.29/5.53 26. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom % 5.29/5.53 27. (apart_point_and_line (intersection_point T_2 T_1) T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) ### Axiom % 5.29/5.53 28. ((convergent_lines T_2 T_1) => (-. (apart_point_and_line (intersection_point T_2 T_1) T_1))) (apart_point_and_line (intersection_point T_2 T_1) T_1) (convergent_lines T_2 T_1) ### Imply 26 27 % 5.29/5.53 29. (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) Y)))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_2 T_1) T_1) ### All 28 % 5.29/5.53 30. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_2 T_1) T_1) (convergent_lines T_2 T_1) ### All 29 % 5.29/5.53 31. (apart_point_and_line (intersection_point T_2 T_1) T_0) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) ### Axiom % 5.29/5.53 32. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 5.29/5.53 33. (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (-. (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1))) ### Axiom % 5.29/5.53 34. (distinct_lines T_1 T_0) (-. (distinct_lines T_1 T_0)) ### Axiom % 5.29/5.53 35. (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (apart_point_and_line (intersection_point T_2 T_1) T_1) ### Axiom % 5.29/5.53 36. (apart_point_and_line (intersection_point T_2 T_1) T_0) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) ### Axiom % 5.29/5.53 37. (distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_3)) (-. (distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_3))) ### Axiom % 5.29/5.53 38. (distinct_lines T_2 T_1) (-. (distinct_lines T_2 T_1)) ### Axiom % 5.29/5.53 39. (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (apart_point_and_line (intersection_point T_2 T_1) T_2) ### Axiom % 5.29/5.53 40. (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (apart_point_and_line (intersection_point T_2 T_1) T_1) ### Axiom % 5.29/5.53 41. (convergent_lines T_2 T_3) (-. (convergent_lines T_2 T_3)) ### Axiom % 5.29/5.53 42. (apart_point_and_line (intersection_point T_2 T_3) T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) T_2)) ### Axiom % 5.29/5.53 43. ((convergent_lines T_2 T_3) => (-. (apart_point_and_line (intersection_point T_2 T_3) T_2))) (apart_point_and_line (intersection_point T_2 T_3) T_2) (convergent_lines T_2 T_3) ### Imply 41 42 % 5.29/5.53 44. (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (apart_point_and_line (intersection_point T_2 T_3) T_2) ### All 43 % 5.29/5.53 45. (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (apart_point_and_line (intersection_point T_2 T_3) T_1) ### Axiom % 5.29/5.53 46. (((distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines T_2 T_1)) => ((apart_point_and_line (intersection_point T_2 T_1) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_2) \/ (apart_point_and_line (intersection_point T_2 T_3) T_1))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_3)) ### DisjTree 37 38 39 40 44 45 % 5.58/5.79 47. (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines T_2 V)) => ((apart_point_and_line (intersection_point T_2 T_1) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_2) \/ (apart_point_and_line (intersection_point T_2 T_3) V)))))) (distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_3)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) ### All 46 % 5.58/5.79 48. (All U, (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) U) \/ (apart_point_and_line (intersection_point T_2 T_3) V))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_3)) ### All 47 % 5.58/5.79 49. (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_3)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) ### All 48 % 5.58/5.79 50. (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (apart_point_and_line (intersection_point T_2 T_3) T_0) ### Axiom % 5.58/5.79 51. ((apart_point_and_line (intersection_point T_2 T_1) T_0) => ((distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_3)) \/ (apart_point_and_line (intersection_point T_2 T_3) T_0))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) ### DisjTree 36 49 50 % 5.58/5.79 52. (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_0) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_0)))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) ### All 51 % 5.58/5.79 53. (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) ### All 52 % 5.58/5.79 54. (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### Axiom % 5.58/5.79 55. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (apart_point_and_line (intersection_point T_0 T_1) T_0) (convergent_lines T_0 T_1) ### All 25 % 5.58/5.79 56. (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (distinct_lines T_1 T_0)) => ((apart_point_and_line (intersection_point T_2 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ (apart_point_and_line (intersection_point T_0 T_1) T_0))))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) ### DisjTree 33 34 35 53 54 55 % 5.58/5.79 57. (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (distinct_lines T_1 V)) => ((apart_point_and_line (intersection_point T_2 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ (apart_point_and_line (intersection_point T_0 T_1) V)))))) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) ### All 56 % 5.67/5.84 58. (All U, (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) ### All 57 % 5.67/5.84 59. (convergent_lines T_0 T_0) (-. (convergent_lines T_0 T_0)) ### Axiom % 5.67/5.84 60. (All X, (-. (convergent_lines X X))) (convergent_lines T_0 T_0) ### All 59 % 5.67/5.84 61. ((convergent_lines T_0 T_1) => ((distinct_lines T_1 T_0) \/ (convergent_lines T_0 T_0))) (All X, (-. (convergent_lines X X))) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All U, (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (convergent_lines T_0 T_1) ### DisjTree 32 58 60 % 5.67/5.84 62. (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All U, (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (All X, (-. (convergent_lines X X))) ### All 61 % 5.67/5.84 63. (All X, (-. (convergent_lines X X))) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### All 62 % 5.67/5.84 64. (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (All X, (-. (convergent_lines X X))) ### All 63 % 5.67/5.84 65. (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### Axiom % 5.67/5.84 66. ((apart_point_and_line (intersection_point T_2 T_1) T_0) => ((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) \/ (apart_point_and_line (intersection_point T_0 T_1) T_0))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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 (intersection_point T_2 T_1) T_0) ### DisjTree 31 64 65 % 5.67/5.87 67. (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_0) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_0)))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) ### All 66 % 5.67/5.87 68. (distinct_lines T_1 T_3) (-. (distinct_lines T_1 T_3)) ### Axiom % 5.67/5.87 69. (-. (convergent_lines T_1 T_3)) (convergent_lines T_1 T_3) ### Axiom % 5.67/5.87 70. ((distinct_lines T_1 T_3) => (convergent_lines T_1 T_3)) (-. (convergent_lines T_1 T_3)) (distinct_lines T_1 T_3) ### Imply 68 69 % 5.67/5.87 71. (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (distinct_lines T_1 T_3) (-. (convergent_lines T_1 T_3)) ### All 70 % 5.67/5.87 72. (-. (apart_point_and_line (intersection_point T_2 T_1) T_3)) (apart_point_and_line (intersection_point T_2 T_1) T_3) ### Axiom % 5.67/5.87 73. ((apart_point_and_line (intersection_point T_2 T_1) T_1) => ((distinct_lines T_1 T_3) \/ (apart_point_and_line (intersection_point T_2 T_1) T_3))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_3)) (-. (convergent_lines T_1 T_3)) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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 (intersection_point T_2 T_1) T_0) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_0) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_0)))) ### DisjTree 67 71 72 % 5.67/5.87 74. (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z)))) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_0) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_0)))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (-. (convergent_lines T_1 T_3)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_3)) ### All 73 % 5.67/5.87 75. (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) ### Axiom % 5.67/5.87 76. