TSTP Solution File: GEO222+3 by SuperZenon---0.0.1
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%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : GEO222+3 : TPTP v8.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% Computer : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 600s
% DateTime : Sat Jul 16 06:31:18 EDT 2022
% Result : Theorem 20.51s 20.73s
% Output : Proof 20.70s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GEO222+3 : TPTP v8.1.0. Released v4.0.0.
% 0.03/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.13/0.34 % Computer : n016.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 600
% 0.13/0.34 % DateTime : Sat Jun 18 11:16:52 EDT 2022
% 0.13/0.34 % CPUTime :
% 20.51/20.73 % SZS status Theorem
% 20.51/20.73 (* PROOF-FOUND *)
% 20.51/20.73 (* BEGIN-PROOF *)
% 20.51/20.73 % SZS output start Proof
% 20.51/20.73 1. (convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### Axiom
% 20.51/20.73 2. (convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) ### Axiom
% 20.51/20.73 3. (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_0) ### Axiom
% 20.51/20.73 4. (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) ### Axiom
% 20.51/20.73 5. (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### Axiom
% 20.51/20.73 6. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) ### Axiom
% 20.51/20.73 7. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) ### Axiom
% 20.51/20.73 8. (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) ### All 7
% 20.51/20.73 9. (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) ### Axiom
% 20.51/20.73 10. (convergent_lines (orthogonal_through_point T_0 T_1) T_0) (-. (convergent_lines (orthogonal_through_point T_0 T_1) T_0)) ### Axiom
% 20.51/20.73 11. (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0))) => (-. (convergent_lines (orthogonal_through_point T_0 T_1) T_0))) (convergent_lines (orthogonal_through_point T_0 T_1) T_0) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) ### DisjTree 8 9 10
% 20.51/20.73 12. (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines (orthogonal_through_point T_0 T_1) N)))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) (convergent_lines (orthogonal_through_point T_0 T_1) T_0) ### All 11
% 20.51/20.73 13. (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (convergent_lines (orthogonal_through_point T_0 T_1) T_0) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) ### All 12
% 20.51/20.73 14. (-. (-. (convergent_lines (orthogonal_through_point T_0 T_1) T_0))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) ### NotNot 13
% 20.51/20.73 15. (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0) ### Axiom
% 20.51/20.74 16. (-. (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) ### NotNot 15
% 20.51/20.74 17. (-. ((-. (convergent_lines (orthogonal_through_point T_0 T_1) T_0)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) ### NotAnd 14 16
% 20.51/20.74 18. (All M, (-. ((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) ### All 17
% 20.51/20.74 19. (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) ### All 18
% 20.51/20.74 20. (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) ### All 19
% 20.51/20.74 21. (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All M, (All N, (((-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N))) => (-. (convergent_lines M N))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) ### All 20
% 20.51/20.74 22. (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) ### All 21
% 20.51/20.74 23. (apart_point_and_line T_1 T_0) (-. (apart_point_and_line T_1 T_0)) ### Axiom
% 20.51/20.74 24. (distinct_lines T_0 (orthogonal_through_point T_0 T_1)) (-. (distinct_lines T_0 (orthogonal_through_point T_0 T_1))) ### Axiom
% 20.51/20.74 25. (convergent_lines T_0 (orthogonal_through_point T_0 T_1)) (-. (convergent_lines T_0 (orthogonal_through_point T_0 T_1))) ### Axiom
% 20.51/20.74 26. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) ### Axiom
% 20.51/20.74 27. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) ### Axiom
% 20.58/20.78 28. (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) ### Axiom
% 20.58/20.78 29. ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) ### And 28
% 20.58/20.78 30. (-. (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1))) (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1)) ### Axiom
% 20.58/20.78 31. ((convergent_lines T_0 (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1))) (-. (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1))) ### And 30
% 20.58/20.78 32. (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) \/ ((convergent_lines T_0 (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1))))) (-. (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) ### DisjTree 26 27 29 31
% 20.58/20.78 33. (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (-. (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1))) ### All 32
% 20.58/20.78 34. (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_0) ### Axiom
% 20.58/20.78 35. ((convergent_lines T_0 T_0) /\ (unorthogonal_lines T_0 T_0)) (-. (convergent_lines T_0 T_0)) ### And 34
% 20.58/20.78 36. (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0) ### Axiom
% 20.58/20.78 37. ((convergent_lines (orthogonal_through_point T_0 T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) ### And 36
% 20.58/20.78 38. (((convergent_lines T_0 (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1))) => (((convergent_lines T_0 T_0) /\ (unorthogonal_lines T_0 T_0)) \/ ((convergent_lines (orthogonal_through_point T_0 T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (-. (convergent_lines T_0 T_0)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) (convergent_lines T_0 (orthogonal_through_point T_0 T_1)) ### DisjTree 25 33 35 37
