%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO199+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 : 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 06:30:59 EDT 2022 % Result : Theorem 6.74s 6.97s % Output : Proof 7.12s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.12 % Problem : GEO199+3 : TPTP v8.1.0. Released v4.0.0. % 0.07/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.13/0.34 % Computer : n024.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 18:07:47 EDT 2022 % 0.13/0.34 % CPUTime : % 6.74/6.97 % SZS status Theorem % 6.74/6.97 (* PROOF-FOUND *) % 6.74/6.97 (* BEGIN-PROOF *) % 6.74/6.97 % SZS output start Proof % 6.74/6.97 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 % 6.74/6.97 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 % 6.74/6.97 3. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1))) ### Axiom % 6.74/6.97 4. (All X, (-. (distinct_points X X))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)) ### All 3 % 6.74/6.97 5. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom % 6.74/6.97 6. (distinct_lines T_0 T_1) (-. (distinct_lines T_0 T_1)) ### Axiom % 6.74/6.97 7. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom % 6.74/6.97 8. (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 % 6.74/6.97 9. ((convergent_lines T_1 T_0) => (-. (apart_point_and_line (intersection_point T_1 T_0) T_0))) (apart_point_and_line (intersection_point T_1 T_0) T_0) (convergent_lines T_1 T_0) ### Imply 7 8 % 6.74/6.97 10. (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) Y)))) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_1 T_0) T_0) ### All 9 % 6.74/6.97 11. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_1 T_0) T_0) (convergent_lines T_1 T_0) ### All 10 % 6.74/6.97 12. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom % 6.74/6.97 13. (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 % 6.74/6.97 14. ((convergent_lines T_1 T_0) => (-. (apart_point_and_line (intersection_point T_1 T_0) T_1))) (apart_point_and_line (intersection_point T_1 T_0) T_1) (convergent_lines T_1 T_0) ### Imply 12 13 % 6.74/6.97 15. (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) T_1)))) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_1 T_0) T_1) ### All 14 % 6.74/6.97 16. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (apart_point_and_line (intersection_point T_1 T_0) T_1) (convergent_lines T_1 T_0) ### All 15 % 6.74/6.97 17. (distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) (-. (distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1))) ### Axiom % 6.74/6.97 18. (distinct_lines T_0 T_1) (-. (distinct_lines T_0 T_1)) ### Axiom % 6.74/6.97 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 % 6.74/6.97 20. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 6.74/6.97 21. (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 % 6.74/6.97 22. ((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 20 21 % 6.74/6.97 23. (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 22 % 6.74/6.97 24. (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 23 % 6.74/6.97 25. (((distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) /\ (distinct_lines 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) \/ ((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_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) T_1)))) (convergent_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_0 T_1) (distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) ### DisjTree 17 18 11 15 19 24 % 6.74/6.97 26. (All V, (((distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) /\ (distinct_lines T_0 V)) => ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) V)))))) (distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) (distinct_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_0) (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) T_1)))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (convergent_lines T_0 T_1) ### All 25 % 6.74/6.97 27. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (convergent_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_0 T_1) (distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) (All V, (((distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) /\ (distinct_lines T_0 V)) => ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) V)))))) ### All 26 % 6.74/6.97 28. (All U, (All V, (((distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_0) U) \/ ((apart_point_and_line (intersection_point T_1 T_0) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) (distinct_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### All 27 % 6.74/6.97 29. ((distinct_lines 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) \/ (convergent_lines T_0 T_1)))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) (All U, (All V, (((distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_0) U) \/ ((apart_point_and_line (intersection_point T_1 T_0) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_0 T_1) ### DisjTree 6 11 16 28 % 6.74/6.97 30. (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) Z) \/ (convergent_lines T_0 Z))))) (distinct_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All U, (All V, (((distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_0) U) \/ ((apart_point_and_line (intersection_point T_1 T_0) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) ### All 29 % 7.04/7.29 31. (-. (convergent_lines T_1 T_1)) (convergent_lines T_1 T_1) ### Axiom % 7.04/7.29 32. ((convergent_lines T_1 T_0) => ((distinct_lines T_0 T_1) \/ (convergent_lines T_1 T_1))) (-. (convergent_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) (All U, (All V, (((distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_0) U) \/ ((apart_point_and_line (intersection_point T_1 T_0) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) Z) \/ (convergent_lines T_0 Z))))) (convergent_lines T_1 T_0) ### DisjTree 5 30 31 % 7.04/7.29 33. (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All U, (All V, (((distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_0) U) \/ ((apart_point_and_line (intersection_point T_1 T_0) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (convergent_lines