TSTP Solution File: GEO202+1 by SuperZenon---0.0.1
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%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : GEO202+1 : 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 : n024.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 600s
% DateTime : Sat Jul 16 06:31:01 EDT 2022
% Result : Theorem 0.19s 0.48s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : GEO202+1 : TPTP v8.1.0. Released v3.3.0.
% 0.04/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.13/0.34 % Computer : n024.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 : Fri Jun 17 21:06:02 EDT 2022
% 0.13/0.34 % CPUTime :
% 0.19/0.48 % SZS status Theorem
% 0.19/0.48 (* PROOF-FOUND *)
% 0.19/0.48 (* BEGIN-PROOF *)
% 0.19/0.48 % SZS output start Proof
% 0.19/0.48 1. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (-. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2))) ### Axiom
% 0.19/0.48 2. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 0.19/0.48 3. (distinct_points T_0 T_2) (-. (distinct_points T_0 T_2)) ### Axiom
% 0.19/0.48 4. (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (-. (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0)) ### Axiom
% 0.19/0.48 5. (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (-. (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1))) ### Axiom
% 0.19/0.48 6. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (-. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2))) ### Axiom
% 0.19/0.48 7. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_2)) (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_2))) ### Axiom
% 0.19/0.48 8. ((convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_2)))) (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_2)) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) ### Imply 6 7
% 0.19/0.48 9. (All Y, ((convergent_lines (line_connecting T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) Y)))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_2)) ### All 8
% 0.19/0.48 10. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_2)) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) ### All 9
% 0.19/0.48 11. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (-. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2))) ### Axiom
% 0.19/0.48 12. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_1)) (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_1))) ### Axiom
% 0.19/0.48 13. ((convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_1)))) (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_1)) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) ### Imply 11 12
% 0.19/0.48 14. (All Y, ((convergent_lines (line_connecting T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) (line_connecting T_0 T_1))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_1)) ### All 13
% 0.19/0.48 15. (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (apart_point_and_line T_0 (line_connecting T_0 T_2)) ### Axiom
% 0.19/0.48 16. (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (apart_point_and_line T_0 (line_connecting T_0 T_1)) ### Axiom
% 0.19/0.48 17. (((distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) /\ (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1))) => ((apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_2)) \/ ((apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 (line_connecting T_0 T_2)) \/ (apart_point_and_line T_0 (line_connecting T_0 T_1)))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (All Y, ((convergent_lines (line_connecting T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) (line_connecting T_0 T_1))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) ### DisjTree 4 5 10 14 15 16
% 0.19/0.48 18. (All V, (((distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) /\ (distinct_lines (line_connecting T_0 T_2) V)) => ((apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (line_connecting T_0 T_2)) \/ ((apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) V) \/ ((apart_point_and_line T_0 (line_connecting T_0 T_2)) \/ (apart_point_and_line T_0 V)))))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All Y, ((convergent_lines (line_connecting T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) (line_connecting T_0 T_1))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) ### All 17
% 0.19/0.48 19. (All U, (All V, (((distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) U) \/ ((apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (All Y, ((convergent_lines (line_connecting T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) (line_connecting T_0 T_1))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) ### All 18
% 0.19/0.48 20. (All Y, (All U, (All V, (((distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) Y) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) U) \/ ((apart_point_and_line (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All Y, ((convergent_lines (line_connecting T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) (line_connecting T_0 T_1))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) ### All 19
% 0.19/0.48 21. (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 T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_2))) (All Y, ((convergent_lines (line_connecting T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) (line_connecting T_0 T_1))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) ### All 20
% 0.19/0.49 22. ((distinct_points T_0 T_2) => (-. (apart_point_and_line T_0 (line_connecting T_0 T_2)))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All Y, ((convergent_lines (line_connecting T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) (line_connecting T_0 T_1))))) (-. (apart_point_and_line T_0 (line_connecting 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))))))))) (distinct_points T_0 T_2) ### Imply 3 21
% 0.19/0.49 23. (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_2) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, ((convergent_lines (line_connecting T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point (line_connecting T_0 T_1) Y) (line_connecting T_0 T_1))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) ### All 22
% 0.19/0.49 24. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (-. (apart_point_and_line T_0 (line_connecting 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))))))))) (distinct_points T_0 T_2) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) ### All 23
% 0.19/0.49 25. ((distinct_points T_0 T_1) => (-. (apart_point_and_line T_0 (line_connecting T_0 T_1)))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_2) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points T_0 T_1) ### Imply 2 24
% 0.19/0.49 26. (distinct_points T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (distinct_points T_0 T_2) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) ### All 25
% 0.19/0.49 27. (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_points T_0 T_2) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points T_0 T_1) ### All 26
% 0.19/0.49 28. (-. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_1))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_1)) ### Axiom
% 0.19/0.49 29. ((convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) (line_connecting T_0 T_1)) \/ (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_1)))) (-. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (distinct_points T_0 T_2) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) ### DisjTree 1 27 28
% 0.19/0.49 30. (All Z, ((convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (convergent_lines (line_connecting T_0 T_1) Z)))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_points T_0 T_2) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points T_0 T_1) (-. (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_1))) ### All 29
% 0.19/0.49 31. (All X, (-. (convergent_lines X X))) (distinct_points T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (distinct_points T_0 T_2) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All Z, ((convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) => ((distinct_lines (line_connecting T_0 T_2) Z) \/ (convergent_lines (line_connecting T_0 T_1) Z)))) ### All 30
% 0.19/0.49 32. (All Y, (All Z, ((convergent_lines (line_connecting T_0 T_1) Y) => ((distinct_lines Y Z) \/ (convergent_lines (line_connecting T_0 T_1) Z))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_points T_0 T_2) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points T_0 T_1) (All X, (-. (convergent_lines X X))) ### All 31
% 0.19/0.49 33. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (convergent_lines X X))) (distinct_points T_0 T_1) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (distinct_points T_0 T_2) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)) ### All 32
% 0.19/0.49 34. (-. (((distinct_points T_0 T_1) /\ ((distinct_points T_0 T_2) /\ (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 T_2)))) => (-. (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 T_2)) T_0)))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting 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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### ConjTree 33
% 0.19/0.49 35. (-. (All Z, (((distinct_points T_0 T_1) /\ ((distinct_points T_0 Z) /\ (convergent_lines (line_connecting T_0 T_1) (line_connecting T_0 Z)))) => (-. (distinct_points (intersection_point (line_connecting T_0 T_1) (line_connecting T_0 Z)) T_0))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, (All 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, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) ### NotAllEx 34
% 0.19/0.49 36. (-. (All Y, (All Z, (((distinct_points T_0 Y) /\ ((distinct_points T_0 Z) /\ (convergent_lines (line_connecting T_0 Y) (line_connecting T_0 Z)))) => (-. (distinct_points (intersection_point (line_connecting T_0 Y) (line_connecting T_0 Z)) T_0)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting 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, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotAllEx 35
% 0.19/0.49 37. (-. (All X, (All Y, (All Z, (((distinct_points X Y) /\ ((distinct_points X Z) /\ (convergent_lines (line_connecting X Y) (line_connecting X Z)))) => (-. (distinct_points (intersection_point (line_connecting X Y) (line_connecting X Z)) X))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All X, (-. (convergent_lines X X))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) Y))))) (All X, (All Y, (All 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, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) ### NotAllEx 36
% 0.19/0.49 % SZS output end Proof
% 0.19/0.49 (* END-PROOF *)
%------------------------------------------------------------------------------