TSTP Solution File: GEO198+3 by SuperZenon---0.0.1
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%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : GEO198+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 : n013.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 600s
% DateTime : Sat Jul 16 06:30:58 EDT 2022
% Result : Theorem 5.26s 5.42s
% Output : Proof 5.56s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GEO198+3 : TPTP v8.1.0. Released v4.0.0.
% 0.00/0.12 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.13/0.33 % Computer : n013.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 01:29:14 EDT 2022
% 0.13/0.33 % CPUTime :
% 5.26/5.42 % SZS status Theorem
% 5.26/5.42 (* PROOF-FOUND *)
% 5.26/5.42 (* BEGIN-PROOF *)
% 5.26/5.42 % SZS output start Proof
% 5.26/5.42 1. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 5.26/5.42 2. (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_0) ### Axiom
% 5.26/5.42 3. (-. (convergent_lines T_1 T_0)) (convergent_lines T_1 T_0) ### Axiom
% 5.26/5.42 4. ((convergent_lines T_0 T_1) => ((convergent_lines T_0 T_0) \/ (convergent_lines T_1 T_0))) (-. (convergent_lines T_1 T_0)) (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_1) ### DisjTree 1 2 3
% 5.26/5.42 5. (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_0)) (-. (convergent_lines T_1 T_0)) ### All 4
% 5.26/5.42 6. (All Y, (All Z, ((convergent_lines T_0 Y) => ((convergent_lines T_0 Z) \/ (convergent_lines Y Z))))) (-. (convergent_lines T_1 T_0)) (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_1) ### All 5
% 5.26/5.42 7. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_0)) (-. (convergent_lines T_1 T_0)) ### All 6
% 5.26/5.42 8. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 5.26/5.42 9. (distinct_lines T_1 T_0) (-. (distinct_lines T_1 T_0)) ### Axiom
% 5.26/5.42 10. (convergent_lines T_2 T_0) (-. (convergent_lines T_2 T_0)) ### Axiom
% 5.26/5.42 11. (distinct_lines T_1 T_0) (-. (distinct_lines T_1 T_0)) ### Axiom
% 5.26/5.42 12. (convergent_lines T_2 T_0) (-. (convergent_lines T_2 T_0)) ### Axiom
% 5.26/5.42 13. (convergent_lines T_2 T_2) (-. (convergent_lines T_2 T_2)) ### Axiom
% 5.26/5.42 14. (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_2) ### All 13
% 5.26/5.42 15. (-. (convergent_lines T_0 T_2)) (convergent_lines T_0 T_2) ### Axiom
% 5.26/5.42 16. ((convergent_lines T_2 T_0) => ((convergent_lines T_2 T_2) \/ (convergent_lines T_0 T_2))) (-. (convergent_lines T_0 T_2)) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_0) ### DisjTree 12 14 15
% 5.26/5.42 17. (All Z, ((convergent_lines T_2 T_0) => ((convergent_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_2 T_0) (All X, (-. (convergent_lines X X))) (-. (convergent_lines T_0 T_2)) ### All 16
% 5.26/5.42 18. (convergent_lines T_0 T_2) (-. (convergent_lines T_0 T_2)) ### Axiom
% 5.26/5.42 19. (apart_point_and_line (intersection_point T_2 T_0) T_1) (-. (apart_point_and_line (intersection_point T_2 T_0) T_1)) ### Axiom
% 5.26/5.42 20. (distinct_points (intersection_point T_2 T_0) (intersection_point T_1 T_0)) (-. (distinct_points (intersection_point T_2 T_0) (intersection_point T_1 T_0))) ### Axiom
% 5.26/5.42 21. (distinct_lines T_2 T_0) (-. (distinct_lines T_2 T_0)) ### Axiom
% 5.26/5.42 22. (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (apart_point_and_line (intersection_point T_2 T_0) T_2) ### Axiom
% 5.26/5.42 23. (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (apart_point_and_line (intersection_point T_2 T_0) T_0) ### Axiom
% 5.26/5.42 24. (apart_point_and_line (intersection_point T_1 T_0) T_2) (-. (apart_point_and_line (intersection_point T_1 T_0) T_2)) ### Axiom
% 5.26/5.42 25. (distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_0)) (-. (distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_0))) ### Axiom
% 5.26/5.42 26. (distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_1)) (-. (distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_1))) ### Axiom
% 5.26/5.42 27. (distinct_lines T_1 T_0) (-. (distinct_lines T_1 T_0)) ### Axiom
% 5.26/5.42 28. (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (apart_point_and_line (intersection_point T_1 T_0) T_1) ### Axiom
% 5.26/5.42 29. (-. (apart_point_and_line (intersection_point T_1 T_0) T_0)) (apart_point_and_line (intersection_point T_1 T_0) T_0) ### Axiom
% 5.26/5.42 30. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom
% 5.26/5.42 31. (apart_point_and_line (intersection_point T_2 T_1) T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) ### Axiom
% 5.26/5.42 32. ((convergent_lines T_2 T_1) => (-. (apart_point_and_line (intersection_point T_2 T_1) T_1))) (apart_point_and_line (intersection_point T_2 T_1) T_1) (convergent_lines T_2 T_1) ### Imply 30 31
% 5.26/5.42 33. (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) Y)))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_2 T_1) T_1) ### All 32
% 5.26/5.42 34. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_2 T_1) T_1) (convergent_lines T_2 T_1) ### All 33
% 5.26/5.42 35. (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (apart_point_and_line (intersection_point T_2 T_1) T_0) ### Axiom
% 5.26/5.42 36. (((distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_1)) /\ (distinct_lines T_1 T_0)) => ((apart_point_and_line (intersection_point T_1 T_0) T_1) \/ ((apart_point_and_line (intersection_point T_1 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) T_1) \/ (apart_point_and_line (intersection_point T_2 T_1) T_0))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_1)) ### DisjTree 26 27 28 29 34 35
