%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP147^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n010.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 : 300s
% DateTime : Wed Feb 25 08:55:56 AM UTC 2026
% Result : Theorem 2.77s 1.75s
% Output : Refutation 2.77s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP147^2 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n010.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 08:27:33 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.29/0.45 ---- Original DTF file ---
% 0.29/0.45 thf(spec,logic,$$dhol).
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % File : ITP147^2 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.45 % Domain : Interactive Theorem Proving
% 0.29/0.45 % Problem : Sledgehammer Poincare_Bendixson problem prob_2118__19618702_1
% 0.29/0.45 % Version : Especial.
% 0.29/0.45 % English :
% 0.29/0.45
% 0.29/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.45 % Source : [Des21]
% 0.29/0.45 % Names : Poincare_Bendixson/prob_2118__19618702_1 [Des21]
% 0.29/0.45
% 0.29/0.45 % Status : Theorem
% 0.29/0.45 % Rating : 0.00 v9.1.0, 0.33 v8.1.0, 0.00 v7.5.0
% 0.29/0.45 % Syntax : Number of formulae : 386 ( 60 unt; 66 typ; 0 def)
% 0.29/0.45 % Number of atoms : 926 ( 300 equ; 0 cnn)
% 0.29/0.45 % Maximal formula atoms : 11 ( 2 avg)
% 0.29/0.45 % Number of connectives : 3917 ( 53 ~; 6 |; 28 &;3425 @)
% 0.29/0.45 % ( 0 <=>; 405 =>; 0 <=; 0 <~>)
% 0.29/0.45 % Maximal formula depth : 23 ( 7 avg)
% 0.29/0.45 % Number of types : 3 ( 2 usr)
% 0.29/0.45 % Number of type conns : 184 ( 184 >; 0 *; 0 +; 0 <<)
% 0.29/0.45 % Number of symbols : 67 ( 64 usr; 8 con; 0-5 aty)
% 0.29/0.45 % Number of variables : 723 ( 8 ^; 656 !; 3 ?; 723 :)
% 0.29/0.45 % ( 56 !>; 0 ?*; 0 @-; 0 @+)
% 0.29/0.45 % SPC : TH1_THM_EQU_NAR
% 0.29/0.45
% 0.29/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 16:34:48.401
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % Could-be-implicit typings (3)
% 0.29/0.45 thf(ty_t_Real_Oreal,type,
% 0.29/0.45 real: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_t_Set_Oset,type,
% 0.29/0.45 set: $tType > $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_tf_a,type,
% 0.29/0.45 a: $tType ).
% 0.29/0.45
% 0.29/0.45 % Explicit typings (63)
% 0.29/0.45 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.29/0.45 finite_finite:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Countable_Ocountable,type,
% 0.29/0.45 countable:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oone,type,
% 0.29/0.45 one:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ozero,type,
% 0.29/0.45 zero:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Rings_Oabs__if,type,
% 0.29/0.45 abs_if:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Num_Oneg__numeral,type,
% 0.29/0.45 neg_numeral:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ogroup__add,type,
% 0.29/0.45 group_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Omonoid__add,type,
% 0.29/0.45 monoid_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Rings_Oidom__abs__sgn,type,
% 0.29/0.45 idom_abs_sgn:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Rings_Ozero__neq__one,type,
% 0.29/0.45 zero_neq_one:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oab__group__add,type,
% 0.29/0.45 ab_group_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Rings_Ozero__less__one,type,
% 0.29/0.45 zero_less_one:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.29/0.45 semigroup_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.29/0.45 linordered_idom:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Rings_Olinordered__ring,type,
% 0.29/0.45 linordered_ring:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.29/0.45 comm_monoid_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.29/0.45 ab_semigroup_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.29/0.45 cancel_semigroup_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.29/0.45 ordered_ab_group_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.29/0.45 cancel1352612707id_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.29/0.45 linord219039673up_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 0.29/0.45 ordere216010020id_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
% 0.29/0.45 ordere142940540dd_abs:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.29/0.45 ordere779506340up_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
% 0.29/0.45 real_V2090557954_space:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.29/0.45 linord1659791738miring:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.29/0.45 canoni770627133id_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
% 0.29/0.45 ordere1818651114id_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.29/0.45 ordere236663937imp_le:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 0.29/0.45 ordere223160158up_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 0.29/0.45 ordere516151231imp_le:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Executable__Euclidean__Space_Oexecutable__euclidean__space,type,
% 0.29/0.45 execut510477386_space:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Elementary__Metric__Spaces_Oball,type,
% 0.29/0.45 elemen1815086676c_ball:
% 0.29/0.45 !>[A: $tType] : ( A > real > ( set @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Flow_Oauto__ll__on__open_Oexistence__ivl0,type,
% 0.29/0.45 auto_l1112008849e_ivl0:
% 0.29/0.45 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > ( set @ real ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Flow_Oauto__ll__on__open_Oflow0,type,
% 0.29/0.45 auto_ll_on_flow0:
% 0.29/0.45 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > real > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Oabs__class_Oabs,type,
% 0.29/0.45 abs_abs:
% 0.29/0.45 !>[A: $tType] : ( A > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.29/0.45 one_one:
% 0.29/0.45 !>[A: $tType] : A ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.29/0.45 plus_plus:
% 0.29/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.29/0.45 uminus_uminus:
% 0.29/0.45 !>[A: $tType] : ( A > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.29/0.45 zero_zero:
% 0.29/0.45 !>[A: $tType] : A ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_If,type,
% 0.29/0.45 if:
% 0.29/0.45 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Initial__Value__Problem_Ointerval,type,
% 0.29/0.45 initia826609931terval: ( set @ real ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Limit__Set_Oauto__ll__on__open_O_092_060alpha_062__limit__point,type,
% 0.29/0.45 limit_1335059408_point:
% 0.29/0.45 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > A > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Limit__Set_Oauto__ll__on__open_O_092_060omega_062__limit__point,type,
% 0.29/0.45 limit_1052139091_point:
% 0.29/0.45 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > A > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Line__Segment_Oopen__segment,type,
% 0.29/0.45 line_open_segment:
% 0.29/0.45 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.29/0.45 bot_bot:
% 0.29/0.45 !>[A: $tType] : A ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.29/0.45 ord_less:
% 0.29/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.29/0.45 ord_less_eq:
% 0.29/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.29/0.45 top_top:
% 0.29/0.45 !>[A: $tType] : A ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Oclosed__orbit,type,
% 0.29/0.45 period385816147_orbit:
% 0.29/0.45 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiod,type,
% 0.29/0.45 period1153813292period:
% 0.29/0.45 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > real ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiodic__orbit,type,
% 0.29/0.45 period862636932_orbit:
% 0.29/0.45 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Otransversal__segment,type,
% 0.29/0.45 poinca272511729egment:
% 0.29/0.45 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > A > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_OCollect,type,
% 0.29/0.45 collect:
% 0.29/0.45 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Topological__Spaces_Ocontinuous__on,type,
% 0.29/0.45 topolo2071040574ous_on:
% 0.29/0.45 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_member,type,
% 0.29/0.45 member:
% 0.29/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_X,type,
% 0.29/0.45 x: set @ a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_a,type,
% 0.29/0.45 a2: a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_b,type,
% 0.29/0.45 b: a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_f,type,
% 0.29/0.45 f: a > a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_p,type,
% 0.29/0.45 p: a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_thesisa____,type,
% 0.29/0.45 thesisa: $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_x,type,
% 0.29/0.45 x2: a ).
% 0.29/0.45
% 0.29/0.45 % Relevant facts (254)
% 0.29/0.45 thf(fact_0_rev_Oopen__segment__trichotomy,axiom,
% 0.29/0.45 ! [X: a,A2: a,B2: a,Y: a] :
% 0.29/0.45 ( ( member @ a @ X @ ( line_open_segment @ a @ A2 @ B2 ) )
% 0.29/0.45 => ( ( member @ a @ Y @ ( line_open_segment @ a @ A2 @ B2 ) )
% 0.29/0.45 => ( ( X = Y )
% 0.29/0.45 | ( member @ a @ Y @ ( line_open_segment @ a @ X @ B2 ) )
% 0.29/0.45 | ( member @ a @ Y @ ( line_open_segment @ a @ A2 @ X ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % rev.open_segment_trichotomy
% 0.29/0.45 thf(fact_1__092_060open_062p_A_092_060in_062_AX_092_060close_062,axiom,
% 0.29/0.45 member @ a @ p @ x ).
% 0.29/0.45
% 0.29/0.45 % \<open>p \<in> X\<close>
% 0.29/0.45 thf(fact_2_assms_I4_J,axiom,
% 0.29/0.45 member @ a @ p @ ( line_open_segment @ a @ a2 @ b ) ).
% 0.29/0.45
% 0.29/0.45 % assms(4)
% 0.29/0.45 thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062d_At_O_A_092_060lbrakk_0620_A_060_Ad_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_Aflow0_Ay_A_It_Ay_J_A_092_060in_062_A_123a_060_N_N_060b_125_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_A_092_060bar_062t_Ay_092_060bar_062_A_060_A1_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_At_Ap_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.29/0.45 ~ ! [D: real] :
% 0.29/0.45 ( ( ord_less @ real @ ( zero_zero @ real ) @ D )
% 0.29/0.45 => ! [T: a > real] :
% 0.29/0.45 ( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ p @ D ) @ T )
% 0.29/0.45 => ( ! [Y2: a] :
% 0.29/0.45 ( ( member @ a @ Y2 @ ( elemen1815086676c_ball @ a @ p @ D ) )
% 0.29/0.45 => ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ Y2 @ ( T @ Y2 ) ) @ ( line_open_segment @ a @ a2 @ b ) ) )
% 0.29/0.45 => ( ! [Y2: a] :
% 0.29/0.45 ( ( member @ a @ Y2 @ ( elemen1815086676c_ball @ a @ p @ D ) )
% 0.29/0.45 => ( ord_less @ real @ ( abs_abs @ real @ ( T @ Y2 ) ) @ ( one_one @ real ) ) )
% 0.29/0.45 => ( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ p @ D ) @ T )
% 0.29/0.45 => ( ( T @ p )
% 0.29/0.45 != ( zero_zero @ real ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % \<open>\<And>thesis. (\<And>d t. \<lbrakk>0 < d; continuous_on (ball p d) t; \<And>y. y \<in> ball p d \<Longrightarrow> flow0 y (t y) \<in> {a<--<b}; \<And>y. y \<in> ball p d \<Longrightarrow> \<bar>t y\<bar> < 1; continuous_on (ball p d) t; t p = 0\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.29/0.45 thf(fact_4_that,axiom,
% 0.29/0.45 ! [Delta: real,Tau: a > real] :
% 0.29/0.45 ( ( ord_less @ real @ ( zero_zero @ real ) @ Delta )
% 0.29/0.45 => ( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ p @ Delta ) @ Tau )
% 0.29/0.45 => ( ( ( Tau @ p )
% 0.29/0.45 = ( zero_zero @ real ) )
% 0.29/0.45 => ( ! [Y3: a] :
% 0.29/0.45 ( ( member @ a @ Y3 @ ( elemen1815086676c_ball @ a @ p @ Delta ) )
% 0.29/0.45 => ( ord_less @ real @ ( abs_abs @ real @ ( Tau @ Y3 ) ) @ ( one_one @ real ) ) )
% 0.29/0.45 => ( ! [Y3: a] :
% 0.29/0.45 ( ( member @ a @ Y3 @ ( elemen1815086676c_ball @ a @ p @ Delta ) )
% 0.29/0.45 => ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ Y3 @ ( Tau @ Y3 ) ) @ ( line_open_segment @ a @ a2 @ b ) ) )
% 0.29/0.45 => thesisa ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % that
% 0.29/0.45 thf(fact_5_assms_I1_J,axiom,
% 0.29/0.45 poinca272511729egment @ a @ f @ x @ a2 @ b ).
% 0.29/0.45
% 0.29/0.45 % assms(1)
% 0.29/0.45 thf(fact_6_assms_I3_J,axiom,
% 0.29/0.45 limit_1052139091_point @ a @ f @ x @ x2 @ p ).
% 0.29/0.45
% 0.29/0.45 % assms(3)
% 0.29/0.45 thf(fact_7_centre__in__ball,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( real_V2090557954_space @ A )
% 0.29/0.45 => ! [X: A,E: real] :
% 0.29/0.45 ( ( member @ A @ X @ ( elemen1815086676c_ball @ A @ X @ E ) )
% 0.29/0.45 = ( ord_less @ real @ ( zero_zero @ real ) @ E ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % centre_in_ball
% 0.29/0.45 thf(fact_8_zero__less__abs__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere142940540dd_abs @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) )
% 0.29/0.45 = ( A2
% 0.29/0.45 != ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_less_abs_iff
% 0.29/0.45 thf(fact_9_cross__time__continuous,axiom,
% 0.29/0.45 ! [A2: a,B2: a,X: a,E: real] :
% 0.29/0.45 ( ( poinca272511729egment @ a @ f @ x @ A2 @ B2 )
% 0.29/0.45 => ( ( member @ a @ X @ ( line_open_segment @ a @ A2 @ B2 ) )
% 0.29/0.45 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
% 0.29/0.45 => ~ ! [D: real] :
% 0.29/0.45 ( ( ord_less @ real @ ( zero_zero @ real ) @ D )
% 0.29/0.45 => ! [T: a > real] :
% 0.29/0.45 ( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ X @ D ) @ T )
% 0.29/0.45 => ( ! [Y2: a] :
% 0.29/0.45 ( ( member @ a @ Y2 @ ( elemen1815086676c_ball @ a @ X @ D ) )
% 0.29/0.45 => ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ Y2 @ ( T @ Y2 ) ) @ ( line_open_segment @ a @ A2 @ B2 ) ) )
% 0.29/0.45 => ( ! [Y2: a] :
% 0.29/0.45 ( ( member @ a @ Y2 @ ( elemen1815086676c_ball @ a @ X @ D ) )
% 0.29/0.45 => ( ord_less @ real @ ( abs_abs @ real @ ( T @ Y2 ) ) @ E ) )
% 0.29/0.45 => ( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ X @ D ) @ T )
% 0.29/0.45 => ( ( T @ X )
% 0.29/0.45 != ( zero_zero @ real ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % cross_time_continuous
% 0.29/0.45 thf(fact_10_abs__1,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( idom_abs_sgn @ A )
% 0.29/0.45 => ( ( abs_abs @ A @ ( one_one @ A ) )
% 0.29/0.45 = ( one_one @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % abs_1
% 0.29/0.45 thf(fact_11_abs__0,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( idom_abs_sgn @ A )
% 0.29/0.45 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 0.29/0.45 = ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % abs_0
% 0.29/0.45 thf(fact_12_abs__0__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere142940540dd_abs @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( ( zero_zero @ A )
% 0.29/0.45 = ( abs_abs @ A @ A2 ) )
% 0.29/0.45 = ( A2
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % abs_0_eq
% 0.29/0.45 thf(fact_13_abs__eq__0,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere142940540dd_abs @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( ( abs_abs @ A @ A2 )
% 0.29/0.45 = ( zero_zero @ A ) )
% 0.29/0.45 = ( A2
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % abs_eq_0
% 0.29/0.45 thf(fact_14_abs__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere142940540dd_abs @ A )
% 0.29/0.45 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 0.29/0.45 = ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % abs_zero
% 0.29/0.45 thf(fact_15_not__gr__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( canoni770627133id_add @ A )
% 0.29/0.45 => ! [N: A] :
% 0.29/0.45 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.29/0.45 = ( N
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not_gr_zero
% 0.29/0.45 thf(fact_16_fixpoint__sol_I2_J,axiom,
% 0.29/0.45 ! [X: a,T2: real] :
% 0.29/0.45 ( ( member @ a @ X @ x )
% 0.29/0.45 => ( ( ( f @ X )
% 0.29/0.45 = ( zero_zero @ a ) )
% 0.29/0.45 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ T2 )
% 0.29/0.45 = X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % fixpoint_sol(2)
% 0.29/0.45 thf(fact_17_gt__one__absI,axiom,
% 0.29/0.45 ! [K: real] :
% 0.29/0.45 ( ( ord_less @ real @ ( abs_abs @ real @ K ) @ ( one_one @ real ) )
% 0.29/0.45 => ( ord_less @ real @ K @ ( one_one @ real ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % gt_one_absI
% 0.29/0.45 thf(fact_18_transversal__segment__reverse,axiom,
% 0.29/0.45 ! [X: a,Y: a] :
% 0.29/0.45 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.29/0.45 => ( poinca272511729egment @ a @ f @ x @ Y @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % transversal_segment_reverse
% 0.29/0.45 thf(fact_19_transversal__segment__commute,axiom,
% 0.29/0.45 ! [X: a,Y: a] :
% 0.29/0.45 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.29/0.45 = ( poinca272511729egment @ a @ f @ x @ Y @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % transversal_segment_commute
% 0.29/0.45 thf(fact_20_abs__idempotent,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere142940540dd_abs @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
% 0.29/0.45 = ( abs_abs @ A @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % abs_idempotent
% 0.29/0.45 thf(fact_21_abs__abs,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( idom_abs_sgn @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
% 0.29/0.45 = ( abs_abs @ A @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % abs_abs
% 0.29/0.45 thf(fact_22_transversal__segment__exists,axiom,
% 0.29/0.45 ! [X: a] :
% 0.29/0.45 ( ( member @ a @ X @ x )
% 0.29/0.45 => ( ( ( f @ X )
% 0.29/0.45 != ( zero_zero @ a ) )
% 0.29/0.45 => ~ ! [A3: a,B3: a] :
% 0.29/0.45 ( ( member @ a @ X @ ( line_open_segment @ a @ A3 @ B3 ) )
% 0.29/0.45 => ~ ( poinca272511729egment @ a @ f @ x @ A3 @ B3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % transversal_segment_exists
% 0.29/0.45 thf(fact_23_c1__on__open__R2_Otransversal__segment_Ocong,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( execut510477386_space @ A )
% 0.29/0.45 => ( ( poinca272511729egment @ A )
% 0.29/0.45 = ( poinca272511729egment @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % c1_on_open_R2.transversal_segment.cong
% 0.29/0.45 thf(fact_24_zero__reorient,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( zero @ A )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ( ( ( zero_zero @ A )
% 0.29/0.45 = X )
% 0.29/0.45 = ( X
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_reorient
% 0.29/0.45 thf(fact_25_linorder__neqE__linordered__idom,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_idom @ A )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( X != Y )
% 0.29/0.45 => ( ~ ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % linorder_neqE_linordered_idom
% 0.29/0.45 thf(fact_26_one__reorient,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( one @ A )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ( ( ( one_one @ A )
% 0.29/0.45 = X )
% 0.29/0.45 = ( X
% 0.29/0.45 = ( one_one @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % one_reorient
% 0.29/0.45 thf(fact_27_zero__less__iff__neq__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( canoni770627133id_add @ A )
% 0.29/0.45 => ! [N: A] :
% 0.29/0.45 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.29/0.45 = ( N
% 0.29/0.45 != ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_less_iff_neq_zero
% 0.29/0.45 thf(fact_28_gr__implies__not__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( canoni770627133id_add @ A )
% 0.29/0.45 => ! [M: A,N: A] :
% 0.29/0.45 ( ( ord_less @ A @ M @ N )
% 0.29/0.45 => ( N
% 0.29/0.45 != ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % gr_implies_not_zero
% 0.29/0.45 thf(fact_29_not__less__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( canoni770627133id_add @ A )
% 0.29/0.45 => ! [N: A] :
% 0.29/0.45 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not_less_zero
% 0.29/0.45 thf(fact_30_gr__zeroI,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( canoni770627133id_add @ A )
% 0.29/0.45 => ! [N: A] :
% 0.29/0.45 ( ( N
% 0.29/0.45 != ( zero_zero @ A ) )
% 0.29/0.45 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % gr_zeroI
% 0.29/0.45 thf(fact_31_zero__neq__one,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( zero_neq_one @ A )
% 0.29/0.45 => ( ( zero_zero @ A )
% 0.29/0.45 != ( one_one @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_neq_one
% 0.29/0.45 thf(fact_32_abs__eq__0__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( idom_abs_sgn @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( ( abs_abs @ A @ A2 )
% 0.29/0.45 = ( zero_zero @ A ) )
% 0.29/0.45 = ( A2
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % abs_eq_0_iff
% 0.29/0.45 thf(fact_33_abs__one,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_idom @ A )
% 0.29/0.45 => ( ( abs_abs @ A @ ( one_one @ A ) )
% 0.29/0.45 = ( one_one @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % abs_one
% 0.29/0.45 thf(fact_34_not__one__less__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linord1659791738miring @ A )
% 0.29/0.45 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not_one_less_zero
% 0.29/0.45 thf(fact_35_zero__less__one,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( zero_less_one @ A )
% 0.29/0.45 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_less_one
% 0.29/0.45 thf(fact_36_abs__not__less__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere142940540dd_abs @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ~ ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % abs_not_less_zero
% 0.29/0.45 thf(fact_37_abs__of__pos,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere142940540dd_abs @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.45 => ( ( abs_abs @ A @ A2 )
% 0.29/0.45 = A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % abs_of_pos
% 0.29/0.45 thf(fact_38_periodic__orbit__imp__flow0__regular,axiom,
% 0.29/0.45 ! [X: a,T2: real] :
% 0.29/0.45 ( ( period862636932_orbit @ a @ f @ x @ X )
% 0.29/0.45 => ( ( f @ ( auto_ll_on_flow0 @ a @ f @ x @ X @ T2 ) )
% 0.29/0.45 != ( zero_zero @ a ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % periodic_orbit_imp_flow0_regular
% 0.29/0.45 thf(fact_39_fixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
% 0.29/0.45 ! [X: a] :
% 0.29/0.45 ( ( member @ a @ X @ x )
% 0.29/0.45 => ( ( ( f @ X )
% 0.29/0.45 = ( zero_zero @ a ) )
% 0.29/0.45 => ( ( period1153813292period @ a @ f @ x @ X )
% 0.29/0.45 = ( zero_zero @ real ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % fixed_point_imp_closed_orbit_period_zero(2)
% 0.29/0.45 thf(fact_40_local_Oflow__initial__time__if,axiom,
% 0.29/0.45 ! [X0: a] :
% 0.29/0.45 ( ( ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.29/0.45 & ( member @ a @ X0 @ x ) )
% 0.29/0.45 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( zero_zero @ real ) )
% 0.29/0.45 = X0 ) )
% 0.29/0.45 & ( ~ ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.29/0.45 & ( member @ a @ X0 @ x ) )
% 0.29/0.45 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( zero_zero @ real ) )
% 0.29/0.45 = ( zero_zero @ a ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % local.flow_initial_time_if
% 0.29/0.45 thf(fact_41_rev_Otransversal__segment__exists,axiom,
% 0.29/0.45 ! [X: a] :
% 0.29/0.45 ( ( member @ a @ X @ x )
% 0.29/0.45 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 0.29/0.45 != ( zero_zero @ a ) )
% 0.29/0.45 => ~ ! [A3: a,B3: a] :
% 0.29/0.45 ( ( member @ a @ X @ ( line_open_segment @ a @ A3 @ B3 ) )
% 0.29/0.45 => ~ ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ A3 @ B3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % rev.transversal_segment_exists
% 0.29/0.45 thf(fact_42_rev_Ofixpoint__sol_I2_J,axiom,
% 0.29/0.45 ! [X: a,T2: real] :
% 0.29/0.45 ( ( member @ a @ X @ x )
% 0.29/0.45 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 0.29/0.45 = ( zero_zero @ a ) )
% 0.29/0.45 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T2 )
% 0.29/0.45 = X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % rev.fixpoint_sol(2)
% 0.29/0.45 thf(fact_43_mem__Collect__eq,axiom,
% 0.29/0.45 ! [A: $tType,A2: A,P: A > $o] :
% 0.29/0.45 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.29/0.45 = ( P @ A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % mem_Collect_eq
% 0.29/0.45 thf(fact_44_Collect__mem__eq,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A] :
% 0.29/0.45 ( ( collect @ A
% 0.29/0.45 @ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
% 0.29/0.45 = A4 ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mem_eq
% 0.29/0.45 thf(fact_45_Collect__cong,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ! [X3: A] :
% 0.29/0.46 ( ( P @ X3 )
% 0.29/0.46 = ( Q @ X3 ) )
% 0.29/0.46 => ( ( collect @ A @ P )
% 0.29/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_cong
% 0.29/0.46 thf(fact_46_ext,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.29/0.46 ( ! [X3: A] :
% 0.29/0.46 ( ( F @ X3 )
% 0.29/0.46 = ( G @ X3 ) )
% 0.29/0.46 => ( F = G ) ) ).
% 0.29/0.46
% 0.29/0.46 % ext
% 0.29/0.46 thf(fact_47_local_Oflow__undefined0,axiom,
% 0.29/0.46 ! [T2: real,X: a] :
% 0.29/0.46 ( ~ ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
% 0.29/0.46 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ T2 )
% 0.29/0.46 = ( zero_zero @ a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.flow_undefined0
% 0.29/0.46 thf(fact_48_rev__transversal__segment,axiom,
% 0.29/0.46 ! [A2: a,B2: a] :
% 0.29/0.46 ( ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ A2 @ B2 )
% 0.29/0.46 = ( poinca272511729egment @ a @ f @ x @ A2 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev_transversal_segment
% 0.29/0.46 thf(fact_49_rev_Otransversal__segment__reverse,axiom,
% 0.29/0.46 ! [X: a,Y: a] :
% 0.29/0.46 ( ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ Y )
% 0.29/0.46 => ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.transversal_segment_reverse
% 0.29/0.46 thf(fact_50_rev_Otransversal__segment__commute,axiom,
% 0.29/0.46 ! [X: a,Y: a] :
% 0.29/0.46 ( ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ Y )
% 0.29/0.46 = ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.transversal_segment_commute
% 0.29/0.46 thf(fact_51_fixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( member @ a @ X @ x )
% 0.29/0.46 => ( ( ( f @ X )
% 0.29/0.46 = ( zero_zero @ a ) )
% 0.29/0.46 => ( period385816147_orbit @ a @ f @ x @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % fixed_point_imp_closed_orbit_period_zero(1)
% 0.29/0.46 thf(fact_52_existence__ivl0__cong,axiom,
% 0.29/0.46 ! [B: $tType,Y4: set @ a,G: a > a,X0: a] :
% 0.29/0.46 ( ( x = Y4 )
% 0.29/0.46 => ( ! [X3: a,T: B] :
% 0.29/0.46 ( ( member @ a @ X3 @ Y4 )
% 0.29/0.46 => ( ( f @ X3 )
% 0.29/0.46 = ( G @ X3 ) ) )
% 0.29/0.46 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
% 0.29/0.46 = ( auto_l1112008849e_ivl0 @ a @ G @ Y4 @ X0 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % existence_ivl0_cong
% 0.29/0.46 thf(fact_53_closed__orbit__in__domain,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.29/0.46 => ( member @ a @ X @ x ) ) ).
% 0.29/0.46
% 0.29/0.46 % closed_orbit_in_domain
% 0.29/0.46 thf(fact_54_add_Oinverse__inverse,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.inverse_inverse
% 0.29/0.46 thf(fact_55_neg__equal__iff__equal,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ( uminus_uminus @ A @ A2 )
% 0.29/0.46 = ( uminus_uminus @ A @ B2 ) )
% 0.29/0.46 = ( A2 = B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_equal_iff_equal
% 0.29/0.46 thf(fact_56_existence__ivl__zero,axiom,
% 0.29/0.46 ! [X0: a] :
% 0.29/0.46 ( ( member @ a @ X0 @ x )
% 0.29/0.46 => ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % existence_ivl_zero
% 0.29/0.46 thf(fact_57_rev_Oexistence__ivl0__cong,axiom,
% 0.29/0.46 ! [B: $tType,Y4: set @ a,G: a > a,X0: a] :
% 0.29/0.46 ( ( x = Y4 )
% 0.29/0.46 => ( ! [X3: a,T: B] :
% 0.29/0.46 ( ( member @ a @ X3 @ Y4 )
% 0.29/0.46 => ( ( uminus_uminus @ ( a > a ) @ f @ X3 )
% 0.29/0.46 = ( G @ X3 ) ) )
% 0.29/0.46 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
% 0.29/0.46 = ( auto_l1112008849e_ivl0 @ a @ G @ Y4 @ X0 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.existence_ivl0_cong
% 0.29/0.46 thf(fact_58_local_Omem__existence__ivl__subset,axiom,
% 0.29/0.46 ! [T2: real,X0: a] :
% 0.29/0.46 ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.29/0.46 => ( member @ real @ T2 @ ( top_top @ ( set @ real ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.mem_existence_ivl_subset
% 0.29/0.46 thf(fact_59_flow0__cong,axiom,
% 0.29/0.46 ! [B: $tType,Y4: set @ a,G: a > a,T2: real,X0: a] :
% 0.29/0.46 ( ( x = Y4 )
% 0.29/0.46 => ( ! [X3: a,T: B] :
% 0.29/0.46 ( ( member @ a @ X3 @ Y4 )
% 0.29/0.46 => ( ( f @ X3 )
% 0.29/0.46 = ( G @ X3 ) ) )
% 0.29/0.46 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.29/0.46 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ T2 )
% 0.29/0.46 = ( auto_ll_on_flow0 @ a @ G @ Y4 @ X0 @ T2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % flow0_cong
% 0.29/0.46 thf(fact_60_rev_Oclosed__orbit__eq__rev,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 = ( period385816147_orbit @ a @ f @ x @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.closed_orbit_eq_rev
% 0.29/0.46 thf(fact_61_rev_Oclosed__orbit__in__domain,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 => ( member @ a @ X @ x ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.closed_orbit_in_domain
% 0.29/0.46 thf(fact_62_closed__orbit__flow0,axiom,
% 0.29/0.46 ! [X: a,T2: real] :
% 0.29/0.46 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.29/0.46 => ( period385816147_orbit @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X @ T2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % closed_orbit_flow0
% 0.29/0.46 thf(fact_63_rev_Oexistence__ivl__zero,axiom,
% 0.29/0.46 ! [X0: a] :
% 0.29/0.46 ( ( member @ a @ X0 @ x )
% 0.29/0.46 => ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.existence_ivl_zero
% 0.29/0.46 thf(fact_64_local_Oexistence__ivl__initial__time,axiom,
% 0.29/0.46 ! [X0: a] :
% 0.29/0.46 ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.29/0.46 => ( ( member @ a @ X0 @ x )
% 0.29/0.46 => ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.existence_ivl_initial_time
% 0.29/0.46 thf(fact_65_local_Orev_Omem__existence__ivl__subset,axiom,
% 0.29/0.46 ! [T2: real,X0: a] :
% 0.29/0.46 ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.29/0.46 => ( member @ real @ T2 @ ( top_top @ ( set @ real ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.rev.mem_existence_ivl_subset
% 0.29/0.46 thf(fact_66_rev_Oflow0__cong,axiom,
% 0.29/0.46 ! [B: $tType,Y4: set @ a,G: a > a,T2: real,X0: a] :
% 0.29/0.46 ( ( x = Y4 )
% 0.29/0.46 => ( ! [X3: a,T: B] :
% 0.29/0.46 ( ( member @ a @ X3 @ Y4 )
% 0.29/0.46 => ( ( uminus_uminus @ ( a > a ) @ f @ X3 )
% 0.29/0.46 = ( G @ X3 ) ) )
% 0.29/0.46 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.29/0.46 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ T2 )
% 0.29/0.46 = ( auto_ll_on_flow0 @ a @ G @ Y4 @ X0 @ T2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.flow0_cong
% 0.29/0.46 thf(fact_67_fixpoint__sol_I1_J,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( member @ a @ X @ x )
% 0.29/0.46 => ( ( ( f @ X )
% 0.29/0.46 = ( zero_zero @ a ) )
% 0.29/0.46 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X )
% 0.29/0.46 = ( top_top @ ( set @ real ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % fixpoint_sol(1)
% 0.29/0.46 thf(fact_68_rev_Oclosed__orbit__flow0,axiom,
% 0.29/0.46 ! [X: a,T2: real] :
% 0.29/0.46 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 => ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.closed_orbit_flow0
% 0.29/0.46 thf(fact_69_closed__orbit__global__existence,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.29/0.46 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X )
% 0.29/0.46 = ( top_top @ ( set @ real ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % closed_orbit_global_existence
% 0.29/0.46 thf(fact_70_closed__orbitI,axiom,
% 0.29/0.46 ! [T2: real,T3: real,Y: a] :
% 0.29/0.46 ( ( T2 != T3 )
% 0.29/0.46 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ Y ) )
% 0.29/0.46 => ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ Y ) )
% 0.29/0.46 => ( ( ( auto_ll_on_flow0 @ a @ f @ x @ Y @ T2 )
% 0.29/0.46 = ( auto_ll_on_flow0 @ a @ f @ x @ Y @ T3 ) )
% 0.29/0.46 => ( period385816147_orbit @ a @ f @ x @ Y ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % closed_orbitI
% 0.29/0.46 thf(fact_71_rev_Ofixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( member @ a @ X @ x )
% 0.29/0.46 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 0.29/0.46 = ( zero_zero @ a ) )
% 0.29/0.46 => ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.fixed_point_imp_closed_orbit_period_zero(1)
% 0.29/0.46 thf(fact_72_periodic__orbit__period_I2_J,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period862636932_orbit @ a @ f @ x @ X )
% 0.29/0.46 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ ( period1153813292period @ a @ f @ x @ X ) )
% 0.29/0.46 = X ) ) ).