% 20.58/20.78 39. (All N, (((convergent_lines T_0 (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1))) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))))) (convergent_lines T_0 (orthogonal_through_point T_0 T_1)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (-. (convergent_lines T_0 T_0)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) ### All 38
% 20.58/20.78 40. (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (-. (convergent_lines T_0 T_0)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) (convergent_lines T_0 (orthogonal_through_point T_0 T_1)) ### All 39
% 20.58/20.78 41. ((distinct_lines T_0 (orthogonal_through_point T_0 T_1)) => (convergent_lines T_0 (orthogonal_through_point T_0 T_1))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (-. (convergent_lines T_0 T_0)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (distinct_lines T_0 (orthogonal_through_point T_0 T_1)) ### Imply 24 40
% 20.58/20.81 42. (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (distinct_lines T_0 (orthogonal_through_point T_0 T_1)) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (-. (convergent_lines T_0 T_0)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) ### All 41
% 20.58/20.81 43. (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1)) ### Axiom
% 20.58/20.81 44. ((apart_point_and_line T_1 T_0) => ((distinct_lines T_0 (orthogonal_through_point T_0 T_1)) \/ (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1)))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (-. (convergent_lines T_0 T_0)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (apart_point_and_line T_1 T_0) ### DisjTree 23 42 43
% 20.58/20.81 45. (All Z, ((apart_point_and_line T_1 T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 T_0) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (-. (convergent_lines T_0 T_0)) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) ### All 44
% 20.58/20.81 46. (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (apart_point_and_line T_1 (orthogonal_through_point T_0 T_1))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines T_0 T_0)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (apart_point_and_line T_1 T_0) (All Z, ((apart_point_and_line T_1 T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line T_1 Z)))) ### All 45
% 20.58/20.81 47. (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All Z, ((apart_point_and_line T_1 T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 T_0) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (-. (convergent_lines T_0 T_0)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) ### All 46
% 20.58/20.81 48. (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines T_0 T_0)) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (apart_point_and_line T_1 T_0) (All Z, ((apart_point_and_line T_1 T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line T_1 Z)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) ### All 47
% 20.58/20.81 49. (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All Z, ((apart_point_and_line T_1 T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 T_0) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines T_0 T_0)) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) ### All 48
% 20.58/20.82 50. ((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (convergent_lines T_0 T_0)) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (apart_point_and_line T_1 T_0) (All Z, ((apart_point_and_line T_1 T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line T_1 Z)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) ### And 49
% 20.58/20.82 51. (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) \/ ((convergent_lines T_0 T_0) /\ (unorthogonal_lines T_0 T_0)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All Z, ((apart_point_and_line T_1 T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 T_0) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines T_0 T_0)) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) ### DisjTree 6 22 50 35
% 20.58/20.82 52. (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N))))) (-. (convergent_lines T_0 T_0)) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (apart_point_and_line T_1 T_0) (All Z, ((apart_point_and_line T_1 T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line T_1 Z)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) ### All 51
% 20.58/20.82 53. (All M, (All N, (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M)) => (((convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) /\ (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All Z, ((apart_point_and_line T_1 T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line T_1 Z)))) (apart_point_and_line T_1 T_0) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines T_0 T_0)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) ### All 52
% 20.58/20.82 54. (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines T_0 T_0)) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (apart_point_and_line T_1 T_0) (All Z, ((apart_point_and_line T_1 T_0) => ((distinct_lines T_0 Z) \/ (apart_point_and_line T_1 Z)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) ### All 53
% 20.58/20.82 55. (All Y, (All Z, ((apart_point_and_line T_1 Y) => ((distinct_lines Y Z) \/ (apart_point_and_line T_1 Z))))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (apart_point_and_line T_1 T_0) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines T_0 T_0)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) ### All 54
% 20.67/20.85 56. (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines T_0 T_0)) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (apart_point_and_line T_1 T_0) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) ### All 55
% 20.67/20.85 57. (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) ### Axiom
% 20.67/20.85 58. (convergent_lines (orthogonal_through_point T_0 T_1) T_0) (-. (convergent_lines (orthogonal_through_point T_0 T_1) T_0)) ### Axiom
% 20.67/20.85 59. (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) ### Axiom
