%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO177+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 : n029.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 06:30:40 EDT 2022 % Result : Theorem 65.10s 65.27s % Output : Proof 66.01s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.03/0.12 % Problem : GEO177+2 : TPTP v8.1.0. Released v3.3.0. % 0.03/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.13/0.34 % Computer : n029.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 17:51:14 EDT 2022 % 0.13/0.35 % CPUTime : % 65.10/65.27 % SZS status Theorem % 65.10/65.27 (* PROOF-FOUND *) % 65.10/65.27 (* BEGIN-PROOF *) % 65.10/65.27 % SZS output start Proof % 65.10/65.27 1. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 65.10/65.27 2. (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 % 65.10/65.27 3. (-. (distinct_points T_2 T_1)) (distinct_points T_2 T_1) ### Axiom % 65.10/65.27 4. ((distinct_points T_2 T_0) /\ (distinct_points T_2 T_1)) (-. (distinct_points T_2 T_1)) ### And 3 % 65.10/65.27 5. ((distinct_points T_0 T_1) => ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_points T_2 T_0) /\ (distinct_points T_2 T_1)))) (-. (distinct_points T_2 T_1)) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### DisjTree 1 2 4 % 65.10/65.27 6. (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_2 (line_connecting T_0 T_1)) (-. (distinct_points T_2 T_1)) ### All 5 % 65.10/65.27 7. (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_1)) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### All 6 % 65.10/65.27 8. (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_0 T_1) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (distinct_points T_2 T_1)) ### All 7 % 65.10/65.27 9. (distinct_points T_2 T_1) (-. (distinct_points T_2 T_1)) ### Axiom % 65.10/65.27 10. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 65.10/65.27 11. (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 % 65.10/65.27 12. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 65.10/65.27 13. (distinct_points T_2 T_1) (-. (distinct_points T_2 T_1)) ### Axiom % 65.10/65.27 14. (distinct_points T_2 T_3) (-. (distinct_points T_2 T_3)) ### Axiom % 65.10/65.27 15. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 65.10/65.27 16. (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_1)) (-. (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_1))) ### Axiom % 65.10/65.27 17. (-. (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 % 65.10/65.27 18. (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 % 65.10/65.27 19. (distinct_points T_2 T_1) (-. (distinct_points T_2 T_1)) ### Axiom % 65.10/65.27 20. (distinct_lines (line_connecting T_2 T_1) (line_connecting T_2 T_3)) (-. (distinct_lines (line_connecting T_2 T_1) (line_connecting T_2 T_3))) ### Axiom % 65.10/65.27 21. (-. (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 % 65.10/65.27 22. (-. (apart_point_and_line T_2 (line_connecting T_2 T_3))) (apart_point_and_line T_2 (line_connecting T_2 T_3)) ### Axiom % 65.10/65.27 23. (distinct_points T_2 T_1) (-. (distinct_points T_2 T_1)) ### Axiom % 65.10/65.27 24. (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 % 65.10/65.27 25. (distinct_points T_1 T_1) ### Extension/test/apart1 % 65.10/65.27 26. ((distinct_points T_1 T_2) /\ (distinct_points T_1 T_1)) ### And 25 % 65.10/65.27 27. ((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)))) (apart_point_and_line T_1 (line_connecting T_2 T_1)) (distinct_points T_2 T_1) ### DisjTree 23 24 26 % 65.10/65.27 28. (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_1 (line_connecting T_2 T_1)) ### All 27 % 65.10/65.27 29. (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (apart_point_and_line T_1 (line_connecting T_2 T_3)) ### Axiom % 65.10/65.27 30. (((distinct_points T_2 T_1) /\ (distinct_lines (line_connecting T_2 T_1) (line_connecting T_2 T_3))) => ((apart_point_and_line T_2 (line_connecting T_2 T_1)) \/ ((apart_point_and_line T_2 (line_connecting T_2 T_3)) \/ ((apart_point_and_line T_1 (line_connecting T_2 T_1)) \/ (apart_point_and_line T_1 (line_connecting T_2 T_3)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_1))) (distinct_lines (line_connecting T_2 T_1) (line_connecting T_2 T_3)) (distinct_points T_2 T_1) ### DisjTree 19 20 21 22 28 29 % 65.10/65.27 31. (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_2 T_3)) (-. (apart_point_and_line T_2 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_3))) (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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) ### All 30 % 65.10/65.27 32. (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_0 (line_connecting T_2 T_3)) ### Axiom % 65.10/65.27 33. ((apart_point_and_line T_0 (line_connecting T_2 T_1)) => ((distinct_lines (line_connecting T_2 T_1) (line_connecting T_2 T_3)) \/ (apart_point_and_line T_0 (line_connecting T_2 T_3)))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_3))) (-. (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 18 31 32 % 65.10/65.27 34. (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_2 T_3))) (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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) ### All 33 % 65.10/65.27 35. (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_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_1))) (distinct_points T_2 T_1) (apart_point_and_line T_0 (line_connecting T_2 T_1)) (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)))) ### All 34 % 65.38/65.60 36. (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_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_2 T_3))) (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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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 35 % 65.38/65.60 37. