%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO199+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 : n025.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 0.19s 0.54s % Output : Proof 0.19s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.10/0.12 % Problem : GEO199+2 : TPTP v8.1.0. Released v3.3.0. % 0.10/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.13/0.34 % Computer : n025.cluster.edu % 0.13/0.34 % Model : x86_64 x86_64 % 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.13/0.34 % Memory : 8042.1875MB % 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.13/0.34 % CPULimit : 300 % 0.13/0.34 % WCLimit : 600 % 0.13/0.34 % DateTime : Sat Jun 18 01:48:12 EDT 2022 % 0.13/0.34 % CPUTime : % 0.19/0.54 % SZS status Theorem % 0.19/0.54 (* PROOF-FOUND *) % 0.19/0.54 (* BEGIN-PROOF *) % 0.19/0.54 % SZS output start Proof % 0.19/0.54 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 % 0.19/0.54 2. (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 % 0.19/0.54 3. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom % 0.19/0.54 4. (-. (distinct_lines T_1 T_0)) (convergent_lines T_1 T_0) ### Extension/test/ceq3ctrp 3 % 0.19/0.54 5. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom % 0.19/0.54 6. (convergent_lines T_1 T_1) (-. (convergent_lines T_1 T_1)) ### Axiom % 0.19/0.54 7. (-. (distinct_lines T_1 T_1)) (convergent_lines T_1 T_1) ### Extension/test/ceq3ctrp 6 % 0.19/0.54 8. (All X, (-. (distinct_lines X X))) (convergent_lines T_1 T_1) ### All 7 % 0.19/0.54 9. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 0.19/0.54 10. (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 % 0.19/0.54 11. (-. ((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 10 % 0.19/0.54 12. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)) ### Extension/test/apart1 % 0.19/0.54 13. ((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 9 11 12 % 0.19/0.54 14. (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 13 % 0.19/0.54 15. ((convergent_lines T_1 T_0) => ((convergent_lines T_1 T_1) \/ (convergent_lines T_0 T_1))) (apart_point_and_line (intersection_point T_0 T_1) T_1) (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))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_1 T_0) ### DisjTree 5 8 14 % 0.19/0.54 16. (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, (-. (distinct_lines X X))) (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))))) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### All 15 % 0.19/0.54 17. (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) (All X, (-. (distinct_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 16 % 0.19/0.54 18. (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 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, (-. (distinct_lines X X))) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### All 17 % 0.19/0.54 19. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom % 0.19/0.54 20. (-. (convergent_lines T_0 T_1)) (convergent_lines T_0 T_1) ### Axiom % 0.19/0.54 21. ((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, (-. (distinct_lines X X))) (convergent_lines T_1 T_0) ### DisjTree 19 8 20 % 0.19/0.54 22. (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, (-. (distinct_lines X X))) (-. (convergent_lines T_0 T_1)) ### All 21 % 0.19/0.54 23. (apart_point_and_line (intersection_point T_0 T_1) T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) ### Axiom % 0.19/0.54 24. (-. ((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 23 % 0.19/0.54 25. ((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) (All X, (-. (distinct_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)))) ### DisjTree 22 24 12 % 0.19/0.54 26. (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))))) (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, (-. (distinct_lines X X))) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### All 25 % 0.19/0.54 27. (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) (All X, (-. (distinct_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 26 % 0.19/0.54 28. (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 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, (-. (distinct_lines X X))) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### All 27 % 0.19/0.54 29. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom % 0.19/0.54 30. (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 % 0.19/0.54 31. (-. ((apart_point_and_line (intersection_point T_1 T_0) 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) ### NotOr 30 % 0.19/0.54 32. (distinct_points (intersection_point T_1 T_0) (intersection_point T_1 T_0)) ### Extension/test/apart1 % 0.19/0.54 33. ((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_0)) => (distinct_points (intersection_point T_1 T_0) (intersection_point T_1 T_0)))) (apart_point_and_line (intersection_point T_1 T_0) T_1) (convergent_lines T_1 T_0) ### DisjTree 29 31 32 % 0.19/0.54 34. (All Z, ((convergent_lines T_1 T_0) => (((apart_point_and_line Z T_1) \/ (apart_point_and_line Z T_0)) => (distinct_points Z (intersection_point T_1 T_0))))) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_1 T_0) T_1) ### All 33 % 0.19/0.54 35. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom % 0.19/0.54 36. (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 % 0.19/0.54 37. (-. ((apart_point_and_line (intersection_point T_1 T_0) 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_0) ### NotOr 36 % 0.19/0.54 38. ((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_0)) => (distinct_points (intersection_point T_1 T_0) (intersection_point T_1 T_0)))) (apart_point_and_line (intersection_point T_1 T_0) T_0) (convergent_lines T_1 T_0) ### DisjTree 35 37 32 % 0.19/0.54 39. (All Z, ((convergent_lines T_1 T_0) => (((apart_point_and_line Z T_1) \/ (apart_point_and_line Z T_0)) => (distinct_points Z (intersection_point T_1 T_0))))) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_1 T_0) T_0) ### All 38 % 0.19/0.55 40. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) /\ (distinct_lines T_1 T_0)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) T_1) \/ (apart_point_and_line (intersection_point T_1 T_0) T_0))))) (All Z, ((convergent_lines T_1 T_0) => (((apart_point_and_line Z T_1) \/ (apart_point_and_line Z T_0)) => (distinct_points Z (intersection_point T_1 T_0))))) (All X, (-. (distinct_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, (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_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) ### DisjTree 2 4 18 28 34 39 % 0.19/0.55 41. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) /\ (distinct_lines T_1 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_1 T_0) T_1) \/ (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)) (convergent_lines T_1 T_0) (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 Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_1 T_0) => (((apart_point_and_line Z T_1) \/ (apart_point_and_line Z T_0)) => (distinct_points Z (intersection_point T_1 T_0))))) ### All 40 % 0.19/0.55 42. (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 Z, ((convergent_lines T_1 T_0) => (((apart_point_and_line Z T_1) \/ (apart_point_and_line Z T_0)) => (distinct_points Z (intersection_point T_1 T_0))))) (All X, (-. (distinct_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, (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_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) ### All 41 % 0.19/0.55 43. (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (convergent_lines T_1 T_0) (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 Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_1 T_0) => (((apart_point_and_line Z T_1) \/ (apart_point_and_line Z T_0)) => (distinct_points Z (intersection_point T_1 T_0))))) ### All 42 % 0.19/0.55 44. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((convergent_lines T_1 T_0) => (((apart_point_and_line Z T_1) \/ (apart_point_and_line Z T_0)) => (distinct_points Z (intersection_point T_1 T_0))))) (All X, (-. (distinct_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, (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_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) ### All 43 % 0.19/0.55 45. (All Y, (All Z, ((convergent_lines T_1 Y) => (((apart_point_and_line Z T_1) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_1 Y)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (convergent_lines T_1 T_0) (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 Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 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 44 % 0.19/0.55 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 X, (-. (distinct_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, (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_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) ### All 45 % 0.19/0.55 47. (-. (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 % 0.19/0.55 48. ((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)) (convergent_lines T_1 T_0) (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 Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 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))))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### DisjTree 1 46 47 % 0.19/0.55 49. (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, (-. (distinct_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, (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_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) ### All 48 % 0.19/0.55 50. (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)) (convergent_lines T_1 T_0) (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 Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 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))))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### All 49 % 0.19/0.55 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_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, (-. (distinct_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, (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_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) ### All 50 % 0.19/0.55 52. (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (convergent_lines T_1 T_0) (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, (-. (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))))))))) (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 51 % 0.19/0.55 53. (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, (-. (distinct_lines X X))) (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_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) ### All 52 % 0.19/0.55 54. (-. (((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) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (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, ((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)))))) ### ConjTree 53 % 0.19/0.55 55. (-. (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) => ((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, (-. (distinct_lines X X))) (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))))))) ### NotAllEx 54 % 0.19/0.55 56. (-. (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) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (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, ((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 55 % 0.19/0.55 57. (-. (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) => ((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, (-. (distinct_lines X X))) (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))))))) ### NotAllEx 56 % 0.19/0.55 % SZS output end Proof % 0.19/0.55 (* END-PROOF *) %------------------------------------------------------------------------------