% 20.67/20.85 60. (-. (convergent_lines T_0 (orthogonal_through_point T_0 T_1))) (convergent_lines T_0 (orthogonal_through_point T_0 T_1)) ### Axiom
% 20.67/20.85 61. ((convergent_lines (orthogonal_through_point T_0 T_1) T_0) => ((convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1)) \/ (convergent_lines T_0 (orthogonal_through_point T_0 T_1)))) (-. (convergent_lines T_0 (orthogonal_through_point T_0 T_1))) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (convergent_lines (orthogonal_through_point T_0 T_1) T_0) ### DisjTree 58 59 60
% 20.67/20.85 62. (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) T_0) => ((convergent_lines (orthogonal_through_point T_0 T_1) Z) \/ (convergent_lines T_0 Z)))) (convergent_lines (orthogonal_through_point T_0 T_1) T_0) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (-. (convergent_lines T_0 (orthogonal_through_point T_0 T_1))) ### All 61
% 20.67/20.85 63. (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1)) (-. (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1))) ### Axiom
% 20.67/20.85 64. (((convergent_lines T_0 (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1))) => (((convergent_lines T_0 T_0) /\ (unorthogonal_lines T_0 T_0)) \/ ((convergent_lines (orthogonal_through_point T_0 T_1) T_0) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (-. (convergent_lines T_0 T_0)) (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (convergent_lines (orthogonal_through_point T_0 T_1) T_0) (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) T_0) => ((convergent_lines (orthogonal_through_point T_0 T_1) Z) \/ (convergent_lines T_0 Z)))) ### DisjTree 62 63 35 37
% 20.67/20.85 65. (All N, (((convergent_lines T_0 (orthogonal_through_point T_0 T_1)) /\ (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1))) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines (orthogonal_through_point T_0 T_1) N) /\ (unorthogonal_lines (orthogonal_through_point T_0 T_1) N))))) (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) T_0) => ((convergent_lines (orthogonal_through_point T_0 T_1) Z) \/ (convergent_lines T_0 Z)))) (convergent_lines (orthogonal_through_point T_0 T_1) T_0) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1)) (-. (convergent_lines T_0 T_0)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) ### All 64
% 20.67/20.85 66. (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (-. (convergent_lines T_0 T_0)) (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1)) (-. (convergent_lines (orthogonal_through_point T_0 T_1) (orthogonal_through_point T_0 T_1))) (convergent_lines (orthogonal_through_point T_0 T_1) T_0) (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) T_0) => ((convergent_lines (orthogonal_through_point T_0 T_1) Z) \/ (convergent_lines T_0 Z)))) ### All 65
% 20.67/20.85 67. (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) T_0) => ((convergent_lines (orthogonal_through_point T_0 T_1) Z) \/ (convergent_lines T_0 Z)))) (convergent_lines (orthogonal_through_point T_0 T_1) T_0) (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1)) (-. (convergent_lines T_0 T_0)) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) ### All 66
% 20.67/20.85 68. ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 T_0) \/ ((unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1)) \/ (unorthogonal_lines T_0 (orthogonal_through_point T_0 T_1)))))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (convergent_lines (orthogonal_through_point T_0 T_1) T_0) (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) T_0) => ((convergent_lines (orthogonal_through_point T_0 T_1) Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines T_0 T_0)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (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 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) ### DisjTree 4 5 56 57 67
% 20.67/20.86 69. (All N, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 T_0) \/ ((unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) \/ (unorthogonal_lines T_0 N)))))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) 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 L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines T_0 T_0)) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (-. (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) (orthogonal_through_point T_0 T_1))) (All X, (-. (convergent_lines X X))) (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) T_0) => ((convergent_lines (orthogonal_through_point T_0 T_1) Z) \/ (convergent_lines T_0 Z)))) (convergent_lines (orthogonal_through_point T_0 T_1) T_0) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) ### All 68
% 20.67/20.86 70. (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (convergent_lines (orthogonal_through_point T_0 T_1) T_0) (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) T_0) => ((convergent_lines (orthogonal_through_point T_0 T_1) Z) \/ (convergent_lines T_0 Z)))) (All X, (-. (convergent_lines X X))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines T_0 T_0)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (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 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All N, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 T_0) \/ ((unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) \/ (unorthogonal_lines T_0 N)))))) ### All 69
% 20.67/20.86 71. (All Y, (All Z, ((convergent_lines (orthogonal_through_point T_0 T_1) Y) => ((convergent_lines (orthogonal_through_point T_0 T_1) Z) \/ (convergent_lines Y Z))))) (All N, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 T_0) \/ ((unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) \/ (unorthogonal_lines T_0 N)))))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) 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 L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines T_0 T_0)) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All X, (-. (convergent_lines X X))) (convergent_lines (orthogonal_through_point T_0 T_1) T_0) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) ### All 70
