%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO195+2 : 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 : n018.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 0.38s 0.56s % Output : Proof 0.38s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.12 % Problem : GEO195+2 : TPTP v8.1.0. Released v3.3.0. % 0.07/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.12/0.34 % Computer : n018.cluster.edu % 0.12/0.34 % Model : x86_64 x86_64 % 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.34 % Memory : 8042.1875MB % 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.34 % CPULimit : 300 % 0.12/0.34 % WCLimit : 600 % 0.12/0.34 % DateTime : Fri Jun 17 18:53:47 EDT 2022 % 0.12/0.34 % CPUTime : % 0.38/0.56 % SZS status Theorem % 0.38/0.56 (* PROOF-FOUND *) % 0.38/0.56 (* BEGIN-PROOF *) % 0.38/0.56 % SZS output start Proof % 0.38/0.56 1. (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 % 0.38/0.56 2. (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0))) ### Axiom % 0.38/0.56 3. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 0.38/0.56 4. (-. (distinct_lines T_0 T_1)) (convergent_lines T_0 T_1) ### Extension/test/ceq3ctrp 3 % 0.38/0.56 5. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 0.38/0.56 6. (apart_point_and_line (intersection_point T_0 T_1) T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) ### Axiom % 0.38/0.56 7. (-. ((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_0) ### NotOr 6 % 0.38/0.56 8. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)) ### Extension/test/apart1 % 0.38/0.56 9. ((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_1)) => (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)))) (apart_point_and_line (intersection_point T_0 T_1) T_0) (convergent_lines T_0 T_1) ### DisjTree 5 7 8 % 0.38/0.56 10. (All Z, ((convergent_lines T_0 T_1) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_1)) => (distinct_points Z (intersection_point T_0 T_1))))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### All 9 % 0.38/0.56 11. (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) (apart_point_and_line (intersection_point T_0 T_1) T_0) (convergent_lines T_0 T_1) ### All 10 % 0.38/0.56 12. (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### All 11 % 0.38/0.56 13. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 0.38/0.56 14. (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 % 0.38/0.56 15. (-. ((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) ### NotOr 14 % 0.38/0.56 16. ((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_1)) => (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)))) (apart_point_and_line (intersection_point T_0 T_1) T_1) (convergent_lines T_0 T_1) ### DisjTree 13 15 8 % 0.38/0.56 17. (All Z, ((convergent_lines T_0 T_1) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_1)) => (distinct_points Z (intersection_point T_0 T_1))))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### All 16 % 0.38/0.56 18. (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) (apart_point_and_line (intersection_point T_0 T_1) T_1) (convergent_lines T_0 T_1) ### All 17 % 0.38/0.56 19. (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### All 18 % 0.38/0.56 20. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 0.38/0.56 21. (convergent_lines T_0 T_0) (-. (convergent_lines T_0 T_0)) ### Axiom % 0.38/0.56 22. (-. (distinct_lines T_0 T_0)) (convergent_lines T_0 T_0) ### Extension/test/ceq3ctrp 21 % 0.38/0.56 23. (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_0) ### All 22 % 0.38/0.56 24. (-. (convergent_lines T_1 T_0)) (convergent_lines T_1 T_0) ### Axiom % 0.38/0.56 25. ((convergent_lines T_0 T_1) => ((convergent_lines T_0 T_0) \/ (convergent_lines T_1 T_0))) (-. (convergent_lines T_1 T_0)) (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_1) ### DisjTree 20 23 24 % 0.38/0.56 26. (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 X, (-. (distinct_lines X X))) (-. (convergent_lines T_1 T_0)) ### All 25 % 0.38/0.56 27. (apart_point_and_line (intersection_point T_1 T_0) T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_0)) ### Axiom % 0.38/0.56 28. (-. ((apart_point_and_line (intersection_point T_1 T_0) T_1) \/ (apart_point_and_line (intersection_point T_1 T_0) T_0))) (apart_point_and_line (intersection_point T_1 T_0) T_0) ### NotOr 27 % 0.38/0.56 29. (distinct_points (intersection_point T_1 T_0) (intersection_point T_1 T_0)) ### Extension/test/apart1 % 0.38/0.56 30. ((convergent_lines T_1 T_0) => (((apart_point_and_line (intersection_point T_1 T_0) T_1) \/ (apart_point_and_line (intersection_point T_1 T_0) T_0)) => (distinct_points (intersection_point T_1 T_0) (intersection_point T_1 T_0)))) (apart_point_and_line (intersection_point T_1 T_0) T_0) (All X, (-. (distinct_lines X X))) (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 26 28 29 % 0.38/0.56 31. (All Z, ((convergent_lines T_1 T_0) => (((apart_point_and_line Z T_1) \/ (apart_point_and_line Z T_0)) => (distinct_points Z (intersection_point T_1 T_0))))) (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 X, (-. (distinct_lines X X))) (apart_point_and_line (intersection_point T_1 T_0) T_0) ### All 30 % 0.38/0.56 32. (apart_point_and_line (intersection_point T_1 T_0) T_1) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) ### Axiom % 0.38/0.56 33. (-. ((apart_point_and_line (intersection_point T_1 T_0) T_1) \/ (apart_point_and_line (intersection_point T_1 T_0) T_0))) (apart_point_and_line (intersection_point T_1 T_0) T_1) ### NotOr 32 % 0.38/0.56 34. ((convergent_lines T_1 T_0) => (((apart_point_and_line (intersection_point T_1 T_0) T_1) \/ (apart_point_and_line (intersection_point T_1 T_0) T_0)) => (distinct_points (intersection_point T_1 T_0) (intersection_point T_1 T_0)))) (apart_point_and_line (intersection_point T_1 T_0) T_1) (All X, (-. (distinct_lines X X))) (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 26 33 29 % 0.38/0.56 35. (All Z, ((convergent_lines T_1 T_0) => (((apart_point_and_line Z T_1) \/ (apart_point_and_line Z T_0)) => (distinct_points Z (intersection_point T_1 T_0))))) (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 X, (-. (distinct_lines X X))) (apart_point_and_line (intersection_point T_1 T_0) T_1) ### All 34 % 0.38/0.56 36. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) /\ (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_1 T_0) T_0) \/ (apart_point_and_line (intersection_point T_1 T_0) T_1))))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All Z, ((convergent_lines T_1 T_0) => (((apart_point_and_line Z T_1) \/ (apart_point_and_line Z T_0)) => (distinct_points Z (intersection_point T_1 T_0))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) ### DisjTree 2 4 12 19 31 35 % 0.38/0.56 37. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) /\ (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_1 T_0) T_0) \/ (apart_point_and_line (intersection_point T_1 T_0) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All Z, ((convergent_lines T_1 T_0) => (((apart_point_and_line Z T_1) \/ (apart_point_and_line Z T_0)) => (distinct_points Z (intersection_point T_1 T_0))))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) ### All 36 % 0.38/0.56 38. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) /\ (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_1 T_0) U) \/ (apart_point_and_line (intersection_point T_1 T_0) V))))))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All Z, ((convergent_lines T_1 T_0) => (((apart_point_and_line Z T_1) \/ (apart_point_and_line Z T_0)) => (distinct_points Z (intersection_point T_1 T_0))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) ### All 37 % 0.38/0.56 39. (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_1 T_0)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All Z, ((convergent_lines T_1 T_0) => (((apart_point_and_line Z T_1) \/ (apart_point_and_line Z T_0)) => (distinct_points Z (intersection_point T_1 T_0))))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) ### All 38 % 0.38/0.56 40. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All Z, ((convergent_lines T_1 T_0) => (((apart_point_and_line Z T_1) \/ (apart_point_and_line Z T_0)) => (distinct_points Z (intersection_point T_1 T_0))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) ### All 39 % 0.38/0.56 41. (All Y, (All Z, ((convergent_lines T_1 Y) => (((apart_point_and_line Z T_1) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_1 Y)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### All 40 % 0.38/0.56 42. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) ### All 41 % 0.38/0.56 43. (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (apart_point_and_line (intersection_point T_1 T_0) T_2) ### Axiom % 0.38/0.56 44. ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_1 T_0)) \/ (apart_point_and_line (intersection_point T_1 T_0) T_2))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### DisjTree 1 42 43 % 0.38/0.56 45. (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, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) ### All 44 % 0.38/0.56 46. (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))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### All 45 % 0.38/0.56 47. (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_0 T_1) T_2) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (-. (distinct_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) ### All 46 % 0.38/0.56 48. (All Y, (All Z, ((convergent_lines T_0 Y) => ((convergent_lines T_0 Z) \/ (convergent_lines Y Z))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (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_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 47 % 0.38/0.56 49. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_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)))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) ### All 48 % 0.38/0.56 50. (-. (((convergent_lines T_0 T_1) /\ ((convergent_lines T_2 T_1) /\ (convergent_lines T_0 T_2))) => ((apart_point_and_line (intersection_point T_0 T_1) T_2) => (apart_point_and_line (intersection_point T_1 T_0) T_2)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### ConjTree 49 % 0.38/0.56 51. (-. (All Z, (((convergent_lines T_0 T_1) /\ ((convergent_lines Z T_1) /\ (convergent_lines T_0 Z))) => ((apart_point_and_line (intersection_point T_0 T_1) Z) => (apart_point_and_line (intersection_point T_1 T_0) Z))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_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, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) ### NotAllEx 50 % 0.38/0.56 52. (-. (All Y, (All Z, (((convergent_lines T_0 Y) /\ ((convergent_lines Z Y) /\ (convergent_lines T_0 Z))) => ((apart_point_and_line (intersection_point T_0 Y) Z) => (apart_point_and_line (intersection_point Y T_0) Z)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 51 % 0.38/0.56 53. (-. (All X, (All Y, (All Z, (((convergent_lines X Y) /\ ((convergent_lines Z Y) /\ (convergent_lines X Z))) => ((apart_point_and_line (intersection_point X Y) Z) => (apart_point_and_line (intersection_point Y X) Z))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_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, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) ### NotAllEx 52 % 0.38/0.56 % SZS output end Proof % 0.38/0.56 (* END-PROOF *) %------------------------------------------------------------------------------