TSTP Solution File: GEO196+2 by SuperZenon---0.0.1
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- Process Solution
%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : GEO196+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 : n019.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 06:30:56 EDT 2022
% Result : Theorem 1.39s 1.63s
% Output : Proof 1.52s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GEO196+2 : TPTP v8.1.0. Released v3.3.0.
% 0.03/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.13/0.34 % Computer : n019.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 600
% 0.13/0.34 % DateTime : Sat Jun 18 04:13:54 EDT 2022
% 0.13/0.34 % CPUTime :
% 1.39/1.63 % SZS status Theorem
% 1.39/1.63 (* PROOF-FOUND *)
% 1.39/1.63 (* BEGIN-PROOF *)
% 1.39/1.63 % SZS output start Proof
% 1.39/1.63 1. (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
% 1.39/1.63 2. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3))) ### Axiom
% 1.39/1.63 3. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 1.39/1.63 4. (-. (distinct_lines T_0 T_1)) (convergent_lines T_0 T_1) ### Extension/test/ceq3ctrp 3
% 1.39/1.63 5. (-. (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
% 1.39/1.63 6. (-. (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
% 1.39/1.63 7. (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (apart_point_and_line (intersection_point T_2 T_3) T_0) ### Axiom
% 1.39/1.63 8. (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (apart_point_and_line (intersection_point T_2 T_3) T_1) ### Axiom
% 1.39/1.63 9. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (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_2 T_3) T_0) \/ (apart_point_and_line (intersection_point T_2 T_3) T_1))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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)) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) ### DisjTree 2 4 5 6 7 8
% 1.39/1.63 10. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (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_2 T_3) T_0) \/ (apart_point_and_line (intersection_point T_2 T_3) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (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)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) ### All 9
% 1.39/1.63 11. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (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_3) U) \/ (apart_point_and_line (intersection_point T_2 T_3) V))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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)) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) ### All 10
% 1.39/1.63 12. (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_3)) (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)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) ### All 11
% 1.39/1.63 13. (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_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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)) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) ### All 12
% 1.39/1.63 14. (-. (apart_point_and_line (intersection_point T_2 T_3) T_2)) (apart_point_and_line (intersection_point T_2 T_3) T_2) ### Axiom
% 1.39/1.63 15. ((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_3)) \/ (apart_point_and_line (intersection_point T_2 T_3) T_2))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_2)) (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)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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) ### DisjTree 1 13 14
% 1.39/1.63 16. (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 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_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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)) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_2)) ### All 15
% 1.39/1.63 17. (-. (distinct_lines T_2 T_2)) (distinct_lines T_2 T_2) ### Axiom
% 1.39/1.63 18. (convergent_lines T_2 T_3) (-. (convergent_lines T_2 T_3)) ### Axiom
% 1.39/1.63 19. (apart_point_and_line (intersection_point T_2 T_3) T_2) (-. (apart_point_and_line (intersection_point T_2 T_3) T_2)) ### Axiom
% 1.39/1.63 20. (-. ((apart_point_and_line (intersection_point T_2 T_3) T_2) \/ (apart_point_and_line (intersection_point T_2 T_3) T_3))) (apart_point_and_line (intersection_point T_2 T_3) T_2) ### NotOr 19
% 1.39/1.63 21. (distinct_points (intersection_point T_2 T_3) (intersection_point T_2 T_3)) ### Extension/test/apart1
% 1.39/1.63 22. ((convergent_lines T_2 T_3) => (((apart_point_and_line (intersection_point T_2 T_3) T_2) \/ (apart_point_and_line (intersection_point T_2 T_3) T_3)) => (distinct_points (intersection_point T_2 T_3) (intersection_point T_2 T_3)))) (apart_point_and_line (intersection_point T_2 T_3) T_2) (convergent_lines T_2 T_3) ### DisjTree 18 20 21
% 1.39/1.63 23. (All Z, ((convergent_lines T_2 T_3) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_3)) => (distinct_points Z (intersection_point T_2 T_3))))) (convergent_lines T_2 T_3) (apart_point_and_line (intersection_point T_2 T_3) T_2) ### All 22
% 1.47/1.63 24. ((apart_point_and_line (intersection_point T_2 T_3) T_2) => ((distinct_lines T_2 T_2) \/ (apart_point_and_line (intersection_point T_2 T_3) T_2))) (convergent_lines T_2 T_3) (All Z, ((convergent_lines T_2 T_3) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_3)) => (distinct_points Z (intersection_point T_2 T_3))))) (-. (distinct_lines T_2 T_2)) (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)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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) (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 16 17 23
