%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO202+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 : n003.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 06:31:02 EDT 2022 % Result : Theorem 10.92s 11.10s % Output : Proof 11.23s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.12 % Problem : GEO202+3 : TPTP v8.1.0. Released v4.0.0. % 0.07/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.13/0.34 % Computer : n003.cluster.edu % 0.13/0.34 % Model : x86_64 x86_64 % 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.13/0.34 % Memory : 8042.1875MB % 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.13/0.34 % CPULimit : 300 % 0.13/0.34 % WCLimit : 600 % 0.13/0.34 % DateTime : Fri Jun 17 17:05:12 EDT 2022 % 0.13/0.34 % CPUTime : % 10.92/11.10 % SZS status Theorem % 10.92/11.10 (* PROOF-FOUND *) % 10.92/11.10 (* BEGIN-PROOF *) % 10.92/11.10 % SZS output start Proof % 10.92/11.10 1. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 10.92/11.10 2. (distinct_points T_0 T_2) (-. (distinct_points T_0 T_2)) ### Axiom % 10.92/11.10 3. (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (-. (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1))) ### Axiom % 10.92/11.10 4. (distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) (-. (distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0)) ### Axiom % 10.92/11.10 5. (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 % 10.92/11.10 6. (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (-. (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1))) ### Axiom % 10.92/11.10 7. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (line_connecting T_0 T_1)) (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (line_connecting T_0 T_1))) ### Axiom % 10.92/11.10 8. ((convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (line_connecting T_0 T_1)))) (apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (line_connecting T_0 T_1)) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) ### Imply 6 7 % 10.92/11.10 9. (All Y, ((convergent_lines (line_connecting T_0 T_2) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) Y) Y)))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (line_connecting T_0 T_1)) ### All 8 % 10.92/11.10 10. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (line_connecting T_0 T_1)) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) ### All 9 % 10.92/11.10 11. (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (-. (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1))) ### Axiom % 10.92/11.10 12. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (line_connecting T_0 T_2)) (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (line_connecting T_0 T_2))) ### Axiom % 10.92/11.10 13. ((convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (line_connecting T_0 T_2)))) (apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (line_connecting T_0 T_2)) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) ### Imply 11 12 % 10.92/11.10 14. (All Y, ((convergent_lines (line_connecting T_0 T_2) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) Y) (line_connecting T_0 T_2))))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (line_connecting T_0 T_2)) ### All 13 % 10.92/11.10 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 % 10.92/11.10 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 % 10.92/11.10 17. (((distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) /\ (distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2))) => ((apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (line_connecting T_0 T_1)) \/ ((apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (line_connecting T_0 T_2)) \/ ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_0 (line_connecting T_0 T_2)))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, ((convergent_lines (line_connecting T_0 T_2) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) Y) (line_connecting T_0 T_2))))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) ### DisjTree 4 5 10 14 15 16 % 10.92/11.10 18. (All V, (((distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (line_connecting T_0 T_1)) \/ ((apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) V) \/ ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_0 V)))))) (distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (All Y, ((convergent_lines (line_connecting T_0 T_2) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) Y) (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 % 10.92/11.10 19. (All U, (All V, (((distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) U) \/ ((apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 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))) (All Y, ((convergent_lines (line_connecting T_0 T_2) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) Y) (line_connecting T_0 T_2))))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) ### All 18 % 10.92/11.10 20. (All Y, (All U, (All V, (((distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) U) \/ ((apart_point_and_line (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (All Y, ((convergent_lines (line_connecting T_0 T_2) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) Y) (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 19 % 10.92/11.10 21. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, ((convergent_lines (line_connecting T_0 T_2) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) Y) (line_connecting T_0 T_2))))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) ### All 20 % 10.96/11.15 22. (-. (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_2))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_2)) ### Axiom % 10.96/11.15 23. ((convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) \/ (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_2)))) (-. (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_2))) (distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All Y, ((convergent_lines (line_connecting T_0 T_2) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) Y) (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 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))))))))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) ### DisjTree 3 21 22 % 10.96/11.15 24. (All Z, ((convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (convergent_lines (line_connecting T_0 T_2) Z)))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 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 T_0 (line_connecting T_0 T_2))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, ((convergent_lines (line_connecting T_0 T_2) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) Y) (line_connecting 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 (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) (-. (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_2))) ### All 23 % 10.96/11.15 25. (All X, (-. (convergent_lines X