T_1 T_1)) ### All 32 % 7.04/7.29 34. (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) (All U, (All V, (((distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_0) U) \/ ((apart_point_and_line (intersection_point T_1 T_0) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) Z) \/ (convergent_lines T_0 Z))))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) ### All 33 % 7.04/7.29 35. (All Y, (All U, (All V, (((distinct_points (intersection_point T_1 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_0) U) \/ ((apart_point_and_line (intersection_point T_1 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (-. (convergent_lines X X))) ### All 34 % 7.04/7.29 36. (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, (-. (convergent_lines X X))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) Z) \/ (convergent_lines T_0 Z))))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) ### All 35 % 7.04/7.29 37. ((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_0 T_1)) \/ (distinct_points (intersection_point T_1 T_0) (intersection_point T_0 T_1)))) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (-. (convergent_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, (-. (distinct_points X X))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) ### DisjTree 2 4 36 % 7.04/7.29 38. (All Z, ((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points (intersection_point T_1 T_0) Z)))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (All X, (-. (distinct_points 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, (-. (convergent_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) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) Z) \/ (convergent_lines T_0 Z))))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) ### All 37 % 7.04/7.29 39. (distinct_lines T_0 T_0) (-. (distinct_lines T_0 T_0)) ### Axiom % 7.04/7.29 40. (All X, (-. (distinct_lines X X))) (distinct_lines T_0 T_0) ### All 39 % 7.04/7.29 41. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom % 7.04/7.29 42. (convergent_lines T_1 T_1) (-. (convergent_lines T_1 T_1)) ### Axiom % 7.12/7.30 43. (All X, (-. (convergent_lines X X))) (convergent_lines T_1 T_1) ### All 42 % 7.12/7.30 44. (-. (convergent_lines T_0 T_1)) (convergent_lines T_0 T_1) ### Axiom % 7.12/7.30 45. ((convergent_lines T_1 T_0) => ((convergent_lines T_1 T_1) \/ (convergent_lines T_0 T_1))) (-. (convergent_lines T_0 T_1)) (All X, (-. (convergent_lines X X))) (convergent_lines T_1 T_0) ### DisjTree 41 43 44 % 7.12/7.30 46. (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_1 T_0) (All X, (-. (convergent_lines X X))) (-. (convergent_lines T_0 T_1)) ### All 45 % 7.12/7.30 47. (apart_point_and_line (intersection_point T_0 T_1) T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) ### Axiom % 7.12/7.30 48. ((convergent_lines T_0 T_1) => (-. (apart_point_and_line (intersection_point T_0 T_1) T_0))) (apart_point_and_line (intersection_point T_0 T_1) T_0) (All X, (-. (convergent_lines X X))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### Imply 46 47 % 7.12/7.30 49. (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_1 T_0) (All X, (-. (convergent_lines X X))) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### All 48 % 7.12/7.30 50. ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_lines T_0 T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) T_0))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_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, (-. (distinct_points X X))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (All Z, ((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points (intersection_point T_1 T_0) Z)))) ### DisjTree 38 40 49 % 7.12/7.30 51. (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 Z, ((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points (intersection_point T_1 T_0) Z)))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (All X, (-. (distinct_points 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, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) Z) \/ (convergent_lines T_0 Z))))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### All 50 % 7.12/7.30 52. (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_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, (-. (distinct_points X X))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (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 51 % 7.12/7.30 53. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (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)))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (All X, (-. (distinct_points 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, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) Z) \/ (convergent_lines T_0 Z))))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### All 52 % 7.12/7.30 54. (-. (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 % 7.12/7.30 55. ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) \/ (apart_point_and_line (intersection_point T_1 T_0) T_2))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_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, (-. (distinct_points X X))) (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 Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### DisjTree 1 53 54 % 7.12/7.32 56. (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 Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (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, (-. (distinct_points 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, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) Z) \/ (convergent_lines T_0 Z))))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) ### All 55 % 7.12/7.32 57. (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_1 T_0) Y) \/ ((apart_point_and_line (intersection_point T_1 T_0) Z) \/ (convergent_lines Y Z)))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_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, (-. (distinct_points X X))) (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 Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (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)))) ### All 56 % 7.12/7.32 58. (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All Z, ((apart_point_and_line (intersection_point T_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 Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (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, (-. (distinct_points 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, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) ### All 57 % 7.12/7.32 59. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_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, (-. (distinct_points X X))) (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 Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) ### All 58 % 7.12/7.32 60. (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, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (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, (-. (distinct_points 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, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) ### All 59 % 7.12/7.35 61. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_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, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 60 % 7.12/7.35 62. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points 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, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) ### All 61 % 7.12/7.35 63. (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (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) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (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_points X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All 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 62 % 7.12/7.35 64. (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points 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, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) ### All 63 % 7.12/7.35 65. (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (All X, (-. (distinct_lines X X))) (convergent_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (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_points X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All 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 Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) ### All 64 % 7.12/7.35 66. (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_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points 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, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_0) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) ### All 65 % 7.12/7.35 67. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (All X, (-. (distinct_lines X X))) (convergent_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (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_points X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All 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 Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 66 % 7.12/7.35 68. (-. (-. (apart_point_and_line (intersection_point T_0 T_1) T_2))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points 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, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_0) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotNot 67 % 7.12/7.35 69. (-. (incident_point_and_line (intersection_point T_0 T_1) T_2)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (All X, (-. (distinct_lines X X))) (convergent_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (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_points X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (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 Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### Definition-Pseudo(incident_point_and_line) 68 % 7.12/7.35 70. (incident_point_and_line (intersection_point T_1 T_0) T_2) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points 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, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_0) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (incident_point_and_line (intersection_point T_0 T_1) T_2)) ### Definition-Pseudo(incident_point_and_line) 69 % 7.12/7.35 71. (-. (((convergent_lines T_1 T_0) /\ ((convergent_lines T_2 T_0) /\ ((convergent_lines T_1 T_2) /\ (incident_point_and_line (intersection_point T_1 T_0) T_2)))) => (incident_point_and_line (intersection_point T_0 T_1) T_2))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (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_points X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (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 Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### ConjTree 70 % 7.12/7.35 72. (-. (All Z, (((convergent_lines T_1 T_0) /\ ((convergent_lines Z T_0) /\ ((convergent_lines T_1 Z) /\ (incident_point_and_line (intersection_point T_1 T_0) Z)))) => (incident_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_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points 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, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotAllEx 71 % 7.12/7.36 73. (-. (All Y, (All Z, (((convergent_lines T_1 Y) /\ ((convergent_lines Z Y) /\ ((convergent_lines T_1 Z) /\ (incident_point_and_line (intersection_point T_1 Y) Z)))) => (incident_point_and_line (intersection_point Y T_1) Z))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (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_points X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (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 Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 72 % 7.12/7.36 74. (-. (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 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_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points 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, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotAllEx 73 % 7.12/7.36 % SZS output end Proof % 7.12/7.36 (* END-PROOF *) %------------------------------------------------------------------------------