TSTP Solution File: GEO173+2 by SuperZenon---0.0.1
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%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : GEO173+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 06:30:36 EDT 2022
% Result : Theorem 10.51s 10.77s
% Output : Proof 10.72s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.09/0.11 % Problem : GEO173+2 : TPTP v8.1.0. Released v3.3.0.
% 0.09/0.12 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.11/0.33 % Computer : n019.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 600
% 0.11/0.33 % DateTime : Sat Jun 18 05:36:39 EDT 2022
% 0.11/0.33 % CPUTime :
% 10.51/10.77 % SZS status Theorem
% 10.51/10.77 (* PROOF-FOUND *)
% 10.51/10.77 (* BEGIN-PROOF *)
% 10.51/10.77 % SZS output start Proof
% 10.51/10.77 1. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 10.51/10.77 2. (distinct_points T_0 T_0) ### Extension/test/apart1
% 10.51/10.77 3. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 10.51/10.77 4. (distinct_lines T_2 (line_connecting T_0 T_1)) (-. (distinct_lines T_2 (line_connecting T_0 T_1))) ### Axiom
% 10.51/10.77 5. (-. (distinct_lines T_2 T_2)) (distinct_lines T_2 T_2) ### Axiom
% 10.51/10.77 6. (-. (distinct_lines (line_connecting T_0 T_1) T_2)) (distinct_lines (line_connecting T_0 T_1) T_2) ### Axiom
% 10.51/10.77 7. ((distinct_lines T_2 (line_connecting T_0 T_1)) => ((distinct_lines T_2 T_2) \/ (distinct_lines (line_connecting T_0 T_1) T_2))) (-. (distinct_lines (line_connecting T_0 T_1) T_2)) (-. (distinct_lines T_2 T_2)) (distinct_lines T_2 (line_connecting T_0 T_1)) ### DisjTree 4 5 6
% 10.51/10.77 8. (All Z, ((distinct_lines T_2 (line_connecting T_0 T_1)) => ((distinct_lines T_2 Z) \/ (distinct_lines (line_connecting T_0 T_1) Z)))) (distinct_lines T_2 (line_connecting T_0 T_1)) (-. (distinct_lines T_2 T_2)) (-. (distinct_lines (line_connecting T_0 T_1) T_2)) ### All 7
% 10.51/10.77 9. (All Y, (All Z, ((distinct_lines T_2 Y) => ((distinct_lines T_2 Z) \/ (distinct_lines Y Z))))) (-. (distinct_lines (line_connecting T_0 T_1) T_2)) (-. (distinct_lines T_2 T_2)) (distinct_lines T_2 (line_connecting T_0 T_1)) ### All 8
% 10.51/10.77 10. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 10.51/10.77 11. (distinct_points T_1 T_0) (-. (distinct_points T_1 T_0)) ### Axiom
% 10.51/10.77 12. (distinct_lines (line_connecting T_0 T_1) T_2) (-. (distinct_lines (line_connecting T_0 T_1) T_2)) ### Axiom
% 10.51/10.77 13. (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (apart_point_and_line T_1 (line_connecting T_0 T_1)) ### Axiom
% 10.51/10.77 14. (-. (apart_point_and_line T_1 T_2)) (apart_point_and_line T_1 T_2) ### Axiom
% 10.51/10.77 15. (-. (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
% 10.51/10.77 16. (-. (apart_point_and_line T_0 T_2)) (apart_point_and_line T_0 T_2) ### Axiom
% 10.51/10.77 17. (((distinct_points T_1 T_0) /\ (distinct_lines (line_connecting T_0 T_1) T_2)) => ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_1 T_2) \/ ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_0 T_2))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 T_2)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) T_2) (distinct_points T_1 T_0) ### DisjTree 11 12 13 14 15 16
% 10.51/10.77 18. (All V, (((distinct_points T_1 T_0) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_0 V)))))) (distinct_points T_1 T_0) (distinct_lines (line_connecting T_0 T_1) T_2) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 T_2)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 T_2)) ### All 17
% 10.51/10.77 19. ((distinct_points T_0 T_0) /\ (distinct_points T_0 T_1)) ### And 2
% 10.51/10.77 20. ((distinct_points T_0 T_1) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) => ((distinct_points T_0 T_0) /\ (distinct_points T_0 T_1)))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_1 T_2)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) T_2) (distinct_points T_1 T_0) (All V, (((distinct_points T_1 T_0) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_0 V)))))) (distinct_points T_0 T_1) ### DisjTree 10 18 19
% 10.51/10.77 21. (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) (All V, (((distinct_points T_1 T_0) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_0 V)))))) (distinct_points T_1 T_0) (distinct_lines (line_connecting T_0 T_1) T_2) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 T_2)) (-. (apart_point_and_line T_0 T_2)) ### All 20
% 10.51/10.77 22. (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_1 T_2)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) T_2) (distinct_points T_1 T_0) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) ### All 21
% 10.51/10.77 23. (-. (distinct_lines T_2 T_2)) (distinct_lines T_2 T_2) ### Axiom
