%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO182+1 : 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 : n032.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 06:30:44 EDT 2022 % Result : Theorem 0.13s 0.37s % Output : Proof 0.13s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.02/0.09 % Problem : GEO182+1 : TPTP v8.1.0. Released v3.3.0. % 0.02/0.09 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.09/0.28 % Computer : n032.cluster.edu % 0.09/0.28 % Model : x86_64 x86_64 % 0.09/0.28 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.09/0.28 % Memory : 8042.1875MB % 0.09/0.28 % OS : Linux 3.10.0-693.el7.x86_64 % 0.09/0.28 % CPULimit : 300 % 0.09/0.28 % WCLimit : 600 % 0.09/0.28 % DateTime : Sat Jun 18 02:32:16 EDT 2022 % 0.09/0.28 % CPUTime : % 0.13/0.37 % SZS status Theorem % 0.13/0.37 (* PROOF-FOUND *) % 0.13/0.37 (* BEGIN-PROOF *) % 0.13/0.37 % SZS output start Proof % 0.13/0.37 1. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 0.13/0.37 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 % 0.13/0.37 3. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 0.13/0.37 4. (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 % 0.13/0.37 5. (-. (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 % 0.13/0.37 6. (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 % 0.13/0.37 7. (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 % 0.13/0.37 8. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 0.13/0.37 9. (distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (-. (distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2))) ### Axiom % 0.13/0.37 10. (-. (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 % 0.13/0.37 11. (distinct_points T_0 T_2) (-. (distinct_points T_0 T_2)) ### Axiom % 0.13/0.37 12. (apart_point_and_line T_0 (line_connecting T_0 T_2)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) ### Axiom % 0.13/0.37 13. ((distinct_points T_0 T_2) => (-. (apart_point_and_line T_0 (line_connecting T_0 T_2)))) (apart_point_and_line T_0 (line_connecting T_0 T_2)) (distinct_points T_0 T_2) ### Imply 11 12 % 0.13/0.37 14. (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_2) (apart_point_and_line T_0 (line_connecting T_0 T_2)) ### All 13 % 0.13/0.37 15. (-. (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 % 0.13/0.37 16. (distinct_points T_0 T_2) (-. (distinct_points T_0 T_2)) ### Axiom % 0.13/0.37 17. (apart_point_and_line T_1 (line_connecting T_0 T_2)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) ### Axiom % 0.13/0.37 18. (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 % 0.13/0.37 19. (distinct_points T_1 T_0) (apart_point_and_line T_0 (line_connecting T_1 T_0)) ### Extension/test/ci2 18 % 0.13/0.37 20. (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (apart_point_and_line T_0 (line_connecting T_0 T_2)) ### Axiom % 0.13/0.37 21. ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_points T_1 T_0) \/ (apart_point_and_line T_0 (line_connecting T_0 T_2)))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) (apart_point_and_line T_1 (line_connecting T_0 T_2)) ### DisjTree 17 19 20 % 0.13/0.37 22. (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z (line_connecting T_0 T_2))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (apart_point_and_line T_0 (line_connecting T_1 T_0)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) ### All 21 % 0.13/0.37 23. ((distinct_points T_0 T_2) => (-. (apart_point_and_line T_0 (line_connecting T_0 T_2)))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z (line_connecting T_0 T_2))))) (distinct_points T_0 T_2) ### Imply 16 22 % 0.13/0.37 24. (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_2) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z (line_connecting T_0 T_2))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (apart_point_and_line T_0 (line_connecting T_1 T_0)) ### All 23 % 0.13/0.37 25. (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (distinct_points T_0 T_2) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) ### All 24 % 0.13/0.37 26. (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2))) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 (line_connecting T_0 T_2)) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 (line_connecting T_0 T_2)))))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_0 T_2) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (distinct_points T_0 T_1) ### DisjTree 8 9 10 14 15 25 % 0.13/0.37 27. (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_0 T_2)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_2) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) ### All 26 % 0.13/0.37 28. (-. (apart_point_and_line T_2 (line_connecting T_0 T_2))) (apart_point_and_line T_2 (line_connecting T_0 T_2)) ### Axiom % 0.13/0.37 29. ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) \/ (apart_point_and_line T_2 (line_connecting T_0 T_2)))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_2))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_0 T_2) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) ### DisjTree 7 27 28 % 0.13/0.37 30. (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_2) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) (-. (apart_point_and_line T_2 (line_connecting T_0 T_2))) ### All 29 % 0.13/0.37 31. (apart_point_and_line T_0 (line_connecting T_1 T_0)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_0 T_2) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) ### Extension/test/ci2 30 % 0.13/0.38 32. (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) ### Axiom % 0.13/0.38 33. ((apart_point_and_line T_0 (line_connecting T_1 T_0)) => ((distinct_points T_0 T_2) \/ (apart_point_and_line T_2 (line_connecting T_1 T_0)))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) ### DisjTree 6 31 32 % 0.13/0.38 34. (All Z, ((apart_point_and_line T_0 (line_connecting T_1 T_0)) => ((distinct_points T_0 Z) \/ (apart_point_and_line Z (line_connecting T_1 T_0))))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) ### All 33 % 0.13/0.38 35. (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_points T_0 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) ### All 34 % 0.13/0.38 36. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) ### All 35 % 0.13/0.38 37. (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 36 % 0.13/0.38 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 % 0.13/0.38 39. (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 % 0.13/0.38 40. (distinct_points T_1 T_0) (-. (distinct_points T_1 T_0)) ### Axiom % 0.13/0.38 41. (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 % 0.13/0.38 42. ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) (distinct_points T_1 T_0) ### Imply 40 41 % 0.13/0.38 43. (-. (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 % 0.13/0.38 44. ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_points T_1 T_0) \/ (apart_point_and_line T_0 (line_connecting T_1 T_0)))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ### DisjTree 39 42 43 % 0.13/0.38 45. (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z (line_connecting T_1 T_0))))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) ### All 44 % 0.13/0.38 46. (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 % 0.13/0.38 47. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 0.13/0.38 48. (distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (-. (distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2))) ### Axiom % 0.13/0.38 49. (-. (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 % 0.13/0.38 50. (-. (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 % 0.13/0.39 51. (distinct_points T_0 T_2) (-. (distinct_points T_0 T_2)) ### Axiom % 0.13/0.39 52. (apart_point_and_line T_1 (line_connecting T_0 T_2)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) ### Axiom % 0.13/0.39 53. (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (apart_point_and_line T_0 (line_connecting T_0 T_2)) ### Axiom % 0.13/0.39 54. ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_points T_1 T_0) \/ (apart_point_and_line T_0 (line_connecting T_0 T_2)))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) ### DisjTree 52 42 53 % 0.13/0.39 55. (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z (line_connecting T_0 T_2))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) ### All 54 % 0.13/0.39 56. ((distinct_points T_0 T_2) => (-. (apart_point_and_line T_0 (line_connecting T_0 T_2)))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z (line_connecting T_0 T_2))))) (distinct_points T_0 T_2) ### Imply 51 55 % 0.13/0.39 57. (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_2) (All Z, ((apart_point_and_line T_1 (line_connecting T_0 T_2)) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z (line_connecting T_0 T_2))))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ### All 56 % 0.13/0.39 58. (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) (distinct_points T_0 T_2) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) ### All 57 % 0.13/0.39 59. (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2))) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 (line_connecting T_0 T_2)) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 (line_connecting T_0 T_2)))))) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_0 T_2) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (distinct_points T_0 T_1) ### DisjTree 47 48 49 14 50 58 % 0.13/0.39 60. (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_0 T_2)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_2) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) ### All 59 % 0.13/0.39 61. (-. (apart_point_and_line T_2 (line_connecting T_0 T_2))) (apart_point_and_line T_2 (line_connecting T_0 T_2)) ### Axiom % 0.13/0.39 62. ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) \/ (apart_point_and_line T_2 (line_connecting T_0 T_2)))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_2))) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_0 T_2) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) ### DisjTree 46 60 61 % 0.13/0.39 63. (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_2) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_2))) ### All 62 % 0.13/0.39 64. ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_0 T_2) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) ### Extension/test/ci2 63 % 0.13/0.39 65. (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) ### Axiom % 0.13/0.39 66. ((apart_point_and_line T_0 (line_connecting T_1 T_0)) => ((distinct_points T_0 T_2) \/ (apart_point_and_line T_2 (line_connecting T_1 T_0)))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z (line_connecting T_1 T_0))))) ### DisjTree 45 64 65 % 0.13/0.39 67. (All Z, ((apart_point_and_line T_0 (line_connecting T_1 T_0)) => ((distinct_points T_0 Z) \/ (apart_point_and_line Z (line_connecting T_1 T_0))))) (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z (line_connecting T_1 T_0))))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) ### All 66 % 0.13/0.39 68. (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_points T_0 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z (line_connecting T_1 T_0))))) ### All 67 % 0.13/0.39 69. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z (line_connecting T_1 T_0))))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) ### All 68 % 0.13/0.39 70. (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z (line_connecting T_1 T_0))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 69 % 0.13/0.39 71. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_points T_1 Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) ### All 70 % 0.13/0.39 72. (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 71 % 0.13/0.39 73. (All Y, ((distinct_points T_1 Y) => (-. (apart_point_and_line T_1 (line_connecting T_1 Y))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) ### All 72 % 0.13/0.40 74. (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 73 % 0.13/0.40 75. (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0))) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 (line_connecting T_1 T_0)))))) (All 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, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0)) (distinct_points T_0 T_1) ### DisjTree 3 4 5 37 38 74 % 0.13/0.40 76. (distinct_points T_0 T_1) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All 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)))))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (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 75 % 0.13/0.40 77. (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, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (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)))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0)) (distinct_points T_0 T_1) ### All 76 % 0.13/0.40 78. (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) ### Axiom % 0.13/0.40 79. ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0)) \/ (apart_point_and_line T_2 (line_connecting T_1 T_0)))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) ### DisjTree 2 77 78 % 0.13/0.40 80. (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (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_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) ### All 79 % 0.13/0.40 81. (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) ### All 80 % 0.13/0.40 82. (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)) (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, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) ### All 81 % 0.13/0.40 83. ((distinct_points T_0 T_1) => (-. (apart_point_and_line T_0 (line_connecting T_0 T_1)))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All 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) ### Imply 1 82 % 0.13/0.40 84. (distinct_points T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) ### All 83 % 0.13/0.40 85. (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)))))))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (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)))))) (distinct_points T_0 T_1) ### All 84 % 0.13/0.40 86. (distinct_points T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (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 85 % 0.13/0.40 87. (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_2 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (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)))))) (distinct_points T_0 T_1) ### All 86 % 0.13/0.40 88. (distinct_points T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### Extension/test/ci2 87 % 0.13/0.40 89. (-. ((distinct_points T_0 T_1) => ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => (apart_point_and_line T_2 (line_connecting T_1 T_0))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### ConjTree 88 % 0.13/0.40 90. (-. (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => (apart_point_and_line Z (line_connecting T_1 T_0)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### NotAllEx 89 % 0.13/0.40 91. (-. (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => (apart_point_and_line Z (line_connecting Y T_0))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### NotAllEx 90 % 0.13/0.41 92. (-. (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => (apart_point_and_line Z (line_connecting Y X)))))))) (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, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### NotAllEx 91 % 0.13/0.41 % SZS output end Proof % 0.13/0.41 (* END-PROOF *) %------------------------------------------------------------------------------