%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO202+1 : 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 : n024.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 06:31:01 EDT 2022 % Result : Theorem 0.19s 0.48s % Output : Proof 0.19s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.04/0.12 % Problem : GEO202+1 : TPTP v8.1.0. Released v3.3.0. % 0.04/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.13/0.34 % Computer : n024.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 21:06:02 EDT 2022 % 0.13/0.34 % CPUTime : % 0.19/0.48 % SZS status Theorem % 0.19/0.48 (* PROOF-FOUND *) % 0.19/0.48 (* BEGIN-PROOF *) % 0.19/0.48 % SZS output start Proof % 0.19/0.48 1. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (-. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2))) ### Axiom % 0.19/0.48 2. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 0.19/0.48 3. (distinct_points T_0 T_2) (-. (distinct_points T_0 T_2)) ### Axiom % 0.19/0.48 4. (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (-. (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0)) ### Axiom % 0.19/0.48 5. (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (-. (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1))) ### Axiom % 0.19/0.48 6. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (-. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2))) ### Axiom % 0.19/0.48 7. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_2)) (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_2))) ### Axiom % 0.19/0.48 8. ((convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_2)))) (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_2)) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) ### Imply 6 7 % 0.19/0.48 9. (All Y, ((convergent_lines (line_connecting T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) Y)))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_2)) ### All 8 % 0.19/0.48 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_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_2)) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) ### All 9 % 0.19/0.48 11. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (-. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2))) ### Axiom % 0.19/0.48 12. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_1)) (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_1))) ### Axiom % 0.19/0.48 13. ((convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_1)))) (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_1)) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) ### Imply 11 12 % 0.19/0.48 14. (All Y, ((convergent_lines (line_connecting T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) (line_connecting T_0 T_1))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_1)) ### All 13 % 0.19/0.48 15. (-. (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 % 0.19/0.48 16. (-. (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 % 0.19/0.48 17. (((distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) /\ (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1))) => ((apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_2)) \/ ((apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (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_1)))))) (-. (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 Y, ((convergent_lines (line_connecting T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) (line_connecting T_0 T_1))))) (convergent_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))))) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) ### DisjTree 4 5 10 14 15 16 % 0.19/0.48 18. (All V, (((distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) /\ (distinct_lines (line_connecting T_0 T_2) V)) => ((apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_2)) \/ ((apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) V) \/ ((apart_point_and_line T_0 (line_connecting T_0 T_2)) \/ (apart_point_and_line T_0 V)))))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (distinct_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))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All Y, ((convergent_lines (line_connecting T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) (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_1))) ### All 17 % 0.19/0.48 19. (All U, (All V, (((distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) U) \/ ((apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (All Y, ((convergent_lines (line_connecting T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) (line_connecting T_0 T_1))))) (convergent_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))))) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) ### All 18 % 0.19/0.48 20. (All Y, (All U, (All V, (((distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) U) \/ ((apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (distinct_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))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All Y, ((convergent_lines (line_connecting T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) (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_1))) ### All 19 % 0.19/0.48 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_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (All Y, ((convergent_lines (line_connecting T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) (line_connecting T_0 T_1))))) (convergent_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))))) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) ### All 20 % 0.19/0.49 22. ((distinct_points T_0 T_2) => (-. (apart_point_and_line T_0 (line_connecting T_0 T_2)))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (distinct_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))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All Y, ((convergent_lines (line_connecting T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) (line_connecting T_0 T_1))))) (-. (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))))))))) (distinct_points T_0 T_2) ### Imply 3 21 % 0.19/0.49 23. (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, (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_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) (line_connecting T_0 T_1))))) (convergent_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))))) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) ### All 22 % 0.19/0.49 24. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (distinct_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))))) (convergent_lines (line_connecting T_0 T_1) (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))))))))) (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 23 % 0.19/0.49 25. ((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, (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_1) (line_connecting T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points T_0 T_1) ### Imply 2 24 % 0.19/0.49 26. (distinct_points T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (distinct_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))))) (convergent_lines (line_connecting T_0 T_1) (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))))))))) (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 25 % 0.19/0.49 27. (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, (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_1) (line_connecting T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points T_0 T_1) ### All 26 % 0.19/0.49 28. (-. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_1))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_1)) ### Axiom % 0.19/0.49 29. ((convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) \/ (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_1)))) (-. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) 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))))))))) (distinct_points T_0 T_2) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) ### DisjTree 1 27 28 % 0.19/0.49 30. (All Z, ((convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (convergent_lines (line_connecting T_0 T_1) Z)))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (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, (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_1) (line_connecting T_0 T_2)) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points T_0 T_1) (-. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_1))) ### All 29 % 0.19/0.49 31. (All X, (-. (convergent_lines X X))) (distinct_points T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) 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))))))))) (distinct_points T_0 T_2) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All Z, ((convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (convergent_lines (line_connecting T_0 T_1) Z)))) ### All 30 % 0.19/0.49 32. (All Y, (All Z, ((convergent_lines (line_connecting T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (line_connecting T_0 T_1) Z))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (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, (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_1) (line_connecting T_0 T_2)) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points T_0 T_1) (All X, (-. (convergent_lines X X))) ### All 31 % 0.19/0.49 33. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (convergent_lines X X))) (distinct_points T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) 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))))))))) (distinct_points T_0 T_2) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) ### All 32 % 0.19/0.49 34. (-. (((distinct_points T_0 T_1) /\ ((distinct_points T_0 T_2) /\ (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)))) => (-. (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0)))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X 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) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### ConjTree 33 % 0.19/0.49 35. (-. (All Z, (((distinct_points T_0 T_1) /\ ((distinct_points T_0 Z) /\ (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 Z)))) => (-. (distinct_points (intersection_point (line_connecting T_0 T_1) (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, (-. (convergent_lines X X))) (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, (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, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) ### NotAllEx 34 % 0.19/0.49 36. (-. (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)))) => (-. (distinct_points (intersection_point (line_connecting T_0 Y) (line_connecting T_0 Z)) T_0)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X 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) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotAllEx 35 % 0.19/0.49 37. (-. (All X, (All Y, (All Z, (((distinct_points X Y) /\ ((distinct_points X Z) /\ (convergent_lines (line_connecting X Y) (line_connecting X Z)))) => (-. (distinct_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, (-. (convergent_lines X X))) (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, (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, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) ### NotAllEx 36 % 0.19/0.49 % SZS output end Proof % 0.19/0.49 (* END-PROOF *) %------------------------------------------------------------------------------