%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO190+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 : n004.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 06:30:51 EDT 2022 % Result : Theorem 5.07s 5.26s % Output : Proof 5.31s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.08/0.13 % Problem : GEO190+3 : TPTP v8.1.0. Released v4.0.0. % 0.08/0.14 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.13/0.35 % Computer : n004.cluster.edu % 0.13/0.35 % Model : x86_64 x86_64 % 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.13/0.35 % Memory : 8042.1875MB % 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64 % 0.13/0.35 % CPULimit : 300 % 0.13/0.35 % WCLimit : 600 % 0.13/0.35 % DateTime : Fri Jun 17 20:43:08 EDT 2022 % 0.13/0.35 % CPUTime : % 5.07/5.26 % SZS status Theorem % 5.07/5.26 (* PROOF-FOUND *) % 5.07/5.26 (* BEGIN-PROOF *) % 5.07/5.26 % SZS output start Proof % 5.07/5.26 1. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 5.07/5.26 2. (distinct_points T_0 T_0) (-. (distinct_points T_0 T_0)) ### Axiom % 5.07/5.26 3. (All X, (-. (distinct_points X X))) (distinct_points T_0 T_0) ### All 2 % 5.07/5.26 4. (-. (distinct_points T_1 T_0)) (distinct_points T_1 T_0) ### Axiom % 5.07/5.26 5. ((distinct_points T_0 T_1) => ((distinct_points T_0 T_0) \/ (distinct_points T_1 T_0))) (-. (distinct_points T_1 T_0)) (All X, (-. (distinct_points X X))) (distinct_points T_0 T_1) ### DisjTree 1 3 4 % 5.07/5.26 6. (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (All X, (-. (distinct_points X X))) (-. (distinct_points T_1 T_0)) ### All 5 % 5.07/5.26 7. (apart_point_and_line T_2 (line_connecting T_1 T_0)) (-. (apart_point_and_line T_2 (line_connecting T_1 T_0))) ### Axiom % 5.07/5.26 8. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 5.07/5.26 9. (distinct_lines (line_connecting T_1 T_0) (line_connecting T_0 T_1)) (-. (distinct_lines (line_connecting T_1 T_0) (line_connecting T_0 T_1))) ### Axiom % 5.07/5.26 10. (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) ### Axiom % 5.07/5.26 11. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 5.07/5.26 12. (apart_point_and_line T_0 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) ### Axiom % 5.07/5.26 13. ((distinct_points T_0 T_1) => (-. (apart_point_and_line T_0 (line_connecting T_0 T_1)))) (apart_point_and_line T_0 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### Imply 11 12 % 5.07/5.26 14. (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_1) (apart_point_and_line T_0 (line_connecting T_0 T_1)) ### All 13 % 5.07/5.26 15. (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (apart_point_and_line T_0 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### All 14 % 5.07/5.26 16. (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ### Axiom % 5.07/5.26 17. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 5.07/5.26 18. (apart_point_and_line T_1 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) ### Axiom % 5.07/5.26 19. ((distinct_points T_0 T_1) => (-. (apart_point_and_line T_1 (line_connecting T_0 T_1)))) (apart_point_and_line T_1 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### Imply 17 18 % 5.07/5.26 20. (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line Y (line_connecting T_0 Y))))) (distinct_points T_0 T_1) (apart_point_and_line T_1 (line_connecting T_0 T_1)) ### All 19 % 5.07/5.26 21. (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (apart_point_and_line T_1 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### All 20 % 5.07/5.26 22. (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_1 T_0) (line_connecting T_0 T_1))) => ((apart_point_and_line T_0 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_1 (line_connecting T_1 T_0)) \/ (apart_point_and_line T_1 (line_connecting T_0 T_1)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### DisjTree 8 9 10 15 16 21 % 5.07/5.26 23. (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_1 T_0) V)) => ((apart_point_and_line T_0 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_1 T_0)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) ### All 22 % 5.07/5.26 24. (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### All 23 % 5.07/5.26 25. (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) ### Axiom % 5.07/5.26 26. ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) (line_connecting T_0 T_1)) \/ (apart_point_and_line T_2 (line_connecting T_0 T_1)))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) ### DisjTree 7 24 25 % 5.07/5.26 27. (All Z, ((apart_point_and_line T_2 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) ### All 26 % 5.07/5.26 28. (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) ### All 27 % 5.07/5.26 29. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) ### All 28 % 5.31/5.51 30. ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_0 (line_connecting T_1 T_0)))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (distinct_points X X))) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) ### Imply 6 29 % 5.31/5.51 31. (All Y, ((distinct_points T_1 Y) => (-. (apart_point_and_line Y (line_connecting T_1 Y))))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) ### All 30 % 5.31/5.51 32. ((distinct_points T_1 T_0) => (-. (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, ((distinct_points T_1 Y) => (-. (apart_point_and_line Y (line_connecting T_1 Y))))) (All X, (-. (distinct_points X X))) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) ### Imply 6 31 % 5.31/5.51 33. (All Y, ((distinct_points T_1 Y) => (-. (apart_point_and_line T_1 (line_connecting T_1 Y))))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (All X, (-. (distinct_points X X))) (All Y, ((distinct_points T_1 Y) => (-. (apart_point_and_line Y (line_connecting T_1 Y))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) ### All 32 % 5.31/5.51 34. (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (distinct_points X X))) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (All Y, ((distinct_points T_1 Y) => (-. (apart_point_and_line T_1 (line_connecting T_1 Y))))) ### All 33 % 5.31/5.51 35. (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (All Y, ((distinct_points T_1 Y) => (-. (apart_point_and_line T_1 (line_connecting T_1 Y))))) (distinct_points T_0 T_1) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) ### All 34 % 5.31/5.51 36. (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (distinct_points X X))) (distinct_points T_0 T_1) (All Y, ((distinct_points T_1 Y) => (-. (apart_point_and_line T_1 (line_connecting T_1 Y))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) ### All 35 % 5.31/5.51 37. (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (distinct_points T_0 T_1) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) ### All 36 % 5.31/5.51 38. (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 T_2 (line_connecting T_0 T_1))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (distinct_points X X))) (distinct_points T_0 T_1) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) ### All 37 % 5.31/5.53 39. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_points T_0 T_1) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line T_2 (line_connecting T_1 T_0)) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### All 38 % 5.31/5.53 40. (-. (-. (apart_point_and_line T_2 (line_connecting T_1 T_0)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting 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, (-. (distinct_points X X))) (distinct_points T_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotNot 39 % 5.31/5.53 41. (-. (incident_point_and_line T_2 (line_connecting T_1 T_0))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_points T_0 T_1) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line T_2 (line_connecting 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))))))))) ### Definition-Pseudo(incident_point_and_line) 40 % 5.31/5.53 42. (incident_point_and_line T_2 (line_connecting T_0 T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting 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, (-. (distinct_points X X))) (distinct_points T_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (incident_point_and_line T_2 (line_connecting T_1 T_0))) ### Definition-Pseudo(incident_point_and_line) 41 % 5.31/5.54 43. (-. (((distinct_points T_0 T_1) /\ ((distinct_points T_0 T_2) /\ ((distinct_points T_1 T_2) /\ (incident_point_and_line T_2 (line_connecting T_0 T_1))))) => (incident_point_and_line T_2 (line_connecting T_1 T_0)))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting 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))))))))) ### ConjTree 42 % 5.31/5.54 44. (-. (All Z, (((distinct_points T_0 T_1) /\ ((distinct_points T_0 Z) /\ ((distinct_points T_1 Z) /\ (incident_point_and_line Z (line_connecting T_0 T_1))))) => (incident_point_and_line Z (line_connecting T_1 T_0))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting 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, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotAllEx 43 % 5.31/5.54 45. (-. (All Y, (All Z, (((distinct_points T_0 Y) /\ ((distinct_points T_0 Z) /\ ((distinct_points Y Z) /\ (incident_point_and_line Z (line_connecting T_0 Y))))) => (incident_point_and_line Z (line_connecting Y T_0)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting 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))))))))) ### NotAllEx 44 % 5.31/5.54 46. (-. (All X, (All Y, (All Z, (((distinct_points X Y) /\ ((distinct_points X Z) /\ ((distinct_points Y Z) /\ (incident_point_and_line Z (line_connecting X Y))))) => (incident_point_and_line Z (line_connecting Y X))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting 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, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotAllEx 45 % 5.31/5.54 % SZS output end Proof % 5.31/5.54 (* END-PROOF *) %------------------------------------------------------------------------------