% 5.26/5.42 37. (All V, (((distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_1)) /\ (distinct_lines T_1 V)) => ((apart_point_and_line (intersection_point T_1 T_0) T_1) \/ ((apart_point_and_line (intersection_point T_1 T_0) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) T_1) \/ (apart_point_and_line (intersection_point T_2 T_1) V)))))) (distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_1)) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (-. (apart_point_and_line (intersection_point T_1 T_0) T_0)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) ### All 36
% 5.26/5.42 38. (All U, (All V, (((distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_0) U) \/ ((apart_point_and_line (intersection_point T_1 T_0) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) U) \/ (apart_point_and_line (intersection_point T_2 T_1) V))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_1)) ### All 37
% 5.26/5.42 39. (All Y, (All U, (All V, (((distinct_points (intersection_point T_1 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_0) U) \/ ((apart_point_and_line (intersection_point T_1 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_1)) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (-. (apart_point_and_line (intersection_point T_1 T_0) T_0)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) ### All 38
% 5.26/5.42 40. (distinct_points (intersection_point T_2 T_0) (intersection_point T_2 T_1)) (-. (distinct_points (intersection_point T_2 T_0) (intersection_point T_2 T_1))) ### Axiom
% 5.26/5.42 41. (distinct_lines T_2 T_0) (-. (distinct_lines T_2 T_0)) ### Axiom
% 5.26/5.42 42. (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (apart_point_and_line (intersection_point T_2 T_0) T_2) ### Axiom
% 5.26/5.42 43. (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (apart_point_and_line (intersection_point T_2 T_0) T_0) ### Axiom
% 5.26/5.44 44. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom
% 5.26/5.44 45. (apart_point_and_line (intersection_point T_2 T_1) T_2) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) ### Axiom
% 5.26/5.44 46. ((convergent_lines T_2 T_1) => (-. (apart_point_and_line (intersection_point T_2 T_1) T_2))) (apart_point_and_line (intersection_point T_2 T_1) T_2) (convergent_lines T_2 T_1) ### Imply 44 45
% 5.26/5.44 47. (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_2 T_1) T_2) ### All 46
% 5.26/5.44 48. (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (apart_point_and_line (intersection_point T_2 T_1) T_0) ### Axiom
% 5.26/5.44 49. (((distinct_points (intersection_point T_2 T_0) (intersection_point T_2 T_1)) /\ (distinct_lines T_2 T_0)) => ((apart_point_and_line (intersection_point T_2 T_0) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) T_2) \/ (apart_point_and_line (intersection_point T_2 T_1) T_0))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (distinct_lines T_2 T_0) (distinct_points (intersection_point T_2 T_0) (intersection_point T_2 T_1)) ### DisjTree 40 41 42 43 47 48
% 5.26/5.44 50. (All V, (((distinct_points (intersection_point T_2 T_0) (intersection_point T_2 T_1)) /\ (distinct_lines T_2 V)) => ((apart_point_and_line (intersection_point T_2 T_0) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) T_2) \/ (apart_point_and_line (intersection_point T_2 T_1) V)))))) (distinct_points (intersection_point T_2 T_0) (intersection_point T_2 T_1)) (distinct_lines T_2 T_0) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) ### All 49
% 5.26/5.44 51. (All U, (All V, (((distinct_points (intersection_point T_2 T_0) (intersection_point T_2 T_1)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) U) \/ (apart_point_and_line (intersection_point T_2 T_1) V))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (distinct_lines T_2 T_0) (distinct_points (intersection_point T_2 T_0) (intersection_point T_2 T_1)) ### All 50
% 5.26/5.44 52. (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point T_2 T_0) (intersection_point T_2 T_1)) (distinct_lines T_2 T_0) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) ### All 51
% 5.26/5.44 53. ((distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_0)) => ((distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_1)) \/ (distinct_points (intersection_point T_2 T_0) (intersection_point T_2 T_1)))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (distinct_lines T_2 T_0) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All Y, (All U, (All V, (((distinct_points (intersection_point T_1 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_0) U) \/ ((apart_point_and_line (intersection_point T_1 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_0)) ### DisjTree 25 39 52
% 5.26/5.44 54. (All Z, ((distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_0)) => ((distinct_points (intersection_point T_1 T_0) Z) \/ (distinct_points (intersection_point T_2 T_0) Z)))) (distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_0)) (All Y, (All U, (All V, (((distinct_points (intersection_point T_1 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_0) U) \/ ((apart_point_and_line (intersection_point T_1 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (-. (apart_point_and_line (intersection_point T_1 T_0) T_0)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_2 T_0) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) ### All 53