% 1.47/1.65 25. (All Z, ((apart_point_and_line (intersection_point T_2 T_3) T_2) => ((distinct_lines T_2 Z) \/ (apart_point_and_line (intersection_point T_2 T_3) 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 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_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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)) (convergent_lines T_0 T_1) (-. (distinct_lines T_2 T_2)) (All Z, ((convergent_lines T_2 T_3) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_3)) => (distinct_points Z (intersection_point T_2 T_3))))) (convergent_lines T_2 T_3) ### All 24
% 1.47/1.65 26. (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_3) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_3) Z))))) (convergent_lines T_2 T_3) (All Z, ((convergent_lines T_2 T_3) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_3)) => (distinct_points Z (intersection_point T_2 T_3))))) (-. (distinct_lines T_2 T_2)) (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)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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) (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 25
% 1.47/1.65 27. (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_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 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_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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)) (convergent_lines T_0 T_1) (-. (distinct_lines T_2 T_2)) (All Z, ((convergent_lines T_2 T_3) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_3)) => (distinct_points Z (intersection_point T_2 T_3))))) (convergent_lines T_2 T_3) ### All 26
% 1.47/1.65 28. (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)))))) (convergent_lines T_2 T_3) (-. (distinct_lines T_2 T_2)) (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)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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) (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 Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 27
% 1.47/1.65 29. (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 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 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_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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)) (convergent_lines T_0 T_1) (-. (distinct_lines T_2 T_2)) (convergent_lines T_2 T_3) ### All 28
% 1.47/1.65 30. (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))))) (convergent_lines T_2 T_3) (-. (distinct_lines T_2 T_2)) (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)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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) (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 29
% 1.47/1.65 31. (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, ((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)))))) (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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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)) (convergent_lines T_0 T_1) (-. (distinct_lines T_2 T_2)) (convergent_lines T_2 T_3) ### All 30
% 1.47/1.65 32. (-. (distinct_lines T_0 T_0)) (distinct_lines T_0 T_0) ### Axiom
% 1.47/1.65 33. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 1.47/1.65 34. (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
% 1.47/1.65 35. (-. ((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 34
% 1.47/1.65 36. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_1)) ### Extension/test/apart1
% 1.47/1.65 37. ((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 33 35 36
% 1.47/1.66 38. (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 37
% 1.47/1.66 39. (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 38
% 1.47/1.66 40. (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 39
% 1.47/1.66 41. ((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_2 T_3) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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) (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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### DisjTree 31 32 40
% 1.47/1.66 42. (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 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, ((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)))))) (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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_0 T_1) (-. (distinct_lines T_2 T_2)) (convergent_lines T_2 T_3) (-. (distinct_lines T_0 T_0)) ### All 41
% 1.47/1.66 43. (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_1) ### Axiom
% 1.47/1.66 44. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 1.47/1.66 45. (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
% 1.47/1.66 46. (-. ((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 45
% 1.47/1.66 47. ((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 44 46 36
% 1.47/1.66 48. (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 47
% 1.47/1.66 49. (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 48
% 1.47/1.66 50. (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 49
% 1.47/1.66 51. ((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))) (-. (distinct_lines T_1 T_1)) (-. (distinct_lines T_0 T_0)) (convergent_lines T_2 T_3) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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) (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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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)))) ### DisjTree 42 43 50