% 10.51/10.77 24. ((distinct_lines (line_connecting T_0 T_1) T_2) => ((distinct_lines (line_connecting T_0 T_1) T_2) \/ (distinct_lines T_2 T_2))) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) (distinct_points T_1 T_0) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 T_2)) (-. (apart_point_and_line T_0 T_2)) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (distinct_lines T_2 (line_connecting T_0 T_1)) (-. (distinct_lines T_2 T_2)) (All Y, (All Z, ((distinct_lines T_2 Y) => ((distinct_lines T_2 Z) \/ (distinct_lines Y Z))))) ### DisjTree 9 22 23
% 10.51/10.77 25. (All Z, ((distinct_lines (line_connecting T_0 T_1) T_2) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (distinct_lines T_2 Z)))) (All Y, (All Z, ((distinct_lines T_2 Y) => ((distinct_lines T_2 Z) \/ (distinct_lines Y Z))))) (-. (distinct_lines T_2 T_2)) (distinct_lines T_2 (line_connecting T_0 T_1)) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_1 T_2)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_1 T_0) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) ### All 24
% 10.51/10.77 26. (All Y, (All Z, ((distinct_lines (line_connecting T_0 T_1) Y) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (distinct_lines Y Z))))) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) (distinct_points T_1 T_0) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 T_2)) (-. (apart_point_and_line T_0 T_2)) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (distinct_lines T_2 (line_connecting T_0 T_1)) (-. (distinct_lines T_2 T_2)) (All Y, (All Z, ((distinct_lines T_2 Y) => ((distinct_lines T_2 Z) \/ (distinct_lines Y Z))))) ### All 25
% 10.51/10.77 27. (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((distinct_lines T_2 Y) => ((distinct_lines T_2 Z) \/ (distinct_lines Y Z))))) (distinct_lines T_2 (line_connecting T_0 T_1)) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_1 T_2)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_1 T_0) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (All Y, (All Z, ((distinct_lines (line_connecting T_0 T_1) Y) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (distinct_lines Y Z))))) ### All 26
% 10.72/10.96 28. (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_0 T_1) (distinct_points T_1 T_0) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 T_2)) (-. (apart_point_and_line T_0 T_2)) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (distinct_lines T_2 (line_connecting T_0 T_1)) (All Y, (All Z, ((distinct_lines T_2 Y) => ((distinct_lines T_2 Z) \/ (distinct_lines Y Z))))) (All X, (-. (distinct_lines X X))) ### All 27
% 10.72/10.96 29. (distinct_points T_1 T_1) ### Extension/test/apart1
% 10.72/10.96 30. ((distinct_points T_1 T_0) /\ (distinct_points T_1 T_1)) ### And 29
% 10.72/10.96 31. ((distinct_points T_0 T_1) => ((apart_point_and_line T_1 (line_connecting T_0 T_1)) => ((distinct_points T_1 T_0) /\ (distinct_points T_1 T_1)))) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((distinct_lines T_2 Y) => ((distinct_lines T_2 Z) \/ (distinct_lines Y Z))))) (distinct_lines T_2 (line_connecting T_0 T_1)) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_1 T_2)) (distinct_points T_1 T_0) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (distinct_points T_0 T_1) ### DisjTree 3 28 30
% 10.72/10.96 32. (distinct_points T_0 T_1) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Z, ((distinct_points T_0 T_1) => ((apart_point_and_line Z (line_connecting T_0 T_1)) => ((distinct_points Z T_0) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_1 T_2)) (-. (apart_point_and_line T_0 T_2)) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (distinct_lines T_2 (line_connecting T_0 T_1)) (All Y, (All Z, ((distinct_lines T_2 Y) => ((distinct_lines T_2 Z) \/ (distinct_lines Y Z))))) (All X, (-. (distinct_lines X X))) ### All 31
% 10.72/10.96 33. (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((distinct_lines T_2 Y) => ((distinct_lines T_2 Z) \/ (distinct_lines Y Z))))) (distinct_lines T_2 (line_connecting T_0 T_1)) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_1 T_2)) (distinct_points T_1 T_0) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (distinct_points T_0 T_1) ### All 32
% 10.72/10.96 34. ((distinct_points T_0 T_1) => ((distinct_points T_0 T_0) \/ (distinct_points T_1 T_0))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (-. (apart_point_and_line T_1 T_2)) (-. (apart_point_and_line T_0 T_2)) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (distinct_lines T_2 (line_connecting T_0 T_1)) (All Y, (All Z, ((distinct_lines T_2 Y) => ((distinct_lines T_2 Z) \/ (distinct_lines Y Z))))) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (distinct_points T_0 T_1) ### DisjTree 1 2 33