% 5.26/5.44 55. (All Y, (All Z, ((distinct_points (intersection_point T_1 T_0) Y) => ((distinct_points (intersection_point T_1 T_0) Z) \/ (distinct_points Y Z))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (distinct_lines T_2 T_0) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All Y, (All U, (All V, (((distinct_points (intersection_point T_1 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_0) U) \/ ((apart_point_and_line (intersection_point T_1 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_0)) ### All 54
% 5.26/5.44 56. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_0)) (All Y, (All U, (All V, (((distinct_points (intersection_point T_1 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_1 T_0) U) \/ ((apart_point_and_line (intersection_point T_1 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (-. (apart_point_and_line (intersection_point T_1 T_0) T_0)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_2 T_0) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) ### All 55
% 5.26/5.48 57. (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 Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (distinct_lines T_2 T_0) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 56
% 5.26/5.48 58. (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (apart_point_and_line (intersection_point T_2 T_0) T_2) ### Axiom
% 5.26/5.48 59. ((apart_point_and_line (intersection_point T_1 T_0) T_2) => ((distinct_points (intersection_point T_1 T_0) (intersection_point T_2 T_0)) \/ (apart_point_and_line (intersection_point T_2 T_0) T_2))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (-. (apart_point_and_line (intersection_point T_1 T_0) T_0)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_2 T_0) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_1 T_0) T_2) ### DisjTree 24 57 58
% 5.26/5.48 60. (All Z, ((apart_point_and_line (intersection_point T_1 T_0) T_2) => ((distinct_points (intersection_point T_1 T_0) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_1 T_0) T_2) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (distinct_lines T_2 T_0) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 59
% 5.26/5.48 61. (distinct_lines T_0 T_0) (-. (distinct_lines T_0 T_0)) ### Axiom
% 5.26/5.48 62. (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (apart_point_and_line (intersection_point T_2 T_1) T_0) ### Axiom
% 5.26/5.48 63. (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (apart_point_and_line (intersection_point T_2 T_1) T_0) ### Axiom
% 5.26/5.48 64. (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_0) ### Axiom
% 5.26/5.48 65. ((distinct_lines T_0 T_0) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ (convergent_lines T_0 T_0)))) (-. (convergent_lines T_0 T_0)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (distinct_lines T_0 T_0) ### DisjTree 61 62 63 64
% 5.26/5.48 66. (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (distinct_lines T_0 T_0) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (convergent_lines T_0 T_0)) ### All 65
% 5.26/5.48 67. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom
% 5.26/5.48 68. (apart_point_and_line (intersection_point T_1 T_0) T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_0)) ### Axiom
% 5.26/5.48 69. ((convergent_lines T_1 T_0) => (-. (apart_point_and_line (intersection_point T_1 T_0) T_0))) (apart_point_and_line (intersection_point T_1 T_0) T_0) (convergent_lines T_1 T_0) ### Imply 67 68
% 5.26/5.48 70. (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) Y)))) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_1 T_0) T_0) ### All 69
% 5.26/5.48 71. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_1 T_0) T_0) (convergent_lines T_1 T_0) ### All 70
% 5.26/5.48 72. ((apart_point_and_line (intersection_point T_1 T_0) T_0) => ((distinct_lines T_0 T_0) \/ (apart_point_and_line (intersection_point T_1 T_0) T_0))) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_2 T_0) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_1 T_0) T_2) (All Z, ((apart_point_and_line (intersection_point T_1 T_0) T_2) => ((distinct_points (intersection_point T_1 T_0) Z) \/ (apart_point_and_line Z T_2)))) ### DisjTree 60 66 71
% 5.26/5.50 73. (All Z, ((apart_point_and_line (intersection_point T_1 T_0) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_1 T_0) Z)))) (All Z, ((apart_point_and_line (intersection_point T_1 T_0) T_2) => ((distinct_points (intersection_point T_1 T_0) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_1 T_0) T_2) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (distinct_lines T_2 T_0) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) ### All 72
% 5.26/5.50 74. (All Y, (All Z, ((apart_point_and_line (intersection_point T_1 T_0) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_1 T_0) Z))))) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_2 T_0) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_1 T_0) T_2) (All Z, ((apart_point_and_line (intersection_point T_1 T_0) T_2) => ((distinct_points (intersection_point T_1 T_0) Z) \/ (apart_point_and_line Z T_2)))) ### All 73
% 5.26/5.50 75. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line (intersection_point T_1 T_0) T_2) => ((distinct_points (intersection_point T_1 T_0) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_1 T_0) T_2) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (distinct_lines T_2 T_0) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) ### All 74
% 5.26/5.50 76. (All Y, (All Z, ((apart_point_and_line (intersection_point T_1 T_0) Y) => ((distinct_points (intersection_point T_1 T_0) Z) \/ (apart_point_and_line Z Y))))) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_lines T_2 T_0) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_1 T_0) 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 75
