%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO189+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 : n004.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 06:30:50 EDT 2022 % Result : Theorem 12.38s 12.60s % Output : Proof 12.46s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.12 % Problem : GEO189+2 : TPTP v8.1.0. Released v3.3.0. % 0.07/0.12 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.12/0.33 % Computer : n004.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 08:23:23 EDT 2022 % 0.12/0.33 % CPUTime : % 12.38/12.60 % SZS status Theorem % 12.38/12.60 (* PROOF-FOUND *) % 12.38/12.60 (* BEGIN-PROOF *) % 12.38/12.60 % SZS output start Proof % 12.38/12.60 1. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 12.38/12.60 2. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 12.38/12.60 3. (distinct_points T_0 T_0) ### Extension/test/apart1 % 12.38/12.60 4. (-. (distinct_points T_1 T_0)) (distinct_points T_1 T_0) ### Axiom % 12.38/12.60 5. ((distinct_points T_0 T_1) => ((distinct_points T_0 T_0) \/ (distinct_points T_1 T_0))) (-. (distinct_points T_1 T_0)) (distinct_points T_0 T_1) ### DisjTree 2 3 4 % 12.38/12.60 6. (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (-. (distinct_points T_1 T_0)) ### All 5 % 12.38/12.60 7. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 12.38/12.60 8. (distinct_points T_0 T_2) (-. (distinct_points T_0 T_2)) ### Axiom % 12.38/12.60 9. (distinct_points T_0 T_2) (-. (distinct_points T_0 T_2)) ### Axiom % 12.38/12.60 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 % 12.38/12.60 11. (distinct_points T_1 T_0) (-. (distinct_points T_1 T_0)) ### Axiom % 12.38/12.60 12. (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 % 12.38/12.60 13. (-. (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 % 12.38/12.60 14. (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) (apart_point_and_line T_1 (line_connecting T_0 T_2)) ### Axiom % 12.38/12.60 15. (-. (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 % 12.38/12.60 16. (-. (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 % 12.38/12.60 17. (((distinct_points T_1 T_0) /\ (distinct_lines (line_connecting T_0 T_1) (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_0 T_1)) \/ (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_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (distinct_points T_1 T_0) ### DisjTree 11 12 13 14 15 16 % 12.38/12.60 18. (All V, (((distinct_points T_1 T_0) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_0 V)))))) (distinct_points T_1 T_0) (distinct_lines (line_connecting T_0 T_1) (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_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) ### All 17 % 12.38/12.60 19. (-. (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 % 12.38/12.60 20. ((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_0 T_2))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_1 T_0) (All V, (((distinct_points T_1 T_0) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_0 V)))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) ### DisjTree 10 18 19 % 12.38/12.60 21. (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_1 T_0) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_0 V)))))) (distinct_points T_1 T_0) (-. (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_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_2))) ### All 20 % 12.38/12.60 22. (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_2 (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_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_1 T_0) (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 21 % 12.38/12.60 23. (distinct_points T_2 T_2) ### Extension/test/apart1 % 12.38/12.60 24. ((distinct_points T_2 T_0) /\ (distinct_points T_2 T_2)) ### And 23 % 12.38/12.60 25. ((distinct_points T_0 T_2) => ((apart_point_and_line T_2 (line_connecting T_0 T_2)) => ((distinct_points T_2 T_0) /\ (distinct_points T_2 T_2)))) (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_1 T_0) (-. (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_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (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_2) ### DisjTree 9 22 24 % 12.38/12.60 26. (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (distinct_points T_0 T_2) (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_2))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_1 T_0) (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 25 % 12.38/12.60 27. ((distinct_points T_0 T_0) /\ (distinct_points T_0 T_2)) ### And 3 % 12.38/12.60 28. ((distinct_points T_0 T_2) => ((apart_point_and_line T_0 (line_connecting T_0 T_2)) => ((distinct_points T_0 T_0) /\ (distinct_points T_0 T_2)))) (All Z, ((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_1 T_0) (-. (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_0 T_1))) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (distinct_points T_0 T_2) ### DisjTree 8 26 27 % 12.46/12.63 29. (distinct_points T_0 T_2) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (All U, (All V, (((distinct_points T_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_1 (line_connecting T_0 T_2))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_1 T_0) (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 28 % 12.46/12.63 30. ((distinct_points T_0 T_0) /\ (distinct_points T_0 T_1)) ### And 3 % 12.46/12.63 31. ((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)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points 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_0 T_2))) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (distinct_points T_0 T_2) (distinct_points T_0 T_1) ### DisjTree 7 29 30 % 12.46/12.63 32. (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) (distinct_points T_0 T_2) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (All U, (All V, (((distinct_points T_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_1 (line_connecting T_0 T_2))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_1 