TSTP Solution File: GEO187+2 by SuperZenon---0.0.1
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%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : GEO187+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 : n026.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 06:30:48 EDT 2022
% Result : Theorem 13.23s 13.45s
% Output : Proof 13.84s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.11 % Problem : GEO187+2 : TPTP v8.1.0. Released v3.3.0.
% 0.10/0.12 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.12/0.33 % Computer : n026.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 600
% 0.12/0.33 % DateTime : Sat Jun 18 17:08:09 EDT 2022
% 0.12/0.33 % CPUTime :
% 13.23/13.45 % SZS status Theorem
% 13.23/13.45 (* PROOF-FOUND *)
% 13.23/13.45 (* BEGIN-PROOF *)
% 13.23/13.45 % SZS output start Proof
% 13.23/13.45 1. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 13.23/13.45 2. (distinct_points T_0 T_0) ### Extension/test/apart1
% 13.23/13.45 3. (distinct_points T_2 T_3) (-. (distinct_points T_2 T_3)) ### Axiom
% 13.23/13.45 4. (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
% 13.23/13.45 5. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 13.23/13.45 6. (distinct_points T_1 T_0) (-. (distinct_points T_1 T_0)) ### Axiom
% 13.23/13.45 7. (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
% 13.23/13.45 8. (-. (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
% 13.23/13.45 9. (-. (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
% 13.23/13.45 10. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 13.23/13.45 11. (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
% 13.23/13.45 12. ((distinct_points T_0 T_0) /\ (distinct_points T_0 T_1)) ### And 2
% 13.23/13.45 13. ((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 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### DisjTree 10 11 12
% 13.23/13.45 14. (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) (apart_point_and_line T_0 (line_connecting T_0 T_1)) ### All 13
% 13.23/13.45 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
% 13.23/13.45 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))) (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))))) (-. (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 6 7 8 9 14 15
% 13.23/13.45 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))) (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) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) ### All 16
% 13.23/13.45 18. (distinct_points T_1 T_1) ### Extension/test/apart1
% 13.23/13.45 19. ((distinct_points T_1 T_0) /\ (distinct_points T_1 T_1)) ### And 18
% 13.23/13.45 20. ((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)))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_3)) (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 5 17 19
% 13.23/13.45 21. (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) (line_connecting T_2 T_3)) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) ### All 20
% 13.23/13.45 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 (line_connecting T_2 T_3))) (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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_2 T_3)) (distinct_points T_1 T_0) (distinct_points T_0 T_1) ### All 21
% 13.23/13.45 23. (-. (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
% 13.23/13.45 24. ((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_0 T_1) (distinct_points T_1 T_0) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))) (-. (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 4 22 23
% 13.23/13.45 25. (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))) (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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (distinct_points T_1 T_0) (distinct_points T_0 T_1) (-. (apart_point_and_line T_2 (line_connecting T_2 T_3))) ### All 24
% 13.23/13.45 26. (distinct_points T_2 T_2) ### Extension/test/apart1
% 13.23/13.45 27. ((distinct_points T_2 T_2) /\ (distinct_points T_2 T_3)) ### And 26
% 13.23/13.45 28. ((distinct_points T_2 T_3) => ((apart_point_and_line T_2 (line_connecting T_2 T_3)) => ((distinct_points T_2 T_2) /\ (distinct_points T_2 T_3)))) (distinct_points T_0 T_1) (distinct_points T_1 T_0) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))) (-. (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) ### DisjTree 3 25 27
% 13.23/13.48 29. (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (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))) (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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (distinct_points T_1 T_0) (distinct_points T_0 T_1) ### All 28
% 13.23/13.48 30. ((distinct_points T_0 T_1) => ((distinct_points T_0 T_0) \/ (distinct_points T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))) (-. (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 Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_0 T_1) ### DisjTree 1 2 29
% 13.23/13.48 31. (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 Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (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))) (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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) ### All 30
% 13.23/13.48 32. (All Y, (All Z, ((distinct_points T_2 Y) => ((apart_point_and_line Z (line_connecting T_2 Y)) => ((distinct_points Z T_2) /\ (distinct_points Z Y)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (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))))) (-. (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) (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 31
% 13.23/13.48 33. (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) (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))) (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))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) ### All 32
% 13.23/13.48 34. (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (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) (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 33
% 13.23/13.48 35. (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_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) (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)) (-. (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 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 34
% 13.23/13.48 36. (-. (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)) (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) (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 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
% 13.23/13.48 37. (All Y, (All Z, ((apart_point_and_line T_2 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_2 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, (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) (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 36
% 13.35/13.60 38. (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 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) (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 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 37
% 13.35/13.60 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, (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) (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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 38
% 13.35/13.60 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (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) (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 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
% 13.35/13.60 41. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points 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 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) (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, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### All 40
% 13.35/13.60 42. (-. (-. (apart_point_and_line T_2 (line_connecting T_0 T_1)))) (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 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) (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 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_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotNot 41
% 13.35/13.60 43. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 13.35/13.60 44. (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
% 13.35/13.60 45. (-. (distinct_points T_3 T_1)) (distinct_points T_3 T_1) ### Axiom
% 13.35/13.60 46. ((distinct_points T_3 T_0) /\ (distinct_points T_3 T_1)) (-. (distinct_points T_3 T_1)) ### And 45
% 13.35/13.60 47. ((distinct_points T_0 T_1) => ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_points T_3 T_0) /\ (distinct_points T_3 T_1)))) (-. (distinct_points T_3 T_1)) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### DisjTree 43 44 46
% 13.35/13.60 48. (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) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (distinct_points T_3 T_1)) ### All 47
% 13.35/13.60 49. (distinct_points T_3 T_1) (-. (distinct_points T_3 T_1)) ### Axiom
% 13.35/13.60 50. (distinct_points T_3 T_3) ### Extension/test/apart1
% 13.35/13.60 51. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 13.35/13.60 52. (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