% 1.47/1.66 52. (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 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 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, ((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)))))) (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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_0 T_1) (-. (distinct_lines T_2 T_2)) (convergent_lines T_2 T_3) (-. (distinct_lines T_0 T_0)) (-. (distinct_lines T_1 T_1)) ### All 51
% 1.47/1.66 53. (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_1 T_1)) (-. (distinct_lines T_0 T_0)) (convergent_lines T_2 T_3) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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) (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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (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 52
% 1.47/1.67 54. (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, ((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)))))) (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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_0 T_1) (-. (distinct_lines T_2 T_2)) (convergent_lines T_2 T_3) (-. (distinct_lines T_0 T_0)) (-. (distinct_lines T_1 T_1)) (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 53
% 1.47/1.67 55. (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_1 T_1)) (convergent_lines T_2 T_3) (-. (distinct_lines T_2 T_2)) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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) (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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 54
% 1.47/1.67 56. (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, ((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)))))) (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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_0 T_1) (convergent_lines T_2 T_3) (-. (distinct_lines T_1 T_1)) (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 55
% 1.47/1.67 57. (All X, (-. (distinct_lines X X))) (-. (distinct_lines T_1 T_1)) (convergent_lines T_2 T_3) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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) (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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 56
% 1.47/1.67 58. (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, ((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)))))) (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))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_0 T_1) (convergent_lines T_2 T_3) (All X, (-. (distinct_lines X X))) ### All 57
% 1.47/1.67 59. (-. (-. (apart_point_and_line (intersection_point T_0 T_1) T_2))) (All X, (-. (distinct_lines X X))) (convergent_lines T_2 T_3) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### NotNot 58
% 1.47/1.67 60. (apart_point_and_line (intersection_point T_0 T_1) T_3) (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)) ### Axiom
% 1.47/1.67 61. (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (apart_point_and_line (intersection_point T_2 T_3) T_3) ### Axiom
% 1.47/1.67 62. ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) \/ (apart_point_and_line (intersection_point T_2 T_3) T_3))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) (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)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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_3) ### DisjTree 60 13 61
% 1.47/1.67 63. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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)) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) ### All 62
% 1.52/1.68 64. (-. (distinct_lines T_3 T_3)) (distinct_lines T_3 T_3) ### Axiom
% 1.52/1.68 65. (convergent_lines T_2 T_3) (-. (convergent_lines T_2 T_3)) ### Axiom
% 1.52/1.68 66. (apart_point_and_line (intersection_point T_2 T_3) T_3) (-. (apart_point_and_line (intersection_point T_2 T_3) T_3)) ### Axiom
% 1.52/1.68 67. (-. ((apart_point_and_line (intersection_point T_2 T_3) T_2) \/ (apart_point_and_line (intersection_point T_2 T_3) T_3))) (apart_point_and_line (intersection_point T_2 T_3) T_3) ### NotOr 66
% 1.52/1.68 68. ((convergent_lines T_2 T_3) => (((apart_point_and_line (intersection_point T_2 T_3) T_2) \/ (apart_point_and_line (intersection_point T_2 T_3) T_3)) => (distinct_points (intersection_point T_2 T_3) (intersection_point T_2 T_3)))) (apart_point_and_line (intersection_point T_2 T_3) T_3) (convergent_lines T_2 T_3) ### DisjTree 65 67 21
% 1.52/1.68 69. (All Z, ((convergent_lines T_2 T_3) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_3)) => (distinct_points Z (intersection_point T_2 T_3))))) (convergent_lines T_2 T_3) (apart_point_and_line (intersection_point T_2 T_3) T_3) ### All 68
% 1.52/1.68 70. ((apart_point_and_line (intersection_point T_2 T_3) T_3) => ((distinct_lines T_3 T_3) \/ (apart_point_and_line (intersection_point T_2 T_3) T_3))) (convergent_lines T_2 T_3) (All Z, ((convergent_lines T_2 T_3) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_3)) => (distinct_points Z (intersection_point T_2 T_3))))) (-. (distinct_lines T_3 T_3)) (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)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) ### DisjTree 63 64 69
% 1.52/1.68 71. (All Z, ((apart_point_and_line (intersection_point T_2 T_3) T_3) => ((distinct_lines T_3 Z) \/ (apart_point_and_line (intersection_point T_2 T_3) Z)))) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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)) (convergent_lines T_0 T_1) (-. (distinct_lines T_3 T_3)) (All Z, ((convergent_lines T_2 T_3) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_3)) => (distinct_points Z (intersection_point T_2 T_3))))) (convergent_lines T_2 T_3) ### All 70
