%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO180+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 : n020.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 06:30:43 EDT 2022 % Result : Theorem 1.39s 1.56s % Output : Proof 1.39s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.03/0.12 % Problem : GEO180+2 : TPTP v8.1.0. Released v3.3.0. % 0.03/0.12 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.12/0.33 % Computer : n020.cluster.edu % 0.12/0.33 % Model : x86_64 x86_64 % 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.33 % Memory : 8042.1875MB % 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.33 % CPULimit : 300 % 0.12/0.33 % WCLimit : 600 % 0.12/0.33 % DateTime : Sat Jun 18 06:02:18 EDT 2022 % 0.12/0.33 % CPUTime : % 1.39/1.56 % SZS status Theorem % 1.39/1.56 (* PROOF-FOUND *) % 1.39/1.56 (* BEGIN-PROOF *) % 1.39/1.56 % SZS output start Proof % 1.39/1.56 1. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 1.39/1.56 2. (distinct_points T_0 T_0) ### Extension/test/apart1 % 1.39/1.56 3. (distinct_points T_1 T_0) (-. (distinct_points T_1 T_0)) ### Axiom % 1.39/1.56 4. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 1.39/1.56 5. (distinct_points T_1 T_0) (-. (distinct_points T_1 T_0)) ### Axiom % 1.39/1.56 6. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 1.39/1.56 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 % 1.39/1.56 8. (distinct_points T_2 T_1) (-. (distinct_points T_2 T_1)) ### Axiom % 1.39/1.56 9. (distinct_points T_2 T_1) (-. (distinct_points T_2 T_1)) ### Axiom % 1.39/1.56 10. (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 % 1.39/1.56 11. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 1.39/1.56 12. (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 % 1.39/1.56 13. (-. (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 % 1.39/1.56 14. (-. (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 % 1.39/1.56 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 % 1.39/1.56 16. (-. (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 % 1.39/1.56 17. (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0))) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 (line_connecting T_1 T_0)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0)) (distinct_points T_0 T_1) ### DisjTree 11 12 13 14 15 16 % 1.39/1.56 18. (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) ### All 17 % 1.39/1.56 19. (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0)) (distinct_points T_0 T_1) ### All 18 % 1.39/1.56 20. (-. (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 % 1.39/1.56 21. ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) (line_connecting T_1 T_0)) \/ (apart_point_and_line T_2 (line_connecting T_1 T_0)))) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) ### DisjTree 10 19 20 % 1.39/1.56 22. (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) ### All 21 % 1.39/1.56 23. (distinct_points T_1 T_0) (-. (distinct_points T_1 T_0)) ### Axiom % 1.39/1.56 24. (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_1)) (-. (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_1))) ### Axiom % 1.39/1.56 25. (-. (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 % 1.39/1.56 26. (-. (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 % 1.39/1.56 27. (-. (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 % 1.39/1.56 28. (-. (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 % 1.39/1.56 29. (((distinct_points T_1 T_0) /\ (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_1))) => ((apart_point_and_line T_1 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_1 (line_connecting T_2 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_1)))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_1)) (distinct_points T_1 T_0) ### DisjTree 23 24 25 26 27 28 % 1.39/1.56 30. (All V, (((distinct_points T_1 T_0) /\ (distinct_lines (line_connecting T_1 T_0) V)) => ((apart_point_and_line T_1 (line_connecting T_1 T_0)) \/ ((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 V)))))) (distinct_points T_1 T_0) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_1)) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 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_1))) ### All 29 % 1.39/1.56 31. (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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_1)) (distinct_points T_1 T_0) ### All 30 % 1.39/1.56 32. (-. (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 % 1.39/1.56 33. ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_1)) \/ (apart_point_and_line T_2 (line_connecting T_2 T_1)))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_1))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (All U, (All V, (((distinct_points T_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) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) ### DisjTree 22 31 32 % 1.39/1.57 34. (All Z, ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All 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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_2 (line_connecting T_2 T_1))) ### All 33 % 1.39/1.57 35. (distinct_points T_2 T_2) ### Extension/test/apart1 % 1.39/1.57 36. ((distinct_points T_2 T_2) /\ (distinct_points T_2 T_1)) ### And 35 % 1.39/1.57 37. ((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_1 T_0) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (All U, (All V, (((distinct_points T_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) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_2 Z)))) (distinct_points T_2 T_1) ### DisjTree 9 34 36 % 1.39/1.57 38. (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (distinct_points T_2 T_1) (All Z, ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All 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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (distinct_points T_1 T_0) ### All 37 % 1.39/1.57 39. (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_1 (line_connecting T_2 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (All U, (All V, (((distinct_points T_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) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (distinct_points T_2 T_1) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) ### All 38 % 1.39/1.57 40. (distinct_points T_1 T_1) ### Extension/test/apart1 % 1.39/1.57 41. ((distinct_points T_1 T_2) /\ (distinct_points T_1 T_1)) ### And 40 % 1.39/1.57 42. ((distinct_points T_2 T_1) => ((apart_point_and_line T_1 (line_connecting T_2 T_1)) => ((distinct_points T_1 T_2) /\ (distinct_points T_1 T_1)))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All 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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (distinct_points T_1 T_0) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_2 T_1) ### DisjTree 8 39 41 % 1.39/1.57 43. (distinct_points T_2 T_1) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_0 (line_connecting T_2 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))))))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All Z, ((distinct_points T_2 T_1) => ((apart_point_and_line Z (line_connecting T_2 T_1)) => ((distinct_points Z T_2) /\ (distinct_points Z T_1))))) ### All 42 % 1.39/1.59 44. (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, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All 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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (distinct_points T_1 T_0) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_2 T_1) ### All 43 % 1.39/1.59 45. (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) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_0 (line_connecting T_2 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))))))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) ### All 44 % 1.39/1.59 46. (-. (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 % 1.39/1.59 47. ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_points T_2 T_1) \/ (apart_point_and_line T_1 (line_connecting T_0 T_1)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All 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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (distinct_points T_1 T_0) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (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))))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) ### DisjTree 7 45 46 % 1.39/1.59 48. (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z (line_connecting T_0 T_1))))) (apart_point_and_line T_2 (line_connecting 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_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_0 (line_connecting T_2 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))))))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (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 47 % 1.39/1.59 49. (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_points T_2 Z) \/ (apart_point_and_line Z 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)))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All 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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (distinct_points T_1 T_0) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (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))))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) ### All 48 % 1.39/1.59 50. (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_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_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_0 (line_connecting T_2 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))))))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (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 49 % 1.39/1.60 51. ((distinct_points T_1 T_0) /\ (distinct_points T_1 T_1)) ### And 40 % 1.39/1.60 52. ((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, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting 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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (distinct_points T_1 T_0) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (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))))))) (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_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_points T_0 T_1) ### DisjTree 6 50 51 % 1.39/1.60 