%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO196+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 : n021.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 06:30:56 EDT 2022 % Result : Theorem 20.27s 20.47s % Output : Proof 21.11s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.12 % Problem : GEO196+1 : TPTP v8.1.0. Released v3.3.0. % 0.12/0.12 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.12/0.33 % Computer : n021.cluster.edu % 0.12/0.33 % Model : x86_64 x86_64 % 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.33 % Memory : 8042.1875MB % 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.33 % CPULimit : 300 % 0.12/0.33 % WCLimit : 600 % 0.12/0.33 % DateTime : Fri Jun 17 17:58:01 EDT 2022 % 0.12/0.33 % CPUTime : % 20.27/20.47 % SZS status Theorem % 20.27/20.47 (* PROOF-FOUND *) % 20.27/20.47 (* BEGIN-PROOF *) % 20.27/20.47 % SZS output start Proof % 20.27/20.47 1. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 20.27/20.47 2. (distinct_lines T_1 T_0) (-. (distinct_lines T_1 T_0)) ### Axiom % 20.27/20.47 3. (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_1) ### Axiom % 20.27/20.47 4. (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) ### Axiom % 20.27/20.47 5. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3))) ### Axiom % 20.27/20.47 6. (distinct_lines T_0 T_1) (-. (distinct_lines T_0 T_1)) ### Axiom % 20.27/20.47 7. (-. (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 % 20.27/20.47 8. (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### Axiom % 20.27/20.47 9. (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (apart_point_and_line (intersection_point T_2 T_3) T_0) ### Axiom % 20.27/20.47 10. (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (apart_point_and_line (intersection_point T_2 T_3) T_1) ### Axiom % 20.27/20.47 11. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines T_0 T_1)) => ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_0) \/ (apart_point_and_line (intersection_point T_2 T_3) T_1))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) ### DisjTree 5 6 7 8 9 10 % 20.27/20.47 12. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines T_0 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_0) \/ (apart_point_and_line (intersection_point T_2 T_3) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (distinct_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) ### All 11 % 20.27/20.47 13. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (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_2 T_3) U) \/ (apart_point_and_line (intersection_point T_2 T_3) V))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) ### All 12 % 20.27/20.47 14. (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)))))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (distinct_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) ### All 13 % 20.27/20.47 15. (-. (apart_point_and_line (intersection_point T_2 T_3) T_2)) (apart_point_and_line (intersection_point T_2 T_3) T_2) ### Axiom % 20.27/20.47 16. ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) \/ (apart_point_and_line (intersection_point T_2 T_3) T_2))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_0 T_1) (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)))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### DisjTree 4 14 15 % 20.27/20.47 17. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (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)))))))) (distinct_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_2)) ### All 16 % 20.27/20.47 18. (-. (distinct_lines T_2 T_2)) (distinct_lines T_2 T_2) ### Axiom % 20.27/20.47 19. (convergent_lines T_2 T_3) (-. (convergent_lines T_2 T_3)) ### Axiom % 20.27/20.47 20. (apart_point_and_line (intersection_point T_2 T_3) T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) T_2)) ### Axiom % 20.27/20.47 21. ((convergent_lines T_2 T_3) => (-. (apart_point_and_line (intersection_point T_2 T_3) T_2))) (apart_point_and_line (intersection_point T_2 T_3) T_2) (convergent_lines T_2 T_3) ### Imply 19 20 % 20.27/20.47 22. (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (apart_point_and_line (intersection_point T_2 T_3) T_2) ### All 21 % 20.27/20.47 23. ((apart_point_and_line (intersection_point T_2 T_3) T_2) => ((distinct_lines T_2 T_2) \/ (apart_point_and_line (intersection_point T_2 T_3) T_2))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (distinct_lines T_2 T_2)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_0 T_1) (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)))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) ### DisjTree 17 18 22 % 20.27/20.47 24. (All Z, ((apart_point_and_line (intersection_point T_2 T_3) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_2 T_3) Z)))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (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)))))))) (distinct_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (distinct_lines T_2 T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) ### All 23 % 20.27/20.49 25. (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_3) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_3) Z))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (distinct_lines T_2 T_2)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_0 T_1) (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)))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) ### All 24 % 20.27/20.49 26. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (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)))))))) (distinct_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (distinct_lines T_2 T_2)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) ### All 25 % 20.27/20.49 27. (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (distinct_lines T_0 T_1) (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)))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 26 % 20.27/20.49 28. (-. (distinct_lines T_0 T_0)) (distinct_lines T_0 T_0) ### Axiom % 20.27/20.49 29. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 20.27/20.49 30. (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 % 20.27/20.49 31. ((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 29 30 % 20.27/20.49 32. (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 31 % 20.27/20.49 33. ((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_lines T_0 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 Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (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)))))))) (distinct_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (-. (distinct_lines X X))) ### DisjTree 27 28 32 % 20.27/20.49 34. (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 X, (-. (distinct_lines X X))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_0 T_1) (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)))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_0 T_0)) (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 33 % 20.27/20.49 35. (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_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (-. (distinct_lines T_0 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 Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (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)))))))) (distinct_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (-. (distinct_lines X X))) ### All 34 % 20.32/20.53 36. (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_0 T_1) (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)))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (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 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 35 % 20.32/20.53 37. ((distinct_lines T_1 T_0) => ((distinct_lines T_1 T_1) \/ (distinct_lines T_0 T_1))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (convergent_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) 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 Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (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)))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (-. (distinct_lines X X))) (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_0) ### DisjTree 2 3 36 % 20.32/20.53 38. (All Z, ((distinct_lines T_1 T_0) => ((distinct_lines T_1 Z) \/ (distinct_lines T_0 Z)))) (distinct_lines T_1 T_0) (-. (distinct_lines T_1 T_1)) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (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)))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (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 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 37 % 20.32/20.53 39. (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (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_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) 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 Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (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)))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (-. (distinct_lines X X))) (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_0) ### All 38 % 20.32/20.53 40. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_lines T_1 T_0) (-. (distinct_lines T_1 T_1)) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (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)))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) ### All 39 % 20.32/20.53 41. (convergent_lines T_0 T_0) (-. (convergent_lines T_0 T_0)) ### Axiom % 20.32/20.53 42. (All X, (-. (convergent_lines X X))) (convergent_lines T_0 T_0) ### All 41 % 20.32/20.53 43. ((convergent_lines T_0 T_1) => ((distinct_lines T_1 T_0) \/ (convergent_lines T_0 T_0))) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (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)))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (-. (distinct_lines X X))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) ### DisjTree 1 40 42 % 20.35/20.54 44. (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (distinct_lines T_1 T_1)) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (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)))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (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 Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (All X, (-. (convergent_lines X X))) ### All 43 % 20.35/20.54 45. (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_1) ### Axiom % 20.35/20.54 46. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 20.35/20.54 47. (apart_point_and_line (intersection_point T_0 T_1) T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) ### Axiom % 20.35/20.54 48. ((convergent_lines T_0 T_1) => (-. (apart_point_and_line (intersection_point T_0 T_1) T_1))) (apart_point_and_line (intersection_point T_0 T_1) T_1) (convergent_lines T_0 T_1) ### Imply 46 47 % 20.35/20.54 49. (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) Y)))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### All 48 % 20.35/20.54 50. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_1) (convergent_lines T_0 T_1) ### All 49 % 20.35/20.54 51. ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 T_1) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1))) (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_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (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)))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (-. (distinct_lines X X))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### DisjTree 44 45 50 % 20.35/20.54 52. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (distinct_lines T_1 T_1)) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (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)))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (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 Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines 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 51 % 20.35/20.54 53. (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_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (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)))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (-. (distinct_lines X X))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### All 52 % 20.35/20.54 54. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (distinct_lines T_1 T_1)) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (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 Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines 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 53 % 20.35/20.56 55. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (All 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_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (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, (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_1) T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (-. (distinct_lines X X))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### All 54 % 20.35/20.57 56. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (distinct_lines T_1 T_1)) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (apart_point_and_line (intersection_point T_0 T_1) 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 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 Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines 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 55 % 20.35/20.57 57. (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_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (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_1) T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (-. (distinct_lines X X))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 56 % 20.35/20.57 58. (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (-. (distinct_lines T_1 T_1)) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) ### All 57 % 20.35/20.57 59. (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, ((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_1) T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### All 58 % 20.35/20.57 60. (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) ### All 59 % 20.35/20.60 61. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((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_1) T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines 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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### All 60 % 20.35/20.60 62. (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (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 Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 61 % 20.35/20.60 63. (-. (-. (apart_point_and_line (intersection_point T_0 T_1) T_2))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (All X, (-. (distinct_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines 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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### NotNot 62 % 20.35/20.60 64. (convergent_lines T_2 T_3) (-. (convergent_lines T_2 T_3)) ### Axiom % 20.35/20.60 65. (convergent_lines T_2 T_3) (-. (convergent_lines T_2 T_3)) ### Axiom % 20.35/20.60 66. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom % 20.35/20.60 67. (convergent_lines T_2 T_3) (-. (convergent_lines T_2 T_3)) ### Axiom % 20.35/20.60 68. (-. (convergent_lines T_2 T_2)) (convergent_lines T_2 T_2) ### Axiom % 20.35/20.60 69. (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) ### Axiom % 20.35/20.60 70. (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) ### Axiom % 20.35/20.60 71. (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (apart_point_and_line (intersection_point T_2 T_3) T_3) ### Axiom % 20.35/20.60 72. ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) \/ (apart_point_and_line (intersection_point T_2 T_3) T_3))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3))) (apart_point_and_line (intersection_point T_0 T_1) T_3) ### DisjTree 69 70 71 % 20.35/20.60 73. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) ### All 72 % 20.35/20.60 74. (-. (distinct_lines T_3 T_3)) (distinct_lines T_3 T_3) ### Axiom % 20.35/20.60 75. (convergent_lines T_2 T_3) (-. (convergent_lines T_2 T_3)) ### Axiom % 20.35/20.60 76. (apart_point_and_line (intersection_point T_2 T_3) T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) ### Axiom % 20.35/20.60 77. ((convergent_lines T_2 T_3) => (-. (apart_point_and_line (intersection_point T_2 T_3) T_3))) (apart_point_and_line (intersection_point T_2 T_3) T_3) (convergent_lines T_2 T_3) ### Imply 75 76 % 20.35/20.60 78. (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) Y)))) (convergent_lines T_2 T_3) (apart_point_and_line (intersection_point T_2 T_3) T_3) ### All 77 % 20.35/20.60 79. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_2 T_3) T_3) (convergent_lines T_2 T_3) ### All 78 % 20.35/20.60 80. ((apart_point_and_line (intersection_point T_2 T_3) T_3) => ((distinct_lines T_3 T_3) \/ (apart_point_and_line (intersection_point T_2 T_3) T_3))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_3 T_3)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) ### DisjTree 73 74 79 % 20.35/20.60 81. (All Z, ((apart_point_and_line (intersection_point T_2 T_3) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_2 T_3) Z)))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3))) (-. (distinct_lines T_3 T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) ### All 80 % 20.35/20.60 82. (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_3) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_3) Z))))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_3 T_3)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) ### All 81 % 20.44/20.64 83. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3))) (-. (distinct_lines T_3 T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) ### All 82 % 20.44/20.64 84. (distinct_lines T_1 T_2) (-. (distinct_lines T_1 T_2)) ### Axiom % 20.44/20.64 85. (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### Axiom % 20.44/20.64 86. (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### Axiom % 20.44/20.64 87. (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (apart_point_and_line (intersection_point T_2 T_3) T_1) ### Axiom % 20.44/20.64 88. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines T_1 T_2)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_1) \/ (apart_point_and_line (intersection_point T_2 T_3) T_2))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_1 T_2) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_3 T_3)) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### DisjTree 83 84 85 86 87 22 % 20.44/20.64 89. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines T_1 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_1) \/ (apart_point_and_line (intersection_point T_2 T_3) V)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (distinct_lines T_3 T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) ### All 88 % 20.44/20.64 90. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (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_2 T_3) U) \/ (apart_point_and_line (intersection_point T_2 T_3) V))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_1 T_2) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_3 T_3)) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 89 % 20.44/20.64 91. (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)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (distinct_lines T_3 T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) ### All 90 % 20.44/20.64 92. (All X, (-. (distinct_lines X X))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (distinct_lines T_1 T_2) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) ### All 91 % 20.44/20.64 93. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3))) ### Axiom % 20.44/20.64 94. (distinct_lines T_1 T_2) (-. (distinct_lines T_1 T_2)) ### Axiom % 20.44/20.64 95. (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### Axiom % 20.44/20.64 96. (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (apart_point_and_line (intersection_point T_2 T_3) T_1) ### Axiom % 20.44/20.64 97. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines T_1 T_2)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_1) \/ (apart_point_and_line (intersection_point T_2 T_3) T_2))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_1 T_2) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) ### DisjTree 93 94 50 95 96 22 % 20.44/20.66 98. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines T_1 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_1) \/ (apart_point_and_line (intersection_point T_2 T_3) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (distinct_lines T_1 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) ### All 97 % 20.44/20.66 99. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (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_2 T_3) U) \/ (apart_point_and_line (intersection_point T_2 T_3) V))))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_1 T_2) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) ### All 98 % 20.44/20.66 100. (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)))))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (distinct_lines T_1 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) ### All 99 % 20.44/20.66 101. (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (apart_point_and_line (intersection_point T_2 T_3) T_1) ### Axiom % 20.44/20.66 102. ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) \/ (apart_point_and_line (intersection_point T_2 T_3) T_1))) (convergent_lines T_0 T_1) (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)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (-. (distinct_lines X X))) ### DisjTree 92 100 101 % 20.44/20.66 103. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_1)))) (All X, (-. (distinct_lines X X))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (distinct_lines T_1 T_2) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (convergent_lines T_0 T_1) ### All 102 % 20.44/20.66 104. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (convergent_lines T_0 T_1) (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)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (-. (distinct_lines X X))) ### All 103 % 20.44/20.66 105. (-. (distinct_lines T_2 T_2)) (distinct_lines T_2 T_2) ### Axiom % 20.44/20.66 106. (convergent_lines T_3 T_2) (-. (convergent_lines T_3 T_2)) ### Axiom % 20.44/20.66 107. (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) ### Axiom % 20.44/20.66 108. (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_2)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_2))) ### Axiom % 20.44/20.66 109. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1))) ### Axiom % 20.44/20.66 110. (All X, (-. (distinct_points X X))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)) ### All 109 % 20.44/20.66 111. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 20.44/20.66 112. (distinct_lines T_1 T_0) (-. (distinct_lines T_1 T_0)) ### Axiom % 20.44/20.66 113. (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_1) ### Axiom % 20.44/20.66 114. (distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) (-. (distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1))) ### Axiom % 20.44/20.66 115. (distinct_lines T_0 T_1) (-. (distinct_lines T_0 T_1)) ### Axiom % 20.44/20.66 116. (apart_point_and_line (intersection_point T_3 T_2) T_0) (-. (apart_point_and_line (intersection_point T_3 T_2) T_0)) ### Axiom % 20.44/20.66 117. (distinct_points (intersection_point T_3 T_2) (intersection_point T_2 T_3)) (-. (distinct_points (intersection_point T_3 T_2) (intersection_point T_2 T_3))) ### Axiom % 20.49/20.70 118. (distinct_lines T_3 T_2) (-. (distinct_lines T_3 T_2)) ### Axiom % 20.49/20.70 119. (-. (apart_point_and_line (intersection_point T_3 T_2) T_3)) (apart_point_and_line (intersection_point T_3 T_2) T_3) ### Axiom % 20.49/20.70 120. (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (apart_point_and_line (intersection_point T_3 T_2) T_2) ### Axiom % 20.49/20.70 121. (((distinct_points (intersection_point T_3 T_2) (intersection_point T_2 T_3)) /\ (distinct_lines T_3 T_2)) => ((apart_point_and_line (intersection_point T_3 T_2) T_3) \/ ((apart_point_and_line (intersection_point T_3 T_2) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_3) \/ (apart_point_and_line (intersection_point T_2 T_3) T_2))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (-. (apart_point_and_line (intersection_point T_3 T_2) T_3)) (distinct_lines T_3 T_2) (distinct_points (intersection_point T_3 T_2) (intersection_point T_2 T_3)) ### DisjTree 117 118 119 120 79 22 % 20.49/20.70 122. (All V, (((distinct_points (intersection_point T_3 T_2) (intersection_point T_2 T_3)) /\ (distinct_lines T_3 V)) => ((apart_point_and_line (intersection_point T_3 T_2) T_3) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_3) \/ (apart_point_and_line (intersection_point T_2 T_3) V)))))) (distinct_points (intersection_point T_3 T_2) (intersection_point T_2 T_3)) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_3)) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) ### All 121 % 20.49/20.70 123. (All U, (All V, (((distinct_points (intersection_point T_3 T_2) (intersection_point T_2 T_3)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) U) \/ (apart_point_and_line (intersection_point T_2 T_3) V))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (-. (apart_point_and_line (intersection_point T_3 T_2) T_3)) (distinct_lines T_3 T_2) (distinct_points (intersection_point T_3 T_2) (intersection_point T_2 T_3)) ### All 122 % 20.49/20.70 124. (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point T_3 T_2) (intersection_point T_2 T_3)) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_3)) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) ### All 123 % 20.49/20.70 125. (-. (distinct_lines T_3 T_3)) (distinct_lines T_3 T_3) ### Axiom % 20.49/20.70 126. (convergent_lines T_3 T_2) (-. (convergent_lines T_3 T_2)) ### Axiom % 20.49/20.70 127. (apart_point_and_line (intersection_point T_3 T_2) T_3) (-. (apart_point_and_line (intersection_point T_3 T_2) T_3)) ### Axiom % 20.49/20.70 128. ((convergent_lines T_3 T_2) => (-. (apart_point_and_line (intersection_point T_3 T_2) T_3))) (apart_point_and_line (intersection_point T_3 T_2) T_3) (convergent_lines T_3 T_2) ### Imply 126 127 % 20.49/20.70 129. (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) (apart_point_and_line (intersection_point T_3 T_2) T_3) ### All 128 % 20.49/20.70 130. ((apart_point_and_line (intersection_point T_3 T_2) T_3) => ((distinct_lines T_3 T_3) \/ (apart_point_and_line (intersection_point T_3 T_2) T_3))) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (distinct_lines T_3 T_3)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (distinct_lines T_3 T_2) (distinct_points (intersection_point T_3 T_2) (intersection_point T_2 T_3)) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) ### DisjTree 124 125 129 % 20.49/20.70 131. (All Z, ((apart_point_and_line (intersection_point T_3 T_2) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_3 T_2) Z)))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point T_3 T_2) (intersection_point T_2 T_3)) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (distinct_lines T_3 T_3)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) ### All 130 % 20.49/20.70 132. (All Y, (All Z, ((apart_point_and_line (intersection_point T_3 T_2) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_3 T_2) Z))))) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (distinct_lines T_3 T_3)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (distinct_lines T_3 T_2) (distinct_points (intersection_point T_3 T_2) (intersection_point T_2 T_3)) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) ### All 131 % 20.49/20.70 133. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point T_3 T_2) (intersection_point T_2 T_3)) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (distinct_lines T_3 T_3)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) ### All 132 % 20.49/20.72 134. (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (apart_point_and_line (intersection_point T_2 T_3) T_0) ### Axiom % 20.49/20.72 135. ((apart_point_and_line (intersection_point T_3 T_2) T_0) => ((distinct_points (intersection_point T_3 T_2) (intersection_point T_2 T_3)) \/ (apart_point_and_line (intersection_point T_2 T_3) T_0))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (distinct_lines T_3 T_3)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (distinct_lines T_3 T_2) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) 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_3 T_2) T_0) ### DisjTree 116 133 134 % 20.49/20.72 136. (All Z, ((apart_point_and_line (intersection_point T_3 T_2) T_0) => ((distinct_points (intersection_point T_3 T_2) Z) \/ (apart_point_and_line Z T_0)))) (apart_point_and_line (intersection_point T_3 T_2) 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 Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (distinct_lines T_3 T_3)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) ### All 135 % 20.49/20.72 137. (All Y, (All Z, ((apart_point_and_line (intersection_point T_3 T_2) Y) => ((distinct_points (intersection_point T_3 T_2) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (distinct_lines T_3 T_3)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (distinct_lines T_3 T_2) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) 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_3 T_2) T_0) ### All 136 % 20.49/20.72 138. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (apart_point_and_line (intersection_point T_3 T_2) 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 Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (distinct_lines T_3 T_3)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) ### All 137 % 20.49/20.72 139. (apart_point_and_line (intersection_point T_3 T_2) T_1) (-. (apart_point_and_line (intersection_point T_3 T_2) T_1)) ### Axiom % 20.49/20.72 140. (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (apart_point_and_line (intersection_point T_2 T_3) T_1) ### Axiom % 20.49/20.72 141. ((apart_point_and_line (intersection_point T_3 T_2) T_1) => ((distinct_points (intersection_point T_3 T_2) (intersection_point T_2 T_3)) \/ (apart_point_and_line (intersection_point T_2 T_3) T_1))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (-. (apart_point_and_line (intersection_point T_3 T_2) T_3)) (distinct_lines T_3 T_2) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (apart_point_and_line (intersection_point T_3 T_2) T_1) ### DisjTree 139 124 140 % 20.49/20.72 142. (All Z, ((apart_point_and_line (intersection_point T_3 T_2) T_1) => ((distinct_points (intersection_point T_3 T_2) Z) \/ (apart_point_and_line Z T_1)))) (apart_point_and_line (intersection_point T_3 T_2) T_1) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_3)) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) ### All 141 % 20.49/20.72 143. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (-. (apart_point_and_line (intersection_point T_3 T_2) T_3)) (distinct_lines T_3 T_2) (apart_point_and_line (intersection_point T_3 T_2) T_1) (All Z, ((apart_point_and_line (intersection_point T_3 T_2) T_1) => ((distinct_points (intersection_point T_3 T_2) Z) \/ (apart_point_and_line Z T_1)))) ### All 142 % 20.49/20.72 144. (distinct_lines T_3 T_3) (-. (distinct_lines T_3 T_3)) ### Axiom % 20.49/20.72 145. (All X, (-. (distinct_lines X X))) (distinct_lines T_3 T_3) ### All 144 % 20.49/20.72 146. ((apart_point_and_line (intersection_point T_3 T_2) T_3) => ((distinct_lines T_3 T_3) \/ (apart_point_and_line (intersection_point T_3 T_2) T_3))) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (-. (distinct_lines X X))) (All Z, ((apart_point_and_line (intersection_point T_3 T_2) T_1) => ((distinct_points (intersection_point T_3 T_2) Z) \/ (apart_point_and_line Z T_1)))) (apart_point_and_line (intersection_point T_3 T_2) T_1) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) 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))))))))) ### DisjTree 143 145 129 % 20.49/20.76 147. (All Z, ((apart_point_and_line (intersection_point T_3 T_2) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_3 T_2) Z)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (distinct_lines T_3 T_2) (apart_point_and_line (intersection_point T_3 T_2) T_1) (All Z, ((apart_point_and_line (intersection_point T_3 T_2) T_1) => ((distinct_points (intersection_point T_3 T_2) Z) \/ (apart_point_and_line Z T_1)))) (All X, (-. (distinct_lines X X))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) ### All 146 % 20.49/20.76 148. (All Y, (All Z, ((apart_point_and_line (intersection_point T_3 T_2) Y) => ((distinct_points (intersection_point T_3 T_2) Z) \/ (apart_point_and_line Z Y))))) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (-. (distinct_lines X X))) (apart_point_and_line (intersection_point T_3 T_2) T_1) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((apart_point_and_line (intersection_point T_3 T_2) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_3 T_2) Z)))) ### All 147 % 20.49/20.76 149. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Z, ((apart_point_and_line (intersection_point T_3 T_2) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_3 T_2) Z)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (distinct_lines T_3 T_2) (apart_point_and_line (intersection_point T_3 T_2) T_1) (All X, (-. (distinct_lines X X))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) ### All 148 % 20.49/20.76 150. (All Y, (All Z, ((apart_point_and_line (intersection_point T_3 T_2) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_3 T_2) Z))))) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (-. (distinct_lines X X))) (apart_point_and_line (intersection_point T_3 T_2) T_1) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 149 % 20.49/20.76 151. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (distinct_lines T_3 T_2) (apart_point_and_line (intersection_point T_3 T_2) T_1) (All X, (-. (distinct_lines X X))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) ### All 150 % 20.49/20.76 152. (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### Axiom % 20.49/20.76 153. (((distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) /\ (distinct_lines T_0 T_1)) => ((apart_point_and_line (intersection_point T_3 T_2) T_0) \/ ((apart_point_and_line (intersection_point T_3 T_2) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (distinct_lines T_3 T_3)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (distinct_lines T_3 T_2) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_lines T_0 T_1) (distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) ### DisjTree 114 115 138 151 32 152 % 20.58/20.77 154. (All V, (((distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) /\ (distinct_lines T_0 V)) => ((apart_point_and_line (intersection_point T_3 T_2) T_0) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) V)))))) (distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) (distinct_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (distinct_lines T_3 T_3)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) 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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (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_1)) ### All 153 % 20.58/20.77 155. (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (distinct_lines T_3 T_2) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_lines T_0 T_1) (distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) (All V, (((distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) /\ (distinct_lines T_0 V)) => ((apart_point_and_line (intersection_point T_3 T_2) T_0) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) V)))))) ### All 154 % 20.58/20.77 156. (All U, (All V, (((distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) (distinct_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) 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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (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_1)) ### All 155 % 20.58/20.77 157. ((distinct_lines T_1 T_0) => ((distinct_lines T_1 T_1) \/ (distinct_lines T_0 T_1))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (distinct_lines T_3 T_2) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) (All U, (All V, (((distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_0) ### DisjTree 112 113 156 % 20.59/20.82 158. (All Z, ((distinct_lines T_1 T_0) => ((distinct_lines T_1 Z) \/ (distinct_lines T_0 Z)))) (distinct_lines T_1 T_0) (-. (distinct_lines T_1 T_1)) (All U, (All V, (((distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) 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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (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_1)) ### All 157 % 20.59/20.82 159. (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (distinct_lines T_3 T_2) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) (All U, (All V, (((distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_0) ### All 158 % 20.59/20.82 160. ((convergent_lines T_0 T_1) => ((distinct_lines T_1 T_0) \/ (convergent_lines T_0 T_0))) (All X, (-. (convergent_lines X X))) (-. (distinct_lines T_1 T_1)) (All U, (All V, (((distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) 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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (convergent_lines T_0 T_1) ### DisjTree 111 159 42 % 20.59/20.82 161. (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (distinct_lines T_3 T_2) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) (All U, (All V, (((distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (-. (distinct_lines T_1 T_1)) (All X, (-. (convergent_lines X X))) ### All 160 % 20.64/20.83 162. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (-. (distinct_lines T_1 T_1)) (All U, (All V, (((distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) 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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### All 161 % 20.64/20.83 163. (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (distinct_lines T_3 T_2) (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_2) U) \/ ((apart_point_and_line (intersection_point T_3 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) (-. (distinct_lines T_1 T_1)) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### All 162 % 20.64/20.83 164. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (-. (distinct_lines T_1 T_1)) (distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) 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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### All 163 % 20.64/20.83 165. ((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_2)) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)) \/ (distinct_points (intersection_point T_3 T_2) (intersection_point T_0 T_1)))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (distinct_lines T_3 T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (distinct_lines T_1 T_1)) (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, (-. (distinct_points X X))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_2)) ### DisjTree 108 110 164 % 20.66/20.87 166. (All Z, ((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_2)) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points (intersection_point T_3 T_2) Z)))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_2)) (All X, (-. (distinct_points X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (-. (distinct_lines T_1 T_1)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) 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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### All 165 % 20.66/20.87 167. (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (distinct_lines T_3 T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (distinct_lines T_1 T_1)) (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, (-. (distinct_points X X))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_2)) ### All 166 % 20.66/20.87 168. (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_2)) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) 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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) ### All 167 % 20.66/20.87 169. (distinct_lines T_1 T_1) (-. (distinct_lines T_1 T_1)) ### Axiom % 20.66/20.87 170. (All X, (-. (distinct_lines X X))) (distinct_lines T_1 T_1) ### All 169 % 20.66/20.87 171. ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 T_1) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1))) (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (distinct_lines T_3 T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_2)) ### DisjTree 168 170 50 % 20.66/20.87 172. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_2)) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) 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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) ### All 171 % 20.66/20.89 173. (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (apart_point_and_line (intersection_point T_3 T_2) T_2) ### Axiom % 20.66/20.89 174. ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_2)) \/ (apart_point_and_line (intersection_point T_3 T_2) T_2))) (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (distinct_lines T_3 T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### DisjTree 107 172 173 % 20.66/20.89 175. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (distinct_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_3 T_2) T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (convergent_lines T_3 T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) 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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) ### All 174 % 20.66/20.89 176. ((convergent_lines T_3 T_2) => (-. (apart_point_and_line (intersection_point T_3 T_2) T_2))) (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_3 T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (convergent_lines T_3 T_2) ### Imply 106 175 % 20.66/20.89 177. (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) Y)))) (convergent_lines T_3 T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (distinct_lines T_3 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) 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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) ### All 176 % 20.73/20.93 178. (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 Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_3 T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (convergent_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) Y)))) ### All 177 % 20.73/20.93 179. (convergent_lines T_3 T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (distinct_lines T_3 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) 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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (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 178 % 20.73/20.93 180. ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 T_2) \/ (apart_point_and_line (intersection_point T_0 T_1) 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 Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (distinct_lines T_3 T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (convergent_lines T_3 T_2) (-. (distinct_lines T_2 T_2)) (All X, (-. (distinct_lines X X))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (distinct_lines T_1 T_2) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (convergent_lines T_0 T_1) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) ### DisjTree 104 105 179 % 20.73/20.93 181. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (convergent_lines T_0 T_1) (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)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (-. (distinct_lines X X))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_3 T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) 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 Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (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 180 % 20.73/20.95 182. (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 Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (distinct_lines T_3 T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (convergent_lines T_3 T_2) (-. (distinct_lines T_2 T_2)) (All X, (-. (distinct_lines X X))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (distinct_lines T_1 T_2) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (convergent_lines T_0 T_1) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) ### All 181 % 20.73/20.95 183. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (convergent_lines T_0 T_1) (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)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_3) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_3 T_2) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) 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 Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (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 182 % 20.73/20.95 184. (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 Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (distinct_lines T_3 T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (convergent_lines T_3 T_2) (-. (distinct_lines T_2 T_2)) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (distinct_lines T_1 T_2) (convergent_lines T_2 T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (convergent_lines T_0 T_1) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) ### All 183 % 20.73/20.95 185. ((convergent_lines T_2 T_3) => ((convergent_lines T_2 T_2) \/ (convergent_lines T_3 T_2))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (convergent_lines T_0 T_1) (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)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (-. (distinct_lines T_2 T_2)) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) 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 Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (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_2 T_2)) (convergent_lines T_2 T_3) ### DisjTree 67 68 184 % 20.73/20.95 186. (All Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (convergent_lines T_2 T_3) (-. (convergent_lines T_2 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 Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, (All Z, ((distinct_lines T_1 Y) => ((distinct_lines T_1 Z) \/ (distinct_lines Y Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (distinct_lines T_3 T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (-. (distinct_lines T_2 T_2)) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (distinct_lines T_1 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (convergent_lines T_0 T_1) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) ### All 185 % 20.73/20.95 187. (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (convergent_lines T_0 T_1) (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)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_1 T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (-. (distinct_lines T_2 T_2)) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (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_2 T_2)) (convergent_lines T_2 T_3) (All Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) ### All 186 % 20.78/21.00 188. (All Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (convergent_lines T_2 T_3) (-. (convergent_lines T_2 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 Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (distinct_lines T_3 T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (distinct_lines T_1 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (convergent_lines T_0 T_1) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### All 187 % 20.78/21.00 189. (-. (convergent_lines T_2 T_2)) (convergent_lines T_2 T_2) ### Axiom % 20.78/21.00 190. ((convergent_lines T_2 T_1) => ((distinct_lines T_1 T_2) \/ (convergent_lines T_2 T_2))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (convergent_lines T_0 T_1) (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)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_lines T_3 T_2) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (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_2 T_2)) (convergent_lines T_2 T_3) (All Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (convergent_lines T_2 T_1) ### DisjTree 66 188 189 % 20.78/21.00 191. (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (convergent_lines T_2 T_3) (-. (convergent_lines T_2 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 Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (distinct_lines T_3 T_2) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (convergent_lines T_0 T_1) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### All 190 % 20.78/21.00 192. (-. (convergent_lines T_2 T_2)) (convergent_lines T_2 T_2) ### Axiom % 20.78/21.00 193. ((convergent_lines T_2 T_3) => ((distinct_lines T_3 T_2) \/ (convergent_lines T_2 T_2))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (convergent_lines T_0 T_1) (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)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (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_2 T_2)) (All Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_2 T_3) ### DisjTree 65 191 192 % 20.78/21.01 194. (All Z, ((convergent_lines T_2 T_3) => ((distinct_lines T_3 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_2 T_3) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (-. (convergent_lines T_2 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 Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (convergent_lines T_0 T_1) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### All 193 % 20.78/21.01 195. (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (convergent_lines T_0 T_1) (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)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Y, (All Z, ((distinct_points (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (distinct_points Y Z))))) (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 Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_2 T_3) (All Z, ((convergent_lines T_2 T_3) => ((distinct_lines T_3 Z) \/ (convergent_lines T_2 Z)))) ### All 194 % 20.78/21.01 196. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Z, ((convergent_lines T_2 T_3) => ((distinct_lines T_3 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_2 T_3) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (convergent_lines T_0 T_1) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### All 195 % 20.78/21.03 197. (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (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 Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_2 T_3) (All Z, ((convergent_lines T_2 T_3) => ((distinct_lines T_3 Z) \/ (convergent_lines T_2 Z)))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 196 % 20.78/21.03 198. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Z, ((convergent_lines T_2 T_3) => ((distinct_lines T_3 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_2 T_3) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (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 Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### All 197 % 20.78/21.03 199. (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (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 Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_2 T_3) (All Z, ((convergent_lines T_2 T_3) => ((distinct_lines T_3 Z) \/ (convergent_lines T_2 Z)))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 198 % 20.78/21.03 200. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Z, ((convergent_lines T_2 T_3) => ((distinct_lines T_3 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_2 T_3) (All Z, ((convergent_lines T_2 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_2 Z)))) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### All 199 % 20.87/21.06 201. (All Y, (All Z, ((convergent_lines T_2 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_2 Z))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (convergent_lines T_2 T_1) (convergent_lines T_2 T_3) (All Z, ((convergent_lines T_2 T_3) => ((distinct_lines T_3 Z) \/ (convergent_lines T_2 Z)))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 200 % 20.87/21.06 202. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_2 T_3) (convergent_lines T_2 T_1) (All Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, (All Z, ((convergent_lines T_2 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_2 Z))))) ### All 201 % 20.87/21.06 203. (-. (convergent_lines T_3 T_1)) (convergent_lines T_3 T_1) ### Axiom % 20.87/21.06 204. ((convergent_lines T_2 T_3) => ((convergent_lines T_2 T_1) \/ (convergent_lines T_3 T_1))) (-. (convergent_lines T_3 T_1)) (All Y, (All Z, ((convergent_lines T_2 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_2 Z))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_2 T_3) ### DisjTree 64 202 203 % 20.87/21.06 205. (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, (All Z, ((convergent_lines T_2 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_2 Z))))) (-. (convergent_lines T_3 T_1)) ### All 204 % 20.87/21.06 206. (All Y, (All Z, ((convergent_lines T_2 Y) => ((convergent_lines T_2 Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines T_3 T_1)) (All Y, (All Z, ((convergent_lines T_2 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_2 Z))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_2 T_3) ### All 205 % 20.87/21.07 207. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (-. (convergent_lines T_3 T_1)) (All Y, (All Z, ((convergent_lines T_2 Y) => ((convergent_lines T_2 Z) \/ (convergent_lines Y Z))))) ### All 206 % 20.87/21.07 208. (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All Y, (All Z, ((convergent_lines T_2 Y) => ((convergent_lines T_2 Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines T_3 T_1)) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 207 % 20.87/21.07 209. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (-. (convergent_lines T_3 T_1)) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) ### All 208 % 20.87/21.07 210. (-. (convergent_lines T_3 T_1)) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 209 % 20.87/21.07 211. (convergent_lines T_2 T_3) (-. (convergent_lines T_2 T_3)) ### Axiom % 20.87/21.07 212. (convergent_lines T_3 T_1) (-. (convergent_lines T_3 T_1)) ### Axiom % 20.87/21.07 213. (-. (convergent_lines T_3 T_3)) (convergent_lines T_3 T_3) ### Axiom % 20.87/21.07 214. (convergent_lines T_1 T_3) (-. (convergent_lines T_1 T_3)) ### Axiom % 20.87/21.07 215. (-. (convergent_lines T_1 T_1)) (convergent_lines T_1 T_1) ### Axiom % 20.87/21.07 216. (convergent_lines T_1 T_3) (-. (convergent_lines T_1 T_3)) ### Axiom % 20.87/21.07 217. (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) ### Axiom % 20.87/21.07 218. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 20.87/21.07 219. (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_0) ### Axiom % 20.87/21.07 220. (-. (convergent_lines T_1 T_0)) (convergent_lines T_1 T_0) ### Axiom % 20.87/21.07 221. ((convergent_lines T_0 T_1) => ((convergent_lines T_0 T_0) \/ (convergent_lines T_1 T_0))) (-. (convergent_lines T_1 T_0)) (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_1) ### DisjTree 218 219 220 % 20.91/21.11 222. (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_0)) (-. (convergent_lines T_1 T_0)) ### All 221 % 20.91/21.11 223. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom % 20.91/21.11 224. (-. (distinct_lines T_0 T_1)) (distinct_lines T_0 T_1) ### Axiom % 20.91/21.11 225. (-. (convergent_lines T_1 T_1)) (convergent_lines T_1 T_1) ### Axiom % 20.91/21.11 226. ((convergent_lines T_1 T_0) => ((distinct_lines T_0 T_1) \/ (convergent_lines T_1 T_1))) (-. (convergent_lines T_1 T_1)) (-. (distinct_lines T_0 T_1)) (convergent_lines T_1 T_0) ### DisjTree 223 224 225 % 20.91/21.11 227. (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (-. (distinct_lines T_0 T_1)) (-. (convergent_lines T_1 T_1)) ### All 226 % 20.91/21.11 228. (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (-. (distinct_lines T_0 T_1)) (convergent_lines T_1 T_0) ### All 227 % 20.91/21.11 229. (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_0) ### Axiom % 20.91/21.11 230. ((convergent_lines T_1 T_0) => ((convergent_lines T_1 T_0) \/ (convergent_lines T_0 T_0))) (-. (distinct_lines T_0 T_1)) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) ### DisjTree 222 228 229 % 20.91/21.11 231. (All Z, ((convergent_lines T_1 T_0) => ((convergent_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_0)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (-. (distinct_lines T_0 T_1)) ### All 230 % 20.91/21.11 232. (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (distinct_lines T_0 T_1)) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) ### All 231 % 20.91/21.11 233. (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (-. (distinct_lines T_0 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) ### All 232 % 20.91/21.11 234. (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1))) ### Axiom % 20.91/21.11 235. (distinct_lines T_0 T_1) (-. (distinct_lines T_0 T_1)) ### Axiom % 20.91/21.11 236. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (apart_point_and_line (intersection_point T_0 T_1) T_0) (convergent_lines T_0 T_1) ### All 32 % 20.91/21.11 237. (apart_point_and_line (intersection_point T_3 T_1) T_0) (-. (apart_point_and_line (intersection_point T_3 T_1) T_0)) ### Axiom % 20.91/21.11 238. (distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) (-. (distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3))) ### Axiom % 20.91/21.11 239. (distinct_lines T_3 T_1) (-. (distinct_lines T_3 T_1)) ### Axiom % 20.91/21.11 240. (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (apart_point_and_line (intersection_point T_3 T_1) T_3) ### Axiom % 20.91/21.11 241. (convergent_lines T_3 T_1) (-. (convergent_lines T_3 T_1)) ### Axiom % 20.91/21.11 242. (apart_point_and_line (intersection_point T_3 T_1) T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_1)) ### Axiom % 20.91/21.11 243. ((convergent_lines T_3 T_1) => (-. (apart_point_and_line (intersection_point T_3 T_1) T_1))) (apart_point_and_line (intersection_point T_3 T_1) T_1) (convergent_lines T_3 T_1) ### Imply 241 242 % 20.91/21.11 244. (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) Y)))) (convergent_lines T_3 T_1) (apart_point_and_line (intersection_point T_3 T_1) T_1) ### All 243 % 20.91/21.11 245. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_3 T_1) T_1) (convergent_lines T_3 T_1) ### All 244 % 20.91/21.11 246. (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (apart_point_and_line (intersection_point T_2 T_3) T_1) ### Axiom % 20.91/21.11 247. (((distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines T_3 T_1)) => ((apart_point_and_line (intersection_point T_3 T_1) T_3) \/ ((apart_point_and_line (intersection_point T_3 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_3) \/ (apart_point_and_line (intersection_point T_2 T_3) T_1))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (convergent_lines T_3 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) ### DisjTree 238 239 240 245 79 246 % 20.91/21.11 248. (All V, (((distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines T_3 V)) => ((apart_point_and_line (intersection_point T_3 T_1) T_3) \/ ((apart_point_and_line (intersection_point T_3 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) T_3) \/ (apart_point_and_line (intersection_point T_2 T_3) V)))))) (distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_3 T_1) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) ### All 247 % 20.91/21.11 249. (All U, (All V, (((distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_1) U) \/ ((apart_point_and_line (intersection_point T_3 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_3) U) \/ (apart_point_and_line (intersection_point T_2 T_3) V))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (convergent_lines T_3 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) ### All 248 % 20.91/21.11 250. (All Y, (All U, (All V, (((distinct_points (intersection_point T_3 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_3 T_1) U) \/ ((apart_point_and_line (intersection_point T_3 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_3 T_1) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) ### All 249 % 20.91/21.11 251. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (convergent_lines T_3 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) ### All 250 % 20.91/21.12 252. (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (apart_point_and_line (intersection_point T_2 T_3) T_0) ### Axiom % 20.91/21.12 253. ((apart_point_and_line (intersection_point T_3 T_1) T_0) => ((distinct_points (intersection_point T_3 T_1) (intersection_point T_2 T_3)) \/ (apart_point_and_line (intersection_point T_2 T_3) T_0))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_3 T_1) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_3 T_1) T_0) ### DisjTree 237 251 252 % 20.91/21.12 254. (All Z, ((apart_point_and_line (intersection_point T_3 T_1) T_0) => ((distinct_points (intersection_point T_3 T_1) Z) \/ (apart_point_and_line Z T_0)))) (apart_point_and_line (intersection_point T_3 T_1) T_0) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (convergent_lines T_3 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) ### All 253 % 20.91/21.12 255. (All Y, (All Z, ((apart_point_and_line (intersection_point T_3 T_1) Y) => ((distinct_points (intersection_point T_3 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_3 T_1) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_3 T_1) T_0) ### All 254 % 20.91/21.12 256. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (apart_point_and_line (intersection_point T_3 T_1) T_0) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (convergent_lines T_3 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) ### All 255 % 20.91/21.12 257. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (distinct_lines T_0 T_1)) => ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_3 T_1) T_0) \/ (apart_point_and_line (intersection_point T_3 T_1) T_1))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (convergent_lines T_3 T_1) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) ### DisjTree 234 235 236 50 256 245 % 20.91/21.12 258. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (distinct_lines T_0 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_3 T_1) T_0) \/ (apart_point_and_line (intersection_point T_3 T_1) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (distinct_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (convergent_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) ### All 257 % 20.91/21.12 259. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (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_3 T_1) U) \/ (apart_point_and_line (intersection_point T_3 T_1) V))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (convergent_lines T_3 T_1) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) ### All 258 % 20.91/21.12 260. (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_1) ### Axiom % 20.91/21.12 261. ((distinct_lines T_0 T_1) => ((distinct_lines T_0 T_1) \/ (distinct_lines T_1 T_1))) (-. (distinct_lines T_1 T_1)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (convergent_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (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_3 T_1) U) \/ (apart_point_and_line (intersection_point T_3 T_1) V))))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (convergent_lines X X))) ### DisjTree 233 259 260 % 20.97/21.16 262. (All Z, ((distinct_lines T_0 T_1) => ((distinct_lines T_0 Z) \/ (distinct_lines T_1 Z)))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (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_3 T_1) U) \/ (apart_point_and_line (intersection_point T_3 T_1) V))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (convergent_lines T_3 T_1) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (-. (distinct_lines T_1 T_1)) ### All 261 % 20.97/21.16 263. (All Y, (All Z, ((distinct_lines T_0 Y) => ((distinct_lines T_0 Z) \/ (distinct_lines Y Z))))) (-. (distinct_lines T_1 T_1)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (convergent_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (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_3 T_1) U) \/ (apart_point_and_line (intersection_point T_3 T_1) V))))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (convergent_lines X X))) ### All 262 % 20.97/21.16 264. (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) /\ (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_3 T_1) U) \/ (apart_point_and_line (intersection_point T_3 T_1) V))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (convergent_lines T_3 T_1) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (-. (distinct_lines T_1 T_1)) ### All 263 % 20.97/21.16 265. (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)))))))) (-. (distinct_lines T_1 T_1)) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (convergent_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### All 264 % 20.97/21.18 266. (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (convergent_lines T_3 T_1) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) (-. (distinct_lines T_1 T_1)) ### All 265 % 20.97/21.18 267. (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (apart_point_and_line (intersection_point T_3 T_1) T_3) ### Axiom % 20.97/21.18 268. ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_3 T_1)) \/ (apart_point_and_line (intersection_point T_3 T_1) T_3))) (-. (distinct_lines T_1 T_1)) (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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (convergent_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) ### DisjTree 217 266 267 % 20.97/21.18 269. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (convergent_lines T_3 T_1) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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))))) (-. (distinct_lines T_1 T_1)) ### All 268 % 20.97/21.18 270. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z Y))))) (-. (distinct_lines T_1 T_1)) (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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (convergent_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) ### All 269 % 20.97/21.18 271. (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (convergent_lines T_3 T_1) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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))))) (-. (distinct_lines T_1 T_1)) ### All 270 % 20.97/21.18 272. (All X, (-. (distinct_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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (convergent_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (distinct_lines T_3 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) ### All 271 % 21.04/21.21 273. (-. (convergent_lines T_1 T_1)) (convergent_lines T_1 T_1) ### Axiom % 21.04/21.21 274. ((convergent_lines T_1 T_3) => ((distinct_lines T_3 T_1) \/ (convergent_lines T_1 T_1))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (convergent_lines T_1 T_1)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (convergent_lines T_3 T_1) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_lines X X))) (convergent_lines T_1 T_3) ### DisjTree 216 272 273 % 21.04/21.21 275. ((convergent_lines T_1 T_3) => ((convergent_lines T_1 T_1) \/ (convergent_lines T_3 T_1))) (All X, (-. (distinct_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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) ((convergent_lines T_1 T_3) => ((distinct_lines T_3 T_1) \/ (convergent_lines T_1 T_1))) (-. (convergent_lines T_1 T_1)) (convergent_lines T_1 T_3) ### DisjTree 214 215 274 % 21.04/21.21 276. (All Z, ((convergent_lines T_1 T_3) => ((convergent_lines T_1 Z) \/ (convergent_lines T_3 Z)))) (convergent_lines T_1 T_3) (-. (convergent_lines T_1 T_1)) ((convergent_lines T_1 T_3) => ((distinct_lines T_3 T_1) \/ (convergent_lines T_1 T_1))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_lines X X))) ### All 275 % 21.04/21.21 277. (All Z, ((convergent_lines T_1 T_3) => ((distinct_lines T_3 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_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) => (-. 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(convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (convergent_lines T_1 T_1)) (convergent_lines T_1 T_3) (All Z, ((convergent_lines T_1 T_3) => ((convergent_lines T_1 Z) \/ (convergent_lines T_3 Z)))) ### All 276 % 21.04/21.21 278. (convergent_lines T_1 T_3) (-. (convergent_lines T_1 T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_1 T_3) => ((distinct_lines T_3 Z) \/ (convergent_lines T_1 Z)))) ### All 277 % 21.04/21.23 279. (All X, (-. (distinct_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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, (All Z, ((convergent_lines T_1 Y) => ((convergent_lines T_1 Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (convergent_lines T_1 T_1)) (convergent_lines T_1 T_3) ### All 278 % 21.04/21.23 280. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (convergent_lines T_1 T_3) (-. (convergent_lines T_1 T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_lines X X))) ### All 279 % 21.04/21.23 281. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (convergent_lines T_1 T_1)) (convergent_lines T_1 T_3) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 280 % 21.04/21.23 282. ((convergent_lines T_3 T_1) => ((convergent_lines T_3 T_3) \/ (convergent_lines T_1 T_3))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_1 T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (convergent_lines T_3 T_3)) (convergent_lines T_3 T_1) ### DisjTree 212 213 281 % 21.04/21.23 283. (All Z, ((convergent_lines T_3 T_1) => ((convergent_lines T_3 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_3 T_1) (-. (convergent_lines T_3 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (distinct_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) => (-. 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(convergent_lines T_1 T_1)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 282 % 21.04/21.23 284. (All Y, (All Z, ((convergent_lines T_3 Y) => ((convergent_lines T_3 Z) \/ (convergent_lines Y Z))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_1 T_1)) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (convergent_lines T_3 T_3)) (convergent_lines T_3 T_1) ### All 283 % 21.04/21.23 285. (convergent_lines T_3 T_1) (-. (convergent_lines T_3 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. 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(convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (convergent_lines T_1 T_1)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 284 % 21.04/21.23 286. ((convergent_lines T_2 T_3) => ((convergent_lines T_2 T_1) \/ (convergent_lines T_3 T_1))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_1 T_1)) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (convergent_lines T_3 T_3)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Z, ((convergent_lines T_2 T_3) => ((distinct_lines T_3 Z) \/ (convergent_lines T_2 Z)))) (All Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, (All Z, ((convergent_lines T_2 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_2 Z))))) (convergent_lines T_2 T_3) ### DisjTree 211 201 285 % 21.04/21.23 287. (convergent_lines T_2 T_3) (All Y, (All Z, ((convergent_lines T_2 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_2 Z))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All Z, ((convergent_lines T_2 T_3) => ((convergent_lines T_2 Z) \/ (convergent_lines T_3 Z)))) (All Z, ((convergent_lines T_2 T_3) => ((distinct_lines T_3 Z) \/ (convergent_lines T_2 Z)))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_3 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 286 % 21.04/21.23 288. (All Y, (All Z, ((convergent_lines T_2 Y) => ((convergent_lines T_2 Z) \/ (convergent_lines Y Z))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_1 T_1)) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (convergent_lines T_3 T_3)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Z, ((convergent_lines T_2 T_3) => ((distinct_lines T_3 Z) \/ (convergent_lines T_2 Z)))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, (All Z, ((convergent_lines T_2 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_2 Z))))) (convergent_lines T_2 T_3) ### All 287 % 21.04/21.27 289. (convergent_lines T_2 T_3) (All Y, (All Z, ((convergent_lines T_2 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_2 Z))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_3 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All Y, (All Z, ((convergent_lines T_2 Y) => ((convergent_lines T_2 Z) \/ (convergent_lines Y Z))))) ### All 288 % 21.04/21.27 290. (All Y, (All Z, ((convergent_lines T_2 Y) => ((convergent_lines T_2 Z) \/ (convergent_lines Y Z))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_1 T_1)) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (convergent_lines T_3 T_3)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_2 T_3) ### All 289 % 21.04/21.27 291. (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_3 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All Y, (All Z, ((convergent_lines T_2 Y) => ((convergent_lines T_2 Z) \/ (convergent_lines Y Z))))) ### All 290 % 21.04/21.27 292. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_1 T_1)) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (convergent_lines T_3 T_3)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_2 T_3) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) ### All 291 % 21.04/21.27 293. (All Y, (All Z, ((convergent_lines T_0 Y) => ((convergent_lines T_0 Z) \/ (convergent_lines Y Z))))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_3 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (-. (convergent_lines T_1 T_1)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 292 % 21.11/21.28 294. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (convergent_lines T_3 T_3)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_2 T_3) (All Y, (All Z, ((convergent_lines T_0 Y) => ((convergent_lines T_0 Z) \/ (convergent_lines Y Z))))) ### All 293 % 21.11/21.28 295. (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_3 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (-. (apart_point_and_line (intersection_point T_3 T_1) T_3)) (-. (convergent_lines T_1 T_1)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 294 % 21.11/21.28 296. ((convergent_lines T_3 T_1) => (-. (apart_point_and_line (intersection_point T_3 T_1) T_3))) (-. (convergent_lines T_1 T_1)) (-. (convergent_lines T_3 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### Imply 210 295 % 21.11/21.28 297. (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, ((convergent_lines T_3 Y) => (-. (apart_point_and_line (intersection_point T_3 Y) T_3)))) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_3 T_3)) (-. (convergent_lines T_1 T_1)) ### All 296 % 21.11/21.29 298. (-. (convergent_lines T_1 T_1)) (-. (convergent_lines T_3 T_3)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### All 297 % 21.11/21.29 299. (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (convergent_lines T_1 T_1)) ### All 298 % 21.11/21.29 300. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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, (-. (distinct_points X X))) (All X, (-. (distinct_lines X X))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### All 299 % 21.11/21.29 301. (-. (-. (apart_point_and_line (intersection_point T_0 T_1) T_3))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (-. (distinct_lines X X))) (All X, (-. (distinct_points X X))) (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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotNot 300 % 21.11/21.29 302. (-. ((-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) /\ (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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, (-. (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 Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotAnd 63 301 % 21.11/21.29 303. (-. (((convergent_lines T_2 T_3) /\ ((convergent_lines T_0 T_1) /\ ((-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) /\ (-. (apart_point_and_line (intersection_point T_2 T_3) T_1))))) => ((-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) /\ (-. (apart_point_and_line (intersection_point T_0 T_1) T_3))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (distinct_lines X X))) (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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### ConjTree 302 % 21.11/21.29 304. (-. (All V, (((convergent_lines T_2 T_3) /\ ((convergent_lines T_0 V) /\ ((-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) /\ (-. (apart_point_and_line (intersection_point T_2 T_3) V))))) => ((-. (apart_point_and_line (intersection_point T_0 V) T_2)) /\ (-. (apart_point_and_line (intersection_point T_0 V) T_3)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X 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, (-. (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 Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotAllEx 303 % 21.11/21.29 305. (-. (All U, (All V, (((convergent_lines T_2 T_3) /\ ((convergent_lines U V) /\ ((-. (apart_point_and_line (intersection_point T_2 T_3) U)) /\ (-. (apart_point_and_line (intersection_point T_2 T_3) V))))) => ((-. (apart_point_and_line (intersection_point U V) T_2)) /\ (-. (apart_point_and_line (intersection_point U V) T_3))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (distinct_lines X X))) (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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 304 % 21.11/21.29 306. (-. (All Y, (All U, (All V, (((convergent_lines T_2 Y) /\ ((convergent_lines U V) /\ ((-. (apart_point_and_line (intersection_point T_2 Y) U)) /\ (-. (apart_point_and_line (intersection_point T_2 Y) V))))) => ((-. (apart_point_and_line (intersection_point U V) T_2)) /\ (-. (apart_point_and_line (intersection_point U V) Y)))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (distinct_lines X 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, (-. (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 Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotAllEx 305 % 21.11/21.29 307. (-. (All X, (All Y, (All U, (All V, (((convergent_lines X Y) /\ ((convergent_lines U V) /\ ((-. (apart_point_and_line (intersection_point X Y) U)) /\ (-. (apart_point_and_line (intersection_point X Y) V))))) => ((-. (apart_point_and_line (intersection_point U V) X)) /\ (-. (apart_point_and_line (intersection_point U V) Y))))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (distinct_lines X X))) (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, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 306 % 21.11/21.29 % SZS output end Proof % 21.11/21.29 (* END-PROOF *) %------------------------------------------------------------------------------