% 1.52/1.68 72. (All Y, (All Z, ((apart_point_and_line (intersection_point T_2 T_3) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_2 T_3) Z))))) (convergent_lines T_2 T_3) (All Z, ((convergent_lines T_2 T_3) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_3)) => (distinct_points Z (intersection_point T_2 T_3))))) (-. (distinct_lines T_3 T_3)) (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)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) ### All 71
% 1.52/1.68 73. (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_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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)) (convergent_lines T_0 T_1) (-. (distinct_lines T_3 T_3)) (All Z, ((convergent_lines T_2 T_3) => (((apart_point_and_line Z T_2) \/ (apart_point_and_line Z T_3)) => (distinct_points Z (intersection_point T_2 T_3))))) (convergent_lines T_2 T_3) ### All 72
% 1.52/1.68 74. (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)))))) (convergent_lines T_2 T_3) (-. (distinct_lines T_3 T_3)) (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)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 73
% 1.52/1.68 75. (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 Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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)) (convergent_lines T_0 T_1) (-. (distinct_lines T_3 T_3)) (convergent_lines T_2 T_3) ### All 74
% 1.52/1.68 76. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3))) ### Axiom
% 1.52/1.68 77. (-. (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
% 1.52/1.68 78. (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (apart_point_and_line (intersection_point T_2 T_3) T_0) ### Axiom
% 1.52/1.68 79. (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (apart_point_and_line (intersection_point T_2 T_3) T_1) ### Axiom
% 1.52/1.68 80. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (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_2 T_3) T_0) \/ (apart_point_and_line (intersection_point T_2 T_3) T_1))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_0 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_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) ### DisjTree 76 4 40 77 78 79
% 1.52/1.69 81. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (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_2 T_3) T_0) \/ (apart_point_and_line (intersection_point T_2 T_3) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) (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))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) ### All 80
% 1.52/1.69 82. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) /\ (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_3) U) \/ (apart_point_and_line (intersection_point T_2 T_3) V))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_0 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_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) ### All 81
% 1.52/1.69 83. (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_3)) (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))))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) ### All 82
% 1.52/1.69 84. (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_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_0 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_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) ### All 83
% 1.52/1.69 85. (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (apart_point_and_line (intersection_point T_2 T_3) T_0) ### Axiom
% 1.52/1.69 86. ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_points (intersection_point T_0 T_1) (intersection_point T_2 T_3)) \/ (apart_point_and_line (intersection_point T_2 T_3) T_0))) (convergent_lines T_2 T_3) (-. (distinct_lines T_3 T_3)) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (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))))))) ### DisjTree 75 84 85
% 1.52/1.69 87. (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_0) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z 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))))))) (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_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_0 T_1) (-. (distinct_lines T_3 T_3)) (convergent_lines T_2 T_3) ### All 86
% 1.52/1.69 88. (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))))) (convergent_lines T_2 T_3) (-. (distinct_lines T_3 T_3)) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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_3) (All Z, ((apart_point_and_line (intersection_point T_0 T_1) T_3) => ((distinct_points (intersection_point T_0 T_1) Z) \/ (apart_point_and_line Z T_3)))) (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 87
% 1.52/1.69 89. (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)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (convergent_lines T_0 T_1) (-. (distinct_lines T_3 T_3)) (convergent_lines T_2 T_3) (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 88
% 1.52/1.70 90. (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_3) (-. (distinct_lines T_3 T_3)) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_0 T_1) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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_3) (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 89
% 1.52/1.70 91. (-. (distinct_lines T_1 T_1)) (distinct_lines T_1 T_1) ### Axiom
