%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO194+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 : n012.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 06:30:55 EDT 2022 % Result : Theorem 4.32s 4.55s % Output : Proof 4.78s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.03/0.13 % Problem : GEO194+3 : TPTP v8.1.0. Released v4.0.0. % 0.03/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.14/0.34 % Computer : n012.cluster.edu % 0.14/0.34 % Model : x86_64 x86_64 % 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.14/0.34 % Memory : 8042.1875MB % 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.14/0.34 % CPULimit : 300 % 0.14/0.34 % WCLimit : 600 % 0.14/0.34 % DateTime : Sat Jun 18 06:40:38 EDT 2022 % 0.14/0.35 % CPUTime : % 4.32/4.55 % SZS status Theorem % 4.32/4.55 (* PROOF-FOUND *) % 4.32/4.55 (* BEGIN-PROOF *) % 4.32/4.55 % SZS output start Proof % 4.32/4.55 1. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 4.32/4.55 2. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 4.32/4.55 3. (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 % 4.32/4.55 4. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom % 4.32/4.55 5. (convergent_lines T_2 T_2) (-. (convergent_lines T_2 T_2)) ### Axiom % 4.32/4.55 6. (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_2) ### All 5 % 4.32/4.55 7. (convergent_lines T_1 T_2) (-. (convergent_lines T_1 T_2)) ### Axiom % 4.32/4.55 8. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom % 4.32/4.55 9. (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 % 4.32/4.55 10. (distinct_lines T_2 T_1) (-. (distinct_lines T_2 T_1)) ### Axiom % 4.32/4.55 11. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom % 4.32/4.55 12. (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 % 4.32/4.55 13. ((convergent_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_2) (convergent_lines T_2 T_1) ### Imply 11 12 % 4.32/4.55 14. (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_2 T_1) T_2) ### All 13 % 4.32/4.55 15. (-. (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 % 4.32/4.55 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 % 4.32/4.55 17. (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 % 4.32/4.55 18. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_0)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_0))) ### Axiom % 4.32/4.55 19. (distinct_lines T_1 T_0) (-. (distinct_lines T_1 T_0)) ### Axiom % 4.32/4.55 20. (-. (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 % 4.32/4.55 21. (-. (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 % 4.32/4.55 22. (-. (apart_point_and_line (intersection_point T_2 T_0) T_1)) (apart_point_and_line (intersection_point T_2 T_0) T_1) ### Axiom % 4.32/4.55 23. (convergent_lines T_2 T_0) (-. (convergent_lines T_2 T_0)) ### Axiom % 4.32/4.55 24. (apart_point_and_line (intersection_point T_2 T_0) T_0) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) ### Axiom % 4.32/4.55 25. ((convergent_lines T_2 T_0) => (-. (apart_point_and_line (intersection_point T_2 T_0) T_0))) (apart_point_and_line (intersection_point T_2 T_0) T_0) (convergent_lines T_2 T_0) ### Imply 23 24 % 4.32/4.55 26. (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) Y)))) (convergent_lines T_2 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_0) ### All 25 % 4.32/4.55 27. (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_0) T_0) (convergent_lines T_2 T_0) ### All 26 % 4.32/4.55 28. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_0)) /\ (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_0) T_1) \/ (apart_point_and_line (intersection_point T_2 T_0) T_0))))) (convergent_lines T_2 T_0) (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_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)) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_0)) ### DisjTree 18 19 20 21 22 27 % 4.32/4.55 29. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_0)) /\ (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_0) T_1) \/ (apart_point_and_line (intersection_point T_2 T_0) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_0)) (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_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_0) ### All 28 % 4.32/4.55 30. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_0)) /\ (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_0) U) \/ (apart_point_and_line (intersection_point T_2 T_0) V))))))) (convergent_lines T_2 T_0) (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_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)) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_0)) ### All 29 % 4.32/4.55 31. (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_0)) (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_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_0) ### All 30 % 4.32/4.55 32. (distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_0)) (-. (distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_0))) ### Axiom % 4.32/4.55 33. (distinct_lines T_1 T_2) (-. (distinct_lines T_1 T_2)) ### Axiom % 4.32/4.55 34. (-. (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 % 4.32/4.55 35. (-. (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 % 4.32/4.55 36. (-. (apart_point_and_line (intersection_point T_2 T_0) T_1)) (apart_point_and_line (intersection_point T_2 T_0) T_1) ### Axiom % 4.32/4.55 37. (convergent_lines T_2 T_0) (-. (convergent_lines T_2 T_0)) ### Axiom % 4.32/4.55 38. (apart_point_and_line (intersection_point T_2 T_0) T_2) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) ### Axiom % 4.32/4.55 39. ((convergent_lines T_2 T_0) => (-. (apart_point_and_line (intersection_point T_2 T_0) T_2))) (apart_point_and_line (intersection_point T_2 T_0) T_2) (convergent_lines T_2 T_0) ### Imply 37 38 % 4.32/4.55 40. (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_2) ### All 39 % 4.32/4.55 41. (((distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_0)) /\ (distinct_lines 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_1) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_0) T_1) \/ (apart_point_and_line (intersection_point T_2 T_0) T_2))))) (convergent_lines T_2 T_0) (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_0) 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)) (distinct_lines T_1 T_2) (distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_0)) ### DisjTree 32 33 34 35 36 40 % 4.62/4.77 42. (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_0)) /\ (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_2 T_0) T_1) \/ (apart_point_and_line (intersection_point T_2 T_0) V)))))) (distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_0)) (distinct_lines 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_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_0) 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_0) ### All 41 % 4.62/4.77 43. (All U, (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_0)) /\ (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_0) U) \/ (apart_point_and_line (intersection_point T_2 T_0) V))))))) (convergent_lines T_2 T_0) (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_0) 