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1))) ### Axiom % 5.67/5.87 77. (distinct_lines T_3 T_1) (-. (distinct_lines T_3 T_1)) ### Axiom % 5.67/5.87 78. (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (apart_point_and_line (intersection_point T_0 T_1) T_3) ### Axiom % 5.67/5.87 79. (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### Axiom % 5.67/5.87 80. (apart_point_and_line (intersection_point T_2 T_1) T_3) (-. (apart_point_and_line (intersection_point T_2 T_1) T_3)) ### Axiom % 5.67/5.87 81. (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (apart_point_and_line (intersection_point T_0 T_1) T_3) ### Axiom % 5.67/5.87 82. ((apart_point_and_line (intersection_point T_2 T_1) T_3) => ((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) \/ (apart_point_and_line (intersection_point T_0 T_1) T_3))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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 (intersection_point T_2 T_1) T_3) ### DisjTree 80 64 81 % 5.67/5.88 83. (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_3) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_2 T_1) T_3) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) ### All 82 % 5.67/5.88 84. (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_1) ### Axiom % 5.67/5.88 85. ((apart_point_and_line (intersection_point T_2 T_1) T_1) => ((distinct_lines T_1 T_1) \/ (apart_point_and_line (intersection_point T_2 T_1) T_1))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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 (intersection_point T_2 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_3) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_3)))) ### DisjTree 83 84 30 % 5.67/5.88 86. (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z)))) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_3) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_2 T_1) T_3) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) ### All 85 % 5.67/5.88 87. (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z))))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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 (intersection_point T_2 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_3) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_3)))) ### All 86 % 5.67/5.88 88. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_3) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_2 T_1) T_3) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) ### All 87 % 5.73/5.91 89. (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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 (intersection_point T_2 T_1) T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 88 % 5.73/5.91 90. (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_2 T_1) T_3) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) ### All 89 % 5.73/5.91 91. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines T_3 T_1)) => ((apart_point_and_line (intersection_point T_0 T_1) T_3) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_2 T_1) T_3) \/ (apart_point_and_line (intersection_point T_2 T_1) T_1))))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 T_1)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (distinct_lines T_3 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) ### DisjTree 76 77 78 79 90 30 % 5.73/5.91 92. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines T_3 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_3) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) T_3) \/ (apart_point_and_line (intersection_point T_2 T_1) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) ### All 91 % 5.73/5.91 93. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) U) \/ (apart_point_and_line (intersection_point T_2 T_1) V))))))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 T_1)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (distinct_lines T_3 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) ### All 92 % 5.73/5.93 94. (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) ### All 93 % 5.73/5.93 95. (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 T_1)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (distinct_lines T_3 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) ### All 94 % 5.73/5.93 96. (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (apart_point_and_line (intersection_point T_2 T_1) T_2) ### Axiom % 5.73/5.93 97. ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) \/ (apart_point_and_line (intersection_point T_2 T_1) T_2))) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### DisjTree 75 95 96 % 5.73/5.93 98. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 T_1)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (distinct_lines T_3 T_1) ### All 97 % 5.73/5.93 99. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### All 98 % 5.80/5.96 100. (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 T_1)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (distinct_lines T_3 T_1) ### All 99 % 5.80/5.96 101. (-. (distinct_lines T_3 T_3)) (distinct_lines T_3 T_3) ### Axiom % 5.80/5.96 102. (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) ### Axiom % 5.80/5.96 103. (distinct_lines T_3 T_1) (-. (distinct_lines T_3 T_1)) ### Axiom % 5.80/5.96 104. (-. (convergent_lines T_3 T_1)) (convergent_lines T_3 T_1) ### Axiom % 5.80/5.96 105. ((distinct_lines T_3 T_1) => (convergent_lines T_3 T_1)) (-. (convergent_lines T_3 T_1)) (distinct_lines T_3 T_1) ### Imply 103 104 % 5.80/5.96 106. (All Y, ((distinct_lines T_3 Y) => (convergent_lines T_3 Y))) (distinct_lines T_3 T_1) (-. (convergent_lines T_3 T_1)) ### All 105 % 5.80/5.96 107. (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_3 T_1)) (distinct_lines T_3 T_1) ### All 106 % 5.80/5.96 108. (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### Axiom % 5.80/5.96 109. ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 T_1) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (convergent_lines T_3 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) ### DisjTree 102 107 108 % 5.80/5.96 110. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_3 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) ### All 109 % 5.80/5.97 111. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 5.80/5.97 112. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3))) ### Axiom % 5.80/5.97 113. (distinct_lines T_1 T_0) (-. (distinct_lines T_1 T_0)) ### Axiom % 5.80/5.97 114. (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### Axiom % 5.80/5.97 115. (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (apart_point_and_line (intersection_point T_2 T_3) T_1) ### Axiom % 5.80/5.97 116. (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (apart_point_and_line (intersection_point T_2 T_3) T_0) ### Axiom % 5.80/5.97 117. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines T_1 T_0)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_1) \/ (apart_point_and_line (intersection_point T_2 T_3) T_0))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) ### DisjTree 112 113 10 114 115 116 % 5.80/5.97 118. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines T_1 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_1) \/ (apart_point_and_line (intersection_point T_2 T_3) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (distinct_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) ### All 117 % 5.80/5.97 119. (-. (distinct_lines T_0 T_0)) (distinct_lines T_0 T_0) ### Axiom % 5.80/5.97 120. ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_lines T_0 T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) T_0))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (-. (distinct_lines T_0 T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines T_1 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_1) \/ (apart_point_and_line (intersection_point T_2 T_3) V)))))) ### DisjTree 118 119 25 % 5.80/5.97 121. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines T_1 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_1) \/ (apart_point_and_line (intersection_point T_2 T_3) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (distinct_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (distinct_lines T_0 T_0)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) ### All 120 % 5.80/5.97 122. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) U) \/ (apart_point_and_line (intersection_point T_2 T_3) V))))))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (-. (distinct_lines T_0 T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) ### All 121 % 5.80/5.98 123. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (distinct_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (distinct_lines T_0 T_0)) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) U) \/ (apart_point_and_line (intersection_point T_2 T_3) V))))))) ### All 122 % 5.80/5.98 124. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) U) \/ (apart_point_and_line (intersection_point T_2 T_3) V))))))) (-. (distinct_lines T_0 T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### All 123 % 5.80/5.98 125. (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (distinct_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) U) \/ (apart_point_and_line (intersection_point T_2 T_3) V))))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) ### All 124 % 5.80/5.98 126. (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_0) ### Axiom % 5.80/5.98 127. ((convergent_lines T_0 T_1) => ((distinct_lines T_1 T_0) \/ (convergent_lines T_0 T_0))) (-. (convergent_lines T_0 T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) U) \/ (apart_point_and_line (intersection_point T_2 T_3) V))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_1) ### DisjTree 111 125 126 % 5.80/5.98 128. (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) U) \/ (apart_point_and_line (intersection_point T_2 T_3) V))))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (convergent_lines T_0 T_0)) ### All 127 % 5.80/5.98 129. (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) U) \/ (apart_point_and_line (intersection_point T_2 T_3) V))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### All 128 % 5.80/5.98 130. (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All X, (-. (convergent_lines X X))) ### All 129 % 5.80/5.98 131. (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, (-. (convergent_lines X X))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### All 130 % 5.80/6.01 132. (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (apart_point_and_line (intersection_point T_2 T_3) T_1) ### Axiom % 5.80/6.01 133. ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) \/ (apart_point_and_line (intersection_point T_2 T_3) T_1))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All X, (-. (convergent_lines X X))) (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 T_3 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) ### DisjTree 110 131 132 % 5.80/6.01 134. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_1)))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_3 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, (-. (convergent_lines X X))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### All 133 % 5.80/6.01 135. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All X, (-. (convergent_lines X X))) (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 T_3 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) ### All 134 % 5.80/6.01 136. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_3 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, (-. (convergent_lines X X))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### All 135 % 5.80/6.01 137. (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All X, (-. (convergent_lines X X))) (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 T_3 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 136 % 5.80/6.01 138. (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_3 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, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### All 137 % 5.85/6.02 139. (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (convergent_lines T_0 T_1) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (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 T_3 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 138 % 5.85/6.02 140. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_3 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, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_1) ### All 139 % 5.85/6.02 141. (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) ### Axiom % 5.85/6.02 142. (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) ### Axiom % 5.85/6.02 143. (distinct_lines T_3 T_1) (-. (distinct_lines T_3 T_1)) ### Axiom % 5.85/6.02 144. (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (apart_point_and_line (intersection_point T_3 T_1) T_3) ### Axiom % 5.85/6.02 145. (apart_point_and_line (intersection_point T_3 T_1) T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_1)) ### Axiom % 5.85/6.02 146. (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_3)) (distinct_lines T_1 T_3) ### All 71 % 5.85/6.02 147. (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (apart_point_and_line (intersection_point T_3 T_1) T_3) ### Axiom % 5.85/6.02 148. ((apart_point_and_line (intersection_point T_3 T_1) T_1) => ((distinct_lines T_1 T_3) \/ (apart_point_and_line (intersection_point T_3 T_1) T_3))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (-. (convergent_lines T_1 T_3)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (apart_point_and_line (intersection_point T_3 T_1) T_1) ### DisjTree 145 146 147 % 5.85/6.02 149. (All Z, ((apart_point_and_line (intersection_point T_3 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_3 T_1) Z)))) (apart_point_and_line (intersection_point T_3 T_1) T_1) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_3)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) ### All 148 % 5.85/6.02 150. (All Y, (All Z, ((apart_point_and_line (intersection_point T_3 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_3 T_1) Z))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (-. (convergent_lines T_1 T_3)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (apart_point_and_line (intersection_point T_3 T_1) T_1) ### All 149 % 5.85/6.02 151. (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 (intersection_point T_3 T_1) T_1) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_3)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) ### All 150 % 5.85/6.02 152. (-. (convergent_lines T_1 T_1)) (convergent_lines T_1 T_1) ### Axiom % 5.85/6.02 153. (convergent_lines T_3 T_1) (-. (convergent_lines T_3 T_1)) ### Axiom % 5.85/6.02 154. (apart_point_and_line (intersection_point T_3 T_1) T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_1)) ### Axiom % 5.85/6.02 155. ((convergent_lines T_3 T_1) => (-. (apart_point_and_line (intersection_point T_3 T_1) T_1))) (apart_point_and_line (intersection_point T_3 T_1) T_1) (convergent_lines T_3 T_1) ### Imply 153 154 % 5.85/6.02 156. (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) Y)))) (convergent_lines T_3 T_1) (apart_point_and_line (intersection_point T_3 T_1) T_1) ### All 155 % 5.85/6.02 157. ((convergent_lines T_1 T_3) => ((convergent_lines T_1 T_1) \/ (convergent_lines T_3 T_1))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) Y)))) (-. (convergent_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (apart_point_and_line (intersection_point T_3 T_1) T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### DisjTree 151 152 156 % 5.85/6.02 158. (All Z, ((convergent_lines T_1 T_3) => ((convergent_lines T_1 Z) \/ (convergent_lines T_3 Z)))) (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 (intersection_point T_3 T_1) T_1) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (-. (convergent_lines T_1 T_1)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) Y)))) ### All 157 % 5.85/6.02 159. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (apart_point_and_line (intersection_point T_3 T_1) 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 Z, ((convergent_lines T_1 T_3) => ((convergent_lines T_1 Z) \/ (convergent_lines T_3 Z)))) ### All 158 % 5.85/6.02 160. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 5.85/6.02 161. (distinct_lines T_1 T_0) (-. (distinct_lines T_1 T_0)) ### Axiom % 5.85/6.02 162. (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_1) ### Axiom % 5.85/6.02 163. (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1))) ### Axiom % 5.85/6.02 164. (distinct_lines T_0 T_1) (-. (distinct_lines T_0 T_1)) ### Axiom % 5.85/6.02 165. (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### Axiom % 5.85/6.02 166. (apart_point_and_line (intersection_point T_3 T_1) T_0) (-. (apart_point_and_line (intersection_point T_3 T_1) T_0)) ### Axiom % 5.85/6.02 167. (distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) (-. (distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3))) ### Axiom % 5.89/6.06 168. (distinct_lines T_3 T_1) (-. (distinct_lines T_3 T_1)) ### Axiom % 5.89/6.06 169. (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (apart_point_and_line (intersection_point T_3 T_1) T_3) ### Axiom % 5.89/6.06 170. (-. (apart_point_and_line (intersection_point T_3 T_1) T_1)) (apart_point_and_line (intersection_point T_3 T_1) T_1) ### Axiom % 5.89/6.06 171. (convergent_lines T_2 T_3) (-. (convergent_lines T_2 T_3)) ### Axiom % 5.89/6.06 172. (apart_point_and_line (intersection_point T_2 T_3) T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) ### Axiom % 5.89/6.06 173. ((convergent_lines T_2 T_3) => (-. (apart_point_and_line (intersection_point T_2 T_3) T_3))) (apart_point_and_line (intersection_point T_2 T_3) T_3) (convergent_lines T_2 T_3) ### Imply 171 172 % 5.89/6.06 174. (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) Y)))) (convergent_lines T_2 T_3) (apart_point_and_line (intersection_point T_2 T_3) T_3) ### All 173 % 5.89/6.06 175. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_2 T_3) T_3) (convergent_lines T_2 T_3) ### All 174 % 5.89/6.06 176. (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (apart_point_and_line (intersection_point T_2 T_3) T_1) ### Axiom % 5.89/6.06 177. (((distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines T_3 T_1)) => ((apart_point_and_line (intersection_point T_3 T_1) T_3) \/ ((apart_point_and_line (intersection_point T_3 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_3) \/ (apart_point_and_line (intersection_point T_2 T_3) T_1))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) ### DisjTree 167 168 169 170 175 176 % 5.89/6.06 178. (All V, (((distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines T_3 V)) => ((apart_point_and_line (intersection_point T_3 T_1) T_3) \/ ((apart_point_and_line (intersection_point T_3 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_3) \/ (apart_point_and_line (intersection_point T_2 T_3) V)))))) (distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) ### All 177 % 5.89/6.06 179. (All U, (All V, (((distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_1) U) \/ ((apart_point_and_line (intersection_point T_3 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) U) \/ (apart_point_and_line (intersection_point T_2 T_3) V))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) ### All 178 % 5.89/6.06 