%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO190+2 : TPTP v8.1.0. Released v3.3.0. % Transfm : none % Format : tptp:raw % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % Computer : n017.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 06:30:51 EDT 2022 % Result : Theorem 3.82s 4.00s % Output : Proof 4.18s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.12 % Problem : GEO190+2 : TPTP v8.1.0. Released v3.3.0. % 0.12/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.13/0.34 % Computer : n017.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 06:14:42 EDT 2022 % 0.13/0.34 % CPUTime : % 3.82/4.00 % SZS status Theorem % 3.82/4.00 (* PROOF-FOUND *) % 3.82/4.00 (* BEGIN-PROOF *) % 3.82/4.00 % SZS output start Proof % 3.82/4.00 1. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 3.82/4.00 2. (distinct_points T_0 T_0) ### Extension/test/apart1 % 3.82/4.00 3. (distinct_points T_2 T_0) (-. (distinct_points T_2 T_0)) ### Axiom % 3.82/4.00 4. (distinct_points T_2 T_2) ### Extension/test/apart1 % 3.82/4.00 5. (-. (distinct_points T_0 T_2)) (distinct_points T_0 T_2) ### Axiom % 3.82/4.00 6. ((distinct_points T_2 T_0) => ((distinct_points T_2 T_2) \/ (distinct_points T_0 T_2))) (-. (distinct_points T_0 T_2)) (distinct_points T_2 T_0) ### DisjTree 3 4 5 % 3.82/4.00 7. (All Z, ((distinct_points T_2 T_0) => ((distinct_points T_2 Z) \/ (distinct_points T_0 Z)))) (distinct_points T_2 T_0) (-. (distinct_points T_0 T_2)) ### All 6 % 3.82/4.00 8. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 3.82/4.00 9. (distinct_points T_1 T_0) (-. (distinct_points T_1 T_0)) ### Axiom % 3.82/4.00 10. (distinct_points T_2 T_0) (-. (distinct_points T_2 T_0)) ### Axiom % 3.82/4.00 11. (distinct_points T_0 T_2) (-. (distinct_points T_0 T_2)) ### Axiom % 3.82/4.00 12. (distinct_points T_1 T_0) (-. (distinct_points T_1 T_0)) ### Axiom % 3.82/4.00 13. (distinct_points T_2 T_1) (-. (distinct_points T_2 T_1)) ### Axiom % 3.82/4.00 14. (apart_point_and_line T_1 (line_connecting T_0 T_2)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) ### Axiom % 3.82/4.00 15. (distinct_points T_0 T_2) (-. (distinct_points T_0 T_2)) ### Axiom % 3.82/4.00 16. (distinct_lines (line_connecting T_0 T_2) (line_connecting T_2 T_1)) (-. (distinct_lines (line_connecting T_0 T_2) (line_connecting T_2 T_1))) ### Axiom % 3.82/4.00 17. (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (apart_point_and_line T_0 (line_connecting T_0 T_2)) ### Axiom % 3.82/4.00 18. (distinct_points T_2 T_0) (-. (distinct_points T_2 T_0)) ### Axiom % 3.82/4.00 19. (distinct_points T_2 T_1) (-. (distinct_points T_2 T_1)) ### Axiom % 3.82/4.00 20. (apart_point_and_line T_0 (line_connecting T_2 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) ### Axiom % 3.82/4.00 21. (distinct_points T_2 T_1) (-. (distinct_points T_2 T_1)) ### Axiom % 3.82/4.00 22. (distinct_lines (line_connecting T_2 T_1) (line_connecting T_0 T_1)) (-. (distinct_lines (line_connecting T_2 T_1) (line_connecting T_0 T_1))) ### Axiom % 3.82/4.00 23. (-. (apart_point_and_line T_2 (line_connecting T_2 T_1))) (apart_point_and_line T_2 (line_connecting T_2 T_1)) ### Axiom % 3.82/4.00 24. (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) ### Axiom % 3.82/4.00 25. (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (apart_point_and_line T_1 (line_connecting T_2 T_1)) ### Axiom % 3.82/4.00 26. (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (apart_point_and_line T_1 (line_connecting T_0 T_1)) ### Axiom % 3.82/4.00 27. (((distinct_points T_2 T_1) /\ (distinct_lines (line_connecting T_2 T_1) (line_connecting T_0 T_1))) => ((apart_point_and_line T_2 (line_connecting T_2 T_1)) \/ ((apart_point_and_line T_2 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_1 (line_connecting T_2 T_1)) \/ (apart_point_and_line T_1 (line_connecting T_0 T_1)))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_1))) (distinct_lines (line_connecting T_2 T_1) (line_connecting T_0 T_1)) (distinct_points T_2 T_1) ### DisjTree 21 22 23 24 25 26 % 3.82/4.00 28. (All V, (((distinct_points T_2 T_1) /\ (distinct_lines (line_connecting T_2 T_1) V)) => ((apart_point_and_line T_2 (line_connecting T_2 T_1)) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 (line_connecting T_2 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_2 T_1) (distinct_lines (line_connecting T_2 T_1) (line_connecting T_0 T_1)) (-. (apart_point_and_line T_2 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) ### All 27 % 3.82/4.00 29. (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (apart_point_and_line T_0 (line_connecting T_0 T_1)) ### Axiom % 3.82/4.00 30. ((apart_point_and_line T_0 (line_connecting T_2 T_1)) => ((distinct_lines (line_connecting T_2 T_1) (line_connecting T_0 T_1)) \/ (apart_point_and_line T_0 (line_connecting T_0 T_1)))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_1))) (distinct_points T_2 T_1) (All V, (((distinct_points T_2 T_1) /\ (distinct_lines (line_connecting T_2 T_1) V)) => ((apart_point_and_line T_2 (line_connecting T_2 T_1)) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 (line_connecting T_2 T_1)) \/ (apart_point_and_line T_1 V)))))) (apart_point_and_line T_0 (line_connecting T_2 T_1)) ### DisjTree 20 28 29 % 3.82/4.00 31. (All Z, ((apart_point_and_line T_0 (line_connecting T_2 T_1)) => ((distinct_lines (line_connecting T_2 T_1) Z) \/ (apart_point_and_line T_0 Z)))) (apart_point_and_line T_0 (line_connecting T_2 T_1)) (All V, (((distinct_points T_2 T_1) /\ (distinct_lines (line_connecting T_2 T_1) V)) => ((apart_point_and_line T_2 (line_connecting T_2 T_1)) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 (line_connecting T_2 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_2 T_1) (-. (apart_point_and_line T_2 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) ### All 30 % 3.82/4.00 32. (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_1))) (distinct_points T_2 T_1) (All V, (((distinct_points T_2 T_1) /\ (distinct_lines (line_connecting T_2 T_1) V)) => ((apart_point_and_line T_2 (line_connecting T_2 T_1)) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 (line_connecting T_2 T_1)) \/ (apart_point_and_line T_1 V)))))) (apart_point_and_line T_0 (line_connecting T_2 T_1)) ### All 31 % 3.82/4.00 33. (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_0 (line_connecting T_2 T_1)) (distinct_points T_2 T_1) (-. (apart_point_and_line T_2 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) ### All 32 % 3.82/4.00 34. ((distinct_points T_2 T_2) /\ (distinct_points T_2 T_1)) ### And 4 % 3.82/4.00 35. ((distinct_points T_2 T_1) => ((apart_point_and_line T_2 (line_connecting T_2 T_1)) => ((distinct_points T_2 T_2) /\ (distinct_points T_2 T_1)))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (apart_point_and_line T_0 (line_connecting T_2 T_1)) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_2 T_1) ### DisjTree 19 33 34 % 3.82/4.00 36. (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_1) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_0 (line_connecting T_2 T_1)) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) ### All 35 % 3.87/4.10 37. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 3.87/4.10 38. (distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0)) (-. (distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0))) ### Axiom % 3.87/4.10 39. (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (apart_point_and_line T_0 (line_connecting T_0 T_1)) ### Axiom % 3.87/4.10 40. (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) ### Axiom % 3.87/4.10 41. (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (apart_point_and_line T_1 (line_connecting T_0 T_1)) ### Axiom % 3.87/4.10 42. (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ### Axiom % 3.87/4.10 43. (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0))) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 (line_connecting T_1 T_0)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0)) (distinct_points T_0 T_1) ### DisjTree 37 38 39 40 41 42 % 3.87/4.10 44. (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) ### All 43 % 3.87/4.10 45. (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0)) (distinct_points T_0 T_1) ### All 44 % 3.87/4.10 46. (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) ### Axiom % 3.87/4.10 47. ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0)) \/ (apart_point_and_line T_2 (line_connecting T_1 T_0)))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (apart_point_and_line T_0 (line_connecting T_2 T_1)) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_2 T_1) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) ### DisjTree 36 45 46 % 3.87/4.10 48. (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_1) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_0 (line_connecting T_2 T_1)) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) ### All 47 % 3.87/4.10 49. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 3.87/4.10 50. (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_0)) (-. (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_0))) ### Axiom % 3.87/4.10 51. (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) ### Axiom % 3.87/4.10 52. (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (apart_point_and_line T_0 (line_connecting T_2 T_0)) ### Axiom % 3.87/4.10 53. (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ### Axiom % 3.87/4.10 54. (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (apart_point_and_line T_1 (line_connecting T_2 T_0)) ### Axiom % 3.87/4.10 55. (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_0))) => ((apart_point_and_line T_0 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_0 (line_connecting T_2 T_0)) \/ ((apart_point_and_line T_1 (line_connecting T_1 T_0)) \/ (apart_point_and_line T_1 (line_connecting T_2 T_0)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_0)) (distinct_points T_0 T_1) ### DisjTree 49 50 51 52 53 54 % 3.87/4.10 56. (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_1 T_0) V)) => ((apart_point_and_line T_0 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_1 T_0)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_0)) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) ### All 55 % 3.87/4.10 57. (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_0)) (distinct_points T_0 T_1) ### All 56 % 3.95/4.12 58. (-. (apart_point_and_line T_2 (line_connecting T_2 T_0))) (apart_point_and_line T_2 (line_connecting T_2 T_0)) ### Axiom % 3.95/4.12 59. ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_0)) \/ (apart_point_and_line T_2 (line_connecting T_2 T_0)))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (apart_point_and_line T_0 (line_connecting T_2 T_1)) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_2 T_1) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) ### DisjTree 48 57 58 % 3.95/4.12 60. (All Z, ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_1) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_0 (line_connecting T_2 T_1)) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_0))) ### All 59 % 3.95/4.12 61. ((distinct_points T_2 T_2) /\ (distinct_points T_2 T_0)) ### And 4 % 3.95/4.12 62. ((distinct_points T_2 T_0) => ((apart_point_and_line T_2 (line_connecting T_2 T_0)) => ((distinct_points T_2 T_2) /\ (distinct_points T_2 T_0)))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (apart_point_and_line T_0 (line_connecting T_2 T_1)) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_2 T_1) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_2 Z)))) (distinct_points T_2 T_0) ### DisjTree 18 60 61 % 3.95/4.12 63. (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (distinct_points T_2 T_0) (All Z, ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_1) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_0 (line_connecting T_2 T_1)) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) ### All 62 % 3.95/4.12 64. (distinct_points T_0 T_2) (-. (distinct_points T_0 T_2)) ### Axiom % 3.95/4.12 65. (apart_point_and_line T_2 (line_connecting T_0 T_2)) (-. (apart_point_and_line T_2 (line_connecting T_0 T_2))) ### Axiom % 3.95/4.12 66. ((distinct_points T_2 T_0) /\ (distinct_points T_2 T_2)) ### And 4 % 3.95/4.12 67. ((distinct_points T_0 T_2) => ((apart_point_and_line T_2 (line_connecting T_0 T_2)) => ((distinct_points T_2 T_0) /\ (distinct_points T_2 T_2)))) (apart_point_and_line T_2 (line_connecting T_0 T_2)) (distinct_points T_0 T_2) ### DisjTree 64 65 66 % 3.95/4.12 68. (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (distinct_points T_0 T_2) (apart_point_and_line T_2 (line_connecting T_0 T_2)) ### All 67 % 3.95/4.12 69. (distinct_points T_2 T_1) (-. (distinct_points T_2 T_1)) ### Axiom % 3.95/4.12 70. (apart_point_and_line T_2 (line_connecting T_2 T_1)) (-. (apart_point_and_line T_2 (line_connecting T_2 T_1))) ### Axiom % 3.95/4.12 71. ((distinct_points T_2 T_1) => ((apart_point_and_line T_2 (line_connecting T_2 T_1)) => ((distinct_points T_2 T_2) /\ (distinct_points T_2 T_1)))) (apart_point_and_line T_2 (line_connecting T_2 T_1)) (distinct_points T_2 T_1) ### DisjTree 69 70 34 % 3.95/4.14 72. (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_1) (apart_point_and_line T_2 (line_connecting T_2 T_1)) ### All 71 % 3.95/4.14 73. (((distinct_points T_0 T_2) /\ (distinct_lines (line_connecting T_0 T_2) (line_connecting T_2 T_1))) => ((apart_point_and_line T_0 (line_connecting T_0 T_2)) \/ ((apart_point_and_line T_0 (line_connecting T_2 T_1)) \/ ((apart_point_and_line T_2 (line_connecting T_0 T_2)) \/ (apart_point_and_line T_2 (line_connecting T_2 T_1)))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_2 T_1) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_2 Z)))) (distinct_points T_2 T_0) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_2 T_1)) (distinct_points T_0 T_2) ### DisjTree 15 16 17 63 68 72 % 3.95/4.14 74. (All V, (((distinct_points T_0 T_2) /\ (distinct_lines (line_connecting T_0 T_2) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_2)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 (line_connecting T_0 T_2)) \/ (apart_point_and_line T_2 V)))))) (distinct_points T_0 T_2) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_2 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (distinct_points T_2 T_0) (All Z, ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_1) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) ### All 73 % 3.95/4.14 75. (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (apart_point_and_line T_1 (line_connecting T_2 T_1)) ### Axiom % 3.95/4.14 76. ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) (line_connecting T_2 T_1)) \/ (apart_point_and_line T_1 (line_connecting T_2 T_1)))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_2 T_1) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_2 Z)))) (distinct_points T_2 T_0) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (distinct_points T_0 T_2) (All V, (((distinct_points T_0 T_2) /\ (distinct_lines (line_connecting T_0 T_2) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_2)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 (line_connecting T_0 T_2)) \/ (apart_point_and_line T_2 V)))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) ### DisjTree 14 74 75 % 3.95/4.14 77. (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All V, (((distinct_points T_0 T_2) /\ (distinct_lines (line_connecting T_0 T_2) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_2)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 (line_connecting T_0 T_2)) \/ (apart_point_and_line T_2 V)))))) (distinct_points T_0 T_2) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (distinct_points T_2 T_0) (All Z, ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_1) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) ### All 76 % 3.95/4.15 78. (distinct_points T_1 T_1) ### Extension/test/apart1 % 3.95/4.15 79. ((distinct_points T_1 T_2) /\ (distinct_points T_1 T_1)) ### And 78 % 3.95/4.15 80. ((distinct_points T_2 T_1) => ((apart_point_and_line T_1 (line_connecting T_2 T_1)) => ((distinct_points T_1 T_2) /\ (distinct_points T_1 T_1)))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_2 Z)))) (distinct_points T_2 T_0) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (distinct_points T_0 T_2) (All V, (((distinct_points T_0 T_2) /\ (distinct_lines (line_connecting T_0 T_2) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_2)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 (line_connecting T_0 T_2)) \/ (apart_point_and_line T_2 V)))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (distinct_points T_2 T_1) ### DisjTree 13 77 79 % 3.95/4.15 81. (distinct_points T_2 T_1) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All V, (((distinct_points T_0 T_2) /\ (distinct_lines (line_connecting T_0 