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) (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_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_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_0 (line_connecting T_2 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))))))) (-. (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_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 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 52 % 1.39/1.60 54. ((distinct_points T_1 T_1) /\ (distinct_points T_1 T_0)) ### And 40 % 1.39/1.60 55. ((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, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting 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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (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))))))) (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_points X Z) \/ (apart_point_and_line 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))))) (distinct_points T_1 T_0) ### DisjTree 5 53 54 % 1.39/1.60 56. (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All 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, ((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_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_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_2 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))))))) (-. (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 U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) ### All 55 % 1.39/1.60 57. ((distinct_points T_0 T_0) /\ (distinct_points T_0 T_1)) ### And 2 % 1.39/1.60 58. ((distinct_points T_0 T_1) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) => ((distinct_points T_0 T_0) /\ (distinct_points T_0 T_1)))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (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))))))) (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_points X Z) \/ (apart_point_and_line 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))))) (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_0 T_1) ### DisjTree 4 56 57 % 1.39/1.60 59. (distinct_points T_0 T_1) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (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, ((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_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_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_2 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))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All 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 58 % 1.39/1.60 60. (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (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))))))) (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_points X Z) \/ (apart_point_and_line Z Y)))))) (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_0 T_1) ### All 59 % 1.39/1.60 61. ((distinct_points T_0 T_1) /\ (distinct_points T_0 T_0)) ### And 2 % 1.39/1.60 62. ((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)))) (distinct_points T_0 T_1) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All 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_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_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_2 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 U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All 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_1 T_0) ### DisjTree 3 60 61 % 1.39/1.60 63. (distinct_points 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 Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (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))))))) (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_points X Z) \/ (apart_point_and_line Z Y)))))) (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_0 T_1) ### All 62 % 1.39/1.60 64. (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)))))) (distinct_points 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)))))) (apart_point_and_line T_2 (line_connecting 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_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_2 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 U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All 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_1 T_0) ### All 63 % 1.39/1.60 65. (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)))))))) (distinct_points 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 Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (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))))))) (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_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_points T_0 T_1) (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 64 % 1.39/1.61 66. (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)))))) (distinct_points 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)))))) (apart_point_and_line T_2 (line_connecting 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_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (-. (apart_point_and_line T_0 (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))))))) (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, (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_1 T_0) ### All 65 % 1.39/1.61 67. (distinct_points 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 U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (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))))))) (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_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_points T_0 T_1) (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))))))))) ### All 66 % 1.39/1.61 68. ((distinct_points T_0 T_1) => ((distinct_points T_0 T_0) \/ (distinct_points 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 Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (apart_point_and_line T_2 (line_connecting 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_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (-. (apart_point_and_line T_0 (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))))))) (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 1 2 67 % 1.39/1.61 69. (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, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (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))))))) (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_points X Z) \/ (apart_point_and_line 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))))))))) ### All 68 % 1.39/1.61 70. (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, ((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_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_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (-. (apart_point_and_line T_0 (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_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) ### All 69 % 1.39/1.61 71. (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)))) (distinct_points T_0 T_1) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting 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))))))) (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_points X Z) \/ (apart_point_and_line 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))))))))) ### All 70 % 1.39/1.62 72. (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (apart_point_and_line T_2 (line_connecting 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))))))) (-. (apart_point_and_line T_0 (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_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 71 % 1.39/1.62 73. (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting 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))))))) (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_points X Z) \/ (apart_point_and_line 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))))))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) ### All 72 % 1.39/1.62 74. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (apart_point_and_line T_2 (line_connecting 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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (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)))))) (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 73 % 1.39/1.62 75. (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) (-. (apart_point_and_line T_0 (line_connecting 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))))))) (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_points X Z) \/ (apart_point_and_line 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))))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 74 % 1.39/1.62 76. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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_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))))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_1))) (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)))))) ### All 75 % 1.39/1.62 77. (-. ((distinct_points T_0 T_1) => ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => (apart_point_and_line T_0 (line_connecting T_2 T_1))))) (All 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 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))))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### ConjTree 76 % 1.39/1.62 78. (-. (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => (apart_point_and_line T_0 (line_connecting Z T_1)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((distinct_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)))))) ### NotAllEx 77 % 1.39/1.62 79. (-. (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => (apart_point_and_line T_0 (line_connecting 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) => ((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_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))))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotAllEx 78 % 1.39/1.62 80. (-. (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => (apart_point_and_line X (line_connecting 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 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, (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)))))) ### NotAllEx 79 % 1.39/1.62 % SZS output end Proof % 1.39/1.62 (* END-PROOF *) %------------------------------------------------------------------------------