180. (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_1) U) \/ ((apart_point_and_line (intersection_point T_3 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) ### All 179 % 5.89/6.06 181. (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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) ### All 180 % 5.89/6.06 182. (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (apart_point_and_line (intersection_point T_2 T_3) T_0) ### Axiom % 5.89/6.06 183. ((apart_point_and_line (intersection_point T_3 T_1) T_0) => ((distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) \/ (apart_point_and_line (intersection_point T_2 T_3) T_0))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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))))))))) (apart_point_and_line (intersection_point T_3 T_1) T_0) ### DisjTree 166 181 182 % 5.89/6.06 184. (All Z, ((apart_point_and_line (intersection_point T_3 T_1) T_0) => ((distinct_points (intersection_point T_3 T_1) Z) \/ (apart_point_and_line Z T_0)))) (apart_point_and_line (intersection_point T_3 T_1) T_0) (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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) ### All 183 % 5.89/6.06 185. (All Y, (All Z, ((apart_point_and_line (intersection_point T_3 T_1) Y) => ((distinct_points (intersection_point T_3 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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))))))))) (apart_point_and_line (intersection_point T_3 T_1) T_0) ### All 184 % 5.89/6.06 186. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (apart_point_and_line (intersection_point T_3 T_1) T_0) (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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) ### All 185 % 5.89/6.07 187. (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_1) ### Axiom % 5.89/6.07 188. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_3 T_1) T_1) (convergent_lines T_3 T_1) ### All 156 % 5.89/6.07 189. ((apart_point_and_line (intersection_point T_3 T_1) T_1) => ((distinct_lines T_1 T_1) \/ (apart_point_and_line (intersection_point T_3 T_1) T_1))) (convergent_lines T_3 T_1) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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))))))))) (apart_point_and_line (intersection_point T_3 T_1) T_0) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### DisjTree 186 187 188 % 5.89/6.07 190. (All Z, ((apart_point_and_line (intersection_point T_3 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_3 T_1) Z)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (apart_point_and_line (intersection_point T_3 T_1) T_0) (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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (distinct_lines T_1 T_1)) (convergent_lines T_3 T_1) ### All 189 % 5.89/6.07 191. (All Y, (All Z, ((apart_point_and_line (intersection_point T_3 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_3 T_1) Z))))) (convergent_lines T_3 T_1) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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))))))))) (apart_point_and_line (intersection_point T_3 T_1) T_0) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 190 % 5.89/6.07 192. (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (apart_point_and_line (intersection_point T_3 T_1) T_0) (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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (distinct_lines T_1 T_1)) (convergent_lines T_3 T_1) ### All 191 % 5.89/6.07 193. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (distinct_lines T_0 T_1)) => ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_3 T_1) T_0) \/ (apart_point_and_line (intersection_point T_3 T_1) T_1))))) (convergent_lines T_3 T_1) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z 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 (intersection_point T_0 T_1) T_1)) (convergent_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (distinct_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) ### DisjTree 163 164 25 165 192 188 % 5.89/6.07 194. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (distinct_lines T_0 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_3 T_1) T_0) \/ (apart_point_and_line (intersection_point T_3 T_1) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (distinct_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (distinct_lines T_1 T_1)) (convergent_lines T_3 T_1) ### All 193 % 5.89/6.07 195. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_3 T_1) U) \/ (apart_point_and_line (intersection_point T_3 T_1) V))))))) (convergent_lines T_3 T_1) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z 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 (intersection_point T_0 T_1) T_1)) (convergent_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (distinct_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) ### All 194 % 5.89/6.10 196. ((distinct_lines T_1 T_0) => ((distinct_lines T_1 T_1) \/ (distinct_lines T_0 T_1))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_3 T_1) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_3 T_1) U) \/ (apart_point_and_line (intersection_point T_3 T_1) V))))))) (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_0) ### DisjTree 161 162 195 % 5.89/6.10 197. (All Z, ((distinct_lines T_1 T_0) => ((distinct_lines T_1 Z) \/ (distinct_lines T_0 Z)))) (distinct_lines T_1 T_0) (-. (distinct_lines T_1 T_1)) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_3 T_1) U) \/ (apart_point_and_line (intersection_point T_3 T_1) V))))))) (convergent_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z 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 (intersection_point T_0 T_1) T_1)) (convergent_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) ### All 196 % 5.89/6.10 198. (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_3 T_1) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_3 T_1) U) \/ (apart_point_and_line (intersection_point T_3 T_1) V))))))) (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_0) ### All 197 % 5.89/6.10 199. (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_0) ### Axiom % 5.89/6.10 200. ((convergent_lines T_0 T_1) => ((distinct_lines T_1 T_0) \/ (convergent_lines T_0 T_0))) (-. (convergent_lines T_0 T_0)) (-. (distinct_lines T_1 T_1)) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_3 T_1) U) \/ (apart_point_and_line (intersection_point T_3 T_1) V))))))) (convergent_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z 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 (intersection_point T_0 T_1) T_1)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (convergent_lines T_0 T_1) ### DisjTree 160 198 199 % 5.89/6.10 201. (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (-. (apart_point_and_line (intersection_point T_0 T_1) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_3 T_1) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_3 T_1) U) \/ (apart_point_and_line (intersection_point T_3 T_1) V))))))) (-. (distinct_lines T_1 T_1)) (-. (convergent_lines T_0 T_0)) ### All 200 % 5.89/6.11 202. (All X, (-. (distinct_lines X X))) (-. (convergent_lines T_0 T_0)) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_3 T_1) U) \/ (apart_point_and_line (intersection_point T_3 T_1) V))))))) (convergent_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z 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 (intersection_point T_0 T_1) T_1)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### All 201 % 5.89/6.11 203. (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (-. (apart_point_and_line (intersection_point T_0 T_1) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_3 T_1) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_3 T_1) U) \/ (apart_point_and_line (intersection_point T_3 T_1) V))))))) (All X, (-. (distinct_lines X X))) ### All 202 % 5.89/6.11 204. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_3 T_1) U) \/ (apart_point_and_line (intersection_point T_3 T_1) V))))))) (convergent_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z 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 (intersection_point T_0 T_1) T_1)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) ### All 203 % 5.89/6.11 205. (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_3 T_1) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### All 204 % 5.89/6.11 206. ((distinct_lines T_3 T_1) => ((apart_point_and_line (intersection_point T_3 T_1) T_3) \/ ((apart_point_and_line (intersection_point T_3 T_1) T_1) \/ (convergent_lines T_3 T_1)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Z, ((convergent_lines T_1 T_3) => ((convergent_lines T_1 Z) \/ (convergent_lines T_3 Z)))) (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, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) ### DisjTree 143 144 159 205 % 5.89/6.15 207. (All Z, ((distinct_lines T_3 Z) => ((apart_point_and_line (intersection_point T_3 T_1) T_3) \/ ((apart_point_and_line (intersection_point T_3 T_1) Z) \/ (convergent_lines T_3 Z))))) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((convergent_lines T_1 T_3) => ((convergent_lines T_1 Z) \/ (convergent_lines T_3 Z)))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (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 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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### All 206 % 5.89/6.15 208. (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### Axiom % 5.89/6.15 209. ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 T_1) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Z, ((convergent_lines T_1 T_3) => ((convergent_lines T_1 Z) \/ (convergent_lines T_3 Z)))) (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, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All Z, ((distinct_lines T_3 Z) => ((apart_point_and_line (intersection_point T_3 T_1) T_3) \/ ((apart_point_and_line (intersection_point T_3 T_1) Z) \/ (convergent_lines T_3 Z))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) ### DisjTree 142 207 208 % 5.89/6.15 210. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((distinct_lines T_3 Z) => ((apart_point_and_line (intersection_point T_3 T_1) T_3) \/ ((apart_point_and_line (intersection_point T_3 T_1) Z) \/ (convergent_lines T_3 Z))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((convergent_lines T_1 T_3) => ((convergent_lines T_1 Z) \/ (convergent_lines T_3 Z)))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (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 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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### All 209 % 5.89/6.15 211. (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (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, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All Z, ((distinct_lines T_3 Z) => ((apart_point_and_line (intersection_point T_3 T_1) T_3) \/ ((apart_point_and_line (intersection_point T_3 T_1) Z) \/ (convergent_lines T_3 Z))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) ### All 210 % 5.99/6.16 212. (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_3 T_1) Y) \/ ((apart_point_and_line (intersection_point T_3 T_1) Z) \/ (convergent_lines Y Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (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 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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) ### All 211 % 5.99/6.16 213. (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (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, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_3 T_1) Y) \/ ((apart_point_and_line (intersection_point T_3 T_1) Z) \/ (convergent_lines Y Z)))))) ### All 212 % 5.99/6.16 214. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_3 T_1) Y) \/ ((apart_point_and_line (intersection_point T_3 T_1) Z) \/ (convergent_lines Y Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (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 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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### All 213 % 5.99/6.16 215. (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (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, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_3 T_1) Y) \/ ((apart_point_and_line (intersection_point T_3 T_1) Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 214 % 5.99/6.16 216. (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (apart_point_and_line (intersection_point T_2 T_3) T_1) ### Axiom % 5.99/6.16 217. ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) \/ (apart_point_and_line (intersection_point T_2 T_3) T_1))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_3 T_1) Y) \/ ((apart_point_and_line (intersection_point T_3 T_1) Z) \/ (convergent_lines Y Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (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 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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### DisjTree 215 131 216 % 5.99/6.16 218. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_1)))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (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, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_3 T_1) Y) \/ ((apart_point_and_line (intersection_point T_3 T_1) Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) ### All 217 % 5.99/6.16 219. (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (apart_point_and_line (intersection_point T_3 T_1) T_3) ### Axiom % 5.99/6.16 220. ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) \/ (apart_point_and_line (intersection_point T_3 T_1) T_3))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_3 T_1) Y) \/ ((apart_point_and_line (intersection_point T_3 T_1) Z) \/ (convergent_lines Y Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (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 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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_1)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) ### DisjTree 141 218 219 % 6.05/6.20 221. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_1)))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (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, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_3 T_1) Y) \/ ((apart_point_and_line (intersection_point T_3 T_1) Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) ### All 220 % 6.05/6.20 222. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_3 T_1) Y) \/ ((apart_point_and_line (intersection_point T_3 T_1) Z) \/ (convergent_lines Y Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (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 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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) ### All 221 % 6.05/6.20 223. (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (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, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_3 T_1) Y) \/ ((apart_point_and_line (intersection_point T_3 T_1) Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) ### All 222 % 6.05/6.20 224. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_3 T_1) Y) \/ ((apart_point_and_line (intersection_point T_3 T_1) Z) \/ (convergent_lines Y Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (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 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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) ### All 223 % 6.05/6.20 225. (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (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, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_3 T_1) Y) \/ ((apart_point_and_line (intersection_point T_3 T_1) Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) ### All 224 % 6.05/6.20 226. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_3 T_1) Y) \/ ((apart_point_and_line (intersection_point T_3 T_1) Z) \/ (convergent_lines Y Z)))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (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 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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) ### All 225 % 6.05/6.20 227. (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))))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (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, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 226 % 6.05/6.20 228. (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (convergent_lines X X))) (convergent_lines T_0 T_1) (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 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 (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (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 227 % 6.05/6.20 229. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (convergent_lines T_0 T_1) (All X, (-. (convergent_lines X X))) (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, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 228 % 6.05/6.21 230. ((convergent_lines T_3 T_1) => (-. (apart_point_and_line (intersection_point T_3 T_1) T_3))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_1 T_1)) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (convergent_lines T_0 T_1) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (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, ((distinct_lines X Y) => (convergent_lines X Y)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### Imply 140 229 % 6.05/6.21 231. (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (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, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_2 T_3) (-. (convergent_lines T_1 T_1)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 230 % 6.05/6.21 232. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_1 T_1)) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (convergent_lines T_0 T_1) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (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, ((distinct_lines X Y) => (convergent_lines X Y)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 231 % 6.05/6.21 233. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (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, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. 