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_1))) (distinct_points T_2 T_1) (apart_point_and_line T_0 (line_connecting T_2 T_1)) ### All 36 % 65.38/65.60 38. (-. (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 % 65.38/65.60 39. (-. (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 % 65.38/65.60 40. (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_1))) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 (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_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))) (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_2 T_3))) (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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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 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_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_1)) (distinct_points T_0 T_1) ### DisjTree 15 16 17 37 38 39 % 65.38/65.60 41. (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_2 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_1))) (distinct_points 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_2 T_1))) ### All 40 % 65.38/65.60 42. (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_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 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_2 T_3))) (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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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 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_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_1)) (distinct_points T_0 T_1) ### All 41 % 65.38/65.60 43. (distinct_points T_2 T_2) ### Extension/test/apart1 % 65.38/65.60 44. ((distinct_points T_2 T_2) /\ (distinct_points T_2 T_3)) ### And 43 % 65.38/65.60 45. ((distinct_points T_2 T_3) => ((apart_point_and_line T_2 (line_connecting T_2 T_3)) => ((distinct_points T_2 T_2) /\ (distinct_points T_2 T_3)))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_1))) (distinct_points 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_2 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))))))) (distinct_points T_2 T_3) ### DisjTree 14 42 44 % 65.38/65.60 46. (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (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_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_2 T_1) (-. (apart_point_and_line T_2 (line_connecting 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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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 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_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_1)) (distinct_points T_0 T_1) ### All 45 % 65.38/65.62 47. ((distinct_points T_2 T_2) /\ (distinct_points T_2 T_1)) ### And 43 % 65.38/65.62 48. ((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)))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))) (-. (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_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_2 T_3) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_1) ### DisjTree 13 46 47 % 65.38/65.62 49. (distinct_points T_2 T_1) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (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_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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 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_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_1)) (distinct_points T_0 T_1) ### All 48 % 65.38/65.62 50. (distinct_points T_0 T_0) ### Extension/test/apart1 % 65.38/65.62 51. ((distinct_points T_0 T_0) /\ (distinct_points T_0 T_1)) ### And 50 % 65.38/65.62 52. ((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)))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_1)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))) (-. (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_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_2 T_3) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_1) (distinct_points T_0 T_1) ### DisjTree 12 49 51 % 65.38/65.62 53. (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_2 T_1) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (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_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_1)) ### All 52 % 65.38/65.62 54. (distinct_points T_2 T_1) (-. (distinct_points T_2 T_1)) ### Axiom % 65.38/65.62 55. (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 % 65.38/65.62 56. ((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 54 55 47 % 65.38/65.62 57. (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 56 % 65.38/65.62 58. ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_1)) \/ (apart_point_and_line T_2 (line_connecting T_2 T_1)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))) (-. (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_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_2 T_3) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_1) (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))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) ### DisjTree 11 53 57 % 65.47/65.68 59. (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 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_2 T_1) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (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_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 58 % 65.47/65.68 60. ((distinct_points T_1 T_0) /\ (distinct_points T_1 T_1)) ### And 25 % 65.47/65.68 61. ((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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 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))))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 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))))) (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_0 T_1) ### DisjTree 10 59 60 % 65.47/65.68 62. (distinct_points 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)))) (apart_point_and_line T_2 (line_connecting 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_1) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (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_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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 61 % 65.47/65.68 63. ((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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_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_2 T_3) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (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))))) (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_0 T_1) (distinct_points T_2 T_1) ### DisjTree 9 62 26 % 65.47/65.68 64. (distinct_points T_2 T_1) (distinct_points 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)))) (apart_point_and_line T_2 (line_connecting 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 Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (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, ((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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 63 % 65.47/65.68 65. (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_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_2 T_3) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (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_0 T_1) (distinct_points T_2 T_1) ### All 64 % 65.54/65.72 66. (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_1) (distinct_points 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)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (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, ((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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 65 % 65.54/65.72 67. (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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_2 T_3) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (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_0 T_1) (distinct_points T_2 T_1) (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 66 % 65.54/65.72 68. (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_1) (distinct_points 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)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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 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_2 Y) => ((apart_point_and_line Z (line_connecting T_2 Y)) => ((distinct_points Z T_2) /\ (distinct_points Z Y)))))) ### All 67 % 65.54/65.72 69. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_2 T_3) (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_0 T_1) (distinct_points T_2 T_1) (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 68 % 65.54/65.72 70. (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 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))))))) (distinct_points T_2 T_1) (distinct_points T_0 T_1) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 69 % 65.54/65.72 71. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_2 T_3) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) (distinct_points T_2 T_1) (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 70 % 65.54/65.72 72. ((distinct_points T_2 T_1) => ((distinct_points T_2 T_1) \/ (distinct_points T_1 T_1))) (distinct_points T_2 T_3) (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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 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_2 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) (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))))))) ### DisjTree 8 71 25 % 65.54/65.73 73. (All Z, ((distinct_points T_2 T_1) => ((distinct_points T_2 Z) \/ (distinct_points T_1 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))))))) (distinct_points T_0 T_1) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_2 T_3) ### All 72 % 65.54/65.73 74. (All Y, (All Z, ((distinct_points T_2 Y) => ((distinct_points T_2 Z) \/ (distinct_points Y Z))))) (distinct_points T_2 T_3) (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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 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_2 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) (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 73 % 65.54/65.73 75. (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_0 T_1) (apart_point_and_line T_2 (line_connecting 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_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_2 T_3) (All Y, (All Z, ((distinct_points T_2 Y) => ((distinct_points T_2 Z) \/ (distinct_points Y Z))))) ### All 74 % 65.54/65.73 76. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_points T_2 T_3) (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_2 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) (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 75 % 65.54/65.73 77. (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) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (distinct_points T_0 T_1) (apart_point_and_line T_2 (line_connecting 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_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_2 T_3) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 76 % 65.54/65.73 78. (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 X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_points T_2 T_3) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_2 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) (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 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 77 % 65.54/65.73 79. (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) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (distinct_points T_0 T_1) (apart_point_and_line T_2 (line_connecting 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_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (distinct_points T_2 T_3) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 78 % 65.54/65.73 80. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 65.54/65.73 81. (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_3 (line_connecting T_0 T_1))) ### Axiom % 65.54/65.73 82. (-. (distinct_points T_3 T_1)) (distinct_points T_3 T_1) ### Axiom % 65.54/65.73 83. ((distinct_points T_3 T_0) /\ (distinct_points T_3 T_1)) (-. (distinct_points T_3 T_1)) ### And 82 % 65.54/65.73 84. ((distinct_points T_0 T_1) => ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_points T_3 T_0) /\ (distinct_points T_3 T_1)))) (-. (distinct_points T_3 T_1)) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### DisjTree 80 81 83 % 65.62/65.79 85. (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_3 (line_connecting T_0 T_1)) (-. (distinct_points T_3 T_1)) ### All 84 % 65.62/65.79 86. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 65.62/65.79 87. (-. (distinct_points T_1 T_0)) (distinct_points T_1 T_0) ### Axiom % 65.62/65.79 88. ((distinct_points T_0 T_1) => ((distinct_points T_0 T_0) \/ (distinct_points T_1 T_0))) (-. (distinct_points T_1 T_0)) (distinct_points T_0 T_1) ### DisjTree 86 50 87 % 65.62/65.79 89. (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (-. (distinct_points T_1 T_0)) ### All 88 % 65.62/65.79 90. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 65.62/65.79 91. (distinct_points T_1 T_0) (-. (distinct_points T_1 T_0)) ### Axiom % 65.62/65.79 92. (distinct_points T_3 T_1) (-. (distinct_points T_3 T_1)) ### Axiom % 65.62/65.79 93. (distinct_points T_1 T_0) (-. (distinct_points T_1 T_0)) ### Axiom % 65.62/65.79 94. (distinct_points T_3 T_1) (-. (distinct_points T_3 T_1)) ### Axiom % 65.62/65.79 95. (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_3 (line_connecting T_0 T_1))) ### Axiom % 65.62/65.79 96. (-. (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 % 65.62/65.79 97. (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (apart_point_and_line T_3 (line_connecting T_1 T_0)) ### Axiom % 65.62/65.79 98. ((apart_point_and_line T_3 (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_3 (line_connecting T_1 T_0)))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) ### DisjTree 95 96 97 % 65.62/65.79 99. (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (apart_point_and_line T_3 (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_3 (line_connecting T_1 T_0))) ### All 98 % 65.62/65.79 100. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 65.62/65.79 101. (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_3)) (-. (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_3))) ### Axiom % 65.62/65.79 102. (-. (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 % 65.62/65.79 103. (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_0 (line_connecting T_2 T_3)) ### Axiom % 65.62/65.79 104. (-. (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 % 65.62/65.79 105. (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (apart_point_and_line T_1 (line_connecting T_2 T_3)) ### Axiom % 65.62/65.79 106. (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_3))) => ((apart_point_and_line T_0 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_0 (line_connecting T_2 T_3)) \/ ((apart_point_and_line T_1 (line_connecting T_1 T_0)) \/ (apart_point_and_line T_1 (line_connecting T_2 T_3)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (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_3)) (distinct_points T_0 T_1) ### DisjTree 100 101 102 103 104 105 % 65.62/65.79 107. (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_3)) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) ### All 106 % 65.62/65.79 108. (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_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (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_3)) (distinct_points T_0 T_1) ### All 107 % 65.62/65.79 109. (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (apart_point_and_line T_3 (line_connecting T_2 T_3)) ### Axiom % 65.62/65.79 110. ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_3)) \/ (apart_point_and_line T_3 (line_connecting T_2 T_3)))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (distinct_points 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_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) ### DisjTree 99 108 109 % 65.62/65.79 111. (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (distinct_lines (line_connecting T_0 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) ### All 110 % 65.62/65.79 112. (distinct_points T_3 T_1) (-. (distinct_points T_3 T_1)) ### Axiom % 65.62/65.79 113. (distinct_lines (line_connecting T_2 T_3) (line_connecting T_3 T_1)) (-. (distinct_lines (line_connecting T_2 T_3) (line_connecting T_3 T_1))) ### Axiom % 65.62/65.79 114. (distinct_points T_2 T_3) (-. (distinct_points T_2 T_3)) ### Axiom % 65.62/65.79 115. (apart_point_and_line T_3 (line_connecting T_2 T_3)) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) ### Axiom % 65.62/65.79 116. (distinct_points T_3 T_3) ### Extension/test/apart1 % 65.62/65.79 117. ((distinct_points T_3 T_2) /\ (distinct_points T_3 T_3)) ### And 116 % 65.62/65.79 118. ((distinct_points T_2 T_3) => ((apart_point_and_line T_3 (line_connecting T_2 T_3)) => ((distinct_points T_3 T_2) /\ (distinct_points T_3 T_3)))) (apart_point_and_line T_3 (line_connecting T_2 T_3)) (distinct_points T_2 T_3) ### DisjTree 114 115 117 % 65.62/65.79 119. (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (apart_point_and_line T_3 (line_connecting T_2 T_3)) ### All 118 % 65.67/65.84 120. (distinct_points T_3 T_1) (-. (distinct_points T_3 T_1)) ### Axiom % 65.67/65.84 121. (apart_point_and_line T_3 (line_connecting T_3 T_1)) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) ### Axiom % 65.67/65.84 122. ((distinct_points T_3 T_3) /\ (distinct_points T_3 T_1)) ### And 116 % 65.67/65.84 123. ((distinct_points T_3 T_1) => ((apart_point_and_line T_3 (line_connecting T_3 T_1)) => ((distinct_points T_3 T_3) /\ (distinct_points T_3 T_1)))) (apart_point_and_line T_3 (line_connecting T_3 T_1)) (distinct_points T_3 T_1) ### DisjTree 120 121 122 % 65.67/65.84 124. (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_3 T_1) (apart_point_and_line T_3 (line_connecting T_3 T_1)) ### All 123 % 65.67/65.84 125. (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (apart_point_and_line T_1 (line_connecting T_2 T_3)) ### Axiom % 65.67/65.84 126. (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) (apart_point_and_line T_1 (line_connecting T_3 T_1)) ### Axiom % 65.67/65.84 127. (((distinct_points T_3 T_1) /\ (distinct_lines (line_connecting T_2 T_3) (line_connecting T_3 T_1))) => ((apart_point_and_line T_3 (line_connecting T_2 T_3)) \/ ((apart_point_and_line T_3 (line_connecting T_3 T_1)) \/ ((apart_point_and_line T_1 (line_connecting T_2 T_3)) \/ (apart_point_and_line T_1 (line_connecting T_3 T_1)))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_lines (line_connecting T_2 T_3) (line_connecting T_3 T_1)) (distinct_points T_3 T_1) ### DisjTree 112 113 119 124 125 126 % 65.67/65.84 128. (All V, (((distinct_points T_3 T_1) /\ (distinct_lines (line_connecting T_2 T_3) V)) => ((apart_point_and_line T_3 (line_connecting T_2 T_3)) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 (line_connecting T_2 T_3)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_3 T_1) (distinct_lines (line_connecting T_2 T_3) (line_connecting T_3 T_1)) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) ### All 127 % 65.67/65.84 129. (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_lines (line_connecting T_2 T_3) (line_connecting T_3 T_1)) (distinct_points T_3 T_1) ### All 128 % 65.67/65.84 130. (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (apart_point_and_line T_3 (line_connecting T_3 T_1)) ### Axiom % 