%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : GEO187+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 : n022.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 06:30:49 EDT 2022 % Result : Theorem 5.03s 5.22s % Output : Proof 5.26s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.12 % Problem : GEO187+3 : TPTP v8.1.0. Released v4.0.0. % 0.12/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.13/0.33 % Computer : n022.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 % 0.13/0.33 % CPULimit : 300 % 0.13/0.33 % WCLimit : 600 % 0.13/0.33 % DateTime : Sat Jun 18 05:11:32 EDT 2022 % 0.13/0.33 % CPUTime : % 5.03/5.22 % SZS status Theorem % 5.03/5.22 (* PROOF-FOUND *) % 5.03/5.22 (* BEGIN-PROOF *) % 5.03/5.22 % SZS output start Proof % 5.03/5.22 1. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 5.03/5.22 2. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 5.03/5.22 3. (distinct_points T_2 T_3) (-. (distinct_points T_2 T_3)) ### Axiom % 5.03/5.22 4. (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 % 5.03/5.22 5. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 5.03/5.22 6. (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_3)) (-. (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_3))) ### Axiom % 5.03/5.22 7. (-. (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 % 5.03/5.22 8. (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_0 (line_connecting T_2 T_3)) ### Axiom % 5.03/5.22 9. (-. (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 % 5.03/5.22 10. (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (apart_point_and_line T_1 (line_connecting T_2 T_3)) ### Axiom % 5.03/5.22 11. (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_3))) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 (line_connecting T_2 T_3)) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 (line_connecting T_2 T_3)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_3)) (distinct_points T_0 T_1) ### DisjTree 5 6 7 8 9 10 % 5.03/5.22 12. (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_3)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) ### All 11 % 5.03/5.22 13. (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_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_3)) (distinct_points T_0 T_1) ### All 12 % 5.03/5.22 14. (-. (apart_point_and_line T_2 (line_connecting T_2 T_3))) (apart_point_and_line T_2 (line_connecting T_2 T_3)) ### Axiom % 5.03/5.22 15. ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_3)) \/ (apart_point_and_line T_2 (line_connecting T_2 T_3)))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_3))) (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_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_0 T_1)) ### DisjTree 4 13 14 % 5.03/5.22 16. (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (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_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_2 (line_connecting T_2 T_3))) ### All 15 % 5.03/5.22 17. ((distinct_points T_2 T_3) => (-. (apart_point_and_line T_2 (line_connecting T_2 T_3)))) (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_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (distinct_points T_2 T_3) ### Imply 3 16 % 5.03/5.22 18. (All Y, ((distinct_points T_2 Y) => (-. (apart_point_and_line T_2 (line_connecting T_2 Y))))) (distinct_points T_2 T_3) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (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_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) ### All 17 % 5.03/5.22 19. (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (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_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (distinct_points T_2 T_3) ### All 18 % 5.03/5.22 20. (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_2 T_3) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (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_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) ### All 19 % 5.03/5.22 21. (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 X (line_connecting X Y)))))) (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_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_0 T_1)) (distinct_points T_2 T_3) ### All 20 % 5.17/5.40 22. ((distinct_points T_0 T_1) => (-. (apart_point_and_line T_1 (line_connecting T_0 T_1)))) (distinct_points T_2 T_3) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (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_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (distinct_points T_0 T_1) ### Imply 2 21 % 5.17/5.40 23. (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line Y (line_connecting T_0 Y))))) (distinct_points 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 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))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_0 T_1)) (distinct_points T_2 T_3) ### All 22 % 5.17/5.40 24. ((distinct_points T_0 T_1) => (-. (apart_point_and_line T_0 (line_connecting T_0 T_1)))) (distinct_points T_2 T_3) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (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_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (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 Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line Y (line_connecting T_0 Y))))) (distinct_points T_0 T_1) ### Imply 1 23 % 5.17/5.40 25. (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_1) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line Y (line_connecting T_0 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, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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_0 T_1)) (distinct_points T_2 T_3) ### All 24 % 5.17/5.40 26. (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (distinct_points T_2 T_3) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (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_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (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)))))) (distinct_points T_0 T_1) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) ### All 25 % 5.17/5.40 27. (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 Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points 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 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_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) ### All 26 % 5.17/5.40 28. (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 Y (line_connecting X Y)))))) (distinct_points T_2 T_3) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (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)))))) (distinct_points T_0 T_1) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) ### All 27 % 5.17/5.40 29. (distinct_points 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 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_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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))))))))) ### All 28 % 5.17/5.40 30. (-. (-. (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 X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (distinct_points T_2 T_3) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (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)))))) (distinct_points T_0 T_1) ### NotNot 29 % 5.17/5.40 31. (-. (incident_point_and_line T_2 (line_connecting T_0 T_1))) (distinct_points 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 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_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (distinct_points T_2 T_3) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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))))))))) ### Definition-Pseudo(incident_point_and_line) 30 % 5.26/5.44 32. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 5.26/5.44 33. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom % 5.26/5.44 34. (distinct_points T_2 T_3) (-. (distinct_points T_2 T_3)) ### Axiom % 5.26/5.44 35. (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_3 (line_connecting T_0 T_1))) ### Axiom % 5.26/5.44 36. (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (apart_point_and_line T_3 (line_connecting T_2 T_3)) ### Axiom % 5.26/5.44 37. ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_3)) \/ (apart_point_and_line T_3 (line_connecting T_2 T_3)))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) ### DisjTree 35 12 36 % 5.26/5.44 38. (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (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_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) ### All 37 % 5.26/5.44 39. (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_3 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) ### All 38 % 5.26/5.44 40. ((distinct_points T_2 T_3) => (-. (apart_point_and_line T_3 (line_connecting T_2 T_3)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))))) (distinct_points T_2 T_3) ### Imply 34 39 % 5.26/5.44 41. (All Y, ((distinct_points T_2 Y) => (-. (apart_point_and_line Y (line_connecting T_2 Y))))) (distinct_points T_2 T_3) (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_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) ### All 40 % 5.26/5.44 42. (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))))) (distinct_points T_2 T_3) ### All 41 % 5.26/5.44 43. (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (distinct_points T_2 T_3) (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_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) ### All 42 % 5.26/5.44 44. (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)))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))))) (distinct_points T_2 T_3) ### All 43 % 5.26/5.44 45. ((distinct_points T_0 T_1) => (-. (apart_point_and_line T_1 (line_connecting T_0 T_1)))) (distinct_points T_2 T_3) (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_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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)))))) (distinct_points T_0 T_1) ### Imply 33 44 % 5.26/5.44 46. (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line Y (line_connecting T_0 Y))))) (distinct_points 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 X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (apart_point_and_line T_3 (line_connecting 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_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))))) (distinct_points T_2 T_3) ### All 45 % 5.26/5.45 47. ((distinct_points T_0 T_1) => (-. (apart_point_and_line T_0 (line_connecting T_0 T_1)))) (distinct_points T_2 T_3) (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_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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 Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line Y (line_connecting T_0 Y))))) (distinct_points T_0 T_1) ### Imply 32 46 % 5.26/5.45 48. (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_1) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line Y (line_connecting T_0 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, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))))) (distinct_points T_2 T_3) ### All 47 % 5.26/5.45 49. (distinct_points T_2 T_3) (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_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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)))))) (distinct_points T_0 T_1) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) ### All 48 % 5.26/5.45 50. (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 Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points 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 X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (distinct_points T_2 T_3) ### All 49 % 5.26/5.45 51. (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_2 T_3) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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)))))) (distinct_points T_0 T_1) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) ### All 50 % 5.26/5.45 52. (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_points 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 X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (distinct_points T_2 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))))))))) ### All 51 % 5.26/5.45 53. (-. (-. (apart_point_and_line T_3 (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))))))))) (distinct_points T_2 T_3) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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)))))) (distinct_points T_0 T_1) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) ### NotNot 52 % 5.26/5.45 54. (-. (incident_point_and_line T_3 (line_connecting T_0 T_1))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_points 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 X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (distinct_points T_2 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))))))))) ### Definition-Pseudo(incident_point_and_line) 53 % 5.26/5.45 55. (-. ((incident_point_and_line T_2 (line_connecting T_0 T_1)) /\ (incident_point_and_line T_3 (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 Y (line_connecting X Y)))))) (distinct_points T_2 T_3) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (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)))))) (distinct_points T_0 T_1) ### NotAnd 31 54 % 5.26/5.45 56. (incident_point_and_line T_1 (line_connecting T_2 T_3)) (distinct_points 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 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_2 T_3))) (distinct_points T_2 T_3) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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))))))))) (-. ((incident_point_and_line T_2 (line_connecting T_0 T_1)) /\ (incident_point_and_line T_3 (line_connecting T_0 T_1)))) ### Definition-Pseudo(incident_point_and_line) 55 % 5.26/5.47 57. (incident_point_and_line T_0 (line_connecting T_2 T_3)) (-. ((incident_point_and_line T_2 (line_connecting T_0 T_1)) /\ (incident_point_and_line T_3 (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 Y (line_connecting X Y)))))) (distinct_points T_2 T_3) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (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)))))) (distinct_points T_0 T_1) (incident_point_and_line T_1 (line_connecting T_2 T_3)) ### Definition-Pseudo(incident_point_and_line) 56 % 5.26/5.47 58. (-. (((distinct_points T_0 T_1) /\ ((distinct_points T_2 T_3) /\ ((incident_point_and_line T_0 (line_connecting T_2 T_3)) /\ (incident_point_and_line T_1 (line_connecting T_2 T_3))))) => ((incident_point_and_line T_2 (line_connecting T_0 T_1)) /\ (incident_point_and_line T_3 (line_connecting 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 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 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 57 % 5.26/5.47 59. (-. (All V, (((distinct_points T_0 T_1) /\ ((distinct_points T_2 V) /\ ((incident_point_and_line T_0 (line_connecting T_2 V)) /\ (incident_point_and_line T_1 (line_connecting T_2 V))))) => ((incident_point_and_line T_2 (line_connecting T_0 T_1)) /\ (incident_point_and_line V (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 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 Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### NotAllEx 58 % 5.26/5.47 60. (-. (All U, (All V, (((distinct_points T_0 T_1) /\ ((distinct_points U V) /\ ((incident_point_and_line T_0 (line_connecting U V)) /\ (incident_point_and_line T_1 (line_connecting U V))))) => ((incident_point_and_line U (line_connecting T_0 T_1)) /\ (incident_point_and_line V (line_connecting 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 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 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 59 % 5.26/5.47 61. (-. (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ ((distinct_points U V) /\ ((incident_point_and_line T_0 (line_connecting U V)) /\ (incident_point_and_line Y (line_connecting U V))))) => ((incident_point_and_line U (line_connecting T_0 Y)) /\ (incident_point_and_line V (line_connecting T_0 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, ((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 Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### NotAllEx 60 % 5.26/5.47 62. (-. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ ((distinct_points U V) /\ ((incident_point_and_line X (line_connecting U V)) /\ (incident_point_and_line Y (line_connecting U V))))) => ((incident_point_and_line U (line_connecting X Y)) /\ (incident_point_and_line V (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, (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 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 61 % 5.26/5.47 % SZS output end Proof % 5.26/5.47 (* END-PROOF *) %------------------------------------------------------------------------------