% 0.29/0.46
% 0.29/0.46 % periodic_orbit_period(2)
% 0.29/0.46 thf(fact_73_closed__orbit__periodic,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.29/0.46 => ( ( ( f @ X )
% 0.29/0.46 != ( zero_zero @ a ) )
% 0.29/0.46 => ( period862636932_orbit @ a @ f @ x @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % closed_orbit_periodic
% 0.29/0.46 thf(fact_74_local_Orev_Oexistence__ivl__initial__time,axiom,
% 0.29/0.46 ! [X0: a] :
% 0.29/0.46 ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.29/0.46 => ( ( member @ a @ X0 @ x )
% 0.29/0.46 => ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.rev.existence_ivl_initial_time
% 0.29/0.46 thf(fact_75_rev_Ofixpoint__sol_I1_J,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( member @ a @ X @ x )
% 0.29/0.46 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 0.29/0.46 = ( zero_zero @ a ) )
% 0.29/0.46 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 = ( top_top @ ( set @ real ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.fixpoint_sol(1)
% 0.29/0.46 thf(fact_76_local_Orev_Oflow__undefined0,axiom,
% 0.29/0.46 ! [T2: real,X: a] :
% 0.29/0.46 ( ~ ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
% 0.29/0.46 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T2 )
% 0.29/0.46 = ( zero_zero @ a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.rev.flow_undefined0
% 0.29/0.46 thf(fact_77_closed__orbit__def,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.29/0.46 = ( ? [X2: real] :
% 0.29/0.46 ( ( member @ real @ X2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
% 0.29/0.46 & ( X2
% 0.29/0.46 != ( zero_zero @ real ) )
% 0.29/0.46 & ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ X2 )
% 0.29/0.46 = X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % closed_orbit_def
% 0.29/0.46 thf(fact_78_rev_Oclosed__orbit__global__existence,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 = ( top_top @ ( set @ real ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.closed_orbit_global_existence
% 0.29/0.46 thf(fact_79_rev_Oclosed__orbitI,axiom,
% 0.29/0.46 ! [T2: real,T3: real,Y: a] :
% 0.29/0.46 ( ( T2 != T3 )
% 0.29/0.46 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) )
% 0.29/0.46 => ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) )
% 0.29/0.46 => ( ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ T2 )
% 0.29/0.46 = ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ T3 ) )
% 0.29/0.46 => ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.closed_orbitI
% 0.29/0.46 thf(fact_80_rev_Ofixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( member @ a @ X @ x )
% 0.29/0.46 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 0.29/0.46 = ( zero_zero @ a ) )
% 0.29/0.46 => ( ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 = ( zero_zero @ real ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.fixed_point_imp_closed_orbit_period_zero(2)
% 0.29/0.46 thf(fact_81_rev_Operiodic__orbit__imp__flow0__regular,axiom,
% 0.29/0.46 ! [X: a,T2: real] :
% 0.29/0.46 ( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 => ( ( uminus_uminus @ ( a > a ) @ f @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T2 ) )
% 0.29/0.46 != ( zero_zero @ a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.periodic_orbit_imp_flow0_regular
% 0.29/0.46 thf(fact_82_periodic__orbitI,axiom,
% 0.29/0.46 ! [T2: real,T3: real,Y: a] :
% 0.29/0.46 ( ( T2 != T3 )
% 0.29/0.46 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ Y ) )
% 0.29/0.46 => ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ Y ) )
% 0.29/0.46 => ( ( ( auto_ll_on_flow0 @ a @ f @ x @ Y @ T2 )
% 0.29/0.46 = ( auto_ll_on_flow0 @ a @ f @ x @ Y @ T3 ) )
% 0.29/0.46 => ( ( ( f @ Y )
% 0.29/0.46 != ( zero_zero @ a ) )
% 0.29/0.46 => ( period862636932_orbit @ a @ f @ x @ Y ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % periodic_orbitI
% 0.29/0.46 thf(fact_83_periodic__orbit__period_I1_J,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period862636932_orbit @ a @ f @ x @ X )
% 0.29/0.46 => ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % periodic_orbit_period(1)
% 0.29/0.46 thf(fact_84_closed__orbit__period__zero__fixed__point,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.29/0.46 => ( ( ( period1153813292period @ a @ f @ x @ X )
% 0.29/0.46 = ( zero_zero @ real ) )
% 0.29/0.46 => ( ( f @ X )
% 0.29/0.46 = ( zero_zero @ a ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % closed_orbit_period_zero_fixed_point
% 0.29/0.46 thf(fact_85_rev_Operiodic__orbit__period_I2_J,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
% 0.29/0.46 = X ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.periodic_orbit_period(2)
% 0.29/0.46 thf(fact_86_rev_Oclosed__orbit__periodic,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 0.29/0.46 != ( zero_zero @ a ) )
% 0.29/0.46 => ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.closed_orbit_periodic
% 0.29/0.46 thf(fact_87_add_Oinverse__neutral,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 0.29/0.46 = ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.inverse_neutral
% 0.29/0.46 thf(fact_88_neg__0__equal__iff__equal,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ( zero_zero @ A )
% 0.29/0.46 = ( uminus_uminus @ A @ A2 ) )
% 0.29/0.46 = ( ( zero_zero @ A )
% 0.29/0.46 = A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_0_equal_iff_equal
% 0.29/0.46 thf(fact_89_neg__equal__0__iff__equal,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ( uminus_uminus @ A @ A2 )
% 0.29/0.46 = ( zero_zero @ A ) )
% 0.29/0.46 = ( A2
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_equal_0_iff_equal
% 0.29/0.46 thf(fact_90_equal__neg__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord219039673up_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( uminus_uminus @ A @ A2 ) )
% 0.29/0.46 = ( A2
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % equal_neg_zero
% 0.29/0.46 thf(fact_91_neg__equal__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord219039673up_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ( uminus_uminus @ A @ A2 )
% 0.29/0.46 = A2 )
% 0.29/0.46 = ( A2
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_equal_zero
% 0.29/0.46 thf(fact_92_local_Orev_Oflow__initial__time__if,axiom,
% 0.29/0.46 ! [X0: a] :
% 0.29/0.46 ( ( ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.29/0.46 & ( member @ a @ X0 @ x ) )
% 0.29/0.46 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( zero_zero @ real ) )
% 0.29/0.46 = X0 ) )
% 0.29/0.46 & ( ~ ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.29/0.46 & ( member @ a @ X0 @ x ) )
% 0.29/0.46 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( zero_zero @ real ) )
% 0.29/0.46 = ( zero_zero @ a ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.rev.flow_initial_time_if
% 0.29/0.46 thf(fact_93_neg__less__iff__less,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [B2: A,A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 0.29/0.46 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_less_iff_less
% 0.29/0.46 thf(fact_94_rev_Oclosed__orbit__def,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 = ( ? [X2: real] :
% 0.29/0.46 ( ( member @ real @ X2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
% 0.29/0.46 & ( X2
% 0.29/0.46 != ( zero_zero @ real ) )
% 0.29/0.46 & ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ X2 )
% 0.29/0.46 = X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.closed_orbit_def
% 0.29/0.46 thf(fact_95_abs__minus,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( idom_abs_sgn @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
% 0.29/0.46 = ( abs_abs @ A @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % abs_minus
% 0.29/0.46 thf(fact_96_abs__minus__cancel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere142940540dd_abs @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
% 0.29/0.46 = ( abs_abs @ A @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % abs_minus_cancel
% 0.29/0.46 thf(fact_97_rev_Operiodic__orbitI,axiom,
% 0.29/0.46 ! [T2: real,T3: real,Y: a] :
% 0.29/0.46 ( ( T2 != T3 )
% 0.29/0.46 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) )
% 0.29/0.46 => ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) )
% 0.29/0.46 => ( ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ T2 )
% 0.29/0.46 = ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ T3 ) )
% 0.29/0.46 => ( ( ( uminus_uminus @ ( a > a ) @ f @ Y )
% 0.29/0.46 != ( zero_zero @ a ) )
% 0.29/0.46 => ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.periodic_orbitI
% 0.29/0.46 thf(fact_98_rev_Operiodic__orbit__period_I1_J,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 => ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.periodic_orbit_period(1)
% 0.29/0.46 thf(fact_99_rev_Oclosed__orbit__period__zero__fixed__point,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 => ( ( ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 = ( zero_zero @ real ) )
% 0.29/0.46 => ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 0.29/0.46 = ( zero_zero @ a ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.closed_orbit_period_zero_fixed_point
% 0.29/0.46 thf(fact_100_periodic__orbit__def,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period862636932_orbit @ a @ f @ x @ X )
% 0.29/0.46 = ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.29/0.46 & ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % periodic_orbit_def
% 0.29/0.46 thf(fact_101_rev_Operiodic__orbit__def,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 = ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 & ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.periodic_orbit_def
% 0.29/0.46 thf(fact_102_less__neg__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord219039673up_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 0.29/0.46 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_neg_neg
% 0.29/0.46 thf(fact_103_neg__less__pos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord219039673up_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 0.29/0.46 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_less_pos
% 0.29/0.46 thf(fact_104_neg__0__less__iff__less,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 0.29/0.46 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_0_less_iff_less
% 0.29/0.46 thf(fact_105_neg__less__0__iff__less,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.29/0.46 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_less_0_iff_less
% 0.29/0.46 thf(fact_106_local_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.29/0.46 ! [T2: real,X0: a] :
% 0.29/0.46 ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.29/0.46 => ( member @ a @ X0 @ x ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.mem_existence_ivl_iv_defined(2)
% 0.29/0.46 thf(fact_107_local_Orev_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.29/0.46 ! [T2: real,X0: a] :
% 0.29/0.46 ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.29/0.46 => ( member @ a @ X0 @ x ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.rev.mem_existence_ivl_iv_defined(2)
% 0.29/0.46 thf(fact_108_flow0__defined,axiom,
% 0.29/0.46 ! [Xa: real,X0: a] :
% 0.29/0.46 ( ( member @ real @ Xa @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.29/0.46 => ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ Xa ) @ x ) ) ).
% 0.29/0.46
% 0.29/0.46 % flow0_defined
% 0.29/0.46 thf(fact_109_local_Oflow__in__domain,axiom,
% 0.29/0.46 ! [T2: real,X0: a] :
% 0.29/0.46 ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.29/0.46 => ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ T2 ) @ x ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.flow_in_domain
% 0.29/0.46 thf(fact_110_local_Omem__existence__ivl__iv__defined_I1_J,axiom,
% 0.29/0.46 ! [T2: real,X0: a] :
% 0.29/0.46 ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.29/0.46 => ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.mem_existence_ivl_iv_defined(1)
% 0.29/0.46 thf(fact_111_local_Oexistence__ivl__initial__time__iff,axiom,
% 0.29/0.46 ! [X0: a] :
% 0.29/0.46 ( ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.29/0.46 = ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.29/0.46 & ( member @ a @ X0 @ x ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.existence_ivl_initial_time_iff
% 0.29/0.46 thf(fact_112_local_Oflow__initial__time,axiom,
% 0.29/0.46 ! [X0: a] :
% 0.29/0.46 ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.29/0.46 => ( ( member @ a @ X0 @ x )
% 0.29/0.46 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( zero_zero @ real ) )
% 0.29/0.46 = X0 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.flow_initial_time
% 0.29/0.46 thf(fact_113_rev_Oflow0__defined,axiom,
% 0.29/0.46 ! [Xa: real,X0: a] :
% 0.29/0.46 ( ( member @ real @ Xa @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.29/0.46 => ( member @ a @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ Xa ) @ x ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.flow0_defined
% 0.29/0.46 thf(fact_114_local_Orev_Oflow__in__domain,axiom,
% 0.29/0.46 ! [T2: real,X0: a] :
% 0.29/0.46 ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.29/0.46 => ( member @ a @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ T2 ) @ x ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.rev.flow_in_domain
% 0.29/0.46 thf(fact_115_local_Orev_Omem__existence__ivl__iv__defined_I1_J,axiom,
% 0.29/0.46 ! [T2: real,X0: a] :
% 0.29/0.46 ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.29/0.46 => ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.rev.mem_existence_ivl_iv_defined(1)
% 0.29/0.46 thf(fact_116_local_Orev_Oexistence__ivl__initial__time__iff,axiom,
% 0.29/0.46 ! [X0: a] :
% 0.29/0.46 ( ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.29/0.46 = ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.29/0.46 & ( member @ a @ X0 @ x ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.rev.existence_ivl_initial_time_iff
% 0.29/0.46 thf(fact_117_local_Orev_Oflow__initial__time,axiom,
% 0.29/0.46 ! [X0: a] :
% 0.29/0.46 ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.29/0.46 => ( ( member @ a @ X0 @ x )
% 0.29/0.46 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( zero_zero @ real ) )
% 0.29/0.46 = X0 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.rev.flow_initial_time
% 0.29/0.46 thf(fact_118_equation__minus__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( uminus_uminus @ A @ B2 ) )
% 0.29/0.46 = ( B2
% 0.29/0.46 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % equation_minus_iff
% 0.29/0.46 thf(fact_119_minus__equation__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ( uminus_uminus @ A @ A2 )
% 0.29/0.46 = B2 )
% 0.29/0.46 = ( ( uminus_uminus @ A @ B2 )
% 0.29/0.46 = A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minus_equation_iff
% 0.29/0.46 thf(fact_120_less__minus__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 0.29/0.46 = ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_minus_iff
% 0.29/0.46 thf(fact_121_minus__less__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 0.29/0.46 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minus_less_iff
% 0.29/0.46 thf(fact_122_abs__eq__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_ring @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ( abs_abs @ A @ X )
% 0.29/0.46 = ( abs_abs @ A @ Y ) )
% 0.29/0.46 = ( ( X = Y )
% 0.29/0.46 | ( X
% 0.29/0.46 = ( uminus_uminus @ A @ Y ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % abs_eq_iff
% 0.29/0.46 thf(fact_123_abs__less__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ B2 )
% 0.29/0.46 = ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.46 & ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % abs_less_iff
% 0.29/0.46 thf(fact_124_abs__if,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( abs_if @ A )
% 0.29/0.46 => ( ( abs_abs @ A )
% 0.29/0.46 = ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % abs_if
% 0.29/0.46 thf(fact_125_abs__of__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere142940540dd_abs @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( abs_abs @ A @ A2 )
% 0.29/0.46 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % abs_of_neg
% 0.29/0.46 thf(fact_126__092_060alpha_062__limit__point__eq__rev,axiom,
% 0.29/0.46 ! [X: a,P2: a] :
% 0.29/0.46 ( ( limit_1335059408_point @ a @ f @ x @ X @ P2 )
% 0.29/0.46 = ( limit_1052139091_point @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ P2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % \<alpha>_limit_point_eq_rev
% 0.29/0.46 thf(fact_127_rev_O_092_060alpha_062__limit__point__eq__rev,axiom,
% 0.29/0.46 ! [X: a,P2: a] :
% 0.29/0.46 ( ( limit_1335059408_point @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ P2 )
% 0.29/0.46 = ( limit_1052139091_point @ a @ f @ x @ X @ P2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.\<alpha>_limit_point_eq_rev
% 0.29/0.46 thf(fact_128_rev_Omvar_Ointerval__axioms,axiom,
% 0.29/0.46 ! [X0: a] : ( initia826609931terval @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.mvar.interval_axioms
% 0.29/0.46 thf(fact_129_mvar_Ointerval__axioms,axiom,
% 0.29/0.46 ! [X0: a] : ( initia826609931terval @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ).
% 0.29/0.46
% 0.29/0.46 % mvar.interval_axioms
% 0.29/0.46 thf(fact_130_abs__neg__one,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A )
% 0.29/0.46 => ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.29/0.46 = ( one_one @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % abs_neg_one
% 0.29/0.46 thf(fact_131_rev_Oclosed__orbitE,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 => ~ ! [T4: real] :
% 0.29/0.46 ( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
% 0.29/0.46 => ~ ! [T5: real] :
% 0.29/0.46 ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ ( plus_plus @ real @ T5 @ T4 ) )
% 0.29/0.46 = ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.closed_orbitE
% 0.29/0.46 thf(fact_132_local_Orev_Oexistence__ivl__notempty,axiom,
% 0.29/0.46 ! [X0: a] :
% 0.29/0.46 ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.29/0.46 => ( ( member @ a @ X0 @ x )
% 0.29/0.46 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
% 0.29/0.46 != ( bot_bot @ ( set @ real ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.rev.existence_ivl_notempty
% 0.29/0.46 thf(fact_133_rev_Oclosed__orbit__period__nonneg,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.29/0.46 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.closed_orbit_period_nonneg
% 0.29/0.46 thf(fact_134_rev_Ovec__simps_I11_J,axiom,
% 0.29/0.46 ! [Ag: $tType] :
% 0.29/0.46 ( ( semigroup_add @ Ag )
% 0.29/0.46 => ! [Ae: Ag,Be: Ag,Cd: Ag] :
% 0.29/0.46 ( ( plus_plus @ Ag @ ( plus_plus @ Ag @ Ae @ Be ) @ Cd )
% 0.29/0.46 = ( plus_plus @ Ag @ Ae @ ( plus_plus @ Ag @ Be @ Cd ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.vec_simps(11)
% 0.29/0.46 thf(fact_135_rev_Ovec__simps_I12_J,axiom,
% 0.29/0.46 ! [Ai: $tType] :
% 0.29/0.46 ( ( ab_semigroup_add @ Ai )
% 0.29/0.46 => ( ( plus_plus @ Ai )
% 0.29/0.46 = ( ^ [Ag2: Ai,Bg: Ai] : ( plus_plus @ Ai @ Bg @ Ag2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.vec_simps(12)
% 0.29/0.46 thf(fact_136_rev_Ovec__simps_I13_J,axiom,
% 0.29/0.46 ! [Ai: $tType] :
% 0.29/0.46 ( ( ab_semigroup_add @ Ai )
% 0.29/0.46 => ! [Bg2: Ai,Ag3: Ai,Cf: Ai] :
% 0.29/0.46 ( ( plus_plus @ Ai @ Bg2 @ ( plus_plus @ Ai @ Ag3 @ Cf ) )
% 0.29/0.46 = ( plus_plus @ Ai @ Ag3 @ ( plus_plus @ Ai @ Bg2 @ Cf ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.vec_simps(13)
% 0.29/0.46 thf(fact_137_interval__axioms,axiom,
% 0.29/0.46 initia826609931terval @ ( top_top @ ( set @ real ) ) ).
% 0.29/0.46
% 0.29/0.46 % interval_axioms
% 0.29/0.46 thf(fact_138_add__right__cancel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel_semigroup_add @ A )
% 0.29/0.46 => ! [B2: A,A2: A,C: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.29/0.46 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/0.46 = ( B2 = C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_right_cancel
% 0.29/0.46 thf(fact_139_add__left__cancel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel_semigroup_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.46 = ( B2 = C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_left_cancel
% 0.29/0.46 thf(fact_140_rev__eq__flow,axiom,
% 0.29/0.46 ! [Y: a,T2: real] :
% 0.29/0.46 ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ T2 )
% 0.29/0.46 = ( auto_ll_on_flow0 @ a @ f @ x @ Y @ ( uminus_uminus @ real @ T2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev_eq_flow
% 0.29/0.46 thf(fact_141_rev_Orev__eq__flow,axiom,
% 0.29/0.46 ! [Y: a,T2: real] :
% 0.29/0.46 ( ( auto_ll_on_flow0 @ a @ f @ x @ Y @ T2 )
% 0.29/0.46 = ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ ( uminus_uminus @ real @ T2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.rev_eq_flow
% 0.29/0.46 thf(fact_142_recurrence__time__flip__sign_I1_J,axiom,
% 0.29/0.46 ! [T6: real,X: a] :
% 0.29/0.46 ( ( member @ real @ T6 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
% 0.29/0.46 => ( ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ T6 )
% 0.29/0.46 = X )
% 0.29/0.46 => ( member @ real @ ( uminus_uminus @ real @ T6 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % recurrence_time_flip_sign(1)
% 0.29/0.46 thf(fact_143_recurrence__time__flip__sign_I2_J,axiom,
% 0.29/0.46 ! [T6: real,X: a] :
% 0.29/0.46 ( ( member @ real @ T6 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
% 0.29/0.46 => ( ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ T6 )
% 0.29/0.46 = X )
% 0.29/0.46 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ ( uminus_uminus @ real @ T6 ) )
% 0.29/0.46 = X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % recurrence_time_flip_sign(2)
% 0.29/0.46 thf(fact_144_rev_Orecurrence__time__flip__sign_I1_J,axiom,
% 0.29/0.46 ! [T6: real,X: a] :
% 0.29/0.46 ( ( member @ real @ T6 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
% 0.29/0.46 => ( ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T6 )
% 0.29/0.46 = X )
% 0.29/0.46 => ( member @ real @ ( uminus_uminus @ real @ T6 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.recurrence_time_flip_sign(1)
% 0.29/0.46 thf(fact_145_rev_Orecurrence__time__flip__sign_I2_J,axiom,
% 0.29/0.46 ! [T6: real,X: a] :
% 0.29/0.46 ( ( member @ real @ T6 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
% 0.29/0.46 => ( ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T6 )
% 0.29/0.46 = X )
% 0.29/0.46 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ ( uminus_uminus @ real @ T6 ) )
% 0.29/0.46 = X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev.recurrence_time_flip_sign(2)
% 0.29/0.46 thf(fact_146_local_Oflow__trans,axiom,
% 0.29/0.46 ! [S: real,X0: a,T2: real] :
% 0.29/0.46 ( ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.29/0.46 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ S ) ) )
% 0.29/0.46 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( plus_plus @ real @ S @ T2 ) )
% 0.29/0.46 = ( auto_ll_on_flow0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ S ) @ T2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.flow_trans
% 0.29/0.46 thf(fact_147_local_Oexistence__ivl__trans_H,axiom,
% 0.29/0.46 ! [T2: real,S: real,X0: a] :
% 0.29/0.46 ( ( member @ real @ ( plus_plus @ real @ T2 @ S ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.29/0.46 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.29/0.46 => ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ T2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.existence_ivl_trans'
% 0.29/0.46 thf(fact_148_local_Oexistence__ivl__trans,axiom,
% 0.29/0.46 ! [S: real,X0: a,T2: real] :
% 0.29/0.46 ( ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.29/0.46 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ S ) ) )
% 0.29/0.46 => ( member @ real @ ( plus_plus @ real @ S @ T2 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.existence_ivl_trans
% 0.29/0.46 thf(fact_149_local_Orev_Oflow__trans,axiom,
% 0.29/0.46 ! [S: real,X0: a,T2: real] :
% 0.29/0.46 ( ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.29/0.46 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ S ) ) )
% 0.29/0.46 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( plus_plus @ real @ S @ T2 ) )
% 0.29/0.46 = ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ S ) @ T2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.rev.flow_trans
% 0.29/0.46 thf(fact_150_local_Orev_Oexistence__ivl__trans_H,axiom,
% 0.29/0.46 ! [T2: real,S: real,X0: a] :
% 0.29/0.46 ( ( member @ real @ ( plus_plus @ real @ T2 @ S ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.29/0.46 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.29/0.46 => ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ T2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.rev.existence_ivl_trans'
% 0.29/0.46 thf(fact_151_local_Orev_Oexistence__ivl__trans,axiom,
% 0.29/0.46 ! [S: real,X0: a,T2: real] :
% 0.29/0.46 ( ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.29/0.46 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ S ) ) )
% 0.29/0.46 => ( member @ real @ ( plus_plus @ real @ S @ T2 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.rev.existence_ivl_trans
% 0.29/0.46 thf(fact_152_local_Oexistence__ivl__notempty,axiom,
% 0.29/0.46 ! [X0: a] :
% 0.29/0.46 ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.29/0.46 => ( ( member @ a @ X0 @ x )
% 0.29/0.46 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
% 0.29/0.46 != ( bot_bot @ ( set @ real ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.existence_ivl_notempty
% 0.29/0.46 thf(fact_153_closed__orbit__period__nonneg,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.29/0.46 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % closed_orbit_period_nonneg
% 0.29/0.46 thf(fact_154_le__zero__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ! [N: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.29/0.46 = ( N
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_zero_eq
% 0.29/0.46 thf(fact_155_neg__le__iff__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [B2: A,A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_le_iff_le
% 0.29/0.46 thf(fact_156_zero__eq__add__iff__both__eq__0,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ( zero_zero @ A )
% 0.29/0.46 = ( plus_plus @ A @ X @ Y ) )
% 0.29/0.46 = ( ( X
% 0.29/0.46 = ( zero_zero @ A ) )
% 0.29/0.46 & ( Y
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_eq_add_iff_both_eq_0
% 0.29/0.46 thf(fact_157_add__eq__0__iff__both__eq__0,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ X @ Y )
% 0.29/0.46 = ( zero_zero @ A ) )
% 0.29/0.46 = ( ( X
% 0.29/0.46 = ( zero_zero @ A ) )
% 0.29/0.46 & ( Y
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_eq_0_iff_both_eq_0
% 0.29/0.46 thf(fact_158_add__cancel__right__right,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel1352612707id_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( plus_plus @ A @ A2 @ B2 ) )
% 0.29/0.46 = ( B2
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_cancel_right_right
% 0.29/0.46 thf(fact_159_add__cancel__right__left,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel1352612707id_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( plus_plus @ A @ B2 @ A2 ) )
% 0.29/0.46 = ( B2
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_cancel_right_left
% 0.29/0.46 thf(fact_160_add__cancel__left__right,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel1352612707id_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.29/0.46 = A2 )
% 0.29/0.46 = ( B2
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_cancel_left_right
% 0.29/0.46 thf(fact_161_add__cancel__left__left,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel1352612707id_add @ A )
% 0.29/0.46 => ! [B2: A,A2: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.29/0.46 = A2 )
% 0.29/0.46 = ( B2
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_cancel_left_left
% 0.29/0.46 thf(fact_162_double__zero__sym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord219039673up_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ( zero_zero @ A )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ A2 ) )
% 0.29/0.46 = ( A2
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % double_zero_sym
% 0.29/0.46 thf(fact_163_linordered__ab__group__add__class_Odouble__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord219039673up_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ A2 @ A2 )
% 0.29/0.46 = ( zero_zero @ A ) )
% 0.29/0.46 = ( A2
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linordered_ab_group_add_class.double_zero
% 0.29/0.46 thf(fact_164_add_Oright__neutral,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( monoid_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.right_neutral
% 0.29/0.46 thf(fact_165_add_Oleft__neutral,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( monoid_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.left_neutral
% 0.29/0.46 thf(fact_166_add__le__cancel__right,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [A2: A,C: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 0.29/0.46 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_cancel_right
% 0.29/0.46 thf(fact_167_add__le__cancel__left,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [C: A,A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_cancel_left
% 0.29/0.46 thf(fact_168_add__less__cancel__right,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [A2: A,C: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 0.29/0.46 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_cancel_right
% 0.29/0.46 thf(fact_169_add__less__cancel__left,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [C: A,A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 0.29/0.46 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_cancel_left
% 0.29/0.46 thf(fact_170_closed__orbitE,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.29/0.46 => ~ ! [T4: real] :
% 0.29/0.46 ( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
% 0.29/0.46 => ~ ! [T5: real] :
% 0.29/0.46 ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ ( plus_plus @ real @ T5 @ T4 ) )
% 0.29/0.46 = ( auto_ll_on_flow0 @ a @ f @ x @ X @ T5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % closed_orbitE
% 0.29/0.46 thf(fact_171_minus__add__distrib,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ab_group_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.29/0.46 = ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minus_add_distrib
% 0.29/0.46 thf(fact_172_minus__add__cancel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.29/0.46 = B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % minus_add_cancel
% 0.29/0.46 thf(fact_173_add__minus__cancel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
% 0.29/0.46 = B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_minus_cancel
% 0.29/0.46 thf(fact_174_abs__add__abs,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere142940540dd_abs @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) )
% 0.29/0.46 = ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % abs_add_abs
% 0.29/0.46 thf(fact_175_neg__0__le__iff__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_0_le_iff_le
% 0.29/0.46 thf(fact_176_neg__le__0__iff__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.29/0.46 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_le_0_iff_le
% 0.29/0.46 thf(fact_177_less__eq__neg__nonpos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord219039673up_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_eq_neg_nonpos
% 0.29/0.46 thf(fact_178_neg__less__eq__nonneg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord219039673up_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 0.29/0.46 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_less_eq_nonneg
% 0.29/0.46 thf(fact_179_linordered__ab__group__add__class_Ozero__le__double__add__iff__zero__le__single__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord219039673up_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linordered_ab_group_add_class.zero_le_double_add_iff_zero_le_single_add
% 0.29/0.46 thf(fact_180_linordered__ab__group__add__class_Odouble__add__le__zero__iff__single__add__le__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord219039673up_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 0.29/0.46 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linordered_ab_group_add_class.double_add_le_zero_iff_single_add_le_zero
% 0.29/0.46 thf(fact_181_le__add__same__cancel2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere516151231imp_le @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_add_same_cancel2
% 0.29/0.46 thf(fact_182_le__add__same__cancel1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere516151231imp_le @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_add_same_cancel1
% 0.29/0.46 thf(fact_183_add__le__same__cancel2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere516151231imp_le @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 0.29/0.46 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_same_cancel2
% 0.29/0.46 thf(fact_184_add__le__same__cancel1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere516151231imp_le @ A )
% 0.29/0.46 => ! [B2: A,A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
% 0.29/0.46 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_same_cancel1
% 0.29/0.46 thf(fact_185_linordered__ab__group__add__class_Ozero__less__double__add__iff__zero__less__single__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord219039673up_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 0.29/0.46 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linordered_ab_group_add_class.zero_less_double_add_iff_zero_less_single_add
% 0.29/0.46 thf(fact_186_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord219039673up_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 0.29/0.46 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % double_add_less_zero_iff_single_add_less_zero
% 0.29/0.46 thf(fact_187_less__add__same__cancel2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere516151231imp_le @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
% 0.29/0.46 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_add_same_cancel2
% 0.29/0.46 thf(fact_188_less__add__same__cancel1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere516151231imp_le @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.29/0.46 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_add_same_cancel1
% 0.29/0.46 thf(fact_189_add__less__same__cancel2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere516151231imp_le @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 0.29/0.46 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_same_cancel2
% 0.29/0.46 thf(fact_190_add__less__same__cancel1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere516151231imp_le @ A )
% 0.29/0.46 => ! [B2: A,A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
% 0.29/0.46 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_same_cancel1
% 0.29/0.46 thf(fact_191_add_Oleft__inverse,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 0.29/0.46 = ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.left_inverse
% 0.29/0.46 thf(fact_192_add_Oright__inverse,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( plus_plus @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 0.29/0.46 = ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.right_inverse
% 0.29/0.46 thf(fact_193_abs__le__zero__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere142940540dd_abs @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.29/0.46 = ( A2
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % abs_le_zero_iff
% 0.29/0.46 thf(fact_194_abs__le__self__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere142940540dd_abs @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 )
% 0.29/0.46 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % abs_le_self_iff
% 0.29/0.46 thf(fact_195_abs__of__nonneg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere142940540dd_abs @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 => ( ( abs_abs @ A @ A2 )
% 0.29/0.46 = A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % abs_of_nonneg
% 0.29/0.46 thf(fact_196_ball__trivial,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( real_V2090557954_space @ A )
% 0.29/0.46 => ! [X: A] :
% 0.29/0.46 ( ( elemen1815086676c_ball @ A @ X @ ( zero_zero @ real ) )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ball_trivial
% 0.29/0.46 thf(fact_197_add__neg__numeral__special_I7_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( neg_numeral @ A )
% 0.29/0.46 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.29/0.46 = ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_neg_numeral_special(7)
% 0.29/0.46 thf(fact_198_add__neg__numeral__special_I8_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( neg_numeral @ A )
% 0.29/0.46 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
% 0.29/0.46 = ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_neg_numeral_special(8)
% 0.29/0.46 thf(fact_199_abs__of__nonpos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere142940540dd_abs @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( abs_abs @ A @ A2 )
% 0.29/0.46 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % abs_of_nonpos
% 0.29/0.46 thf(fact_200_ball__eq__empty,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( real_V2090557954_space @ A )
% 0.29/0.46 => ! [X: A,E: real] :
% 0.29/0.46 ( ( ( elemen1815086676c_ball @ A @ X @ E )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( ord_less_eq @ real @ E @ ( zero_zero @ real ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ball_eq_empty