65.67/65.84 131. ((apart_point_and_line T_3 (line_connecting T_2 T_3)) => ((distinct_lines (line_connecting T_2 T_3) (line_connecting T_3 T_1)) \/ (apart_point_and_line T_3 (line_connecting T_3 T_1)))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 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_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) ### DisjTree 111 129 130 % 65.67/65.84 132. (All Z, ((apart_point_and_line T_3 (line_connecting T_2 T_3)) => ((distinct_lines (line_connecting T_2 T_3) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (distinct_lines (line_connecting T_0 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_3 T_1) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) ### All 131 % 65.67/65.84 133. ((distinct_points T_3 T_1) => ((apart_point_and_line T_3 (line_connecting T_3 T_1)) => ((distinct_points T_3 T_3) /\ (distinct_points T_3 T_1)))) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 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_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_2 T_3)) => ((distinct_lines (line_connecting T_2 T_3) Z) \/ (apart_point_and_line T_3 Z)))) (distinct_points T_3 T_1) ### DisjTree 94 132 122 % 65.70/65.86 134. (distinct_points T_3 T_1) (All Z, ((apart_point_and_line T_3 (line_connecting T_2 T_3)) => ((distinct_lines (line_connecting T_2 T_3) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (distinct_lines (line_connecting T_0 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) ### All 133 % 65.70/65.86 135. ((distinct_points T_1 T_1) /\ (distinct_points T_1 T_0)) ### And 25 % 65.70/65.86 136. ((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)))) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 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_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_2 T_3)) => ((distinct_lines (line_connecting T_2 T_3) Z) \/ (apart_point_and_line T_3 Z)))) (distinct_points T_3 T_1) (distinct_points T_1 T_0) ### DisjTree 93 134 135 % 65.70/65.86 137. (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) (distinct_points T_3 T_1) (All Z, ((apart_point_and_line T_3 (line_connecting T_2 T_3)) => ((distinct_lines (line_connecting T_2 T_3) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (distinct_lines (line_connecting T_0 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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) ### All 136 % 65.70/65.86 138. ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)) ### And 25 % 65.70/65.86 139. ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 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_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_2 T_3)) => ((distinct_lines (line_connecting T_2 T_3) Z) \/ (apart_point_and_line T_3 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))))) (distinct_points T_3 T_1) ### DisjTree 92 137 138 % 65.70/65.86 140. (-. (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 % 65.70/65.91 141. (-. (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 % 65.70/65.91 142. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 65.70/65.91 143. (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 % 65.70/65.91 144. ((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)))) (apart_point_and_line T_0 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### DisjTree 142 143 51 % 65.70/65.91 145. (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_0 (line_connecting T_0 T_1)) ### All 144 % 65.70/65.91 146. (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)))))) (apart_point_and_line T_0 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### All 145 % 65.70/65.91 147. (-. (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 % 65.70/65.91 148. (((distinct_points T_1 T_0) /\ (distinct_lines (line_connecting T_0 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_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)))))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_3 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 Z, ((apart_point_and_line T_3 (line_connecting T_2 T_3)) => ((distinct_lines (line_connecting T_2 T_3) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (apart_point_and_line T_3 (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (distinct_points T_1 T_0) ### DisjTree 91 139 140 141 146 147 % 65.70/65.91 149. (All V, (((distinct_points T_1 T_0) /\ (distinct_lines (line_connecting T_0 T_1) 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_0 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_0 V)))))) (distinct_points T_1 T_0) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 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_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_3 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_2 T_3)) => ((distinct_lines (line_connecting T_2 T_3) Z) \/ (apart_point_and_line T_3 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_3 T_1) (-. (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 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 148 % 65.70/65.91 150. (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_3 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 Z, ((apart_point_and_line T_3 (line_connecting T_2 T_3)) => ((distinct_lines (line_connecting T_2 T_3) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (apart_point_and_line T_3 (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (distinct_points T_1 T_0) ### All 149 % 65.80/65.96 151. (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (distinct_points T_1 T_0) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 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_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_3 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_2 T_3)) => ((distinct_lines (line_connecting T_2 T_3) Z) \/ (apart_point_and_line T_3 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_3 T_1) (-. (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 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 U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) ### All 150 % 65.80/65.96 152. (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_3 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 Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (apart_point_and_line T_3 (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (distinct_points T_1 T_0) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) ### All 