%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO186+3 : TPTP v8.1.0. Released v4.0.0. % Transfm : none % Format : tptp:raw % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % Computer : n013.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 06:30:48 EDT 2022 % Result : Theorem 50.13s 50.37s % Output : Proof 50.24s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.08/0.14 % Problem : GEO186+3 : TPTP v8.1.0. Released v4.0.0. % 0.08/0.14 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.15/0.36 % Computer : n013.cluster.edu % 0.15/0.36 % Model : x86_64 x86_64 % 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.15/0.36 % Memory : 8042.1875MB % 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64 % 0.15/0.36 % CPULimit : 300 % 0.15/0.36 % WCLimit : 600 % 0.15/0.36 % DateTime : Fri Jun 17 15:30:14 EDT 2022 % 0.15/0.36 % CPUTime : % 50.13/50.37 % SZS status Theorem % 50.13/50.37 (* PROOF-FOUND *) % 50.13/50.37 (* BEGIN-PROOF *) % 50.13/50.37 % SZS output start Proof % 50.13/50.37 1. (distinct_points T_0 (intersection_point T_1 T_2)) (-. (distinct_points T_0 (intersection_point T_1 T_2))) ### Axiom % 50.13/50.37 2. (-. (distinct_points T_0 T_0)) (distinct_points T_0 T_0) ### Axiom % 50.13/50.37 3. (convergent_lines T_1 T_2) (-. (convergent_lines T_1 T_2)) ### Axiom % 50.13/50.37 4. (distinct_points (intersection_point T_1 T_2) T_0) (-. (distinct_points (intersection_point T_1 T_2) T_0)) ### Axiom % 50.13/50.37 5. (distinct_lines T_2 T_1) (-. (distinct_lines T_2 T_1)) ### Axiom % 50.13/50.37 6. (convergent_lines T_1 T_2) (-. (convergent_lines T_1 T_2)) ### Axiom % 50.13/50.37 7. (apart_point_and_line (intersection_point T_1 T_2) T_2) (-. (apart_point_and_line (intersection_point T_1 T_2) T_2)) ### Axiom % 50.13/50.37 8. ((convergent_lines T_1 T_2) => (-. (apart_point_and_line (intersection_point T_1 T_2) T_2))) (apart_point_and_line (intersection_point T_1 T_2) T_2) (convergent_lines T_1 T_2) ### Imply 6 7 % 50.13/50.37 9. (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) Y)))) (convergent_lines T_1 T_2) (apart_point_and_line (intersection_point T_1 T_2) T_2) ### All 8 % 50.13/50.37 10. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_1 T_2) T_2) (convergent_lines T_1 T_2) ### All 9 % 50.13/50.37 11. (convergent_lines T_1 T_2) (-. (convergent_lines T_1 T_2)) ### Axiom % 50.13/50.37 12. (apart_point_and_line (intersection_point T_1 T_2) T_1) (-. (apart_point_and_line (intersection_point T_1 T_2) T_1)) ### Axiom % 50.13/50.37 13. ((convergent_lines T_1 T_2) => (-. (apart_point_and_line (intersection_point T_1 T_2) T_1))) (apart_point_and_line (intersection_point T_1 T_2) T_1) (convergent_lines T_1 T_2) ### Imply 11 12 % 50.13/50.37 14. (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) T_1)))) (convergent_lines T_1 T_2) (apart_point_and_line (intersection_point T_1 T_2) T_1) ### All 13 % 50.13/50.37 15. (-. (apart_point_and_line T_0 T_2)) (apart_point_and_line T_0 T_2) ### Axiom % 50.13/50.37 16. (-. (apart_point_and_line T_0 T_1)) (apart_point_and_line T_0 T_1) ### Axiom % 50.13/50.37 17. (((distinct_points (intersection_point T_1 T_2) T_0) /\ (distinct_lines T_2 T_1)) => ((apart_point_and_line (intersection_point T_1 T_2) T_2) \/ ((apart_point_and_line (intersection_point T_1 T_2) T_1) \/ ((apart_point_and_line T_0 T_2) \/ (apart_point_and_line T_0 T_1))))) (-. (apart_point_and_line T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) T_1)))) (convergent_lines T_1 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_2 T_1) (distinct_points (intersection_point T_1 T_2) T_0) ### DisjTree 4 5 10 14 15 16 % 50.13/50.37 18. (All V, (((distinct_points (intersection_point T_1 T_2) T_0) /\ (distinct_lines T_2 V)) => ((apart_point_and_line (intersection_point T_1 T_2) T_2) \/ ((apart_point_and_line (intersection_point T_1 T_2) V) \/ ((apart_point_and_line T_0 T_2) \/ (apart_point_and_line T_0 V)))))) (distinct_points (intersection_point T_1 T_2) T_0) (distinct_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_2) (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) T_1)))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_0 T_1)) ### All 17 % 50.13/50.37 19. (All U, (All V, (((distinct_points (intersection_point T_1 T_2) T_0) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_2) U) \/ ((apart_point_and_line (intersection_point T_1 T_2) V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (-. (apart_point_and_line