% 13.35/13.60 53. (-. (distinct_points T_3 T_0)) (distinct_points T_3 T_0) ### Axiom
% 13.35/13.60 54. ((distinct_points T_3 T_0) /\ (distinct_points T_3 T_1)) (-. (distinct_points T_3 T_0)) ### And 53
% 13.35/13.60 55. ((distinct_points T_0 T_1) => ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_points T_3 T_0) /\ (distinct_points T_3 T_1)))) (-. (distinct_points T_3 T_0)) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### DisjTree 51 52 54
% 13.35/13.60 56. (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) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (distinct_points T_3 T_0)) ### All 55
% 13.35/13.60 57. (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_3 T_0)) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### All 56
% 13.35/13.60 58. (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) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (distinct_points T_3 T_0)) ### All 57
% 13.35/13.60 59. (-. (distinct_points T_0 T_3)) (distinct_points T_0 T_3) ### Axiom
% 13.35/13.60 60. ((distinct_points T_3 T_0) => ((distinct_points T_3 T_3) \/ (distinct_points T_0 T_3))) (-. (distinct_points T_0 T_3)) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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))))))) ### DisjTree 58 50 59
% 13.35/13.60 61. (All Z, ((distinct_points T_3 T_0) => ((distinct_points T_3 Z) \/ (distinct_points T_0 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) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (-. (distinct_points T_0 T_3)) ### All 60
% 13.35/13.60 62. (All Y, (All Z, ((distinct_points T_3 Y) => ((distinct_points T_3 Z) \/ (distinct_points Y Z))))) (-. (distinct_points T_0 T_3)) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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 61
% 13.45/13.63 63. (distinct_points T_0 T_3) (-. (distinct_points T_0 T_3)) ### Axiom
% 13.45/13.63 64. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 13.45/13.63 65. (-. (distinct_points T_1 T_0)) (distinct_points T_1 T_0) ### Axiom
% 13.45/13.63 66. ((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_1) ### DisjTree 64 2 65
% 13.45/13.63 67. (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_1 T_0)) ### All 66
% 13.45/13.63 68. (distinct_points T_1 T_0) (-. (distinct_points T_1 T_0)) ### Axiom
% 13.45/13.63 69. (distinct_points T_3 T_0) (-. (distinct_points T_3 T_0)) ### Axiom
% 13.45/13.63 70. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 13.45/13.63 71. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 13.45/13.63 72. (distinct_points T_3 T_0) (-. (distinct_points T_3 T_0)) ### Axiom
% 13.45/13.63 73. (distinct_points T_3 T_1) (-. (distinct_points T_3 T_1)) ### Axiom
% 13.45/13.63 74. (distinct_points T_2 T_3) (-. (distinct_points T_2 T_3)) ### Axiom
% 13.45/13.63 75. (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
% 13.45/13.63 76. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 13.45/13.63 77. (distinct_lines (line_connecting T_0 T_1) (line_connecting T_3 T_1)) (-. (distinct_lines (line_connecting T_0 T_1) (line_connecting T_3 T_1))) ### Axiom
% 13.45/13.63 78. (-. (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
% 13.45/13.63 79. (apart_point_and_line T_0 (line_connecting T_3 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_3 T_1))) ### Axiom
% 13.45/13.63 80. (distinct_points T_1 T_3) (-. (distinct_points T_1 T_3)) ### Axiom
% 13.45/13.63 81. (distinct_lines (line_connecting T_3 T_1) (line_connecting T_1 T_0)) (-. (distinct_lines (line_connecting T_3 T_1) (line_connecting T_1 T_0))) ### Axiom
% 13.45/13.63 82. (distinct_points T_3 T_1) (-. (distinct_points T_3 T_1)) ### Axiom
% 13.45/13.63 83. (apart_point_and_line T_1 (line_connecting T_3 T_1)) (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) ### Axiom
% 13.45/13.63 84. ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)) ### And 18
% 13.45/13.63 85. ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (apart_point_and_line T_1 (line_connecting T_3 T_1)) (distinct_points T_3 T_1) ### DisjTree 82 83 84
% 13.45/13.63 86. (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_3 T_1) (apart_point_and_line T_1 (line_connecting T_3 T_1)) ### All 85
% 13.45/13.63 87. (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ### Axiom
% 13.45/13.63 88. (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (apart_point_and_line T_3 (line_connecting T_3 T_1)) ### Axiom
% 13.45/13.63 89. (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (apart_point_and_line T_3 (line_connecting T_1 T_0)) ### Axiom
% 13.45/13.63 90. (((distinct_points T_1 T_3) /\ (distinct_lines (line_connecting T_3 T_1) (line_connecting T_1 T_0))) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) \/ ((apart_point_and_line T_1 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_3 (line_connecting T_3 T_1)) \/ (apart_point_and_line T_3 (line_connecting T_1 T_0)))))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_lines (line_connecting T_3 T_1) (line_connecting T_1 T_0)) (distinct_points T_1 T_3) ### DisjTree 80 81 86 87 88 89
% 13.45/13.63 91. (All V, (((distinct_points T_1 T_3) /\ (distinct_lines (line_connecting T_3 T_1) V)) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 (line_connecting T_3 T_1)) \/ (apart_point_and_line T_3 V)))))) (distinct_points T_1 T_3) (distinct_lines (line_connecting T_3 T_1) (line_connecting T_1 T_0)) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_3 T_1) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) ### All 90
% 13.45/13.63 92. (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) ### Axiom
% 13.45/13.63 93. ((apart_point_and_line T_0 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) (line_connecting T_1 T_0)) \/ (apart_point_and_line T_0 (line_connecting T_1 T_0)))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All V, (((distinct_points T_1 T_3) /\ (distinct_lines (line_connecting T_3 T_1) V)) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 (line_connecting T_3 T_1)) \/ (apart_point_and_line T_3 V)))))) (apart_point_and_line T_0 (line_connecting T_3 T_1)) ### DisjTree 79 91 92
% 13.45/13.63 94. (All Z, ((apart_point_and_line T_0 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_0 Z)))) (apart_point_and_line T_0 (line_connecting T_3 T_1)) (All V, (((distinct_points T_1 T_3) /\ (distinct_lines (line_connecting T_3 T_1) V)) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 (line_connecting T_3 T_1)) \/ (apart_point_and_line T_3 V)))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_3 T_1) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) ### All 93
% 13.45/13.63 95. (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (apart_point_and_line T_0 (line_connecting T_3 T_1)) (All Z, ((apart_point_and_line T_0 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_0 Z)))) ### All 94
% 13.45/13.63 96. (distinct_points T_1 T_0) (-. (distinct_points T_1 T_0)) ### Axiom
% 13.45/13.63 97. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 13.45/13.63 98. (distinct_lines (line_connecting T_1 T_0) (line_connecting T_3 T_0)) (-. (distinct_lines (line_connecting T_1 T_0) (line_connecting T_3 T_0))) ### Axiom
% 13.45/13.63 99. (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) ### Axiom
% 13.45/13.63 100. (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (apart_point_and_line T_0 (line_connecting T_3 T_0)) ### Axiom
% 13.45/13.65 101. (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ### Axiom
% 13.45/13.65 102. (-. (apart_point_and_line T_1 (line_connecting T_3 T_0))) (apart_point_and_line T_1 (line_connecting T_3 T_0)) ### Axiom
% 13.45/13.65 103. (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_1 T_0) (line_connecting T_3 T_0))) => ((apart_point_and_line T_0 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_0 (line_connecting T_3 T_0)) \/ ((apart_point_and_line T_1 (line_connecting T_1 T_0)) \/ (apart_point_and_line T_1 (line_connecting T_3 T_0)))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_3 T_0)) (distinct_points T_0 T_1) ### DisjTree 97 98 99 100 101 102
% 13.45/13.65 104. (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_1 T_0) V)) => ((apart_point_and_line T_0 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_1 T_0)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_3 T_0)) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_0))) ### All 103
% 13.45/13.65 105. (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_3 T_0)) (distinct_points T_0 T_1) ### All 104
% 13.45/13.65 106. ((distinct_points T_1 T_1) /\ (distinct_points T_1 T_0)) ### And 18
% 13.45/13.65 107. ((distinct_points T_1 T_0) => ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_points T_1 T_1) /\ (distinct_points T_1 T_0)))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_3 T_0)) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) ### DisjTree 96 105 106
% 13.45/13.65 108. (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_3 T_0)) (distinct_points T_0 T_1) ### All 107
% 13.45/13.65 109. (-. (apart_point_and_line T_1 (line_connecting T_3 T_0))) (apart_point_and_line T_1 (line_connecting T_3 T_0)) ### Axiom
