TSTP Solution File: GEO205+3 by SuperZenon---0.0.1
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%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : GEO205+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 : n027.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 06:31:04 EDT 2022
% Result : Theorem 31.36s 31.57s
% Output : Proof 31.55s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : GEO205+3 : TPTP v8.1.0. Released v4.0.0.
% 0.07/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.12/0.34 % Computer : n027.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 600
% 0.12/0.34 % DateTime : Sat Jun 18 00:11:37 EDT 2022
% 0.12/0.34 % CPUTime :
% 31.36/31.57 % SZS status Theorem
% 31.36/31.57 (* PROOF-FOUND *)
% 31.36/31.57 (* BEGIN-PROOF *)
% 31.36/31.57 % SZS output start Proof
% 31.36/31.57 1. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 31.36/31.57 2. (-. (distinct_lines T_1 T_2)) (distinct_lines T_1 T_2) ### Axiom
% 31.36/31.57 3. (-. (convergent_lines T_0 T_2)) (convergent_lines T_0 T_2) ### Axiom
% 31.36/31.57 4. ((convergent_lines T_0 T_1) => ((distinct_lines T_1 T_2) \/ (convergent_lines T_0 T_2))) (-. (convergent_lines T_0 T_2)) (-. (distinct_lines T_1 T_2)) (convergent_lines T_0 T_1) ### DisjTree 1 2 3
% 31.36/31.57 5. (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (-. (distinct_lines T_1 T_2)) (-. (convergent_lines T_0 T_2)) ### All 4
% 31.36/31.57 6. (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (-. (convergent_lines T_0 T_2)) (-. (distinct_lines T_1 T_2)) (convergent_lines T_0 T_1) ### All 5
% 31.36/31.57 7. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (convergent_lines T_0 T_1) (-. (distinct_lines T_1 T_2)) (-. (convergent_lines T_0 T_2)) ### All 6
% 31.36/31.57 8. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 31.36/31.57 9. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 31.36/31.57 10. (convergent_lines T_0 T_0) (-. (convergent_lines T_0 T_0)) ### Axiom
% 31.36/31.57 11. (All X, (-. (convergent_lines X X))) (convergent_lines T_0 T_0) ### All 10
% 31.36/31.57 12. (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
% 31.36/31.57 13. (distinct_lines T_1 T_0) (-. (distinct_lines T_1 T_0)) ### Axiom
% 31.36/31.57 14. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 31.36/31.57 15. (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
% 31.36/31.57 16. ((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_1) T_1) (convergent_lines T_0 T_1) ### Imply 14 15
% 31.36/31.57 17. (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) Y)))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### All 16
% 31.36/31.57 18. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_1) T_1) (convergent_lines T_0 T_1) ### All 17
% 31.36/31.57 19. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 31.36/31.57 20. (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
% 31.36/31.57 21. ((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_0) (convergent_lines T_0 T_1) ### Imply 19 20
% 31.36/31.57 22. (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### All 21
% 31.36/31.57 23. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom
% 31.36/31.57 24. (distinct_lines T_0 T_1) (-. (distinct_lines T_0 T_1)) ### Axiom
% 31.36/31.57 25. (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
% 31.36/31.57 26. (-. (distinct_lines T_1 T_2)) (distinct_lines T_1 T_2) ### Axiom
% 31.36/31.57 27. (distinct_lines T_0 T_2) (-. (distinct_lines T_0 T_2)) ### Axiom
% 31.36/31.57 28. (convergent_lines T_0 T_2) (-. (convergent_lines T_0 T_2)) ### Axiom
% 31.36/31.57 29. (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
% 31.36/31.57 30. ((convergent_lines 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_2) ### Imply 28 29
% 31.36/31.57 31. (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) Y)))) (convergent_lines T_0 T_2) (apart_point_and_line (intersection_point T_0 T_2) T_2) ### All 30
% 31.36/31.57 32. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_2) T_2) (convergent_lines T_0 T_2) ### All 31
% 31.36/31.57 33. ((distinct_lines T_0 T_2) => (convergent_lines T_0 T_2)) (apart_point_and_line (intersection_point T_0 T_2) T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_0 T_2) ### Imply 27 32
% 31.36/31.57 34. (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (distinct_lines T_0 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_0 T_2) T_2) ### All 33
% 31.36/31.57 35. ((apart_point_and_line (intersection_point T_0 T_2) T_1) => ((distinct_lines T_1 T_2) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_0 T_2) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (-. (distinct_lines T_1 T_2)) (apart_point_and_line (intersection_point T_0 T_2) T_1) ### DisjTree 25 26 34
% 31.36/31.57 36. (All Z, ((apart_point_and_line (intersection_point T_0 T_2) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z)))) (apart_point_and_line (intersection_point T_0 T_2) T_1) (-. (distinct_lines T_1 T_2)) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (distinct_lines T_0 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) ### All 35
% 31.36/31.57 37. (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 X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_0 T_2) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (-. (distinct_lines T_1 T_2)) (apart_point_and_line (intersection_point T_0 T_2) T_1) ### All 36
% 31.36/31.57 38. (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 (intersection_point T_0 T_2) T_1) (-. (distinct_lines T_1 T_2)) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (distinct_lines T_0 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) ### All 37
% 31.36/31.57 39. (-. (distinct_lines T_1 T_2)) (distinct_lines T_1 T_2) ### Axiom
% 31.36/31.57 40. ((distinct_lines T_0 T_1) => ((distinct_lines T_0 T_2) \/ (distinct_lines T_1 T_2))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (-. (distinct_lines T_1 T_2)) (apart_point_and_line (intersection_point T_0 T_2) T_1) (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_0 T_1) ### DisjTree 24 38 39
