%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO199+1 : TPTP v8.1.0. Released v3.3.0. % Transfm : none % Format : tptp:raw % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % Computer : n003.cluster.edu % Model : x86_64 x86_64 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz % Memory : 8042.1875MB % OS : Linux 3.10.0-693.el7.x86_64 % CPULimit : 300s % WCLimit : 600s % DateTime : Sat Jul 16 06:30:59 EDT 2022 % Result : Theorem 22.11s 22.33s % Output : Proof 22.26s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.12 % Problem : GEO199+1 : TPTP v8.1.0. Released v3.3.0. % 0.07/0.12 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.12/0.33 % Computer : n003.cluster.edu % 0.12/0.33 % Model : x86_64 x86_64 % 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.33 % Memory : 8042.1875MB % 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.33 % CPULimit : 300 % 0.12/0.33 % WCLimit : 600 % 0.12/0.33 % DateTime : Sat Jun 18 01:31:27 EDT 2022 % 0.12/0.33 % CPUTime : % 22.11/22.33 % SZS status Theorem % 22.11/22.33 (* PROOF-FOUND *) % 22.11/22.33 (* BEGIN-PROOF *) % 22.11/22.33 % SZS output start Proof % 22.11/22.33 1. (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) ### Axiom % 22.11/22.33 2. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom % 22.11/22.33 3. (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0))) ### Axiom % 22.11/22.33 4. (distinct_lines T_0 T_1) (-. (distinct_lines T_0 T_1)) ### Axiom % 22.11/22.33 5. (-. (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 % 22.11/22.33 6. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom % 22.11/22.33 7. (convergent_lines T_1 T_1) (-. (convergent_lines T_1 T_1)) ### Axiom % 22.11/22.33 8. (All X, (-. (convergent_lines X X))) (convergent_lines T_1 T_1) ### All 7 % 22.11/22.33 9. (-. (convergent_lines T_0 T_1)) (convergent_lines T_0 T_1) ### Axiom % 22.11/22.33 10. ((convergent_lines T_1 T_0) => ((convergent_lines T_1 T_1) \/ (convergent_lines T_0 T_1))) (-. (convergent_lines T_0 T_1)) (All X, (-. (convergent_lines X X))) (convergent_lines T_1 T_0) ### DisjTree 6 8 9 % 22.11/22.33 11. (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_1 T_0) (All X, (-. (convergent_lines X X))) (-. (convergent_lines T_0 T_1)) ### All 10 % 22.11/22.33 12. (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 % 22.11/22.33 13. ((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) (All X, (-. (convergent_lines X X))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### Imply 11 12 % 22.11/22.33 14. (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) Y)))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_1 T_0) (All X, (-. (convergent_lines X X))) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### All 13 % 22.11/22.33 15. (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) (All X, (-. (convergent_lines X X))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### All 14 % 22.11/22.33 16. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom % 22.11/22.33 17. (apart_point_and_line (intersection_point T_1 T_0) T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_0)) ### Axiom % 22.11/22.33 18. ((convergent_lines T_1 T_0) => (-. (apart_point_and_line (intersection_point T_1 T_0) T_0))) (apart_point_and_line (intersection_point T_1 T_0) T_0) (convergent_lines T_1 T_0) ### Imply 16 17 % 22.11/22.33 19. (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) Y)))) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_1 T_0) T_0) ### All 18 % 22.11/22.33 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_1 T_0) T_0) (convergent_lines T_1 T_0) ### All 19 % 22.11/22.33 21. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom % 22.11/22.33 22. (apart_point_and_line (intersection_point T_1 T_0) T_1) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) ### Axiom % 22.11/22.33 23. ((convergent_lines T_1 T_0) => (-. (apart_point_and_line (intersection_point T_1 T_0) T_1))) (apart_point_and_line (intersection_point T_1 T_0) T_1) (convergent_lines T_1 T_0) ### Imply 21 22 % 22.11/22.33 24. (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) T_1)))) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_1 T_0) T_1) ### All 23 % 22.11/22.33 25. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (apart_point_and_line (intersection_point T_1 T_0) T_1) (convergent_lines T_1 T_0) ### All 24 % 22.11/22.33 26. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) /\ (distinct_lines T_0 T_1)) => ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ (apart_point_and_line (intersection_point T_1 T_0) T_1))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_1 T_0) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) ### DisjTree 3 4 5 15 20 25 % 22.11/22.33 27. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) /\ (distinct_lines T_0 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ (apart_point_and_line (intersection_point T_1 T_0) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (distinct_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### All 26 % 22.11/22.33 28. ((convergent_lines T_1 T_0) => ((distinct_lines T_0 T_1) \/ (convergent_lines T_1 T_1))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 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_0)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) /\ (distinct_lines T_0 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ (apart_point_and_line (intersection_point T_1 T_0) V)))))) (convergent_lines T_1 T_0) ### DisjTree 2 27 8 % 22.11/22.33 29. (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) /\ (distinct_lines T_0 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ (apart_point_and_line (intersection_point T_1 T_0) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### All 28 % 22.11/22.33 30. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 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_1 T_0) U) \/ (apart_point_and_line (intersection_point T_1 T_0) V))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 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_0)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) ### All 29 % 22.11/22.38 31. (distinct_lines T_0 T_0) (-. (distinct_lines T_0 T_0)) ### Axiom % 22.11/22.38 32. (All X, (-. (distinct_lines X X))) (distinct_lines T_0 T_0) ### All 31 % 22.11/22.38 33. (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 % 22.11/22.38 34. ((convergent_lines T_0 T_1) => (-. (apart_point_and_line (intersection_point T_0 T_1) T_0))) (apart_point_and_line (intersection_point T_0 T_1) T_0) (All X, (-. (convergent_lines X X))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### Imply 11 33 % 22.11/22.38 35. (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_1 T_0) (All X, (-. (convergent_lines X X))) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### All 34 % 22.11/22.38 36. ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_lines T_0 T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) T_0))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 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_1 T_0) U) \/ (apart_point_and_line (intersection_point T_1 T_0) V))))))) ### DisjTree 30 32 35 % 22.11/22.38 37. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 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_1 T_0) U) \/ (apart_point_and_line (intersection_point T_1 T_0) V))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) ### All 36 % 22.11/22.38 38. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 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_1 T_0) U) \/ (apart_point_and_line (intersection_point T_1 T_0) V))))))) ### All 37 % 22.11/22.38 39. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 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_1 T_0) U) \/ (apart_point_and_line (intersection_point T_1 T_0) V))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) ### All 38 % 22.11/22.38 40. (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### All 39 % 22.11/22.38 41. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) ### All 40 % 22.24/22.42 42. (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (apart_point_and_line (intersection_point T_1 T_0) T_2) ### Axiom % 22.24/22.42 43. ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) \/ (apart_point_and_line (intersection_point T_1 T_0) T_2))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((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 1 41 42 % 22.24/22.42 44. (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 X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) ### All 43 % 22.24/22.42 45. (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))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((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 44 % 22.24/22.42 46. (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 X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) ### All 45 % 22.24/22.42 47. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((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 46 % 22.24/22.42 48. (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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 X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 47 % 22.24/22.42 49. (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines 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)))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (All X, (-. (distinct_lines X X))) (convergent_lines T_1 T_0) (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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 Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) ### All 48 % 22.24/22.42 50. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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 X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_0) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) ### All 49 % 22.26/22.44 51. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (All X, (-. (distinct_lines X X))) (convergent_lines T_1 T_0) (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 50 % 22.26/22.44 52. (-. (((convergent_lines T_1 T_0) /\ ((convergent_lines T_2 T_0) /\ ((convergent_lines T_1 T_2) /\ (-. (apart_point_and_line (intersection_point T_1 T_0) T_2))))) => (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((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) X))))) (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, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### ConjTree 51 % 22.26/22.44 53. (-. (All Z, (((convergent_lines T_1 T_0) /\ ((convergent_lines Z T_0) /\ ((convergent_lines T_1 Z) /\ (-. (apart_point_and_line (intersection_point T_1 T_0) Z))))) => (-. (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (distinct_lines X 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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 52 % 22.26/22.44 54. (-. (All Y, (All Z, (((convergent_lines T_1 Y) /\ ((convergent_lines Z Y) /\ ((convergent_lines T_1 Z) /\ (-. (apart_point_and_line (intersection_point T_1 Y) Z))))) => (-. (apart_point_and_line (intersection_point Y T_1) Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotAllEx 53 % 22.26/22.44 55. (-. (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 Y X) Z))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (distinct_lines X 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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 54 % 22.26/22.44 % SZS output end Proof % 22.26/22.44 (* END-PROOF *) %------------------------------------------------------------------------------