% 13.45/13.65 110. ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) (line_connecting T_3 T_0)) \/ (apart_point_and_line T_1 (line_connecting T_3 T_0)))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Z, ((apart_point_and_line T_0 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_0 Z)))) (apart_point_and_line T_0 (line_connecting T_3 T_1)) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_3 T_1) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) ### DisjTree 95 108 109
% 13.45/13.65 111. (-. (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
% 13.45/13.65 112. (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) (apart_point_and_line T_1 (line_connecting T_3 T_1)) ### Axiom
% 13.45/13.65 113. (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) (line_connecting T_3 T_1))) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 (line_connecting T_3 T_1)) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 (line_connecting T_3 T_1)))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((apart_point_and_line T_0 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_0 Z)))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) (line_connecting T_3 T_0)) \/ (apart_point_and_line T_1 (line_connecting T_3 T_0)))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_3 T_1)) (distinct_points T_0 T_1) ### DisjTree 76 77 78 110 111 112
% 13.45/13.65 114. (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_3 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) (line_connecting T_3 T_0)) \/ (apart_point_and_line T_1 (line_connecting T_3 T_0)))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Z, ((apart_point_and_line T_0 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_0 Z)))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_3 T_1) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) ### All 113
% 13.45/13.68 115. (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((apart_point_and_line T_0 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_0 Z)))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_3 T_1)) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) ### All 114
% 13.45/13.68 116. (distinct_points T_3 T_1) (-. (distinct_points T_3 T_1)) ### Axiom
% 13.45/13.68 117. (distinct_points T_3 T_1) (-. (distinct_points T_3 T_1)) ### Axiom
% 13.45/13.68 118. (distinct_lines (line_connecting T_3 T_1) (line_connecting T_1 T_0)) (-. (distinct_lines (line_connecting T_3 T_1) (line_connecting T_1 T_0))) ### Axiom
% 13.45/13.68 119. (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (apart_point_and_line T_3 (line_connecting T_3 T_1)) ### Axiom
% 13.45/13.68 120. (apart_point_and_line T_3 (line_connecting T_1 T_0)) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) ### Axiom
% 13.45/13.68 121. (distinct_points T_0 T_1) (-. (distinct_points T_0 T_1)) ### Axiom
% 13.45/13.68 122. (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_3)) (-. (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_3))) ### Axiom
% 13.45/13.68 123. (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (apart_point_and_line T_0 (line_connecting T_1 T_0)) ### Axiom
% 13.45/13.68 124. (-. (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
% 13.45/13.68 125. (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ### Axiom
% 13.45/13.68 126. (-. (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
% 13.45/13.68 127. (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_3))) => ((apart_point_and_line T_0 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_0 (line_connecting T_2 T_3)) \/ ((apart_point_and_line T_1 (line_connecting T_1 T_0)) \/ (apart_point_and_line T_1 (line_connecting T_2 T_3)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_3)) (distinct_points T_0 T_1) ### DisjTree 121 122 123 124 125 126
% 13.45/13.68 128. (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_1 T_0) V)) => ((apart_point_and_line T_0 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_1 T_0)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_2 T_3)) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) ### All 127
% 13.45/13.68 129. (-. (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
% 13.45/13.68 130. ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) (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))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_1 T_0) V)) => ((apart_point_and_line T_0 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_1 T_0)) \/ (apart_point_and_line T_1 V)))))) (apart_point_and_line T_3 (line_connecting T_1 T_0)) ### DisjTree 120 128 129
% 13.45/13.68 131. (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) (apart_point_and_line T_3 (line_connecting T_1 T_0)) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_1 T_0) V)) => ((apart_point_and_line T_0 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_1 T_0)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) ### All 130
% 13.45/13.68 132. (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (apart_point_and_line T_3 (line_connecting T_1 T_0)) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) ### All 131
% 13.45/13.70 133. (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (apart_point_and_line T_3 (line_connecting T_1 T_0)) (distinct_points T_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) ### All 132
% 13.45/13.70 134. (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) (apart_point_and_line T_1 (line_connecting T_3 T_1)) ### Axiom
% 13.45/13.70 135. (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ### Axiom
% 13.45/13.70 136. (((distinct_points T_3 T_1) /\ (distinct_lines (line_connecting T_3 T_1) (line_connecting T_1 T_0))) => ((apart_point_and_line T_3 (line_connecting T_3 T_1)) \/ ((apart_point_and_line T_3 (line_connecting T_1 T_0)) \/ ((apart_point_and_line T_1 (line_connecting T_3 T_1)) \/ (apart_point_and_line T_1 (line_connecting T_1 T_0)))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (distinct_lines (line_connecting T_3 T_1) (line_connecting T_1 T_0)) (distinct_points T_3 T_1) ### DisjTree 117 118 119 133 134 135
% 13.45/13.70 137. (All V, (((distinct_points T_3 T_1) /\ (distinct_lines (line_connecting T_3 T_1) V)) => ((apart_point_and_line T_3 (line_connecting T_3 T_1)) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 (line_connecting T_3 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_3 T_1) (distinct_lines (line_connecting T_3 T_1) (line_connecting T_1 T_0)) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (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_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_1))) ### All 136
% 13.45/13.70 138. ((distinct_points T_3 T_1) => ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_points T_1 T_3) /\ (distinct_points T_1 T_1)))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_points T_0 T_1) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (distinct_lines (line_connecting T_3 T_1) (line_connecting T_1 T_0)) (All V, (((distinct_points T_3 T_1) /\ (distinct_lines (line_connecting T_3 T_1) V)) => ((apart_point_and_line T_3 (line_connecting T_3 T_1)) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 (line_connecting T_3 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_3 T_1) ### DisjTree 116 137 84
% 13.45/13.70 139. (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_3 T_1) (All V, (((distinct_points T_3 T_1) /\ (distinct_lines (line_connecting T_3 T_1) V)) => ((apart_point_and_line T_3 (line_connecting T_3 T_1)) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 (line_connecting T_3 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_lines (line_connecting T_3 T_1) (line_connecting T_1 T_0)) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (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_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) ### All 138
% 13.45/13.70 140. (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (apart_point_and_line T_1 (line_connecting T_1 T_0)) ### Axiom
% 13.45/13.70 141. ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) (line_connecting T_1 T_0)) \/ (apart_point_and_line T_1 (line_connecting T_1 T_0)))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (All V, (((distinct_points T_3 T_1) /\ (distinct_lines (line_connecting T_3 T_1) V)) => ((apart_point_and_line T_3 (line_connecting T_3 T_1)) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 (line_connecting T_3 T_1)) \/ (apart_point_and_line T_1 V)))))) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_3 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Z, ((apart_point_and_line T_0 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_0 Z)))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_3 T_1) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_1 Z)))) ### DisjTree 115 139 140
% 13.55/13.72 142. (All Z, ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_1 Z)))) (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((apart_point_and_line T_0 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_0 Z)))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_3 T_1)) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (All V, (((distinct_points T_3 T_1) /\ (distinct_lines (line_connecting T_3 T_1) V)) => ((apart_point_and_line T_3 (line_connecting T_3 T_1)) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 (line_connecting T_3 T_1)) \/ (apart_point_and_line T_1 V)))))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) ### All 141
% 13.55/13.72 143. (-. (apart_point_and_line T_1 (line_connecting T_3 T_0))) (apart_point_and_line T_1 (line_connecting T_3 T_0)) ### Axiom
