TSTP Solution File: GEO197+2 by SuperZenon---0.0.1
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : GEO197+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 : n029.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 06:30:57 EDT 2022
% Result : Theorem 0.77s 0.94s
% Output : Proof 0.77s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : GEO197+2 : TPTP v8.1.0. Released v3.3.0.
% 0.11/0.12 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.12/0.33 % Computer : n029.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 600
% 0.12/0.33 % DateTime : Sat Jun 18 10:08:29 EDT 2022
% 0.12/0.33 % CPUTime :
% 0.77/0.94 % SZS status Theorem
% 0.77/0.94 (* PROOF-FOUND *)
% 0.77/0.94 (* BEGIN-PROOF *)
% 0.77/0.94 % SZS output start Proof
% 0.77/0.94 1. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 0.77/0.94 2. (-. (distinct_lines T_0 T_0)) (distinct_lines T_0 T_0) ### Axiom
% 0.77/0.94 3. (convergent_lines T_0 T_0) (-. (distinct_lines T_0 T_0)) ### Extension/test/ceq3 2
% 0.77/0.94 4. (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) ### Axiom
% 0.77/0.94 5. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1))) ### Axiom
% 0.77/0.94 6. (convergent_lines T_1 T_0) (-. (convergent_lines T_1 T_0)) ### Axiom
% 0.77/0.94 7. (-. (distinct_lines T_1 T_0)) (convergent_lines T_1 T_0) ### Extension/test/ceq3ctrp 6
% 0.77/0.94 8. (-. (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
% 0.77/0.94 9. (-. (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
% 0.77/0.94 10. (-. (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
% 0.77/0.94 11. (-. (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
% 0.77/0.94 12. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines T_1 T_0)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_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)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) ### DisjTree 5 7 8 9 10 11
% 0.77/0.94 13. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (distinct_lines T_1 V)) => ((apart_point_and_line (intersection_point T_0 T_1) T_1) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_2 T_1) T_1) \/ (apart_point_and_line (intersection_point T_2 T_1) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) ### All 12
% 0.77/0.94 14. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) /\ (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_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)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_1)) ### All 13
% 0.77/0.94 15. (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_2 T_1)) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) ### All 14
% 0.77/0.94 16. (-. (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
% 0.77/0.94 17. ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) (intersection_point 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)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (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)))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) ### DisjTree 4 15 16
% 0.77/0.94 18. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_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_2)) ### All 17
% 0.77/0.94 19. (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_1) ### Axiom
% 0.77/0.94 20. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom
% 0.77/0.94 21. (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
% 0.77/0.94 22. (-. ((apart_point_and_line (intersection_point T_2 T_1) T_2) \/ (apart_point_and_line (intersection_point T_2 T_1) T_1))) (apart_point_and_line (intersection_point T_2 T_1) T_1) ### NotOr 21
% 0.77/0.94 23. (distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_1)) ### Extension/test/apart1
% 0.77/0.94 24. ((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_1)) => (distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_1)))) (apart_point_and_line (intersection_point T_2 T_1) T_1) (convergent_lines T_2 T_1) ### DisjTree 20 22 23
% 0.77/0.94 25. (All Z, ((convergent_lines T_2 T_1) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_1)) => (distinct_points Z (intersection_point T_2 T_1))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_2 T_1) T_1) ### All 24
% 0.77/0.94 26. (All Y, (All Z, ((convergent_lines T_2 Y) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_2 Y)))))) (apart_point_and_line (intersection_point T_2 T_1) T_1) (convergent_lines T_2 T_1) ### All 25
% 0.77/0.94 27. (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_2 T_1) (apart_point_and_line (intersection_point T_2 T_1) T_1) ### All 26
% 0.77/0.94 28. ((apart_point_and_line (intersection_point T_2 T_1) T_1) => ((distinct_lines T_1 T_1) \/ (apart_point_and_line (intersection_point T_2 T_1) T_1))) (convergent_lines T_2 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))))))) (-. (distinct_lines T_1 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_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (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)))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) ### DisjTree 18 19 27
% 0.77/0.95 29. (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z)))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_2)) (-. (distinct_lines T_1 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))))))) (convergent_lines T_2 T_1) ### All 28
% 0.77/0.95 30. (-. (distinct_lines T_2 T_2)) (distinct_lines T_2 T_2) ### Axiom
% 0.77/0.95 31. (convergent_lines T_2 T_1) (-. (convergent_lines T_2 T_1)) ### Axiom
% 0.77/0.95 32. (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
% 0.77/0.95 33. (-. ((apart_point_and_line (intersection_point T_2 T_1) T_2) \/ (apart_point_and_line (intersection_point T_2 T_1) T_1))) (apart_point_and_line (intersection_point T_2 T_1) T_2) ### NotOr 32
% 0.77/0.95 34. ((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_1)) => (distinct_points (intersection_point T_2 T_1) (intersection_point T_2 T_1)))) (apart_point_and_line (intersection_point T_2 T_1) T_2) (convergent_lines T_2 T_1) ### DisjTree 31 33 23
% 0.77/0.95 35. (All Z, ((convergent_lines T_2 T_1) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_1)) => (distinct_points Z (intersection_point T_2 T_1))))) (convergent_lines T_2 T_1) (apart_point_and_line (intersection_point T_2 T_1) T_2) ### All 34
% 0.77/0.95 36. (All Y, (All Z, ((convergent_lines T_2 Y) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z Y)) => (distinct_points Z (intersection_point T_2 Y)))))) (apart_point_and_line (intersection_point T_2 T_1) T_2) (convergent_lines T_2 T_1) ### All 35