T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) T_1)))) (convergent_lines T_1 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_2 T_1) (distinct_points (intersection_point T_1 T_2) T_0) ### All 18 % 50.13/50.37 20. (-. (convergent_lines T_1 T_1)) (convergent_lines T_1 T_1) ### Axiom % 50.13/50.37 21. ((convergent_lines T_1 T_2) => ((distinct_lines T_2 T_1) \/ (convergent_lines T_1 T_1))) (-. (convergent_lines T_1 T_1)) (distinct_points (intersection_point T_1 T_2) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) T_1)))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_0 T_1)) (All U, (All V, (((distinct_points (intersection_point T_1 T_2) T_0) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_2) U) \/ ((apart_point_and_line (intersection_point T_1 T_2) V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (convergent_lines T_1 T_2) ### DisjTree 3 19 20 % 50.13/50.37 22. (All Z, ((convergent_lines T_1 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_2) (All U, (All V, (((distinct_points (intersection_point T_1 T_2) T_0) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_2) U) \/ ((apart_point_and_line (intersection_point T_1 T_2) V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (-. (apart_point_and_line T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) T_1)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point T_1 T_2) T_0) (-. (convergent_lines T_1 T_1)) ### All 21 % 50.13/50.37 23. (All X, (-. (convergent_lines X X))) (distinct_points (intersection_point T_1 T_2) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) T_1)))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_0 T_1)) (All U, (All V, (((distinct_points (intersection_point T_1 T_2) T_0) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_2) U) \/ ((apart_point_and_line (intersection_point T_1 T_2) V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (convergent_lines T_1 T_2) (All Z, ((convergent_lines T_1 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_1 Z)))) ### All 22 % 50.13/50.37 24. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_1 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_2) (All U, (All V, (((distinct_points (intersection_point T_1 T_2) T_0) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_2) U) \/ ((apart_point_and_line (intersection_point T_1 T_2) V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (-. (apart_point_and_line T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point T_1 T_2) T_0) (All X, (-. (convergent_lines X X))) ### All 23 % 50.13/50.37 25. (All Y, (All U, (All V, (((distinct_points (intersection_point T_1 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_2) U) \/ ((apart_point_and_line (intersection_point T_1 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (-. (convergent_lines X X))) (distinct_points (intersection_point T_1 T_2) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_0 T_1)) (convergent_lines T_1 T_2) (All Z, ((convergent_lines T_1 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### All 24 % 50.13/50.37 26. (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_1 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_2) (-. (apart_point_and_line T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point T_1 T_2) T_0) (All X, (-. (convergent_lines X X))) ### All 25 % 50.24/50.41 27. ((distinct_points T_0 (intersection_point T_1 T_2)) => ((distinct_points T_0 T_0) \/ (distinct_points (intersection_point T_1 T_2) T_0))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_0 T_1)) (convergent_lines T_1 T_2) (All Z, ((convergent_lines T_1 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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_points T_0 T_0)) (distinct_points T_0 (intersection_point T_1 T_2)) ### DisjTree 1 2 26 % 50.24/50.41 28. (All Z, ((distinct_points T_0 (intersection_point T_1 T_2)) => ((distinct_points T_0 Z) \/ (distinct_points (intersection_point T_1 T_2) Z)))) (distinct_points T_0 (intersection_point T_1 T_2)) (-. (distinct_points T_0 T_0)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_1 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_2) (-. (apart_point_and_line T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) ### All 27 % 50.24/50.41 29. (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_0 T_1)) (convergent_lines T_1 T_2) (All Z, ((convergent_lines T_1 