% 31.36/31.57 41. (All Z, ((distinct_lines T_0 T_1) => ((distinct_lines T_0 Z) \/ (distinct_lines T_1 Z)))) (distinct_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_2) T_1) (-. (distinct_lines T_1 T_2)) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) ### All 40
% 31.36/31.57 42. (All Y, (All Z, ((distinct_lines T_0 Y) => ((distinct_lines T_0 Z) \/ (distinct_lines Y Z))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (-. (distinct_lines T_1 T_2)) (apart_point_and_line (intersection_point T_0 T_2) T_1) (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_0 T_1) ### All 41
% 31.36/31.57 43. (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (distinct_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_2) T_1) (-. (distinct_lines T_1 T_2)) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) ### All 42
% 31.55/31.76 44. (-. (convergent_lines T_1 T_1)) (convergent_lines T_1 T_1) ### Axiom
% 31.55/31.76 45. ((convergent_lines T_1 T_0) => ((distinct_lines T_0 T_1) \/ (convergent_lines T_1 T_1))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (-. (distinct_lines T_1 T_2)) (apart_point_and_line (intersection_point T_0 T_2) 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, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_1 T_0) ### DisjTree 23 43 44
% 31.55/31.76 46. (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (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 (intersection_point T_0 T_2) T_1) (-. (distinct_lines T_1 T_2)) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_1 T_1)) ### All 45
% 31.55/31.76 47. (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 T_2)) (apart_point_and_line (intersection_point T_0 T_2) 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, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_1 T_0) => ((distinct_lines T_0 Z) \/ (convergent_lines T_1 Z)))) ### All 46
% 31.55/31.76 48. (All Y, (All Z, ((convergent_lines T_1 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_1 Z))))) (convergent_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (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 (intersection_point T_0 T_2) T_1) (-. (distinct_lines T_1 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### All 47
% 31.55/31.76 49. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (distinct_lines T_1 T_2)) (apart_point_and_line (intersection_point T_0 T_2) 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, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_1 T_0) ### All 48
% 31.55/31.76 50. (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
% 31.55/31.76 51. ((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_0) (-. (distinct_lines T_1 T_2)) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### Imply 5 50
% 31.55/31.76 52. (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (-. (distinct_lines T_1 T_2)) (apart_point_and_line (intersection_point T_0 T_2) T_0) ### All 51
% 31.55/31.76 53. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines 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) \/ ((apart_point_and_line (intersection_point T_0 T_2) T_1) \/ (apart_point_and_line (intersection_point T_0 T_2) T_0))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (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_1 T_2)) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### DisjTree 12 13 18 22 49 52
% 31.55/31.76 54. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines T_1 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) T_1) \/ (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)) (distinct_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (-. (distinct_lines T_1 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, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (convergent_lines T_1 T_0) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### All 53
% 31.55/31.76 55. (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))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (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_1 T_2)) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### All 54
% 31.55/31.77 56. ((convergent_lines T_0 T_1) => ((convergent_lines T_0 T_0) \/ (convergent_lines T_1 T_0))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (distinct_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (-. (distinct_lines T_1 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, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (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))))))) (All X, (-. (convergent_lines X X))) (convergent_lines T_0 T_1) ### DisjTree 9 11 55
% 31.55/31.77 57. (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (All X, (-. (convergent_lines X X))) (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))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (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_1 T_2)) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### All 56
% 31.55/31.77 58. ((convergent_lines T_0 T_1) => ((distinct_lines T_1 T_0) \/ (convergent_lines T_0 T_0))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (-. (distinct_lines T_1 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, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (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))))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) ### DisjTree 8 57 11
% 31.55/31.77 59. (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (convergent_lines X X))) (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))))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (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_1 T_2)) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### All 58
% 31.55/31.77 60. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point T_0 T_1) (intersection_point 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, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (-. (distinct_lines T_1 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, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (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))))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) ### All 59