% 13.55/13.72 144. ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) (line_connecting T_3 T_0)) \/ (apart_point_and_line T_1 (line_connecting T_3 T_0)))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (All V, (((distinct_points T_3 T_1) /\ (distinct_lines (line_connecting T_3 T_1) V)) => ((apart_point_and_line T_3 (line_connecting T_3 T_1)) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 (line_connecting T_3 T_1)) \/ (apart_point_and_line T_1 V)))))) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_3 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Z, ((apart_point_and_line T_0 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_0 Z)))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_3 T_1) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (All Z, ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_1 Z)))) ### DisjTree 142 108 143
% 13.55/13.72 145. (All Z, ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_1 Z)))) (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((apart_point_and_line T_0 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_0 Z)))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_3 T_1)) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (All V, (((distinct_points T_3 T_1) /\ (distinct_lines (line_connecting T_3 T_1) V)) => ((apart_point_and_line T_3 (line_connecting T_3 T_1)) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 (line_connecting T_3 T_1)) \/ (apart_point_and_line T_1 V)))))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) ### All 144
% 13.55/13.74 146. (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_3 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_1 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Z, ((apart_point_and_line T_0 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_0 Z)))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_3 T_1) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (All Z, ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_1 Z)))) ### All 145
% 13.55/13.75 147. (distinct_points T_0 T_3) (-. (distinct_points T_0 T_3)) ### Axiom
% 13.55/13.75 148. (distinct_lines (line_connecting T_3 T_0) (line_connecting T_2 T_3)) (-. (distinct_lines (line_connecting T_3 T_0) (line_connecting T_2 T_3))) ### Axiom
% 13.55/13.75 149. (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (apart_point_and_line T_0 (line_connecting T_3 T_0)) ### Axiom
% 13.55/13.75 150. (-. (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
% 13.55/13.75 151. (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (apart_point_and_line T_3 (line_connecting T_3 T_0)) ### Axiom
% 13.55/13.75 152. (-. (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
% 13.55/13.75 153. (((distinct_points T_0 T_3) /\ (distinct_lines (line_connecting T_3 T_0) (line_connecting T_2 T_3))) => ((apart_point_and_line T_0 (line_connecting T_3 T_0)) \/ ((apart_point_and_line T_0 (line_connecting T_2 T_3)) \/ ((apart_point_and_line T_3 (line_connecting T_3 T_0)) \/ (apart_point_and_line T_3 (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_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (distinct_lines (line_connecting T_3 T_0) (line_connecting T_2 T_3)) (distinct_points T_0 T_3) ### DisjTree 147 148 149 150 151 152
% 13.55/13.75 154. (All V, (((distinct_points T_0 T_3) /\ (distinct_lines (line_connecting T_3 T_0) V)) => ((apart_point_and_line T_0 (line_connecting T_3 T_0)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 (line_connecting T_3 T_0)) \/ (apart_point_and_line T_3 V)))))) (distinct_points T_0 T_3) (distinct_lines (line_connecting T_3 T_0) (line_connecting T_2 T_3)) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) ### All 153
% 13.55/13.75 155. (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (distinct_lines (line_connecting T_3 T_0) (line_connecting T_2 T_3)) (distinct_points T_0 T_3) ### All 154
% 13.55/13.75 156. (-. (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
% 13.55/13.75 157. ((apart_point_and_line T_1 (line_connecting T_3 T_0)) => ((distinct_lines (line_connecting T_3 T_0) (line_connecting T_2 T_3)) \/ (apart_point_and_line T_1 (line_connecting T_2 T_3)))) (distinct_points T_0 T_3) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All Z, ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_1 Z)))) (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((apart_point_and_line T_0 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_0 Z)))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_3 T_1)) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) ### DisjTree 146 155 156
% 13.55/13.76 158. (All Z, ((apart_point_and_line T_1 (line_connecting T_3 T_0)) => ((distinct_lines (line_connecting T_3 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_3 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All Z, ((apart_point_and_line T_0 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_0 Z)))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_3 T_1) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (All Z, ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_1 Z)))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (distinct_points T_0 T_3) ### All 157
% 13.55/13.76 159. (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (distinct_points T_0 T_3) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All Z, ((apart_point_and_line T_1 (line_connecting T_3 T_1)) => ((distinct_lines (line_connecting T_3 T_1) Z) \/ (apart_point_and_line T_1 Z)))) (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_3 T_1)) (distinct_points T_0 T_1) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((apart_point_and_line T_1 (line_connecting T_3 T_0)) => ((distinct_lines (line_connecting T_3 T_0) Z) \/ (apart_point_and_line T_1 Z)))) ### All 158
% 13.55/13.76 160. (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Z, ((apart_point_and_line T_1 (line_connecting T_3 T_0)) => ((distinct_lines (line_connecting T_3 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (All V, (((distinct_points T_0 T_1) /\ (distinct_lines (line_connecting T_0 T_1) V)) => ((apart_point_and_line T_0 (line_connecting T_0 T_1)) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 (line_connecting T_0 T_1)) \/ (apart_point_and_line T_1 V)))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_3 T_1)) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_3 T_1) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (distinct_points T_0 T_3) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) ### All 159
% 13.55/13.78 161. (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (distinct_points T_0 T_3) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_lines (line_connecting T_0 T_1) (line_connecting T_3 T_1)) (distinct_points T_0 T_1) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((apart_point_and_line T_1 (line_connecting T_3 T_0)) => ((distinct_lines (line_connecting T_3 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) ### All 160
% 13.55/13.78 162. (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (apart_point_and_line T_3 (line_connecting T_3 T_1)) ### Axiom
% 13.55/13.78 163. ((apart_point_and_line T_3 (line_connecting T_0 T_1)) => ((distinct_lines (line_connecting T_0 T_1) (line_connecting T_3 T_1)) \/ (apart_point_and_line T_3 (line_connecting T_3 T_1)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Z, ((apart_point_and_line T_1 (line_connecting T_3 T_0)) => ((distinct_lines (line_connecting T_3 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_3 T_1) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (distinct_points T_0 T_3) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) ### DisjTree 75 161 162
% 13.55/13.78 164. (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 Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (distinct_points T_0 T_3) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All Z, ((apart_point_and_line T_1 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((apart_point_and_line T_1 (line_connecting T_3 T_0)) => ((distinct_lines (line_connecting T_3 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) ### All 163
% 13.55/13.80 165. (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All Z, ((apart_point_and_line T_1 (line_connecting T_3 T_0)) => ((distinct_lines (line_connecting T_3 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_3 T_1) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (distinct_points T_0 T_3) (All Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (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)))) ### All 164
% 13.55/13.80 166. (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 Y, (All Z, ((apart_point_and_line T_0 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_0 Z))))) (distinct_points T_0 T_3) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) ### All 165
% 13.55/13.80 167. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_3 T_1) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_3 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (distinct_points T_0 T_3) (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)))) ### All 166
% 13.55/13.80 168. (-. (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