T_0) (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 31 % 12.46/12.63 33. ((distinct_points T_1 T_0) => ((distinct_points T_1 T_0) \/ (distinct_points T_0 T_0))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (distinct_points T_0 T_2) (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 Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) ### DisjTree 6 32 3 % 12.46/12.63 34. (All Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (All 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_2) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (All U, (All V, (((distinct_points T_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_1 (line_connecting T_0 T_2))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) ### All 33 % 12.46/12.63 35. (distinct_points T_1 T_1) ### Extension/test/apart1 % 12.46/12.63 36. ((distinct_points T_1 T_0) /\ (distinct_points T_1 T_1)) ### And 35 % 12.46/12.63 37. ((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)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (distinct_points T_0 T_2) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (All Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (distinct_points T_0 T_1) ### DisjTree 1 34 36 % 12.46/12.63 38. (distinct_points T_0 T_1) (All Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_2) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (All U, (All V, (((distinct_points T_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_1 (line_connecting T_0 T_2))) (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 37 % 12.46/12.63 39. (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)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (All Z, ((distinct_points T_0 T_2) => ((apart_point_and_line Z (line_connecting T_0 T_2)) => ((distinct_points Z T_0) /\ (distinct_points Z T_2))))) (distinct_points T_0 T_2) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (All Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (distinct_points T_0 T_1) ### All 38 % 12.46/12.65 40. (distinct_points T_0 T_1) (All Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_2) (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_1 (line_connecting T_0 T_2))) (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 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 39 % 12.46/12.65 41. (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) (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_2) (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 40 % 12.46/12.65 42. (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_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_2) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) (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 Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) ### All 41 % 12.46/12.65 43. (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y 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)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) (distinct_points T_0 T_2) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (All Y, (All U, (All V, (((distinct_points T_1 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) ### All 42 % 12.46/12.65 44. (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (All Y, (All U, (All V, (((distinct_points T_1 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_2) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) ### All 43 % 12.46/12.65 45. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) (distinct_points T_0 T_2) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) ### All 44 % 12.46/12.65 46. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (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) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_2) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) (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 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 45 % 12.46/12.65 47. (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 U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) (distinct_points T_0 T_2) (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 X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 46 % 12.46/12.65 48. (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_points T_0 T_1) (distinct_points T_0 T_2) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) (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 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_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 47 % 12.46/12.65 49. (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 U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_2))) (distinct_points T_0 T_2) (distinct_points T_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 48 % 12.46/12.65 50. (-. (((distinct_points T_0 T_2) /\ ((distinct_points T_0 T_1) /\ ((distinct_points T_2 T_1) /\ (-. (apart_point_and_line T_1 (line_connecting T_0 T_2)))))) => (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### ConjTree 49 % 12.46/12.65 51. (-. (All Z, (((distinct_points T_0 T_2) /\ ((distinct_points T_0 Z) /\ ((distinct_points T_2 Z) /\ (-. (apart_point_and_line Z (line_connecting T_0 T_2)))))) => (-. (apart_point_and_line T_2 (line_connecting T_0 Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotAllEx 50 % 12.46/12.65 52. (-. (All Y, (All Z, (((distinct_points T_0 Y) /\ ((distinct_points T_0 Z) /\ ((distinct_points Y Z) /\ (-. (apart_point_and_line Z (line_connecting T_0 Y)))))) => (-. (apart_point_and_line Y (line_connecting T_0 Z))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### NotAllEx 51 % 12.46/12.65 53. (-. (All X, (All Y, (All Z, (((distinct_points X Y) /\ ((distinct_points X Z) /\ ((distinct_points Y Z) /\ (-. (apart_point_and_line Z (line_connecting X Y)))))) => (-. (apart_point_and_line Y (line_connecting X Z)))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotAllEx 52 % 12.46/12.65 % SZS output end Proof % 12.46/12.65 (* END-PROOF *) %------------------------------------------------------------------------------