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)) (distinct_lines T_1 T_2) (distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_0)) ### All 42 % 4.62/4.77 44. (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_0)) (distinct_lines 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_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_0) 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_0) ### All 43 % 4.62/4.77 45. ((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_0)) \/ (distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_0)))) (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)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (distinct_lines T_1 T_2) (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)))))))) (convergent_lines T_2 T_0) (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_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)) (distinct_lines T_1 T_0) (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)) ### DisjTree 17 31 44 % 4.62/4.77 46. (All Z, ((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points (intersection_point T_2 T_1) Z)))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (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_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_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_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_1 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)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) ### All 45 % 4.62/4.77 47. (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (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)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (distinct_lines T_1 T_2) (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)))))))) (convergent_lines T_2 T_0) (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_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)) (distinct_lines T_1 T_0) (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)) ### All 46 % 4.62/4.77 48. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (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_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_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_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_1 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)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) ### All 47 % 4.62/4.80 49. (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, ((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)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (distinct_lines T_1 T_2) (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)))))))) (convergent_lines T_2 T_0) (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_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)) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 48 % 4.62/4.80 50. (-. (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 % 4.62/4.80 51. ((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))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (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_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_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_1 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)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) 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))))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### DisjTree 16 49 50 % 4.62/4.80 52. (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 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, ((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)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (distinct_lines T_1 T_2) (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)))))))) (convergent_lines T_2 T_0) (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_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)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 51 % 4.62/4.80 53. (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 X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (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_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_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_1 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)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) 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))))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### All 52 % 4.62/4.80 54. (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_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 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)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (distinct_lines T_1 T_2) (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)))))))) (convergent_lines T_2 T_0) (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_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)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 53 % 4.62/4.80 55. (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_1) ### Axiom % 4.62/4.80 56. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 4.62/4.80 57. (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 % 4.62/4.80 58. ((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 56 57 % 4.62/4.80 59. (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 58 % 4.62/4.80 60. (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 59 % 4.62/4.80 61. ((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))) (convergent_lines T_0 T_1) (-. (distinct_lines T_1 T_1)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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_1 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)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) 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))))))))) (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_points X Z) \/ (apart_point_and_line Z Y)))))) ### DisjTree 54 55 60 % 4.62/4.81 62. (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 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_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 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)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (distinct_lines T_1 T_2) (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)))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (distinct_lines T_1 T_1)) (convergent_lines T_0 T_1) ### All 61 % 4.62/4.81 63. (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) (-. (distinct_lines T_1 T_1)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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_1 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)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) 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))))))))) (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_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 62 % 4.62/4.81 64. (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_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 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)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (distinct_lines