% 20.67/20.86 72. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (convergent_lines (orthogonal_through_point T_0 T_1) T_0) (All X, (-. (convergent_lines X X))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines T_0 T_0)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (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 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All N, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 T_0) \/ ((unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) \/ (unorthogonal_lines T_0 N)))))) ### All 71
% 20.70/20.89 73. (-. (-. (convergent_lines (orthogonal_through_point T_0 T_1) T_0))) (All N, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 T_0) \/ ((unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) \/ (unorthogonal_lines T_0 N)))))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) 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 L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines T_0 T_0)) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All X, (-. (convergent_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotNot 72
% 20.70/20.89 74. (-. ((-. (convergent_lines (orthogonal_through_point T_0 T_1) T_0)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All X, (-. (convergent_lines X X))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines T_0 T_0)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (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 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All N, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 T_0) \/ ((unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) \/ (unorthogonal_lines T_0 N)))))) ### NotAnd 73 16
% 20.70/20.89 75. (All M, (-. ((-. (convergent_lines (orthogonal_through_point T_0 T_1) M)) /\ (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) M))))) (All N, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 T_0) \/ ((unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) \/ (unorthogonal_lines T_0 N)))))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) 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 L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines T_0 T_0)) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All X, (-. (convergent_lines X X))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 74
% 20.70/20.89 76. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (-. (unorthogonal_lines (orthogonal_through_point T_0 T_1) T_0)) (All X, (-. (convergent_lines X X))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines T_0 T_0)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (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 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All N, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 T_0) \/ ((unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) \/ (unorthogonal_lines T_0 N)))))) ### All 75
% 20.70/20.89 77. (All N, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 T_0) \/ ((unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) \/ (unorthogonal_lines T_0 N)))))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) 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 L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines T_0 T_0)) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All X, (-. (convergent_lines X X))) (All L, (-. (unorthogonal_lines (orthogonal_through_point L T_1) L))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 76
% 20.70/20.90 78. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines T_0 T_0)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (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 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All N, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 T_0) \/ ((unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) \/ (unorthogonal_lines T_0 N)))))) ### All 77
% 20.70/20.90 79. (All M, (All N, ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) M) => ((apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) \/ ((apart_point_and_line T_1 M) \/ ((unorthogonal_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) N) \/ (unorthogonal_lines M N))))))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) 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 L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines T_0 T_0)) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 78
% 20.70/20.90 80. (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line T_1 L) \/ ((apart_point_and_line T_1 M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N)))))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (-. (convergent_lines T_0 T_0)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (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 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) ### All 79
% 20.70/20.90 81. (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) 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 L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines T_0 T_0)) (All M, (All N, (((convergent_lines T_0 M) /\ (unorthogonal_lines T_0 M)) => (((convergent_lines T_0 N) /\ (unorthogonal_lines T_0 N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N)))))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 80
% 20.70/20.90 82. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All Y, ((distinct_lines T_0 Y) => (convergent_lines T_0 Y))) (-. (convergent_lines T_0 T_0)) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (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 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) ### All 81
% 20.70/20.90 83. (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) 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 L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (-. (convergent_lines T_0 T_0)) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 82
