%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO175+3 : TPTP v8.1.0. Released v4.0.0. % Transfm : none % Format : tptp:raw % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % Computer : n014.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 06:30:38 EDT 2022 % Result : Theorem 37.75s 37.97s % Output : Proof 37.84s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.11 % Problem : GEO175+3 : TPTP v8.1.0. Released v4.0.0. % 0.07/0.12 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.12/0.33 % Computer : n014.cluster.edu % 0.12/0.33 % Model : x86_64 x86_64 % 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.33 % Memory : 8042.1875MB % 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.33 % CPULimit : 300 % 0.12/0.33 % WCLimit : 600 % 0.12/0.33 % DateTime : Sat Jun 18 12:28:34 EDT 2022 % 0.12/0.33 % CPUTime : % 37.75/37.97 % SZS status Theorem % 37.75/37.97 (* PROOF-FOUND *) % 37.75/37.97 (* BEGIN-PROOF *) % 37.75/37.97 % SZS output start Proof % 37.75/37.97 1. (distinct_points T_0 (intersection_point T_1 T_2)) (-. (distinct_points T_0 (intersection_point T_1 T_2))) ### Axiom % 37.75/37.97 2. (-. (distinct_points T_0 T_0)) (distinct_points T_0 T_0) ### Axiom % 37.75/37.97 3. (convergent_lines T_1 T_2) (-. (convergent_lines T_1 T_2)) ### Axiom % 37.75/37.97 4. (distinct_points (intersection_point T_1 T_2) T_0) (-. (distinct_points (intersection_point T_1 T_2) T_0)) ### Axiom % 37.75/37.97 5. (distinct_lines T_2 T_1) (-. (distinct_lines T_2 T_1)) ### Axiom % 37.75/37.97 6. (convergent_lines T_1 T_2) (-. (convergent_lines T_1 T_2)) ### Axiom % 37.75/37.97 7. (apart_point_and_line (intersection_point T_1 T_2) T_2) (-. (apart_point_and_line (intersection_point T_1 T_2) T_2)) ### Axiom % 37.75/37.97 8. ((convergent_lines T_1 T_2) => (-. (apart_point_and_line (intersection_point T_1 T_2) T_2))) (apart_point_and_line (intersection_point T_1 T_2) T_2) (convergent_lines T_1 T_2) ### Imply 6 7 % 37.75/37.97 9. (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) Y)))) (convergent_lines T_1 T_2) (apart_point_and_line (intersection_point T_1 T_2) T_2) ### All 8 % 37.75/37.97 10. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_1 T_2) T_2) (convergent_lines T_1 T_2) ### All 9 % 37.75/37.97 11. (convergent_lines T_1 T_2) (-. (convergent_lines T_1 T_2)) ### Axiom % 37.75/37.97 12. (apart_point_and_line (intersection_point T_1 T_2) T_1) (-. (apart_point_and_line (intersection_point T_1 T_2) T_1)) ### Axiom % 37.75/37.97 13. ((convergent_lines T_1 T_2) => (-. (apart_point_and_line (intersection_point T_1 T_2) T_1))) (apart_point_and_line (intersection_point T_1 T_2) T_1) (convergent_lines T_1 T_2) ### Imply 11 12 % 37.75/37.97 14. (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) T_1)))) (convergent_lines T_1 T_2) (apart_point_and_line (intersection_point T_1 T_2) T_1) ### All 13 % 37.75/37.97 15. (-. (apart_point_and_line T_0 T_2)) (apart_point_and_line T_0 T_2) ### Axiom % 37.75/37.97 16. (-. (apart_point_and_line T_0 T_1)) (apart_point_and_line T_0 T_1) ### Axiom % 37.75/37.97 17. (((distinct_points (intersection_point T_1 T_2) T_0) /\ (distinct_lines T_2 T_1)) => ((apart_point_and_line (intersection_point T_1 T_2) T_2) \/ ((apart_point_and_line (intersection_point T_1 T_2) T_1) \/ ((apart_point_and_line T_0 T_2) \/ (apart_point_and_line T_0 T_1))))) (-. (apart_point_and_line T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) T_1)))) (convergent_lines T_1 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_2 T_1) (distinct_points (intersection_point T_1 T_2) T_0) ### DisjTree 4 5 10 14 15 16 % 37.75/37.97 18. (All V, (((distinct_points (intersection_point T_1 T_2) T_0) /\ (distinct_lines T_2 V)) => ((apart_point_and_line (intersection_point T_1 T_2) T_2) \/ ((apart_point_and_line (intersection_point T_1 T_2) V) \/ ((apart_point_and_line T_0 T_2) \/ (apart_point_and_line T_0 V)))))) (distinct_points (intersection_point T_1 T_2) T_0) (distinct_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_2) (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) T_1)))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_0 T_1)) ### All 17 % 37.75/37.97 19. (All U, (All V, (((distinct_points (intersection_point T_1 T_2) T_0) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_2) U) \/ ((apart_point_and_line (intersection_point T_1 T_2) V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (-. (apart_point_and_line T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) T_1)))) (convergent_lines T_1 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_2 T_1) (distinct_points (intersection_point T_1 T_2) T_0) ### All 18 % 37.75/37.97 20. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point T_1 T_2) T_0) (distinct_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_1 T_2) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_0 T_1)) (All U, (All V, (((distinct_points (intersection_point T_1 T_2) T_0) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_2) U) \/ ((apart_point_and_line (intersection_point T_1 T_2) V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) ### All 19 % 37.75/37.97 21. (convergent_lines T_1 T_1) (-. (convergent_lines T_1 T_1)) ### Axiom % 37.75/37.97 22. (All X, (-. (convergent_lines X X))) (convergent_lines T_1 T_1) ### All 21 % 37.75/37.97 23. ((convergent_lines T_1 T_2) => ((distinct_lines T_2 T_1) \/ (convergent_lines T_1 T_1))) (All X, (-. (convergent_lines X X))) (All U, (All V, (((distinct_points (intersection_point T_1 T_2) T_0) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_2) U) \/ ((apart_point_and_line (intersection_point T_1 T_2) V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (-. (apart_point_and_line T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point T_1 T_2) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_1 T_2) ### DisjTree 3 20 22 % 37.75/37.97 24. (All Z, ((convergent_lines T_1 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point T_1 T_2) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_0 T_1)) (All U, (All V, (((distinct_points (intersection_point T_1 T_2) T_0) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_2) U) \/ ((apart_point_and_line (intersection_point T_1 T_2) V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (All X, (-. (convergent_lines X X))) ### All 23 % 37.75/37.97 25. (All Y, (All U, (All V, (((distinct_points (intersection_point T_1 T_2) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_2) U) \/ ((apart_point_and_line (intersection_point T_1 T_2) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point T_1 T_2) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_1 T_2) (All Z, ((convergent_lines T_1 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_1 Z)))) ### All 24 % 37.75/37.97 26. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((convergent_lines T_1 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point T_1 T_2) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_0 T_1)) (All X, (-. (convergent_lines X X))) ### All 25 % 37.75/37.97 27. ((distinct_points T_0 (intersection_point T_1 T_2)) => ((distinct_points T_0 T_0) \/ (distinct_points (intersection_point T_1 T_2) T_0))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_1 T_2) (All Z, ((convergent_lines T_1 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_1 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))))))))) (-. (distinct_points T_0 T_0)) (distinct_points T_0 (intersection_point T_1 T_2)) ### DisjTree 1 2 26 % 37.84/38.01 28. (All Z, ((distinct_points T_0 (intersection_point T_1 T_2)) => ((distinct_points T_0 Z) \/ (distinct_points (intersection_point T_1 T_2) Z)))) (distinct_points T_0 (intersection_point T_1 T_2)) (-. (distinct_points T_0 T_0)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((convergent_lines T_1 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_0 T_1)) (All X, (-. (convergent_lines X X))) ### All 27 % 37.84/38.01 29. (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_1 T_2) (All Z, ((convergent_lines T_1 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_1 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))))))))) (-. (distinct_points T_0 T_0)) (distinct_points T_0 (intersection_point T_1 T_2)) ### All 28 % 37.84/38.01 30. (All X, (-. (distinct_points X X))) (distinct_points T_0 (intersection_point 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 Z, ((convergent_lines T_1 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_0 T_1)) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) ### All 29 % 37.84/38.01 31. (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines 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))))))))) (distinct_points T_0 (intersection_point T_1 T_2)) (All X, (-. (distinct_points X X))) ### All 30 % 37.84/38.01 32. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points X X))) (distinct_points T_0 (intersection_point 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))))))))) (convergent_lines T_1 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_0 T_1)) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) ### All 31 % 37.84/38.01 33. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (convergent_lines X X))) (-. (apart_point_and_line T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines 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))))))))) (distinct_points T_0 (intersection_point T_1 T_2)) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 32 % 37.84/38.01 34. (-. (((distinct_points T_0 T_3) /\ ((convergent_lines T_1 T_2) /\ (distinct_points T_0 (intersection_point T_1 T_2)))) => ((apart_point_and_line T_0 T_1) \/ (apart_point_and_line T_0 T_2)))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points X X))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### ConjTree 33 % 37.84/38.01 35. (-. (All V, (((distinct_points T_0 T_3) /\ ((convergent_lines T_1 V) /\ (distinct_points T_0 (intersection_point T_1 V)))) => ((apart_point_and_line T_0 T_1) \/ (apart_point_and_line T_0 V))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotAllEx 34 % 37.84/38.01 36. (-. (All U, (All V, (((distinct_points T_0 T_3) /\ ((convergent_lines U V) /\ (distinct_points T_0 (intersection_point U V)))) => ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points X X))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotAllEx 35 % 37.84/38.02 37. (-. (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ ((convergent_lines U V) /\ (distinct_points T_0 (intersection_point U V)))) => ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotAllEx 36 % 37.84/38.02 38. (-. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ ((convergent_lines U V) /\ (distinct_points X (intersection_point U V)))) => ((apart_point_and_line X U) \/ (apart_point_and_line X V)))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points X X))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotAllEx 37 % 37.84/38.02 % SZS output end Proof % 37.84/38.02 (* END-PROOF *) %------------------------------------------------------------------------------