T_1 T_2) (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)))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (distinct_lines T_1 T_1)) (convergent_lines T_0 T_1) ### All 63 % 4.62/4.81 65. (distinct_lines T_2 T_2) (-. (distinct_lines T_2 T_2)) ### Axiom % 4.62/4.81 66. (-. (convergent_lines T_2 T_2)) (convergent_lines T_2 T_2) ### Axiom % 4.62/4.81 67. ((distinct_lines T_2 T_2) => ((apart_point_and_line (intersection_point T_2 T_1) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_1) T_2) \/ (convergent_lines T_2 T_2)))) (-. (convergent_lines T_2 T_2)) (convergent_lines T_2 T_1) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (distinct_lines T_2 T_2) ### DisjTree 65 14 14 66 % 4.62/4.81 68. (All Z, ((distinct_lines T_2 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_2 Z))))) (distinct_lines T_2 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_1) (-. (convergent_lines T_2 T_2)) ### All 67 % 4.62/4.81 69. ((apart_point_and_line (intersection_point T_2 T_1) T_2) => ((distinct_lines T_2 T_2) \/ (apart_point_and_line (intersection_point T_2 T_1) T_2))) (-. (convergent_lines T_2 T_2)) (convergent_lines T_2 T_1) (All Z, ((distinct_lines T_2 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_2 Z))))) (convergent_lines T_0 T_1) (-. (distinct_lines T_1 T_1)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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_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)))) (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_0 T_1) T_2) (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)))))) ### DisjTree 64 68 14 % 4.62/4.84 70. (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_2 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)))))) (apart_point_and_line (intersection_point T_0 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 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)) (distinct_lines T_1 T_2) (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)))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (distinct_lines T_1 T_1)) (convergent_lines T_0 T_1) (All Z, ((distinct_lines T_2 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_2 Z))))) (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_2)) ### All 69 % 4.62/4.84 71. (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_2)) (convergent_lines T_2 T_1) (All Z, ((distinct_lines T_2 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_2 Z))))) (convergent_lines T_0 T_1) (-. (distinct_lines T_1 T_1)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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_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)))) (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_0 T_1) T_2) (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 70 % 4.62/4.84 72. (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (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_0 T_1) T_2) \/ (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))))))))) (distinct_lines T_1 T_2) (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)))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (distinct_lines T_1 T_1)) (convergent_lines T_0 T_1) (All Z, ((distinct_lines T_2 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_2 Z))))) (-. (convergent_lines T_2 T_2)) (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_2 T_1) T_1)) (convergent_lines T_2 T_1) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (distinct_lines T_2 T_1) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) ### DisjTree 9 10 14 15 71 60 % 4.62/4.84 73. (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (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_0 T_1) T_2) \/ (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_2 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_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_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z))))) (-. (convergent_lines T_2 T_2)) (All Z, ((distinct_lines T_2 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_2 Z))))) (convergent_lines T_0 T_1) (-. (distinct_lines T_1 T_1)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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_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, (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 72 % 4.62/4.85 74. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) 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, (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))))))))) (distinct_lines T_1 T_2) (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)))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (distinct_lines T_1 T_1)) (convergent_lines T_0 T_1) (All Z, ((distinct_lines T_2 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_2 Z))))) (-. (convergent_lines T_2 T_2)) (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_2 T_1) T_1)) (convergent_lines T_2 T_1) (distinct_lines T_2 T_1) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (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_0 T_1) T_2) \/ (apart_point_and_line (intersection_point T_0 T_1) V)))))) ### All 73 % 4.62/4.86 75. (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_2 T_1) Y) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines Y Z)))))) (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (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_0 T_1) T_2) \/ (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_2 T_1) (convergent_lines T_2 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_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z))))) (-. (convergent_lines T_2 T_2)) (convergent_lines T_0 T_1) (-. (distinct_lines T_1 T_1)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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_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, (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) X))))) ### All 74 % 4.62/4.86 76. (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, ((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))))))))) (distinct_lines T_1 T_2) (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)))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_0 T_1) (-. (convergent_lines T_2 T_2)) (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_2 T_1) T_1)) (convergent_lines T_2 T_1) (distinct_lines T_2 T_1) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (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_0 T_1) T_2) \/ (apart_point_and_line (intersection_point T_0 T_1) V)))))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_2 T_1) Y) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines Y Z)))))) ### All 75 % 4.62/4.86 77. (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 Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_2 T_1) Y) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines Y Z)))))) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (distinct_lines T_2 T_1) (convergent_lines T_2 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_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z))))) (-. (convergent_lines T_2 