% 0.29/0.46 thf(fact_201_local_Oexistence__ivl__undefined,axiom,
% 0.29/0.46 ! [X0: a] :
% 0.29/0.46 ( ~ ( member @ a @ X0 @ x )
% 0.29/0.46 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
% 0.29/0.46 = ( bot_bot @ ( set @ real ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.existence_ivl_undefined
% 0.29/0.46 thf(fact_202_local_Orev_Oexistence__ivl__undefined,axiom,
% 0.29/0.46 ! [X0: a] :
% 0.29/0.46 ( ~ ( member @ a @ X0 @ x )
% 0.29/0.46 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
% 0.29/0.46 = ( bot_bot @ ( set @ real ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.rev.existence_ivl_undefined
% 0.29/0.46 thf(fact_203_local_Oexistence__ivl__empty__iff,axiom,
% 0.29/0.46 ! [X0: a] :
% 0.29/0.46 ( ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
% 0.29/0.46 = ( bot_bot @ ( set @ real ) ) )
% 0.29/0.46 = ( ~ ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.29/0.46 | ~ ( member @ a @ X0 @ x ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.existence_ivl_empty_iff
% 0.29/0.46 thf(fact_204_local_Oexistence__ivl__empty1,axiom,
% 0.29/0.46 ! [X0: a] :
% 0.29/0.46 ( ~ ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.29/0.46 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
% 0.29/0.46 = ( bot_bot @ ( set @ real ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.existence_ivl_empty1
% 0.29/0.46 thf(fact_205_local_Orev_Oexistence__ivl__empty__iff,axiom,
% 0.29/0.46 ! [X0: a] :
% 0.29/0.46 ( ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
% 0.29/0.46 = ( bot_bot @ ( set @ real ) ) )
% 0.29/0.46 = ( ~ ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.29/0.46 | ~ ( member @ a @ X0 @ x ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.rev.existence_ivl_empty_iff
% 0.29/0.46 thf(fact_206_local_Orev_Oexistence__ivl__empty1,axiom,
% 0.29/0.46 ! [X0: a] :
% 0.29/0.46 ( ~ ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.29/0.46 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
% 0.29/0.46 = ( bot_bot @ ( set @ real ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % local.rev.existence_ivl_empty1
% 0.29/0.46 thf(fact_207_is__num__normalize_I8_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( neg_numeral @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.29/0.46 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_num_normalize(8)
% 0.29/0.46 thf(fact_208_abs__triangle__ineq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere142940540dd_abs @ A )
% 0.29/0.46 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % abs_triangle_ineq
% 0.29/0.46 thf(fact_209_le__numeral__extra_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord1659791738miring @ A )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_numeral_extra(3)
% 0.29/0.46 thf(fact_210_le__numeral__extra_I4_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord1659791738miring @ A )
% 0.29/0.46 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_numeral_extra(4)
% 0.29/0.46 thf(fact_211_is__num__normalize_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( neg_numeral @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_num_normalize(1)
% 0.29/0.46 thf(fact_212_add__le__imp__le__right,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [A2: A,C: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 0.29/0.46 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_imp_le_right
% 0.29/0.46 thf(fact_213_add__le__imp__le__left,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [C: A,A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 0.29/0.46 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_imp_le_left
% 0.29/0.46 thf(fact_214_add__less__le__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A,D2: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ C @ D2 )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_le_mono
% 0.29/0.46 thf(fact_215_add__le__less__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A,D2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less @ A @ C @ D2 )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_less_mono
% 0.29/0.46 thf(fact_216_add__nonpos__eq__0__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere216010020id_add @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ( plus_plus @ A @ X @ Y )
% 0.29/0.46 = ( zero_zero @ A ) )
% 0.29/0.46 = ( ( X
% 0.29/0.46 = ( zero_zero @ A ) )
% 0.29/0.46 & ( Y
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_nonpos_eq_0_iff
% 0.29/0.46 thf(fact_217_add__nonneg__eq__0__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere216010020id_add @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.29/0.46 => ( ( ( plus_plus @ A @ X @ Y )
% 0.29/0.46 = ( zero_zero @ A ) )
% 0.29/0.46 = ( ( X
% 0.29/0.46 = ( zero_zero @ A ) )
% 0.29/0.46 & ( Y
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_nonneg_eq_0_iff
% 0.29/0.46 thf(fact_218_add__nonpos__nonpos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere216010020id_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_nonpos_nonpos
% 0.29/0.46 thf(fact_219_add__nonneg__nonneg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere216010020id_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_nonneg_nonneg
% 0.29/0.46 thf(fact_220_le__iff__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ( ( ord_less_eq @ A )
% 0.29/0.46 = ( ^ [A5: A,B4: A] :
% 0.29/0.46 ? [C2: A] :
% 0.29/0.46 ( B4
% 0.29/0.46 = ( plus_plus @ A @ A5 @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_iff_add
% 0.29/0.46 thf(fact_221_add__increasing2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere216010020id_add @ A )
% 0.29/0.46 => ! [C: A,B2: A,A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.46 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_increasing2
% 0.29/0.46 thf(fact_222_add__decreasing2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere216010020id_add @ A )
% 0.29/0.46 => ! [C: A,A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_decreasing2
% 0.29/0.46 thf(fact_223_add__right__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_right_mono
% 0.29/0.46 thf(fact_224_less__eqE,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.46 => ~ ! [C3: A] :
% 0.29/0.46 ( B2
% 0.29/0.46 != ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_eqE
% 0.29/0.46 thf(fact_225_add__increasing,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere216010020id_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.29/0.46 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_increasing
% 0.29/0.46 thf(fact_226_add__decreasing,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere216010020id_add @ A )
% 0.29/0.46 => ! [A2: A,C: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_decreasing
% 0.29/0.46 thf(fact_227_add__left__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_left_mono
% 0.29/0.46 thf(fact_228_add__right__imp__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel_semigroup_add @ A )
% 0.29/0.46 => ! [B2: A,A2: A,C: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.29/0.46 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/0.46 => ( B2 = C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_right_imp_eq
% 0.29/0.46 thf(fact_229_add__left__imp__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel_semigroup_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.46 => ( B2 = C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_left_imp_eq
% 0.29/0.46 thf(fact_230_add__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A,D2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ C @ D2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono
% 0.29/0.46 thf(fact_231_add_Oleft__commute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ab_semigroup_add @ A )
% 0.29/0.46 => ! [B2: A,A2: A,C: A] :
% 0.29/0.46 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.left_commute
% 0.29/0.46 thf(fact_232_add_Ocommute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ab_semigroup_add @ A )
% 0.29/0.46 => ( ( plus_plus @ A )
% 0.29/0.46 = ( ^ [A5: A,B4: A] : ( plus_plus @ A @ B4 @ A5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.commute
% 0.29/0.46 thf(fact_233_add_Oright__cancel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [B2: A,A2: A,C: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.29/0.46 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/0.46 = ( B2 = C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.right_cancel
% 0.29/0.46 thf(fact_234_add_Oleft__cancel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.46 = ( B2 = C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.left_cancel
% 0.29/0.46 thf(fact_235_add_Oassoc,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semigroup_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.assoc
% 0.29/0.46 thf(fact_236_group__cancel_Oadd2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comm_monoid_add @ A )
% 0.29/0.46 => ! [B5: A,K: A,B2: A,A2: A] :
% 0.29/0.46 ( ( B5
% 0.29/0.46 = ( plus_plus @ A @ K @ B2 ) )
% 0.29/0.46 => ( ( plus_plus @ A @ A2 @ B5 )
% 0.29/0.46 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % group_cancel.add2
% 0.29/0.46 thf(fact_237_group__cancel_Oadd1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comm_monoid_add @ A )
% 0.29/0.46 => ! [A4: A,K: A,A2: A,B2: A] :
% 0.29/0.46 ( ( A4
% 0.29/0.46 = ( plus_plus @ A @ K @ A2 ) )
% 0.29/0.46 => ( ( plus_plus @ A @ A4 @ B2 )
% 0.29/0.46 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % group_cancel.add1
% 0.29/0.46 thf(fact_238_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ I @ J )
% 0.29/0.46 & ( ord_less_eq @ A @ K @ L ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_semiring(1)
% 0.29/0.46 thf(fact_239_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( I = J )
% 0.29/0.46 & ( ord_less_eq @ A @ K @ L ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_semiring(2)
% 0.29/0.46 thf(fact_240_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ I @ J )
% 0.29/0.46 & ( K = L ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_semiring(3)
% 0.29/0.46 thf(fact_241_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( I = J )
% 0.29/0.46 & ( K = L ) )
% 0.29/0.46 => ( ( plus_plus @ A @ I @ K )
% 0.29/0.46 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_semiring(4)
% 0.29/0.46 thf(fact_242_add__mono__thms__linordered__field_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( ord_less @ A @ I @ J )
% 0.29/0.46 & ( ord_less_eq @ A @ K @ L ) )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_field(3)
% 0.29/0.46 thf(fact_243_add__mono__thms__linordered__field_I4_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ I @ J )
% 0.29/0.46 & ( ord_less @ A @ K @ L ) )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_field(4)
% 0.29/0.46 thf(fact_244_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ab_semigroup_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.29/0.46 thf(fact_245_add__strict__increasing2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere1818651114id_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 => ( ( ord_less @ A @ B2 @ C )
% 0.29/0.46 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_strict_increasing2
% 0.29/0.46 thf(fact_246_add__strict__increasing,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere1818651114id_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.29/0.46 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_strict_increasing
% 0.29/0.46 thf(fact_247_add__pos__nonneg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere216010020id_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.29/0.46 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_pos_nonneg
% 0.29/0.46 thf(fact_248_add__nonpos__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere216010020id_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_nonpos_neg
% 0.29/0.46 thf(fact_249_add__nonneg__pos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere216010020id_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.29/0.46 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_nonneg_pos
% 0.29/0.46 thf(fact_250_add__neg__nonpos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere216010020id_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_neg_nonpos
% 0.29/0.46 thf(fact_251_Elementary__Metric__Spaces_Oball__empty,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( real_V2090557954_space @ A )
% 0.29/0.46 => ! [E: real,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ real @ E @ ( zero_zero @ real ) )
% 0.29/0.46 => ( ( elemen1815086676c_ball @ A @ X @ E )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Elementary_Metric_Spaces.ball_empty
% 0.29/0.46 thf(fact_252_add_Ogroup__left__neutral,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.group_left_neutral
% 0.29/0.46 thf(fact_253_add_Ocomm__neutral,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comm_monoid_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.comm_neutral
% 0.29/0.46
% 0.29/0.46 % Subclasses (18)
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ozero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( zero @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ogroup__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( group_add @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Omonoid__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( monoid_add @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oab__group__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( ab_group_add @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Osemigroup__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( semigroup_add @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocomm__monoid__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( comm_monoid_add @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oab__semigroup__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( ab_semigroup_add @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocancel__semigroup__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( cancel_semigroup_add @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__group__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( ordered_ab_group_add @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocancel__comm__monoid__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( cancel1352612707id_add @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__comm__monoid__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( ordere216010020id_add @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__group__add__abs,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( ordere142940540dd_abs @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__semigroup__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( ordere779506340up_add @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Real__Vector__Spaces_Ometric__space,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( real_V2090557954_space @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__cancel__comm__monoid__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( ordere1818651114id_add @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( ordere236663937imp_le @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( ordere223160158up_add @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( execut510477386_space @ A )
% 0.29/0.46 => ( ordere516151231imp_le @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % Type constructors (42)
% 0.29/0.46 thf(tcon_HOL_Obool___Countable_Ocountable,axiom,
% 0.29/0.46 countable @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Countable_Ocountable_1,axiom,
% 0.29/0.46 ! [A6: $tType] :
% 0.29/0.46 ( ( finite_finite @ A6 )
% 0.29/0.46 => ( countable @ ( set @ A6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Countable_Ocountable_2,axiom,
% 0.29/0.46 ! [A6: $tType,A7: $tType] :
% 0.29/0.46 ( ( ( finite_finite @ A6 )
% 0.29/0.46 & ( countable @ A7 ) )
% 0.29/0.46 => ( countable @ ( A6 > A7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.29/0.46 ! [A6: $tType,A7: $tType] :
% 0.29/0.46 ( ( ( finite_finite @ A6 )
% 0.29/0.46 & ( finite_finite @ A7 ) )
% 0.29/0.46 => ( finite_finite @ ( A6 > A7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Finite__Set_Ofinite_3,axiom,
% 0.29/0.46 ! [A6: $tType] :
% 0.29/0.46 ( ( finite_finite @ A6 )
% 0.29/0.46 => ( finite_finite @ ( set @ A6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Finite__Set_Ofinite_4,axiom,
% 0.29/0.46 finite_finite @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Real__Vector__Spaces_Ometric__space,axiom,
% 0.29/0.46 ! [A6: $tType,A7: $tType] :
% 0.29/0.46 ( ( ( countable @ A6 )
% 0.29/0.46 & ( real_V2090557954_space @ A7 ) )
% 0.29/0.46 => ( real_V2090557954_space @ ( A6 > A7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Groups_Oab__semigroup__add,axiom,
% 0.29/0.46 ! [A6: $tType] :
% 0.29/0.46 ( ( ab_semigroup_add @ A6 )
% 0.29/0.46 => ( ab_semigroup_add @ ( set @ A6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Groups_Ocomm__monoid__add,axiom,
% 0.29/0.46 ! [A6: $tType] :
% 0.29/0.46 ( ( comm_monoid_add @ A6 )
% 0.29/0.46 => ( comm_monoid_add @ ( set @ A6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Groups_Osemigroup__add,axiom,
% 0.29/0.46 ! [A6: $tType] :
% 0.29/0.46 ( ( semigroup_add @ A6 )
% 0.29/0.46 => ( semigroup_add @ ( set @ A6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Groups_Omonoid__add,axiom,
% 0.29/0.46 ! [A6: $tType] :
% 0.29/0.46 ( ( monoid_add @ A6 )
% 0.29/0.46 => ( monoid_add @ ( set @ A6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Groups_Ozero,axiom,
% 0.29/0.46 ! [A6: $tType] :
% 0.29/0.46 ( ( zero @ A6 )
% 0.29/0.46 => ( zero @ ( set @ A6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Groups_Oone,axiom,
% 0.29/0.46 ! [A6: $tType] :
% 0.29/0.46 ( ( one @ A6 )
% 0.29/0.46 => ( one @ ( set @ A6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Executable__Euclidean__Space_Oexecutable__euclidean__space,axiom,
% 0.29/0.46 execut510477386_space @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 0.29/0.46 ordere516151231imp_le @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 0.29/0.46 ordere223160158up_add @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.29/0.46 ordere236663937imp_le @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add,axiom,
% 0.29/0.46 ordere1818651114id_add @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring,axiom,
% 0.29/0.46 linord1659791738miring @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space_5,axiom,
% 0.29/0.46 real_V2090557954_space @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add,axiom,
% 0.29/0.46 ordere779506340up_add @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs,axiom,
% 0.29/0.46 ordere142940540dd_abs @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add,axiom,
% 0.29/0.46 ordere216010020id_add @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
% 0.29/0.46 linord219039673up_add @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add,axiom,
% 0.29/0.46 cancel1352612707id_add @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
% 0.29/0.46 ordered_ab_group_add @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add,axiom,
% 0.29/0.46 cancel_semigroup_add @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_6,axiom,
% 0.29/0.46 ab_semigroup_add @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_7,axiom,
% 0.29/0.46 comm_monoid_add @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Olinordered__ring,axiom,
% 0.29/0.46 linordered_ring @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Olinordered__idom,axiom,
% 0.29/0.46 linordered_idom @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Osemigroup__add_8,axiom,
% 0.29/0.46 semigroup_add @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Ozero__less__one,axiom,
% 0.29/0.46 zero_less_one @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Oab__group__add,axiom,
% 0.29/0.46 ab_group_add @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Ozero__neq__one,axiom,
% 0.29/0.46 zero_neq_one @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn,axiom,
% 0.29/0.46 idom_abs_sgn @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Omonoid__add_9,axiom,
% 0.29/0.46 monoid_add @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 0.29/0.46 group_add @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Num_Oneg__numeral,axiom,
% 0.29/0.46 neg_numeral @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Oabs__if,axiom,
% 0.29/0.46 abs_if @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Ozero_10,axiom,
% 0.29/0.46 zero @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Oone_11,axiom,
% 0.29/0.46 one @ real ).
% 0.29/0.46
% 0.29/0.46 % Helper facts (3)
% 0.29/0.46 thf(help_If_3_1_T,axiom,
% 0.29/0.46 ! [P: $o] :
% 0.29/0.46 ( ( P = $true )
% 0.29/0.46 | ( P = $false ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(help_If_2_1_T,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Y: A] :
% 0.29/0.46 ( ( if @ A @ $false @ X @ Y )
% 0.29/0.46 = Y ) ).
% 0.29/0.46
% 0.29/0.46 thf(help_If_1_1_T,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Y: A] :
% 0.29/0.46 ( ( if @ A @ $true @ X @ Y )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % Free types (1)
% 0.29/0.46 thf(tfree_0,hypothesis,
% 0.29/0.46 execut510477386_space @ a ).
% 0.29/0.46
% 0.29/0.46 % Conjectures (2)
% 0.29/0.46 thf(conj_0,hypothesis,
% 0.29/0.46 ! [Delta2: real,Tau2: a > real] :
% 0.29/0.46 ( ( ord_less @ real @ ( zero_zero @ real ) @ Delta2 )
% 0.29/0.46 => ( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ p @ Delta2 ) @ Tau2 )
% 0.29/0.46 => ( ( ( Tau2 @ p )
% 0.29/0.46 = ( zero_zero @ real ) )
% 0.29/0.46 => ( ! [Y3: a] :
% 0.29/0.46 ( ( member @ a @ Y3 @ ( elemen1815086676c_ball @ a @ p @ Delta2 ) )
% 0.29/0.46 => ( ord_less @ real @ ( abs_abs @ real @ ( Tau2 @ Y3 ) ) @ ( one_one @ real ) ) )
% 0.29/0.46 => ( ! [Y3: a] :
% 0.29/0.46 ( ( member @ a @ Y3 @ ( elemen1815086676c_ball @ a @ p @ Delta2 ) )
% 0.29/0.46 => ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ Y3 @ ( Tau2 @ Y3 ) ) @ ( line_open_segment @ a @ a2 @ b ) ) )
% 0.29/0.46 => thesisa ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(conj_1,conjecture,
% 0.29/0.46 thesisa ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 2.53/1.58 ---- Embedded in THF ---
% 2.53/1.58 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.53/1.58 %%% Generated on Tue Feb 24 08:27:34 EST 2026
% 2.53/1.58 %%% using '$$dhol' embedding, version 1.3.0.
% 2.53/1.58 %%% Logic specification used:
% 2.53/1.58 %%% thf(spec, logic, $$dhol).
% 2.53/1.58
% 2.53/1.58 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.SqIxTx6pAT/DTF2DTF24163.p
% 2.53/1.58 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.53/1.58 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.53/1.58 thf(ty_tf_a, type, a: $tType).
% 2.53/1.58 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Countable_Ocountable, type, countable: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Oone, type, one: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Rings_Oabs__if, type, abs_if: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Num_Oneg__numeral, type, neg_numeral: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Omonoid__add, type, monoid_add: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Rings_Oidom__abs__sgn, type, idom_abs_sgn: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Rings_Ozero__neq__one, type, zero_neq_one: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Oab__group__add, type, ab_group_add: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Rings_Ozero__less__one, type, zero_less_one: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Rings_Olinordered__ring, type, linordered_ring: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Oordered__comm__monoid__add, type, ordere216010020id_add: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Oordered__ab__group__add__abs, type, ordere142940540dd_abs: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Real__Vector__Spaces_Ometric__space, type, real_V2090557954_space: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add, type, ordere1818651114id_add: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add, type, ordere223160158up_add: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le, type, ordere516151231imp_le: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_cl_Executable__Euclidean__Space_Oexecutable__euclidean__space, type, execut510477386_space: (!> [A:$tType]: ($o))).
% 2.53/1.58 thf(sy_c_Elementary__Metric__Spaces_Oball, type, elemen1815086676c_ball: (!> [A:$tType]: ((A > (real > (set @ A)))))).
% 2.53/1.58 thf(sy_c_Flow_Oauto__ll__on__open_Oexistence__ivl0, type, auto_l1112008849e_ivl0: (!> [A:$tType]: (((A > A) > ((set @ A) > (A > (set @ real))))))).
% 2.53/1.58 thf(sy_c_Flow_Oauto__ll__on__open_Oflow0, type, auto_ll_on_flow0: (!> [A:$tType]: (((A > A) > ((set @ A) > (A > (real > A))))))).
% 2.53/1.58 thf(sy_c_Groups_Oabs__class_Oabs, type, abs_abs: (!> [A:$tType]: ((A > A)))).
% 2.53/1.58 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 2.53/1.58 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.53/1.58 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 2.53/1.58 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.53/1.58 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.53/1.58 thf(sy_c_Initial__Value__Problem_Ointerval, type, initia826609931terval: ((set @ real) > $o)).
% 2.53/1.58 thf(sy_c_Limit__Set_Oauto__ll__on__open_O_092_060alpha_062__limit__point, type, limit_1335059408_point: (!> [A:$tType]: (((A > A) > ((set @ A) > (A > (A > $o))))))).
% 2.53/1.58 thf(sy_c_Limit__Set_Oauto__ll__on__open_O_092_060omega_062__limit__point, type, limit_1052139091_point: (!> [A:$tType]: (((A > A) > ((set @ A) > (A > (A > $o))))))).
% 2.53/1.58 thf(sy_c_Line__Segment_Oopen__segment, type, line_open_segment: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 2.53/1.58 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.53/1.58 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.53/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.53/1.58 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.53/1.58 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Oclosed__orbit, type, period385816147_orbit: (!> [A:$tType]: (((A > A) > ((set @ A) > (A > $o)))))).
% 2.53/1.58 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiod, type, period1153813292period: (!> [A:$tType]: (((A > A) > ((set @ A) > (A > real)))))).
% 2.53/1.58 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiodic__orbit, type, period862636932_orbit: (!> [A:$tType]: (((A > A) > ((set @ A) > (A > $o)))))).
% 2.53/1.58 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Otransversal__segment, type, poinca272511729egment: (!> [A:$tType]: (((A > A) > ((set @ A) > (A > (A > $o))))))).
% 2.53/1.58 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.53/1.58 thf(sy_c_Topological__Spaces_Ocontinuous__on, type, topolo2071040574ous_on: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > $o))))).
% 2.53/1.58 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.53/1.58 thf(sy_v_X, type, x: (set @ a)).
% 2.53/1.58 thf(sy_v_a, type, a2: a).
% 2.53/1.58 thf(sy_v_b, type, b: a).
% 2.53/1.58 thf(sy_v_f, type, f: (a > a)).
% 2.53/1.58 thf(sy_v_p, type, p: a).
% 2.53/1.58 thf(sy_v_thesisa____, type, thesisa: $o).
% 2.53/1.58 thf(sy_v_x, type, x2: a).
% 2.53/1.58 thf(fact_0_rev_Oopen__segment__trichotomy, axiom, (! [X:a,A2:a,B2:a,Y:a]: (((((member @ a) @ X) @ (((line_open_segment @ a) @ A2) @ B2)) => ((((member @ a) @ Y) @ (((line_open_segment @ a) @ A2) @ B2)) => ((X = Y) | ((((member @ a) @ Y) @ (((line_open_segment @ a) @ X) @ B2)) | (((member @ a) @ Y) @ (((line_open_segment @ a) @ A2) @ X))))))))).
% 2.53/1.58 thf(fact_1__092_060open_062p_A_092_060in_062_AX_092_060close_062, axiom, (((member @ a) @ p) @ x)).
% 2.53/1.58 thf(fact_2_assms_I4_J, axiom, (((member @ a) @ p) @ (((line_open_segment @ a) @ a2) @ b))).
% 2.53/1.58 thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062d_At_O_A_092_060lbrakk_0620_A_060_Ad_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_Aflow0_Ay_A_It_Ay_J_A_092_060in_062_A_123a_060_N_N_060b_125_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_A_092_060bar_062t_Ay_092_060bar_062_A_060_A1_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_At_Ap_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [D:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D) => (! [T:(a > real)]: ((((((topolo2071040574ous_on @ a) @ real) @ (((elemen1815086676c_ball @ a) @ p) @ D)) @ T) => ((! [Y2:a]: (((((member @ a) @ Y2) @ (((elemen1815086676c_ball @ a) @ p) @ D)) => (((member @ a) @ (((((auto_ll_on_flow0 @ a) @ f) @ x) @ Y2) @ (T @ Y2))) @ (((line_open_segment @ a) @ a2) @ b))))) => ((! [Y2:a]: (((((member @ a) @ Y2) @ (((elemen1815086676c_ball @ a) @ p) @ D)) => (((ord_less @ real) @ ((abs_abs @ real) @ (T @ Y2))) @ (one_one @ real))))) => (((((topolo2071040574ous_on @ a) @ real) @ (((elemen1815086676c_ball @ a) @ p) @ D)) @ T) => ((T @ p) != (zero_zero @ real)))))))))))))).
% 2.53/1.58 thf(fact_4_that, axiom, (! [Delta:real,Tau:(a > real)]: (((((ord_less @ real) @ (zero_zero @ real)) @ Delta) => (((((topolo2071040574ous_on @ a) @ real) @ (((elemen1815086676c_ball @ a) @ p) @ Delta)) @ Tau) => (((Tau @ p) = (zero_zero @ real)) => ((! [Y3:a]: (((((member @ a) @ Y3) @ (((elemen1815086676c_ball @ a) @ p) @ Delta)) => (((ord_less @ real) @ ((abs_abs @ real) @ (Tau @ Y3))) @ (one_one @ real))))) => ((! [Y3:a]: (((((member @ a) @ Y3) @ (((elemen1815086676c_ball @ a) @ p) @ Delta)) => (((member @ a) @ (((((auto_ll_on_flow0 @ a) @ f) @ x) @ Y3) @ (Tau @ Y3))) @ (((line_open_segment @ a) @ a2) @ b))))) => thesisa)))))))).
% 2.53/1.58 thf(fact_5_assms_I1_J, axiom, (((((poinca272511729egment @ a) @ f) @ x) @ a2) @ b)).
% 2.53/1.58 thf(fact_6_assms_I3_J, axiom, (((((limit_1052139091_point @ a) @ f) @ x) @ x2) @ p)).
% 2.53/1.58 thf(fact_7_centre__in__ball, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X:A,E:real]: (((((member @ A) @ X) @ (((elemen1815086676c_ball @ A) @ X) @ E)) = (((ord_less @ real) @ (zero_zero @ real)) @ E)))))))).
% 2.53/1.58 thf(fact_8_zero__less__abs__iff, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((abs_abs @ A) @ A2)) = (A2 != (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_9_cross__time__continuous, axiom, (! [A2:a,B2:a,X:a,E:real]: (((((((poinca272511729egment @ a) @ f) @ x) @ A2) @ B2) => ((((member @ a) @ X) @ (((line_open_segment @ a) @ A2) @ B2)) => ((((ord_less @ real) @ (zero_zero @ real)) @ E) => (~ ((! [D:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D) => (! [T:(a > real)]: ((((((topolo2071040574ous_on @ a) @ real) @ (((elemen1815086676c_ball @ a) @ X) @ D)) @ T) => ((! [Y2:a]: (((((member @ a) @ Y2) @ (((elemen1815086676c_ball @ a) @ X) @ D)) => (((member @ a) @ (((((auto_ll_on_flow0 @ a) @ f) @ x) @ Y2) @ (T @ Y2))) @ (((line_open_segment @ a) @ A2) @ B2))))) => ((! [Y2:a]: (((((member @ a) @ Y2) @ (((elemen1815086676c_ball @ a) @ X) @ D)) => (((ord_less @ real) @ ((abs_abs @ real) @ (T @ Y2))) @ E)))) => (((((topolo2071040574ous_on @ a) @ real) @ (((elemen1815086676c_ball @ a) @ X) @ D)) @ T) => ((T @ X) != (zero_zero @ real))))))))))))))))))).
% 2.53/1.58 thf(fact_10_abs__1, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (((abs_abs @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 2.53/1.58 thf(fact_11_abs__0, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (((abs_abs @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.53/1.58 thf(fact_12_abs__0__eq, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: ((((zero_zero @ A) = ((abs_abs @ A) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_13_abs__eq__0, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((abs_abs @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_14_abs__zero, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (((abs_abs @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.53/1.58 thf(fact_15_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_16_fixpoint__sol_I2_J, axiom, (! [X:a,T2:real]: (((((member @ a) @ X) @ x) => (((f @ X) = (zero_zero @ a)) => ((((((auto_ll_on_flow0 @ a) @ f) @ x) @ X) @ T2) = X)))))).
% 2.53/1.58 thf(fact_17_gt__one__absI, axiom, (! [K:real]: (((((ord_less @ real) @ ((abs_abs @ real) @ K)) @ (one_one @ real)) => (((ord_less @ real) @ K) @ (one_one @ real)))))).
% 2.53/1.58 thf(fact_18_transversal__segment__reverse, axiom, (! [X:a,Y:a]: (((((((poinca272511729egment @ a) @ f) @ x) @ X) @ Y) => (((((poinca272511729egment @ a) @ f) @ x) @ Y) @ X))))).
% 2.53/1.58 thf(fact_19_transversal__segment__commute, axiom, (! [X:a,Y:a]: (((((((poinca272511729egment @ a) @ f) @ x) @ X) @ Y) = (((((poinca272511729egment @ a) @ f) @ x) @ Y) @ X))))).
% 2.53/1.58 thf(fact_20_abs__idempotent, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: ((((abs_abs @ A) @ ((abs_abs @ A) @ A2)) = ((abs_abs @ A) @ A2)))))))).
% 2.53/1.58 thf(fact_21_abs__abs, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A2:A]: ((((abs_abs @ A) @ ((abs_abs @ A) @ A2)) = ((abs_abs @ A) @ A2)))))))).
% 2.53/1.58 thf(fact_22_transversal__segment__exists, axiom, (! [X:a]: (((((member @ a) @ X) @ x) => (((f @ X) != (zero_zero @ a)) => (~ ((! [A3:a,B3:a]: (((((member @ a) @ X) @ (((line_open_segment @ a) @ A3) @ B3)) => (~ ((((((poinca272511729egment @ a) @ f) @ x) @ A3) @ B3))))))))))))).
% 2.53/1.58 thf(fact_23_c1__on__open__R2_Otransversal__segment_Ocong, axiom, (! [A:$tType]: (((execut510477386_space @ A) => ((poinca272511729egment @ A) = (poinca272511729egment @ A)))))).
% 2.53/1.58 thf(fact_24_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_25_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 2.53/1.58 thf(fact_26_one__reorient, axiom, (! [A:$tType]: (((one @ A) => (! [X:A]: ((((one_one @ A) = X) = (X = (one_one @ A))))))))).
% 2.53/1.58 thf(fact_27_zero__less__iff__neq__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_28_gr__implies__not__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_29_not__less__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 2.53/1.58 thf(fact_30_gr__zeroI, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 2.53/1.58 thf(fact_31_zero__neq__one, axiom, (! [A:$tType]: (((zero_neq_one @ A) => ((zero_zero @ A) != (one_one @ A)))))).
% 2.53/1.58 thf(fact_32_abs__eq__0__iff, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A2:A]: (((((abs_abs @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_33_abs__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((abs_abs @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 2.53/1.58 thf(fact_34_not__one__less__zero, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 2.53/1.58 thf(fact_35_zero__less__one, axiom, (! [A:$tType]: (((zero_less_one @ A) => (((ord_less @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 2.53/1.58 thf(fact_36_abs__not__less__zero, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ ((abs_abs @ A) @ A2)) @ (zero_zero @ A)))))))))).
% 2.53/1.58 thf(fact_37_abs__of__pos, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((abs_abs @ A) @ A2) = A2)))))))).
% 2.53/1.58 thf(fact_38_periodic__orbit__imp__flow0__regular, axiom, (! [X:a,T2:real]: ((((((period862636932_orbit @ a) @ f) @ x) @ X) => ((f @ (((((auto_ll_on_flow0 @ a) @ f) @ x) @ X) @ T2)) != (zero_zero @ a)))))).
% 2.53/1.58 thf(fact_39_fixed__point__imp__closed__orbit__period__zero_I2_J, axiom, (! [X:a]: (((((member @ a) @ X) @ x) => (((f @ X) = (zero_zero @ a)) => (((((period1153813292period @ a) @ f) @ x) @ X) = (zero_zero @ real))))))).
% 2.53/1.58 thf(fact_40_local_Oflow__initial__time__if, axiom, (! [X0:a]: (((((((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))) & (((member @ a) @ X0) @ x)) => ((((((auto_ll_on_flow0 @ a) @ f) @ x) @ X0) @ (zero_zero @ real)) = X0)) & ((~ (((((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))) & (((member @ a) @ X0) @ x)))) => ((((((auto_ll_on_flow0 @ a) @ f) @ x) @ X0) @ (zero_zero @ real)) = (zero_zero @ a))))))).
% 2.53/1.58 thf(fact_41_rev_Otransversal__segment__exists, axiom, (! [X:a]: (((((member @ a) @ X) @ x) => (((((uminus_uminus @ (a > a)) @ f) @ X) != (zero_zero @ a)) => (~ ((! [A3:a,B3:a]: (((((member @ a) @ X) @ (((line_open_segment @ a) @ A3) @ B3)) => (~ ((((((poinca272511729egment @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ A3) @ B3))))))))))))).
% 2.53/1.58 thf(fact_42_rev_Ofixpoint__sol_I2_J, axiom, (! [X:a,T2:real]: (((((member @ a) @ X) @ x) => (((((uminus_uminus @ (a > a)) @ f) @ X) = (zero_zero @ a)) => ((((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) @ T2) = X)))))).
% 2.53/1.58 thf(fact_43_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.53/1.58 thf(fact_44_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A4)))) = A4)))).
% 2.53/1.58 thf(fact_45_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.53/1.58 thf(fact_46_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.53/1.58 thf(fact_47_local_Oflow__undefined0, axiom, (! [T2:real,X:a]: (((~ ((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X)))) => ((((((auto_ll_on_flow0 @ a) @ f) @ x) @ X) @ T2) = (zero_zero @ a)))))).
% 2.53/1.58 thf(fact_48_rev__transversal__segment, axiom, (! [A2:a,B2:a]: (((((((poinca272511729egment @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ A2) @ B2) = (((((poinca272511729egment @ a) @ f) @ x) @ A2) @ B2))))).
% 2.53/1.58 thf(fact_49_rev_Otransversal__segment__reverse, axiom, (! [X:a,Y:a]: (((((((poinca272511729egment @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) @ Y) => (((((poinca272511729egment @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ Y) @ X))))).
% 2.53/1.58 thf(fact_50_rev_Otransversal__segment__commute, axiom, (! [X:a,Y:a]: (((((((poinca272511729egment @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) @ Y) = (((((poinca272511729egment @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ Y) @ X))))).
