%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO193+2 : 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 : n024.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 1.01s 1.19s % Output : Proof 1.04s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.01/0.12 % Problem : GEO193+2 : TPTP v8.1.0. Released v3.3.0. % 0.12/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.12/0.34 % Computer : n024.cluster.edu % 0.12/0.34 % Model : x86_64 x86_64 % 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.34 % Memory : 8042.1875MB % 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.34 % CPULimit : 300 % 0.12/0.34 % WCLimit : 600 % 0.12/0.34 % DateTime : Fri Jun 17 17:56:17 EDT 2022 % 0.18/0.34 % CPUTime : % 1.01/1.19 % SZS status Theorem % 1.01/1.19 (* PROOF-FOUND *) % 1.01/1.19 (* BEGIN-PROOF *) % 1.01/1.19 % SZS output start Proof % 1.01/1.19 1. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 1.01/1.19 2. (-. (distinct_lines T_0 T_0)) (distinct_lines T_0 T_0) ### Axiom % 1.01/1.19 3. (convergent_lines T_0 T_0) (-. (distinct_lines T_0 T_0)) ### Extension/test/ceq3 2 % 1.01/1.19 4. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom % 1.01/1.19 5. (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 % 1.01/1.19 6. (-. ((apart_point_and_line (intersection_point T_0 T_1) T_2) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### NotOr 5 % 1.01/1.19 7. (-. (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 % 1.01/1.19 8. ((convergent_lines T_2 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)) => (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))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) ### DisjTree 4 6 7 % 1.01/1.19 9. (All Z, ((convergent_lines T_2 T_1) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_1)) => (distinct_points Z (intersection_point T_2 T_1))))) (convergent_lines T_2 T_1) (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))) ### All 8 % 1.01/1.19 10. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom % 1.01/1.19 11. (-. (distinct_lines T_1 T_0)) (convergent_lines T_1 T_0) ### Extension/test/ceq3ctrp 10 % 1.01/1.19 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 % 1.01/1.19 13. (-. (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 % 1.01/1.19 14. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom % 1.01/1.19 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 % 1.01/1.19 16. (-. ((apart_point_and_line (intersection_point T_2 T_1) T_2) \/ (apart_point_and_line (intersection_point T_2 T_1) T_1))) (apart_point_and_line (intersection_point T_2 T_1) T_1) ### NotOr 15 % 1.01/1.19 17. (distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_1)) ### Extension/test/apart1 % 1.01/1.19 18. ((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_1)) => (distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_1)))) (apart_point_and_line (intersection_point T_2 T_1) T_1) (convergent_lines T_2 T_1) ### DisjTree 14 16 17 % 1.01/1.19 19. (All Z, ((convergent_lines T_2 T_1) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_1)) => (distinct_points Z (intersection_point T_2 T_1))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_2 T_1) T_1) ### All 18 % 1.01/1.19 20. (-. (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 % 1.01/1.19 21. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines T_1 T_0)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) T_1) \/ (apart_point_and_line (intersection_point T_2 T_1) T_0))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_1)) => (distinct_points Z (intersection_point T_2 T_1))))) ### DisjTree 9 11 12 13 19 20 % 1.01/1.19 22. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines T_1 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) T_1) \/ (apart_point_and_line (intersection_point T_2 T_1) V)))))) (All Z, ((convergent_lines T_2 T_1) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_1)) => (distinct_points Z (intersection_point T_2 T_1))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) ### All 21 % 1.01/1.19 23. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) U) \/ (apart_point_and_line (intersection_point T_2 T_1) V))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_1)) => (distinct_points Z (intersection_point T_2 T_1))))) ### All 22 % 1.01/1.19 24. (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Z, ((convergent_lines T_2 T_1) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_1)) => (distinct_points Z (intersection_point T_2 T_1))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) ### All 23 % 1.01/1.19 25. (All Y, (All Z, ((convergent_lines T_2 Y) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_2 Y)))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines 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)))))))) ### All 24 % 1.01/1.19 26. (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) ### All 25 % 1.01/1.19 27. (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_1) ### Axiom % 1.01/1.19 28. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 1.04/1.21 29. (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 % 1.04/1.21 30. (-. ((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_0 T_1) T_1) ### NotOr 29 % 1.04/1.21 31. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)) ### Extension/test/apart1 % 1.04/1.21 32. ((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_1)) => (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)))) (apart_point_and_line (intersection_point T_0 T_1) T_1) (convergent_lines T_0 T_1) ### DisjTree 28 30 31 % 1.04/1.21 33. (All Z, ((convergent_lines T_0 T_1) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_1)) => (distinct_points Z (intersection_point T_0 T_1))))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### All 32 % 1.04/1.21 34. (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) (apart_point_and_line (intersection_point T_0 T_1) T_1) (convergent_lines T_0 T_1) ### All 33 % 1.04/1.21 35. (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### All 34 % 1.04/1.21 36. ((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)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines 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)))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) ### DisjTree 26 27 35 % 1.04/1.21 37. (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, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_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_1) T_0)) (-. (distinct_lines T_1 T_1)) (convergent_lines T_0 T_1) ### All 36 % 1.04/1.21 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))))) (convergent_lines T_0 T_1) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines 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)))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) ### All 37 % 1.04/1.21 39. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_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_1) T_0)) (-. (distinct_lines T_1 T_1)) (convergent_lines T_0 T_1) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) ### All 38 % 1.04/1.21 40. (-. (distinct_lines T_0 T_0)) (distinct_lines T_0 T_0) ### Axiom % 1.04/1.21 41. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 1.04/1.21 42. (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 % 1.04/1.21 43. (-. ((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_0 T_1) T_0) ### NotOr 42 % 1.04/1.21 44. ((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_1)) => (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)))) (apart_point_and_line (intersection_point T_0 T_1) T_0) (convergent_lines T_0 T_1) ### DisjTree 41 43 31 % 1.04/1.21 45. (All Z, ((convergent_lines T_0 T_1) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_1)) => (distinct_points Z (intersection_point T_0 T_1))))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### All 44 % 1.04/1.21 46. (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) (apart_point_and_line (intersection_point T_0 T_1) T_0) (convergent_lines T_0 T_1) ### All 45 % 1.04/1.21 47. (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### All 46 % 1.04/1.21 48. ((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))) (-. (distinct_lines T_0 T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (convergent_lines T_0 T_1) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### DisjTree 39 40 47 % 1.04/1.21 49. (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 X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (distinct_lines T_1 T_1)) (convergent_lines T_0 T_1) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (distinct_lines T_0 T_0)) ### All 48 % 1.04/1.21 50. (-. (distinct_lines T_0 T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (convergent_lines T_0 T_1) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### All 49 % 1.04/1.21 51. ((convergent_lines T_0 T_1) => ((convergent_lines T_0 T_0) \/ (convergent_lines 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, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (distinct_lines T_1 T_1)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (distinct_lines T_0 T_0)) (convergent_lines T_0 T_1) ### DisjTree 1 3 50 % 1.04/1.21 52. (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (-. (distinct_lines T_0 T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### All 51 % 1.04/1.21 53. (All X, (-. (distinct_lines X X))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (distinct_lines T_1 T_1)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) ### All 52 % 1.04/1.21 54. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (-. (distinct_lines X X))) ### All 53 % 1.04/1.21 55. (All Y, (All Z, ((convergent_lines T_0 Y) => ((convergent_lines T_0 Z) \/ (convergent_lines Y Z))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (distinct_lines T_1 T_1)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 54 % 1.04/1.21 56. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((convergent_lines T_0 Z) \/ (convergent_lines Y Z))))) ### All 55 % 1.04/1.21 57. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 56 % 1.04/1.21 58. (-. (((convergent_lines T_0 T_1) /\ ((convergent_lines T_2 T_1) /\ (convergent_lines T_0 T_2))) => ((apart_point_and_line (intersection_point T_0 T_1) T_2) => (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_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### ConjTree 57 % 1.04/1.21 59. (-. (All Z, (((convergent_lines T_0 T_1) /\ ((convergent_lines Z T_1) /\ (convergent_lines T_0 Z))) => ((apart_point_and_line (intersection_point T_0 T_1) Z) => (apart_point_and_line (intersection_point Z T_1) T_0))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### NotAllEx 58 % 1.04/1.21 60. (-. (All Y, (All Z, (((convergent_lines T_0 Y) /\ ((convergent_lines Z Y) /\ (convergent_lines T_0 Z))) => ((apart_point_and_line (intersection_point T_0 Y) Z) => (apart_point_and_line (intersection_point Z Y) 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, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 59 % 1.04/1.21 61. (-. (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) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### NotAllEx 60 % 1.04/1.21 % SZS output end Proof % 1.04/1.21 (* END-PROOF *) %------------------------------------------------------------------------------