% 13.55/13.80 169. ((apart_point_and_line T_1 (line_connecting T_3 T_0)) => ((distinct_lines (line_connecting T_3 T_0) (line_connecting T_2 T_3)) \/ (apart_point_and_line T_1 (line_connecting T_2 T_3)))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (distinct_points T_0 T_3) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_3 T_0)) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) ### DisjTree 108 155 168
% 13.65/13.82 170. (All Z, ((apart_point_and_line T_1 (line_connecting T_3 T_0)) => ((distinct_lines (line_connecting T_3 T_0) Z) \/ (apart_point_and_line T_1 Z)))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_3 T_0)) (distinct_points T_0 T_1) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (distinct_points T_0 T_3) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) ### All 169
% 13.65/13.82 171. (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (distinct_points T_0 T_3) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_0 T_1) (distinct_lines (line_connecting T_1 T_0) (line_connecting T_3 T_0)) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) ### All 170
% 13.65/13.82 172. (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (apart_point_and_line T_3 (line_connecting T_3 T_0)) ### Axiom
% 13.65/13.82 173. ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) (line_connecting T_3 T_0)) \/ (apart_point_and_line T_3 (line_connecting T_3 T_0)))) (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)) (distinct_points T_0 T_3) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line 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)))))) ### DisjTree 167 171 172
% 13.65/13.82 174. (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_3 T_1) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (distinct_points T_0 T_3) (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)))) ### All 173
% 13.65/13.85 175. (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)) (distinct_points T_0 T_3) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_3 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) ### All 174
% 13.65/13.85 176. ((distinct_points T_3 T_2) /\ (distinct_points T_3 T_3)) ### And 50
% 13.65/13.85 177. ((distinct_points T_2 T_3) => ((apart_point_and_line T_3 (line_connecting T_2 T_3)) => ((distinct_points T_3 T_2) /\ (distinct_points T_3 T_3)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) (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 T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_3 T_1) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (distinct_points T_0 T_3) (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) ### DisjTree 74 175 176
% 13.65/13.85 178. (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (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)) (distinct_points T_0 T_3) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_1))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (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 U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) ### All 177
% 13.65/13.85 179. ((distinct_points T_3 T_3) /\ (distinct_points T_3 T_1)) ### And 50
% 13.65/13.85 180. ((distinct_points T_3 T_1) => ((apart_point_and_line T_3 (line_connecting T_3 T_1)) => ((distinct_points T_3 T_3) /\ (distinct_points T_3 T_1)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) (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 T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (distinct_points T_0 T_3) (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 Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_3 T_1) ### DisjTree 73 178 179
% 13.65/13.87 181. (distinct_points T_3 T_1) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (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)) (distinct_points T_0 T_3) (-. (apart_point_and_line T_3 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (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 U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) ### All 180
% 13.65/13.87 182. ((distinct_points T_3 T_3) /\ (distinct_points T_3 T_0)) ### And 50
% 13.65/13.87 183. ((distinct_points T_3 T_0) => ((apart_point_and_line T_3 (line_connecting T_3 T_0)) => ((distinct_points T_3 T_3) /\ (distinct_points T_3 T_0)))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) (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 T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (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_0 T_1) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_0 T_3) (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 Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_3 T_1) (distinct_points T_3 T_0) ### DisjTree 72 181 182
% 13.65/13.87 184. (All Z, ((distinct_points T_3 T_0) => ((apart_point_and_line Z (line_connecting T_3 T_0)) => ((distinct_points Z T_3) /\ (distinct_points Z T_0))))) (distinct_points T_3 T_0) (distinct_points T_3 T_1) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (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)) (distinct_points T_0 T_3) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_0 T_1))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (distinct_points T_0 T_1) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (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 U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) ### All 183
% 13.65/13.88 185. ((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 Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) (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 T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_0 T_3) (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 Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_3 T_1) (distinct_points T_3 T_0) (All Z, ((distinct_points T_3 T_0) => ((apart_point_and_line Z (line_connecting T_3 T_0)) => ((distinct_points Z T_3) /\ (distinct_points Z T_0))))) (distinct_points T_0 T_1) ### DisjTree 71 184 19
% 13.65/13.88 186. (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 Z, ((distinct_points T_3 T_0) => ((apart_point_and_line Z (line_connecting T_3 T_0)) => ((distinct_points Z T_3) /\ (distinct_points Z T_0))))) (distinct_points T_3 T_0) (distinct_points T_3 T_1) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (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)) (distinct_points T_0 T_3) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (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 U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Z, ((apart_point_and_line T_3 (line_connecting T_1 T_0)) => ((distinct_lines (line_connecting T_1 T_0) Z) \/ (apart_point_and_line T_3 Z)))) ### All 185
% 13.65/13.88 187. (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 T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_0 T_1))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_0 T_3) (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 Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_3 T_1) (distinct_points T_3 T_0) (All Z, ((distinct_points T_3 T_0) => ((apart_point_and_line Z (line_connecting T_3 T_0)) => ((distinct_points Z T_3) /\ (distinct_points Z 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 186
% 13.74/13.90 188. ((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)))) (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 Z, ((distinct_points T_3 T_0) => ((apart_point_and_line Z (line_connecting T_3 T_0)) => ((distinct_points Z T_3) /\ (distinct_points Z T_0))))) (distinct_points T_3 T_0) (distinct_points T_3 T_1) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (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)) (distinct_points T_0 T_3) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (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 U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (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) ### DisjTree 70 187 12
% 13.74/13.90 189. (distinct_points T_0 T_1) (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 T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (apart_point_and_line T_0 (line_connecting T_3 T_0))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_0 T_3) (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 Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_3 T_1) (distinct_points T_3 T_0) (All Z, ((distinct_points T_3 T_0) => ((apart_point_and_line Z (line_connecting T_3 T_0)) => ((distinct_points Z T_3) /\ (distinct_points Z 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 188
% 13.74/13.90 190. ((distinct_points T_0 T_3) /\ (distinct_points T_0 T_0)) ### And 2
% 13.74/13.90 191. ((distinct_points T_3 T_0) => ((apart_point_and_line T_0 (line_connecting T_3 T_0)) => ((distinct_points T_0 T_3) /\ (distinct_points T_0 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 Z, ((distinct_points T_3 T_0) => ((apart_point_and_line Z (line_connecting T_3 T_0)) => ((distinct_points Z T_3) /\ (distinct_points Z T_0))))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (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)) (distinct_points T_0 T_3) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_0) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (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 U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (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) (distinct_points T_3 T_0) ### DisjTree 69 189 190
