%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO203+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 : n007.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 0.20s 0.50s % Output : Proof 0.20s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.08/0.13 % Problem : GEO203+1 : TPTP v8.1.0. Released v3.3.0. % 0.08/0.14 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.14/0.35 % Computer : n007.cluster.edu % 0.14/0.35 % Model : x86_64 x86_64 % 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.14/0.35 % Memory : 8042.1875MB % 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64 % 0.14/0.35 % CPULimit : 300 % 0.14/0.35 % WCLimit : 600 % 0.14/0.35 % DateTime : Sat Jun 18 05:54:58 EDT 2022 % 0.14/0.35 % CPUTime : % 0.20/0.50 % SZS status Theorem % 0.20/0.50 (* PROOF-FOUND *) % 0.20/0.50 (* BEGIN-PROOF *) % 0.20/0.50 % SZS output start Proof % 0.20/0.50 1. (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 % 0.20/0.50 2. (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (-. (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0)) ### Axiom % 0.20/0.50 3. (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 % 0.20/0.50 4. (apart_point_and_line (intersection_point T_0 T_1) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2))) (-. (apart_point_and_line (intersection_point T_0 T_1) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) ### Axiom % 0.20/0.50 5. ((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) => (-. (apart_point_and_line (intersection_point T_0 T_1) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2))))) (apart_point_and_line (intersection_point T_0 T_1) (line_connecting (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)) ### Imply 3 4 % 0.20/0.50 6. (All Y, ((distinct_points (intersection_point T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point T_0 T_1) (line_connecting (intersection_point T_0 T_1) Y))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (apart_point_and_line (intersection_point T_0 T_1) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2))) ### All 5 % 0.20/0.50 7. (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (apart_point_and_line (intersection_point T_0 T_1) (line_connecting (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)) ### All 6 % 0.20/0.50 8. (-. (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 % 0.20/0.50 9. (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2))) ### Axiom % 0.20/0.50 10. (-. (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 % 0.20/0.50 11. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0)) => ((apart_point_and_line (intersection_point T_0 T_1) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2))) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point 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_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### DisjTree 1 2 7 8 9 10 % 0.20/0.50 12. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) V)) => ((apart_point_and_line (intersection_point T_0 T_1) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2))) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2))) \/ (apart_point_and_line (intersection_point T_0 T_2) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) ### All 11 % 0.20/0.50 13. (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) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### All 12 % 0.20/0.50 14. (distinct_lines T_0 T_0) ### Extension/test/apart2 % 0.20/0.50 15. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 0.20/0.50 16. (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 % 0.20/0.50 17. ((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_0) (convergent_lines T_0 T_1) ### Imply 15 16 % 0.20/0.50 18. (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### All 17 % 0.20/0.50 19. ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_lines T_0 T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) T_0))) (convergent_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) (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))))))) ### DisjTree 13 14 18 % 0.20/0.50 20. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (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) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_1) ### All 19 % 0.20/0.52 21. (convergent_lines T_0 T_2) (-. (convergent_lines T_0 T_2)) ### Axiom % 0.20/0.52 22. (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 % 0.20/0.52 23. ((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_0) (convergent_lines T_0 T_2) ### Imply 21 22 % 0.20/0.52 24. (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_2) (apart_point_and_line (intersection_point T_0 T_2) T_0) ### All 23 % 0.20/0.52 25. ((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) (convergent_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (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))))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) ### DisjTree 20 14 24 % 0.20/0.52 26. (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 Z, ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (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) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_1) (convergent_lines T_0 T_2) ### All 25 % 0.20/0.52 27. (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))))) (convergent_lines T_0 T_2) (convergent_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (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))))))) (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 26 % 0.20/0.52 28. (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 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) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_1) (convergent_lines T_0 T_2) (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 27 % 0.20/0.52 29. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (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))))) (convergent_lines T_0 T_2) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (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))))))) (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 28 % 0.20/0.52 30. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (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) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (convergent_lines T_0 T_1) (convergent_lines T_0 T_2) (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 X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### All 29 % 0.20/0.52 31. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_2) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (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))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 30 % 0.20/0.52 32. (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 X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (convergent_lines T_0 T_1) (convergent_lines T_0 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### All 31 % 0.20/0.52 33. (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_0 T_2) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point 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 32 % 0.20/0.52 34. (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, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (convergent_lines T_0 T_1) (convergent_lines T_0 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))))))))) ### Extension/test/ci2 33 % 0.20/0.52 35. (-. (((convergent_lines T_0 T_1) /\ ((convergent_lines T_0 T_2) /\ (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) => (-. (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) 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 X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### ConjTree 34 % 0.20/0.52 36. (-. (All Z, (((convergent_lines T_0 T_1) /\ ((convergent_lines T_0 Z) /\ (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 Z)))) => (-. (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 Z)) T_0))))) (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, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X 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))))))))) ### NotAllEx 35 % 0.20/0.52 37. (-. (All Y, (All Z, (((convergent_lines T_0 Y) /\ ((convergent_lines T_0 Z) /\ (distinct_points (intersection_point T_0 Y) (intersection_point T_0 Z)))) => (-. (distinct_lines (line_connecting (intersection_point T_0 Y) (intersection_point T_0 Z)) 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 X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### NotAllEx 36 % 0.20/0.52 38. (-. (All X, (All Y, (All Z, (((convergent_lines X Y) /\ ((convergent_lines X Z) /\ (distinct_points (intersection_point X Y) (intersection_point X Z)))) => (-. (distinct_lines (line_connecting (intersection_point X Y) (intersection_point X Z)) X))))))) (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, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X 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))))))))) ### NotAllEx 37 % 0.20/0.52 % SZS output end Proof % 0.20/0.52 (* END-PROOF *) %------------------------------------------------------------------------------