TSTP Solution File: GEO177+3 by SuperZenon---0.0.1
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- Process Solution
%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : GEO177+3 : TPTP v8.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% Computer : n029.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 600s
% DateTime : Sat Jul 16 06:30:40 EDT 2022
% Result : Theorem 5.84s 6.03s
% Output : Proof 6.16s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.13 % Problem : GEO177+3 : TPTP v8.1.0. Released v4.0.0.
% 0.03/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.13/0.35 % Computer : n029.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 600
% 0.13/0.35 % DateTime : Sat Jun 18 03:36:58 EDT 2022
% 0.13/0.35 % CPUTime :
% 5.84/6.03 % SZS status Theorem
% 5.84/6.03 (* PROOF-FOUND *)
% 5.84/6.03 (* BEGIN-PROOF *)
% 5.84/6.03 % SZS output start Proof
% 5.84/6.03 1. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 5.84/6.03 2. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 5.84/6.03 3. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 5.84/6.03 4. (-. (distinct_points T_0 T_0)) (distinct_points T_0 T_0) ### Axiom
% 5.84/6.03 5. (-. (distinct_points T_1 T_0)) (distinct_points T_1 T_0) ### Axiom
% 5.84/6.03 6. ((distinct_points T_0 T_1) => ((distinct_points T_0 T_0) \/ (distinct_points T_1 T_0))) (-. (distinct_points T_1 T_0)) (-. (distinct_points T_0 T_0)) (distinct_points T_0 T_1) ### DisjTree 3 4 5
% 5.84/6.03 7. (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (-. (distinct_points T_0 T_0)) (-. (distinct_points T_1 T_0)) ### All 6
% 5.84/6.03 8. (distinct_points T_2 T_3) (-. (distinct_points T_2 T_3)) ### Axiom
% 5.84/6.03 9. (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) ### Axiom
% 5.84/6.03 10. (distinct_points T_1 T_0) (-. (distinct_points T_1 T_0)) ### Axiom
% 5.84/6.03 11. (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_3)) (-. (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_3))) ### Axiom
% 5.84/6.03 12. (-. (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
% 5.84/6.03 13. (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (apart_point_and_line T_1 (line_connecting T_2 T_3)) ### Axiom
% 5.84/6.03 14. (-. (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
% 5.84/6.03 15. (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_0 (line_connecting T_2 T_3)) ### Axiom
% 5.84/6.03 16. (((distinct_points T_1 T_0) /\ (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_3))) => ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_1 (line_connecting T_2 T_3)) \/ ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_0 (line_connecting T_2 T_3)))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_3)) (distinct_points T_1 T_0) ### DisjTree 10 11 12 13 14 15
% 5.84/6.03 17. (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) (line_connecting T_2 T_3)) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) ### All 16
% 5.84/6.03 18. (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 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_3)) (distinct_points T_1 T_0) ### All 17
% 5.84/6.03 19. (-. (apart_point_and_line T_2 (line_connecting T_2 T_3))) (apart_point_and_line T_2 (line_connecting T_2 T_3)) ### Axiom
% 5.84/6.03 20. ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_3)) \/ (apart_point_and_line T_2 (line_connecting T_2 T_3)))) (-. (apart_point_and_line T_2 (line_connecting T_2 T_3))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_2 (line_connecting T_0 T_1)) ### DisjTree 9 18 19
% 5.84/6.03 21. (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line 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 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_2 (line_connecting T_2 T_3))) ### All 20
% 5.84/6.03 22. ((distinct_points T_2 T_3) => (-. (apart_point_and_line T_2 (line_connecting T_2 T_3)))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (distinct_points T_2 T_3) ### Imply 8 21
% 5.84/6.03 23. (All Y, ((distinct_points T_2 Y) => (-. (apart_point_and_line T_2 (line_connecting T_2 Y))))) (distinct_points T_2 T_3) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (apart_point_and_line 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 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_1 T_0) ### All 22
% 5.84/6.03 24. (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_2 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_2 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_2 Z)))) (distinct_points T_2 T_3) ### All 23
% 5.84/6.03 25. (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 Z))))) (distinct_points T_2 T_3) (apart_point_and_line 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 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_1 T_0) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) ### All 24
% 6.06/6.30 26. (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_points T_1 T_0) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_2 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) ### All 25
% 6.06/6.30 27. (-. (distinct_points T_0 T_0)) (distinct_points T_0 T_0) ### Axiom
% 6.06/6.30 28. ((distinct_points T_1 T_0) => ((distinct_points T_1 T_0) \/ (distinct_points T_0 T_0))) (distinct_points T_2 T_3) (apart_point_and_line 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 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (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)))))) (-. (distinct_points T_0 T_0)) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) ### DisjTree 7 26 27
% 6.06/6.30 29. (All Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (-. (distinct_points T_0 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)))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_2 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) ### All 28
% 6.06/6.30 30. (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (distinct_points T_2 T_3) (apart_point_and_line 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 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (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)))))) (-. (distinct_points T_0 T_0)) (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 29
% 6.06/6.30 31. (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 Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (-. (distinct_points T_0 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)))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) ### All 30
% 6.06/6.30 32. (All X, (-. (distinct_points X X))) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (distinct_points T_2 T_3) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (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)))))) (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 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 31
% 6.06/6.30 33. ((distinct_points T_0 T_1) => (-. (apart_point_and_line T_1 (line_connecting T_0 T_1)))) (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 Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (All X, (-. (distinct_points X X))) (distinct_points T_0 T_1) ### Imply 2 32
