TSTP Solution File: GEO205+2 by SuperZenon---0.0.1
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%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : GEO205+2 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% Computer : n008.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 600s
% DateTime : Sat Jul 16 06:31:04 EDT 2022
% Result : Theorem 0.23s 0.53s
% Output : Proof 0.23s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.14 % Problem : GEO205+2 : TPTP v8.1.0. Released v3.3.0.
% 0.12/0.14 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.15/0.36 % Computer : n008.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36 % CPULimit : 300
% 0.15/0.36 % WCLimit : 600
% 0.15/0.36 % DateTime : Sat Jun 18 16:07:37 EDT 2022
% 0.15/0.36 % CPUTime :
% 0.23/0.53 % SZS status Theorem
% 0.23/0.53 (* PROOF-FOUND *)
% 0.23/0.53 (* BEGIN-PROOF *)
% 0.23/0.53 % SZS output start Proof
% 0.23/0.53 1. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 0.23/0.53 2. (-. (convergent_lines T_0 T_2)) (convergent_lines T_0 T_2) ### Axiom
% 0.23/0.53 3. (-. (distinct_lines T_1 T_2)) (distinct_lines T_1 T_2) ### Axiom
% 0.23/0.53 4. (convergent_lines T_1 T_2) (-. (distinct_lines T_1 T_2)) ### Extension/test/ceq3 3
% 0.23/0.53 5. ((convergent_lines T_0 T_1) => ((convergent_lines T_0 T_2) \/ (convergent_lines T_1 T_2))) (-. (distinct_lines T_1 T_2)) (-. (convergent_lines T_0 T_2)) (convergent_lines T_0 T_1) ### DisjTree 1 2 4
% 0.23/0.53 6. (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_2)) (-. (distinct_lines T_1 T_2)) ### All 5
% 0.23/0.53 7. (All Y, (All Z, ((convergent_lines T_0 Y) => ((convergent_lines T_0 Z) \/ (convergent_lines Y Z))))) (-. (distinct_lines T_1 T_2)) (-. (convergent_lines T_0 T_2)) (convergent_lines T_0 T_1) ### All 6
% 0.23/0.53 8. (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_2)) (-. (distinct_lines T_1 T_2)) ### All 7
% 0.23/0.53 9. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 0.23/0.53 10. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2))) ### Axiom
% 0.23/0.53 11. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 0.23/0.53 12. (-. (distinct_lines T_0 T_1)) (convergent_lines T_0 T_1) ### Extension/test/ceq3ctrp 11
% 0.23/0.53 13. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 0.23/0.53 14. (apart_point_and_line (intersection_point T_0 T_1) T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) ### Axiom
% 0.23/0.53 15. (-. ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1))) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### NotOr 14
% 0.23/0.53 16. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)) ### Extension/test/apart1
% 0.23/0.53 17. ((convergent_lines T_0 T_1) => (((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1)) => (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)))) (apart_point_and_line (intersection_point T_0 T_1) T_0) (convergent_lines T_0 T_1) ### DisjTree 13 15 16
% 0.23/0.53 18. (All Z, ((convergent_lines T_0 T_1) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_1)) => (distinct_points Z (intersection_point T_0 T_1))))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### All 17
% 0.23/0.53 19. (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) (apart_point_and_line (intersection_point T_0 T_1) T_0) (convergent_lines T_0 T_1) ### All 18
% 0.23/0.53 20. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 0.23/0.53 21. (apart_point_and_line (intersection_point T_0 T_1) T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) ### Axiom
% 0.23/0.53 22. (-. ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1))) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### NotOr 21
% 0.23/0.53 23. ((convergent_lines T_0 T_1) => (((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1)) => (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)))) (apart_point_and_line (intersection_point T_0 T_1) T_1) (convergent_lines T_0 T_1) ### DisjTree 20 22 16
% 0.23/0.53 24. (All Z, ((convergent_lines T_0 T_1) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_1)) => (distinct_points Z (intersection_point T_0 T_1))))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### All 23
% 0.23/0.53 25. (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) (apart_point_and_line (intersection_point T_0 T_1) T_1) (convergent_lines T_0 T_1) ### All 24
% 0.23/0.53 26. (convergent_lines T_0 T_2) (-. (convergent_lines T_0 T_2)) ### Axiom
% 0.23/0.53 27. (apart_point_and_line (intersection_point T_0 T_2) T_0) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) ### Axiom
% 0.23/0.53 28. (-. ((apart_point_and_line (intersection_point T_0 T_2) T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2))) (apart_point_and_line (intersection_point T_0 T_2) T_0) ### NotOr 27
% 0.23/0.53 29. (distinct_points (intersection_point T_0 T_2) (intersection_point T_0 T_2)) ### Extension/test/apart1
% 0.23/0.53 30. ((convergent_lines T_0 T_2) => (((apart_point_and_line (intersection_point T_0 T_2) T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2)) => (distinct_points (intersection_point T_0 T_2) (intersection_point T_0 T_2)))) (apart_point_and_line (intersection_point T_0 T_2) T_0) (convergent_lines T_0 T_2) ### DisjTree 26 28 29
% 0.23/0.53 31. (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (convergent_lines T_0 T_2) (apart_point_and_line (intersection_point T_0 T_2) T_0) ### All 30
% 0.23/0.53 32. (apart_point_and_line (intersection_point T_0 T_2) T_1) (-. (apart_point_and_line (intersection_point T_0 T_2) T_1)) ### Axiom
% 0.23/0.53 33. (-. (distinct_lines T_1 T_2)) (distinct_lines T_1 T_2) ### Axiom
% 0.23/0.53 34. (convergent_lines T_0 T_2) (-. (convergent_lines T_0 T_2)) ### Axiom
% 0.23/0.53 35. (apart_point_and_line (intersection_point T_0 T_2) T_2) (-. (apart_point_and_line (intersection_point T_0 T_2) T_2)) ### Axiom
% 0.23/0.53 36. (-. ((apart_point_and_line (intersection_point T_0 T_2) T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2))) (apart_point_and_line (intersection_point T_0 T_2) T_2) ### NotOr 35
% 0.23/0.53 37. ((convergent_lines T_0 T_2) => (((apart_point_and_line (intersection_point T_0 T_2) T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2)) => (distinct_points (intersection_point T_0 T_2) (intersection_point T_0 T_2)))) (apart_point_and_line (intersection_point T_0 T_2) T_2) (convergent_lines T_0 T_2) ### DisjTree 34 36 29
% 0.23/0.53 38. (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (convergent_lines T_0 T_2) (apart_point_and_line (intersection_point T_0 T_2) T_2) ### All 37