T_2) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_2)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 (line_connecting T_0 T_2)) \/ (apart_point_and_line T_2 V)))))) (distinct_points T_0 T_2) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (distinct_points T_2 T_0) (All Z, ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) ### All 80 % 3.95/4.15 82. (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_2 Z)))) (distinct_points T_2 T_0) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (distinct_points T_0 T_2) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (distinct_points T_2 T_1) ### All 81 % 3.95/4.16 83. ((distinct_points T_1 T_1) /\ (distinct_points T_1 T_0)) ### And 78 % 3.95/4.16 84. ((distinct_points T_1 T_0) => ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_points T_1 T_1) /\ (distinct_points T_1 T_0)))) (distinct_points T_2 T_1) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (distinct_points T_0 T_2) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (distinct_points T_2 T_0) (All Z, ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_0) ### DisjTree 12 82 83 % 3.95/4.16 85. (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_2 Z)))) (distinct_points T_2 T_0) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (distinct_points T_0 T_2) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (distinct_points T_2 T_1) ### All 84 % 3.95/4.16 86. (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_2 T_1) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (distinct_points T_0 T_2) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (distinct_points T_2 T_0) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) ### All 85 % 3.95/4.16 87. ((distinct_points T_0 T_0) /\ (distinct_points T_0 T_2)) ### And 2 % 3.95/4.16 88. ((distinct_points T_0 T_2) => ((apart_point_and_line T_0 (line_connecting T_0 T_2)) => ((distinct_points T_0 T_0) /\ (distinct_points T_0 T_2)))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (distinct_points T_2 T_0) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (distinct_points T_2 T_1) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_0 T_2) ### DisjTree 11 86 87 % 4.01/4.18 89. (distinct_points T_0 T_2) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_2 T_1) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (distinct_points T_2 T_0) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) ### All 88 % 4.01/4.18 90. ((distinct_points T_0 T_2) /\ (distinct_points T_0 T_0)) ### And 2 % 4.01/4.18 91. ((distinct_points T_2 T_0) => ((apart_point_and_line T_0 (line_connecting T_2 T_0)) => ((distinct_points T_0 T_2) /\ (distinct_points T_0 T_0)))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (distinct_points T_2 T_1) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_0 T_2) (distinct_points T_2 T_0) ### DisjTree 10 89 90 % 4.01/4.18 92. (distinct_points T_2 T_0) (distinct_points T_0 T_2) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_2 T_1) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) ### All 91 % 4.01/4.20 93. ((distinct_points T_0 T_1) /\ (distinct_points T_0 T_0)) ### And 2 % 4.01/4.20 94. ((distinct_points T_1 T_0) => ((apart_point_and_line T_0 (line_connecting T_1 T_0)) => ((distinct_points T_0 T_1) /\ (distinct_points T_0 T_0)))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (distinct_points T_0 T_1) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (distinct_points T_2 T_1) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_0 T_2) (distinct_points T_2 T_0) (distinct_points T_1 T_0) ### DisjTree 9 92 93 % 4.01/4.20 95. (distinct_points T_1 T_0) (distinct_points T_2 T_0) (distinct_points T_0 T_2) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_2 T_1) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) ### All 94 % 4.01/4.20 96. ((distinct_points T_0 T_0) /\ (distinct_points T_0 T_1)) ### And 2 % 4.01/4.20 97. ((distinct_points T_0 T_1) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) => ((distinct_points T_0 T_0) /\ (distinct_points T_0 T_1)))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (distinct_points T_2 T_1) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_0 T_2) (distinct_points T_2 T_0) (distinct_points T_1 T_0) (distinct_points T_0 T_1) ### DisjTree 8 95 96 % 4.01/4.20 98. (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) (distinct_points T_1 T_0) (distinct_points T_2 T_0) (distinct_points T_0 T_2) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_2 T_1) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) ### All 97 % 4.01/4.22 99. (distinct_points T_2 T_1) (-. (distinct_points T_2 T_1)) ### Axiom % 4.01/4.22 100. (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_0)) (-. (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_0))) ### Axiom % 4.01/4.22 101. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 4.01/4.22 102. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 4.01/4.22 103. (distinct_points T_1 T_0) (-. (distinct_points T_1 T_0)) ### Axiom % 4.01/4.22 104. (distinct_points T_2 T_0) (-. (distinct_points T_2 T_0)) ### Axiom % 4.01/4.22 105. (distinct_points T_2 T_0) (-. (distinct_points T_2 T_0)) ### Axiom % 4.01/4.22 106. (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) ### Axiom % 4.01/4.22 107. (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) ### Axiom % 4.01/4.22 108. ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0)) \/ (apart_point_and_line T_2 (line_connecting T_1 T_0)))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) ### DisjTree 106 44 107 % 4.01/4.22 109. (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) ### All 108 % 4.01/4.22 110. (-. (apart_point_and_line T_2 (line_connecting T_2 T_0))) (apart_point_and_line T_2 (line_connecting T_2 T_0)) ### Axiom % 4.01/4.22 111. ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_0)) \/ (apart_point_and_line T_2 (line_connecting T_2 T_0)))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) ### DisjTree 109 57 110 % 4.01/4.22 112. (All Z, ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_0))) ### All 111 % 4.01/4.22 113. (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) ### All 112 % 4.01/4.22 114. (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_0))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) ### All 113 % 4.01/4.22 115. ((distinct_points T_2 T_0) => ((apart_point_and_line T_2 (line_connecting T_2 T_0)) => ((distinct_points T_2 T_2) /\ (distinct_points T_2 T_0)))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (distinct_points T_2 T_0) ### DisjTree 105 114 61 % 4.01/4.23 116. (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (distinct_points T_2 T_0) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) ### All 115 % 4.01/4.23 117. (distinct_points T_1 T_0) (-. (distinct_points T_1 T_0)) ### Axiom % 4.01/4.23 118. ((distinct_points T_1 T_0) => ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_points T_1 T_1) /\ (distinct_points T_1 T_0)))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_0)) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_1 T_0) V)) => ((apart_point_and_line T_0 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_1 T_0)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_1 T_0) ### DisjTree 117 56 83 % 4.01/4.23 119. (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_1 T_0) V)) => ((apart_point_and_line T_0 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_1 T_0)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_0)) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) ### All 118 % 4.01/4.23 120. (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_0)) (distinct_points T_0 T_1) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) ### All 119 % 4.01/4.23 121. (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (apart_point_and_line T_1 (line_connecting T_2 T_0)) ### Axiom % 4.01/4.23 122. ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_0)) \/ (apart_point_and_line T_1 (line_connecting T_2 T_0)))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (distinct_points T_2 T_0) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) ### DisjTree 116 120 121 % 4.01/4.23 123. (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (distinct_points T_2 T_0) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) ### All 122 % 4.01/4.23 124. (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (distinct_points T_2 T_0) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) ### All 123 % 4.01/4.25 125. ((distinct_points T_2 T_0) => ((apart_point_and_line T_0 (line_connecting T_2 T_0)) => ((distinct_points T_0 T_2) /\ (distinct_points T_0 T_0)))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (distinct_points T_2 T_0) ### DisjTree 104 124 90 % 4.01/4.25 126. (distinct_points T_2 T_0) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) ### All 125 % 4.01/4.25 127. ((distinct_points T_1 T_0) => ((apart_point_and_line T_0 (line_connecting T_1 T_0)) => ((distinct_points T_0 T_1) /\ (distinct_points T_0 T_0)))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (distinct_points T_2 T_0) (distinct_points T_1 T_0) ### DisjTree 103 126 93 % 4.01/4.25 128. (distinct_points T_1 T_0) (distinct_points T_2 T_0) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) ### All 127 % 4.01/4.25 129. ((distinct_points T_0 T_1) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) => ((distinct_points T_0 T_0) /\ (distinct_points T_0 T_1)))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (distinct_points T_2 T_0) (distinct_points T_1 T_0) (distinct_points T_0 T_1) ### DisjTree 102 128 96 % 4.01/4.25 130. (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) (distinct_points T_1 T_0) (distinct_points T_2 T_0) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) ### All 129 % 4.01/4.25 131. ((distinct_points T_1 T_0) /\ (distinct_points T_1 T_1)) ### And 78 % 4.01/4.25 132. ((distinct_points T_0 T_1) => ((apart_point_and_line T_1 (line_connecting T_0 T_1)) => ((distinct_points T_1 T_0) /\ (distinct_points T_1 T_1)))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (distinct_points T_2 T_0) (distinct_points T_1 T_0) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) ### DisjTree 101 130 131 % 4.09/4.26 133. (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_0) (distinct_points T_2 T_0) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) ### All 132 % 4.09/4.26 134. (distinct_points T_2 T_0) (-. (distinct_points T_2 T_0)) ### Axiom % 4.09/4.26 135. (apart_point_and_line T_2 (line_connecting T_2 T_0)) (-. (apart_point_and_line T_2 (line_connecting T_2 T_0))) ### Axiom % 4.09/4.26 136. ((distinct_points T_2 T_0) => ((apart_point_and_line T_2 (line_connecting T_2 T_0)) => ((distinct_points T_2 T_2) /\ (distinct_points T_2 T_0)))) (apart_point_and_line T_2 (line_connecting T_2 T_0)) (distinct_points T_2 T_0) ### DisjTree 134 135 61 % 4.09/4.26 137. (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (distinct_points T_2 T_0) (apart_point_and_line T_2 (line_connecting T_2 T_0)) ### All 136 % 4.09/4.26 138. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 4.09/4.26 139. (apart_point_and_line T_1 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) ### Axiom % 4.09/4.26 140. ((distinct_points T_0 T_1) => ((apart_point_and_line T_1 (line_connecting T_0 T_1)) => ((distinct_points T_1 T_0) /\ (distinct_points T_1 T_1)))) (apart_point_and_line T_1 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### DisjTree 138 139 131 % 4.09/4.26 141. (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) (apart_point_and_line T_1 (line_connecting T_0 T_1)) ### All 140 % 4.09/4.26 142. (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (apart_point_and_line T_1 (line_connecting T_2 T_0)) ### Axiom % 4.09/4.26 143. (((distinct_points T_2 T_1) /\ (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_0))) => ((apart_point_and_line T_2 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_2 (line_connecting T_2 T_0)) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 (line_connecting T_2 T_0)))))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (distinct_points T_2 T_0) (distinct_points T_1 T_0) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_0)) (distinct_points T_2 T_1) ### DisjTree 99 100 133 137 141 142 % 4.09/4.26 144. (All V, (((distinct_points T_2 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_2 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_2 T_1) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_0)) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_0) (distinct_points T_2 T_0) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) ### All 143 % 4.09/4.26 145. (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (distinct_points T_2 T_0) (distinct_points T_1 T_0) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_0)) (distinct_points T_2 T_1) ### All 144 % 4.11/4.28 146. (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (apart_point_and_line T_1 (line_connecting T_2 T_0)) ### Axiom % 4.11/4.28 147. ((apart_point_and_line T_1 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_0)) \/ (apart_point_and_line T_1 (line_connecting T_2 T_0)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (distinct_points T_2 T_1) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_0 T_2) (distinct_points T_2 T_0) (distinct_points T_1 T_0) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) ### DisjTree 98 145 146 % 4.11/4.28 148. (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_1 Z)))) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) (distinct_points T_1 T_0) (distinct_points T_2 T_0) (distinct_points T_0 T_2) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_2 T_1) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) ### All 147 % 4.11/4.28 149. (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (distinct_points T_2 T_1) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_0 T_2) (distinct_points T_2 T_0) (distinct_points T_1 T_0) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) ### All 148 % 4.11/4.28 150. (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) (distinct_points T_1 T_0) (distinct_points T_2 T_0) (distinct_points T_0 T_2) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_2 T_1) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) ### All 149 % 4.11/4.30 151. (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, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (distinct_points T_2 T_1) (distinct_points T_0 T_2) (distinct_points T_2 T_0) (distinct_points T_1 T_0) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) ### All 150 % 4.11/4.30 152. (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) (distinct_points T_1 T_0) (distinct_points T_2 T_0) (distinct_points T_0 T_2) (distinct_points T_2 T_1) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 151 % 4.11/4.30 153. ((distinct_points T_0 T_2) => ((distinct_points T_0 T_2) \/ (distinct_points T_2 T_2))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (distinct_points T_2 T_1) (distinct_points T_1 T_0) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_0) (All Z, ((distinct_points T_2 T_0) => ((distinct_points T_2 Z) \/ (distinct_points T_0 Z)))) ### DisjTree 7 152 4 % 4.11/4.30 154. (All Z, ((distinct_points T_0 T_2) => ((distinct_points T_0 Z) \/ (distinct_points T_2 Z)))) (All Z, ((distinct_points T_2 T_0) => ((distinct_points T_2 Z) \/ (distinct_points T_0 Z)))) (distinct_points T_2 T_0) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) (distinct_points T_1 T_0) (distinct_points T_2 T_1) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 153 % 4.11/4.31 155. (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points 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, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (distinct_points T_2 T_1) (distinct_points T_1 T_0) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_0) (All Z, ((distinct_points T_2 T_0) => ((distinct_points T_2 Z) \/ (distinct_points T_0 Z)))) (All Z, ((distinct_points T_0 T_2) => ((distinct_points T_0 Z) \/ (distinct_points T_2 Z)))) ### All 154 % 4.11/4.31 156. (All Y, (All Z, ((distinct_points T_2 Y) => ((apart_point_and_line Z (line_connecting T_2 Y)) => ((distinct_points Z T_2) /\ (distinct_points Z Y)))))) (All Z, ((distinct_points T_0 T_2) => ((distinct_points T_0 Z) \/ (distinct_points T_2 Z)))) (All Z, ((distinct_points T_2 T_0) => ((distinct_points T_2 Z) \/ (distinct_points T_0 Z)))) (distinct_points T_2 T_0) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) (distinct_points T_1 T_0) (distinct_points T_2 T_1) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((distinct_points T_2 T_0) => ((apart_point_and_line Z (line_connecting T_2 T_0)) => ((distinct_points Z T_2) /\ (distinct_points Z T_0))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) ### All 155 % 4.11/4.31 