T_2)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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_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, (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) X))))) (All X, (-. (distinct_lines X X))) ### All 76 % 4.71/4.88 78. (-. (convergent_lines T_2 T_2)) (convergent_lines T_2 T_2) ### Axiom % 4.71/4.88 79. ((convergent_lines T_2 T_1) => ((distinct_lines T_1 T_2) \/ (convergent_lines T_2 T_2))) (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, ((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))))))))) (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)))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_0 T_1) (-. (convergent_lines T_2 T_2)) (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_2 T_1) T_1)) (distinct_lines T_2 T_1) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_2 T_1) Y) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines Y Z)))))) (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_2 T_1) ### DisjTree 8 77 78 % 4.71/4.88 80. (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_2 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 Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_2 T_1) Y) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines Y Z)))))) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (distinct_lines T_2 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_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z))))) (-. (convergent_lines T_2 T_2)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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)))))))) (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)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) ### All 79 % 4.71/4.88 81. (-. (convergent_lines T_1 T_1)) (convergent_lines T_1 T_1) ### Axiom % 4.71/4.88 82. ((convergent_lines T_1 T_2) => ((distinct_lines T_2 T_1) \/ (convergent_lines T_1 T_1))) (-. (convergent_lines T_1 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, (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))))))))) (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)))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_0 T_1) (-. (convergent_lines T_2 T_2)) (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_2 T_1) T_1)) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_2 T_1) Y) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines Y Z)))))) (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_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_1 T_2) ### DisjTree 7 80 81 % 4.71/4.88 83. (All Z, ((convergent_lines T_1 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_2) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_2 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 Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_2 T_1) Y) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines Y Z)))))) (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_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z))))) (-. (convergent_lines T_2 T_2)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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)))))))) (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)))))) (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_1 T_1)) ### All 82 % 4.71/4.89 84. (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 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, (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))))))))) (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)))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_0 T_1) (-. (convergent_lines T_2 T_2)) (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_2 T_1) T_1)) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_2 T_1) Y) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines Y Z)))))) (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_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_1 T_2) ### All 83 % 4.71/4.89 85. (All X, (-. (convergent_lines X X))) (convergent_lines T_1 T_2) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_2 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 Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_2 T_1) Y) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines Y Z)))))) (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_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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)))))))) (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)))))) (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_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) ### All 84 % 4.71/4.89 86. ((convergent_lines T_2 T_1) => ((convergent_lines T_2 T_2) \/ (convergent_lines T_1 T_2))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 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, (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))))))))) (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)))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_0 T_1) (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_2 T_1) T_1)) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_2 T_1) Y) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines Y Z)))))) (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 Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_1) ### DisjTree 4 6 85 % 4.74/4.92 87. (All Z, ((convergent_lines T_2 T_1) => ((convergent_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_2 T_1) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (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 Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_2 T_1) Y) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines Y Z)))))) (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_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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)))))))) (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)))))) (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_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) ### All 86 % 4.74/4.92 88. (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 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, (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))))))))) (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)))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_0 T_1) (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_2 T_1) T_1)) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_2 T_1) Y) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines Y Z)))))) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((convergent_lines T_2 Z) \/ (convergent_lines T_1 Z)))) ### All 87 % 4.74/4.92 89. (All Z, ((convergent_lines T_2 T_1) => ((convergent_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_2 T_1) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_2 T_1) Y) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines Y Z)))))) (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_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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 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) X))))) (All X, (-. (distinct_lines X X))) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) ### All 88 % 4.74/4.92 90. (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 Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 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, (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))))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_0 T_1) (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_2 T_1) T_1)) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((convergent_lines T_2 Z) \/ (convergent_lines T_1 Z)))) ### All 89 % 4.77/4.93 91. (distinct_lines T_1 T_1) (-. (distinct_lines T_1 T_1)) ### Axiom % 4.77/4.93 92. (All X, (-. (distinct_lines X X))) (distinct_lines T_1 T_1) ### All 91 % 4.77/4.93 93. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom % 4.77/4.93 94. (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 % 4.77/4.93 95. ((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 93 94 % 4.77/4.93 96. (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 95 % 4.77/4.93 97. (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 96 % 4.77/4.93 98. ((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))) (All Z, ((convergent_lines T_2 T_1) => ((convergent_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_2 T_1) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (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_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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 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) X))))) (All X, (-. (distinct_lines X X))) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) ### DisjTree 90 92 97 % 4.77/4.93 99. (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 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 Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 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, (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))))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_0 T_1) (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))))) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((convergent_lines T_2 Z) \/ (convergent_lines T_1 Z)))) ### All 98 % 4.77/4.93 100. (-. (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 % 4.77/4.93 101. ((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))) (All Z, ((convergent_lines T_2 T_1) => ((convergent_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_2 T_1) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (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_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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 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) X))))) (All X, (-. (distinct_lines X X))) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All 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)))) (apart_point_and_line (intersection_point T_2 T_1) T_0) ### DisjTree 3 99 100 % 4.78/4.96 102. (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 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 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 Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 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, (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))))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_0 T_1) (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))))) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((convergent_lines T_2 Z) \/ (convergent_lines T_1 Z)))) ### All 101 % 4.78/4.96 103. (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 Z, ((convergent_lines T_2 T_1) => ((convergent_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_2 T_1) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (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_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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 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) X))))) (All X, (-. (distinct_lines X X))) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All 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)))) (apart_point_and_line (intersection_point T_2 T_1) T_0) ### All 102 % 4.78/4.96 104. (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_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 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, (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))))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_0 T_1) (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))))) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((convergent_lines T_2 Z) \/ (convergent_lines T_1 Z)))) ### All 103 % 4.78/4.96 105. (All Z, ((convergent_lines T_2 T_1) => ((convergent_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_2 T_1) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (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_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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 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) X))))) (All X, (-. (distinct_lines X X))) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) ### All 104 % 4.78/4.96 106. (apart_point_and_line (intersection_point T_2 T_1) T_0) (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 Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 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, (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))))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((convergent_lines T_2 Z) \/ (convergent_lines T_1 Z)))) ### All 105 % 4.78/4.97 107. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All Z, ((convergent_lines T_2 T_1) => ((convergent_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_2 T_1) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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 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) X))))) (All X, (-. (distinct_lines X X))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) ### All 106 % 4.78/4.97 108. (All Y, (All Z, ((convergent_lines T_2 Y) => ((convergent_lines T_2 Z) \/ (convergent_lines Y Z))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (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_1 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, (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))))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 107 % 4.78/4.97 109. (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_0) ### Axiom % 4.78/4.97 110. ((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 X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (convergent_lines T_2 T_1) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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 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) X))))) (All X, (-. (distinct_lines X X))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (All Y, (All Z, ((convergent_lines T_2 Y) => ((convergent_lines T_2 Z) \/ (convergent_lines Y Z))))) (convergent_lines T_0 T_1) ### DisjTree 2 108 109 % 4.78/4.97 111. (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, ((convergent_lines