X))) (distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All Y, ((convergent_lines (line_connecting T_0 T_2) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) Y) (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 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))))))))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (All Z, ((convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (convergent_lines (line_connecting T_0 T_2) Z)))) ### All 24 % 10.96/11.15 26. ((distinct_points T_0 T_2) => (-. (apart_point_and_line T_0 (line_connecting T_0 T_2)))) (All Z, ((convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (convergent_lines (line_connecting T_0 T_2) Z)))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 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 T_0 (line_connecting T_0 T_1))) (All Y, ((convergent_lines (line_connecting T_0 T_2) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) Y) (line_connecting 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 (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) (All X, (-. (convergent_lines X X))) (distinct_points T_0 T_2) ### Imply 2 25 % 10.96/11.15 27. (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_2) (All X, (-. (convergent_lines X X))) (distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All Y, ((convergent_lines (line_connecting T_0 T_2) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_2) Y) (line_connecting T_0 T_2))))) (-. (apart_point_and_line T_0 (line_connecting T_0 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))))))))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (All Z, ((convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (convergent_lines (line_connecting T_0 T_2) Z)))) ### All 26 % 10.96/11.15 28. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (convergent_lines (line_connecting T_0 T_2) Z)))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 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 T_0 (line_connecting T_0 T_1))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) (All X, (-. (convergent_lines X X))) (distinct_points T_0 T_2) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) ### All 27 % 10.96/11.15 29. ((distinct_points T_0 T_1) => (-. (apart_point_and_line T_0 (line_connecting T_0 T_1)))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_2) (All X, (-. (convergent_lines X X))) (distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) 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))))))))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (All Z, ((convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (convergent_lines (line_connecting T_0 T_2) Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points T_0 T_1) ### Imply 1 28 % 11.23/11.43 30. (distinct_points T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (convergent_lines (line_connecting T_0 T_2) Z)))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 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 X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) (All X, (-. (convergent_lines X X))) (distinct_points T_0 T_2) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) ### All 29 % 11.23/11.43 31. (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_points T_0 T_2) (All X, (-. (convergent_lines X X))) (distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) 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))))))))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (All Z, ((convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (convergent_lines (line_connecting T_0 T_2) Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points T_0 T_1) ### All 30 % 11.23/11.43 32. (All Y, (All Z, ((convergent_lines (line_connecting T_0 T_2) Y) => ((distinct_lines Y Z) \/ (convergent_lines (line_connecting T_0 T_2) Z))))) (distinct_points T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 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 X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) (All X, (-. (convergent_lines X X))) (distinct_points T_0 T_2) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) ### All 31 % 11.23/11.43 33. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_points T_0 T_2) (All X, (-. (convergent_lines X X))) (distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) 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))))))))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points T_0 T_1) ### All 32 % 11.23/11.43 34. (-. (-. (distinct_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0))) (distinct_points T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 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 X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (distinct_points T_0 T_2) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotNot 33 % 11.23/11.43 35. (-. (equal_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_points T_0 T_2) (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, (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))))))))) (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points T_0 T_1) ### Definition-Pseudo(equal_points) 34 % 11.23/11.43 36. (-. (((distinct_points T_0 T_2) /\ ((distinct_points T_0 T_1) /\ (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)))) => (equal_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 T_1)) T_0))) (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, (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, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### ConjTree 35 % 11.23/11.43 37. (-. (All Z, (((distinct_points T_0 T_2) /\ ((distinct_points T_0 Z) /\ (convergent_lines (line_connecting T_0 T_2) (line_connecting T_0 Z)))) => (equal_points (intersection_point (line_connecting T_0 T_2) (line_connecting T_0 Z)) T_0)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (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, (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))))) ### NotAllEx 36 % 11.23/11.43 38. (-. (All Y, (All Z, (((distinct_points T_0 Y) /\ ((distinct_points T_0 Z) /\ (convergent_lines (line_connecting T_0 Y) (line_connecting T_0 Z)))) => (equal_points (intersection_point (line_connecting T_0 Y) (line_connecting T_0 Z)) T_0))))) (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, (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, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotAllEx 37 % 11.23/11.43 39. (-. (All X, (All Y, (All Z, (((distinct_points X Y) /\ ((distinct_points X Z) /\ (convergent_lines (line_connecting X Y) (line_connecting X Z)))) => (equal_points (intersection_point (line_connecting X Y) (line_connecting X Z)) X)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (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, (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))))) ### NotAllEx 38 % 11.23/11.43 % SZS output end Proof % 11.23/11.43 (* END-PROOF *) %------------------------------------------------------------------------------