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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_points T_0 T_0)) (distinct_points T_0 (intersection_point T_1 T_2)) ### All 28 % 50.24/50.41 30. (All X, (-. (distinct_points X X))) (distinct_points T_0 (intersection_point 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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_1 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_2) (-. (apart_point_and_line T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) ### All 29 % 50.24/50.41 31. (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_0 T_1)) (convergent_lines T_1 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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_points T_0 (intersection_point T_1 T_2)) (All X, (-. (distinct_points X X))) ### All 30 % 50.24/50.41 32. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_points X X))) (distinct_points T_0 (intersection_point 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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_1 T_2) (-. (apart_point_and_line T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) ### All 31 % 50.24/50.41 33. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_0 T_1)) (convergent_lines T_1 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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_points T_0 (intersection_point T_1 T_2)) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 32 % 50.24/50.41 34. (-. (-. (distinct_points T_0 (intersection_point T_1 T_2)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_points X X))) (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_1 T_2) (-. (apart_point_and_line T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotNot 33 % 50.24/50.41 35. (-. (equal_points T_0 (intersection_point T_1 T_2))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_0 T_1)) (convergent_lines T_1 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### Definition-Pseudo(equal_points) 34 % 50.24/50.42 36. (incident_point_and_line T_0 T_2) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_points X X))) (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_1 T_2) (-. (apart_point_and_line T_0 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (equal_points T_0 (intersection_point T_1 T_2))) ### Definition-Pseudo(incident_point_and_line) 35 % 50.24/50.42 37. (incident_point_and_line T_0 T_1) (-. (equal_points T_0 (intersection_point T_1 T_2))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (incident_point_and_line T_0 T_2) ### Definition-Pseudo(incident_point_and_line) 36 % 50.24/50.42 38. (-. (((distinct_points T_0 T_3) /\ ((convergent_lines T_1 T_2) /\ ((incident_point_and_line T_0 T_1) /\ (incident_point_and_line T_0 T_2)))) => (equal_points T_0 (intersection_point T_1 T_2)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_points X X))) (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### ConjTree 37 % 50.24/50.42 39. (-. (All V, (((distinct_points T_0 T_3) /\ ((convergent_lines T_1 V) /\ ((incident_point_and_line T_0 T_1) /\ (incident_point_and_line T_0 V)))) => (equal_points T_0 (intersection_point T_1 V))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotAllEx 38 % 50.24/50.42 40. (-. (All U, (All V, (((distinct_points T_0 T_3) /\ ((convergent_lines U V) /\ ((incident_point_and_line T_0 U) /\ (incident_point_and_line T_0 V)))) => (equal_points T_0 (intersection_point U V)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_points X X))) (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotAllEx 39 % 50.24/50.42 41. (-. (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ ((convergent_lines U V) /\ ((incident_point_and_line T_0 U) /\ (incident_point_and_line T_0 V)))) => (equal_points T_0 (intersection_point U V))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotAllEx 40 % 50.24/50.42 42. (-. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ ((convergent_lines U V) /\ ((incident_point_and_line X U) /\ (incident_point_and_line X V)))) => (equal_points X (intersection_point U V)))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_points X X))) (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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotAllEx 41 % 50.24/50.42 % SZS output end Proof % 50.24/50.42 (* END-PROOF *) %------------------------------------------------------------------------------