% 6.06/6.30 34. (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line Y (line_connecting T_0 Y))))) (distinct_points T_0 T_1) (All X, (-. (distinct_points X X))) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (distinct_points T_2 T_3) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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)))))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 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 33
% 6.06/6.30 35. ((distinct_points T_0 T_1) => (-. (apart_point_and_line T_0 (line_connecting T_0 T_1)))) (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 Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (All X, (-. (distinct_points X X))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line Y (line_connecting T_0 Y))))) (distinct_points T_0 T_1) ### Imply 1 34
% 6.16/6.32 36. (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_1) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line Y (line_connecting T_0 Y))))) (All X, (-. (distinct_points X X))) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (distinct_points T_2 T_3) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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)))))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 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 35
% 6.16/6.32 37. (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (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 Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (All X, (-. (distinct_points X X))) (distinct_points T_0 T_1) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) ### All 36
% 6.16/6.32 38. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_1) (All X, (-. (distinct_points X X))) (distinct_points T_2 T_3) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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)))))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 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 X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) ### All 37
% 6.16/6.32 39. (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 Y (line_connecting X Y)))))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (All X, (-. (distinct_points X X))) (distinct_points T_0 T_1) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 38
% 6.16/6.32 40. (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) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_1) (All X, (-. (distinct_points X X))) (distinct_points T_2 T_3) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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 39
% 6.16/6.32 41. (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 Y (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)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (All X, (-. (distinct_points X X))) (distinct_points T_0 T_1) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 40
% 6.16/6.32 42. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_points T_0 T_1) (All X, (-. (distinct_points X X))) (distinct_points T_2 T_3) (apart_point_and_line T_2 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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 41
% 6.16/6.36 43. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 6.16/6.36 44. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 6.16/6.36 45. (distinct_points T_2 T_3) (-. (distinct_points T_2 T_3)) ### Axiom
% 6.16/6.36 46. (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_3 (line_connecting T_0 T_1))) ### Axiom
% 6.16/6.36 47. (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (apart_point_and_line T_3 (line_connecting T_2 T_3)) ### Axiom
% 6.16/6.36 48. ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_3)) \/ (apart_point_and_line T_3 (line_connecting T_2 T_3)))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_3 (line_connecting T_0 T_1)) ### DisjTree 46 18 47
% 6.16/6.36 49. (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (apart_point_and_line T_3 (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 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) ### All 48
% 6.16/6.36 50. ((distinct_points T_2 T_3) => (-. (apart_point_and_line T_3 (line_connecting T_2 T_3)))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_3 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (distinct_points T_2 T_3) ### Imply 45 49
% 6.16/6.36 51. (All Y, ((distinct_points T_2 Y) => (-. (apart_point_and_line Y (line_connecting T_2 Y))))) (distinct_points T_2 T_3) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (apart_point_and_line T_3 (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 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_1 T_0) ### All 50
% 6.16/6.36 52. (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_3 (line_connecting T_0 T_1)) (All Z, ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) Z) \/ (apart_point_and_line T_3 Z)))) (distinct_points T_2 T_3) ### All 51
% 6.16/6.36 53. (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (distinct_points T_2 T_3) (apart_point_and_line T_3 (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 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (distinct_points T_1 T_0) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) ### All 52
% 6.16/6.36 54. (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 Y (line_connecting X Y)))))) (distinct_points T_1 T_0) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_3 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) ### All 53
% 6.16/6.36 55. (-. (distinct_points T_0 T_0)) (distinct_points T_0 T_0) ### Axiom
% 6.16/6.36 56. ((distinct_points T_1 T_0) => ((distinct_points T_1 T_0) \/ (distinct_points T_0 T_0))) (distinct_points T_2 T_3) (apart_point_and_line T_3 (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 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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)))))) (-. (distinct_points T_0 T_0)) (distinct_points T_0 T_1) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) ### DisjTree 7 54 55
% 6.16/6.36 57. (All Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (-. (distinct_points T_0 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 Y (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_3 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) ### All 56
% 6.16/6.36 58. (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (distinct_points T_2 T_3) (apart_point_and_line T_3 (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 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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)))))) (-. (distinct_points T_0 T_0)) (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 57
% 6.16/6.38 59. (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 Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_0 T_1) (-. (distinct_points T_0 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 Y (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) ### All 58
% 6.16/6.38 60. (All X, (-. (distinct_points X X))) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (distinct_points T_2 T_3) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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)))))) (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 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 59