% 5.26/5.50 77. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z 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 (intersection_point T_1 T_0) T_2) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (distinct_lines T_2 T_0) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) ### All 76
% 5.35/5.53 78. (((distinct_points (intersection_point T_2 T_0) (intersection_point T_1 T_0)) /\ (distinct_lines T_2 T_0)) => ((apart_point_and_line (intersection_point T_2 T_0) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_0) T_0) \/ ((apart_point_and_line (intersection_point T_1 T_0) T_2) \/ (apart_point_and_line (intersection_point T_1 T_0) T_0))))) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (distinct_lines T_2 T_0) (distinct_points (intersection_point T_2 T_0) (intersection_point T_1 T_0)) ### DisjTree 20 21 22 23 77 71
% 5.35/5.53 79. (All V, (((distinct_points (intersection_point T_2 T_0) (intersection_point T_1 T_0)) /\ (distinct_lines T_2 V)) => ((apart_point_and_line (intersection_point T_2 T_0) T_2) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line (intersection_point T_1 T_0) T_2) \/ (apart_point_and_line (intersection_point T_1 T_0) V)))))) (distinct_points (intersection_point T_2 T_0) (intersection_point T_1 T_0)) (distinct_lines T_2 T_0) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) ### All 78
% 5.35/5.53 80. (All U, (All V, (((distinct_points (intersection_point T_2 T_0) (intersection_point T_1 T_0)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line (intersection_point T_1 T_0) U) \/ (apart_point_and_line (intersection_point T_1 T_0) V))))))) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (distinct_lines T_2 T_0) (distinct_points (intersection_point T_2 T_0) (intersection_point T_1 T_0)) ### All 79
% 5.35/5.53 81. (distinct_points (intersection_point T_2 T_0) (intersection_point T_1 T_0)) (distinct_lines T_2 T_0) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All Y, (All U, (All V, (((distinct_points (intersection_point T_2 T_0) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_2 T_0) U) \/ ((apart_point_and_line (intersection_point T_2 T_0) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) ### All 80
% 5.35/5.54 82. (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (distinct_lines T_2 T_0) (distinct_points (intersection_point T_2 T_0) (intersection_point T_1 T_0)) ### All 81
% 5.35/5.54 83. (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (apart_point_and_line (intersection_point T_1 T_0) T_1) ### Axiom
% 5.35/5.54 84. ((apart_point_and_line (intersection_point T_2 T_0) T_1) => ((distinct_points (intersection_point T_2 T_0) (intersection_point T_1 T_0)) \/ (apart_point_and_line (intersection_point T_1 T_0) T_1))) (distinct_lines T_2 T_0) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) ### DisjTree 19 82 83
% 5.35/5.54 85. (All Z, ((apart_point_and_line (intersection_point T_2 T_0) T_1) => ((distinct_points (intersection_point T_2 T_0) Z) \/ (apart_point_and_line Z T_1)))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (distinct_lines T_2 T_0) ### All 84
% 5.35/5.54 86. (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_0) Y) => ((distinct_points (intersection_point T_2 T_0) Z) \/ (apart_point_and_line Z Y))))) (distinct_lines T_2 T_0) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) ### All 85
% 5.35/5.54 87. (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (distinct_lines T_2 T_0) ### All 86
% 5.35/5.58 88. (convergent_lines T_2 T_0) (-. (convergent_lines T_2 T_0)) ### Axiom
% 5.35/5.58 89. (apart_point_and_line (intersection_point T_2 T_0) T_0) (-. (apart_point_and_line (intersection_point T_2 T_0) T_0)) ### Axiom
% 5.35/5.58 90. ((convergent_lines T_2 T_0) => (-. (apart_point_and_line (intersection_point T_2 T_0) T_0))) (apart_point_and_line (intersection_point T_2 T_0) T_0) (convergent_lines T_2 T_0) ### Imply 88 89
% 5.35/5.58 91. (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) Y)))) (convergent_lines T_2 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_0) ### All 90
% 5.35/5.58 92. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point T_2 T_0) T_0) (convergent_lines T_2 T_0) ### All 91
% 5.35/5.58 93. ((apart_point_and_line (intersection_point T_2 T_0) T_0) => ((distinct_lines T_0 T_0) \/ (apart_point_and_line (intersection_point T_2 T_0) T_0))) (convergent_lines T_2 T_0) (distinct_lines T_2 T_0) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) ### DisjTree 87 66 92
% 5.35/5.58 94. (All Z, ((apart_point_and_line (intersection_point T_2 T_0) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_2 T_0) Z)))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All Z, ((distinct_lines T_0 Z) => ((apart_point_and_line (intersection_point T_2 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (distinct_lines T_2 T_0) (convergent_lines T_2 T_0) ### All 93
% 5.35/5.58 95. (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line (intersection_point T_2 T_1) Y) \/ ((apart_point_and_line (intersection_point T_2 T_1) Z) \/ (convergent_lines Y Z)))))) (convergent_lines T_2 T_0) (distinct_lines T_2 T_0) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All Z, ((apart_point_and_line (intersection_point T_2 T_0) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_2 T_0) Z)))) ### All 94
% 5.35/5.58 96. (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All Z, ((apart_point_and_line (intersection_point T_2 T_0) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_2 T_0) Z)))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) (distinct_lines T_2 T_0) (convergent_lines T_2 T_0) ### All 95
% 5.35/5.58 97. (-. (distinct_lines T_2 T_2)) (distinct_lines T_2 T_2) ### Axiom