% 0.77/0.95 37. (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_2 T_1) (apart_point_and_line (intersection_point T_2 T_1) T_2) ### All 36
% 0.77/0.95 38. ((apart_point_and_line (intersection_point T_2 T_1) T_2) => ((distinct_lines T_2 T_2) \/ (apart_point_and_line (intersection_point T_2 T_1) T_2))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_2 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))))))) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (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)))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z)))) ### DisjTree 29 30 37
% 0.77/0.95 39. (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z)))) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z)))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (distinct_lines T_1 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))))))) (convergent_lines T_2 T_1) (-. (distinct_lines T_2 T_2)) ### All 38
% 0.77/0.95 40. (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_2 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))))))) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (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)))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (All Z, ((apart_point_and_line (intersection_point T_2 T_1) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z)))) ### All 39
% 0.77/0.95 41. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Y, (All U, (All V, (((distinct_points (intersection_point T_0 T_1) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (distinct_lines T_1 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))))))) (convergent_lines T_2 T_1) (-. (distinct_lines T_2 T_2)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_1) Z))))) ### All 40
% 0.77/0.95 42. (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_2 T_2)) (convergent_lines T_2 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))))))) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_1 T_0) (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)))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) ### All 41
% 0.77/0.95 43. (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 Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (distinct_lines T_1 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))))))) (convergent_lines T_2 T_1) (-. (distinct_lines T_2 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 42
% 0.77/0.95 44. (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_1) ### Axiom
% 0.77/0.95 45. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 0.77/0.95 46. (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
% 0.77/0.95 47. (-. ((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 46
% 0.77/0.95 48. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)) ### Extension/test/apart1
% 0.77/0.95 49. ((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 45 47 48
% 0.77/0.95 50. (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 49
% 0.77/0.95 51. (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 50
% 0.77/0.95 52. (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) (apart_point_and_line (intersection_point T_0 T_1) T_1) ### All 51
% 0.77/0.95 53. ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 T_1) \/ (apart_point_and_line (intersection_point T_0 T_1) T_1))) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_2 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))))))) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z 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))))))))) ### DisjTree 43 44 52
% 0.77/0.95 54. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) 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 Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (distinct_lines T_1 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))))))) (convergent_lines T_2 T_1) (-. (distinct_lines T_2 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_1) ### All 53
% 0.77/0.95 55. (-. (distinct_lines T_0 T_0)) (distinct_lines T_0 T_0) ### Axiom
% 0.77/0.95 56. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 0.77/0.95 57. (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
% 0.77/0.95 58. (-. ((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 57
% 0.77/0.95 59. ((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 56 58 48
% 0.77/0.95 60. (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 59
% 0.77/0.95 61. (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 60
% 0.77/0.95 62. (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) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### All 61
% 0.77/0.95 63. ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_lines T_0 T_0) \/ (apart_point_and_line (intersection_point T_0 T_1) T_0))) (-. (distinct_lines T_0 T_0)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_2 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))))))) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z 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 Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) ### DisjTree 54 55 62
% 0.77/0.96 64. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_1) => ((distinct_lines T_1 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) 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 Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (distinct_lines T_1 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))))))) (convergent_lines T_2 T_1) (-. (distinct_lines T_2 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_1) (-. (distinct_lines T_0 T_0)) ### All 63
% 0.77/0.96 65. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (distinct_lines T_0 T_0)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (distinct_lines T_2 T_2)) (convergent_lines T_2 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))))))) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z 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 Z, ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z)))) ### All 64
% 0.77/0.96 66. (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 Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_2)))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (distinct_lines T_1 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))))))) (convergent_lines T_2 T_1) (-. (distinct_lines T_2 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_1) (-. (distinct_lines T_0 T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) ### All 65