% 2.53/1.58 thf(fact_51_fixed__point__imp__closed__orbit__period__zero_I1_J, axiom, (! [X:a]: (((((member @ a) @ X) @ x) => (((f @ X) = (zero_zero @ a)) => ((((period385816147_orbit @ a) @ f) @ x) @ X)))))).
% 2.53/1.58 thf(fact_52_existence__ivl0__cong, axiom, (! [B:$tType,Y4:(set @ a),G:(a > a),X0:a]: (((x = Y4) => ((! [X3:a,T:B]: (((((member @ a) @ X3) @ Y4) => ((f @ X3) = (G @ X3))))) => (((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0) = ((((auto_l1112008849e_ivl0 @ a) @ G) @ Y4) @ X0))))))).
% 2.53/1.58 thf(fact_53_closed__orbit__in__domain, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ f) @ x) @ X) => (((member @ a) @ X) @ x))))).
% 2.53/1.58 thf(fact_54_add_Oinverse__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ A2)) = A2))))))).
% 2.53/1.58 thf(fact_55_neg__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((uminus_uminus @ A) @ A2) = ((uminus_uminus @ A) @ B2)) = (A2 = B2)))))))).
% 2.53/1.58 thf(fact_56_existence__ivl__zero, axiom, (! [X0:a]: (((((member @ a) @ X0) @ x) => (((member @ real) @ (zero_zero @ real)) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)))))).
% 2.53/1.58 thf(fact_57_rev_Oexistence__ivl0__cong, axiom, (! [B:$tType,Y4:(set @ a),G:(a > a),X0:a]: (((x = Y4) => ((! [X3:a,T:B]: (((((member @ a) @ X3) @ Y4) => ((((uminus_uminus @ (a > a)) @ f) @ X3) = (G @ X3))))) => (((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0) = ((((auto_l1112008849e_ivl0 @ a) @ G) @ Y4) @ X0))))))).
% 2.53/1.58 thf(fact_58_local_Omem__existence__ivl__subset, axiom, (! [T2:real,X0:a]: (((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) => (((member @ real) @ T2) @ (top_top @ (set @ real))))))).
% 2.53/1.58 thf(fact_59_flow0__cong, axiom, (! [B:$tType,Y4:(set @ a),G:(a > a),T2:real,X0:a]: (((x = Y4) => ((! [X3:a,T:B]: (((((member @ a) @ X3) @ Y4) => ((f @ X3) = (G @ X3))))) => ((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) => ((((((auto_ll_on_flow0 @ a) @ f) @ x) @ X0) @ T2) = (((((auto_ll_on_flow0 @ a) @ G) @ Y4) @ X0) @ T2)))))))).
% 2.53/1.58 thf(fact_60_rev_Oclosed__orbit__eq__rev, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) = ((((period385816147_orbit @ a) @ f) @ x) @ X))))).
% 2.53/1.58 thf(fact_61_rev_Oclosed__orbit__in__domain, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) => (((member @ a) @ X) @ x))))).
% 2.53/1.58 thf(fact_62_closed__orbit__flow0, axiom, (! [X:a,T2:real]: ((((((period385816147_orbit @ a) @ f) @ x) @ X) => ((((period385816147_orbit @ a) @ f) @ x) @ (((((auto_ll_on_flow0 @ a) @ f) @ x) @ X) @ T2)))))).
% 2.53/1.58 thf(fact_63_rev_Oexistence__ivl__zero, axiom, (! [X0:a]: (((((member @ a) @ X0) @ x) => (((member @ real) @ (zero_zero @ real)) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)))))).
% 2.53/1.58 thf(fact_64_local_Oexistence__ivl__initial__time, axiom, (! [X0:a]: (((((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))) => ((((member @ a) @ X0) @ x) => (((member @ real) @ (zero_zero @ real)) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0))))))).
% 2.53/1.58 thf(fact_65_local_Orev_Omem__existence__ivl__subset, axiom, (! [T2:real,X0:a]: (((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)) => (((member @ real) @ T2) @ (top_top @ (set @ real))))))).
% 2.53/1.58 thf(fact_66_rev_Oflow0__cong, axiom, (! [B:$tType,Y4:(set @ a),G:(a > a),T2:real,X0:a]: (((x = Y4) => ((! [X3:a,T:B]: (((((member @ a) @ X3) @ Y4) => ((((uminus_uminus @ (a > a)) @ f) @ X3) = (G @ X3))))) => ((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)) => ((((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0) @ T2) = (((((auto_ll_on_flow0 @ a) @ G) @ Y4) @ X0) @ T2)))))))).
% 2.53/1.58 thf(fact_67_fixpoint__sol_I1_J, axiom, (! [X:a]: (((((member @ a) @ X) @ x) => (((f @ X) = (zero_zero @ a)) => (((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X) = (top_top @ (set @ real)))))))).
% 2.53/1.58 thf(fact_68_rev_Oclosed__orbit__flow0, axiom, (! [X:a,T2:real]: ((((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) => ((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ (((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) @ T2)))))).
% 2.53/1.58 thf(fact_69_closed__orbit__global__existence, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ f) @ x) @ X) => (((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X) = (top_top @ (set @ real))))))).
% 2.53/1.58 thf(fact_70_closed__orbitI, axiom, (! [T2:real,T3:real,Y:a]: (((T2 != T3) => ((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ Y)) => ((((member @ real) @ T3) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ Y)) => (((((((auto_ll_on_flow0 @ a) @ f) @ x) @ Y) @ T2) = (((((auto_ll_on_flow0 @ a) @ f) @ x) @ Y) @ T3)) => ((((period385816147_orbit @ a) @ f) @ x) @ Y)))))))).
% 2.53/1.58 thf(fact_71_rev_Ofixed__point__imp__closed__orbit__period__zero_I1_J, axiom, (! [X:a]: (((((member @ a) @ X) @ x) => (((((uminus_uminus @ (a > a)) @ f) @ X) = (zero_zero @ a)) => ((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X)))))).
% 2.53/1.58 thf(fact_72_periodic__orbit__period_I2_J, axiom, (! [X:a]: ((((((period862636932_orbit @ a) @ f) @ x) @ X) => ((((((auto_ll_on_flow0 @ a) @ f) @ x) @ X) @ ((((period1153813292period @ a) @ f) @ x) @ X)) = X))))).
% 2.53/1.58 thf(fact_73_closed__orbit__periodic, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ f) @ x) @ X) => (((f @ X) != (zero_zero @ a)) => ((((period862636932_orbit @ a) @ f) @ x) @ X)))))).
% 2.53/1.58 thf(fact_74_local_Orev_Oexistence__ivl__initial__time, axiom, (! [X0:a]: (((((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))) => ((((member @ a) @ X0) @ x) => (((member @ real) @ (zero_zero @ real)) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0))))))).
% 2.53/1.58 thf(fact_75_rev_Ofixpoint__sol_I1_J, axiom, (! [X:a]: (((((member @ a) @ X) @ x) => (((((uminus_uminus @ (a > a)) @ f) @ X) = (zero_zero @ a)) => (((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) = (top_top @ (set @ real)))))))).
% 2.53/1.58 thf(fact_76_local_Orev_Oflow__undefined0, axiom, (! [T2:real,X:a]: (((~ ((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X)))) => ((((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) @ T2) = (zero_zero @ a)))))).
% 2.53/1.58 thf(fact_77_closed__orbit__def, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ f) @ x) @ X) = (? [X2:real]: (((((member @ real) @ X2) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X)) & ((X2 != (zero_zero @ real)) & ((((((auto_ll_on_flow0 @ a) @ f) @ x) @ X) @ X2) = X))))))))).
% 2.53/1.58 thf(fact_78_rev_Oclosed__orbit__global__existence, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) => (((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) = (top_top @ (set @ real))))))).
% 2.53/1.58 thf(fact_79_rev_Oclosed__orbitI, axiom, (! [T2:real,T3:real,Y:a]: (((T2 != T3) => ((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ Y)) => ((((member @ real) @ T3) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ Y)) => (((((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ Y) @ T2) = (((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ Y) @ T3)) => ((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ Y)))))))).
% 2.53/1.58 thf(fact_80_rev_Ofixed__point__imp__closed__orbit__period__zero_I2_J, axiom, (! [X:a]: (((((member @ a) @ X) @ x) => (((((uminus_uminus @ (a > a)) @ f) @ X) = (zero_zero @ a)) => (((((period1153813292period @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) = (zero_zero @ real))))))).
% 2.53/1.58 thf(fact_81_rev_Operiodic__orbit__imp__flow0__regular, axiom, (! [X:a,T2:real]: ((((((period862636932_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) => ((((uminus_uminus @ (a > a)) @ f) @ (((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) @ T2)) != (zero_zero @ a)))))).
% 2.53/1.58 thf(fact_82_periodic__orbitI, axiom, (! [T2:real,T3:real,Y:a]: (((T2 != T3) => ((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ Y)) => ((((member @ real) @ T3) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ Y)) => (((((((auto_ll_on_flow0 @ a) @ f) @ x) @ Y) @ T2) = (((((auto_ll_on_flow0 @ a) @ f) @ x) @ Y) @ T3)) => (((f @ Y) != (zero_zero @ a)) => ((((period862636932_orbit @ a) @ f) @ x) @ Y))))))))).
% 2.53/1.58 thf(fact_83_periodic__orbit__period_I1_J, axiom, (! [X:a]: ((((((period862636932_orbit @ a) @ f) @ x) @ X) => (((ord_less @ real) @ (zero_zero @ real)) @ ((((period1153813292period @ a) @ f) @ x) @ X)))))).
% 2.53/1.58 thf(fact_84_closed__orbit__period__zero__fixed__point, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ f) @ x) @ X) => ((((((period1153813292period @ a) @ f) @ x) @ X) = (zero_zero @ real)) => ((f @ X) = (zero_zero @ a))))))).
% 2.53/1.58 thf(fact_85_rev_Operiodic__orbit__period_I2_J, axiom, (! [X:a]: ((((((period862636932_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) => ((((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) @ ((((period1153813292period @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X)) = X))))).
% 2.53/1.58 thf(fact_86_rev_Oclosed__orbit__periodic, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) => (((((uminus_uminus @ (a > a)) @ f) @ X) != (zero_zero @ a)) => ((((period862636932_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X)))))).
% 2.53/1.58 thf(fact_87_add_Oinverse__neutral, axiom, (! [A:$tType]: (((group_add @ A) => (((uminus_uminus @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.53/1.58 thf(fact_88_neg__0__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: ((((zero_zero @ A) = ((uminus_uminus @ A) @ A2)) = ((zero_zero @ A) = A2)))))))).
% 2.53/1.58 thf(fact_89_neg__equal__0__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((uminus_uminus @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_90_equal__neg__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((A2 = ((uminus_uminus @ A) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_91_neg__equal__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((uminus_uminus @ A) @ A2) = A2) = (A2 = (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_92_local_Orev_Oflow__initial__time__if, axiom, (! [X0:a]: (((((((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))) & (((member @ a) @ X0) @ x)) => ((((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0) @ (zero_zero @ real)) = X0)) & ((~ (((((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))) & (((member @ a) @ X0) @ x)))) => ((((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0) @ (zero_zero @ real)) = (zero_zero @ a))))))).
% 2.53/1.58 thf(fact_93_neg__less__iff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less @ A) @ A2) @ B2)))))))).
% 2.53/1.58 thf(fact_94_rev_Oclosed__orbit__def, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) = (? [X2:real]: (((((member @ real) @ X2) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X)) & ((X2 != (zero_zero @ real)) & ((((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) @ X2) = X))))))))).
% 2.53/1.58 thf(fact_95_abs__minus, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A2:A]: ((((abs_abs @ A) @ ((uminus_uminus @ A) @ A2)) = ((abs_abs @ A) @ A2)))))))).
% 2.53/1.58 thf(fact_96_abs__minus__cancel, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: ((((abs_abs @ A) @ ((uminus_uminus @ A) @ A2)) = ((abs_abs @ A) @ A2)))))))).
% 2.53/1.58 thf(fact_97_rev_Operiodic__orbitI, axiom, (! [T2:real,T3:real,Y:a]: (((T2 != T3) => ((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ Y)) => ((((member @ real) @ T3) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ Y)) => (((((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ Y) @ T2) = (((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ Y) @ T3)) => (((((uminus_uminus @ (a > a)) @ f) @ Y) != (zero_zero @ a)) => ((((period862636932_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ Y))))))))).
% 2.53/1.58 thf(fact_98_rev_Operiodic__orbit__period_I1_J, axiom, (! [X:a]: ((((((period862636932_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) => (((ord_less @ real) @ (zero_zero @ real)) @ ((((period1153813292period @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X)))))).
% 2.53/1.58 thf(fact_99_rev_Oclosed__orbit__period__zero__fixed__point, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) => ((((((period1153813292period @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) = (zero_zero @ real)) => ((((uminus_uminus @ (a > a)) @ f) @ X) = (zero_zero @ a))))))).
% 2.53/1.58 thf(fact_100_periodic__orbit__def, axiom, (! [X:a]: ((((((period862636932_orbit @ a) @ f) @ x) @ X) = (((((period385816147_orbit @ a) @ f) @ x) @ X) & (((ord_less @ real) @ (zero_zero @ real)) @ ((((period1153813292period @ a) @ f) @ x) @ X))))))).
% 2.53/1.58 thf(fact_101_rev_Operiodic__orbit__def, axiom, (! [X:a]: ((((((period862636932_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) = (((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) & (((ord_less @ real) @ (zero_zero @ real)) @ ((((period1153813292period @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X))))))).
% 2.53/1.58 thf(fact_102_less__neg__neg, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less @ A) @ A2) @ ((uminus_uminus @ A) @ A2)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_103_neg__less__pos, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.53/1.58 thf(fact_104_neg__0__less__iff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_105_neg__less__0__iff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.53/1.58 thf(fact_106_local_Omem__existence__ivl__iv__defined_I2_J, axiom, (! [T2:real,X0:a]: (((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) => (((member @ a) @ X0) @ x))))).
% 2.53/1.58 thf(fact_107_local_Orev_Omem__existence__ivl__iv__defined_I2_J, axiom, (! [T2:real,X0:a]: (((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)) => (((member @ a) @ X0) @ x))))).
% 2.53/1.58 thf(fact_108_flow0__defined, axiom, (! [Xa:real,X0:a]: (((((member @ real) @ Xa) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) => (((member @ a) @ (((((auto_ll_on_flow0 @ a) @ f) @ x) @ X0) @ Xa)) @ x))))).
% 2.53/1.58 thf(fact_109_local_Oflow__in__domain, axiom, (! [T2:real,X0:a]: (((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) => (((member @ a) @ (((((auto_ll_on_flow0 @ a) @ f) @ x) @ X0) @ T2)) @ x))))).
% 2.53/1.58 thf(fact_110_local_Omem__existence__ivl__iv__defined_I1_J, axiom, (! [T2:real,X0:a]: (((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) => (((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))))))).
% 2.53/1.58 thf(fact_111_local_Oexistence__ivl__initial__time__iff, axiom, (! [X0:a]: (((((member @ real) @ (zero_zero @ real)) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) = ((((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))) & (((member @ a) @ X0) @ x)))))).
% 2.53/1.58 thf(fact_112_local_Oflow__initial__time, axiom, (! [X0:a]: (((((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))) => ((((member @ a) @ X0) @ x) => ((((((auto_ll_on_flow0 @ a) @ f) @ x) @ X0) @ (zero_zero @ real)) = X0)))))).
% 2.53/1.58 thf(fact_113_rev_Oflow0__defined, axiom, (! [Xa:real,X0:a]: (((((member @ real) @ Xa) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)) => (((member @ a) @ (((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0) @ Xa)) @ x))))).
% 2.53/1.58 thf(fact_114_local_Orev_Oflow__in__domain, axiom, (! [T2:real,X0:a]: (((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)) => (((member @ a) @ (((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0) @ T2)) @ x))))).
% 2.53/1.58 thf(fact_115_local_Orev_Omem__existence__ivl__iv__defined_I1_J, axiom, (! [T2:real,X0:a]: (((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)) => (((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))))))).
% 2.53/1.58 thf(fact_116_local_Orev_Oexistence__ivl__initial__time__iff, axiom, (! [X0:a]: (((((member @ real) @ (zero_zero @ real)) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)) = ((((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))) & (((member @ a) @ X0) @ x)))))).
% 2.53/1.58 thf(fact_117_local_Orev_Oflow__initial__time, axiom, (! [X0:a]: (((((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))) => ((((member @ a) @ X0) @ x) => ((((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0) @ (zero_zero @ real)) = X0)))))).
% 2.53/1.58 thf(fact_118_equation__minus__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((A2 = ((uminus_uminus @ A) @ B2)) = (B2 = ((uminus_uminus @ A) @ A2))))))))).
% 2.53/1.58 thf(fact_119_minus__equation__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((uminus_uminus @ A) @ A2) = B2) = (((uminus_uminus @ A) @ B2) = A2)))))))).
% 2.53/1.58 thf(fact_120_less__minus__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ ((uminus_uminus @ A) @ B2)) = (((ord_less @ A) @ B2) @ ((uminus_uminus @ A) @ A2))))))))).
% 2.53/1.58 thf(fact_121_minus__less__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) = (((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ A2)))))))).
% 2.53/1.58 thf(fact_122_abs__eq__iff, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [X:A,Y:A]: (((((abs_abs @ A) @ X) = ((abs_abs @ A) @ Y)) = ((X = Y) | (X = ((uminus_uminus @ A) @ Y)))))))))).
% 2.53/1.58 thf(fact_123_abs__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ ((abs_abs @ A) @ A2)) @ B2) = ((((ord_less @ A) @ A2) @ B2) & (((ord_less @ A) @ ((uminus_uminus @ A) @ A2)) @ B2))))))))).
% 2.53/1.58 thf(fact_124_abs__if, axiom, (! [A:$tType]: (((abs_if @ A) => ((abs_abs @ A) = (^ [A5:A]: (((((if @ A) @ (((ord_less @ A) @ A5) @ (zero_zero @ A))) @ ((uminus_uminus @ A) @ A5)) @ A5)))))))).
% 2.53/1.58 thf(fact_125_abs__of__neg, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => (((abs_abs @ A) @ A2) = ((uminus_uminus @ A) @ A2))))))))).
% 2.53/1.58 thf(fact_126__092_060alpha_062__limit__point__eq__rev, axiom, (! [X:a,P2:a]: (((((((limit_1335059408_point @ a) @ f) @ x) @ X) @ P2) = (((((limit_1052139091_point @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) @ P2))))).
% 2.53/1.58 thf(fact_127_rev_O_092_060alpha_062__limit__point__eq__rev, axiom, (! [X:a,P2:a]: (((((((limit_1335059408_point @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) @ P2) = (((((limit_1052139091_point @ a) @ f) @ x) @ X) @ P2))))).
% 2.53/1.58 thf(fact_128_rev_Omvar_Ointerval__axioms, axiom, (! [X0:a]: ((initia826609931terval @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0))))).
% 2.53/1.58 thf(fact_129_mvar_Ointerval__axioms, axiom, (! [X0:a]: ((initia826609931terval @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0))))).
% 2.53/1.58 thf(fact_130_abs__neg__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((abs_abs @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) = (one_one @ A)))))).
% 2.53/1.58 thf(fact_131_rev_Oclosed__orbitE, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) => (~ ((! [T4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ T4) => (~ ((! [T5:real]: (((((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) @ (((plus_plus @ real) @ T5) @ T4)) = (((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) @ T5))))))))))))))).
% 2.53/1.58 thf(fact_132_local_Orev_Oexistence__ivl__notempty, axiom, (! [X0:a]: (((((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))) => ((((member @ a) @ X0) @ x) => (((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0) != (bot_bot @ (set @ real)))))))).
% 2.53/1.58 thf(fact_133_rev_Oclosed__orbit__period__nonneg, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) => (((ord_less_eq @ real) @ (zero_zero @ real)) @ ((((period1153813292period @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X)))))).
% 2.53/1.58 thf(fact_134_rev_Ovec__simps_I11_J, axiom, (! [Ag:$tType]: (((semigroup_add @ Ag) => (! [Ae:Ag,Be:Ag,Cd:Ag]: (((((plus_plus @ Ag) @ (((plus_plus @ Ag) @ Ae) @ Be)) @ Cd) = (((plus_plus @ Ag) @ Ae) @ (((plus_plus @ Ag) @ Be) @ Cd))))))))).
% 2.53/1.58 thf(fact_135_rev_Ovec__simps_I12_J, axiom, (! [Ai:$tType]: (((ab_semigroup_add @ Ai) => ((plus_plus @ Ai) = (^ [Ag2:Ai,Bg:Ai]: ((((plus_plus @ Ai) @ Bg) @ Ag2)))))))).
% 2.53/1.58 thf(fact_136_rev_Ovec__simps_I13_J, axiom, (! [Ai:$tType]: (((ab_semigroup_add @ Ai) => (! [Bg2:Ai,Ag3:Ai,Cf:Ai]: (((((plus_plus @ Ai) @ Bg2) @ (((plus_plus @ Ai) @ Ag3) @ Cf)) = (((plus_plus @ Ai) @ Ag3) @ (((plus_plus @ Ai) @ Bg2) @ Cf))))))))).
% 2.53/1.58 thf(fact_137_interval__axioms, axiom, (initia826609931terval @ (top_top @ (set @ real)))).
% 2.53/1.58 thf(fact_138_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A2:A,C:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C) @ A2)) = (B2 = C)))))))).
% 2.53/1.58 thf(fact_139_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B2:A,C:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C)) = (B2 = C)))))))).
% 2.53/1.58 thf(fact_140_rev__eq__flow, axiom, (! [Y:a,T2:real]: (((((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ Y) @ T2) = (((((auto_ll_on_flow0 @ a) @ f) @ x) @ Y) @ ((uminus_uminus @ real) @ T2)))))).
% 2.53/1.58 thf(fact_141_rev_Orev__eq__flow, axiom, (! [Y:a,T2:real]: (((((((auto_ll_on_flow0 @ a) @ f) @ x) @ Y) @ T2) = (((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ Y) @ ((uminus_uminus @ real) @ T2)))))).
% 2.53/1.58 thf(fact_142_recurrence__time__flip__sign_I1_J, axiom, (! [T6:real,X:a]: (((((member @ real) @ T6) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X)) => (((((((auto_ll_on_flow0 @ a) @ f) @ x) @ X) @ T6) = X) => (((member @ real) @ ((uminus_uminus @ real) @ T6)) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X))))))).
% 2.53/1.58 thf(fact_143_recurrence__time__flip__sign_I2_J, axiom, (! [T6:real,X:a]: (((((member @ real) @ T6) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X)) => (((((((auto_ll_on_flow0 @ a) @ f) @ x) @ X) @ T6) = X) => ((((((auto_ll_on_flow0 @ a) @ f) @ x) @ X) @ ((uminus_uminus @ real) @ T6)) = X)))))).
% 2.53/1.58 thf(fact_144_rev_Orecurrence__time__flip__sign_I1_J, axiom, (! [T6:real,X:a]: (((((member @ real) @ T6) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X)) => (((((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) @ T6) = X) => (((member @ real) @ ((uminus_uminus @ real) @ T6)) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X))))))).
% 2.53/1.58 thf(fact_145_rev_Orecurrence__time__flip__sign_I2_J, axiom, (! [T6:real,X:a]: (((((member @ real) @ T6) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X)) => (((((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) @ T6) = X) => ((((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) @ ((uminus_uminus @ real) @ T6)) = X)))))).
% 2.53/1.58 thf(fact_146_local_Oflow__trans, axiom, (! [S:real,X0:a,T2:real]: (((((member @ real) @ S) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) => ((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ (((((auto_ll_on_flow0 @ a) @ f) @ x) @ X0) @ S))) => ((((((auto_ll_on_flow0 @ a) @ f) @ x) @ X0) @ (((plus_plus @ real) @ S) @ T2)) = (((((auto_ll_on_flow0 @ a) @ f) @ x) @ (((((auto_ll_on_flow0 @ a) @ f) @ x) @ X0) @ S)) @ T2))))))).
% 2.53/1.58 thf(fact_147_local_Oexistence__ivl__trans_H, axiom, (! [T2:real,S:real,X0:a]: (((((member @ real) @ (((plus_plus @ real) @ T2) @ S)) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) => ((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) => (((member @ real) @ S) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ (((((auto_ll_on_flow0 @ a) @ f) @ x) @ X0) @ T2)))))))).
% 2.53/1.58 thf(fact_148_local_Oexistence__ivl__trans, axiom, (! [S:real,X0:a,T2:real]: (((((member @ real) @ S) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) => ((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ (((((auto_ll_on_flow0 @ a) @ f) @ x) @ X0) @ S))) => (((member @ real) @ (((plus_plus @ real) @ S) @ T2)) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0))))))).
% 2.53/1.58 thf(fact_149_local_Orev_Oflow__trans, axiom, (! [S:real,X0:a,T2:real]: (((((member @ real) @ S) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)) => ((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ (((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0) @ S))) => ((((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0) @ (((plus_plus @ real) @ S) @ T2)) = (((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ (((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0) @ S)) @ T2))))))).
% 2.53/1.58 thf(fact_150_local_Orev_Oexistence__ivl__trans_H, axiom, (! [T2:real,S:real,X0:a]: (((((member @ real) @ (((plus_plus @ real) @ T2) @ S)) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)) => ((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)) => (((member @ real) @ S) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ (((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0) @ T2)))))))).
% 2.53/1.58 thf(fact_151_local_Orev_Oexistence__ivl__trans, axiom, (! [S:real,X0:a,T2:real]: (((((member @ real) @ S) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)) => ((((member @ real) @ T2) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ (((((auto_ll_on_flow0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0) @ S))) => (((member @ real) @ (((plus_plus @ real) @ S) @ T2)) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0))))))).
% 2.53/1.58 thf(fact_152_local_Oexistence__ivl__notempty, axiom, (! [X0:a]: (((((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))) => ((((member @ a) @ X0) @ x) => (((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0) != (bot_bot @ (set @ real)))))))).
% 2.53/1.58 thf(fact_153_closed__orbit__period__nonneg, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ f) @ x) @ X) => (((ord_less_eq @ real) @ (zero_zero @ real)) @ ((((period1153813292period @ a) @ f) @ x) @ X)))))).
% 2.53/1.58 thf(fact_154_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_155_neg__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.53/1.58 thf(fact_156_zero__eq__add__iff__both__eq__0, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A,Y:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ X) @ Y)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))).
% 2.53/1.58 thf(fact_157_add__eq__0__iff__both__eq__0, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A,Y:A]: ((((((plus_plus @ A) @ X) @ Y) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))).
% 2.53/1.58 thf(fact_158_add__cancel__right__right, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A,B2:A]: (((A2 = (((plus_plus @ A) @ A2) @ B2)) = (B2 = (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_159_add__cancel__right__left, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A,B2:A]: (((A2 = (((plus_plus @ A) @ B2) @ A2)) = (B2 = (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_160_add__cancel__left__right, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A,B2:A]: ((((((plus_plus @ A) @ A2) @ B2) = A2) = (B2 = (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_161_add__cancel__left__left, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [B2:A,A2:A]: ((((((plus_plus @ A) @ B2) @ A2) = A2) = (B2 = (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_162_double__zero__sym, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ A2) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_163_linordered__ab__group__add__class_Odouble__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: ((((((plus_plus @ A) @ A2) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_164_add_Oright__neutral, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.53/1.58 thf(fact_165_add_Oleft__neutral, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))).
% 2.53/1.58 thf(fact_166_add__le__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.53/1.58 thf(fact_167_add__le__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.53/1.58 thf(fact_168_add__less__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C)) = (((ord_less @ A) @ A2) @ B2)))))))).
% 2.53/1.58 thf(fact_169_add__less__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2)) = (((ord_less @ A) @ A2) @ B2)))))))).
% 2.53/1.58 thf(fact_170_closed__orbitE, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ f) @ x) @ X) => (~ ((! [T4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ T4) => (~ ((! [T5:real]: (((((((auto_ll_on_flow0 @ a) @ f) @ x) @ X) @ (((plus_plus @ real) @ T5) @ T4)) = (((((auto_ll_on_flow0 @ a) @ f) @ x) @ X) @ T5))))))))))))))).
% 2.53/1.58 thf(fact_171_minus__add__distrib, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,B2:A]: ((((uminus_uminus @ A) @ (((plus_plus @ A) @ A2) @ B2)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ ((uminus_uminus @ A) @ B2))))))))).
% 2.53/1.58 thf(fact_172_minus__add__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ (((plus_plus @ A) @ A2) @ B2)) = B2))))))).
% 2.53/1.58 thf(fact_173_add__minus__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ B2)) = B2))))))).
% 2.53/1.58 thf(fact_174_abs__add__abs, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A,B2:A]: ((((abs_abs @ A) @ (((plus_plus @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B2))) = (((plus_plus @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B2))))))))).
% 2.53/1.58 thf(fact_175_neg__0__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_176_neg__le__0__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.53/1.58 thf(fact_177_less__eq__neg__nonpos, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_178_neg__less__eq__nonneg, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.53/1.58 thf(fact_179_linordered__ab__group__add__class_Ozero__le__double__add__iff__zero__le__single__add, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.53/1.58 thf(fact_180_linordered__ab__group__add__class_Odouble__add__le__zero__iff__single__add__le__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_181_le__add__same__cancel2, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (((plus_plus @ A) @ B2) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)))))))).
% 2.53/1.58 thf(fact_182_le__add__same__cancel1, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)))))))).
% 2.53/1.58 thf(fact_183_add__le__same__cancel2, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_184_add__le__same__cancel1, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ B2) @ A2)) @ B2) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_185_linordered__ab__group__add__class_Ozero__less__double__add__iff__zero__less__single__add, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.53/1.58 thf(fact_186_double__add__less__zero__iff__single__add__less__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_187_less__add__same__cancel2, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (((plus_plus @ A) @ B2) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))).
% 2.53/1.58 thf(fact_188_less__add__same__cancel1, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B2)) = (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))).
% 2.53/1.58 thf(fact_189_add__less__same__cancel2, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_190_add__less__same__cancel1, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ B2) @ A2)) @ B2) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_191_add_Oleft__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (zero_zero @ A)))))))).
% 2.53/1.58 thf(fact_192_add_Oright__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ ((uminus_uminus @ A) @ A2)) = (zero_zero @ A)))))))).
% 2.53/1.58 thf(fact_193_abs__le__zero__iff, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A2)) @ (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.53/1.58 thf(fact_194_abs__le__self__iff, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A2)) @ A2) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.53/1.58 thf(fact_195_abs__of__nonneg, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((abs_abs @ A) @ A2) = A2)))))))).
% 2.53/1.58 thf(fact_196_ball__trivial, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X:A]: (((((elemen1815086676c_ball @ A) @ X) @ (zero_zero @ real)) = (bot_bot @ (set @ A))))))))).
% 2.53/1.58 thf(fact_197_add__neg__numeral__special_I7_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((plus_plus @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))) = (zero_zero @ A)))))).
% 2.53/1.58 thf(fact_198_add__neg__numeral__special_I8_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((plus_plus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (one_one @ A)) = (zero_zero @ A)))))).
% 2.53/1.58 thf(fact_199_abs__of__nonpos, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => (((abs_abs @ A) @ A2) = ((uminus_uminus @ A) @ A2))))))))).
% 2.53/1.58 thf(fact_200_ball__eq__empty, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X:A,E:real]: ((((((elemen1815086676c_ball @ A) @ X) @ E) = (bot_bot @ (set @ A))) = (((ord_less_eq @ real) @ E) @ (zero_zero @ real))))))))).
% 2.53/1.58 thf(fact_201_local_Oexistence__ivl__undefined, axiom, (! [X0:a]: (((~ ((((member @ a) @ X0) @ x))) => (((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0) = (bot_bot @ (set @ real))))))).
% 2.53/1.58 thf(fact_202_local_Orev_Oexistence__ivl__undefined, axiom, (! [X0:a]: (((~ ((((member @ a) @ X0) @ x))) => (((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0) = (bot_bot @ (set @ real))))))).
% 2.53/1.58 thf(fact_203_local_Oexistence__ivl__empty__iff, axiom, (! [X0:a]: (((((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0) = (bot_bot @ (set @ real))) = ((~ ((((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))))) | (~ ((((member @ a) @ X0) @ x)))))))).
% 2.53/1.58 thf(fact_204_local_Oexistence__ivl__empty1, axiom, (! [X0:a]: (((~ ((((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))))) => (((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0) = (bot_bot @ (set @ real))))))).
% 2.53/1.58 thf(fact_205_local_Orev_Oexistence__ivl__empty__iff, axiom, (! [X0:a]: (((((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0) = (bot_bot @ (set @ real))) = ((~ ((((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))))) | (~ ((((member @ a) @ X0) @ x)))))))).
% 2.53/1.58 thf(fact_206_local_Orev_Oexistence__ivl__empty1, axiom, (! [X0:a]: (((~ ((((member @ real) @ (zero_zero @ real)) @ (top_top @ (set @ real))))) => (((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0) = (bot_bot @ (set @ real))))))).
% 2.53/1.58 thf(fact_207_is__num__normalize_I8_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [A2:A,B2:A]: ((((uminus_uminus @ A) @ (((plus_plus @ A) @ A2) @ B2)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2))))))))).
% 2.53/1.58 thf(fact_208_abs__triangle__ineq, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A,B2:A]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((plus_plus @ A) @ A2) @ B2))) @ (((plus_plus @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B2))))))))).
% 2.53/1.58 thf(fact_209_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.53/1.58 thf(fact_210_le__numeral__extra_I4_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (one_one @ A)) @ (one_one @ A)))))).