% 13.74/13.92 192. (distinct_points T_3 T_0) (distinct_points T_0 T_1) (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 T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_0) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (-. (apart_point_and_line T_0 (line_connecting T_1 T_0))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_0 T_3) (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 Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_0) => ((apart_point_and_line Z (line_connecting T_3 T_0)) => ((distinct_points Z T_3) /\ (distinct_points Z 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 191
% 13.74/13.92 193. ((distinct_points T_0 T_1) /\ (distinct_points T_0 T_0)) ### And 2
% 13.74/13.92 194. ((distinct_points T_1 T_0) => ((apart_point_and_line T_0 (line_connecting T_1 T_0)) => ((distinct_points T_0 T_1) /\ (distinct_points T_0 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 Z, ((distinct_points T_3 T_0) => ((apart_point_and_line Z (line_connecting T_3 T_0)) => ((distinct_points Z T_3) /\ (distinct_points Z T_0))))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (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)) (distinct_points T_0 T_3) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (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 U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (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) (distinct_points T_3 T_0) (distinct_points T_1 T_0) ### DisjTree 68 192 193
% 13.74/13.92 195. (distinct_points T_1 T_0) (distinct_points T_3 T_0) (distinct_points T_0 T_1) (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 T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_0 T_3) (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 Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_0) => ((apart_point_and_line Z (line_connecting T_3 T_0)) => ((distinct_points Z T_3) /\ (distinct_points Z 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 194
% 13.74/13.92 196. ((distinct_points T_1 T_0) => ((distinct_points T_1 T_0) \/ (distinct_points T_0 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 Z, ((distinct_points T_3 T_0) => ((apart_point_and_line Z (line_connecting T_3 T_0)) => ((distinct_points Z T_3) /\ (distinct_points Z T_0))))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (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)) (distinct_points T_0 T_3) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (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 U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (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_3 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 67 195 2
% 13.74/13.94 197. (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_3 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 U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_0 T_3) (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 Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_0) => ((apart_point_and_line Z (line_connecting T_3 T_0)) => ((distinct_points Z T_3) /\ (distinct_points Z 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 196
% 13.74/13.94 198. ((distinct_points T_0 T_3) => ((distinct_points T_0 T_0) \/ (distinct_points T_3 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 Z, ((distinct_points T_3 T_0) => ((apart_point_and_line Z (line_connecting T_3 T_0)) => ((distinct_points Z T_3) /\ (distinct_points Z T_0))))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (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_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (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 U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (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 Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (distinct_points T_0 T_3) ### DisjTree 63 2 197
% 13.74/13.94 199. (All Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (distinct_points T_0 T_3) (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) (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 T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 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 Z, ((distinct_points T_2 T_3) => ((apart_point_and_line Z (line_connecting T_2 T_3)) => ((distinct_points Z T_2) /\ (distinct_points Z T_3))))) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_0) => ((apart_point_and_line Z (line_connecting T_3 T_0)) => ((distinct_points Z T_3) /\ (distinct_points Z 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 198
% 13.74/13.95 200. (All Y, (All Z, ((distinct_points T_2 Y) => ((apart_point_and_line Z (line_connecting T_2 Y)) => ((distinct_points Z T_2) /\ (distinct_points Z Y)))))) (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 Z, ((distinct_points T_3 T_0) => ((apart_point_and_line Z (line_connecting T_3 T_0)) => ((distinct_points Z T_3) /\ (distinct_points Z T_0))))) (distinct_points T_3 T_1) (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_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (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 U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (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 Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (distinct_points T_0 T_3) (All Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) ### All 199
% 13.74/13.95 201. (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_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (distinct_points T_0 T_3) (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) (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 T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 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) (distinct_points T_3 T_1) (All Z, ((distinct_points T_3 T_0) => ((apart_point_and_line Z (line_connecting T_3 T_0)) => ((distinct_points Z T_3) /\ (distinct_points Z 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 200
% 13.74/13.95 202. (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (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_3 T_1) (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_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All Z, ((distinct_points T_3 T_1) => ((apart_point_and_line Z (line_connecting T_3 T_1)) => ((distinct_points Z T_3) /\ (distinct_points Z T_1))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (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 U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (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 Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (distinct_points T_0 T_3) (All Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 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 201
% 13.74/13.97 203. (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_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (distinct_points T_0 T_3) (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) (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 T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (distinct_points T_1 T_3) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 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) (distinct_points T_3 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_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) ### All 202
% 13.74/13.97 204. (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 Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (distinct_points T_3 T_1) (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_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_1 T_3) (All Z, ((distinct_points T_1 T_0) => ((apart_point_and_line Z (line_connecting T_1 T_0)) => ((distinct_points Z T_1) /\ (distinct_points Z T_0))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (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 U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (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 Z, ((distinct_points T_1 T_0) => ((distinct_points T_1 Z) \/ (distinct_points T_0 Z)))) (distinct_points T_0 T_3) (All Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 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 203
% 13.74/13.97 205. (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (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_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (distinct_points T_0 T_3) (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) (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 T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_3) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 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) (distinct_points T_3 T_1) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (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 204
% 13.74/13.97 206. (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (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 Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (distinct_points T_3 T_1) (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_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_1 T_3) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (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 U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (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)))) (distinct_points T_0 T_3) (All Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 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 Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) ### All 205