% 0.77/0.96 67. (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (distinct_lines T_0 T_0)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_2 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))))))) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 T_1) T_2) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_2) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z 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 66
% 0.77/0.96 68. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z 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))))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (distinct_lines T_1 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))))))) (convergent_lines T_2 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (-. (distinct_lines T_0 T_0)) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (All X, (-. (distinct_lines X X))) ### All 67
% 0.77/0.96 69. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_1) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_1) Z))))) (-. (distinct_lines T_0 T_0)) (convergent_lines T_0 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_2 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))))))) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (convergent_lines T_1 T_0) (apart_point_and_line (intersection_point T_0 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 68
% 0.77/0.96 70. (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_0 T_1) T_2) (convergent_lines T_1 T_0) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (distinct_lines T_1 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))))))) (convergent_lines T_2 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (convergent_lines T_0 T_1) (-. (distinct_lines T_0 T_0)) (All X, (-. (distinct_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 69
% 0.77/0.96 71. ((convergent_lines T_0 T_1) => ((convergent_lines T_0 T_0) \/ (convergent_lines T_1 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, (-. (distinct_lines X X))) (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_2 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))))))) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (apart_point_and_line (intersection_point T_0 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))))))))) (-. (distinct_lines T_0 T_0)) (convergent_lines T_0 T_1) ### DisjTree 1 3 70
% 0.77/0.96 72. (All Z, ((convergent_lines T_0 T_1) => ((convergent_lines T_0 Z) \/ (convergent_lines T_1 Z)))) (convergent_lines T_0 T_1) (-. (distinct_lines T_0 T_0)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (distinct_lines T_1 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))))))) (convergent_lines T_2 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (distinct_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 71
% 0.77/0.96 73. (All Y, (All Z, ((convergent_lines T_0 Y) => ((convergent_lines T_0 Z) \/ (convergent_lines Y 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, (-. (distinct_lines X X))) (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_2 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))))))) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (apart_point_and_line (intersection_point T_0 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))))))))) (-. (distinct_lines T_0 T_0)) (convergent_lines T_0 T_1) ### All 72
% 0.77/0.96 74. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (convergent_lines T_0 T_1) (-. (distinct_lines T_0 T_0)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line (intersection_point T_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (-. (distinct_lines T_1 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))))))) (convergent_lines T_2 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (distinct_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 73
% 0.77/0.96 75. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (All X, (-. (distinct_lines X X))) (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_2 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))))))) (-. (distinct_lines T_1 T_1)) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (apart_point_and_line (intersection_point T_0 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))))))))) (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 74
% 0.77/0.96 76. (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 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_0 T_1) T_2) (-. (apart_point_and_line (intersection_point T_2 T_1) T_0)) (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_2 T_1) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (distinct_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 75
% 0.77/0.96 77. (-. (((convergent_lines T_2 T_1) /\ ((convergent_lines T_0 T_1) /\ ((convergent_lines T_2 T_0) /\ (-. (apart_point_and_line (intersection_point T_2 T_1) T_0))))) => (-. (apart_point_and_line (intersection_point T_0 T_1) 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, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (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, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### ConjTree 76
% 0.77/0.96 78. (-. (All Z, (((convergent_lines T_2 T_1) /\ ((convergent_lines Z T_1) /\ ((convergent_lines T_2 Z) /\ (-. (apart_point_and_line (intersection_point T_2 T_1) Z))))) => (-. (apart_point_and_line (intersection_point Z T_1) T_2))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (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 Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (distinct_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)))))) ### NotAllEx 77
% 0.77/0.96 79. (-. (All Y, (All Z, (((convergent_lines T_2 Y) /\ ((convergent_lines Z Y) /\ ((convergent_lines T_2 Z) /\ (-. (apart_point_and_line (intersection_point T_2 Y) Z))))) => (-. (apart_point_and_line (intersection_point Z Y) 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, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (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, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 78
% 0.77/0.96 80. (-. (All X, (All Y, (All Z, (((convergent_lines X Y) /\ ((convergent_lines Z Y) /\ ((convergent_lines X Z) /\ (-. (apart_point_and_line (intersection_point X Y) Z))))) => (-. (apart_point_and_line (intersection_point Z Y) X))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (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 Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (-. (distinct_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)))))) ### NotAllEx 79
% 0.77/0.96 % SZS output end Proof
% 0.77/0.96 (* END-PROOF *)
%------------------------------------------------------------------------------