% 5.35/5.58 98. (convergent_lines T_2 T_0) (-. (convergent_lines T_2 T_0)) ### Axiom
% 5.35/5.58 99. (apart_point_and_line (intersection_point T_2 T_0) T_2) (-. (apart_point_and_line (intersection_point T_2 T_0) T_2)) ### Axiom
% 5.35/5.58 100. ((convergent_lines T_2 T_0) => (-. (apart_point_and_line (intersection_point T_2 T_0) T_2))) (apart_point_and_line (intersection_point T_2 T_0) T_2) (convergent_lines T_2 T_0) ### Imply 98 99
% 5.35/5.58 101. (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (convergent_lines T_2 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_2) ### All 100
% 5.35/5.58 102. ((apart_point_and_line (intersection_point T_2 T_0) T_2) => ((distinct_lines T_2 T_2) \/ (apart_point_and_line (intersection_point T_2 T_0) T_2))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_2 T_0) (distinct_lines T_2 T_0) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All Z, ((apart_point_and_line (intersection_point T_2 T_0) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_2 T_0) Z)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) ### DisjTree 96 97 101
% 5.43/5.59 103. (All Z, ((apart_point_and_line (intersection_point T_2 T_0) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_2 T_0) Z)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All Z, ((apart_point_and_line (intersection_point T_2 T_0) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_2 T_0) Z)))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_lines T_2 T_0) (convergent_lines T_2 T_0) (-. (distinct_lines T_2 T_2)) ### All 102
% 5.43/5.59 104. (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_0) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_0) Z))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_2 T_0) (distinct_lines T_2 T_0) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All Z, ((apart_point_and_line (intersection_point T_2 T_0) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_2 T_0) Z)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) ### All 103
% 5.43/5.59 105. (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_lines T_2 T_0) (convergent_lines T_2 T_0) (-. (distinct_lines T_2 T_2)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_0) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_0) Z))))) ### All 104
% 5.43/5.59 106. (-. (distinct_lines T_2 T_2)) (convergent_lines T_2 T_0) (distinct_lines T_2 T_0) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) ### All 105
% 5.43/5.59 107. (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_0) ### Axiom
% 5.43/5.59 108. ((convergent_lines T_0 T_2) => ((distinct_lines T_2 T_0) \/ (convergent_lines T_0 T_0))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (convergent_lines T_2 T_0) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_2) ### DisjTree 18 106 107
% 5.43/5.63 109. (All Z, ((convergent_lines T_0 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_2) (-. (distinct_lines T_2 T_2)) (convergent_lines T_2 T_0) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) ### All 108
% 5.43/5.63 110. (-. (unorthogonal_lines T_0 T_2)) (unorthogonal_lines T_0 T_2) ### Axiom
% 5.43/5.63 111. ((convergent_lines T_0 T_2) \/ (unorthogonal_lines T_0 T_2)) (-. (unorthogonal_lines T_0 T_2)) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (convergent_lines T_2 T_0) (-. (distinct_lines T_2 T_2)) (All Z, ((convergent_lines T_0 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_0 Z)))) ### Or 109 110
% 5.43/5.63 112. (All M, ((convergent_lines T_0 M) \/ (unorthogonal_lines T_0 M))) (All Z, ((convergent_lines T_0 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_2 T_0) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (-. (unorthogonal_lines T_0 T_2)) ### All 111
% 5.43/5.63 113. (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (unorthogonal_lines T_0 T_2)) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (convergent_lines T_2 T_0) (-. (distinct_lines T_2 T_2)) (All Z, ((convergent_lines T_0 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_0 Z)))) ### All 112
% 5.43/5.63 114. (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_2 T_0) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (-. (unorthogonal_lines T_0 T_2)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) ### All 113
% 5.43/5.63 115. (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_0) ### Axiom
% 5.43/5.63 116. ((convergent_lines T_0 T_0) /\ (unorthogonal_lines T_0 T_0)) (-. (convergent_lines T_0 T_0)) ### And 115
% 5.43/5.63 117. (-. (unorthogonal_lines T_2 T_0)) (unorthogonal_lines T_2 T_0) ### Axiom
% 5.43/5.63 118. ((convergent_lines T_2 T_0) /\ (unorthogonal_lines T_2 T_0)) (-. (unorthogonal_lines T_2 T_0)) ### And 117
% 5.43/5.63 119. (((convergent_lines T_0 T_2) /\ (unorthogonal_lines T_0 T_2)) => (((convergent_lines T_0 T_0) /\ (unorthogonal_lines T_0 T_0)) \/ ((convergent_lines T_2 T_0) /\ (unorthogonal_lines T_2 T_0)))) (-. (unorthogonal_lines T_2 T_0)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (-. (distinct_lines T_2 T_2)) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_0) (All Z, ((convergent_lines T_2 T_0) => ((convergent_lines T_2 Z) \/ (convergent_lines T_0 Z)))) ### DisjTree 17 114 116 118
% 5.43/5.64 120. (All N, (((convergent_lines T_0 T_2) /\ (unorthogonal_lines T_0 T_2)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines T_2 N) /\ (unorthogonal_lines T_2 N))))) (All Z, ((convergent_lines T_2 T_0) => ((convergent_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_2 T_0) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (-. (distinct_lines T_2 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (unorthogonal_lines T_2 T_0)) ### All 119
