%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO197+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 : n016.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 06:30:58 EDT 2022 % Result : Theorem 4.68s 4.84s % Output : Proof 4.82s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.08/0.13 % Problem : GEO197+3 : TPTP v8.1.0. Released v4.0.0. % 0.08/0.14 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.14/0.35 % Computer : n016.cluster.edu % 0.14/0.35 % Model : x86_64 x86_64 % 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.14/0.35 % Memory : 8042.1875MB % 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64 % 0.14/0.35 % CPULimit : 300 % 0.14/0.35 % WCLimit : 600 % 0.14/0.35 % DateTime : Fri Jun 17 17:39:06 EDT 2022 % 0.14/0.35 % CPUTime : % 4.68/4.84 % SZS status Theorem % 4.68/4.84 (* PROOF-FOUND *) % 4.68/4.84 (* BEGIN-PROOF *) % 4.68/4.84 % SZS output start Proof % 4.68/4.84 1. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 4.68/4.84 2. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom % 4.68/4.84 3. (apart_point_and_line (intersection_point T_2 T_1) T_0) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) ### Axiom % 4.68/4.84 4. (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 % 4.68/4.84 5. (distinct_lines T_1 T_2) (-. (distinct_lines T_1 T_2)) ### Axiom % 4.68/4.84 6. (-. (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 % 4.68/4.84 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 % 4.68/4.84 8. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 4.68/4.84 9. (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 % 4.68/4.84 10. ((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 8 9 % 4.68/4.84 11. (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 10 % 4.68/4.84 12. (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 11 % 4.68/4.84 13. (-. (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 % 4.68/4.84 14. (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (distinct_lines T_1 T_2)) => ((apart_point_and_line (intersection_point T_2 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_2 T_1) T_2) \/ ((apart_point_and_line (intersection_point T_0 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_2)) (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_2 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (distinct_lines T_1 T_2) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) ### DisjTree 4 5 6 7 12 13 % 4.68/4.84 15. (All V, (((distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) /\ (distinct_lines T_1 V)) => ((apart_point_and_line (intersection_point T_2 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ (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_1 T_2) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) ### All 14 % 4.68/4.84 16. (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))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (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_2 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (distinct_lines T_1 T_2) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) ### All 15 % 4.68/4.84 17. (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_1) U) \/ ((apart_point_and_line (intersection_point T_2 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) ### All 16 % 4.68/4.84 18. (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)) (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_2 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (distinct_lines T_1 T_2) (distinct_points (intersection_point T_2 T_1) (intersection_point T_0 T_1)) ### All 17 % 4.68/4.84 19. (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### Axiom % 4.68/4.84 20. ((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)) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (-. (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))))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) ### DisjTree 3 18 19 % 4.68/4.84 21. (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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (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_2 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) ### All 20 % 4.68/4.84 22. (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_1) ### Axiom % 4.68/4.84 23. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom % 4.68/4.84 24. (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 % 4.68/4.84 25. ((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 23 24 % 4.68/4.84 26. (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 25 % 4.68/4.84 27. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_2 T_1) T_1) (convergent_lines T_2 T_1) ### All 26 % 4.68/4.90 28. ((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))) (convergent_lines T_2 T_1) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_2 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) (-. (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))))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_0) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_0)))) ### DisjTree 21 22 27 % 4.68/4.90 29. (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 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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (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_2 T_1) T_2)) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (distinct_lines T_1 T_1)) (convergent_lines T_2 T_1) ### All 28 % 4.68/4.90 30. (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_2 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) (-. (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))))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_0) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_0)))) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z)))) ### All 29 % 4.68/4.90 31. (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z))))) (All Z, ((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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (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_2 T_1) T_2)) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (-. (distinct_lines X X))) ### All 30 % 4.68/4.90 32. (-. (distinct_lines T_2 T_2)) (distinct_lines T_2 T_2) ### Axiom % 4.68/4.90 33. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom % 4.68/4.90 34. (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 % 4.68/4.90 35. ((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 33 34 % 4.68/4.90 36. (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 35 % 4.68/4.90 37. ((apart_point_and_line (intersection_point T_2 T_1) T_2) => ((distinct_lines T_2 T_2) \/ (apart_point_and_line (intersection_point T_2 T_1) T_2))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (distinct_lines T_2 T_2)) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines 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) (-. (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))))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_0) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_0)))) (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))))) ### DisjTree 31 32 36 % 4.68/4.90 38. (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z)))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z))))) (All Z, ((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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (-. (distinct_lines X X))) (-. (distinct_lines T_2 T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) ### All 37 % 4.68/4.90 39. (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (distinct_lines T_2 T_2)) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines 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) (-. (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))))))))) (apart_point_and_line (intersection_point T_2 T_1) T_0) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_0) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z T_0)))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z))))) ### All 38 % 4.68/4.90 40. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (-. (distinct_lines X X))) (-. (distinct_lines T_2 T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) ### All 39 % 4.75/4.96 41. (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_points (intersection_point T_2 T_1) Z) \/ (apart_point_and_line Z Y))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (distinct_lines T_2 T_2)) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines 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) (-. (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))))))))) (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 40 % 4.75/4.96 42. (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)))))) (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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (-. (distinct_lines X X))) (-. (distinct_lines T_2 T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) ### All 41 % 4.75/4.96 43. (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines 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) (-. (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))))))))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 42 % 4.75/4.96 44. (-. (convergent_lines T_2 T_2)) (convergent_lines T_2 T_2) ### Axiom % 4.75/4.96 45. ((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 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)))))) (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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (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_0)) (All X, (-. (distinct_lines X X))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_1) ### DisjTree 2 43 44 % 4.75/4.96 46. (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 Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (-. (distinct_lines X X))) (-. (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))))) (convergent_lines T_0 T_1) (-. (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))))))))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (convergent_lines T_2 T_2)) ### All 45 % 4.75/4.96 47. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (convergent_lines T_2 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (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_0)) (All X, (-. (distinct_lines X 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 46 % 4.75/4.96 48. (All Y, (All Z, ((convergent_lines T_2 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_2 Z))))) (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (-. (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))))))))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (convergent_lines T_2 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### All 47 % 4.82/4.98 49. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (convergent_lines T_2 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (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_0)) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_1) ### All 48 % 4.82/4.98 50. ((convergent_lines T_0 T_1) => (-. (apart_point_and_line (intersection_point T_0 T_1) T_0))) (convergent_lines T_2 T_1) (All X, (-. (distinct_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)) (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_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)))))) (-. (convergent_lines T_2 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) 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 49 % 4.82/4.98 51. (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, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (convergent_lines T_2 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (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))))))))) (-. (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, (-. (distinct_lines X X))) (convergent_lines T_2 T_1) ### All 50 % 4.82/4.98 52. (convergent_lines T_2 T_1) (All X, (-. (distinct_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)) (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_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)))))) (-. (convergent_lines T_2 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) 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 51 % 4.82/4.98 53. (All X, (-. (convergent_lines X X))) (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, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_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)))))) (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))))))))) (-. (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, (-. (distinct_lines X X))) (convergent_lines T_2 T_1) ### All 52 % 4.82/4.98 54. (-. (-. (apart_point_and_line (intersection_point T_2 T_1) T_0))) (convergent_lines T_2 T_1) (All X, (-. (distinct_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)) (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_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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) 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 X, (-. (convergent_lines X X))) ### NotNot 53 % 4.82/4.98 55. (-. (incident_point_and_line (intersection_point T_2 T_1) T_0)) (All X, (-. (convergent_lines X X))) (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, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_1) ### Definition-Pseudo(incident_point_and_line) 54 % 4.82/4.98 56. (incident_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_2 T_1) (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, (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_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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) 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 X, (-. (convergent_lines X X))) (-. (incident_point_and_line (intersection_point T_2 T_1) T_0)) ### Definition-Pseudo(incident_point_and_line) 55 % 4.82/4.99 57. (-. (((convergent_lines T_0 T_1) /\ ((convergent_lines T_2 T_1) /\ ((convergent_lines T_0 T_2) /\ (incident_point_and_line (intersection_point T_0 T_1) T_2)))) => (incident_point_and_line (intersection_point T_2 T_1) T_0))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (distinct_lines X X))) ### ConjTree 56 % 4.82/4.99 58. (-. (All Z, (((convergent_lines T_0 T_1) /\ ((convergent_lines Z T_1) /\ ((convergent_lines T_0 Z) /\ (incident_point_and_line (intersection_point T_0 T_1) Z)))) => (incident_point_and_line (intersection_point Z T_1) T_0)))) (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, (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_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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (convergent_lines X X))) ### NotAllEx 57 % 4.82/4.99 59. (-. (All Y, (All Z, (((convergent_lines T_0 Y) /\ ((convergent_lines Z Y) /\ ((convergent_lines T_0 Z) /\ (incident_point_and_line (intersection_point T_0 Y) Z)))) => (incident_point_and_line (intersection_point Z Y) T_0))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (distinct_lines X X))) ### NotAllEx 58 % 4.82/4.99 60. (-. (All X, (All Y, (All Z, (((convergent_lines X Y) /\ ((convergent_lines Z Y) /\ ((convergent_lines X Z) /\ (incident_point_and_line (intersection_point X Y) Z)))) => (incident_point_and_line (intersection_point Z Y) X)))))) (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, (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_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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (convergent_lines X X))) ### NotAllEx 59 % 4.82/4.99 % SZS output end Proof % 4.82/4.99 (* END-PROOF *) %------------------------------------------------------------------------------