% 2.53/1.58 thf(fact_211_is__num__normalize_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [A2:A,B2:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.53/1.58 thf(fact_212_add__le__imp__le__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C)) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.53/1.58 thf(fact_213_add__le__imp__le__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2)) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.53/1.58 thf(fact_214_add__less__le__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A2:A,B2:A,C:A,D2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C) @ D2) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ D2)))))))))).
% 2.53/1.58 thf(fact_215_add__le__less__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A2:A,B2:A,C:A,D2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ A) @ C) @ D2) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ D2)))))))))).
% 2.53/1.58 thf(fact_216_add__nonpos__eq__0__iff, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ Y) @ (zero_zero @ A)) => (((((plus_plus @ A) @ X) @ Y) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))))).
% 2.53/1.58 thf(fact_217_add__nonneg__eq__0__iff, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (((((plus_plus @ A) @ X) @ Y) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))))).
% 2.53/1.58 thf(fact_218_add__nonpos__nonpos, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (zero_zero @ A)))))))))).
% 2.53/1.58 thf(fact_219_add__nonneg__nonneg, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.53/1.58 thf(fact_220_le__iff__add, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => ((ord_less_eq @ A) = (^ [A5:A,B4:A]: ((? [C2:A]: ((B4 = (((plus_plus @ A) @ A5) @ C2))))))))))).
% 2.53/1.58 thf(fact_221_add__increasing2, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [C:A,B2:A,A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C) => ((((ord_less_eq @ A) @ B2) @ A2) => (((ord_less_eq @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C)))))))))).
% 2.53/1.58 thf(fact_222_add__decreasing2, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [C:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ C) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ B2))))))))).
% 2.53/1.58 thf(fact_223_add__right__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.53/1.58 thf(fact_224_less__eqE, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (~ ((! [C3:A]: ((B2 != (((plus_plus @ A) @ A2) @ C3))))))))))))).
% 2.53/1.58 thf(fact_225_add__increasing, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C)))))))))).
% 2.53/1.58 thf(fact_226_add__decreasing, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,C:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ B2))))))))).
% 2.53/1.58 thf(fact_227_add__left__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2))))))))).
% 2.53/1.58 thf(fact_228_add__right__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A2:A,C:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C) @ A2)) => (B2 = C)))))))).
% 2.53/1.58 thf(fact_229_add__left__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B2:A,C:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C)) => (B2 = C)))))))).
% 2.53/1.58 thf(fact_230_add__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B2:A,C:A,D2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C) @ D2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ D2)))))))))).
% 2.53/1.58 thf(fact_231_add_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [B2:A,A2:A,C:A]: (((((plus_plus @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C)) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.53/1.58 thf(fact_232_add_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((plus_plus @ A) = (^ [A5:A,B4:A]: ((((plus_plus @ A) @ B4) @ A5)))))))).
% 2.53/1.58 thf(fact_233_add_Oright__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [B2:A,A2:A,C:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C) @ A2)) = (B2 = C)))))))).
% 2.53/1.58 thf(fact_234_add_Oleft__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C)) = (B2 = C)))))))).
% 2.53/1.58 thf(fact_235_add_Oassoc, axiom, (! [A:$tType]: (((semigroup_add @ A) => (! [A2:A,B2:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.53/1.58 thf(fact_236_group__cancel_Oadd2, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [B5:A,K:A,B2:A,A2:A]: (((B5 = (((plus_plus @ A) @ K) @ B2)) => ((((plus_plus @ A) @ A2) @ B5) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.53/1.58 thf(fact_237_group__cancel_Oadd1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A4:A,K:A,A2:A,B2:A]: (((A4 = (((plus_plus @ A) @ K) @ A2)) => ((((plus_plus @ A) @ A4) @ B2) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.53/1.58 thf(fact_238_add__mono__thms__linordered__semiring_I1_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.53/1.58 thf(fact_239_add__mono__thms__linordered__semiring_I2_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.53/1.58 thf(fact_240_add__mono__thms__linordered__semiring_I3_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (K = L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.53/1.58 thf(fact_241_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (K = L)) => ((((plus_plus @ A) @ I) @ K) = (((plus_plus @ A) @ J) @ L))))))))).
% 2.53/1.58 thf(fact_242_add__mono__thms__linordered__field_I3_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.53/1.58 thf(fact_243_add__mono__thms__linordered__field_I4_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.53/1.58 thf(fact_244_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A2:A,B2:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.53/1.58 thf(fact_245_add__strict__increasing2, axiom, (! [A:$tType]: (((ordere1818651114id_add @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C)))))))))).
% 2.53/1.58 thf(fact_246_add__strict__increasing, axiom, (! [A:$tType]: (((ordere1818651114id_add @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C)))))))))).
% 2.53/1.58 thf(fact_247_add__pos__nonneg, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.53/1.58 thf(fact_248_add__nonpos__neg, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (zero_zero @ A)))))))))).
% 2.53/1.58 thf(fact_249_add__nonneg__pos, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.53/1.58 thf(fact_250_add__neg__nonpos, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (zero_zero @ A)))))))))).
% 2.53/1.58 thf(fact_251_Elementary__Metric__Spaces_Oball__empty, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [E:real,X:A]: (((((ord_less_eq @ real) @ E) @ (zero_zero @ real)) => ((((elemen1815086676c_ball @ A) @ X) @ E) = (bot_bot @ (set @ A)))))))))).
% 2.53/1.58 thf(fact_252_add_Ogroup__left__neutral, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))).
% 2.53/1.58 thf(fact_253_add_Ocomm__neutral, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ozero, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (zero @ A))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ogroup__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (group_add @ A))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Omonoid__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (monoid_add @ A))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oab__group__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (ab_group_add @ A))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Osemigroup__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (semigroup_add @ A))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocomm__monoid__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (comm_monoid_add @ A))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oab__semigroup__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (ab_semigroup_add @ A))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocancel__semigroup__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (cancel_semigroup_add @ A))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__group__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (ordered_ab_group_add @ A))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocancel__comm__monoid__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (cancel1352612707id_add @ A))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__comm__monoid__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (ordere216010020id_add @ A))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__group__add__abs, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (ordere142940540dd_abs @ A))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__semigroup__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (ordere779506340up_add @ A))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Real__Vector__Spaces_Ometric__space, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (real_V2090557954_space @ A))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__cancel__comm__monoid__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (ordere1818651114id_add @ A))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__semigroup__add__imp__le, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (ordere236663937imp_le @ A))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__cancel__ab__semigroup__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (ordere223160158up_add @ A))))).
% 2.53/1.58 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__semigroup__monoid__add__imp__le, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (ordere516151231imp_le @ A))))).
% 2.53/1.58 thf(tcon_HOL_Obool___Countable_Ocountable, axiom, (countable @ $o)).
% 2.53/1.58 thf(tcon_Set_Oset___Countable_Ocountable_1, axiom, (! [A6:$tType]: (((finite_finite @ A6) => (countable @ (set @ A6)))))).
% 2.53/1.58 thf(tcon_fun___Countable_Ocountable_2, axiom, (! [A6:$tType,A7:$tType]: ((((finite_finite @ A6) & (countable @ A7)) => (countable @ (A6 > A7)))))).
% 2.53/1.58 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A6:$tType,A7:$tType]: ((((finite_finite @ A6) & (finite_finite @ A7)) => (finite_finite @ (A6 > A7)))))).
% 2.53/1.58 thf(tcon_Set_Oset___Finite__Set_Ofinite_3, axiom, (! [A6:$tType]: (((finite_finite @ A6) => (finite_finite @ (set @ A6)))))).
% 2.53/1.58 thf(tcon_HOL_Obool___Finite__Set_Ofinite_4, axiom, (finite_finite @ $o)).
% 2.53/1.58 thf(tcon_fun___Real__Vector__Spaces_Ometric__space, axiom, (! [A6:$tType,A7:$tType]: ((((countable @ A6) & (real_V2090557954_space @ A7)) => (real_V2090557954_space @ (A6 > A7)))))).
% 2.53/1.58 thf(tcon_Set_Oset___Groups_Oab__semigroup__add, axiom, (! [A6:$tType]: (((ab_semigroup_add @ A6) => (ab_semigroup_add @ (set @ A6)))))).
% 2.53/1.58 thf(tcon_Set_Oset___Groups_Ocomm__monoid__add, axiom, (! [A6:$tType]: (((comm_monoid_add @ A6) => (comm_monoid_add @ (set @ A6)))))).
% 2.53/1.58 thf(tcon_Set_Oset___Groups_Osemigroup__add, axiom, (! [A6:$tType]: (((semigroup_add @ A6) => (semigroup_add @ (set @ A6)))))).
% 2.53/1.58 thf(tcon_Set_Oset___Groups_Omonoid__add, axiom, (! [A6:$tType]: (((monoid_add @ A6) => (monoid_add @ (set @ A6)))))).
% 2.53/1.58 thf(tcon_Set_Oset___Groups_Ozero, axiom, (! [A6:$tType]: (((zero @ A6) => (zero @ (set @ A6)))))).
% 2.53/1.58 thf(tcon_Set_Oset___Groups_Oone, axiom, (! [A6:$tType]: (((one @ A6) => (one @ (set @ A6)))))).
% 2.53/1.58 thf(tcon_Real_Oreal___Executable__Euclidean__Space_Oexecutable__euclidean__space, axiom, (execut510477386_space @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le, axiom, (ordere516151231imp_le @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add, axiom, (ordere223160158up_add @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le, axiom, (ordere236663937imp_le @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add, axiom, (ordere1818651114id_add @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space_5, axiom, (real_V2090557954_space @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add, axiom, (ordere779506340up_add @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs, axiom, (ordere142940540dd_abs @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add, axiom, (ordere216010020id_add @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add, axiom, (linord219039673up_add @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add, axiom, (cancel1352612707id_add @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add, axiom, (ordered_ab_group_add @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add, axiom, (cancel_semigroup_add @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_6, axiom, (ab_semigroup_add @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_7, axiom, (comm_monoid_add @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Rings_Olinordered__ring, axiom, (linordered_ring @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Rings_Olinordered__idom, axiom, (linordered_idom @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Osemigroup__add_8, axiom, (semigroup_add @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Rings_Ozero__less__one, axiom, (zero_less_one @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Oab__group__add, axiom, (ab_group_add @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Rings_Ozero__neq__one, axiom, (zero_neq_one @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn, axiom, (idom_abs_sgn @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Omonoid__add_9, axiom, (monoid_add @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Ogroup__add, axiom, (group_add @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Num_Oneg__numeral, axiom, (neg_numeral @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Rings_Oabs__if, axiom, (abs_if @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Ozero_10, axiom, (zero @ real)).
% 2.53/1.58 thf(tcon_Real_Oreal___Groups_Oone_11, axiom, (one @ real)).
% 2.53/1.58 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.53/1.58 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.53/1.58 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.53/1.58 thf(tfree_0, hypothesis, (execut510477386_space @ a)).
% 2.53/1.58 thf(conj_0, hypothesis, (! [Delta2:real,Tau2:(a > real)]: (((((ord_less @ real) @ (zero_zero @ real)) @ Delta2) => (((((topolo2071040574ous_on @ a) @ real) @ (((elemen1815086676c_ball @ a) @ p) @ Delta2)) @ Tau2) => (((Tau2 @ p) = (zero_zero @ real)) => ((! [Y3:a]: (((((member @ a) @ Y3) @ (((elemen1815086676c_ball @ a) @ p) @ Delta2)) => (((ord_less @ real) @ ((abs_abs @ real) @ (Tau2 @ Y3))) @ (one_one @ real))))) => ((! [Y3:a]: (((((member @ a) @ Y3) @ (((elemen1815086676c_ball @ a) @ p) @ Delta2)) => (((member @ a) @ (((((auto_ll_on_flow0 @ a) @ f) @ x) @ Y3) @ (Tau2 @ Y3))) @ (((line_open_segment @ a) @ a2) @ b))))) => thesisa)))))))).
% 2.53/1.58 thf(conj_1, conjecture, thesisa).
% 2.53/1.58 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.SqIxTx6pAT/DTF2DTF24163.p
% 2.53/1.58 ------------------------
% 2.72/1.63 ---- Cleaned THF ---
% 2.72/1.63 thf(ty_t_Real_Oreal,type,
% 2.72/1.63 real: $tType ).
% 2.72/1.63
% 2.72/1.63 thf(ty_t_Set_Oset,type,
% 2.72/1.63 set: $tType > $tType ).
% 2.72/1.63
% 2.72/1.63 thf(ty_tf_a,type,
% 2.72/1.63 a: $tType ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.72/1.63 finite_finite:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Countable_Ocountable,type,
% 2.72/1.63 countable:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Oone,type,
% 2.72/1.63 one:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Ozero,type,
% 2.72/1.63 zero:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Rings_Oabs__if,type,
% 2.72/1.63 abs_if:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Num_Oneg__numeral,type,
% 2.72/1.63 neg_numeral:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Ogroup__add,type,
% 2.72/1.63 group_add:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Omonoid__add,type,
% 2.72/1.63 monoid_add:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Rings_Oidom__abs__sgn,type,
% 2.72/1.63 idom_abs_sgn:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Rings_Ozero__neq__one,type,
% 2.72/1.63 zero_neq_one:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Oab__group__add,type,
% 2.72/1.63 ab_group_add:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Rings_Ozero__less__one,type,
% 2.72/1.63 zero_less_one:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Osemigroup__add,type,
% 2.72/1.63 semigroup_add:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.72/1.63 linordered_idom:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Rings_Olinordered__ring,type,
% 2.72/1.63 linordered_ring:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 2.72/1.63 comm_monoid_add:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 2.72/1.63 ab_semigroup_add:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 2.72/1.63 cancel_semigroup_add:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.72/1.63 ordered_ab_group_add:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.72/1.63 cancel1352612707id_add:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 2.72/1.63 linord219039673up_add:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 2.72/1.63 ordere216010020id_add:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
% 2.72/1.63 ordere142940540dd_abs:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 2.72/1.63 ordere779506340up_add:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
% 2.72/1.63 real_V2090557954_space:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.72/1.63 linord1659791738miring:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.72/1.63 canoni770627133id_add:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
% 2.72/1.63 ordere1818651114id_add:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 2.72/1.63 ordere236663937imp_le:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 2.72/1.63 ordere223160158up_add:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 2.72/1.63 ordere516151231imp_le:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_cl_Executable__Euclidean__Space_Oexecutable__euclidean__space,type,
% 2.72/1.63 execut510477386_space:
% 2.72/1.63 !>[A: $tType] : $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Elementary__Metric__Spaces_Oball,type,
% 2.72/1.63 elemen1815086676c_ball:
% 2.72/1.63 !>[A: $tType] : ( A > real > ( set @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Flow_Oauto__ll__on__open_Oexistence__ivl0,type,
% 2.72/1.63 auto_l1112008849e_ivl0:
% 2.72/1.63 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > ( set @ real ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Flow_Oauto__ll__on__open_Oflow0,type,
% 2.72/1.63 auto_ll_on_flow0:
% 2.72/1.63 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > real > A ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Groups_Oabs__class_Oabs,type,
% 2.72/1.63 abs_abs:
% 2.72/1.63 !>[A: $tType] : ( A > A ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Groups_Oone__class_Oone,type,
% 2.72/1.63 one_one:
% 2.72/1.63 !>[A: $tType] : A ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.72/1.63 plus_plus:
% 2.72/1.63 !>[A: $tType] : ( A > A > A ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 2.72/1.63 uminus_uminus:
% 2.72/1.63 !>[A: $tType] : ( A > A ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.72/1.63 zero_zero:
% 2.72/1.63 !>[A: $tType] : A ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_If,type,
% 2.72/1.63 if:
% 2.72/1.63 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Initial__Value__Problem_Ointerval,type,
% 2.72/1.63 initia826609931terval: ( set @ real ) > $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Limit__Set_Oauto__ll__on__open_O_092_060alpha_062__limit__point,type,
% 2.72/1.63 limit_1335059408_point:
% 2.72/1.63 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > A > $o ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Limit__Set_Oauto__ll__on__open_O_092_060omega_062__limit__point,type,
% 2.72/1.63 limit_1052139091_point:
% 2.72/1.63 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > A > $o ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Line__Segment_Oopen__segment,type,
% 2.72/1.63 line_open_segment:
% 2.72/1.63 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.72/1.63 bot_bot:
% 2.72/1.63 !>[A: $tType] : A ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.72/1.63 ord_less:
% 2.72/1.63 !>[A: $tType] : ( A > A > $o ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.72/1.63 ord_less_eq:
% 2.72/1.63 !>[A: $tType] : ( A > A > $o ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.72/1.63 top_top:
% 2.72/1.63 !>[A: $tType] : A ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Oclosed__orbit,type,
% 2.72/1.63 period385816147_orbit:
% 2.72/1.63 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > $o ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiod,type,
% 2.72/1.63 period1153813292period:
% 2.72/1.63 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > real ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiodic__orbit,type,
% 2.72/1.63 period862636932_orbit:
% 2.72/1.63 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > $o ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Otransversal__segment,type,
% 2.72/1.63 poinca272511729egment:
% 2.72/1.63 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > A > $o ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Set_OCollect,type,
% 2.72/1.63 collect:
% 2.72/1.63 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_Topological__Spaces_Ocontinuous__on,type,
% 2.72/1.63 topolo2071040574ous_on:
% 2.72/1.63 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > $o ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_c_member,type,
% 2.72/1.63 member:
% 2.72/1.63 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.72/1.63
% 2.72/1.63 thf(sy_v_X,type,
% 2.72/1.63 x: set @ a ).
% 2.72/1.63
% 2.72/1.63 thf(sy_v_a,type,
% 2.72/1.63 a2: a ).
% 2.72/1.63
% 2.72/1.63 thf(sy_v_b,type,
% 2.72/1.63 b: a ).
% 2.72/1.63
% 2.72/1.63 thf(sy_v_f,type,
% 2.72/1.63 f: a > a ).
% 2.72/1.63
% 2.72/1.63 thf(sy_v_p,type,
% 2.72/1.63 p: a ).
% 2.72/1.63
% 2.72/1.63 thf(sy_v_thesisa____,type,
% 2.72/1.63 thesisa: $o ).
% 2.72/1.63
% 2.72/1.63 thf(sy_v_x,type,
% 2.72/1.63 x2: a ).
% 2.72/1.63
% 2.72/1.63 thf(fact_0_rev_Oopen__segment__trichotomy,axiom,
% 2.72/1.63 ! [X: a,A2: a,B2: a,Y: a] :
% 2.72/1.63 ( ( member @ a @ X @ ( line_open_segment @ a @ A2 @ B2 ) )
% 2.72/1.63 => ( ( member @ a @ Y @ ( line_open_segment @ a @ A2 @ B2 ) )
% 2.72/1.63 => ( ( X = Y )
% 2.72/1.63 | ( member @ a @ Y @ ( line_open_segment @ a @ X @ B2 ) )
% 2.72/1.63 | ( member @ a @ Y @ ( line_open_segment @ a @ A2 @ X ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_1__092_060open_062p_A_092_060in_062_AX_092_060close_062,axiom,
% 2.72/1.63 member @ a @ p @ x ).
% 2.72/1.63
% 2.72/1.63 thf(fact_2_assms_I4_J,axiom,
% 2.72/1.63 member @ a @ p @ ( line_open_segment @ a @ a2 @ b ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062d_At_O_A_092_060lbrakk_0620_A_060_Ad_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_Aflow0_Ay_A_It_Ay_J_A_092_060in_062_A_123a_060_N_N_060b_125_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_A_092_060bar_062t_Ay_092_060bar_062_A_060_A1_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_At_Ap_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 2.72/1.63 ~ ! [D: real] :
% 2.72/1.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ D )
% 2.72/1.63 => ! [T: a > real] :
% 2.72/1.63 ( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ p @ D ) @ T )
% 2.72/1.63 => ( ! [Y2: a] :
% 2.72/1.63 ( ( member @ a @ Y2 @ ( elemen1815086676c_ball @ a @ p @ D ) )
% 2.72/1.63 => ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ Y2 @ ( T @ Y2 ) ) @ ( line_open_segment @ a @ a2 @ b ) ) )
% 2.72/1.63 => ( ! [Y2: a] :
% 2.72/1.63 ( ( member @ a @ Y2 @ ( elemen1815086676c_ball @ a @ p @ D ) )
% 2.72/1.63 => ( ord_less @ real @ ( abs_abs @ real @ ( T @ Y2 ) ) @ ( one_one @ real ) ) )
% 2.72/1.63 => ( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ p @ D ) @ T )
% 2.72/1.63 => ( ( T @ p )
% 2.72/1.63 != ( zero_zero @ real ) ) ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_4_that,axiom,
% 2.72/1.63 ! [Delta: real,Tau: a > real] :
% 2.72/1.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ Delta )
% 2.72/1.63 => ( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ p @ Delta ) @ Tau )
% 2.72/1.63 => ( ( ( Tau @ p )
% 2.72/1.63 = ( zero_zero @ real ) )
% 2.72/1.63 => ( ! [Y3: a] :
% 2.72/1.63 ( ( member @ a @ Y3 @ ( elemen1815086676c_ball @ a @ p @ Delta ) )
% 2.72/1.63 => ( ord_less @ real @ ( abs_abs @ real @ ( Tau @ Y3 ) ) @ ( one_one @ real ) ) )
% 2.72/1.63 => ( ! [Y3: a] :
% 2.72/1.63 ( ( member @ a @ Y3 @ ( elemen1815086676c_ball @ a @ p @ Delta ) )
% 2.72/1.63 => ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ Y3 @ ( Tau @ Y3 ) ) @ ( line_open_segment @ a @ a2 @ b ) ) )
% 2.72/1.63 => thesisa ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_5_assms_I1_J,axiom,
% 2.72/1.63 poinca272511729egment @ a @ f @ x @ a2 @ b ).
% 2.72/1.63
% 2.72/1.63 thf(fact_6_assms_I3_J,axiom,
% 2.72/1.63 limit_1052139091_point @ a @ f @ x @ x2 @ p ).
% 2.72/1.63
% 2.72/1.63 thf(fact_7_centre__in__ball,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( real_V2090557954_space @ A )
% 2.72/1.63 => ! [X: A,E: real] :
% 2.72/1.63 ( ( member @ A @ X @ ( elemen1815086676c_ball @ A @ X @ E ) )
% 2.72/1.63 = ( ord_less @ real @ ( zero_zero @ real ) @ E ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_8_zero__less__abs__iff,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere142940540dd_abs @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) )
% 2.72/1.63 = ( A2
% 2.72/1.63 != ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_9_cross__time__continuous,axiom,
% 2.72/1.63 ! [A2: a,B2: a,X: a,E: real] :
% 2.72/1.63 ( ( poinca272511729egment @ a @ f @ x @ A2 @ B2 )
% 2.72/1.63 => ( ( member @ a @ X @ ( line_open_segment @ a @ A2 @ B2 ) )
% 2.72/1.63 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
% 2.72/1.63 => ~ ! [D: real] :
% 2.72/1.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ D )
% 2.72/1.63 => ! [T: a > real] :
% 2.72/1.63 ( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ X @ D ) @ T )
% 2.72/1.63 => ( ! [Y2: a] :
% 2.72/1.63 ( ( member @ a @ Y2 @ ( elemen1815086676c_ball @ a @ X @ D ) )
% 2.72/1.63 => ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ Y2 @ ( T @ Y2 ) ) @ ( line_open_segment @ a @ A2 @ B2 ) ) )
% 2.72/1.63 => ( ! [Y2: a] :
% 2.72/1.63 ( ( member @ a @ Y2 @ ( elemen1815086676c_ball @ a @ X @ D ) )
% 2.72/1.63 => ( ord_less @ real @ ( abs_abs @ real @ ( T @ Y2 ) ) @ E ) )
% 2.72/1.63 => ( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ X @ D ) @ T )
% 2.72/1.63 => ( ( T @ X )
% 2.72/1.63 != ( zero_zero @ real ) ) ) ) ) ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_10_abs__1,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( idom_abs_sgn @ A )
% 2.72/1.63 => ( ( abs_abs @ A @ ( one_one @ A ) )
% 2.72/1.63 = ( one_one @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_11_abs__0,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( idom_abs_sgn @ A )
% 2.72/1.63 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 2.72/1.63 = ( zero_zero @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_12_abs__0__eq,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere142940540dd_abs @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ( zero_zero @ A )
% 2.72/1.63 = ( abs_abs @ A @ A2 ) )
% 2.72/1.63 = ( A2
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_13_abs__eq__0,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere142940540dd_abs @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ( abs_abs @ A @ A2 )
% 2.72/1.63 = ( zero_zero @ A ) )
% 2.72/1.63 = ( A2
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_14_abs__zero,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere142940540dd_abs @ A )
% 2.72/1.63 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 2.72/1.63 = ( zero_zero @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_15_not__gr__zero,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( canoni770627133id_add @ A )
% 2.72/1.63 => ! [N: A] :
% 2.72/1.63 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 2.72/1.63 = ( N
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_16_fixpoint__sol_I2_J,axiom,
% 2.72/1.63 ! [X: a,T2: real] :
% 2.72/1.63 ( ( member @ a @ X @ x )
% 2.72/1.63 => ( ( ( f @ X )
% 2.72/1.63 = ( zero_zero @ a ) )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ T2 )
% 2.72/1.63 = X ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_17_gt__one__absI,axiom,
% 2.72/1.63 ! [K: real] :
% 2.72/1.63 ( ( ord_less @ real @ ( abs_abs @ real @ K ) @ ( one_one @ real ) )
% 2.72/1.63 => ( ord_less @ real @ K @ ( one_one @ real ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_18_transversal__segment__reverse,axiom,
% 2.72/1.63 ! [X: a,Y: a] :
% 2.72/1.63 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 2.72/1.63 => ( poinca272511729egment @ a @ f @ x @ Y @ X ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_19_transversal__segment__commute,axiom,
% 2.72/1.63 ! [X: a,Y: a] :
% 2.72/1.63 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 2.72/1.63 = ( poinca272511729egment @ a @ f @ x @ Y @ X ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_20_abs__idempotent,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere142940540dd_abs @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
% 2.72/1.63 = ( abs_abs @ A @ A2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_21_abs__abs,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( idom_abs_sgn @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
% 2.72/1.63 = ( abs_abs @ A @ A2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_22_transversal__segment__exists,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( member @ a @ X @ x )
% 2.72/1.63 => ( ( ( f @ X )
% 2.72/1.63 != ( zero_zero @ a ) )
% 2.72/1.63 => ~ ! [A3: a,B3: a] :
% 2.72/1.63 ( ( member @ a @ X @ ( line_open_segment @ a @ A3 @ B3 ) )
% 2.72/1.63 => ~ ( poinca272511729egment @ a @ f @ x @ A3 @ B3 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_23_c1__on__open__R2_Otransversal__segment_Ocong,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( ( poinca272511729egment @ A )
% 2.72/1.63 = ( poinca272511729egment @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_24_zero__reorient,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( zero @ A )
% 2.72/1.63 => ! [X: A] :
% 2.72/1.63 ( ( ( zero_zero @ A )
% 2.72/1.63 = X )
% 2.72/1.63 = ( X
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_25_linorder__neqE__linordered__idom,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linordered_idom @ A )
% 2.72/1.63 => ! [X: A,Y: A] :
% 2.72/1.63 ( ( X != Y )
% 2.72/1.63 => ( ~ ( ord_less @ A @ X @ Y )
% 2.72/1.63 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_26_one__reorient,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( one @ A )
% 2.72/1.63 => ! [X: A] :
% 2.72/1.63 ( ( ( one_one @ A )
% 2.72/1.63 = X )
% 2.72/1.63 = ( X
% 2.72/1.63 = ( one_one @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_27_zero__less__iff__neq__zero,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( canoni770627133id_add @ A )
% 2.72/1.63 => ! [N: A] :
% 2.72/1.63 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 2.72/1.63 = ( N
% 2.72/1.63 != ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_28_gr__implies__not__zero,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( canoni770627133id_add @ A )
% 2.72/1.63 => ! [M: A,N: A] :
% 2.72/1.63 ( ( ord_less @ A @ M @ N )
% 2.72/1.63 => ( N
% 2.72/1.63 != ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_29_not__less__zero,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( canoni770627133id_add @ A )
% 2.72/1.63 => ! [N: A] :
% 2.72/1.63 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_30_gr__zeroI,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( canoni770627133id_add @ A )
% 2.72/1.63 => ! [N: A] :
% 2.72/1.63 ( ( N
% 2.72/1.63 != ( zero_zero @ A ) )
% 2.72/1.63 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_31_zero__neq__one,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( zero_neq_one @ A )
% 2.72/1.63 => ( ( zero_zero @ A )
% 2.72/1.63 != ( one_one @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_32_abs__eq__0__iff,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( idom_abs_sgn @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ( abs_abs @ A @ A2 )
% 2.72/1.63 = ( zero_zero @ A ) )
% 2.72/1.63 = ( A2
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_33_abs__one,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linordered_idom @ A )
% 2.72/1.63 => ( ( abs_abs @ A @ ( one_one @ A ) )
% 2.72/1.63 = ( one_one @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_34_not__one__less__zero,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linord1659791738miring @ A )
% 2.72/1.63 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_35_zero__less__one,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( zero_less_one @ A )
% 2.72/1.63 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_36_abs__not__less__zero,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere142940540dd_abs @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ~ ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_37_abs__of__pos,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere142940540dd_abs @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.63 => ( ( abs_abs @ A @ A2 )
% 2.72/1.63 = A2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_38_periodic__orbit__imp__flow0__regular,axiom,
% 2.72/1.63 ! [X: a,T2: real] :
% 2.72/1.63 ( ( period862636932_orbit @ a @ f @ x @ X )
% 2.72/1.63 => ( ( f @ ( auto_ll_on_flow0 @ a @ f @ x @ X @ T2 ) )
% 2.72/1.63 != ( zero_zero @ a ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_39_fixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( member @ a @ X @ x )
% 2.72/1.63 => ( ( ( f @ X )
% 2.72/1.63 = ( zero_zero @ a ) )
% 2.72/1.63 => ( ( period1153813292period @ a @ f @ x @ X )
% 2.72/1.63 = ( zero_zero @ real ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_40_local_Oflow__initial__time__if,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ( ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 2.72/1.63 & ( member @ a @ X0 @ x ) )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( zero_zero @ real ) )
% 2.72/1.63 = X0 ) )
% 2.72/1.63 & ( ~ ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 2.72/1.63 & ( member @ a @ X0 @ x ) )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( zero_zero @ real ) )
% 2.72/1.63 = ( zero_zero @ a ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_41_rev_Otransversal__segment__exists,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( member @ a @ X @ x )
% 2.72/1.63 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 2.72/1.63 != ( zero_zero @ a ) )
% 2.72/1.63 => ~ ! [A3: a,B3: a] :
% 2.72/1.63 ( ( member @ a @ X @ ( line_open_segment @ a @ A3 @ B3 ) )
% 2.72/1.63 => ~ ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ A3 @ B3 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_42_rev_Ofixpoint__sol_I2_J,axiom,
% 2.72/1.63 ! [X: a,T2: real] :
% 2.72/1.63 ( ( member @ a @ X @ x )
% 2.72/1.63 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 2.72/1.63 = ( zero_zero @ a ) )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T2 )
% 2.72/1.63 = X ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_43_mem__Collect__eq,axiom,
% 2.72/1.63 ! [A: $tType,A2: A,P: A > $o] :
% 2.72/1.63 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.72/1.63 = ( P @ A2 ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_44_Collect__mem__eq,axiom,
% 2.72/1.63 ! [A: $tType,A4: set @ A] :
% 2.72/1.63 ( ( collect @ A
% 2.72/1.63 @ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
% 2.72/1.63 = A4 ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_45_Collect__cong,axiom,
% 2.72/1.63 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.72/1.63 ( ! [X3: A] :
% 2.72/1.63 ( ( P @ X3 )
% 2.72/1.63 = ( Q @ X3 ) )
% 2.72/1.63 => ( ( collect @ A @ P )
% 2.72/1.63 = ( collect @ A @ Q ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_46_ext,axiom,
% 2.72/1.63 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.72/1.63 ( ! [X3: A] :
% 2.72/1.63 ( ( F @ X3 )
% 2.72/1.63 = ( G @ X3 ) )
% 2.72/1.63 => ( F = G ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_47_local_Oflow__undefined0,axiom,
% 2.72/1.63 ! [T2: real,X: a] :
% 2.72/1.63 ( ~ ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ T2 )
% 2.72/1.63 = ( zero_zero @ a ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_48_rev__transversal__segment,axiom,
% 2.72/1.63 ! [A2: a,B2: a] :
% 2.72/1.63 ( ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ A2 @ B2 )
% 2.72/1.63 = ( poinca272511729egment @ a @ f @ x @ A2 @ B2 ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_49_rev_Otransversal__segment__reverse,axiom,
% 2.72/1.63 ! [X: a,Y: a] :
% 2.72/1.63 ( ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ Y )
% 2.72/1.63 => ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ X ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_50_rev_Otransversal__segment__commute,axiom,
% 2.72/1.63 ! [X: a,Y: a] :
% 2.72/1.63 ( ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ Y )
% 2.72/1.63 = ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ X ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_51_fixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( member @ a @ X @ x )
% 2.72/1.63 => ( ( ( f @ X )
% 2.72/1.63 = ( zero_zero @ a ) )
% 2.72/1.63 => ( period385816147_orbit @ a @ f @ x @ X ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_52_existence__ivl0__cong,axiom,
% 2.72/1.63 ! [B: $tType,Y4: set @ a,G: a > a,X0: a] :
% 2.72/1.63 ( ( x = Y4 )
% 2.72/1.63 => ( ! [X3: a,T: B] :
% 2.72/1.63 ( ( member @ a @ X3 @ Y4 )
% 2.72/1.63 => ( ( f @ X3 )
% 2.72/1.63 = ( G @ X3 ) ) )
% 2.72/1.63 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
% 2.72/1.63 = ( auto_l1112008849e_ivl0 @ a @ G @ Y4 @ X0 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_53_closed__orbit__in__domain,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period385816147_orbit @ a @ f @ x @ X )
% 2.72/1.63 => ( member @ a @ X @ x ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_54_add_Oinverse__inverse,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( group_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 2.72/1.63 = A2 ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_55_neg__equal__iff__equal,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( group_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ( uminus_uminus @ A @ A2 )
% 2.72/1.63 = ( uminus_uminus @ A @ B2 ) )
% 2.72/1.63 = ( A2 = B2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_56_existence__ivl__zero,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ( member @ a @ X0 @ x )
% 2.72/1.63 => ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_57_rev_Oexistence__ivl0__cong,axiom,
% 2.72/1.63 ! [B: $tType,Y4: set @ a,G: a > a,X0: a] :
% 2.72/1.63 ( ( x = Y4 )
% 2.72/1.63 => ( ! [X3: a,T: B] :
% 2.72/1.63 ( ( member @ a @ X3 @ Y4 )
% 2.72/1.63 => ( ( uminus_uminus @ ( a > a ) @ f @ X3 )
% 2.72/1.63 = ( G @ X3 ) ) )
% 2.72/1.63 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
% 2.72/1.63 = ( auto_l1112008849e_ivl0 @ a @ G @ Y4 @ X0 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_58_local_Omem__existence__ivl__subset,axiom,
% 2.72/1.63 ! [T2: real,X0: a] :
% 2.72/1.63 ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 2.72/1.63 => ( member @ real @ T2 @ ( top_top @ ( set @ real ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_59_flow0__cong,axiom,
% 2.72/1.63 ! [B: $tType,Y4: set @ a,G: a > a,T2: real,X0: a] :
% 2.72/1.63 ( ( x = Y4 )