% 5.43/5.64 121. (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines T_2 T_0)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_0) (All Z, ((convergent_lines T_2 T_0) => ((convergent_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (All N, (((convergent_lines T_0 T_2) /\ (unorthogonal_lines T_0 T_2)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines T_2 N) /\ (unorthogonal_lines T_2 N))))) ### All 120
% 5.43/5.64 122. (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Z, ((convergent_lines T_2 T_0) => ((convergent_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_2 T_0) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (unorthogonal_lines T_2 T_0)) (All X, (-. (distinct_lines X X))) ### All 121
% 5.43/5.64 123. (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines T_2 T_0)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_0) (All Z, ((convergent_lines T_2 T_0) => ((convergent_lines T_2 Z) \/ (convergent_lines T_0 Z)))) ### All 122
% 5.43/5.64 124. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_2 T_0) => ((convergent_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_2 T_0) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (unorthogonal_lines T_2 T_0)) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) ### All 123
% 5.43/5.64 125. ((distinct_lines T_1 T_0) => (convergent_lines T_1 T_0)) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (-. (distinct_lines X X))) (-. (unorthogonal_lines T_2 T_0)) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (-. (convergent_lines X X))) (convergent_lines T_2 T_0) (All Z, ((convergent_lines T_2 T_0) => ((convergent_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_lines T_1 T_0) ### Imply 11 124
% 5.43/5.64 126. (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (distinct_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_2 T_0) => ((convergent_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_2 T_0) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (-. (unorthogonal_lines T_2 T_0)) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) ### All 125
% 5.43/5.64 127. (convergent_lines T_0 T_2) (-. (convergent_lines T_0 T_2)) ### Axiom
% 5.43/5.64 128. (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (distinct_lines T_2 T_0) (convergent_lines T_2 T_0) ### All 106
% 5.43/5.64 129. (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_0) ### Axiom
% 5.43/5.64 130. ((convergent_lines T_0 T_2) => ((distinct_lines T_2 T_0) \/ (convergent_lines T_0 T_0))) (convergent_lines T_2 T_0) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_2) ### DisjTree 127 128 129
% 5.52/5.68 131. (All Z, ((convergent_lines T_0 T_2) => ((distinct_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_2) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (convergent_lines T_2 T_0) ### All 130
% 5.52/5.68 132. (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (convergent_lines T_2 T_0) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (convergent_lines T_0 T_2) ### All 131
% 5.52/5.68 133. (-. (-. (convergent_lines T_0 T_2))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (convergent_lines T_2 T_0) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) ### NotNot 132
% 5.52/5.68 134. (-. ((-. (convergent_lines T_0 T_2)) \/ (-. (unorthogonal_lines T_0 T_2)))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (convergent_lines T_2 T_0) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) ### NotOr 133
% 5.52/5.68 135. (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_0) ### Axiom
% 5.52/5.68 136. (-. (-. (convergent_lines T_0 T_0))) (-. (convergent_lines T_0 T_0)) ### NotNot 135
% 5.52/5.68 137. (-. ((-. (convergent_lines T_0 T_0)) \/ (-. (unorthogonal_lines T_0 T_0)))) (-. (convergent_lines T_0 T_0)) ### NotOr 136
% 5.52/5.68 138. (convergent_lines T_2 T_0) (-. (convergent_lines T_2 T_0)) ### Axiom
% 5.52/5.68 139. (unorthogonal_lines T_2 T_0) (-. (unorthogonal_lines T_2 T_0)) ### Axiom
% 5.52/5.68 140. ((((-. (convergent_lines T_0 T_2)) \/ (-. (unorthogonal_lines T_0 T_2))) /\ ((-. (convergent_lines T_0 T_0)) \/ (-. (unorthogonal_lines T_0 T_0)))) => ((-. (convergent_lines T_2 T_0)) \/ (-. (unorthogonal_lines T_2 T_0)))) (unorthogonal_lines T_2 T_0) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (convergent_lines T_2 T_0) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) ### DisjTree 134 137 138 139
% 5.52/5.68 141. (All N, ((((-. (convergent_lines T_0 T_2)) \/ (-. (unorthogonal_lines T_0 T_2))) /\ ((-. (convergent_lines T_0 N)) \/ (-. (unorthogonal_lines T_0 N)))) => ((-. (convergent_lines T_2 N)) \/ (-. (unorthogonal_lines T_2 N))))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (convergent_lines T_2 T_0) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (unorthogonal_lines T_2 T_0) ### All 140
% 5.53/5.69 142. (All M, (All N, ((((-. (convergent_lines T_0 M)) \/ (-. (unorthogonal_lines T_0 M))) /\ ((-. (convergent_lines T_0 N)) \/ (-. (unorthogonal_lines T_0 N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) (unorthogonal_lines T_2 T_0) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (All Y, ((convergent_lines T_2 Y) => (-. (apart_point_and_line (intersection_point T_2 Y) T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (convergent_lines T_2 T_0) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) ### All 141