% 13.74/13.99 207. (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (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_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (distinct_points T_0 T_3) (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, (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 T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_3) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 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) (distinct_points T_3 T_1) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) ### All 206
% 13.74/13.99 208. (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (distinct_points T_3 T_1) (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_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_1 T_3) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All Y, (All Z, ((apart_point_and_line T_3 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_3 Z))))) (-. (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 U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (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)))) (distinct_points T_0 T_3) (All Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 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 Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) ### All 207
% 13.74/13.99 209. (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (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_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (distinct_points T_0 T_3) (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, (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 T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_3) (All U, (All V, (((distinct_points T_1 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_1 U) \/ ((apart_point_and_line T_1 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (distinct_points T_3 T_1) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) ### All 208
% 13.84/13.99 210. (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 Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (distinct_points T_3 T_1) (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_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_1 T_3) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (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 U, (All V, (((distinct_points T_3 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (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)))) (distinct_points T_0 T_3) (All Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 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 Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) ### All 209
% 13.84/13.99 211. (All Y, (All U, (All V, (((distinct_points T_3 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (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_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (distinct_points T_0 T_3) (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, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_3) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_2 T_3) (distinct_points T_3 T_1) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points 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 210
% 13.84/13.99 212. ((distinct_points T_0 T_3) => ((distinct_points T_0 T_3) \/ (distinct_points T_3 T_3))) (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 Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points Y Z))))) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (distinct_points T_3 T_1) (distinct_points T_2 T_3) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_1 T_3) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (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, ((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 Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Y, (All U, (All V, (((distinct_points T_3 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (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) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (All Y, (All Z, ((distinct_points T_3 Y) => ((distinct_points T_3 Z) \/ (distinct_points Y Z))))) ### DisjTree 62 211 50
% 13.84/13.99 213. (All Y, (All Z, ((distinct_points T_3 Y) => ((distinct_points T_3 Z) \/ (distinct_points Y Z))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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 U, (All V, (((distinct_points T_3 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_3) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_2 T_3) (distinct_points T_3 T_1) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((distinct_points T_1 Z) \/ (distinct_points 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 212
% 13.84/14.02 214. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points 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 Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (distinct_points T_3 T_1) (distinct_points T_2 T_3) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_1 T_3) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (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, ((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 Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Y, (All U, (All V, (((distinct_points T_3 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (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) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (All Y, (All Z, ((distinct_points T_3 Y) => ((distinct_points T_3 Z) \/ (distinct_points Y Z))))) ### All 213
% 13.84/14.02 215. (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 Y, (All Z, ((distinct_points T_3 Y) => ((distinct_points T_3 Z) \/ (distinct_points Y Z))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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 U, (All V, (((distinct_points T_3 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_1 T_3) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_2 T_3) (distinct_points T_3 T_1) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 214
% 13.84/14.02 216. ((distinct_points T_3 T_1) => ((distinct_points T_3 T_3) \/ (distinct_points T_1 T_3))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (distinct_points T_2 T_3) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (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, ((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 Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Y, (All U, (All V, (((distinct_points T_3 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (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) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (All Y, (All Z, ((distinct_points T_3 Y) => ((distinct_points T_3 Z) \/ (distinct_points 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))))))))) (distinct_points T_3 T_1) ### DisjTree 49 50 215
% 13.84/14.02 217. (All Z, ((distinct_points T_3 T_1) => ((distinct_points T_3 Z) \/ (distinct_points T_1 Z)))) (distinct_points T_3 T_1) (All X, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (All Y, (All Z, ((distinct_points T_3 Y) => ((distinct_points T_3 Z) \/ (distinct_points Y Z))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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 U, (All V, (((distinct_points T_3 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_2 T_3) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 216
% 13.84/14.02 218. ((distinct_points T_3 T_1) => ((distinct_points T_3 T_1) \/ (distinct_points T_1 T_1))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (distinct_points T_2 T_3) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (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, ((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 Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Y, (All U, (All V, (((distinct_points T_3 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (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_3 Y) => ((distinct_points T_3 Z) \/ (distinct_points 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 Z, ((distinct_points T_3 T_1) => ((distinct_points T_3 Z) \/ (distinct_points T_1 Z)))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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))))) ### DisjTree 48 217 18
% 13.84/14.02 219. (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) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (All Z, ((distinct_points T_3 T_1) => ((distinct_points T_3 Z) \/ (distinct_points T_1 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 Y, (All Z, ((distinct_points T_3 Y) => ((distinct_points T_3 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 Y, (All U, (All V, (((distinct_points T_3 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_2 T_3) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 218