% 2.72/1.63 => ( ! [X3: a,T: B] :
% 2.72/1.63 ( ( member @ a @ X3 @ Y4 )
% 2.72/1.63 => ( ( f @ X3 )
% 2.72/1.63 = ( G @ X3 ) ) )
% 2.72/1.63 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ T2 )
% 2.72/1.63 = ( auto_ll_on_flow0 @ a @ G @ Y4 @ X0 @ T2 ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_60_rev_Oclosed__orbit__eq__rev,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 = ( period385816147_orbit @ a @ f @ x @ X ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_61_rev_Oclosed__orbit__in__domain,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 => ( member @ a @ X @ x ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_62_closed__orbit__flow0,axiom,
% 2.72/1.63 ! [X: a,T2: real] :
% 2.72/1.63 ( ( period385816147_orbit @ a @ f @ x @ X )
% 2.72/1.63 => ( period385816147_orbit @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X @ T2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_63_rev_Oexistence__ivl__zero,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ( member @ a @ X0 @ x )
% 2.72/1.63 => ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_64_local_Oexistence__ivl__initial__time,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 2.72/1.63 => ( ( member @ a @ X0 @ x )
% 2.72/1.63 => ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_65_local_Orev_Omem__existence__ivl__subset,axiom,
% 2.72/1.63 ! [T2: real,X0: a] :
% 2.72/1.63 ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 2.72/1.63 => ( member @ real @ T2 @ ( top_top @ ( set @ real ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_66_rev_Oflow0__cong,axiom,
% 2.72/1.63 ! [B: $tType,Y4: set @ a,G: a > a,T2: real,X0: a] :
% 2.72/1.63 ( ( x = Y4 )
% 2.72/1.63 => ( ! [X3: a,T: B] :
% 2.72/1.63 ( ( member @ a @ X3 @ Y4 )
% 2.72/1.63 => ( ( uminus_uminus @ ( a > a ) @ f @ X3 )
% 2.72/1.63 = ( G @ X3 ) ) )
% 2.72/1.63 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ T2 )
% 2.72/1.63 = ( auto_ll_on_flow0 @ a @ G @ Y4 @ X0 @ T2 ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_67_fixpoint__sol_I1_J,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( member @ a @ X @ x )
% 2.72/1.63 => ( ( ( f @ X )
% 2.72/1.63 = ( zero_zero @ a ) )
% 2.72/1.63 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X )
% 2.72/1.63 = ( top_top @ ( set @ real ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_68_rev_Oclosed__orbit__flow0,axiom,
% 2.72/1.63 ! [X: a,T2: real] :
% 2.72/1.63 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 => ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_69_closed__orbit__global__existence,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period385816147_orbit @ a @ f @ x @ X )
% 2.72/1.63 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X )
% 2.72/1.63 = ( top_top @ ( set @ real ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_70_closed__orbitI,axiom,
% 2.72/1.63 ! [T2: real,T3: real,Y: a] :
% 2.72/1.63 ( ( T2 != T3 )
% 2.72/1.63 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ Y ) )
% 2.72/1.63 => ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ Y ) )
% 2.72/1.63 => ( ( ( auto_ll_on_flow0 @ a @ f @ x @ Y @ T2 )
% 2.72/1.63 = ( auto_ll_on_flow0 @ a @ f @ x @ Y @ T3 ) )
% 2.72/1.63 => ( period385816147_orbit @ a @ f @ x @ Y ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_71_rev_Ofixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( member @ a @ X @ x )
% 2.72/1.63 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 2.72/1.63 = ( zero_zero @ a ) )
% 2.72/1.63 => ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_72_periodic__orbit__period_I2_J,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period862636932_orbit @ a @ f @ x @ X )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ ( period1153813292period @ a @ f @ x @ X ) )
% 2.72/1.63 = X ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_73_closed__orbit__periodic,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period385816147_orbit @ a @ f @ x @ X )
% 2.72/1.63 => ( ( ( f @ X )
% 2.72/1.63 != ( zero_zero @ a ) )
% 2.72/1.63 => ( period862636932_orbit @ a @ f @ x @ X ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_74_local_Orev_Oexistence__ivl__initial__time,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 2.72/1.63 => ( ( member @ a @ X0 @ x )
% 2.72/1.63 => ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_75_rev_Ofixpoint__sol_I1_J,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( member @ a @ X @ x )
% 2.72/1.63 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 2.72/1.63 = ( zero_zero @ a ) )
% 2.72/1.63 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 = ( top_top @ ( set @ real ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_76_local_Orev_Oflow__undefined0,axiom,
% 2.72/1.63 ! [T2: real,X: a] :
% 2.72/1.63 ( ~ ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T2 )
% 2.72/1.63 = ( zero_zero @ a ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_77_closed__orbit__def,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period385816147_orbit @ a @ f @ x @ X )
% 2.72/1.63 = ( ? [X2: real] :
% 2.72/1.63 ( ( member @ real @ X2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
% 2.72/1.63 & ( X2
% 2.72/1.63 != ( zero_zero @ real ) )
% 2.72/1.63 & ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ X2 )
% 2.72/1.63 = X ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_78_rev_Oclosed__orbit__global__existence,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 = ( top_top @ ( set @ real ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_79_rev_Oclosed__orbitI,axiom,
% 2.72/1.63 ! [T2: real,T3: real,Y: a] :
% 2.72/1.63 ( ( T2 != T3 )
% 2.72/1.63 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) )
% 2.72/1.63 => ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) )
% 2.72/1.63 => ( ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ T2 )
% 2.72/1.63 = ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ T3 ) )
% 2.72/1.63 => ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_80_rev_Ofixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( member @ a @ X @ x )
% 2.72/1.63 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 2.72/1.63 = ( zero_zero @ a ) )
% 2.72/1.63 => ( ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 = ( zero_zero @ real ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_81_rev_Operiodic__orbit__imp__flow0__regular,axiom,
% 2.72/1.63 ! [X: a,T2: real] :
% 2.72/1.63 ( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 => ( ( uminus_uminus @ ( a > a ) @ f @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T2 ) )
% 2.72/1.63 != ( zero_zero @ a ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_82_periodic__orbitI,axiom,
% 2.72/1.63 ! [T2: real,T3: real,Y: a] :
% 2.72/1.63 ( ( T2 != T3 )
% 2.72/1.63 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ Y ) )
% 2.72/1.63 => ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ Y ) )
% 2.72/1.63 => ( ( ( auto_ll_on_flow0 @ a @ f @ x @ Y @ T2 )
% 2.72/1.63 = ( auto_ll_on_flow0 @ a @ f @ x @ Y @ T3 ) )
% 2.72/1.63 => ( ( ( f @ Y )
% 2.72/1.63 != ( zero_zero @ a ) )
% 2.72/1.63 => ( period862636932_orbit @ a @ f @ x @ Y ) ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_83_periodic__orbit__period_I1_J,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period862636932_orbit @ a @ f @ x @ X )
% 2.72/1.63 => ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_84_closed__orbit__period__zero__fixed__point,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period385816147_orbit @ a @ f @ x @ X )
% 2.72/1.63 => ( ( ( period1153813292period @ a @ f @ x @ X )
% 2.72/1.63 = ( zero_zero @ real ) )
% 2.72/1.63 => ( ( f @ X )
% 2.72/1.63 = ( zero_zero @ a ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_85_rev_Operiodic__orbit__period_I2_J,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
% 2.72/1.63 = X ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_86_rev_Oclosed__orbit__periodic,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 2.72/1.63 != ( zero_zero @ a ) )
% 2.72/1.63 => ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_87_add_Oinverse__neutral,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( group_add @ A )
% 2.72/1.63 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 2.72/1.63 = ( zero_zero @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_88_neg__0__equal__iff__equal,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( group_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ( zero_zero @ A )
% 2.72/1.63 = ( uminus_uminus @ A @ A2 ) )
% 2.72/1.63 = ( ( zero_zero @ A )
% 2.72/1.63 = A2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_89_neg__equal__0__iff__equal,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( group_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ( uminus_uminus @ A @ A2 )
% 2.72/1.63 = ( zero_zero @ A ) )
% 2.72/1.63 = ( A2
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_90_equal__neg__zero,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linord219039673up_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( A2
% 2.72/1.63 = ( uminus_uminus @ A @ A2 ) )
% 2.72/1.63 = ( A2
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_91_neg__equal__zero,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linord219039673up_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ( uminus_uminus @ A @ A2 )
% 2.72/1.63 = A2 )
% 2.72/1.63 = ( A2
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_92_local_Orev_Oflow__initial__time__if,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ( ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 2.72/1.63 & ( member @ a @ X0 @ x ) )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( zero_zero @ real ) )
% 2.72/1.63 = X0 ) )
% 2.72/1.63 & ( ~ ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 2.72/1.63 & ( member @ a @ X0 @ x ) )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( zero_zero @ real ) )
% 2.72/1.63 = ( zero_zero @ a ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_93_neg__less__iff__less,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordered_ab_group_add @ A )
% 2.72/1.63 => ! [B2: A,A2: A] :
% 2.72/1.63 ( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 2.72/1.63 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_94_rev_Oclosed__orbit__def,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 = ( ? [X2: real] :
% 2.72/1.63 ( ( member @ real @ X2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
% 2.72/1.63 & ( X2
% 2.72/1.63 != ( zero_zero @ real ) )
% 2.72/1.63 & ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ X2 )
% 2.72/1.63 = X ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_95_abs__minus,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( idom_abs_sgn @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
% 2.72/1.63 = ( abs_abs @ A @ A2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_96_abs__minus__cancel,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere142940540dd_abs @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
% 2.72/1.63 = ( abs_abs @ A @ A2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_97_rev_Operiodic__orbitI,axiom,
% 2.72/1.63 ! [T2: real,T3: real,Y: a] :
% 2.72/1.63 ( ( T2 != T3 )
% 2.72/1.63 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) )
% 2.72/1.63 => ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) )
% 2.72/1.63 => ( ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ T2 )
% 2.72/1.63 = ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ T3 ) )
% 2.72/1.63 => ( ( ( uminus_uminus @ ( a > a ) @ f @ Y )
% 2.72/1.63 != ( zero_zero @ a ) )
% 2.72/1.63 => ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_98_rev_Operiodic__orbit__period_I1_J,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 => ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_99_rev_Oclosed__orbit__period__zero__fixed__point,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 => ( ( ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 = ( zero_zero @ real ) )
% 2.72/1.63 => ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 2.72/1.63 = ( zero_zero @ a ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_100_periodic__orbit__def,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period862636932_orbit @ a @ f @ x @ X )
% 2.72/1.63 = ( ( period385816147_orbit @ a @ f @ x @ X )
% 2.72/1.63 & ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_101_rev_Operiodic__orbit__def,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 = ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 & ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_102_less__neg__neg,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linord219039673up_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 2.72/1.63 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_103_neg__less__pos,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linord219039673up_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 2.72/1.63 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_104_neg__0__less__iff__less,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordered_ab_group_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 2.72/1.63 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_105_neg__less__0__iff__less,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordered_ab_group_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.72/1.63 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_106_local_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 2.72/1.63 ! [T2: real,X0: a] :
% 2.72/1.63 ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 2.72/1.63 => ( member @ a @ X0 @ x ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_107_local_Orev_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 2.72/1.63 ! [T2: real,X0: a] :
% 2.72/1.63 ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 2.72/1.63 => ( member @ a @ X0 @ x ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_108_flow0__defined,axiom,
% 2.72/1.63 ! [Xa: real,X0: a] :
% 2.72/1.63 ( ( member @ real @ Xa @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 2.72/1.63 => ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ Xa ) @ x ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_109_local_Oflow__in__domain,axiom,
% 2.72/1.63 ! [T2: real,X0: a] :
% 2.72/1.63 ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 2.72/1.63 => ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ T2 ) @ x ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_110_local_Omem__existence__ivl__iv__defined_I1_J,axiom,
% 2.72/1.63 ! [T2: real,X0: a] :
% 2.72/1.63 ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 2.72/1.63 => ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_111_local_Oexistence__ivl__initial__time__iff,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 2.72/1.63 = ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 2.72/1.63 & ( member @ a @ X0 @ x ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_112_local_Oflow__initial__time,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 2.72/1.63 => ( ( member @ a @ X0 @ x )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( zero_zero @ real ) )
% 2.72/1.63 = X0 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_113_rev_Oflow0__defined,axiom,
% 2.72/1.63 ! [Xa: real,X0: a] :
% 2.72/1.63 ( ( member @ real @ Xa @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 2.72/1.63 => ( member @ a @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ Xa ) @ x ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_114_local_Orev_Oflow__in__domain,axiom,
% 2.72/1.63 ! [T2: real,X0: a] :
% 2.72/1.63 ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 2.72/1.63 => ( member @ a @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ T2 ) @ x ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_115_local_Orev_Omem__existence__ivl__iv__defined_I1_J,axiom,
% 2.72/1.63 ! [T2: real,X0: a] :
% 2.72/1.63 ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 2.72/1.63 => ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_116_local_Orev_Oexistence__ivl__initial__time__iff,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 2.72/1.63 = ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 2.72/1.63 & ( member @ a @ X0 @ x ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_117_local_Orev_Oflow__initial__time,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 2.72/1.63 => ( ( member @ a @ X0 @ x )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( zero_zero @ real ) )
% 2.72/1.63 = X0 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_118_equation__minus__iff,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( group_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( A2
% 2.72/1.63 = ( uminus_uminus @ A @ B2 ) )
% 2.72/1.63 = ( B2
% 2.72/1.63 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_119_minus__equation__iff,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( group_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ( uminus_uminus @ A @ A2 )
% 2.72/1.63 = B2 )
% 2.72/1.63 = ( ( uminus_uminus @ A @ B2 )
% 2.72/1.63 = A2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_120_less__minus__iff,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordered_ab_group_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 2.72/1.63 = ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_121_minus__less__iff,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordered_ab_group_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 2.72/1.63 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_122_abs__eq__iff,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linordered_ring @ A )
% 2.72/1.63 => ! [X: A,Y: A] :
% 2.72/1.63 ( ( ( abs_abs @ A @ X )
% 2.72/1.63 = ( abs_abs @ A @ Y ) )
% 2.72/1.63 = ( ( X = Y )
% 2.72/1.63 | ( X
% 2.72/1.63 = ( uminus_uminus @ A @ Y ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_123_abs__less__iff,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linordered_idom @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ B2 )
% 2.72/1.63 = ( ( ord_less @ A @ A2 @ B2 )
% 2.72/1.63 & ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_124_abs__if,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( abs_if @ A )
% 2.72/1.63 => ( ( abs_abs @ A )
% 2.72/1.63 = ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_125_abs__of__neg,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere142940540dd_abs @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.72/1.63 => ( ( abs_abs @ A @ A2 )
% 2.72/1.63 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_126__092_060alpha_062__limit__point__eq__rev,axiom,
% 2.72/1.63 ! [X: a,P2: a] :
% 2.72/1.63 ( ( limit_1335059408_point @ a @ f @ x @ X @ P2 )
% 2.72/1.63 = ( limit_1052139091_point @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ P2 ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_127_rev_O_092_060alpha_062__limit__point__eq__rev,axiom,
% 2.72/1.63 ! [X: a,P2: a] :
% 2.72/1.63 ( ( limit_1335059408_point @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ P2 )
% 2.72/1.63 = ( limit_1052139091_point @ a @ f @ x @ X @ P2 ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_128_rev_Omvar_Ointerval__axioms,axiom,
% 2.72/1.63 ! [X0: a] : ( initia826609931terval @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_129_mvar_Ointerval__axioms,axiom,
% 2.72/1.63 ! [X0: a] : ( initia826609931terval @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_130_abs__neg__one,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linordered_idom @ A )
% 2.72/1.63 => ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 2.72/1.63 = ( one_one @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_131_rev_Oclosed__orbitE,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 => ~ ! [T4: real] :
% 2.72/1.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
% 2.72/1.63 => ~ ! [T5: real] :
% 2.72/1.63 ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ ( plus_plus @ real @ T5 @ T4 ) )
% 2.72/1.63 = ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T5 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_132_local_Orev_Oexistence__ivl__notempty,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 2.72/1.63 => ( ( member @ a @ X0 @ x )
% 2.72/1.63 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
% 2.72/1.63 != ( bot_bot @ ( set @ real ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_133_rev_Oclosed__orbit__period__nonneg,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 2.72/1.63 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_134_rev_Ovec__simps_I11_J,axiom,
% 2.72/1.63 ! [Ag: $tType] :
% 2.72/1.63 ( ( semigroup_add @ Ag )
% 2.72/1.63 => ! [Ae: Ag,Be: Ag,Cd: Ag] :
% 2.72/1.63 ( ( plus_plus @ Ag @ ( plus_plus @ Ag @ Ae @ Be ) @ Cd )
% 2.72/1.63 = ( plus_plus @ Ag @ Ae @ ( plus_plus @ Ag @ Be @ Cd ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_135_rev_Ovec__simps_I12_J,axiom,
% 2.72/1.63 ! [Ai: $tType] :
% 2.72/1.63 ( ( ab_semigroup_add @ Ai )
% 2.72/1.63 => ( ( plus_plus @ Ai )
% 2.72/1.63 = ( ^ [Ag2: Ai,Bg: Ai] : ( plus_plus @ Ai @ Bg @ Ag2 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_136_rev_Ovec__simps_I13_J,axiom,
% 2.72/1.63 ! [Ai: $tType] :
% 2.72/1.63 ( ( ab_semigroup_add @ Ai )
% 2.72/1.63 => ! [Bg2: Ai,Ag3: Ai,Cf: Ai] :
% 2.72/1.63 ( ( plus_plus @ Ai @ Bg2 @ ( plus_plus @ Ai @ Ag3 @ Cf ) )
% 2.72/1.63 = ( plus_plus @ Ai @ Ag3 @ ( plus_plus @ Ai @ Bg2 @ Cf ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_137_interval__axioms,axiom,
% 2.72/1.63 initia826609931terval @ ( top_top @ ( set @ real ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_138_add__right__cancel,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( cancel_semigroup_add @ A )
% 2.72/1.63 => ! [B2: A,A2: A,C: A] :
% 2.72/1.63 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.72/1.63 = ( plus_plus @ A @ C @ A2 ) )
% 2.72/1.63 = ( B2 = C ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_139_add__left__cancel,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( cancel_semigroup_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A,C: A] :
% 2.72/1.63 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.72/1.63 = ( plus_plus @ A @ A2 @ C ) )
% 2.72/1.63 = ( B2 = C ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_140_rev__eq__flow,axiom,
% 2.72/1.63 ! [Y: a,T2: real] :
% 2.72/1.63 ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ T2 )
% 2.72/1.63 = ( auto_ll_on_flow0 @ a @ f @ x @ Y @ ( uminus_uminus @ real @ T2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_141_rev_Orev__eq__flow,axiom,
% 2.72/1.63 ! [Y: a,T2: real] :
% 2.72/1.63 ( ( auto_ll_on_flow0 @ a @ f @ x @ Y @ T2 )
% 2.72/1.63 = ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ ( uminus_uminus @ real @ T2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_142_recurrence__time__flip__sign_I1_J,axiom,
% 2.72/1.63 ! [T6: real,X: a] :
% 2.72/1.63 ( ( member @ real @ T6 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
% 2.72/1.63 => ( ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ T6 )
% 2.72/1.63 = X )
% 2.72/1.63 => ( member @ real @ ( uminus_uminus @ real @ T6 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_143_recurrence__time__flip__sign_I2_J,axiom,
% 2.72/1.63 ! [T6: real,X: a] :
% 2.72/1.63 ( ( member @ real @ T6 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
% 2.72/1.63 => ( ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ T6 )
% 2.72/1.63 = X )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ ( uminus_uminus @ real @ T6 ) )
% 2.72/1.63 = X ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_144_rev_Orecurrence__time__flip__sign_I1_J,axiom,
% 2.72/1.63 ! [T6: real,X: a] :
% 2.72/1.63 ( ( member @ real @ T6 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
% 2.72/1.63 => ( ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T6 )
% 2.72/1.63 = X )
% 2.72/1.63 => ( member @ real @ ( uminus_uminus @ real @ T6 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_145_rev_Orecurrence__time__flip__sign_I2_J,axiom,
% 2.72/1.63 ! [T6: real,X: a] :
% 2.72/1.63 ( ( member @ real @ T6 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
% 2.72/1.63 => ( ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T6 )
% 2.72/1.63 = X )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ ( uminus_uminus @ real @ T6 ) )
% 2.72/1.63 = X ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_146_local_Oflow__trans,axiom,
% 2.72/1.63 ! [S: real,X0: a,T2: real] :
% 2.72/1.63 ( ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 2.72/1.63 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ S ) ) )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( plus_plus @ real @ S @ T2 ) )
% 2.72/1.63 = ( auto_ll_on_flow0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ S ) @ T2 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_147_local_Oexistence__ivl__trans_H,axiom,
% 2.72/1.63 ! [T2: real,S: real,X0: a] :
% 2.72/1.63 ( ( member @ real @ ( plus_plus @ real @ T2 @ S ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 2.72/1.63 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 2.72/1.63 => ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ T2 ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_148_local_Oexistence__ivl__trans,axiom,
% 2.72/1.63 ! [S: real,X0: a,T2: real] :
% 2.72/1.63 ( ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 2.72/1.63 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ S ) ) )
% 2.72/1.63 => ( member @ real @ ( plus_plus @ real @ S @ T2 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_149_local_Orev_Oflow__trans,axiom,
% 2.72/1.63 ! [S: real,X0: a,T2: real] :
% 2.72/1.63 ( ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 2.72/1.63 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ S ) ) )
% 2.72/1.63 => ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( plus_plus @ real @ S @ T2 ) )
% 2.72/1.63 = ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ S ) @ T2 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_150_local_Orev_Oexistence__ivl__trans_H,axiom,
% 2.72/1.63 ! [T2: real,S: real,X0: a] :
% 2.72/1.63 ( ( member @ real @ ( plus_plus @ real @ T2 @ S ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 2.72/1.63 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 2.72/1.63 => ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ T2 ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_151_local_Orev_Oexistence__ivl__trans,axiom,
% 2.72/1.63 ! [S: real,X0: a,T2: real] :
% 2.72/1.63 ( ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 2.72/1.63 => ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ S ) ) )
% 2.72/1.63 => ( member @ real @ ( plus_plus @ real @ S @ T2 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_152_local_Oexistence__ivl__notempty,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 2.72/1.63 => ( ( member @ a @ X0 @ x )
% 2.72/1.63 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
% 2.72/1.63 != ( bot_bot @ ( set @ real ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_153_closed__orbit__period__nonneg,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period385816147_orbit @ a @ f @ x @ X )
% 2.72/1.63 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_154_le__zero__eq,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( canoni770627133id_add @ A )
% 2.72/1.63 => ! [N: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.72/1.63 = ( N
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_155_neg__le__iff__le,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordered_ab_group_add @ A )
% 2.72/1.63 => ! [B2: A,A2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 2.72/1.63 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_156_zero__eq__add__iff__both__eq__0,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( canoni770627133id_add @ A )
% 2.72/1.63 => ! [X: A,Y: A] :
% 2.72/1.63 ( ( ( zero_zero @ A )
% 2.72/1.63 = ( plus_plus @ A @ X @ Y ) )
% 2.72/1.63 = ( ( X
% 2.72/1.63 = ( zero_zero @ A ) )
% 2.72/1.63 & ( Y
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_157_add__eq__0__iff__both__eq__0,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( canoni770627133id_add @ A )
% 2.72/1.63 => ! [X: A,Y: A] :
% 2.72/1.63 ( ( ( plus_plus @ A @ X @ Y )
% 2.72/1.63 = ( zero_zero @ A ) )
% 2.72/1.63 = ( ( X
% 2.72/1.63 = ( zero_zero @ A ) )
% 2.72/1.63 & ( Y
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_158_add__cancel__right__right,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( cancel1352612707id_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( A2
% 2.72/1.63 = ( plus_plus @ A @ A2 @ B2 ) )
% 2.72/1.63 = ( B2
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_159_add__cancel__right__left,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( cancel1352612707id_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( A2
% 2.72/1.63 = ( plus_plus @ A @ B2 @ A2 ) )
% 2.72/1.63 = ( B2
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_160_add__cancel__left__right,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( cancel1352612707id_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.72/1.63 = A2 )
% 2.72/1.63 = ( B2
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_161_add__cancel__left__left,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( cancel1352612707id_add @ A )
% 2.72/1.63 => ! [B2: A,A2: A] :
% 2.72/1.63 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.72/1.63 = A2 )
% 2.72/1.63 = ( B2
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_162_double__zero__sym,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linord219039673up_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ( zero_zero @ A )
% 2.72/1.63 = ( plus_plus @ A @ A2 @ A2 ) )
% 2.72/1.63 = ( A2
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_163_linordered__ab__group__add__class_Odouble__zero,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linord219039673up_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ( plus_plus @ A @ A2 @ A2 )
% 2.72/1.63 = ( zero_zero @ A ) )
% 2.72/1.63 = ( A2
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_164_add_Oright__neutral,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( monoid_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 2.72/1.63 = A2 ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_165_add_Oleft__neutral,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( monoid_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.63 = A2 ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_166_add__le__cancel__right,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere236663937imp_le @ A )
% 2.72/1.63 => ! [A2: A,C: A,B2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 2.72/1.63 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_167_add__le__cancel__left,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere236663937imp_le @ A )
% 2.72/1.63 => ! [C: A,A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 2.72/1.63 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_168_add__less__cancel__right,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere236663937imp_le @ A )
% 2.72/1.63 => ! [A2: A,C: A,B2: A] :
% 2.72/1.63 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 2.72/1.63 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_169_add__less__cancel__left,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere236663937imp_le @ A )
% 2.72/1.63 => ! [C: A,A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 2.72/1.63 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_170_closed__orbitE,axiom,
% 2.72/1.63 ! [X: a] :
% 2.72/1.63 ( ( period385816147_orbit @ a @ f @ x @ X )
% 2.72/1.63 => ~ ! [T4: real] :
% 2.72/1.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
% 2.72/1.63 => ~ ! [T5: real] :
% 2.72/1.63 ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ ( plus_plus @ real @ T5 @ T4 ) )
% 2.72/1.63 = ( auto_ll_on_flow0 @ a @ f @ x @ X @ T5 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_171_minus__add__distrib,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ab_group_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.72/1.63 = ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_172_minus__add__cancel,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( group_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.72/1.63 = B2 ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_173_add__minus__cancel,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( group_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
% 2.72/1.63 = B2 ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_174_abs__add__abs,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere142940540dd_abs @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) )
% 2.72/1.63 = ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_175_neg__0__le__iff__le,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordered_ab_group_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 2.72/1.63 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_176_neg__le__0__iff__le,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordered_ab_group_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.72/1.63 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_177_less__eq__neg__nonpos,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linord219039673up_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 2.72/1.63 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_178_neg__less__eq__nonneg,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linord219039673up_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 2.72/1.63 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_179_linordered__ab__group__add__class_Ozero__le__double__add__iff__zero__le__single__add,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linord219039673up_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 2.72/1.63 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_180_linordered__ab__group__add__class_Odouble__add__le__zero__iff__single__add__le__zero,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linord219039673up_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 2.72/1.63 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_181_le__add__same__cancel2,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere516151231imp_le @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
% 2.72/1.63 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_182_le__add__same__cancel1,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere516151231imp_le @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.72/1.63 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_183_add__le__same__cancel2,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere516151231imp_le @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 2.72/1.63 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_184_add__le__same__cancel1,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere516151231imp_le @ A )
% 2.72/1.63 => ! [B2: A,A2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
% 2.72/1.63 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_185_linordered__ab__group__add__class_Ozero__less__double__add__iff__zero__less__single__add,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linord219039673up_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 2.72/1.63 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_186_double__add__less__zero__iff__single__add__less__zero,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linord219039673up_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 2.72/1.63 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_187_less__add__same__cancel2,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere516151231imp_le @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
% 2.72/1.63 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_188_less__add__same__cancel1,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere516151231imp_le @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.72/1.63 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_189_add__less__same__cancel2,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere516151231imp_le @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 2.72/1.63 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_190_add__less__same__cancel1,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere516151231imp_le @ A )
% 2.72/1.63 => ! [B2: A,A2: A] :
% 2.72/1.63 ( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
% 2.72/1.63 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_191_add_Oleft__inverse,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( group_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 2.72/1.63 = ( zero_zero @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_192_add_Oright__inverse,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( group_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( plus_plus @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 2.72/1.63 = ( zero_zero @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_193_abs__le__zero__iff,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere142940540dd_abs @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.72/1.63 = ( A2
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_194_abs__le__self__iff,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere142940540dd_abs @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 )
% 2.72/1.63 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_195_abs__of__nonneg,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere142940540dd_abs @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.63 => ( ( abs_abs @ A @ A2 )
% 2.72/1.63 = A2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_196_ball__trivial,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( real_V2090557954_space @ A )
% 2.72/1.63 => ! [X: A] :
% 2.72/1.63 ( ( elemen1815086676c_ball @ A @ X @ ( zero_zero @ real ) )
% 2.72/1.63 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_197_add__neg__numeral__special_I7_J,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( neg_numeral @ A )
% 2.72/1.63 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 2.72/1.63 = ( zero_zero @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_198_add__neg__numeral__special_I8_J,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( neg_numeral @ A )
% 2.72/1.63 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
% 2.72/1.63 = ( zero_zero @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_199_abs__of__nonpos,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere142940540dd_abs @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.72/1.63 => ( ( abs_abs @ A @ A2 )
% 2.72/1.63 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_200_ball__eq__empty,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( real_V2090557954_space @ A )
% 2.72/1.63 => ! [X: A,E: real] :
% 2.72/1.63 ( ( ( elemen1815086676c_ball @ A @ X @ E )
% 2.72/1.63 = ( bot_bot @ ( set @ A ) ) )
% 2.72/1.63 = ( ord_less_eq @ real @ E @ ( zero_zero @ real ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_201_local_Oexistence__ivl__undefined,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ~ ( member @ a @ X0 @ x )
% 2.72/1.63 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
% 2.72/1.63 = ( bot_bot @ ( set @ real ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_202_local_Orev_Oexistence__ivl__undefined,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ~ ( member @ a @ X0 @ x )
% 2.72/1.63 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
% 2.72/1.63 = ( bot_bot @ ( set @ real ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_203_local_Oexistence__ivl__empty__iff,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
% 2.72/1.63 = ( bot_bot @ ( set @ real ) ) )
