%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO198+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 : n023.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 06:30:58 EDT 2022 % Result : Theorem 0.91s 1.10s % Output : Proof 0.91s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.10/0.12 % Problem : GEO198+2 : TPTP v8.1.0. Released v3.3.0. % 0.10/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.13/0.33 % Computer : n023.cluster.edu % 0.13/0.33 % Model : x86_64 x86_64 % 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.13/0.33 % Memory : 8042.1875MB % 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64 % 0.13/0.34 % CPULimit : 300 % 0.13/0.34 % WCLimit : 600 % 0.13/0.34 % DateTime : Sat Jun 18 07:32:19 EDT 2022 % 0.13/0.34 % CPUTime : % 0.91/1.10 % SZS status Theorem % 0.91/1.10 (* PROOF-FOUND *) % 0.91/1.10 (* BEGIN-PROOF *) % 0.91/1.10 % SZS output start Proof % 0.91/1.10 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 % 0.91/1.10 2. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2))) ### Axiom % 0.91/1.10 3. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 0.91/1.10 4. (-. (distinct_lines T_0 T_1)) (convergent_lines T_0 T_1) ### Extension/test/ceq3ctrp 3 % 0.91/1.10 5. (-. (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 % 0.91/1.10 6. (-. (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 % 0.91/1.10 7. (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) (apart_point_and_line (intersection_point T_0 T_2) T_0) ### Axiom % 0.91/1.10 8. (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (apart_point_and_line (intersection_point T_0 T_2) T_1) ### Axiom % 0.91/1.10 9. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (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_0 T_2) T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) T_1))))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_2) 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)) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### DisjTree 2 4 5 6 7 8 % 0.91/1.10 10. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (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_0 T_2) T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (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)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) ### All 9 % 0.91/1.10 11. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) U) \/ (apart_point_and_line (intersection_point T_0 T_2) V))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_2) 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)) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### All 10 % 0.91/1.10 12. (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_0 T_2)) (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)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) ### All 11 % 0.91/1.10 13. (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_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_2) 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)) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### All 12 % 0.91/1.10 14. (-. (apart_point_and_line (intersection_point T_0 T_2) T_2)) (apart_point_and_line (intersection_point T_0 T_2) T_2) ### Axiom % 0.91/1.10 15. ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_2)) (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)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) 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_2) ### DisjTree 1 13 14 % 0.91/1.10 16. (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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_2) 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)) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_2) T_2)) ### All 15 % 0.91/1.10 17. (-. (distinct_lines T_0 T_0)) (distinct_lines T_0 T_0) ### Axiom % 0.91/1.10 18. (convergent_lines T_0 T_2) (-. (convergent_lines T_0 T_2)) ### Axiom % 0.91/1.10 19. (apart_point_and_line (intersection_point T_0 T_2) T_0) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) ### Axiom % 0.91/1.10 20. (-. ((apart_point_and_line (intersection_point T_0 T_2) T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2))) (apart_point_and_line (intersection_point T_0 T_2) T_0) ### NotOr 19 % 0.91/1.10 21. (distinct_points (intersection_point T_0 T_2) (intersection_point T_0 T_2)) ### Extension/test/apart1 % 0.91/1.10 22. ((convergent_lines T_0 T_2) => (((apart_point_and_line (intersection_point T_0 T_2) T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2)) => (distinct_points (intersection_point T_0 T_2) (intersection_point T_0 T_2)))) (apart_point_and_line (intersection_point T_0 T_2) T_0) (convergent_lines T_0 T_2) ### DisjTree 18 20 21 % 0.91/1.10 23. (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (convergent_lines T_0 T_2) (apart_point_and_line (intersection_point T_0 T_2) T_0) ### All 22 % 0.91/1.10 24. ((apart_point_and_line (intersection_point T_0 T_2) T_0) => ((distinct_lines T_0 T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) T_0))) (convergent_lines T_0 T_2) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (-. (distinct_lines T_0 T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) T_2)) (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)) (-. (apart_point_and_line (intersection_point T_0 T_2) 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_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 16 17 23 % 0.91/1.11 25. (All Z, ((apart_point_and_line (intersection_point T_0 T_2) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z)))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) 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_0)) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_2) T_2)) (-. (distinct_lines T_0 T_0)) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (convergent_lines T_0 T_2) ### All 24 % 0.91/1.11 26. (-. (distinct_lines T_2 T_2)) (distinct_lines T_2 T_2) ### Axiom % 0.91/1.11 27. (convergent_lines T_0 T_2) (-. (convergent_lines T_0 T_2)) ### Axiom % 0.91/1.11 28. (apart_point_and_line (intersection_point T_0 T_2) T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_2)) ### Axiom % 0.91/1.11 29. (-. ((apart_point_and_line (intersection_point T_0 T_2) T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2))) (apart_point_and_line (intersection_point T_0 T_2) T_2) ### NotOr 28 % 0.91/1.11 30. ((convergent_lines T_0 T_2) => (((apart_point_and_line (intersection_point T_0 T_2) T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2)) => (distinct_points (intersection_point T_0 T_2) (intersection_point T_0 T_2)))) (apart_point_and_line (intersection_point T_0 T_2) T_2) (convergent_lines T_0 T_2) ### DisjTree 27 29 21 % 0.91/1.11 31. (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (convergent_lines T_0 T_2) (apart_point_and_line (intersection_point T_0 T_2) T_2) ### All 30 % 0.91/1.11 32. ((apart_point_and_line (intersection_point T_0 T_2) T_2) => ((distinct_lines T_2 T_2) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_2) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (-. (distinct_lines T_0 T_0)) (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)) (-. (apart_point_and_line (intersection_point T_0 T_2) 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_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 Z, ((apart_point_and_line (intersection_point T_0 T_2) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z)))) ### DisjTree 25 26 31 % 0.91/1.11 33. (All Z, ((apart_point_and_line (intersection_point T_0 T_2) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z)))) (All Z, ((apart_point_and_line (intersection_point T_0 T_2) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z)))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) 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_0)) (convergent_lines T_0 T_1) (-. (distinct_lines T_0 T_0)) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 T_2)) ### All 32 % 0.91/1.11 34. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_2) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_2) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (-. (distinct_lines T_0 T_0)) (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)) (-. (apart_point_and_line (intersection_point T_0 T_2) 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_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 Z, ((apart_point_and_line (intersection_point T_0 T_2) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z)))) ### All 33 % 0.91/1.11 35. (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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) 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_0)) (convergent_lines T_0 T_1) (-. (distinct_lines T_0 T_0)) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 T_2)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_2) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z))))) ### All 34 % 0.91/1.11 36. (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_2 T_2)) (convergent_lines T_0 T_2) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (-. (distinct_lines T_0 T_0)) (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)) (-. (apart_point_and_line (intersection_point T_0 T_2) 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_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 35 % 0.91/1.11 37. (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)))))) (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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) 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_0)) (convergent_lines T_0 T_1) (-. (distinct_lines T_0 T_0)) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 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 36 % 0.91/1.11 38. (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 Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_2) (-. (distinct_lines T_0 T_0)) (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)) (-. (apart_point_and_line (intersection_point T_0 T_2) 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_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 37 % 0.91/1.11 39. (-. (distinct_lines T_0 T_0)) (distinct_lines T_0 T_0) ### Axiom % 0.91/1.11 40. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 0.91/1.11 41. (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 % 0.91/1.11 42. (-. ((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 41 % 0.91/1.11 43. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)) ### Extension/test/apart1 % 0.91/1.11 44. ((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 40 42 43 % 0.91/1.11 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_0 T_1) T_0) ### All 44 % 0.91/1.11 46. (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 45 % 0.91/1.11 47. (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 46 % 0.91/1.11 48. ((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))) (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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_0 T_1) (-. (distinct_lines T_0 T_0)) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 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, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) ### DisjTree 38 39 47 % 0.91/1.11 49. (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, (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 Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_2) (-. (distinct_lines T_0 T_0)) (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_2) 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_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 48 % 0.91/1.11 50. (All X, (-. (distinct_lines 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)))) (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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_0 T_1) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 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, ((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, ((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 49 % 0.91/1.11 51. (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, ((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 Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_2) (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_2) 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_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_lines X X))) ### All 50 % 0.91/1.11 52. (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_1) ### Axiom % 0.91/1.11 53. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom % 0.91/1.11 54. (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 % 0.91/1.12 55. (-. ((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 54 % 0.91/1.12 56. ((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 53 55 43 % 0.91/1.12 57. (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 56 % 0.91/1.12 58. (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 57 % 0.91/1.12 59. (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 58 % 0.91/1.12 60. ((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))) (-. (distinct_lines T_1 T_1)) (All X, (-. (distinct_lines 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)))) (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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (convergent_lines T_0 T_1) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 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, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (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))))) ### DisjTree 51 52 59 % 0.91/1.12 61. (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 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, ((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 Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_2) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_2) 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_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_lines X X))) (-. (distinct_lines T_1 T_1)) ### All 60 % 0.91/1.12 62. (-. (distinct_lines T_1 T_1)) (All X, (-. (distinct_lines 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)))) (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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (convergent_lines T_0 T_1) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 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, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (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 61 % 0.91/1.12 63. (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 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, ((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 Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_2) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_2) 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_2) (All X, (-. (distinct_lines X X))) (-. (distinct_lines T_1 T_1)) ### All 62 % 0.91/1.12 64. (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, (-. (distinct_lines X X))) (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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (convergent_lines T_0 T_1) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 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, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (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 63 % 0.91/1.12 65. (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 Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_2) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_2) 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_2) (All X, (-. (distinct_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 64 % 0.91/1.12 66. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (-. (distinct_lines X X))) (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))))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) (convergent_lines T_0 T_1) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 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, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) ### All 65 % 0.91/1.12 67. (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 Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_2) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_2) 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_2) (All X, (-. (distinct_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 66 % 0.91/1.12 68. (-. (((convergent_lines T_0 T_2) /\ ((convergent_lines T_1 T_2) /\ ((convergent_lines T_0 T_1) /\ (-. (apart_point_and_line (intersection_point T_0 T_2) T_1))))) => (-. (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 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_lines Y Z) \/ (apart_point_and_line X 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))))))) ### ConjTree 67 % 0.91/1.12 69. (-. (All Z, (((convergent_lines T_0 T_2) /\ ((convergent_lines Z T_2) /\ ((convergent_lines T_0 Z) /\ (-. (apart_point_and_line (intersection_point T_0 T_2) Z))))) => (-. (apart_point_and_line (intersection_point T_0 Z) 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, (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, (-. (distinct_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)))))) ### NotAllEx 68 % 0.91/1.12 70. (-. (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 T_0 Z) 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, (-. (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_lines Y Z) \/ (apart_point_and_line X 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))))))) ### NotAllEx 69 % 0.91/1.12 71. (-. (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 X Z) Y))))))) (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 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, (-. (distinct_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)))))) ### NotAllEx 70 % 0.91/1.12 % SZS output end Proof % 0.91/1.12 (* END-PROOF *) %------------------------------------------------------------------------------