% 1.52/1.70 92. ((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))) (-. (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))))))) (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_1) T_3) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_0 T_1) (-. (distinct_lines T_3 T_3)) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### DisjTree 90 91 50
% 1.52/1.70 93. (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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) (convergent_lines T_2 T_3) (-. (distinct_lines T_3 T_3)) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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_3) (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))))))) (-. (distinct_lines T_1 T_1)) ### All 92
% 1.52/1.70 94. (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_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))))))) (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_1) T_3) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_0 T_1) (-. (distinct_lines T_3 T_3)) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 93
% 1.52/1.70 95. (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_3) (-. (distinct_lines T_3 T_3)) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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_3) (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))))))) (-. (distinct_lines T_1 T_1)) ### All 94
% 1.52/1.70 96. (All X, (-. (distinct_lines X X))) (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)))))) (apart_point_and_line (intersection_point T_0 T_1) T_3) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line (intersection_point T_2 T_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_0 T_1) (-. (distinct_lines T_3 T_3)) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### All 95
% 1.52/1.70 97. (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_3) (convergent_lines T_0 T_1) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (-. (apart_point_and_line (intersection_point T_2 T_3) 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_3) (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, (-. (distinct_lines X X))) ### All 96
% 1.52/1.70 98. (-. (-. (apart_point_and_line (intersection_point T_0 T_1) T_3))) (All X, (-. (distinct_lines X X))) (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, (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_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_0 T_1) (convergent_lines T_2 T_3) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### NotNot 97
% 1.52/1.70 99. (-. ((-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) /\ (-. (apart_point_and_line (intersection_point T_0 T_1) T_3)))) (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, ((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, (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_3) T_1)) (-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) (convergent_lines T_0 T_1) (convergent_lines T_2 T_3) (All X, (-. (distinct_lines X X))) ### NotAnd 59 98
% 1.52/1.70 100. (-. (((convergent_lines T_2 T_3) /\ ((convergent_lines T_0 T_1) /\ ((-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) /\ (-. (apart_point_and_line (intersection_point T_2 T_3) T_1))))) => ((-. (apart_point_and_line (intersection_point T_0 T_1) T_2)) /\ (-. (apart_point_and_line (intersection_point T_0 T_1) T_3))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### ConjTree 99
% 1.52/1.70 101. (-. (All V, (((convergent_lines T_2 T_3) /\ ((convergent_lines T_0 V) /\ ((-. (apart_point_and_line (intersection_point T_2 T_3) T_0)) /\ (-. (apart_point_and_line (intersection_point T_2 T_3) V))))) => ((-. (apart_point_and_line (intersection_point T_0 V) T_2)) /\ (-. (apart_point_and_line (intersection_point T_0 V) T_3)))))) (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, ((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, (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, (-. (distinct_lines X X))) ### NotAllEx 100
% 1.52/1.70 102. (-. (All U, (All V, (((convergent_lines T_2 T_3) /\ ((convergent_lines U V) /\ ((-. (apart_point_and_line (intersection_point T_2 T_3) U)) /\ (-. (apart_point_and_line (intersection_point T_2 T_3) V))))) => ((-. (apart_point_and_line (intersection_point U V) T_2)) /\ (-. (apart_point_and_line (intersection_point U V) T_3))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### NotAllEx 101
% 1.52/1.70 103. (-. (All Y, (All U, (All V, (((convergent_lines T_2 Y) /\ ((convergent_lines U V) /\ ((-. (apart_point_and_line (intersection_point T_2 Y) U)) /\ (-. (apart_point_and_line (intersection_point T_2 Y) V))))) => ((-. (apart_point_and_line (intersection_point U V) T_2)) /\ (-. (apart_point_and_line (intersection_point U V) Y)))))))) (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, ((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, (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, (-. (distinct_lines X X))) ### NotAllEx 102
% 1.52/1.70 104. (-. (All X, (All Y, (All U, (All V, (((convergent_lines X Y) /\ ((convergent_lines U V) /\ ((-. (apart_point_and_line (intersection_point X Y) U)) /\ (-. (apart_point_and_line (intersection_point X Y) V))))) => ((-. (apart_point_and_line (intersection_point U V) X)) /\ (-. (apart_point_and_line (intersection_point U V) Y))))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((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 Z, ((apart_point_and_line X Y) => ((distinct_points X Z) \/ (apart_point_and_line Z Y)))))) ### NotAllEx 103
% 1.52/1.70 % SZS output end Proof
% 1.52/1.70 (* END-PROOF *)
%------------------------------------------------------------------------------