% 2.72/1.63 = ( ~ ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 2.72/1.63 | ~ ( member @ a @ X0 @ x ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_204_local_Oexistence__ivl__empty1,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ~ ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 2.72/1.63 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
% 2.72/1.63 = ( bot_bot @ ( set @ real ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_205_local_Orev_Oexistence__ivl__empty__iff,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
% 2.72/1.63 = ( bot_bot @ ( set @ real ) ) )
% 2.72/1.63 = ( ~ ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 2.72/1.63 | ~ ( member @ a @ X0 @ x ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_206_local_Orev_Oexistence__ivl__empty1,axiom,
% 2.72/1.63 ! [X0: a] :
% 2.72/1.63 ( ~ ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 2.72/1.63 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
% 2.72/1.63 = ( bot_bot @ ( set @ real ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_207_is__num__normalize_I8_J,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( neg_numeral @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.72/1.63 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_208_abs__triangle__ineq,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere142940540dd_abs @ A )
% 2.72/1.63 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_209_le__numeral__extra_I3_J,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linord1659791738miring @ A )
% 2.72/1.63 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_210_le__numeral__extra_I4_J,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( linord1659791738miring @ A )
% 2.72/1.63 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_211_is__num__normalize_I1_J,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( neg_numeral @ A )
% 2.72/1.63 => ! [A2: A,B2: A,C: A] :
% 2.72/1.63 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 2.72/1.63 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_212_add__le__imp__le__right,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere236663937imp_le @ A )
% 2.72/1.63 => ! [A2: A,C: A,B2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 2.72/1.63 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_213_add__le__imp__le__left,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere236663937imp_le @ A )
% 2.72/1.63 => ! [C: A,A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 2.72/1.63 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_214_add__less__le__mono,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere223160158up_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A,C: A,D2: A] :
% 2.72/1.63 ( ( ord_less @ A @ A2 @ B2 )
% 2.72/1.63 => ( ( ord_less_eq @ A @ C @ D2 )
% 2.72/1.63 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_215_add__le__less__mono,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere223160158up_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A,C: A,D2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.72/1.63 => ( ( ord_less @ A @ C @ D2 )
% 2.72/1.63 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_216_add__nonpos__eq__0__iff,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere216010020id_add @ A )
% 2.72/1.63 => ! [X: A,Y: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 2.72/1.63 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 2.72/1.63 => ( ( ( plus_plus @ A @ X @ Y )
% 2.72/1.63 = ( zero_zero @ A ) )
% 2.72/1.63 = ( ( X
% 2.72/1.63 = ( zero_zero @ A ) )
% 2.72/1.63 & ( Y
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_217_add__nonneg__eq__0__iff,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere216010020id_add @ A )
% 2.72/1.63 => ! [X: A,Y: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.72/1.63 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 2.72/1.63 => ( ( ( plus_plus @ A @ X @ Y )
% 2.72/1.63 = ( zero_zero @ A ) )
% 2.72/1.63 = ( ( X
% 2.72/1.63 = ( zero_zero @ A ) )
% 2.72/1.63 & ( Y
% 2.72/1.63 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_218_add__nonpos__nonpos,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere216010020id_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.72/1.63 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 2.72/1.63 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_219_add__nonneg__nonneg,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere216010020id_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.63 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 2.72/1.63 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_220_le__iff__add,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( canoni770627133id_add @ A )
% 2.72/1.63 => ( ( ord_less_eq @ A )
% 2.72/1.63 = ( ^ [A5: A,B4: A] :
% 2.72/1.63 ? [C2: A] :
% 2.72/1.63 ( B4
% 2.72/1.63 = ( plus_plus @ A @ A5 @ C2 ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_221_add__increasing2,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere216010020id_add @ A )
% 2.72/1.63 => ! [C: A,B2: A,A2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 2.72/1.63 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.72/1.63 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_222_add__decreasing2,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere216010020id_add @ A )
% 2.72/1.63 => ! [C: A,A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
% 2.72/1.63 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.72/1.63 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_223_add__right__mono,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere779506340up_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A,C: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.72/1.63 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_224_less__eqE,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( canoni770627133id_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.72/1.63 => ~ ! [C3: A] :
% 2.72/1.63 ( B2
% 2.72/1.63 != ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_225_add__increasing,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere216010020id_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A,C: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.63 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.72/1.63 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_226_add__decreasing,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere216010020id_add @ A )
% 2.72/1.63 => ! [A2: A,C: A,B2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.72/1.63 => ( ( ord_less_eq @ A @ C @ B2 )
% 2.72/1.63 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_227_add__left__mono,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere779506340up_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A,C: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.72/1.63 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_228_add__right__imp__eq,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( cancel_semigroup_add @ A )
% 2.72/1.63 => ! [B2: A,A2: A,C: A] :
% 2.72/1.63 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.72/1.63 = ( plus_plus @ A @ C @ A2 ) )
% 2.72/1.63 => ( B2 = C ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_229_add__left__imp__eq,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( cancel_semigroup_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A,C: A] :
% 2.72/1.63 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.72/1.63 = ( plus_plus @ A @ A2 @ C ) )
% 2.72/1.63 => ( B2 = C ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_230_add__mono,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere779506340up_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A,C: A,D2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.72/1.63 => ( ( ord_less_eq @ A @ C @ D2 )
% 2.72/1.63 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_231_add_Oleft__commute,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ab_semigroup_add @ A )
% 2.72/1.63 => ! [B2: A,A2: A,C: A] :
% 2.72/1.63 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
% 2.72/1.63 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_232_add_Ocommute,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ab_semigroup_add @ A )
% 2.72/1.63 => ( ( plus_plus @ A )
% 2.72/1.63 = ( ^ [A5: A,B4: A] : ( plus_plus @ A @ B4 @ A5 ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_233_add_Oright__cancel,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( group_add @ A )
% 2.72/1.63 => ! [B2: A,A2: A,C: A] :
% 2.72/1.63 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.72/1.63 = ( plus_plus @ A @ C @ A2 ) )
% 2.72/1.63 = ( B2 = C ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_234_add_Oleft__cancel,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( group_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A,C: A] :
% 2.72/1.63 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.72/1.63 = ( plus_plus @ A @ A2 @ C ) )
% 2.72/1.63 = ( B2 = C ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_235_add_Oassoc,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( semigroup_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A,C: A] :
% 2.72/1.63 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 2.72/1.63 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_236_group__cancel_Oadd2,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( comm_monoid_add @ A )
% 2.72/1.63 => ! [B5: A,K: A,B2: A,A2: A] :
% 2.72/1.63 ( ( B5
% 2.72/1.63 = ( plus_plus @ A @ K @ B2 ) )
% 2.72/1.63 => ( ( plus_plus @ A @ A2 @ B5 )
% 2.72/1.63 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_237_group__cancel_Oadd1,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( comm_monoid_add @ A )
% 2.72/1.63 => ! [A4: A,K: A,A2: A,B2: A] :
% 2.72/1.63 ( ( A4
% 2.72/1.63 = ( plus_plus @ A @ K @ A2 ) )
% 2.72/1.63 => ( ( plus_plus @ A @ A4 @ B2 )
% 2.72/1.63 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_238_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere779506340up_add @ A )
% 2.72/1.63 => ! [I: A,J: A,K: A,L: A] :
% 2.72/1.63 ( ( ( ord_less_eq @ A @ I @ J )
% 2.72/1.63 & ( ord_less_eq @ A @ K @ L ) )
% 2.72/1.63 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_239_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere779506340up_add @ A )
% 2.72/1.63 => ! [I: A,J: A,K: A,L: A] :
% 2.72/1.63 ( ( ( I = J )
% 2.72/1.63 & ( ord_less_eq @ A @ K @ L ) )
% 2.72/1.63 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_240_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere779506340up_add @ A )
% 2.72/1.63 => ! [I: A,J: A,K: A,L: A] :
% 2.72/1.63 ( ( ( ord_less_eq @ A @ I @ J )
% 2.72/1.63 & ( K = L ) )
% 2.72/1.63 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_241_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere779506340up_add @ A )
% 2.72/1.63 => ! [I: A,J: A,K: A,L: A] :
% 2.72/1.63 ( ( ( I = J )
% 2.72/1.63 & ( K = L ) )
% 2.72/1.63 => ( ( plus_plus @ A @ I @ K )
% 2.72/1.63 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_242_add__mono__thms__linordered__field_I3_J,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere223160158up_add @ A )
% 2.72/1.63 => ! [I: A,J: A,K: A,L: A] :
% 2.72/1.63 ( ( ( ord_less @ A @ I @ J )
% 2.72/1.63 & ( ord_less_eq @ A @ K @ L ) )
% 2.72/1.63 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_243_add__mono__thms__linordered__field_I4_J,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere223160158up_add @ A )
% 2.72/1.63 => ! [I: A,J: A,K: A,L: A] :
% 2.72/1.63 ( ( ( ord_less_eq @ A @ I @ J )
% 2.72/1.63 & ( ord_less @ A @ K @ L ) )
% 2.72/1.63 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_244_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ab_semigroup_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A,C: A] :
% 2.72/1.63 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 2.72/1.63 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_245_add__strict__increasing2,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere1818651114id_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A,C: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.63 => ( ( ord_less @ A @ B2 @ C )
% 2.72/1.63 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_246_add__strict__increasing,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere1818651114id_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A,C: A] :
% 2.72/1.63 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.63 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.72/1.63 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_247_add__pos__nonneg,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere216010020id_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.63 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 2.72/1.63 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_248_add__nonpos__neg,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere216010020id_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.72/1.63 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.72/1.63 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_249_add__nonneg__pos,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere216010020id_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.63 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 2.72/1.63 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_250_add__neg__nonpos,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( ordere216010020id_add @ A )
% 2.72/1.63 => ! [A2: A,B2: A] :
% 2.72/1.63 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.72/1.63 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 2.72/1.63 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_251_Elementary__Metric__Spaces_Oball__empty,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( real_V2090557954_space @ A )
% 2.72/1.63 => ! [E: real,X: A] :
% 2.72/1.63 ( ( ord_less_eq @ real @ E @ ( zero_zero @ real ) )
% 2.72/1.63 => ( ( elemen1815086676c_ball @ A @ X @ E )
% 2.72/1.63 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_252_add_Ogroup__left__neutral,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( group_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.63 = A2 ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(fact_253_add_Ocomm__neutral,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( comm_monoid_add @ A )
% 2.72/1.63 => ! [A2: A] :
% 2.72/1.63 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 2.72/1.63 = A2 ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ozero,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( zero @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ogroup__add,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( group_add @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Omonoid__add,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( monoid_add @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oab__group__add,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( ab_group_add @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Osemigroup__add,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( semigroup_add @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocomm__monoid__add,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( comm_monoid_add @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oab__semigroup__add,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( ab_semigroup_add @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocancel__semigroup__add,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( cancel_semigroup_add @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__group__add,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( ordered_ab_group_add @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocancel__comm__monoid__add,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( cancel1352612707id_add @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__comm__monoid__add,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( ordere216010020id_add @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__group__add__abs,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( ordere142940540dd_abs @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__semigroup__add,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( ordere779506340up_add @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Real__Vector__Spaces_Ometric__space,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( real_V2090557954_space @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__cancel__comm__monoid__add,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( ordere1818651114id_add @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( ordere236663937imp_le @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( ordere223160158up_add @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 2.72/1.63 ! [A: $tType] :
% 2.72/1.63 ( ( execut510477386_space @ A )
% 2.72/1.63 => ( ordere516151231imp_le @ A ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_HOL_Obool___Countable_Ocountable,axiom,
% 2.72/1.63 countable @ $o ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Set_Oset___Countable_Ocountable_1,axiom,
% 2.72/1.63 ! [A6: $tType] :
% 2.72/1.63 ( ( finite_finite @ A6 )
% 2.72/1.63 => ( countable @ ( set @ A6 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_fun___Countable_Ocountable_2,axiom,
% 2.72/1.63 ! [A6: $tType,A7: $tType] :
% 2.72/1.63 ( ( ( finite_finite @ A6 )
% 2.72/1.63 & ( countable @ A7 ) )
% 2.72/1.63 => ( countable @ ( A6 > A7 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 2.72/1.63 ! [A6: $tType,A7: $tType] :
% 2.72/1.63 ( ( ( finite_finite @ A6 )
% 2.72/1.63 & ( finite_finite @ A7 ) )
% 2.72/1.63 => ( finite_finite @ ( A6 > A7 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Set_Oset___Finite__Set_Ofinite_3,axiom,
% 2.72/1.63 ! [A6: $tType] :
% 2.72/1.63 ( ( finite_finite @ A6 )
% 2.72/1.63 => ( finite_finite @ ( set @ A6 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_HOL_Obool___Finite__Set_Ofinite_4,axiom,
% 2.72/1.63 finite_finite @ $o ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_fun___Real__Vector__Spaces_Ometric__space,axiom,
% 2.72/1.63 ! [A6: $tType,A7: $tType] :
% 2.72/1.63 ( ( ( countable @ A6 )
% 2.72/1.63 & ( real_V2090557954_space @ A7 ) )
% 2.72/1.63 => ( real_V2090557954_space @ ( A6 > A7 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Set_Oset___Groups_Oab__semigroup__add,axiom,
% 2.72/1.63 ! [A6: $tType] :
% 2.72/1.63 ( ( ab_semigroup_add @ A6 )
% 2.72/1.63 => ( ab_semigroup_add @ ( set @ A6 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Set_Oset___Groups_Ocomm__monoid__add,axiom,
% 2.72/1.63 ! [A6: $tType] :
% 2.72/1.63 ( ( comm_monoid_add @ A6 )
% 2.72/1.63 => ( comm_monoid_add @ ( set @ A6 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Set_Oset___Groups_Osemigroup__add,axiom,
% 2.72/1.63 ! [A6: $tType] :
% 2.72/1.63 ( ( semigroup_add @ A6 )
% 2.72/1.63 => ( semigroup_add @ ( set @ A6 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Set_Oset___Groups_Omonoid__add,axiom,
% 2.72/1.63 ! [A6: $tType] :
% 2.72/1.63 ( ( monoid_add @ A6 )
% 2.72/1.63 => ( monoid_add @ ( set @ A6 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Set_Oset___Groups_Ozero,axiom,
% 2.72/1.63 ! [A6: $tType] :
% 2.72/1.63 ( ( zero @ A6 )
% 2.72/1.63 => ( zero @ ( set @ A6 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Set_Oset___Groups_Oone,axiom,
% 2.72/1.63 ! [A6: $tType] :
% 2.72/1.63 ( ( one @ A6 )
% 2.72/1.63 => ( one @ ( set @ A6 ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Executable__Euclidean__Space_Oexecutable__euclidean__space,axiom,
% 2.72/1.63 execut510477386_space @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 2.72/1.63 ordere516151231imp_le @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 2.72/1.63 ordere223160158up_add @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 2.72/1.63 ordere236663937imp_le @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add,axiom,
% 2.72/1.63 ordere1818651114id_add @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring,axiom,
% 2.72/1.63 linord1659791738miring @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space_5,axiom,
% 2.72/1.63 real_V2090557954_space @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add,axiom,
% 2.72/1.63 ordere779506340up_add @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs,axiom,
% 2.72/1.63 ordere142940540dd_abs @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add,axiom,
% 2.72/1.63 ordere216010020id_add @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
% 2.72/1.63 linord219039673up_add @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add,axiom,
% 2.72/1.63 cancel1352612707id_add @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
% 2.72/1.63 ordered_ab_group_add @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add,axiom,
% 2.72/1.63 cancel_semigroup_add @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_6,axiom,
% 2.72/1.63 ab_semigroup_add @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_7,axiom,
% 2.72/1.63 comm_monoid_add @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Rings_Olinordered__ring,axiom,
% 2.72/1.63 linordered_ring @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Rings_Olinordered__idom,axiom,
% 2.72/1.63 linordered_idom @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Osemigroup__add_8,axiom,
% 2.72/1.63 semigroup_add @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Rings_Ozero__less__one,axiom,
% 2.72/1.63 zero_less_one @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Oab__group__add,axiom,
% 2.72/1.63 ab_group_add @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Rings_Ozero__neq__one,axiom,
% 2.72/1.63 zero_neq_one @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn,axiom,
% 2.72/1.63 idom_abs_sgn @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Omonoid__add_9,axiom,
% 2.72/1.63 monoid_add @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 2.72/1.63 group_add @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Num_Oneg__numeral,axiom,
% 2.72/1.63 neg_numeral @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Rings_Oabs__if,axiom,
% 2.72/1.63 abs_if @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Ozero_10,axiom,
% 2.72/1.63 zero @ real ).
% 2.72/1.63
% 2.72/1.63 thf(tcon_Real_Oreal___Groups_Oone_11,axiom,
% 2.72/1.63 one @ real ).
% 2.72/1.63
% 2.72/1.63 thf(help_If_3_1_T,axiom,
% 2.72/1.63 ! [P: $o] :
% 2.72/1.63 ( ( P = $true )
% 2.72/1.63 | ( P = $false ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(help_If_2_1_T,axiom,
% 2.72/1.63 ! [A: $tType,X: A,Y: A] :
% 2.72/1.63 ( ( if @ A @ $false @ X @ Y )
% 2.72/1.63 = Y ) ).
% 2.72/1.63
% 2.72/1.63 thf(help_If_1_1_T,axiom,
% 2.72/1.63 ! [A: $tType,X: A,Y: A] :
% 2.72/1.63 ( ( if @ A @ $true @ X @ Y )
% 2.72/1.63 = X ) ).
% 2.72/1.63
% 2.72/1.63 thf(tfree_0,hypothesis,
% 2.72/1.63 execut510477386_space @ a ).
% 2.72/1.63
% 2.72/1.63 thf(conj_0,hypothesis,
% 2.72/1.63 ! [Delta2: real,Tau2: a > real] :
% 2.72/1.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ Delta2 )
% 2.72/1.63 => ( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ p @ Delta2 ) @ Tau2 )
% 2.72/1.63 => ( ( ( Tau2 @ p )
% 2.72/1.63 = ( zero_zero @ real ) )
% 2.72/1.63 => ( ! [Y3: a] :
% 2.72/1.63 ( ( member @ a @ Y3 @ ( elemen1815086676c_ball @ a @ p @ Delta2 ) )
% 2.72/1.63 => ( ord_less @ real @ ( abs_abs @ real @ ( Tau2 @ Y3 ) ) @ ( one_one @ real ) ) )
% 2.72/1.63 => ( ! [Y3: a] :
% 2.72/1.63 ( ( member @ a @ Y3 @ ( elemen1815086676c_ball @ a @ p @ Delta2 ) )
% 2.72/1.63 => ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ Y3 @ ( Tau2 @ Y3 ) ) @ ( line_open_segment @ a @ a2 @ b ) ) )
% 2.72/1.63 => thesisa ) ) ) ) ) ).
% 2.72/1.63
% 2.72/1.63 thf(conj_1,conjecture,
% 2.72/1.63 thesisa ).
% 2.72/1.63 ------------------------
% 2.72/1.66 % (24371)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/183Mi)
% 2.72/1.66 % (24374)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/2Mi)
% 2.72/1.66 % (24372)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/4Mi)
% 2.72/1.66 % (24376)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/275Mi)
% 2.72/1.66 % (24375)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/2Mi)
% 2.72/1.66 % (24377)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/18Mi)
% 2.72/1.66 % (24373)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/27Mi)
% 2.72/1.66 % (24374)Instruction limit reached!
% 2.72/1.66 % (24374)------------------------------
% 2.72/1.66 % (24374)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.66 % (24374)Termination reason: Unknown
% 2.72/1.66 % (24374)Termination phase: shuffling
% 2.72/1.66 % (24375)Instruction limit reached!
% 2.72/1.66 % (24375)------------------------------
% 2.72/1.66 % (24375)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.66
% 2.72/1.66 % (24374)Memory used [KB]: 1279
% 2.72/1.66 % (24374)Time elapsed: 0.003 s
% 2.72/1.66 % (24374)Instructions burned: 3 (million)
% 2.72/1.66 % (24374)------------------------------
% 2.72/1.66 % (24374)------------------------------
% 2.72/1.66 % (24375)Termination reason: Unknown
% 2.72/1.66 % (24375)Termination phase: shuffling
% 2.72/1.66
% 2.72/1.66 % (24375)Memory used [KB]: 1279
% 2.72/1.66 % (24375)Time elapsed: 0.003 s
% 2.72/1.66 % (24375)Instructions burned: 3 (million)
% 2.72/1.66 % (24375)------------------------------
% 2.72/1.66 % (24375)------------------------------
% 2.72/1.66 % (24372)Instruction limit reached!
% 2.72/1.66 % (24372)------------------------------
% 2.72/1.66 % (24372)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.66 % (24372)Termination reason: Unknown
% 2.72/1.66 % (24372)Termination phase: shuffling
% 2.72/1.66
% 2.72/1.66 % (24372)Memory used [KB]: 1407
% 2.72/1.66 % (24372)Time elapsed: 0.004 s
% 2.72/1.66 % (24372)Instructions burned: 5 (million)
% 2.72/1.66 % (24372)------------------------------
% 2.72/1.66 % (24372)------------------------------
% 2.72/1.67 % (24377)Instruction limit reached!
% 2.72/1.67 % (24377)------------------------------
% 2.72/1.67 % (24377)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.67 % (24377)Termination reason: Unknown
% 2.72/1.67 % (24377)Termination phase: shuffling
% 2.72/1.67
% 2.72/1.67 % (24377)Memory used [KB]: 1535
% 2.72/1.67 % (24377)Time elapsed: 0.011 s
% 2.72/1.67 % (24377)Instructions burned: 19 (million)
% 2.72/1.67 % (24377)------------------------------
% 2.72/1.67 % (24377)------------------------------
% 2.72/1.68 % (24373)Instruction limit reached!
% 2.72/1.68 % (24373)------------------------------
% 2.72/1.68 % (24373)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.68 % (24373)Termination reason: Unknown
% 2.72/1.68 % (24373)Termination phase: Property scanning
% 2.72/1.68
% 2.72/1.68 % (24373)Memory used [KB]: 1663
% 2.72/1.68 % (24373)Time elapsed: 0.016 s
% 2.72/1.68 % (24373)Instructions burned: 28 (million)
% 2.72/1.68 % (24373)------------------------------
% 2.72/1.68 % (24373)------------------------------
% 2.72/1.68 % (24378)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/3Mi)
% 2.72/1.68 % (24379)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/37Mi)
% 2.72/1.68 % (24380)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/15Mi)
% 2.72/1.68 % (24378)Instruction limit reached!
% 2.72/1.68 % (24378)------------------------------
% 2.72/1.68 % (24378)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.68 % (24378)Termination reason: Unknown
% 2.72/1.68 % (24378)Termination phase: shuffling
% 2.72/1.68
% 2.72/1.68 % (24378)Memory used [KB]: 1279
% 2.72/1.68 % (24378)Time elapsed: 0.003 s
% 2.72/1.68 % (24378)Instructions burned: 3 (million)
% 2.72/1.68 % (24378)------------------------------
% 2.72/1.68 % (24378)------------------------------
% 2.72/1.69 % (24380)Instruction limit reached!
% 2.72/1.69 % (24380)------------------------------
% 2.72/1.69 % (24380)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.69 % (24380)Termination reason: Unknown
% 2.72/1.69 % (24380)Termination phase: shuffling
% 2.72/1.69
% 2.72/1.69 % (24380)Memory used [KB]: 1535
% 2.72/1.69 % (24380)Time elapsed: 0.009 s
% 2.72/1.69 % (24380)Instructions burned: 15 (million)
% 2.72/1.69 % (24380)------------------------------
% 2.72/1.69 % (24380)------------------------------
% 2.72/1.69 % (24381)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/3Mi)
% 2.72/1.69 % (24382)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/1041Mi)
% 2.72/1.69 % (24381)Instruction limit reached!
% 2.72/1.69 % (24381)------------------------------
% 2.72/1.69 % (24381)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.69 % (24381)Termination reason: Unknown
% 2.72/1.69 % (24381)Termination phase: shuffling
% 2.72/1.69
% 2.72/1.69 % (24381)Memory used [KB]: 1279
% 2.72/1.69 % (24381)Time elapsed: 0.003 s
% 2.72/1.69 % (24381)Instructions burned: 3 (million)
% 2.72/1.69 % (24381)------------------------------
% 2.72/1.69 % (24381)------------------------------
% 2.77/1.70 % (24383)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/7Mi)
% 2.77/1.70 % (24379)Instruction limit reached!
% 2.77/1.70 % (24379)------------------------------
% 2.77/1.70 % (24379)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.70 % (24379)Termination reason: Unknown
% 2.77/1.70 % (24379)Termination phase: Property scanning
% 2.77/1.70
% 2.77/1.70 % (24379)Memory used [KB]: 2046
% 2.77/1.70 % (24379)Time elapsed: 0.019 s
% 2.77/1.70 % (24379)Instructions burned: 39 (million)
% 2.77/1.70 % (24379)------------------------------
% 2.77/1.70 % (24379)------------------------------
% 2.77/1.70 % (24383)Instruction limit reached!
% 2.77/1.70 % (24383)------------------------------
% 2.77/1.70 % (24383)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.70 % (24383)Termination reason: Unknown
% 2.77/1.70 % (24383)Termination phase: shuffling
% 2.77/1.70
% 2.77/1.70 % (24383)Memory used [KB]: 1407
% 2.77/1.70 % (24383)Time elapsed: 0.005 s
% 2.77/1.70 % (24383)Instructions burned: 8 (million)
% 2.77/1.70 % (24383)------------------------------
% 2.77/1.70 % (24383)------------------------------
% 2.77/1.70 % (24384)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/16Mi)
% 2.77/1.71 % (24385)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/3Mi)
% 2.77/1.71 % (24385)Instruction limit reached!
% 2.77/1.71 % (24385)------------------------------
% 2.77/1.71 % (24385)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.71 % (24385)Termination reason: Unknown
% 2.77/1.71 % (24385)Termination phase: shuffling
% 2.77/1.71
% 2.77/1.71 % (24385)Memory used [KB]: 1279
% 2.77/1.71 % (24385)Time elapsed: 0.003 s
% 2.77/1.71 % (24385)Instructions burned: 3 (million)
% 2.77/1.71 % (24385)------------------------------
% 2.77/1.71 % (24385)------------------------------
% 2.77/1.71 % (24384)Instruction limit reached!
% 2.77/1.71 % (24384)------------------------------
% 2.77/1.71 % (24384)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.71 % (24384)Termination reason: Unknown
% 2.77/1.71 % (24384)Termination phase: shuffling
% 2.77/1.71
% 2.77/1.71 % (24384)Memory used [KB]: 1535
% 2.77/1.71 % (24384)Time elapsed: 0.010 s
% 2.77/1.71 % (24384)Instructions burned: 17 (million)
% 2.77/1.71 % (24384)------------------------------
% 2.77/1.71 % (24384)------------------------------
% 2.77/1.71 % (24386)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/3Mi)
% 2.77/1.71 % (24386)Instruction limit reached!
% 2.77/1.71 % (24386)------------------------------
% 2.77/1.71 % (24386)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.71 % (24386)Termination reason: Unknown
% 2.77/1.71 % (24386)Termination phase: shuffling
% 2.77/1.71
% 2.77/1.71 % (24386)Memory used [KB]: 1279
% 2.77/1.72 % (24386)Time elapsed: 0.003 s
% 2.77/1.72 % (24386)Instructions burned: 3 (million)
% 2.77/1.72 % (24386)------------------------------
% 2.77/1.72 % (24386)------------------------------
% 2.77/1.72 % (24387)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/7Mi)
% 2.77/1.72 % (24371)Instruction limit reached!
% 2.77/1.72 % (24371)------------------------------
% 2.77/1.72 % (24371)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.72 % (24371)Termination reason: Unknown
% 2.77/1.72 % (24371)Termination phase: Saturation
% 2.77/1.72
% 2.77/1.72 % (24371)Memory used [KB]: 6780
% 2.77/1.72 % (24371)Time elapsed: 0.065 s
% 2.77/1.72 % (24371)Instructions burned: 183 (million)
% 2.77/1.72 % (24371)------------------------------
% 2.77/1.72 % (24371)------------------------------
% 2.77/1.72 % (24387)Instruction limit reached!
% 2.77/1.72 % (24387)------------------------------
% 2.77/1.72 % (24387)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.72 % (24390)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/4Mi)
% 2.77/1.72 % (24387)Termination reason: Unknown
% 2.77/1.72 % (24387)Termination phase: shuffling
% 2.77/1.72
% 2.77/1.72 % (24387)Memory used [KB]: 1407
% 2.77/1.72 % (24387)Time elapsed: 0.005 s
% 2.77/1.72 % (24387)Instructions burned: 7 (million)
% 2.77/1.72 % (24387)------------------------------
% 2.77/1.72 % (24387)------------------------------
% 2.77/1.72 % (24390)Instruction limit reached!
% 2.77/1.72 % (24390)------------------------------
% 2.77/1.72 % (24390)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.72 % (24390)Termination reason: Unknown
% 2.77/1.72 % (24390)Termination phase: shuffling
% 2.77/1.72
% 2.77/1.72 % (24390)Memory used [KB]: 1279
% 2.77/1.72 % (24390)Time elapsed: 0.002 s
% 2.77/1.72 % (24390)Instructions burned: 4 (million)
% 2.77/1.72 % (24390)------------------------------
% 2.77/1.72 % (24390)------------------------------
% 2.77/1.73 % (24388)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/3Mi)
% 2.77/1.73 % (24388)Instruction limit reached!
% 2.77/1.73 % (24388)------------------------------
% 2.77/1.73 % (24388)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.73 % (24388)Termination reason: Unknown
% 2.77/1.73 % (24388)Termination phase: shuffling
% 2.77/1.73
% 2.77/1.73 % (24388)Memory used [KB]: 1407
% 2.77/1.73 % (24388)Time elapsed: 0.003 s
% 2.77/1.73 % (24388)Instructions burned: 6 (million)
% 2.77/1.73 % (24388)------------------------------
% 2.77/1.73 % (24388)------------------------------
% 2.77/1.73 % (24391)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/18Mi)
% 2.77/1.74 % (24392)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/710Mi)
% 2.77/1.74 % (24396)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/21Mi)
% 2.77/1.74 % (24393)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/6Mi)
% 2.77/1.74 % (24394)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_24163 for (2999ds/902Mi)
% 2.77/1.74 % (24391)Instruction limit reached!
% 2.77/1.74 % (24391)------------------------------
% 2.77/1.74 % (24391)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.74 % (24391)Termination reason: Unknown
% 2.77/1.74 % (24391)Termination phase: shuffling
% 2.77/1.74
% 2.77/1.74 % (24391)Memory used [KB]: 1535
% 2.77/1.74 % (24391)Time elapsed: 0.011 s
% 2.77/1.74 % (24393)Instruction limit reached!
% 2.77/1.74 % (24393)------------------------------
% 2.77/1.74 % (24393)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.74 % (24393)Termination reason: Unknown
% 2.77/1.74 % (24393)Termination phase: shuffling
% 2.77/1.74
% 2.77/1.74 % (24393)Memory used [KB]: 1407
% 2.77/1.74 % (24393)Time elapsed: 0.005 s
% 2.77/1.74 % (24393)Instructions burned: 7 (million)
% 2.77/1.74 % (24393)------------------------------
% 2.77/1.74 % (24393)------------------------------
% 2.77/1.74 % (24391)Instructions burned: 19 (million)
% 2.77/1.74 % (24391)------------------------------
% 2.77/1.74 % (24391)------------------------------
% 2.77/1.74 % (24396)Instruction limit reached!
% 2.77/1.74 % (24396)------------------------------
% 2.77/1.74 % (24396)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.74 % (24396)Termination reason: Unknown
% 2.77/1.74 % (24396)Termination phase: shuffling
% 2.77/1.74
% 2.77/1.74 % (24396)Memory used [KB]: 1535
% 2.77/1.74 % (24396)Time elapsed: 0.008 s
% 2.77/1.74 % (24396)Instructions burned: 21 (million)
% 2.77/1.74 % (24396)------------------------------
% 2.77/1.74 % (24396)------------------------------
% 2.77/1.75 % (24376)First to succeed.
% 2.77/1.75 % (24376)Refutation found. Thanks to Tanya!
% 2.77/1.75 % SZS status Theorem for DTF2THF_24163
% 2.77/1.75 % SZS output start Proof for DTF2THF_24163
% 2.77/1.75 thf(type_def_5, type, real: $tType).
% 2.77/1.75 thf(type_def_6, type, set: ($tType) > $tType).
% 2.77/1.75 thf(type_def_8, type, a: $tType).
% 2.77/1.75 thf(func_def_0, type, real: $tType).
% 2.77/1.75 thf(func_def_1, type, set: ($tType) > $tType).
% 2.77/1.75 thf(func_def_2, type, a: $tType).
% 2.77/1.75 thf(func_def_3, type, finite_finite: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_4, type, countable: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_5, type, one: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_6, type, zero: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_7, type, abs_if: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_8, type, neg_numeral: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_9, type, group_add: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_10, type, monoid_add: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_11, type, idom_abs_sgn: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_12, type, zero_neq_one: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_13, type, ab_group_add: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_14, type, zero_less_one: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_15, type, semigroup_add: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_16, type, linordered_idom: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_17, type, linordered_ring: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_18, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_19, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_20, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_21, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_22, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_23, type, linord219039673up_add: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_24, type, ordere216010020id_add: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_25, type, ordere142940540dd_abs: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_26, type, ordere779506340up_add: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_27, type, real_V2090557954_space: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_28, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_29, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_30, type, ordere1818651114id_add: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_31, type, ordere236663937imp_le: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_32, type, ordere223160158up_add: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_33, type, ordere516151231imp_le: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_34, type, execut510477386_space: !>[X0: $tType]:($o)).
% 2.77/1.75 thf(func_def_35, type, elemen1815086676c_ball: !>[X0: $tType]:(X0 > real > set @ X0)).
% 2.77/1.75 thf(func_def_36, type, auto_l1112008849e_ivl0: !>[X0: $tType]:((X0 > X0) > set @ X0 > X0 > set @ real)).
% 2.77/1.75 thf(func_def_37, type, auto_ll_on_flow0: !>[X0: $tType]:((X0 > X0) > set @ X0 > X0 > real > X0)).
% 2.77/1.75 thf(func_def_38, type, abs_abs: !>[X0: $tType]:(X0 > X0)).
% 2.77/1.75 thf(func_def_39, type, one_one: !>[X0: $tType]:(X0)).
% 2.77/1.75 thf(func_def_40, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.77/1.75 thf(func_def_41, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 2.77/1.75 thf(func_def_42, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.77/1.75 thf(func_def_43, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.77/1.75 thf(func_def_44, type, initia826609931terval: set @ real > $o).
