TSTP Solution File: GEO203+1 by SuperZenon---0.0.1
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%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : GEO203+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 : n007.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 06:31:02 EDT 2022
% Result : Theorem 0.20s 0.50s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.13 % Problem : GEO203+1 : TPTP v8.1.0. Released v3.3.0.
% 0.08/0.14 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.14/0.35 % Computer : n007.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 600
% 0.14/0.35 % DateTime : Sat Jun 18 05:54:58 EDT 2022
% 0.14/0.35 % CPUTime :
% 0.20/0.50 % SZS status Theorem
% 0.20/0.50 (* PROOF-FOUND *)
% 0.20/0.50 (* BEGIN-PROOF *)
% 0.20/0.50 % SZS output start Proof
% 0.20/0.50 1. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2))) ### Axiom
% 0.20/0.50 2. (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (-. (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0)) ### Axiom
% 0.20/0.50 3. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (-. (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2))) ### Axiom
% 0.20/0.50 4. (apart_point_and_line (intersection_point T_0 T_1) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2))) (-. (apart_point_and_line (intersection_point T_0 T_1) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) ### Axiom
% 0.20/0.50 5. ((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) => (-. (apart_point_and_line (intersection_point T_0 T_1) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2))))) (apart_point_and_line (intersection_point T_0 T_1) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### Imply 3 4
% 0.20/0.50 6. (All Y, ((distinct_points (intersection_point T_0 T_1) Y) => (-. (apart_point_and_line (intersection_point T_0 T_1) (line_connecting (intersection_point T_0 T_1) Y))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (apart_point_and_line (intersection_point T_0 T_1) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2))) ### All 5
% 0.20/0.50 7. (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (apart_point_and_line (intersection_point T_0 T_1) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### All 6
% 0.20/0.50 8. (-. (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
% 0.20/0.50 9. (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2))) ### Axiom
% 0.20/0.50 10. (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) (apart_point_and_line (intersection_point T_0 T_2) T_0) ### Axiom
% 0.20/0.50 11. (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0)) => ((apart_point_and_line (intersection_point T_0 T_1) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2))) \/ ((apart_point_and_line (intersection_point T_0 T_1) T_0) \/ ((apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2))) \/ (apart_point_and_line (intersection_point T_0 T_2) T_0))))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### DisjTree 1 2 7 8 9 10
% 0.20/0.50 12. (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) V)) => ((apart_point_and_line (intersection_point T_0 T_1) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2))) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2))) \/ (apart_point_and_line (intersection_point T_0 T_2) V)))))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) ### All 11
% 0.20/0.50 13. (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) U) \/ (apart_point_and_line (intersection_point T_0 T_2) V))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (-. (apart_point_and_line (intersection_point T_0 T_1) T_0)) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) ### All 12
% 0.20/0.50 14. (distinct_lines T_0 T_0) ### Extension/test/apart2
% 0.20/0.50 15. (convergent_lines T_0 T_1) (-. (convergent_lines T_0 T_1)) ### Axiom
% 0.20/0.50 16. (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
% 0.20/0.50 17. ((convergent_lines T_0 T_1) => (-. (apart_point_and_line (intersection_point T_0 T_1) T_0))) (apart_point_and_line (intersection_point T_0 T_1) T_0) (convergent_lines T_0 T_1) ### Imply 15 16
% 0.20/0.50 18. (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_1) (apart_point_and_line (intersection_point T_0 T_1) T_0) ### All 17
% 0.20/0.50 19. ((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))) (convergent_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) U) \/ (apart_point_and_line (intersection_point T_0 T_2) V))))))) ### DisjTree 13 14 18
% 0.20/0.50 20. (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 U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) U) \/ (apart_point_and_line (intersection_point T_0 T_2) V))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_1) ### All 19
% 0.20/0.52 21. (convergent_lines T_0 T_2) (-. (convergent_lines T_0 T_2)) ### Axiom
% 0.20/0.52 22. (apart_point_and_line (intersection_point T_0 T_2) T_0) (-. (apart_point_and_line (intersection_point T_0 T_2) T_0)) ### Axiom
% 0.20/0.52 23. ((convergent_lines T_0 T_2) => (-. (apart_point_and_line (intersection_point T_0 T_2) T_0))) (apart_point_and_line (intersection_point T_0 T_2) T_0) (convergent_lines T_0 T_2) ### Imply 21 22
% 0.20/0.52 24. (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_2) (apart_point_and_line (intersection_point T_0 T_2) T_0) ### All 23
% 0.20/0.52 25. ((apart_point_and_line (intersection_point T_0 T_2) T_0) => ((distinct_lines T_0 T_0) \/ (apart_point_and_line (intersection_point T_0 T_2) T_0))) (convergent_lines T_0 T_2) (convergent_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) U) \/ (apart_point_and_line (intersection_point T_0 T_2) V))))))) (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 20 14 24
