%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO191+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 : n029.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 06:30:52 EDT 2022 % Result : Theorem 0.66s 0.85s % Output : Proof 0.69s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.13/0.14 % Problem : GEO191+2 : TPTP v8.1.0. Released v3.3.0. % 0.13/0.14 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.14/0.36 % Computer : n029.cluster.edu % 0.14/0.36 % Model : x86_64 x86_64 % 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.14/0.36 % Memory : 8042.1875MB % 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64 % 0.14/0.36 % CPULimit : 300 % 0.14/0.36 % WCLimit : 600 % 0.14/0.36 % DateTime : Sat Jun 18 01:46:28 EDT 2022 % 0.14/0.36 % CPUTime : % 0.66/0.85 % SZS status Theorem % 0.66/0.85 (* PROOF-FOUND *) % 0.66/0.85 (* BEGIN-PROOF *) % 0.66/0.85 % SZS output start Proof % 0.66/0.85 1. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 0.66/0.85 2. (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 % 0.66/0.85 3. (-. ((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_0) ### NotOr 2 % 0.66/0.85 4. (distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1)) (-. (distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1))) ### Axiom % 0.66/0.85 5. (convergent_lines T_2 T_3) (-. (convergent_lines T_2 T_3)) ### Axiom % 0.66/0.85 6. (-. (distinct_lines T_2 T_3)) (convergent_lines T_2 T_3) ### Extension/test/ceq3ctrp 5 % 0.66/0.85 7. (convergent_lines T_2 T_3) (-. (convergent_lines T_2 T_3)) ### Axiom % 0.66/0.85 8. (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 % 0.66/0.85 9. (-. ((apart_point_and_line (intersection_point T_2 T_3) 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) ### NotOr 8 % 0.66/0.85 10. (distinct_points (intersection_point T_2 T_3) (intersection_point T_2 T_3)) ### Extension/test/apart1 % 0.66/0.85 11. ((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_3)) => (distinct_points (intersection_point T_2 T_3) (intersection_point T_2 T_3)))) (apart_point_and_line (intersection_point T_2 T_3) T_2) (convergent_lines T_2 T_3) ### DisjTree 7 9 10 % 0.66/0.85 12. (All Z, ((convergent_lines T_2 T_3) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_3)) => (distinct_points Z (intersection_point T_2 T_3))))) (convergent_lines T_2 T_3) (apart_point_and_line (intersection_point T_2 T_3) T_2) ### All 11 % 0.66/0.85 13. (All Y, (All Z, ((convergent_lines T_2 Y) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_2 Y)))))) (apart_point_and_line (intersection_point T_2 T_3) T_2) (convergent_lines T_2 T_3) ### All 12 % 0.66/0.85 14. (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_2 T_3) (apart_point_and_line (intersection_point T_2 T_3) T_2) ### All 13 % 0.66/0.85 15. (-. (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 % 0.66/0.85 16. (-. (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 % 0.66/0.85 17. (-. (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 % 0.66/0.85 18. (((distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1)) /\ (distinct_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_3) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_2) \/ (apart_point_and_line (intersection_point T_0 T_1) T_3))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (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_2 T_3) (distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1)) ### DisjTree 4 6 14 15 16 17 % 0.66/0.85 19. (All V, (((distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1)) /\ (distinct_lines T_2 V)) => ((apart_point_and_line (intersection_point T_2 T_3) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_3) V) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_2) \/ (apart_point_and_line (intersection_point T_0 T_1) V)))))) (distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1)) (convergent_lines T_2 T_3) (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))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (-. (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 18 % 0.66/0.86 20. (All U, (All V, (((distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1)) /\ (distinct_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 T_0 T_1) U) \/ (apart_point_and_line (intersection_point T_0 T_1) V))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (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_2 T_3) (distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1)) ### All 19 % 0.66/0.86 21. (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_3) Y) /\ (distinct_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 Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1)) (convergent_lines T_2 T_3) (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))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (-. (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 20 % 0.66/0.86 22. (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_3)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (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_2 T_3) (distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1)) ### All 21 % 0.66/0.86 23. ((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)) => (distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1)))) (convergent_lines T_2 T_3) (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))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (-. (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 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_0 T_1) ### DisjTree 1 3 22 % 0.66/0.86 24. (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_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_0 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (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_2 T_3) ### All 23 % 0.66/0.86 25. (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)))))) (convergent_lines T_2 T_3) (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))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (-. (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 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_0 T_1) ### All 24 % 0.66/0.86 26. (convergent_lines T_0 T_1) (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_0 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (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_2 T_3) ### All 25 % 0.66/0.86 27. (-. (distinct_lines T_3 T_3)) (distinct_lines T_3 T_3) ### Axiom % 0.66/0.86 28. (convergent_lines T_2 T_3) (-. (convergent_lines T_2 T_3)) ### Axiom % 0.66/0.86 29. (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 % 0.66/0.86 30. (-. ((apart_point_and_line (intersection_point T_2 T_3) 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_3) ### NotOr 29 % 0.66/0.86 31. ((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_3)) => (distinct_points (intersection_point T_2 T_3) (intersection_point T_2 T_3)))) (apart_point_and_line (intersection_point T_2 T_3) T_3) (convergent_lines T_2 T_3) ### DisjTree 28 30 10 % 0.66/0.86 32. (All Z, ((convergent_lines T_2 T_3) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_3)) => (distinct_points Z (intersection_point T_2 T_3))))) (convergent_lines T_2 T_3) (apart_point_and_line (intersection_point T_2 T_3) T_3) ### All 31 % 0.66/0.86 33. (All Y, (All Z, ((convergent_lines T_2 Y) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_2 Y)))))) (apart_point_and_line (intersection_point T_2 T_3) T_3) (convergent_lines T_2 T_3) ### All 32 % 0.66/0.86 34. (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_2 T_3) (apart_point_and_line (intersection_point T_2 T_3) T_3) ### All 33 % 0.66/0.86 35. ((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))) (-. (distinct_lines T_3 T_3)) (convergent_lines T_2 T_3) (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))))))) (-. (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 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_0 T_1) ### DisjTree 26 27 34 % 0.66/0.86 36. (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)))) (convergent_lines T_0 T_1) (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_0 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_0 T_1) 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))))))) (convergent_lines T_2 T_3) (-. (distinct_lines T_3 T_3)) ### All 35 % 0.66/0.86 37. (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))))) (-. (distinct_lines T_3 T_3)) (convergent_lines T_2 T_3) (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))))))) (-. (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 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_0 T_1) ### All 36 % 0.66/0.86 38. (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) (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_0 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_0 T_1) 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))))))) (convergent_lines T_2 T_3) (-. (distinct_lines T_3 T_3)) ### All 37 % 0.66/0.86 39. (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_3) (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))))))) (-. (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 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_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 38 % 0.66/0.86 40. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 0.66/0.86 41. (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 % 0.66/0.86 42. (-. ((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) ### NotOr 41 % 0.66/0.86 43. (-. (distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1))) (distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1)) ### Axiom % 0.66/0.86 44. ((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)) => (distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1)))) (-. (distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1))) (apart_point_and_line (intersection_point T_2 T_3) T_1) (convergent_lines T_0 T_1) ### DisjTree 40 42 43 % 0.69/0.87 45. (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_2 T_3) T_1) (-. (distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1))) ### All 44 % 0.69/0.87 46. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)) ### Extension/test/apart1 % 0.69/0.87 47. ((distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1)) => ((distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1)) \/ (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)))) (convergent_lines T_2 T_3) (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))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (-. (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 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_0 T_1) (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))))) ### DisjTree 45 22 46 % 0.69/0.87 48. (All Z, ((distinct_points (intersection_point T_2 T_3) (intersection_point T_0 T_1)) => ((distinct_points (intersection_point T_2 T_3) Z) \/ (distinct_points (intersection_point T_0 T_1) Z)))) (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_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_0 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (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_2 T_3) ### All 47 % 0.69/0.87 49. (All Y, (All Z, ((distinct_points (intersection_point T_2 T_3) Y) => ((distinct_points (intersection_point T_2 T_3) Z) \/ (distinct_points Y Z))))) (convergent_lines T_2 T_3) (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))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (-. (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 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_0 T_1) (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))))) ### All 48 % 0.69/0.87 50. (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) => (((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_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_0 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (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_2 T_3) ### All 49 % 0.69/0.87 51. (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)))))) (convergent_lines T_2 T_3) (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))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (-. (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 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_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 50 % 0.69/0.87 52. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_0 T_1) (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_0 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (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_2 T_3) ### All 51 % 0.69/0.87 53. (-. (distinct_lines T_3 T_3)) (distinct_lines T_3 T_3) ### Axiom % 0.69/0.87 54. ((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))) (-. (distinct_lines T_3 T_3)) (convergent_lines T_2 T_3) (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))))))) (-. (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 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_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### DisjTree 52 53 34 % 0.69/0.87 55. (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 X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_0 T_1) (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_0 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_0 T_1) 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))))))) (convergent_lines T_2 T_3) (-. (distinct_lines T_3 T_3)) ### All 54 % 0.69/0.87 56. (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))))) (-. (distinct_lines T_3 T_3)) (convergent_lines T_2 T_3) (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))))))) (-. (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 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_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 55 % 0.69/0.87 57. (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, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (convergent_lines T_0 T_1) (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_0 T_1) T_3)) (-. (apart_point_and_line (intersection_point T_0 T_1) 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))))))) (convergent_lines T_2 T_3) (-. (distinct_lines T_3 T_3)) ### All 56 % 0.69/0.87 58. (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_3) (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))))))) (-. (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 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_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 57 % 0.69/0.87 59. ((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 X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (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 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_3)) (-. (apart_point_and_line (intersection_point T_0 T_1) 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))))))) (convergent_lines T_2 T_3) (All X, (-. (distinct_lines X X))) ### Or 39 58 % 0.69/0.87 60. (-. (((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, (-. (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))))))) (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_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### ConjTree 59 % 0.69/0.87 61. (-. (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, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (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 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, ((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))) ### NotAllEx 60 % 0.69/0.87 62. (-. (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, (-. (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))))))) (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_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotAllEx 61 % 0.69/0.88 63. (-. (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, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (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 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, ((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))) ### NotAllEx 62 % 0.69/0.88 64. (-. (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, (-. (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))))))) (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_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotAllEx 63 % 0.69/0.88 % SZS output end Proof % 0.69/0.88 (* END-PROOF *) %------------------------------------------------------------------------------