151 % 65.80/65.96 153. ((distinct_points T_1 T_0) => ((distinct_points T_1 T_0) \/ (distinct_points T_0 T_0))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 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))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_3 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 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_3 T_1) (-. (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 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 U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 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)))) ### DisjTree 89 152 50 % 65.80/65.96 154. (All Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (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 U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_3 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 Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (apart_point_and_line T_3 (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) ### All 153 % 65.80/65.98 155. ((distinct_points T_0 T_1) /\ (distinct_points T_0 T_0)) ### And 50 % 65.80/65.98 156. ((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 Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_3 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 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_3 T_1) (-. (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 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 U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (All Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) ### DisjTree 89 154 155 % 65.80/65.98 157. (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 Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_3 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 Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (apart_point_and_line T_3 (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) ### All 156 % 65.80/65.98 158. (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_3 (line_connecting 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))))) (distinct_points T_3 T_1) (-. (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 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 U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (All Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (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 157 % 65.80/65.98 159. (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_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (All Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_3 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))))) (apart_point_and_line T_3 (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_3) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) ### All 158 % 65.85/66.03 160. (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, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_3 (line_connecting 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))))) (distinct_points T_3 T_1) (-. (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 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 U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (All Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (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 159 % 65.85/66.03 161. ((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_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (All Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (distinct_points T_3 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))))) (apart_point_and_line T_3 (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_3) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 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))))))) (distinct_points T_0 T_1) ### DisjTree 90 160 60 % 65.85/66.03 162. (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) (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, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_3 (line_connecting 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))))) (distinct_points T_3 T_1) (-. (apart_point_and_line T_1 (line_connecting T_1 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 U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (All Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) ### All 161 % 65.85/66.03 163. (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (distinct_points T_3 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))))) (apart_point_and_line T_3 (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_3) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 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))))))) (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 162 % 65.91/66.07 164. ((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)))) (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 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, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_3 (line_connecting 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))))) (distinct_points T_3 T_1) (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 U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) ### DisjTree 89 163 135 % 65.91/66.07 165. (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 Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (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_3 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))))) (apart_point_and_line T_3 (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_3) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 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 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 164 % 65.91/66.07 166. (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, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_3 (line_connecting 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))))) (distinct_points T_3 T_1) (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 U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (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 % 65.91/66.07 167. (All Y, (All U, (All V, (((distinct_points T_1 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (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 Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y 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)))))) (distinct_points T_3 