% 5.53/5.69 143. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (convergent_lines T_2 T_0) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (unorthogonal_lines T_2 T_0) (All M, (All N, ((((-. (convergent_lines T_0 M)) \/ (-. (unorthogonal_lines T_0 M))) /\ ((-. (convergent_lines T_0 N)) \/ (-. (unorthogonal_lines T_0 N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N)))))) ### All 142
% 5.53/5.69 144. (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (unorthogonal_lines T_2 T_0) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (convergent_lines T_1 T_0) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (convergent_lines T_2 T_0) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### All 143
% 5.53/5.69 145. ((convergent_lines T_2 T_0) /\ (unorthogonal_lines T_2 T_0)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (convergent_lines T_2 T_0) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (distinct_lines T_1 T_0) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) ### And 144
% 5.53/5.69 146. (((convergent_lines T_2 T_0) /\ (unorthogonal_lines T_2 T_0)) => (((convergent_lines T_2 T_0) /\ (unorthogonal_lines T_2 T_0)) \/ ((convergent_lines T_0 T_0) /\ (unorthogonal_lines T_0 T_0)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (convergent_lines T_1 T_0) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (-. (distinct_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_2 T_0) => ((convergent_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_lines T_1 T_0) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (convergent_lines T_2 T_0) ### DisjTree 10 126 145 116
% 5.53/5.70 147. (All N, (((convergent_lines T_2 T_0) /\ (unorthogonal_lines T_2 T_0)) => (((convergent_lines T_2 N) /\ (unorthogonal_lines T_2 N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) (convergent_lines T_2 T_0) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (distinct_lines T_1 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_2 T_0) => ((convergent_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (convergent_lines T_1 T_0) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) ### All 146
% 5.53/5.70 148. ((distinct_lines T_1 T_0) => (convergent_lines T_1 T_0)) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (-. (distinct_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_2 T_0) => ((convergent_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (convergent_lines T_2 T_0) (All N, (((convergent_lines T_2 T_0) /\ (unorthogonal_lines T_2 T_0)) => (((convergent_lines T_2 N) /\ (unorthogonal_lines T_2 N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) (distinct_lines T_1 T_0) ### Imply 9 147
% 5.53/5.70 149. (distinct_lines T_1 T_0) (All N, (((convergent_lines T_2 T_0) /\ (unorthogonal_lines T_2 T_0)) => (((convergent_lines T_2 N) /\ (unorthogonal_lines T_2 N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) (convergent_lines T_2 T_0) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_2 T_0) => ((convergent_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) ### All 148
% 5.56/5.71 150. (All M, (All N, (((convergent_lines T_2 M) /\ (unorthogonal_lines T_2 M)) => (((convergent_lines T_2 N) /\ (unorthogonal_lines T_2 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (-. (distinct_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_2 T_0) => ((convergent_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (convergent_lines T_2 T_0) (distinct_lines T_1 T_0) ### All 149
% 5.56/5.71 151. (distinct_lines T_1 T_0) (convergent_lines T_2 T_0) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_2 T_0) => ((convergent_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) ### All 150
% 5.56/5.71 152. (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_0) ### Axiom
% 5.56/5.71 153. ((convergent_lines T_0 T_1) => ((distinct_lines T_1 T_0) \/ (convergent_lines T_0 T_0))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (-. (distinct_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines T_2 T_0) => ((convergent_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (convergent_lines T_2 T_0) (convergent_lines T_0 T_1) ### DisjTree 8 151 152
% 5.56/5.71 154. (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 T_1) (convergent_lines T_2 T_0) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Z, ((convergent_lines T_2 T_0) => ((convergent_lines T_2 Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) ### All 153
% 5.56/5.73 155. (All Y, (All Z, ((convergent_lines T_2 Y) => ((convergent_lines T_2 Z) \/ (convergent_lines Y Z))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (-. (distinct_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (convergent_lines T_2 T_0) (convergent_lines T_0 T_1) (All Z, ((convergent_lines T_0 T_1) => ((distinct_lines T_1 Z) \/ (convergent_lines T_0 Z)))) ### All 154
% 5.56/5.73 156. (convergent_lines T_0 T_1) (convergent_lines T_2 T_0) (All Y, ((distinct_lines T_1 Y) => (convergent_lines T_1 Y))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All Y, (All Z, ((convergent_lines T_2 Y) => ((convergent_lines T_2 Z) \/ (convergent_lines Y Z))))) ### All 155
% 5.56/5.73 157. (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All Y, (All Z, ((convergent_lines T_2 Y) => ((convergent_lines T_2 Z) \/ (convergent_lines Y Z))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (-. (distinct_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_2 T_0) (convergent_lines T_0 T_1) ### All 156
% 5.56/5.73 158. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (convergent_lines T_0 T_1) (convergent_lines T_2 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (convergent_lines T_0 T_0)) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) ### All 157