% 13.84/14.02 220. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (distinct_points T_2 T_3) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (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, ((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 Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Y, (All U, (All V, (((distinct_points T_3 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (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_3 Y) => ((distinct_points T_3 Z) \/ (distinct_points 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))))))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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 219
% 13.84/14.02 221. (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) (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, (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 U, (All V, (((distinct_points T_3 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_3 U) \/ ((apart_point_and_line T_3 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_2 T_3) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 220
% 13.84/14.04 222. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (distinct_points T_2 T_3) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (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, ((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 Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (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 U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (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 221
% 13.84/14.04 223. (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) (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, (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_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (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)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_2 T_3) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 222
% 13.84/14.04 224. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (distinct_points T_2 T_3) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (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, ((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 Z, ((distinct_points T_0 T_3) => ((distinct_points T_0 Z) \/ (distinct_points T_3 Z)))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (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 U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### All 223
% 13.84/14.04 225. (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (distinct_points 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, (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_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z 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)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (distinct_points T_2 T_3) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 224
% 13.84/14.04 226. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (distinct_points T_2 T_3) (All U, (All V, (((distinct_points T_0 T_3) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_3 U) \/ (apart_point_and_line T_3 V))))))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (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 U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) ### All 225
% 13.84/14.04 227. (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (All Y, (All Z, ((distinct_points T_0 Y) => ((distinct_points T_0 Z) \/ (distinct_points Y Z))))) (distinct_points 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, (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_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (apart_point_and_line T_1 (line_connecting T_2 T_3))) (-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (distinct_points T_2 T_3) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 226
% 13.84/14.04 228. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (distinct_points T_2 T_3) (All U, (All V, (((distinct_points T_0 T_1) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line T_1 U) \/ (apart_point_and_line T_1 V))))))) (-. (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (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 U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) ### All 227
% 13.84/14.04 229. (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ (distinct_lines U V)) => ((apart_point_and_line T_0 U) \/ ((apart_point_and_line T_0 V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V)))))))) (distinct_points 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, (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_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (apart_point_and_line 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 Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 228
% 13.84/14.04 230. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (All Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (distinct_points T_2 T_3) (-. (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All Y, (All Z, ((distinct_points T_1 Y) => ((apart_point_and_line Z (line_connecting T_1 Y)) => ((distinct_points Z T_1) /\ (distinct_points Z Y)))))) (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 U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### All 229
% 13.84/14.04 231. (distinct_points 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, (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (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 Y, (All Z, ((distinct_points T_3 Y) => ((apart_point_and_line Z (line_connecting T_3 Y)) => ((distinct_points Z T_3) /\ (distinct_points Z Y)))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### All 230
% 13.84/14.05 232. (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) (distinct_points T_2 T_3) (-. (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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, (All Y, (All U, (All V, (((distinct_points X Y) /\ (distinct_lines U V)) => ((apart_point_and_line X U) \/ ((apart_point_and_line X V) \/ ((apart_point_and_line Y U) \/ (apart_point_and_line Y V))))))))) (apart_point_and_line T_3 (line_connecting T_0 T_1)) (distinct_points T_0 T_1) ### All 231
% 13.84/14.05 233. (-. (-. (apart_point_and_line T_3 (line_connecting T_0 T_1)))) (distinct_points 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, (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (-. (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, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotNot 232
% 13.84/14.05 234. (-. ((-. (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 Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points 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 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) (distinct_points T_2 T_3) (-. (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### NotAnd 42 233
% 13.84/14.05 235. (-. (((distinct_points T_0 T_1) /\ ((distinct_points T_2 T_3) /\ ((-. (apart_point_and_line T_0 (line_connecting T_2 T_3))) /\ (-. (apart_point_and_line T_1 (line_connecting T_2 T_3)))))) => ((-. (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 Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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, (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_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### ConjTree 234
% 13.84/14.05 236. (-. (All V, (((distinct_points T_0 T_1) /\ ((distinct_points T_2 V) /\ ((-. (apart_point_and_line T_0 (line_connecting T_2 V))) /\ (-. (apart_point_and_line T_1 (line_connecting T_2 V)))))) => ((-. (apart_point_and_line T_2 (line_connecting T_0 T_1))) /\ (-. (apart_point_and_line V (line_connecting T_0 T_1))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points 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 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### NotAllEx 235
% 13.84/14.05 237. (-. (All U, (All V, (((distinct_points T_0 T_1) /\ ((distinct_points U V) /\ ((-. (apart_point_and_line T_0 (line_connecting U V))) /\ (-. (apart_point_and_line T_1 (line_connecting U V)))))) => ((-. (apart_point_and_line U (line_connecting T_0 T_1))) /\ (-. (apart_point_and_line V (line_connecting T_0 T_1)))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All X, (All Y, (All Z, ((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 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_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotAllEx 236
% 13.84/14.05 238. (-. (All Y, (All U, (All V, (((distinct_points T_0 Y) /\ ((distinct_points U V) /\ ((-. (apart_point_and_line T_0 (line_connecting U V))) /\ (-. (apart_point_and_line Y (line_connecting U V)))))) => ((-. (apart_point_and_line U (line_connecting T_0 Y))) /\ (-. (apart_point_and_line V (line_connecting T_0 Y))))))))) (All X, (All Y, (All Z, ((distinct_points X Y) => ((distinct_points X Z) \/ (distinct_points 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 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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) ### NotAllEx 237
% 13.84/14.05 239. (-. (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, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X 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, (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_points X Y) => ((distinct_points X Z) \/ (distinct_points Y Z)))))) ### NotAllEx 238
% 13.84/14.05 % SZS output end Proof
% 13.84/14.05 (* END-PROOF *)
%------------------------------------------------------------------------------