% 0.23/0.53 39. ((apart_point_and_line (intersection_point T_0 T_2) T_1) => ((distinct_lines T_1 T_2) \/ (apart_point_and_line (intersection_point T_0 T_2) T_2))) (convergent_lines T_0 T_2) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (-. (distinct_lines T_1 T_2)) (apart_point_and_line (intersection_point T_0 T_2) T_1) ### DisjTree 32 33 38
% 0.23/0.53 40. (All Z, ((apart_point_and_line (intersection_point T_0 T_2) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z)))) (apart_point_and_line (intersection_point T_0 T_2) T_1) (-. (distinct_lines T_1 T_2)) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (convergent_lines T_0 T_2) ### All 39
% 0.23/0.53 41. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_2) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z))))) (convergent_lines T_0 T_2) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (-. (distinct_lines T_1 T_2)) (apart_point_and_line (intersection_point T_0 T_2) T_1) ### All 40
% 0.23/0.53 42. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (apart_point_and_line (intersection_point T_0 T_2) T_1) (-. (distinct_lines T_1 T_2)) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (convergent_lines T_0 T_2) ### All 41
% 0.23/0.54 43. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines T_0 T_1)) => ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_2) T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) T_1))))) (-. (distinct_lines T_1 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_2) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### DisjTree 10 12 19 25 31 42
% 0.23/0.54 44. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines T_0 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (convergent_lines T_0 T_1) (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (convergent_lines 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)))))) (-. (distinct_lines T_1 T_2)) ### All 43
% 0.23/0.54 45. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) U) \/ (apart_point_and_line (intersection_point T_0 T_2) V))))))) (-. (distinct_lines T_1 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_2) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### All 44
% 0.23/0.54 46. (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (convergent_lines T_0 T_1) (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (convergent_lines 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)))))) (-. (distinct_lines T_1 T_2)) ### All 45
% 0.23/0.54 47. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (distinct_lines T_1 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_2) (All Z, ((convergent_lines T_0 T_2) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z T_2)) => (distinct_points Z (intersection_point T_0 T_2))))) (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### All 46
% 0.23/0.54 48. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (convergent_lines T_0 T_1) (All Y, (All Z, ((convergent_lines T_0 Y) => (((apart_point_and_line Z T_0) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_0 Y)))))) (convergent_lines 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)))))) (-. (distinct_lines T_1 T_2)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### All 47
% 0.23/0.54 49. (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (distinct_lines T_1 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_2) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### All 48
% 0.23/0.54 50. ((convergent_lines T_0 T_1) => ((convergent_lines T_0 T_2) \/ (convergent_lines T_1 T_2))) (distinct_points (intersection_point T_0 T_1) (intersection_point 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)))))) (-. (distinct_lines T_1 T_2)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_0 T_1) ### DisjTree 9 49 4
% 0.23/0.54 51. (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (distinct_lines T_1 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### All 50
% 0.23/0.54 52. (All Y, (All Z, ((convergent_lines T_0 Y) => ((convergent_lines T_0 Z) \/ (convergent_lines Y Z))))) (distinct_points (intersection_point T_0 T_1) (intersection_point 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)))))) (-. (distinct_lines T_1 T_2)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (convergent_lines T_0 T_1) ### All 51
% 0.23/0.54 53. (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) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (distinct_lines T_1 T_2)) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### All 52
% 0.23/0.54 54. (-. (-. (distinct_points (intersection_point T_0 T_1) (intersection_point 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)))))) (-. (distinct_lines T_1 T_2)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (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 53
% 0.23/0.54 55. (-. ((convergent_lines T_0 T_2) /\ (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_1 T_2)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAnd 8 54
% 0.23/0.54 56. (-. (((convergent_lines T_0 T_1) /\ (-. (distinct_lines T_1 T_2))) => ((convergent_lines T_0 T_2) /\ (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All 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, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) ### ConjTree 55
% 0.23/0.54 57. (-. (All Z, (((convergent_lines T_0 T_1) /\ (-. (distinct_lines T_1 Z))) => ((convergent_lines T_0 Z) /\ (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 Z))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 56
% 0.23/0.54 58. (-. (All Y, (All Z, (((convergent_lines T_0 Y) /\ (-. (distinct_lines Y Z))) => ((convergent_lines T_0 Z) /\ (-. (distinct_points (intersection_point T_0 Y) (intersection_point T_0 Z)))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All 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, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) ### NotAllEx 57
% 0.23/0.54 59. (-. (All X, (All Y, (All Z, (((convergent_lines X Y) /\ (-. (distinct_lines Y Z))) => ((convergent_lines X Z) /\ (-. (distinct_points (intersection_point X Y) (intersection_point X Z))))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => (((apart_point_and_line Z X) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point X Y))))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All 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, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 58
% 0.23/0.54 % SZS output end Proof
% 0.23/0.54 (* END-PROOF *)
%------------------------------------------------------------------------------