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((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 T_3) \/ (apart_point_and_line (intersection_point T_0 T_1) T_3))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (distinct_lines T_3 T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### DisjTree 100 101 233 % 6.05/6.24 235. ((apart_point_and_line (intersection_point T_2 T_1) T_3) => ((distinct_lines T_3 T_1) \/ (apart_point_and_line (intersection_point T_2 T_1) T_1))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (distinct_lines T_3 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 T_3) \/ (apart_point_and_line (intersection_point T_0 T_1) T_3))) (-. (convergent_lines T_1 T_3)) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (-. 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(All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z)))) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z)))) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_0) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_0)))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. 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(convergent_lines T_1 T_3)) ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 T_3) \/ (apart_point_and_line (intersection_point T_0 T_1) T_3))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (distinct_lines T_3 T_3)) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### All 235 % 6.05/6.24 237. (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (distinct_lines T_3 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 T_3) \/ (apart_point_and_line (intersection_point T_0 T_1) T_3))) (-. (convergent_lines T_1 T_3)) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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 (intersection_point T_2 T_1) T_0) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_0) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_0)))) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z)))) ### All 236 % 6.11/6.25 238. (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_0) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_0)))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. 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(apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (distinct_lines T_3 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 T_3) \/ (apart_point_and_line (intersection_point T_0 T_1) T_3))) (-. (convergent_lines T_1 T_3)) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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 (intersection_point T_2 T_1) T_0) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_0) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_0)))) ### All 238 % 6.11/6.25 240. (apart_point_and_line (intersection_point T_2 T_1) T_0) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (-. (convergent_lines T_1 T_3)) ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 T_3) \/ (apart_point_and_line (intersection_point T_0 T_1) T_3))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (distinct_lines T_3 T_3)) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### All 239 % 6.11/6.26 241. (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (distinct_lines T_3 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 T_3) \/ (apart_point_and_line (intersection_point T_0 T_1) T_3))) (-. (convergent_lines T_1 T_3)) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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 (intersection_point T_2 T_1) T_0) ### All 240 % 6.11/6.26 242. (-. (distinct_lines T_0 T_0)) (distinct_lines T_0 T_0) ### Axiom % 6.11/6.26 243. ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_lines T_0 T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) T_0))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (-. (distinct_lines T_0 T_0)) (apart_point_and_line (intersection_point T_2 T_1) T_0) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (-. (convergent_lines T_1 T_3)) ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 T_3) \/ (apart_point_and_line (intersection_point T_0 T_1) T_3))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (distinct_lines T_3 T_3)) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### DisjTree 241 242 25 % 6.11/6.26 244. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (distinct_lines T_3 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 T_3) \/ (apart_point_and_line (intersection_point T_0 T_1) T_3))) (-. (convergent_lines T_1 T_3)) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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 (intersection_point T_2 T_1) T_0) (-. (distinct_lines T_0 T_0)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) ### All 243 % 6.11/6.30 245. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (-. (distinct_lines T_0 T_0)) (apart_point_and_line (intersection_point T_2 T_1) T_0) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (-. (convergent_lines T_1 T_3)) ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 T_3) \/ (apart_point_and_line (intersection_point T_0 T_1) T_3))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (distinct_lines T_3 T_3)) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### All 244 % 6.11/6.30 246. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (distinct_lines T_3 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_1 T_3)) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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 (intersection_point T_2 T_1) T_0) (-. (distinct_lines T_0 T_0)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) ### All 245 % 6.11/6.30 247. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (-. (distinct_lines T_0 T_0)) (apart_point_and_line (intersection_point T_2 T_1) T_0) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (-. (convergent_lines T_1 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) ### All 246 % 6.11/6.30 248. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_1 T_3)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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 (intersection_point T_2 T_1) T_0) (-. (distinct_lines T_0 T_0)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) ### All 247 % 6.16/6.30 249. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines T_1 T_0)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) T_1) \/ (apart_point_and_line (intersection_point T_2 T_1) T_0))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (distinct_lines T_0 T_0)) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (-. (convergent_lines T_1 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_1 T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) ### DisjTree 19 20 21 25 30 248 % 6.16/6.30 250. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines T_1 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) T_1) \/ (apart_point_and_line (intersection_point T_2 T_1) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (distinct_lines T_1 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_1 T_3)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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))))))))) (-. (distinct_lines T_0 T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) ### All 249 % 6.16/6.30 251. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) U) \/ (apart_point_and_line (intersection_point T_2 T_1) V))))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (distinct_lines T_0 T_0)) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (-. (convergent_lines T_1 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_1 T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) ### All 250 % 6.16/6.34 252. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_1 T_3)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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))))))))) (-. (distinct_lines T_0 T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) U) \/ (apart_point_and_line (intersection_point T_2 T_1) V))))))) ### All 251 % 6.16/6.34 253. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) U) \/ (apart_point_and_line (intersection_point T_2 T_1) V))))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (distinct_lines T_0 T_0)) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (-. (convergent_lines T_1 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) ### All 252 % 6.16/6.34 254. (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### Axiom % 6.16/6.34 255. ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 T_1) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_1 T_3)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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))))))))) (-. (distinct_lines T_0 T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) U) \/ (apart_point_and_line (intersection_point T_2 T_1) V))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### DisjTree 18 253 254 % 6.21/6.35 256. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) U) \/ (apart_point_and_line (intersection_point T_2 T_1) V))))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (distinct_lines T_0 T_0)) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (-. (convergent_lines T_1 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) ### All 255 % 6.21/6.35 257. (-. (convergent_lines T_1 T_1)) (convergent_lines T_1 T_1) ### Axiom % 6.21/6.35 258. (convergent_lines T_3 T_1) (-. (convergent_lines T_3 T_1)) ### Axiom % 6.21/6.35 259. (-. (convergent_lines T_3 T_3)) (convergent_lines T_3 T_3) ### Axiom % 6.21/6.35 260. ((convergent_lines T_3 T_1) => ((distinct_lines T_1 T_3) \/ (convergent_lines T_3 T_3))) (-. (convergent_lines T_3 T_3)) (-. (convergent_lines T_1 T_3)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (convergent_lines T_3 T_1) ### DisjTree 258 146 259 % 6.21/6.35 261. (All Z, ((convergent_lines T_3 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_3 Z)))) (convergent_lines T_3 T_1) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_3)) (-. (convergent_lines T_3 T_3)) ### All 260 % 6.21/6.35 262. (convergent_lines T_1 T_3) (-. (convergent_lines T_1 T_3)) ### Axiom % 6.21/6.35 263. (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) ### Axiom % 6.21/6.35 264. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 T_1)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_3 T_1) (-. (distinct_lines T_3 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 234 % 6.21/6.35 265. (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_1) ### Axiom % 6.21/6.35 266. ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 T_1) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (distinct_lines T_3 T_3)) (distinct_lines T_3 T_1) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) ### DisjTree 264 265 10 % 6.21/6.35 267. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 T_1)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (distinct_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_lines T_3 T_1) (-. (distinct_lines T_3 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 266 % 6.21/6.35 268. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (distinct_lines T_3 T_1) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) ### All 267 % 6.21/6.35 269. ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 T_1) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 T_1)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_lines T_3 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### DisjTree 263 268 10 % 6.21/6.39 270. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (distinct_lines T_3 T_1) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) ### All 269 % 6.21/6.39 271. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 T_1)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_lines T_3 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) ### All 270 % 6.21/6.39 272. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (distinct_lines T_3 T_1) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) ### All 271 % 6.21/6.39 273. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 T_1)) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_lines T_3 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### All 272 % 6.21/6.40 274. (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (distinct_lines T_3 T_1) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) ### All 273 % 6.21/6.40 275. (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_lines T_3 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### All 274 % 6.21/6.40 276. (-. (convergent_lines T_1 T_1)) (convergent_lines T_1 T_1) ### Axiom % 6.21/6.40 277. ((convergent_lines T_1 T_3) => ((distinct_lines T_3 T_1) \/ (convergent_lines T_1 T_1))) (-. (convergent_lines T_1 T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (convergent_lines T_1 T_3) ### DisjTree 262 275 276 % 6.21/6.40 278. (All Z, ((convergent_lines T_1 T_3) => ((distinct_lines T_3 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_3) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (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, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (convergent_lines T_1 T_1)) ### All 277 % 6.21/6.40 279. (-. (convergent_lines T_3 T_3)) (convergent_lines T_3 T_3) ### Axiom % 6.21/6.40 280. ((convergent_lines T_1 T_3) => ((convergent_lines T_1 T_3) \/ (convergent_lines T_3 T_3))) (-. (convergent_lines T_1 T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_1 T_3) => ((distinct_lines T_3 Z) \/ (convergent_lines T_1 Z)))) (-. (convergent_lines T_3 T_3)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (convergent_lines T_3 T_1) (All Z, ((convergent_lines T_3 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_3 Z)))) ### DisjTree 261 278 279 % 6.21/6.40 281. (All Z, ((convergent_lines T_1 T_3) => ((convergent_lines T_1 Z) \/ (convergent_lines T_3 Z)))) (All Z, ((convergent_lines T_3 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_3 Z)))) (convergent_lines T_3 T_1) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_3 T_3)) (All Z, ((convergent_lines T_1 T_3) => ((distinct_lines T_3 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (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, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (convergent_lines T_1 T_1)) ### All 280 % 6.21/6.40 282. (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines T_1 T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_1 T_3) => ((distinct_lines T_3 Z) \/ (convergent_lines T_1 Z)))) (-. (convergent_lines T_3 T_3)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (convergent_lines T_3 T_1) (All Z, ((convergent_lines T_3 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_3 Z)))) ### All 281 % 6.21/6.40 283. (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (All Z, ((convergent_lines T_3 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_3 Z)))) (convergent_lines T_3 T_1) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_3 T_3)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (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, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) ### All 282 % 6.21/6.44 284. (All Y, (All Z, ((convergent_lines T_3 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_3 Z))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines T_1 T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (convergent_lines T_3 T_3)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (convergent_lines T_3 T_1) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) ### All 283 % 6.21/6.44 285. ((convergent_lines T_1 T_3) => ((convergent_lines T_1 T_1) \/ (convergent_lines T_3 T_1))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_3 T_3)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines T_3 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_3 Z))))) (-. (convergent_lines T_1 T_1)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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))))))))) (-. (distinct_lines T_0 T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) U) \/ (apart_point_and_line (intersection_point T_2 T_1) V))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) ### DisjTree 256 257 284 % 6.21/6.44 286. (All Z, ((convergent_lines T_1 T_3) => ((convergent_lines T_1 Z) \/ (convergent_lines T_3 Z)))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) U) \/ (apart_point_and_line (intersection_point T_2 T_1) V))))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (distinct_lines T_0 T_0)) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. 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(convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_3 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_3 Z))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines T_3 T_3)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) ### All 285 % 6.21/6.45 287. (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_3 T_3)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines T_3 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_3 Z))))) (-. (convergent_lines T_1 T_1)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. 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(distinct_lines T_0 T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) U) \/ (apart_point_and_line (intersection_point T_2 T_1) V))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) ### All 286 % 6.21/6.45 288. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) U) \/ (apart_point_and_line (intersection_point T_2 T_1) V))))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. 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(distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_3 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_3 Z))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines T_3 T_3)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) ### All 287 % 6.21/6.45 289. ((convergent_lines T_0 T_1) => ((distinct_lines T_1 T_0) \/ (convergent_lines T_0 T_0))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_3 T_3)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines T_3 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_3 Z))))) (-. (convergent_lines T_1 T_1)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. 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(convergent_lines T_0 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) U) \/ (apart_point_and_line (intersection_point T_2 T_1) V))))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. 