% 31.55/31.77 61. (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)))))))) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (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_1 T_2)) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### All 60
% 31.55/31.78 62. (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))))) (distinct_points (intersection_point T_0 T_1) (intersection_point 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, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (-. (distinct_lines T_1 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, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) ### All 61
% 31.55/31.78 63. (All Y, (All Z, ((convergent_lines T_0 Y) => ((convergent_lines T_0 Z) \/ (convergent_lines Y Z))))) (convergent_lines T_0 T_1) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (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_1 T_2)) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (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 62
% 31.55/31.78 64. (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point T_0 T_1) (intersection_point 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, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (-. (distinct_lines T_1 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, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (convergent_lines T_0 T_1) (All Y, (All Z, ((convergent_lines T_0 Y) => ((convergent_lines T_0 Z) \/ (convergent_lines Y Z))))) ### All 63
% 31.55/31.78 65. (All Y, (All Z, ((convergent_lines T_0 Y) => ((convergent_lines T_0 Z) \/ (convergent_lines Y Z))))) (convergent_lines T_0 T_1) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (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_1 T_2)) (-. (convergent_lines T_1 T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (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 64
% 31.55/31.78 66. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point T_0 T_1) (intersection_point 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, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_1 T_1)) (-. (distinct_lines T_1 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, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (convergent_lines T_0 T_1) ### All 65
% 31.55/31.78 67. (convergent_lines T_0 T_1) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (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_1 T_2)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (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, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 66
% 31.55/31.78 68. (-. (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((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, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (distinct_lines T_1 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, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (convergent_lines T_0 T_1) ### NotNot 67
% 31.55/31.78 69. (-. (equal_points (intersection_point T_0 T_1) (intersection_point T_0 T_2))) (convergent_lines T_0 T_1) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (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_1 T_2)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (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, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### Definition-Pseudo(equal_points) 68
% 31.55/31.78 70. (-. ((convergent_lines T_0 T_2) /\ (equal_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((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, (All Y, ((distinct_lines X Y) => (convergent_lines 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 Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (-. (distinct_lines T_1 T_2)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotAnd 7 69
% 31.55/31.78 71. (equal_lines T_1 T_2) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (convergent_lines T_0 T_1) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (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, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. ((convergent_lines T_0 T_2) /\ (equal_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) ### Definition-Pseudo(equal_lines) 70
% 31.55/31.78 72. (-. (((convergent_lines T_0 T_1) /\ (equal_lines T_1 T_2)) => ((convergent_lines T_0 T_2) /\ (equal_points (intersection_point T_0 T_1) (intersection_point T_0 T_2))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((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, (All Y, ((distinct_lines X Y) => (convergent_lines 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 Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (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 71
% 31.55/31.78 73. (-. (All Z, (((convergent_lines T_0 T_1) /\ (equal_lines T_1 Z)) => ((convergent_lines T_0 Z) /\ (equal_points (intersection_point T_0 T_1) (intersection_point T_0 Z)))))) (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, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (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, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 72
% 31.55/31.78 74. (-. (All Y, (All Z, (((convergent_lines T_0 Y) /\ (equal_lines Y Z)) => ((convergent_lines T_0 Z) /\ (equal_points (intersection_point T_0 Y) (intersection_point T_0 Z))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((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, (All Y, ((distinct_lines X Y) => (convergent_lines 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 Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (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 73
% 31.55/31.78 75. (-. (All X, (All Y, (All Z, (((convergent_lines X Y) /\ (equal_lines Y Z)) => ((convergent_lines X Z) /\ (equal_points (intersection_point X Y) (intersection_point X Z)))))))) (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, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (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, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 74
% 31.55/31.78 % SZS output end Proof
% 31.55/31.78 (* END-PROOF *)
%------------------------------------------------------------------------------