157. (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points 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, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (distinct_points T_2 T_1) (distinct_points T_1 T_0) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_0) (All Z, ((distinct_points T_2 T_0) => ((distinct_points T_2 Z) \/ (distinct_points T_0 Z)))) (All Z, ((distinct_points T_0 T_2) => ((distinct_points T_0 Z) \/ (distinct_points T_2 Z)))) (All Y, (All Z, ((distinct_points T_2 Y) => ((apart_point_and_line Z (line_connecting T_2 Y)) => ((distinct_points Z T_2) /\ (distinct_points Z Y)))))) ### All 156 % 4.11/4.31 158. (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (All Y, (All Z, ((distinct_points T_2 Y) => ((apart_point_and_line Z (line_connecting T_2 Y)) => ((distinct_points Z T_2) /\ (distinct_points Z Y)))))) (All Z, ((distinct_points T_0 T_2) => ((distinct_points T_0 Z) \/ (distinct_points T_2 Z)))) (All Z, ((distinct_points T_2 T_0) => ((distinct_points T_2 Z) \/ (distinct_points T_0 Z)))) (distinct_points T_2 T_0) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) (distinct_points T_1 T_0) (distinct_points T_2 T_1) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) ### All 157 % 4.11/4.32 159. (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points 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, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (distinct_points T_2 T_1) (distinct_points T_1 T_0) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_0) (All Z, ((distinct_points T_2 T_0) => ((distinct_points T_2 Z) \/ (distinct_points T_0 Z)))) (All Y, (All Z, ((distinct_points T_2 Y) => ((apart_point_and_line Z (line_connecting T_2 Y)) => ((distinct_points Z T_2) /\ (distinct_points Z Y)))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) ### All 158 % 4.11/4.32 160. (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (All Z, ((distinct_points T_2 T_0) => ((distinct_points T_2 Z) \/ (distinct_points T_0 Z)))) (distinct_points T_2 T_0) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) (distinct_points T_1 T_0) (distinct_points T_2 T_1) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_2) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_2 U) \/ (apart_point_and_line T_2 V))))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) ### All 159 % 4.11/4.32 161. (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points 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, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (distinct_points T_2 T_1) (distinct_points T_1 T_0) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_0) (All Z, ((distinct_points T_2 T_0) => ((distinct_points T_2 Z) \/ (distinct_points T_0 Z)))) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) ### All 160 % 4.11/4.32 162. (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (All Z, ((distinct_points T_2 T_0) => ((distinct_points T_2 Z) \/ (distinct_points T_0 Z)))) (distinct_points T_2 T_0) (distinct_points T_0 T_1) (distinct_points T_1 T_0) (distinct_points T_2 T_1) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) ### All 161 % 4.16/4.33 163. (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points 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, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (distinct_points T_2 T_1) (distinct_points T_1 T_0) (distinct_points T_0 T_1) (distinct_points T_2 T_0) (All Z, ((distinct_points T_2 T_0) => ((distinct_points T_2 Z) \/ (distinct_points T_0 Z)))) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) ### All 162 % 4.16/4.33 164. ((distinct_points T_0 T_1) => ((distinct_points T_0 T_0) \/ (distinct_points T_1 T_0))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (All Z, ((distinct_points T_2 T_0) => ((distinct_points T_2 Z) \/ (distinct_points T_0 Z)))) (distinct_points T_2 T_0) (distinct_points T_2 T_1) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points T_0 T_1) ### DisjTree 1 2 163 % 4.16/4.33 165. (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points 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, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (distinct_points T_2 T_1) (distinct_points T_2 T_0) (All Z, ((distinct_points T_2 T_0) => ((distinct_points T_2 Z) \/ (distinct_points T_0 Z)))) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) ### All 164 % 4.16/4.33 166. (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (All Z, ((distinct_points T_2 T_0) => ((distinct_points T_2 Z) \/ (distinct_points T_0 Z)))) (distinct_points T_2 T_0) (distinct_points T_2 T_1) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 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 Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) ### All 165 % 4.18/4.34 167. (distinct_points T_0 T_1) (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points 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 T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (distinct_points T_2 T_1) (distinct_points T_2 T_0) (All Z, ((distinct_points T_2 T_0) => ((distinct_points T_2 Z) \/ (distinct_points T_0 Z)))) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) ### All 166 % 4.18/4.34 168. (All Y, (All Z, ((distinct_points T_2 Y) => ((distinct_points T_2 Z) \/ (distinct_points Y Z))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (distinct_points T_2 T_0) (distinct_points T_2 T_1) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 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 Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points T_0 T_1) ### All 167 % 4.18/4.34 169. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_points T_0 T_1) (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points 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 T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (distinct_points T_2 T_1) (distinct_points T_2 T_0) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All Y, (All Z, ((distinct_points T_2 Y) => ((distinct_points T_2 Z) \/ (distinct_points Y Z))))) ### All 168 % 4.18/4.34 170. (All Y, (All Z, ((distinct_points T_2 Y) => ((distinct_points T_2 Z) \/ (distinct_points Y Z))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (distinct_points T_2 T_0) (distinct_points T_2 T_1) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 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 Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points T_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 169 % 4.18/4.34 171. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_points T_0 T_1) (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points 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 T_1 (line_connecting T_2 T_0))) (All U, (All V, (((distinct_points T_2 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (distinct_points T_2 T_1) (distinct_points T_2 T_0) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) ### All 170 % 4.18/4.34 172. (All Y, (All U, (All V, (((distinct_points T_2 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (distinct_points T_2 T_0) (distinct_points T_2 T_1) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (-. (apart_point_and_line T_1 (line_connecting T_2 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 Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points T_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 171 % 4.18/4.34 173. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_points T_0 T_1) (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points 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 T_1 (line_connecting T_2 T_0))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (distinct_points T_2 T_1) (distinct_points T_2 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All Y, (All U, (All V, (((distinct_points T_2 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) ### All 172 % 4.18/4.34 174. (All Y, (All U, (All V, (((distinct_points T_2 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (distinct_points T_2 T_0) (distinct_points T_2 T_1) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (-. (apart_point_and_line T_1 (line_connecting T_2 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 Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points T_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 173 % 4.18/4.34 175. (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, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_points T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (distinct_points T_2 T_1) (distinct_points T_2 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All Y, (All U, (All V, (((distinct_points T_2 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_2 U) \/ ((apart_point_and_line T_2 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) ### All 174 % 4.18/4.34 176. (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (distinct_points T_2 T_0) (distinct_points T_2 T_1) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (-. (apart_point_and_line T_1 (line_connecting T_2 T_0))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (distinct_points T_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y 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 175 % 4.18/4.34 177. (-. (((distinct_points T_2 T_0) /\ ((distinct_points T_2 T_1) /\ ((distinct_points T_0 T_1) /\ (-. (apart_point_and_line T_1 (line_connecting T_2 T_0)))))) => (-. (apart_point_and_line T_1 (line_connecting T_0 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, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points 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, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) ### ConjTree 176 % 4.18/4.34 178. (-. (All Z, (((distinct_points T_2 T_0) /\ ((distinct_points T_2 Z) /\ ((distinct_points T_0 Z) /\ (-. (apart_point_and_line Z (line_connecting T_2 T_0)))))) => (-. (apart_point_and_line Z (line_connecting T_0 T_2)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points 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, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y 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))))))))) ### NotAllEx 177 % 4.18/4.34 179. (-. (All Y, (All Z, (((distinct_points T_2 Y) /\ ((distinct_points T_2 Z) /\ ((distinct_points Y Z) /\ (-. (apart_point_and_line Z (line_connecting T_2 Y)))))) => (-. (apart_point_and_line Z (line_connecting Y 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, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points 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, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) ### NotAllEx 178 % 4.18/4.34 180. (-. (All X, (All Y, (All Z, (((distinct_points X Y) /\ ((distinct_points X Z) /\ ((distinct_points Y Z) /\ (-. (apart_point_and_line Z (line_connecting X Y)))))) => (-. (apart_point_and_line Z (line_connecting Y X)))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points 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, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y 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))))))))) ### NotAllEx 179 % 4.18/4.34 % SZS output end Proof % 4.18/4.34 (* END-PROOF *) %------------------------------------------------------------------------------