T_2 Y) => ((convergent_lines T_2 Z) \/ (convergent_lines Y Z))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (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_1 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, (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))))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (convergent_lines T_0 T_0)) ### All 110 % 4.78/4.97 112. (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (convergent_lines T_2 T_1) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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 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) X))))) (All X, (-. (distinct_lines X X))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (All Y, (All Z, ((convergent_lines T_2 Y) => ((convergent_lines T_2 Z) \/ (convergent_lines Y Z))))) (convergent_lines T_0 T_1) ### All 111 % 4.78/4.97 113. (All Y, (All Z, ((convergent_lines T_2 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_2 Z))))) (convergent_lines T_0 T_1) (All Y, (All Z, ((convergent_lines T_2 Y) => ((convergent_lines T_2 Z) \/ (convergent_lines Y Z))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (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_1 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, (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))))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (convergent_lines T_0 T_0)) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) ### All 112 % 4.78/4.97 114. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (convergent_lines T_2 T_1) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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 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) X))))) (All X, (-. (distinct_lines X X))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (convergent_lines T_0 T_1) (All Y, (All Z, ((convergent_lines T_2 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_2 Z))))) ### All 113 % 4.78/4.97 115. (All Y, (All Z, ((convergent_lines T_2 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_2 Z))))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_2 T_1) T_0) (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_1 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, (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))))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 114 % 4.78/4.97 116. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (convergent_lines T_2 T_1) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 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_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 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) X))))) (All X, (-. (distinct_lines X X))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (convergent_lines T_0 T_1) ### All 115 % 4.78/4.97 117. ((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_1) T_0) (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_1 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, (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))))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (convergent_lines T_0 T_1) ### Imply 1 116 % 4.78/4.97 118. (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, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (convergent_lines T_2 T_1) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (apart_point_and_line (intersection_point T_2 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_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 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) X))))) (All X, (-. (distinct_lines X X))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) ### All 117 % 4.78/4.97 119. (apart_point_and_line (intersection_point T_2 T_1) T_0) (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_1 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, (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))))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (convergent_lines T_0 T_1) ### All 118 % 4.78/4.97 120. (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (convergent_lines T_2 T_1) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (apart_point_and_line (intersection_point T_2 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_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 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) X))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) ### All 119 % 4.78/4.97 121. (apart_point_and_line (intersection_point T_2 T_1) T_0) (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) 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, (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))))))))) (convergent_lines T_2 T_0) (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_0) T_1)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (convergent_lines T_0 T_1) ### All 120 % 4.78/4.97 122. (-. (((convergent_lines T_2 T_1) /\ ((convergent_lines T_0 T_1) /\ (convergent_lines T_2 T_0))) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) => (apart_point_and_line (intersection_point T_2 T_0) 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, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) 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, (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) X))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) ### ConjTree 121 % 4.78/4.97 123. (-. (All Z, (((convergent_lines T_2 T_1) /\ ((convergent_lines Z T_1) /\ (convergent_lines T_2 Z))) => ((apart_point_and_line (intersection_point T_2 T_1) Z) => (apart_point_and_line (intersection_point T_2 Z) 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, (-. (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, ((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))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (convergent_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, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 122 % 4.78/4.97 124. (-. (All Y, (All Z, (((convergent_lines T_2 Y) /\ ((convergent_lines Z Y) /\ (convergent_lines T_2 Z))) => ((apart_point_and_line (intersection_point T_2 Y) Z) => (apart_point_and_line (intersection_point T_2 Z) Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) 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, (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) X))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) ### NotAllEx 123 % 4.78/4.97 125. (-. (All X, (All Y, (All Z, (((convergent_lines X Y) /\ ((convergent_lines Z Y) /\ (convergent_lines X Z))) => ((apart_point_and_line (intersection_point X Y) Z) => (apart_point_and_line (intersection_point X Z) Y))))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (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, ((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))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (convergent_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, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 124 % 4.78/4.97 % SZS output end Proof % 4.78/4.97 (* END-PROOF *) %------------------------------------------------------------------------------