% 5.56/5.74 159. (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)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (-. (distinct_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (-. (convergent_lines T_0 T_0)) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (-. (apart_point_and_line (intersection_point T_1 T_0) T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_2 T_0) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 158
% 5.56/5.74 160. ((convergent_lines T_1 T_0) => (-. (apart_point_and_line (intersection_point T_1 T_0) T_1))) (convergent_lines T_2 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (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)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### Imply 7 159
% 5.56/5.74 161. (All Y, ((convergent_lines T_1 Y) => (-. (apart_point_and_line (intersection_point T_1 Y) T_1)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (convergent_lines T_0 T_1) (-. (convergent_lines T_0 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)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (-. (distinct_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_2 T_0) ### All 160
% 5.56/5.74 162. (convergent_lines T_2 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (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)))))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 161
% 5.56/5.74 163. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (convergent_lines T_0 T_1) (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)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (-. (distinct_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (apart_point_and_line (intersection_point T_2 T_0) T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_2 T_0) ### All 162
% 5.56/5.74 164. (-. (-. (apart_point_and_line (intersection_point T_2 T_0) T_1))) (convergent_lines T_2 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (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)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotNot 163
% 5.56/5.74 165. (-. (incident_point_and_line (intersection_point T_2 T_0) T_1)) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (convergent_lines T_0 T_1) (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)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (-. (distinct_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines T_2 T_1) (-. (apart_point_and_line (intersection_point T_2 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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_2 T_0) ### Definition-Pseudo(incident_point_and_line) 164
% 5.56/5.74 166. (incident_point_and_line (intersection_point T_2 T_1) T_0) (convergent_lines T_2 T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines T_2 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (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)))))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (-. (incident_point_and_line (intersection_point T_2 T_0) T_1)) ### Definition-Pseudo(incident_point_and_line) 165
% 5.56/5.75 167. (-. (((convergent_lines T_2 T_1) /\ ((convergent_lines T_0 T_1) /\ ((convergent_lines T_2 T_0) /\ (incident_point_and_line (intersection_point T_2 T_1) T_0)))) => (incident_point_and_line (intersection_point T_2 T_0) T_1))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (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)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (-. (distinct_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) 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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### ConjTree 166
% 5.56/5.75 168. (-. (All Z, (((convergent_lines T_2 T_1) /\ ((convergent_lines Z T_1) /\ ((convergent_lines T_2 Z) /\ (incident_point_and_line (intersection_point T_2 T_1) Z)))) => (incident_point_and_line (intersection_point T_2 Z) T_1)))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (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, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 167
% 5.56/5.75 169. (-. (All Y, (All Z, (((convergent_lines T_2 Y) /\ ((convergent_lines Z Y) /\ ((convergent_lines T_2 Z) /\ (incident_point_and_line (intersection_point T_2 Y) Z)))) => (incident_point_and_line (intersection_point T_2 Z) Y))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (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)))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (-. (distinct_lines X X))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) 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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### NotAllEx 168
% 5.56/5.75 170. (-. (All X, (All Y, (All Z, (((convergent_lines X Y) /\ ((convergent_lines Z Y) /\ ((convergent_lines X Z) /\ (incident_point_and_line (intersection_point X Y) Z)))) => (incident_point_and_line (intersection_point X Z) Y)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All Z, ((distinct_lines Y Z) => ((apart_point_and_line X Y) \/ ((apart_point_and_line X Z) \/ (convergent_lines Y Z))))))) (All L, (All M, ((convergent_lines L M) \/ (unorthogonal_lines L M)))) (All X, (-. (distinct_lines X X))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, ((((-. (convergent_lines L M)) \/ (-. (unorthogonal_lines L M))) /\ ((-. (convergent_lines L N)) \/ (-. (unorthogonal_lines L N)))) => ((-. (convergent_lines M N)) \/ (-. (unorthogonal_lines M N))))))) (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, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 169
% 5.56/5.75 % SZS output end Proof
% 5.56/5.75 (* END-PROOF *)
%------------------------------------------------------------------------------