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))))) (apart_point_and_line T_3 (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_3) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 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 166 % 65.91/66.09 168. (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, ((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, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_3 (line_connecting T_0 T_1)) (distinct_points T_3 T_1) (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_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (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 Y, (All U, (All V, (((distinct_points T_1 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) ### All 167 % 65.91/66.09 169. (All Y, (All U, (All V, (((distinct_points T_1 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (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 Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y 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)))))) (distinct_points T_3 T_1) (apart_point_and_line T_3 (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_3) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 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_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) ### All 168 % 65.91/66.09 170. (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, ((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, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (distinct_points T_2 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_3 (line_connecting T_0 T_1)) (distinct_points T_3 T_1) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (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 Y, (All U, (All V, (((distinct_points T_1 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) ### All 169 % 65.91/66.09 171. (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_1 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (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 Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (distinct_points T_3 T_1) (apart_point_and_line T_3 (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_3) (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_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) ### All 170 % 65.91/66.10 172. (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (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, ((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_3) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 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_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_3 (line_connecting T_0 T_1)) (distinct_points T_3 T_1) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (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 Y, (All U, (All V, (((distinct_points T_1 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 171 % 65.91/66.10 173. (All Y, (All U, (All V, (((distinct_points T_3 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points T_1 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (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 Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (distinct_points T_3 T_1) (apart_point_and_line T_3 (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (distinct_points T_2 T_3) (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_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_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) ### All 172 % 65.91/66.10 174. (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 Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (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, ((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_3) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_3 (line_connecting T_0 T_1)) (distinct_points T_3 T_1) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (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 Y, (All U, (All V, (((distinct_points T_1 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 173 % 65.91/66.10 175. (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 Y, (All U, (All V, (((distinct_points T_1 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (distinct_points T_3 T_1) (apart_point_and_line T_3 (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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (distinct_points T_2 T_3) (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_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_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### All 174 % 65.91/66.10 176. (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 Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (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, ((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_3) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_3 (line_connecting T_0 T_1)) (distinct_points T_3 T_1) (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 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 175 % 65.96/66.15 177. ((distinct_points T_3 T_1) => ((distinct_points T_3 T_1) \/ (distinct_points T_1 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 Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (distinct_points T_2 T_3) (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_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_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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 85 176 25 % 65.96/66.15 178. (All Z, ((distinct_points T_3 T_1) => ((distinct_points T_3 Z) \/ (distinct_points 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) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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 Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (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, ((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_3) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points 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 X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 177 % 65.96/66.15 179. (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) => ((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 Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (distinct_points T_2 T_3) (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_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_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) (All