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(convergent_lines T_0 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) U) \/ (apart_point_and_line (intersection_point T_2 T_1) V))))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines T_3 T_3)) ### All 291 % 6.31/6.49 293. (-. (convergent_lines T_3 T_3)) (-. (convergent_lines T_1 T_1)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) U) \/ (apart_point_and_line (intersection_point T_2 T_1) V))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_0 T_1) ### All 292 % 6.31/6.49 294. (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (convergent_lines T_0 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (-. (convergent_lines T_1 T_1)) (-. (convergent_lines T_3 T_3)) ### All 293 % 6.35/6.50 295. (-. (convergent_lines T_3 T_3)) (-. (convergent_lines T_1 T_1)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_0 T_1) ### All 294 % 6.35/6.50 296. (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (apart_point_and_line (intersection_point T_2 T_3) T_1) ### Axiom % 6.35/6.50 297. ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) \/ (apart_point_and_line (intersection_point T_2 T_3) T_1))) (convergent_lines T_0 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (-. (convergent_lines T_1 T_1)) (-. (convergent_lines T_3 T_3)) ### DisjTree 295 131 296 % 6.35/6.50 298. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_1)))) (-. (convergent_lines T_3 T_3)) (-. (convergent_lines T_1 T_1)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_0 T_1) ### All 297 % 6.35/6.50 299. (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (apart_point_and_line (intersection_point T_2 T_1) T_2) ### Axiom % 6.35/6.50 300. ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) \/ (apart_point_and_line (intersection_point T_2 T_1) T_2))) (convergent_lines T_0 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_1 T_1)) (-. (convergent_lines T_3 T_3)) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_1)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### DisjTree 16 298 299 % 6.35/6.53 301. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_1)))) (-. (convergent_lines T_3 T_3)) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_0 T_1) ### All 300 % 6.35/6.53 302. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (convergent_lines T_0 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_1 T_1)) (-. (convergent_lines T_3 T_3)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) ### All 301 % 6.35/6.53 303. (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (convergent_lines T_3 T_3)) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) ### All 302 % 6.35/6.54 304. (convergent_lines T_0 T_1) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_1 T_1)) (-. (convergent_lines T_3 T_3)) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### All 303 % 6.35/6.54 305. (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (convergent_lines T_3 T_3)) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (convergent_lines T_0 T_1) ### All 304 % 6.35/6.54 306. (convergent_lines T_0 T_1) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_1 T_1)) (-. (convergent_lines T_3 T_3)) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### All 305 % 6.35/6.54 307. (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (convergent_lines T_3 T_3)) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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 T_0 T_1) ### All 306 % 6.35/6.54 308. (-. (convergent_lines T_1 T_1)) (convergent_lines T_1 T_1) ### Axiom % 6.35/6.54 309. ((convergent_lines T_2 T_1) => ((convergent_lines T_2 T_1) \/ (convergent_lines T_1 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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (convergent_lines T_1 T_1)) (-. (convergent_lines T_3 T_3)) (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 (intersection_point T_0 T_1) T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### DisjTree 15 307 308 % 6.35/6.54 310. (All Z, ((convergent_lines T_2 T_1) => ((convergent_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (convergent_lines T_3 T_3)) (-. (convergent_lines T_1 T_1)) (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, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 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 309 % 6.35/6.54 311. (All Y, (All Z, ((convergent_lines T_2 Y) => ((convergent_lines T_2 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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (convergent_lines T_1 T_1)) (-. (convergent_lines T_3 T_3)) (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 (intersection_point T_0 T_1) T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### All 310 % 6.35/6.54 312. (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (convergent_lines T_3 T_3)) (-. (convergent_lines T_1 T_1)) (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, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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 T_2 Y) => ((convergent_lines T_2 Z) \/ (convergent_lines Y Z))))) ### All 311 % 6.40/6.56 313. (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (convergent_lines T_1 T_1)) (-. (convergent_lines T_3 T_3)) (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 (intersection_point T_0 T_1) T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) ### All 312 % 6.40/6.56 314. (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (convergent_lines T_3 T_3)) (-. (convergent_lines T_1 T_1)) (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, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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 313 % 6.40/6.56 315. ((convergent_lines T_2 T_1) => (-. (apart_point_and_line (intersection_point T_2 T_1) T_2))) (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (convergent_lines T_1 T_1)) (-. (convergent_lines T_3 T_3)) (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 (intersection_point T_0 T_1) T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### Imply 15 314 % 6.40/6.56 316. (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (convergent_lines T_3 T_3)) (-. (convergent_lines T_1 T_1)) (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, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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 315 % 6.42/6.58 317. (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (convergent_lines T_1 T_1)) (-. (convergent_lines T_3 T_3)) (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 (intersection_point T_0 T_1) T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### All 316 % 6.42/6.58 318. (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (convergent_lines T_3 T_3)) (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, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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 317 % 6.42/6.58 319. (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### All 318 % 6.42/6.58 320. ((apart_point_and_line (intersection_point T_0 T_1) T_2) \/ (apart_point_and_line (intersection_point T_0 T_1) T_3)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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))))))))) ### Or 319 233 % 6.42/6.58 321. (-. (((convergent_lines T_0 T_1) /\ ((convergent_lines T_2 T_3) /\ ((apart_point_and_line (intersection_point T_0 T_1) T_2) \/ (apart_point_and_line (intersection_point T_0 T_1) T_3)))) => ((apart_point_and_line (intersection_point T_2 T_3) T_0) \/ (apart_point_and_line (intersection_point T_2 T_3) 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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### ConjTree 320 % 6.42/6.58 322. (-. (All V, (((convergent_lines T_0 T_1) /\ ((convergent_lines T_2 V) /\ ((apart_point_and_line (intersection_point T_0 T_1) T_2) \/ (apart_point_and_line (intersection_point T_0 T_1) V)))) => ((apart_point_and_line (intersection_point T_2 V) T_0) \/ (apart_point_and_line (intersection_point T_2 V) T_1))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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))))))))) ### NotAllEx 321 % 6.42/6.58 323. (-. (All U, (All V, (((convergent_lines T_0 T_1) /\ ((convergent_lines U V) /\ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V)))) => ((apart_point_and_line (intersection_point U V) T_0) \/ (apart_point_and_line (intersection_point U V) 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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### NotAllEx 322 % 6.42/6.58 324. (-. (All Y, (All U, (All V, (((convergent_lines T_0 Y) /\ ((convergent_lines U V) /\ ((apart_point_and_line (intersection_point T_0 Y) U) \/ (apart_point_and_line (intersection_point T_0 Y) V)))) => ((apart_point_and_line (intersection_point U V) T_0) \/ (apart_point_and_line (intersection_point U V) Y))))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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))))))))) ### NotAllEx 323 % 6.42/6.58 325. (-. (All X, (All Y, (All U, (All V, (((convergent_lines X Y) /\ ((convergent_lines U V) /\ ((apart_point_and_line (intersection_point X Y) U) \/ (apart_point_and_line (intersection_point X Y) V)))) => ((apart_point_and_line (intersection_point U V) X) \/ (apart_point_and_line (intersection_point U V) Y)))))))) (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### NotAllEx 324 % 6.42/6.58 % SZS output end Proof % 6.42/6.58 (* END-PROOF *) %------------------------------------------------------------------------------