%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO193+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 : n010.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 06:30:54 EDT 2022 % Result : Theorem 4.86s 5.11s % Output : Proof 5.31s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.08/0.14 % Problem : GEO193+3 : TPTP v8.1.0. Released v4.0.0. % 0.08/0.14 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.14/0.36 % Computer : n010.cluster.edu % 0.14/0.36 % Model : x86_64 x86_64 % 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.14/0.36 % Memory : 8042.1875MB % 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64 % 0.14/0.36 % CPULimit : 300 % 0.14/0.36 % WCLimit : 600 % 0.14/0.36 % DateTime : Fri Jun 17 21:46:38 EDT 2022 % 0.14/0.36 % CPUTime : % 4.86/5.11 % SZS status Theorem % 4.86/5.11 (* PROOF-FOUND *) % 4.86/5.11 (* BEGIN-PROOF *) % 4.86/5.11 % SZS output start Proof % 4.86/5.11 1. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 4.86/5.11 2. (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 % 4.86/5.11 3. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom % 4.86/5.11 4. (distinct_lines T_1 T_2) (-. (distinct_lines T_1 T_2)) ### Axiom % 4.86/5.11 5. (-. (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 % 4.86/5.11 6. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom % 4.86/5.11 7. (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 % 4.86/5.11 8. ((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 6 7 % 4.86/5.11 9. (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 8 % 4.86/5.11 10. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (apart_point_and_line (intersection_point T_2 T_1) T_2) (convergent_lines T_2 T_1) ### All 9 % 4.86/5.11 11. (convergent_lines T_1 T_2) (-. (convergent_lines T_1 T_2)) ### Axiom % 4.86/5.11 12. (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 % 4.86/5.11 13. (distinct_lines T_2 T_1) (-. (distinct_lines T_2 T_1)) ### Axiom % 4.86/5.11 14. (-. (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 % 4.86/5.11 15. (-. (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 % 4.86/5.11 16. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 4.86/5.11 17. (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 % 4.86/5.11 18. ((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 16 17 % 4.86/5.11 19. (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 18 % 4.86/5.11 20. (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 19 % 4.86/5.11 21. (((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))))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_lines T_2 T_1) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) ### DisjTree 12 13 10 14 15 20 % 4.86/5.11 22. (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 X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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_0 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) ### All 21 % 4.86/5.11 23. (All U, (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_lines T_2 T_1) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) ### All 22 % 4.86/5.11 24. (-. (convergent_lines T_1 T_1)) (convergent_lines T_1 T_1) ### Axiom % 4.86/5.11 25. ((convergent_lines T_1 T_2) => ((distinct_lines T_2 T_1) \/ (convergent_lines T_1 T_1))) (-. (convergent_lines T_1 T_1)) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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_0 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All 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_1 T_2) ### DisjTree 11 23 24 % 4.86/5.11 26. (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 U, (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (-. (convergent_lines T_1 T_1)) ### All 25 % 4.86/5.11 27. (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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_0 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All 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_1 T_2) ### All 26 % 5.24/5.46 28. ((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) \/ (convergent_lines T_1 T_2)))) (All U, (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (distinct_lines T_1 T_2) ### DisjTree 4 5 10 27 % 5.24/5.46 29. (All Z, ((distinct_lines T_1 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_1 Z))))) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_2 T_1) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (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_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All 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 28 % 5.24/5.46 30. (-. (convergent_lines T_2 T_2)) (convergent_lines T_2 T_2) ### Axiom % 5.24/5.46 31. ((convergent_lines T_2 T_1) => ((distinct_lines T_1 T_2) \/ (convergent_lines T_2 T_2))) (-. (convergent_lines T_2 T_2)) (All U, (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (-. (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (All Z, ((distinct_lines T_1 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_1 Z))))) (convergent_lines T_2 T_1) ### DisjTree 3 29 30 % 5.24/5.46 32. (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 Z, ((distinct_lines T_1 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_1 Z))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (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_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All 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_2)) ### All 31 % 5.24/5.46 33. (All X, (-. (convergent_lines X X))) (All U, (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (-. (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (All Z, ((distinct_lines T_1 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_1 Z))))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) ### All 32 % 5.24/5.46 34. (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)))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (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_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All U, (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (All X, (-. (convergent_lines X X))) ### All 33 % 5.24/5.46 35. (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (convergent_lines X X))) (All U, (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (-. (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) ### All 34 % 5.31/5.50 36. (distinct_lines T_1 T_1) (-. (distinct_lines T_1 T_1)) ### Axiom % 5.31/5.50 37. (All X, (-. (distinct_lines X X))) (distinct_lines T_1 T_1) ### All 36 % 5.31/5.50 38. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom % 5.31/5.50 39. (apart_point_and_line (intersection_point T_2 T_1) T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) ### Axiom % 5.31/5.50 40. ((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 38 39 % 5.31/5.50 41. (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 40 % 5.31/5.50 42. (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 41 % 5.31/5.50 43. ((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 X, (-. (distinct_lines X X))) (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 X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (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_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All U, (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) ### DisjTree 35 37 42 % 5.31/5.50 44. (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 X, (-. (convergent_lines X X))) (All U, (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (-. (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (All X, (-. (distinct_lines X X))) ### All 43 % 5.31/5.50 45. (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, (-. (distinct_lines X X))) (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 X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (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_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All 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 44 % 5.31/5.50 46. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((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 X, (-. (convergent_lines X X))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (-. (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (All X, (-. (distinct_lines X X))) ### All 45 % 5.31/5.50 47. (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### Axiom % 5.31/5.50 48. ((apart_point_and_line (intersection_point T_2 T_1) T_0) => ((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) \/ (apart_point_and_line (intersection_point T_0 T_1) T_0))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (-. (distinct_lines X X))) (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 X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All 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 U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) ### DisjTree 2 46 47 % 5.31/5.51 49. (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_0) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_0)))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((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 X, (-. (convergent_lines X X))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) ### All 48 % 5.31/5.51 50. (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (-. (distinct_lines X X))) (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 X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All 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 U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) ### All 49 % 5.31/5.51 51. (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_2 T_1) T_0) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((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 X, (-. (convergent_lines X X))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) ### All 50 % 5.31/5.51 52. (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (-. (distinct_lines X X))) (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 X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 51 % 5.31/5.51 53. (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_2 T_1) T_0) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (convergent_lines X X))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) ### All 52 % 5.31/5.51 54. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (-. (distinct_lines X X))) (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 X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (convergent_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (All 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 53 % 5.31/5.51 55. (All Y, (All Z, ((convergent_lines T_2 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_2 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_2 T_1) T_0) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (convergent_lines X X))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_2 T_1) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 54 % 5.31/5.51 56. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (convergent_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (All 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 55 % 5.31/5.51 57. ((convergent_lines T_0 T_1) => (-. (apart_point_and_line (intersection_point T_0 T_1) T_0))) (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_2 T_1) T_0) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_2 T_1) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (convergent_lines T_0 T_1) ### Imply 1 56 % 5.31/5.51 58. (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) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (convergent_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (All 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 57 % 5.31/5.51 59. (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_2 T_1) T_0) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_2 T_1) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (convergent_lines T_0 T_1) ### All 58 % 5.31/5.51 60. (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (All 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 59 % 5.31/5.51 61. (-. (((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_0 T_1) T_2)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((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, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### ConjTree 60 % 5.31/5.52 62. (-. (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 Z T_1) T_2))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((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)))))) ### NotAllEx 61 % 5.31/5.52 63. (-. (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 Z Y) T_2)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((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, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotAllEx 62 % 5.31/5.52 64. (-. (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 Z Y) X))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((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)))))) ### NotAllEx 63 % 5.31/5.52 % SZS output end Proof % 5.31/5.52 (* END-PROOF *) %------------------------------------------------------------------------------