% 6.16/6.38 61. ((distinct_points T_0 T_1) => (-. (apart_point_and_line T_1 (line_connecting T_0 T_1)))) (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 Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (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 Y (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (All X, (-. (distinct_points X X))) (distinct_points T_0 T_1) ### Imply 44 60
% 6.16/6.38 62. (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line Y (line_connecting T_0 Y))))) (distinct_points T_0 T_1) (All X, (-. (distinct_points X X))) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (distinct_points T_2 T_3) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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)))))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 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 61
% 6.16/6.38 63. ((distinct_points T_0 T_1) => (-. (apart_point_and_line T_0 (line_connecting T_0 T_1)))) (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 Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (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 Y (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (All X, (-. (distinct_points X X))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line Y (line_connecting T_0 Y))))) (distinct_points T_0 T_1) ### Imply 43 62
% 6.16/6.38 64. (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_1) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line Y (line_connecting T_0 Y))))) (All X, (-. (distinct_points X X))) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (distinct_points T_2 T_3) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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)))))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 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 63
% 6.16/6.38 65. (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 Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (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 Y (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (All X, (-. (distinct_points X X))) (distinct_points T_0 T_1) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) ### All 64
% 6.16/6.38 66. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_1) (All X, (-. (distinct_points X X))) (distinct_points T_2 T_3) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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)))))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 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 65
% 6.16/6.38 67. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Z, ((distinct_points T_0 T_1) => ((distinct_points T_0 Z) \/ (distinct_points T_1 Z)))) (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 Y (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (All X, (-. (distinct_points X X))) (distinct_points T_0 T_1) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 66
% 6.16/6.38 68. (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) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (distinct_points T_0 T_1) (All X, (-. (distinct_points X X))) (distinct_points T_2 T_3) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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)))))) (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 67
% 6.16/6.38 69. (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, ((distinct_points X Y) => (-. (apart_point_and_line Y (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (All X, (-. (distinct_points X X))) (distinct_points T_0 T_1) (All Y, ((distinct_points T_0 Y) => (-. (apart_point_and_line T_0 (line_connecting T_0 Y))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 68
% 6.16/6.38 70. (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_points T_0 T_1) (All X, (-. (distinct_points X X))) (distinct_points T_2 T_3) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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)))))) (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 69
% 6.16/6.38 71. ((apart_point_and_line T_2 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_3 (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))))))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (distinct_points T_2 T_3) (All X, (-. (distinct_points X X))) (distinct_points T_0 T_1) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### Or 42 70
% 6.16/6.38 72. (-. (((distinct_points T_2 T_3) /\ (distinct_points T_0 T_1)) => (((apart_point_and_line T_2 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_3 (line_connecting T_0 T_1))) => ((apart_point_and_line T_0 (line_connecting T_2 T_3)) \/ (apart_point_and_line T_1 (line_connecting T_2 T_3)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points X 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)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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))))))))) ### ConjTree 71
% 6.16/6.38 73. (-. (All V, (((distinct_points T_2 T_3) /\ (distinct_points T_0 V)) => (((apart_point_and_line T_2 (line_connecting T_0 V)) \/ (apart_point_and_line T_3 (line_connecting T_0 V))) => ((apart_point_and_line T_0 (line_connecting T_2 T_3)) \/ (apart_point_and_line V (line_connecting T_2 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))))))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotAllEx 72
% 6.16/6.38 74. (-. (All U, (All V, (((distinct_points T_2 T_3) /\ (distinct_points U V)) => (((apart_point_and_line T_2 (line_connecting U V)) \/ (apart_point_and_line T_3 (line_connecting U V))) => ((apart_point_and_line U (line_connecting T_2 T_3)) \/ (apart_point_and_line V (line_connecting T_2 T_3)))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points X 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)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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))))))))) ### NotAllEx 73
% 6.16/6.38 75. (-. (All Y, (All U, (All V, (((distinct_points T_2 Y) /\ (distinct_points U V)) => (((apart_point_and_line T_2 (line_connecting U V)) \/ (apart_point_and_line Y (line_connecting U V))) => ((apart_point_and_line U (line_connecting T_2 Y)) \/ (apart_point_and_line V (line_connecting T_2 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 Y (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)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line X (line_connecting X Y)))))) (All X, (-. (distinct_points X X))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotAllEx 74
% 6.16/6.38 76. (-. (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_points U V)) => (((apart_point_and_line X (line_connecting U V)) \/ (apart_point_and_line Y (line_connecting U V))) => ((apart_point_and_line U (line_connecting X Y)) \/ (apart_point_and_line V (line_connecting X Y)))))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All X, (-. (distinct_points X 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)))))) (All X, (All Y, ((distinct_points X Y) => (-. (apart_point_and_line Y (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))))))))) ### NotAllEx 75
% 6.16/6.38 % SZS output end Proof
% 6.16/6.38 (* END-PROOF *)
%------------------------------------------------------------------------------