%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO194+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 : n023.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 06:30:55 EDT 2022 % Result : Theorem 0.53s 0.72s % Output : Proof 0.53s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.12 % Problem : GEO194+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 : n023.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 01:06:49 EDT 2022 % 0.12/0.33 % CPUTime : % 0.53/0.72 % SZS status Theorem % 0.53/0.72 (* PROOF-FOUND *) % 0.53/0.72 (* BEGIN-PROOF *) % 0.53/0.72 % SZS output start Proof % 0.53/0.72 1. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 0.53/0.72 2. (convergent_lines T_0 T_0) (-. (convergent_lines T_0 T_0)) ### Axiom % 0.53/0.72 3. (-. (distinct_lines T_0 T_0)) (convergent_lines T_0 T_0) ### Extension/test/ceq3ctrp 2 % 0.53/0.72 4. (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_0) ### All 3 % 0.53/0.72 5. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 0.53/0.72 6. (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) ### Axiom % 0.53/0.72 7. (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### Axiom % 0.53/0.72 8. (-. (apart_point_and_line (intersection_point T_0 T_2) T_2)) (apart_point_and_line (intersection_point T_0 T_2) T_2) ### Axiom % 0.53/0.72 9. ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_2)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### DisjTree 6 7 8 % 0.53/0.72 10. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_2)) ### All 9 % 0.53/0.72 11. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom % 0.53/0.72 12. (-. (distinct_lines T_1 T_0)) (convergent_lines T_1 T_0) ### Extension/test/ceq3ctrp 11 % 0.53/0.72 13. (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### Axiom % 0.53/0.72 14. (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### Axiom % 0.53/0.72 15. (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (apart_point_and_line (intersection_point T_0 T_2) T_1) ### Axiom % 0.53/0.72 16. (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) (apart_point_and_line (intersection_point T_0 T_2) T_0) ### Axiom % 0.53/0.72 17. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines T_1 T_0)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_0 T_2) T_1) \/ (apart_point_and_line (intersection_point T_0 T_2) T_0))))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_2) T_2)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) ### DisjTree 10 12 13 14 15 16 % 0.53/0.72 18. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines T_1 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) T_1) \/ (apart_point_and_line (intersection_point T_0 T_2) V)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_2)) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) ### All 17 % 0.53/0.72 19. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) U) \/ (apart_point_and_line (intersection_point T_0 T_2) V))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_2) T_2)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) ### All 18 % 0.53/0.72 20. (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_2)) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) ### All 19 % 0.53/0.72 21. (-. (distinct_lines T_0 T_0)) (distinct_lines T_0 T_0) ### Axiom % 0.53/0.72 22. (convergent_lines T_0 T_2) (-. (convergent_lines T_0 T_2)) ### Axiom % 0.53/0.72 23. (apart_point_and_line (intersection_point T_0 T_2) T_0) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) ### Axiom % 0.53/0.72 24. (-. ((apart_point_and_line (intersection_point T_0 T_2) T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2))) (apart_point_and_line (intersection_point T_0 T_2) T_0) ### NotOr 23 % 0.53/0.72 25. (distinct_points (intersection_point T_0 T_2) (intersection_point T_0 T_2)) ### Extension/test/apart1 % 0.53/0.72 26. ((convergent_lines T_0 T_2) => (((apart_point_and_line (intersection_point T_0 T_2) T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2)) => (distinct_points (intersection_point T_0 T_2) (intersection_point T_0 T_2)))) (apart_point_and_line (intersection_point T_0 T_2) T_0) (convergent_lines T_0 T_2) ### DisjTree 22 24 25 % 0.53/0.72 27. (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (convergent_lines T_0 T_2) (apart_point_and_line (intersection_point T_0 T_2) T_0) ### All 26 % 0.53/0.72 28. ((apart_point_and_line (intersection_point T_0 T_2) T_0) => ((distinct_lines T_0 T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) T_0))) (convergent_lines T_0 T_2) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (-. (distinct_lines T_0 T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_2) T_2)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) ### DisjTree 20 21 27 % 0.53/0.74 29. (All Z, ((apart_point_and_line (intersection_point T_0 T_2) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z)))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_2)) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (distinct_lines T_0 T_0)) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (convergent_lines T_0 T_2) ### All 28 % 0.53/0.74 30. (-. (distinct_lines T_2 T_2)) (distinct_lines T_2 T_2) ### Axiom % 0.53/0.74 31. (convergent_lines T_0 T_2) (-. (convergent_lines T_0 T_2)) ### Axiom % 0.53/0.74 32. (apart_point_and_line (intersection_point T_0 T_2) T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_2)) ### Axiom % 0.53/0.74 33. (-. ((apart_point_and_line (intersection_point T_0 T_2) T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2))) (apart_point_and_line (intersection_point T_0 T_2) T_2) ### NotOr 32 % 0.53/0.74 34. ((convergent_lines T_0 T_2) => (((apart_point_and_line (intersection_point T_0 T_2) T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2)) => (distinct_points (intersection_point T_0 T_2) (intersection_point T_0 T_2)))) (apart_point_and_line (intersection_point T_0 T_2) T_2) (convergent_lines T_0 T_2) ### DisjTree 31 33 25 % 0.53/0.74 35. (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (convergent_lines T_0 T_2) (apart_point_and_line (intersection_point T_0 T_2) T_2) ### All 34 % 0.53/0.74 36. ((apart_point_and_line (intersection_point T_0 T_2) T_2) => ((distinct_lines T_2 T_2) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_2) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (-. (distinct_lines T_0 T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_2) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z)))) ### DisjTree 29 30 35 % 0.53/0.74 37. (All Z, ((apart_point_and_line (intersection_point T_0 T_2) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z)))) (All Z, ((apart_point_and_line (intersection_point T_0 T_2) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z)))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (distinct_lines T_0 T_0)) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 T_2)) ### All 36 % 0.53/0.74 38. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_2) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_2) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (-. (distinct_lines T_0 T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_2) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z)))) ### All 37 % 0.53/0.74 39. (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (distinct_lines T_0 T_0)) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 T_2)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_2) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z))))) ### All 38 % 0.53/0.74 40. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_2) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (-. (distinct_lines T_0 T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) ### All 39 % 0.53/0.74 41. (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (distinct_lines T_0 T_0)) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 40 % 0.53/0.74 42. (-. ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_2) (-. (distinct_lines T_0 T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) ### NotOr 41 % 0.53/0.74 43. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)) ### Extension/test/apart1 % 0.53/0.74 44. ((convergent_lines T_0 T_1) => (((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1)) => (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)))) (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (distinct_lines T_0 T_0)) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) ### DisjTree 5 42 43 % 0.53/0.74 45. (All Z, ((convergent_lines T_0 T_1) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_1)) => (distinct_points Z (intersection_point T_0 T_1))))) (convergent_lines 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_lines T_2 T_2)) (convergent_lines T_0 T_2) (-. (distinct_lines T_0 T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) ### All 44 % 0.53/0.74 46. (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (distinct_lines T_0 T_0)) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) ### All 45 % 0.53/0.74 47. (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines 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_lines T_2 T_2)) (convergent_lines T_0 T_2) (-. (distinct_lines T_0 T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) ### All 46 % 0.53/0.74 48. (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 Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (distinct_lines T_0 T_0)) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) ### All 47 % 0.53/0.74 49. (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines 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_lines T_2 T_2)) (convergent_lines T_0 T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### All 48 % 0.53/0.75 50. ((convergent_lines T_0 T_1) => ((convergent_lines T_0 T_0) \/ (convergent_lines 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 Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_1) ### DisjTree 1 4 49 % 0.53/0.75 51. (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### All 50 % 0.53/0.75 52. (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_1) ### Axiom % 0.53/0.75 53. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 0.53/0.75 54. (apart_point_and_line (intersection_point T_0 T_1) T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) ### Axiom % 0.53/0.75 55. (-. ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1))) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### NotOr 54 % 0.53/0.75 56. ((convergent_lines T_0 T_1) => (((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1)) => (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)))) (apart_point_and_line (intersection_point T_0 T_1) T_1) (convergent_lines T_0 T_1) ### DisjTree 53 55 43 % 0.53/0.75 57. (All Z, ((convergent_lines T_0 T_1) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_1)) => (distinct_points Z (intersection_point T_0 T_1))))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### All 56 % 0.53/0.75 58. (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) (apart_point_and_line (intersection_point T_0 T_1) T_1) (convergent_lines T_0 T_1) ### All 57 % 0.53/0.75 59. (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### All 58 % 0.53/0.75 60. ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 T_1) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) ### DisjTree 51 52 59 % 0.53/0.75 61. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (distinct_lines T_1 T_1)) ### All 60 % 0.53/0.75 62. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) ### All 61 % 0.53/0.75 63. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (distinct_lines T_1 T_1)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) ### All 62 % 0.53/0.76 64. (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, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) ### All 63 % 0.53/0.76 65. (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (distinct_lines T_1 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 64 % 0.53/0.76 66. (All Y, (All Z, ((convergent_lines T_0 Y) => ((convergent_lines T_0 Z) \/ (convergent_lines Y Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_1) ### All 65 % 0.53/0.76 67. (convergent_lines T_0 T_1) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (distinct_lines T_1 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, ((convergent_lines T_0 Y) => ((convergent_lines T_0 Z) \/ (convergent_lines Y Z))))) ### All 66 % 0.53/0.76 68. (All Y, (All Z, ((convergent_lines T_0 Y) => ((convergent_lines T_0 Z) \/ (convergent_lines Y Z))))) (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))))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (convergent_lines T_0 T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_1) ### All 67 % 0.53/0.76 69. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (convergent_lines T_0 T_1) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 68 % 0.53/0.76 70. (-. (((convergent_lines T_0 T_1) /\ ((convergent_lines T_2 T_1) /\ (convergent_lines T_0 T_2))) => ((apart_point_and_line (intersection_point T_0 T_1) T_2) => (apart_point_and_line (intersection_point T_0 T_2) 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, (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 X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### ConjTree 69 % 0.53/0.76 71. (-. (All Z, (((convergent_lines T_0 T_1) /\ ((convergent_lines Z T_1) /\ (convergent_lines T_0 Z))) => ((apart_point_and_line (intersection_point T_0 T_1) Z) => (apart_point_and_line (intersection_point T_0 Z) T_1))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X 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 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)))))) ### NotAllEx 70 % 0.53/0.76 72. (-. (All Y, (All Z, (((convergent_lines T_0 Y) /\ ((convergent_lines Z Y) /\ (convergent_lines T_0 Z))) => ((apart_point_and_line (intersection_point T_0 Y) Z) => (apart_point_and_line (intersection_point T_0 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 71 % 0.53/0.76 73. (-. (All X, (All Y, (All Z, (((convergent_lines X Y) /\ ((convergent_lines Z Y) /\ (convergent_lines X Z))) => ((apart_point_and_line (intersection_point X Y) Z) => (apart_point_and_line (intersection_point X Z) Y))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X 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 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)))))) ### NotAllEx 72 % 0.53/0.76 % SZS output end Proof % 0.53/0.76 (* END-PROOF *) %------------------------------------------------------------------------------