% 10.72/10.96 35. (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((distinct_lines T_2 Y) => ((distinct_lines T_2 Z) \/ (distinct_lines Y Z))))) (distinct_lines T_2 (line_connecting T_0 T_1)) (All U, (All V, (((distinct_points T_1 T_0) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_0 V))))))) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_1 T_2)) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) ### All 34
% 10.72/10.96 36. (All Y, (All U, (All V, (((distinct_points T_1 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (-. (apart_point_and_line T_1 T_2)) (-. (apart_point_and_line T_0 T_2)) (distinct_lines T_2 (line_connecting T_0 T_1)) (All Y, (All Z, ((distinct_lines T_2 Y) => ((distinct_lines T_2 Z) \/ (distinct_lines Y Z))))) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((distinct_points T_0 Y) => ((apart_point_and_line Z (line_connecting T_0 Y)) => ((distinct_points Z T_0) /\ (distinct_points Z Y)))))) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) ### All 35
% 10.72/10.96 37. (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((distinct_lines T_2 Y) => ((distinct_lines T_2 Z) \/ (distinct_lines Y Z))))) (distinct_lines T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_1 T_2)) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All Y, (All U, (All V, (((distinct_points T_1 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) ### All 36
% 10.72/10.96 38. (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, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (-. (apart_point_and_line T_1 T_2)) (-. (apart_point_and_line T_0 T_2)) (distinct_lines T_2 (line_connecting T_0 T_1)) (All Y, (All Z, ((distinct_lines T_2 Y) => ((distinct_lines T_2 Z) \/ (distinct_lines Y Z))))) (All X, (-. (distinct_lines X X))) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) ### All 37
% 10.72/10.97 39. (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (distinct_points T_0 T_1) (All X, (-. (distinct_lines X X))) (All Y, (All Z, ((distinct_lines T_2 Y) => ((distinct_lines T_2 Z) \/ (distinct_lines Y Z))))) (distinct_lines T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_1 T_2)) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### All 38
% 10.72/10.97 40. (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, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (-. (apart_point_and_line T_1 T_2)) (-. (apart_point_and_line T_0 T_2)) (distinct_lines T_2 (line_connecting T_0 T_1)) (All X, (-. (distinct_lines X X))) (distinct_points T_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) ### All 39
% 10.72/10.97 41. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (distinct_points T_0 T_1) (All X, (-. (distinct_lines X X))) (distinct_lines T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 T_2)) (-. (apart_point_and_line T_1 T_2)) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### All 40
% 10.72/10.97 42. (-. (((distinct_points T_0 T_1) /\ ((convergent_lines T_2 T_3) /\ (distinct_lines T_2 (line_connecting T_0 T_1)))) => ((apart_point_and_line T_0 T_2) \/ (apart_point_and_line T_1 T_2)))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### ConjTree 41
% 10.72/10.97 43. (-. (All V, (((distinct_points T_0 T_1) /\ ((convergent_lines T_2 V) /\ (distinct_lines T_2 (line_connecting T_0 T_1)))) => ((apart_point_and_line T_0 T_2) \/ (apart_point_and_line T_1 T_2))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### NotAllEx 42
% 10.72/10.97 44. (-. (All U, (All V, (((distinct_points T_0 T_1) /\ ((convergent_lines U V) /\ (distinct_lines U (line_connecting T_0 T_1)))) => ((apart_point_and_line T_0 U) \/ (apart_point_and_line T_1 U)))))) (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, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotAllEx 43
% 10.72/10.97 45. (-. (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ ((convergent_lines U V) /\ (distinct_lines U (line_connecting T_0 Y)))) => ((apart_point_and_line T_0 U) \/ (apart_point_and_line Y U))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) ### NotAllEx 44
% 10.72/10.97 46. (-. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ ((convergent_lines U V) /\ (distinct_lines U (line_connecting X Y)))) => ((apart_point_and_line X U) \/ (apart_point_and_line Y U)))))))) (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, ((distinct_lines X Y) => ((distinct_lines X Z) \/ (distinct_lines Y Z)))))) (All X, (-. (distinct_lines X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((apart_point_and_line Z (line_connecting X Y)) => ((distinct_points Z X) /\ (distinct_points Z Y))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotAllEx 45
% 10.72/10.97 % SZS output end Proof
% 10.72/10.97 (* END-PROOF *)
%------------------------------------------------------------------------------