% 0.20/0.52 26. (All Z, ((apart_point_and_line (intersection_point T_0 T_2) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_2) 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 U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) U) \/ (apart_point_and_line (intersection_point T_0 T_2) V))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_1) (convergent_lines T_0 T_2) ### All 25
% 0.20/0.52 27. (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))))) (convergent_lines T_0 T_2) (convergent_lines T_0 T_1) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) U) \/ (apart_point_and_line (intersection_point T_0 T_2) V))))))) (All Z, ((apart_point_and_line (intersection_point T_0 T_2) T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z)))) ### All 26
% 0.20/0.52 28. (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_2) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z))))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) U) \/ (apart_point_and_line (intersection_point T_0 T_2) V))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (All Y, ((convergent_lines T_0 Y) => (-. (apart_point_and_line (intersection_point T_0 Y) T_0)))) (convergent_lines T_0 T_1) (convergent_lines T_0 T_2) (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 27
% 0.20/0.52 29. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) 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))))) (convergent_lines T_0 T_2) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) U) \/ (apart_point_and_line (intersection_point T_0 T_2) V))))))) (All Y, (All Z, ((apart_point_and_line (intersection_point T_0 T_2) Y) => ((distinct_lines Y Z) \/ (apart_point_and_line (intersection_point T_0 T_2) Z))))) ### All 28
% 0.20/0.52 30. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) U) \/ (apart_point_and_line (intersection_point T_0 T_2) V))))))) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (convergent_lines T_0 T_1) (convergent_lines T_0 T_2) (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, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### All 29
% 0.20/0.52 31. (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (convergent_lines T_0 T_2) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (All U, (All V, (((distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) /\ (distinct_lines U V)) => ((apart_point_and_line (intersection_point T_0 T_1) U) \/ ((apart_point_and_line (intersection_point T_0 T_1) V) \/ ((apart_point_and_line (intersection_point T_0 T_2) U) \/ (apart_point_and_line (intersection_point T_0 T_2) V))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 30
% 0.20/0.52 32. (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)))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (convergent_lines T_0 T_1) (convergent_lines T_0 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) ### All 31
% 0.20/0.52 33. (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) X))))) (convergent_lines T_0 T_2) (convergent_lines T_0 T_1) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line (intersection_point T_0 T_2) (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 32
% 0.20/0.52 34. (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, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0) (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)) (convergent_lines T_0 T_1) (convergent_lines T_0 T_2) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) 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))))))))) ### Extension/test/ci2 33
% 0.20/0.52 35. (-. (((convergent_lines T_0 T_1) /\ ((convergent_lines T_0 T_2) /\ (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 T_2)))) => (-. (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 T_2)) T_0)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### ConjTree 34
% 0.20/0.52 36. (-. (All Z, (((convergent_lines T_0 T_1) /\ ((convergent_lines T_0 Z) /\ (distinct_points (intersection_point T_0 T_1) (intersection_point T_0 Z)))) => (-. (distinct_lines (line_connecting (intersection_point T_0 T_1) (intersection_point T_0 Z)) T_0))))) (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, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) 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))))))))) ### NotAllEx 35
% 0.20/0.52 37. (-. (All Y, (All Z, (((convergent_lines T_0 Y) /\ ((convergent_lines T_0 Z) /\ (distinct_points (intersection_point T_0 Y) (intersection_point T_0 Z)))) => (-. (distinct_lines (line_connecting (intersection_point T_0 Y) (intersection_point T_0 Z)) T_0)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) X))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### NotAllEx 36
% 0.20/0.52 38. (-. (All X, (All Y, (All Z, (((convergent_lines X Y) /\ ((convergent_lines X Z) /\ (distinct_points (intersection_point X Y) (intersection_point X Z)))) => (-. (distinct_lines (line_connecting (intersection_point X Y) (intersection_point X Z)) 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, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, ((convergent_lines X Y) => (-. (apart_point_and_line (intersection_point X Y) 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))))))))) ### NotAllEx 37
% 0.20/0.52 % SZS output end Proof
% 0.20/0.52 (* END-PROOF *)
%------------------------------------------------------------------------------