% 20.70/20.90 84. ((convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines T_0 T_0) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (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 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### DisjTree 2 3 83
% 20.70/20.90 85. (All Z, ((convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines T_0 Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (-. (convergent_lines T_0 T_0)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) (distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) 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 L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 84
% 20.70/20.90 86. (-. (convergent_lines T_0 T_0)) (convergent_lines T_0 T_0) ### Axiom
% 20.70/20.90 87. ((convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) T_0) \/ (convergent_lines T_0 T_0))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (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 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (-. (convergent_lines T_0 T_0)) (All Z, ((convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines T_0 Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) ### DisjTree 1 85 86
% 20.70/20.90 88. (All Z, ((convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z) \/ (convergent_lines T_0 Z)))) (convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All Z, ((convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines T_0 Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (-. (convergent_lines T_0 T_0)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) 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 L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 87
% 20.70/20.92 89. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (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 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All Z, ((convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines T_0 Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All Z, ((convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((distinct_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z) \/ (convergent_lines T_0 Z)))) ### All 88
% 20.70/20.92 90. (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All Z, ((convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines T_0 Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) (-. (apart_point_and_line T_1 (orthogonal_through_point (orthogonal_through_point T_0 T_1) 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 L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 89
% 20.70/20.92 91. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All L, (-. (apart_point_and_line T_1 (orthogonal_through_point L T_1)))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All Z, ((convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines T_0 Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) ### All 90
% 20.70/20.92 92. (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All Z, ((convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) => ((convergent_lines T_0 Z) \/ (convergent_lines (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1) Z)))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### All 91
% 20.70/20.92 93. (All Y, (All Z, ((convergent_lines T_0 Y) => ((convergent_lines T_0 Z) \/ (convergent_lines Y Z))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All Y, (All Z, ((convergent_lines T_0 Y) => ((distinct_lines Y Z) \/ (convergent_lines T_0 Z))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) ### All 92
% 20.70/20.92 94. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All Y, (All Z, ((convergent_lines T_0 Y) => ((convergent_lines T_0 Z) \/ (convergent_lines Y Z))))) ### All 93
% 20.70/20.92 95. (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### All 94
% 20.70/20.92 96. (-. (-. (convergent_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotNot 95
% 20.70/20.92 97. (-. (parallel_lines T_0 (orthogonal_through_point (orthogonal_through_point T_0 T_1) T_1))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### Definition-Pseudo(parallel_lines) 96
% 20.70/20.92 98. (-. (All L, (parallel_lines L (orthogonal_through_point (orthogonal_through_point L T_1) T_1)))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All X, (-. (convergent_lines X X))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) ### NotAllEx 97
% 20.70/20.92 99. (-. (All A, (All L, (parallel_lines L (orthogonal_through_point (orthogonal_through_point L A) A))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((convergent_lines X Z) \/ (convergent_lines Y Z)))))) (All X, (-. (convergent_lines X X))) (All A, (All L, (-. (unorthogonal_lines (orthogonal_through_point L A) L)))) (All L, (All M, (-. ((-. (convergent_lines L M)) /\ (-. (unorthogonal_lines L M)))))) (All L, (All M, (All N, (((-. (unorthogonal_lines L M)) /\ (-. (unorthogonal_lines L N))) => (-. (convergent_lines M N)))))) (All L, (All M, (All N, (((convergent_lines L M) /\ (unorthogonal_lines L M)) => (((convergent_lines L N) /\ (unorthogonal_lines L N)) \/ ((convergent_lines M N) /\ (unorthogonal_lines M N))))))) (All X, (All Y, (All Z, ((apart_point_and_line X Y) => ((distinct_lines Y Z) \/ (apart_point_and_line X Z)))))) (All A, (All L, (All M, (All N, ((distinct_lines L M) => ((apart_point_and_line A L) \/ ((apart_point_and_line A M) \/ ((unorthogonal_lines L N) \/ (unorthogonal_lines M N))))))))) (All X, (All Y, ((distinct_lines X Y) => (convergent_lines X Y)))) (All A, (All L, (-. (apart_point_and_line A (orthogonal_through_point L A))))) (All X, (All Y, (All Z, ((convergent_lines X Y) => ((distinct_lines Y Z) \/ (convergent_lines X Z)))))) ### NotAllEx 98
% 20.70/20.92 % SZS output end Proof
% 20.70/20.92 (* END-PROOF *)
%------------------------------------------------------------------------------