Z, ((distinct_points T_3 T_1) => ((distinct_points T_3 Z) \/ (distinct_points T_1 Z)))) ### All 178 % 65.96/66.15 180. (All Z, ((distinct_points T_3 T_1) => ((distinct_points T_3 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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 Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (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, ((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_3) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points 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 X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 179 % 65.96/66.15 181. (All Y, (All Z, ((distinct_points T_3 Y) => ((distinct_points T_3 Z) \/ (distinct_points Y Z))))) (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 Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (distinct_points T_2 T_3) (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_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_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### All 180 % 65.96/66.15 182. (distinct_points T_0 T_1) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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 Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (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, ((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_3) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points 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 X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 181 % 66.01/66.16 183. (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (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 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_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (distinct_points T_2 T_3) (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_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_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### All 182 % 66.01/66.16 184. (distinct_points T_0 T_1) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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 Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (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, ((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_3) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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 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 183 % 66.01/66.16 185. (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 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)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (distinct_points T_2 T_3) (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_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_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### All 184 % 66.01/66.16 186. (distinct_points T_0 T_1) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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 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, ((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_3) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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 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 185 % 66.01/66.16 187. (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (distinct_points T_2 T_3) (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_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 U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### All 186 % 66.01/66.16 188. (distinct_points T_0 T_1) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (distinct_points T_2 T_3) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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 187 % 66.01/66.17 189. ((apart_point_and_line T_2 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_3 (line_connecting T_0 T_1))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_points T_2 T_3) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### Or 79 188 % 66.01/66.17 190. (-. (((distinct_points T_2 T_3) /\ (distinct_points T_0 T_1)) => (((apart_point_and_line T_2 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_3 (line_connecting T_0 T_1))) => ((apart_point_and_line T_0 (line_connecting T_2 T_3)) \/ (apart_point_and_line T_1 (line_connecting T_2 T_3)))))) (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) => ((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)))))) ### ConjTree 189 % 66.01/66.17 191. (-. (All V, (((distinct_points T_2 T_3) /\ (distinct_points T_0 V)) => (((apart_point_and_line T_2 (line_connecting T_0 V)) \/ (apart_point_and_line T_3 (line_connecting T_0 V))) => ((apart_point_and_line T_0 (line_connecting T_2 T_3)) \/ (apart_point_and_line V (line_connecting T_2 T_3))))))) (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))))))) (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 190 % 66.01/66.17 192. (-. (All U, (All V, (((distinct_points T_2 T_3) /\ (distinct_points U V)) => (((apart_point_and_line T_2 (line_connecting U V)) \/ (apart_point_and_line T_3 (line_connecting U V))) => ((apart_point_and_line U (line_connecting T_2 T_3)) \/ (apart_point_and_line V (line_connecting T_2 T_3)))))))) (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) => ((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)))))) ### NotAllEx 191 % 66.01/66.17 193. (-. (All Y, (All U, (All V, (((distinct_points T_2 Y) /\ (distinct_points U V)) => (((apart_point_and_line T_2 (line_connecting U V)) \/ (apart_point_and_line Y (line_connecting U V))) => ((apart_point_and_line U (line_connecting T_2 Y)) \/ (apart_point_and_line V (line_connecting T_2 Y))))))))) (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))))))) (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 192 % 66.01/66.17 194. (-. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_points U V)) => (((apart_point_and_line X (line_connecting U V)) \/ (apart_point_and_line Y (line_connecting U V))) => ((apart_point_and_line U (line_connecting X Y)) \/ (apart_point_and_line V (line_connecting X Y)))))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (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)))))) ### NotAllEx 193 % 66.01/66.17 % SZS output end Proof % 66.01/66.17 (* END-PROOF *) %------------------------------------------------------------------------------