% 2.77/1.75 thf(func_def_45, type, limit_1335059408_point: !>[X0: $tType]:((X0 > X0) > set @ X0 > X0 > X0 > $o)).
% 2.77/1.75 thf(func_def_46, type, limit_1052139091_point: !>[X0: $tType]:((X0 > X0) > set @ X0 > X0 > X0 > $o)).
% 2.77/1.75 thf(func_def_47, type, line_open_segment: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 2.77/1.75 thf(func_def_48, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.77/1.75 thf(func_def_49, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.77/1.75 thf(func_def_50, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.77/1.75 thf(func_def_51, type, top_top: !>[X0: $tType]:(X0)).
% 2.77/1.75 thf(func_def_52, type, period385816147_orbit: !>[X0: $tType]:((X0 > X0) > set @ X0 > X0 > $o)).
% 2.77/1.75 thf(func_def_53, type, period1153813292period: !>[X0: $tType]:((X0 > X0) > set @ X0 > X0 > real)).
% 2.77/1.75 thf(func_def_54, type, period862636932_orbit: !>[X0: $tType]:((X0 > X0) > set @ X0 > X0 > $o)).
% 2.77/1.75 thf(func_def_55, type, poinca272511729egment: !>[X0: $tType]:((X0 > X0) > set @ X0 > X0 > X0 > $o)).
% 2.77/1.75 thf(func_def_56, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.77/1.75 thf(func_def_57, type, topolo2071040574ous_on: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > $o)).
% 2.77/1.75 thf(func_def_58, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.77/1.75 thf(func_def_59, type, x: set @ a).
% 2.77/1.75 thf(func_def_60, type, a2: a).
% 2.77/1.75 thf(func_def_61, type, b: a).
% 2.77/1.75 thf(func_def_62, type, f: a > a).
% 2.77/1.75 thf(func_def_63, type, p: a).
% 2.77/1.75 thf(func_def_65, type, x2: a).
% 2.77/1.75 thf(func_def_75, type, sK0: (a > a) > set @ a > a).
% 2.77/1.75 thf(func_def_76, type, sK1: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.77/1.75 thf(func_def_77, type, sK2: (a > real) > real > a).
% 2.77/1.75 thf(func_def_78, type, sK3: (a > real) > real > a).
% 2.77/1.75 thf(func_def_79, type, sK4: (a > a) > set @ a > a).
% 2.77/1.75 thf(func_def_80, type, sK5: real > a > a > a > real).
% 2.77/1.75 thf(func_def_81, type, sK6: real > a > a > a > a > real).
% 2.77/1.75 thf(func_def_82, type, sK7: a > a).
% 2.77/1.75 thf(func_def_83, type, sK8: a > a).
% 2.77/1.75 thf(func_def_84, type, sK9: (a > a) > set @ a > a).
% 2.77/1.75 thf(func_def_85, type, sK10: (a > a) > set @ a > a).
% 2.77/1.75 thf(func_def_86, type, sK11: real).
% 2.77/1.75 thf(func_def_87, type, sK12: a > real).
% 2.77/1.75 thf(func_def_88, type, sK13: a > a).
% 2.77/1.75 thf(func_def_89, type, sK14: a > a).
% 2.77/1.75 thf(func_def_90, type, sK15: (a > real) > real > a).
% 2.77/1.75 thf(func_def_91, type, sK16: (a > real) > real > a).
% 2.77/1.75 thf(func_def_93, type, ph18: !>[X0: $tType]:(X0)).
% 2.77/1.75 thf(f1317,plain,(
% 2.77/1.75 $false),
% 2.77/1.75 inference(avatar_sat_refutation,[status(thm)],[f1134,f1138,f1142,f1146,f1147,f1316])).
% 2.77/1.75 thf(f1316,plain,(
% 2.77/1.75 ~spl17_2 | ~spl17_3 | ~spl17_4 | ~spl17_5),
% 2.77/1.75 inference(avatar_contradiction_clause,[status(thm)],[f1315])).
% 2.77/1.75 thf(f1315,plain,(
% 2.77/1.75 $false | (~spl17_2 | ~spl17_3 | ~spl17_4 | ~spl17_5)),
% 2.77/1.75 inference(subsumption_resolution,[status(thm)],[f1314,f1206])).
% 2.77/1.75 thf(f1206,plain,(
% 2.77/1.75 ($true != (ord_less @ real @ (abs_abs @ real @ (sK12 @ (sK15 @ sK12 @ sK11))) @ one_one @ real)) | (~spl17_3 | ~spl17_4)),
% 2.77/1.75 inference(subsumption_resolution,[status(thm)],[f1205,f1102])).
% 2.77/1.75 thf(f1102,plain,(
% 2.77/1.75 ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ sK12) = $true)),
% 2.77/1.75 inference(cnf_transformation,[status(thm)],[f1040])).
% 2.77/1.75 thf(f1040,plain,(
% 2.77/1.75 (! [X2 : a] : (($true = (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ X2 @ (sK12 @ X2)) @ (line_open_segment @ a @ a2 @ b))) | ($true != (member @ a @ X2 @ (elemen1815086676c_ball @ a @ p @ sK11)))) & ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ sK12) = $true) & ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ sK12) = $true) & (zero_zero @ real = (sK12 @ p)) & ! [X3 : a] : (($true != (member @ a @ X3 @ (elemen1815086676c_ball @ a @ p @ sK11))) | ($true = (ord_less @ real @ (abs_abs @ real @ (sK12 @ X3)) @ one_one @ real)))) & ($true = (ord_less @ real @ zero_zero @ real @ sK11))),
% 2.77/1.75 inference(skolemisation,[status(esa),new_symbols(skolem,[sK11,sK12])],[f962,f1039,f1038])).
% 2.77/1.75 thf(f1038,plain,(
% 2.77/1.75 ? [X0 : real] : (? [X1 : a > real] : (! [X2 : a] : (((member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ X2 @ (X1 @ X2)) @ (line_open_segment @ a @ a2 @ b)) = $true) | ($true != (member @ a @ X2 @ (elemen1815086676c_ball @ a @ p @ X0)))) & ($true = (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)) & ($true = (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)) & (zero_zero @ real = (X1 @ p)) & ! [X3 : a] : (($true != (member @ a @ X3 @ (elemen1815086676c_ball @ a @ p @ X0))) | ((ord_less @ real @ (abs_abs @ real @ (X1 @ X3)) @ one_one @ real) = $true))) & ($true = (ord_less @ real @ zero_zero @ real @ X0))) => (? [X1 : a > real] : (! [X2 : a] : (((member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ X2 @ (X1 @ X2)) @ (line_open_segment @ a @ a2 @ b)) = $true) | ($true != (member @ a @ X2 @ (elemen1815086676c_ball @ a @ p @ sK11)))) & ($true = (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ X1)) & ($true = (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ X1)) & (zero_zero @ real = (X1 @ p)) & ! [X3 : a] : (($true != (member @ a @ X3 @ (elemen1815086676c_ball @ a @ p @ sK11))) | ((ord_less @ real @ (abs_abs @ real @ (X1 @ X3)) @ one_one @ real) = $true))) & ($true = (ord_less @ real @ zero_zero @ real @ sK11)))),
% 2.77/1.75 introduced(definition,[],[choice_axiom])).
% 2.77/1.75 thf(f1039,plain,(
% 2.77/1.75 ? [X1 : a > real] : (! [X2 : a] : (((member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ X2 @ (X1 @ X2)) @ (line_open_segment @ a @ a2 @ b)) = $true) | ($true != (member @ a @ X2 @ (elemen1815086676c_ball @ a @ p @ sK11)))) & ($true = (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ X1)) & ($true = (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ X1)) & (zero_zero @ real = (X1 @ p)) & ! [X3 : a] : (($true != (member @ a @ X3 @ (elemen1815086676c_ball @ a @ p @ sK11))) | ((ord_less @ real @ (abs_abs @ real @ (X1 @ X3)) @ one_one @ real) = $true))) => (! [X2 : a] : (($true = (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ X2 @ (sK12 @ X2)) @ (line_open_segment @ a @ a2 @ b))) | ($true != (member @ a @ X2 @ (elemen1815086676c_ball @ a @ p @ sK11)))) & ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ sK12) = $true) & ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ sK12) = $true) & (zero_zero @ real = (sK12 @ p)) & ! [X3 : a] : (($true != (member @ a @ X3 @ (elemen1815086676c_ball @ a @ p @ sK11))) | ($true = (ord_less @ real @ (abs_abs @ real @ (sK12 @ X3)) @ one_one @ real))))),
% 2.77/1.75 introduced(definition,[],[choice_axiom])).
% 2.77/1.75 thf(f962,plain,(
% 2.77/1.75 ? [X0 : real] : (? [X1 : a > real] : (! [X2 : a] : (((member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ X2 @ (X1 @ X2)) @ (line_open_segment @ a @ a2 @ b)) = $true) | ($true != (member @ a @ X2 @ (elemen1815086676c_ball @ a @ p @ X0)))) & ($true = (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)) & ($true = (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)) & (zero_zero @ real = (X1 @ p)) & ! [X3 : a] : (($true != (member @ a @ X3 @ (elemen1815086676c_ball @ a @ p @ X0))) | ((ord_less @ real @ (abs_abs @ real @ (X1 @ X3)) @ one_one @ real) = $true))) & ($true = (ord_less @ real @ zero_zero @ real @ X0)))),
% 2.77/1.75 inference(flattening,[status(thm)],[f961])).
% 2.77/1.75 thf(f961,plain,(
% 2.77/1.75 ? [X0 : real] : (? [X1 : a > real] : (((((zero_zero @ real = (X1 @ p)) & ($true = (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1))) & ! [X3 : a] : (($true != (member @ a @ X3 @ (elemen1815086676c_ball @ a @ p @ X0))) | ((ord_less @ real @ (abs_abs @ real @ (X1 @ X3)) @ one_one @ real) = $true))) & ! [X2 : a] : (((member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ X2 @ (X1 @ X2)) @ (line_open_segment @ a @ a2 @ b)) = $true) | ($true != (member @ a @ X2 @ (elemen1815086676c_ball @ a @ p @ X0))))) & ($true = (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1))) & ($true = (ord_less @ real @ zero_zero @ real @ X0)))),
% 2.77/1.75 inference(ennf_transformation,[status(thm)],[f604])).
% 2.77/1.75 thf(f604,plain,(
% 2.77/1.75 ~! [X0 : real] : (($true = (ord_less @ real @ zero_zero @ real @ X0)) => ! [X1 : a > real] : (($true = (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)) => (! [X2 : a] : (($true = (member @ a @ X2 @ (elemen1815086676c_ball @ a @ p @ X0))) => ((member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ X2 @ (X1 @ X2)) @ (line_open_segment @ a @ a2 @ b)) = $true)) => (! [X3 : a] : (($true = (member @ a @ X3 @ (elemen1815086676c_ball @ a @ p @ X0))) => ((ord_less @ real @ (abs_abs @ real @ (X1 @ X3)) @ one_one @ real) = $true)) => (($true = (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)) => (zero_zero @ real != (X1 @ p)))))))),
% 2.77/1.75 inference(fool_elimination,[status(thm)],[f603])).
% 2.77/1.75 thf(f603,plain,(
% 2.77/1.75 ~! [X0 : real] : ((ord_less @ real @ zero_zero @ real @ X0) => ! [X1 : a > real] : ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1) => (! [X2 : a] : ((member @ a @ X2 @ (elemen1815086676c_ball @ a @ p @ X0)) => (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ X2 @ (X1 @ X2)) @ (line_open_segment @ a @ a2 @ b))) => (! [X3 : a] : ((member @ a @ X3 @ (elemen1815086676c_ball @ a @ p @ X0)) => (ord_less @ real @ (abs_abs @ real @ (X1 @ X3)) @ one_one @ real)) => ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1) => (zero_zero @ real != (X1 @ p)))))))),
% 2.77/1.75 inference(rectify,[status(thm)],[f4])).
% 2.77/1.75 thf(f4,axiom,(
% 2.77/1.75 ~! [X6 : real] : ((ord_less @ real @ zero_zero @ real @ X6) => ! [X7 : a > real] : ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X6) @ X7) => (! [X8 : a] : ((member @ a @ X8 @ (elemen1815086676c_ball @ a @ p @ X6)) => (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ X8 @ (X7 @ X8)) @ (line_open_segment @ a @ a2 @ b))) => (! [X8 : a] : ((member @ a @ X8 @ (elemen1815086676c_ball @ a @ p @ X6)) => (ord_less @ real @ (abs_abs @ real @ (X7 @ X8)) @ one_one @ real)) => ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X6) @ X7) => (zero_zero @ real != (X7 @ p)))))))),
% 2.77/1.75 file('/export/starexec/sandbox2/tmp/tmp.SqIxTx6pAT/DTF2THF_24163.p',fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062d_At_O_A_092_060lbrakk_0620_A_060_Ad_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_Aflow0_Ay_A_It_Ay_J_A_092_060in_062_A_123a_060_N_N_060b_125_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_A_092_060bar_062t_Ay_092_060bar_062_A_060_A1_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_At_Ap_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062)).
% 2.77/1.75 thf(f1205,plain,(
% 2.77/1.75 ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ sK12) != $true) | ($true != (ord_less @ real @ (abs_abs @ real @ (sK12 @ (sK15 @ sK12 @ sK11))) @ one_one @ real)) | (~spl17_3 | ~spl17_4)),
% 2.77/1.75 inference(subsumption_resolution,[status(thm)],[f1204,f1100])).
% 2.77/1.75 thf(f1100,plain,(
% 2.77/1.75 (zero_zero @ real = (sK12 @ p))),
% 2.77/1.75 inference(cnf_transformation,[status(thm)],[f1040])).
% 2.77/1.75 thf(f1204,plain,(
% 2.77/1.75 ($true != (ord_less @ real @ (abs_abs @ real @ (sK12 @ (sK15 @ sK12 @ sK11))) @ one_one @ real)) | (zero_zero @ real != (sK12 @ p)) | ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ sK12) != $true) | (~spl17_3 | ~spl17_4)),
% 2.77/1.75 inference(subsumption_resolution,[status(thm)],[f1203,f1098])).
% 2.77/1.75 thf(f1098,plain,(
% 2.77/1.75 ($true = (ord_less @ real @ zero_zero @ real @ sK11))),
% 2.77/1.75 inference(cnf_transformation,[status(thm)],[f1040])).
% 2.77/1.75 thf(f1203,plain,(
% 2.77/1.75 ($true != (ord_less @ real @ zero_zero @ real @ sK11)) | ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ sK12) != $true) | ($true != (ord_less @ real @ (abs_abs @ real @ (sK12 @ (sK15 @ sK12 @ sK11))) @ one_one @ real)) | (zero_zero @ real != (sK12 @ p)) | (~spl17_3 | ~spl17_4)),
% 2.77/1.75 inference(trivial_inequality_removal,[status(thm)],[f1202])).
% 2.77/1.75 thf(f1202,plain,(
% 2.77/1.75 ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ sK12) != $true) | (zero_zero @ real != (sK12 @ p)) | ($true != (ord_less @ real @ zero_zero @ real @ sK11)) | ($true != $true) | ($true != (ord_less @ real @ (abs_abs @ real @ (sK12 @ (sK15 @ sK12 @ sK11))) @ one_one @ real)) | (~spl17_3 | ~spl17_4)),
% 2.77/1.75 inference(duplicate_literal_removal,[status(thm)],[f1199])).
% 2.77/1.75 thf(f1199,plain,(
% 2.77/1.75 ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ sK12) != $true) | ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ sK12) != $true) | (zero_zero @ real != (sK12 @ p)) | ($true != (ord_less @ real @ zero_zero @ real @ sK11)) | ($true != (ord_less @ real @ (abs_abs @ real @ (sK12 @ (sK15 @ sK12 @ sK11))) @ one_one @ real)) | ($true != $true) | ($true != (ord_less @ real @ (abs_abs @ real @ (sK12 @ (sK15 @ sK12 @ sK11))) @ one_one @ real)) | ($true != (ord_less @ real @ zero_zero @ real @ sK11)) | (~spl17_3 | ~spl17_4)),
% 2.77/1.75 inference(superposition,[status(thm)],[f1165,f1141])).
% 2.77/1.75 thf(f1141,plain,(
% 2.77/1.75 ( ! [X0 : real,X1 : a > real] : (($true = (member @ a @ (sK16 @ X1 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0))) | (zero_zero @ real != (X1 @ p)) | ($true != (ord_less @ real @ (abs_abs @ real @ (X1 @ (sK15 @ X1 @ X0))) @ one_one @ real)) | ($true != (ord_less @ real @ zero_zero @ real @ X0)) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1))) ) | ~spl17_4),
% 2.77/1.75 inference(avatar_component_clause,[status(thm)],[f1140])).
% 2.77/1.75 thf(f1140,definition,(
% 2.77/1.75 spl17_4 <=> ! [X0 : real,X1 : a > real] : (($true = (member @ a @ (sK16 @ X1 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0))) | ($true != (ord_less @ real @ zero_zero @ real @ X0)) | (zero_zero @ real != (X1 @ p)) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)) | ($true != (ord_less @ real @ (abs_abs @ real @ (X1 @ (sK15 @ X1 @ X0))) @ one_one @ real)))),
% 2.77/1.75 introduced(definition,[new_symbols(naming,[spl17_4])],[avatar_definition])).
% 2.77/1.75 thf(f1165,plain,(
% 2.77/1.75 ( ! [X0 : real] : (((member @ a @ (sK16 @ sK12 @ X0) @ (elemen1815086676c_ball @ a @ p @ sK11)) != $true) | ((ord_less @ real @ (abs_abs @ real @ (sK12 @ (sK15 @ sK12 @ X0))) @ one_one @ real) != $true) | ($true != (ord_less @ real @ zero_zero @ real @ X0)) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ sK12))) ) | ~spl17_3),
% 2.77/1.75 inference(subsumption_resolution,[status(thm)],[f1164,f1100])).
% 2.77/1.75 thf(f1164,plain,(
% 2.77/1.75 ( ! [X0 : real] : (($true != (ord_less @ real @ zero_zero @ real @ X0)) | ((ord_less @ real @ (abs_abs @ real @ (sK12 @ (sK15 @ sK12 @ X0))) @ one_one @ real) != $true) | (zero_zero @ real != (sK12 @ p)) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ sK12)) | ((member @ a @ (sK16 @ sK12 @ X0) @ (elemen1815086676c_ball @ a @ p @ sK11)) != $true)) ) | ~spl17_3),
% 2.77/1.75 inference(trivial_inequality_removal,[status(thm)],[f1159])).
% 2.77/1.75 thf(f1159,plain,(
% 2.77/1.75 ( ! [X0 : real] : (($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ sK12)) | (zero_zero @ real != (sK12 @ p)) | ((ord_less @ real @ (abs_abs @ real @ (sK12 @ (sK15 @ sK12 @ X0))) @ one_one @ real) != $true) | ($true != $true) | ((member @ a @ (sK16 @ sK12 @ X0) @ (elemen1815086676c_ball @ a @ p @ sK11)) != $true) | ($true != (ord_less @ real @ zero_zero @ real @ X0))) ) | ~spl17_3),
% 2.77/1.75 inference(superposition,[status(thm)],[f1137,f1103])).
% 2.77/1.75 thf(f1103,plain,(
% 2.77/1.75 ( ! [X2 : a] : (($true = (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ X2 @ (sK12 @ X2)) @ (line_open_segment @ a @ a2 @ b))) | ($true != (member @ a @ X2 @ (elemen1815086676c_ball @ a @ p @ sK11)))) )),
% 2.77/1.75 inference(cnf_transformation,[status(thm)],[f1040])).
% 2.77/1.75 thf(f1137,plain,(
% 2.77/1.75 ( ! [X0 : real,X1 : a > real] : (($true != (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ (sK16 @ X1 @ X0) @ (X1 @ (sK16 @ X1 @ X0))) @ (line_open_segment @ a @ a2 @ b))) | ($true != (ord_less @ real @ (abs_abs @ real @ (X1 @ (sK15 @ X1 @ X0))) @ one_one @ real)) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)) | (zero_zero @ real != (X1 @ p)) | ($true != (ord_less @ real @ zero_zero @ real @ X0))) ) | ~spl17_3),
% 2.77/1.75 inference(avatar_component_clause,[status(thm)],[f1136])).
% 2.77/1.75 thf(f1136,definition,(
% 2.77/1.75 spl17_3 <=> ! [X0 : real,X1 : a > real] : ((zero_zero @ real != (X1 @ p)) | ($true != (ord_less @ real @ (abs_abs @ real @ (X1 @ (sK15 @ X1 @ X0))) @ one_one @ real)) | ($true != (ord_less @ real @ zero_zero @ real @ X0)) | ($true != (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ (sK16 @ X1 @ X0) @ (X1 @ (sK16 @ X1 @ X0))) @ (line_open_segment @ a @ a2 @ b))) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)))),
% 2.77/1.75 introduced(definition,[new_symbols(naming,[spl17_3])],[avatar_definition])).
% 2.77/1.75 thf(f1314,plain,(
% 2.77/1.75 ($true = (ord_less @ real @ (abs_abs @ real @ (sK12 @ (sK15 @ sK12 @ sK11))) @ one_one @ real)) | (~spl17_2 | ~spl17_3 | ~spl17_4 | ~spl17_5)),
% 2.77/1.75 inference(trivial_inequality_removal,[status(thm)],[f1313])).
% 2.77/1.75 thf(f1313,plain,(
% 2.77/1.75 ($true = (ord_less @ real @ (abs_abs @ real @ (sK12 @ (sK15 @ sK12 @ sK11))) @ one_one @ real)) | ($true != $true) | (~spl17_2 | ~spl17_3 | ~spl17_4 | ~spl17_5)),
% 2.77/1.75 inference(superposition,[status(thm)],[f1099,f1229])).
% 2.77/1.75 thf(f1229,plain,(
% 2.77/1.75 ((member @ a @ (sK15 @ sK12 @ sK11) @ (elemen1815086676c_ball @ a @ p @ sK11)) = $true) | (~spl17_2 | ~spl17_3 | ~spl17_4 | ~spl17_5)),
% 2.77/1.75 inference(subsumption_resolution,[status(thm)],[f1228,f1100])).
% 2.77/1.75 thf(f1228,plain,(
% 2.77/1.75 ((member @ a @ (sK15 @ sK12 @ sK11) @ (elemen1815086676c_ball @ a @ p @ sK11)) = $true) | (zero_zero @ real != (sK12 @ p)) | (~spl17_2 | ~spl17_3 | ~spl17_4 | ~spl17_5)),
% 2.77/1.75 inference(subsumption_resolution,[status(thm)],[f1227,f1102])).
% 2.77/1.75 thf(f1227,plain,(
% 2.77/1.75 ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ sK12) != $true) | ((member @ a @ (sK15 @ sK12 @ sK11) @ (elemen1815086676c_ball @ a @ p @ sK11)) = $true) | (zero_zero @ real != (sK12 @ p)) | (~spl17_2 | ~spl17_3 | ~spl17_4 | ~spl17_5)),
% 2.77/1.75 inference(subsumption_resolution,[status(thm)],[f1226,f1098])).
% 2.77/1.75 thf(f1226,plain,(
% 2.77/1.75 ($true != (ord_less @ real @ zero_zero @ real @ sK11)) | ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ sK12) != $true) | (zero_zero @ real != (sK12 @ p)) | ((member @ a @ (sK15 @ sK12 @ sK11) @ (elemen1815086676c_ball @ a @ p @ sK11)) = $true) | (~spl17_2 | ~spl17_3 | ~spl17_4 | ~spl17_5)),
% 2.77/1.75 inference(trivial_inequality_removal,[status(thm)],[f1223])).
% 2.77/1.75 thf(f1223,plain,(
% 2.77/1.75 ((member @ a @ (sK15 @ sK12 @ sK11) @ (elemen1815086676c_ball @ a @ p @ sK11)) = $true) | ($true != $true) | ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ sK12) != $true) | ($true != (ord_less @ real @ zero_zero @ real @ sK11)) | (zero_zero @ real != (sK12 @ p)) | (~spl17_2 | ~spl17_3 | ~spl17_4 | ~spl17_5)),
% 2.77/1.75 inference(superposition,[status(thm)],[f1211,f1145])).
% 2.77/1.75 thf(f1145,plain,(
% 2.77/1.75 ( ! [X0 : real,X1 : a > real] : (($true = (member @ a @ (sK16 @ X1 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0))) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)) | ($true != (ord_less @ real @ zero_zero @ real @ X0)) | ($true = (member @ a @ (sK15 @ X1 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0))) | (zero_zero @ real != (X1 @ p))) ) | ~spl17_5),
% 2.77/1.75 inference(avatar_component_clause,[status(thm)],[f1144])).
% 2.77/1.75 thf(f1144,definition,(
% 2.77/1.75 spl17_5 <=> ! [X0 : real,X1 : a > real] : (($true = (member @ a @ (sK16 @ X1 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0))) | ($true != (ord_less @ real @ zero_zero @ real @ X0)) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)) | (zero_zero @ real != (X1 @ p)) | ($true = (member @ a @ (sK15 @ X1 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0))))),
% 2.77/1.75 introduced(definition,[new_symbols(naming,[spl17_5])],[avatar_definition])).
% 2.77/1.75 thf(f1211,plain,(
% 2.77/1.75 ($true != (member @ a @ (sK16 @ sK12 @ sK11) @ (elemen1815086676c_ball @ a @ p @ sK11))) | (~spl17_2 | ~spl17_3 | ~spl17_4)),
% 2.77/1.75 inference(subsumption_resolution,[status(thm)],[f1210,f1206])).
% 2.77/1.75 thf(f1210,plain,(
% 2.77/1.75 ($true = (ord_less @ real @ (abs_abs @ real @ (sK12 @ (sK15 @ sK12 @ sK11))) @ one_one @ real)) | ($true != (member @ a @ (sK16 @ sK12 @ sK11) @ (elemen1815086676c_ball @ a @ p @ sK11))) | ~spl17_2),
% 2.77/1.75 inference(subsumption_resolution,[status(thm)],[f1209,f1102])).
% 2.77/1.75 thf(f1209,plain,(
% 2.77/1.75 ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ sK12) != $true) | ($true != (member @ a @ (sK16 @ sK12 @ sK11) @ (elemen1815086676c_ball @ a @ p @ sK11))) | ($true = (ord_less @ real @ (abs_abs @ real @ (sK12 @ (sK15 @ sK12 @ sK11))) @ one_one @ real)) | ~spl17_2),
% 2.77/1.75 inference(subsumption_resolution,[status(thm)],[f1208,f1098])).
% 2.77/1.75 thf(f1208,plain,(
% 2.77/1.75 ($true != (ord_less @ real @ zero_zero @ real @ sK11)) | ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ sK12) != $true) | ($true != (member @ a @ (sK16 @ sK12 @ sK11) @ (elemen1815086676c_ball @ a @ p @ sK11))) | ($true = (ord_less @ real @ (abs_abs @ real @ (sK12 @ (sK15 @ sK12 @ sK11))) @ one_one @ real)) | ~spl17_2),
% 2.77/1.75 inference(trivial_inequality_removal,[status(thm)],[f1207])).
% 2.77/1.75 thf(f1207,plain,(
% 2.77/1.75 ($true != (member @ a @ (sK16 @ sK12 @ sK11) @ (elemen1815086676c_ball @ a @ p @ sK11))) | ($true != $true) | ($true != (ord_less @ real @ zero_zero @ real @ sK11)) | ($true = (ord_less @ real @ (abs_abs @ real @ (sK12 @ (sK15 @ sK12 @ sK11))) @ one_one @ real)) | ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ sK11) @ sK12) != $true) | ~spl17_2),
% 2.77/1.75 inference(superposition,[status(thm)],[f1099,f1187])).
% 2.77/1.75 thf(f1187,plain,(
% 2.77/1.75 ( ! [X0 : real] : (($true = (member @ a @ (sK15 @ sK12 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0))) | ((member @ a @ (sK16 @ sK12 @ X0) @ (elemen1815086676c_ball @ a @ p @ sK11)) != $true) | ($true != (ord_less @ real @ zero_zero @ real @ X0)) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ sK12))) ) | ~spl17_2),
% 2.77/1.75 inference(subsumption_resolution,[status(thm)],[f1184,f1100])).
% 2.77/1.75 thf(f1184,plain,(
% 2.77/1.75 ( ! [X0 : real] : ((zero_zero @ real != (sK12 @ p)) | ($true = (member @ a @ (sK15 @ sK12 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0))) | ($true != (ord_less @ real @ zero_zero @ real @ X0)) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ sK12)) | ((member @ a @ (sK16 @ sK12 @ X0) @ (elemen1815086676c_ball @ a @ p @ sK11)) != $true)) ) | ~spl17_2),
% 2.77/1.75 inference(trivial_inequality_removal,[status(thm)],[f1180])).
% 2.77/1.75 thf(f1180,plain,(
% 2.77/1.75 ( ! [X0 : real] : (($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ sK12)) | ($true = (member @ a @ (sK15 @ sK12 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0))) | ((member @ a @ (sK16 @ sK12 @ X0) @ (elemen1815086676c_ball @ a @ p @ sK11)) != $true) | ($true != $true) | ($true != (ord_less @ real @ zero_zero @ real @ X0)) | (zero_zero @ real != (sK12 @ p))) ) | ~spl17_2),
% 2.77/1.75 inference(superposition,[status(thm)],[f1133,f1103])).
% 2.77/1.75 thf(f1133,plain,(
% 2.77/1.75 ( ! [X0 : real,X1 : a > real] : (($true != (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ (sK16 @ X1 @ X0) @ (X1 @ (sK16 @ X1 @ X0))) @ (line_open_segment @ a @ a2 @ b))) | ($true != (ord_less @ real @ zero_zero @ real @ X0)) | (zero_zero @ real != (X1 @ p)) | ($true = (member @ a @ (sK15 @ X1 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0))) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1))) ) | ~spl17_2),
% 2.77/1.75 inference(avatar_component_clause,[status(thm)],[f1132])).
% 2.77/1.75 thf(f1132,definition,(
% 2.77/1.75 spl17_2 <=> ! [X0 : real,X1 : a > real] : (($true = (member @ a @ (sK15 @ X1 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0))) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)) | (zero_zero @ real != (X1 @ p)) | ($true != (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ (sK16 @ X1 @ X0) @ (X1 @ (sK16 @ X1 @ X0))) @ (line_open_segment @ a @ a2 @ b))) | ($true != (ord_less @ real @ zero_zero @ real @ X0)))),
% 2.77/1.75 introduced(definition,[new_symbols(naming,[spl17_2])],[avatar_definition])).
% 2.77/1.75 thf(f1099,plain,(
% 2.77/1.75 ( ! [X3 : a] : (($true != (member @ a @ X3 @ (elemen1815086676c_ball @ a @ p @ sK11))) | ($true = (ord_less @ real @ (abs_abs @ real @ (sK12 @ X3)) @ one_one @ real))) )),
% 2.77/1.75 inference(cnf_transformation,[status(thm)],[f1040])).
% 2.77/1.75 thf(f1147,plain,(
% 2.77/1.75 ~spl17_1),
% 2.77/1.75 inference(avatar_split_clause,[status(thm)],[f1064,f1128])).
% 2.77/1.75 thf(f1128,definition,(
% 2.77/1.75 spl17_1 <=> (thesisa = $true)),
% 2.77/1.75 introduced(definition,[new_symbols(naming,[spl17_1])],[avatar_definition])).
% 2.77/1.75 thf(f1064,plain,(
% 2.77/1.75 (thesisa != $true)),
% 2.77/1.75 inference(cnf_transformation,[status(thm)],[f941])).
% 2.77/1.75 thf(f941,plain,(
% 2.77/1.75 (thesisa != $true)),
% 2.77/1.75 inference(flattening,[status(thm)],[f940])).
% 2.77/1.75 thf(f940,plain,(
% 2.77/1.75 ~(thesisa = $true)),
% 2.77/1.75 inference(fool_elimination,[status(thm)],[f939])).
% 2.77/1.75 thf(f939,plain,(
% 2.77/1.75 ~thesisa),
% 2.77/1.75 inference(rectify,[status(thm)],[f321])).
% 2.77/1.75 thf(f321,negated_conjecture,(
% 2.77/1.75 ~thesisa),
% 2.77/1.75 inference(negated_conjecture,[status(cth)],[f320])).
% 2.77/1.75 thf(f320,conjecture,(
% 2.77/1.75 thesisa),
% 2.77/1.75 file('/export/starexec/sandbox2/tmp/tmp.SqIxTx6pAT/DTF2THF_24163.p',conj_1)).
% 2.77/1.75 thf(f1146,plain,(
% 2.77/1.75 spl17_5 | spl17_1),
% 2.77/1.75 inference(avatar_split_clause,[status(thm)],[f1115,f1128,f1144])).
% 2.77/1.75 thf(f1115,plain,(
% 2.77/1.75 ( ! [X0 : real,X1 : a > real] : (($true = (member @ a @ (sK16 @ X1 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0))) | ($true = (member @ a @ (sK15 @ X1 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0))) | (thesisa = $true) | (zero_zero @ real != (X1 @ p)) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)) | ($true != (ord_less @ real @ zero_zero @ real @ X0))) )),
% 2.77/1.75 inference(cnf_transformation,[status(thm)],[f1045])).
% 2.77/1.75 thf(f1045,plain,(
% 2.77/1.75 ! [X0 : real,X1 : a > real] : ((zero_zero @ real != (X1 @ p)) | (($true != (ord_less @ real @ (abs_abs @ real @ (X1 @ (sK15 @ X1 @ X0))) @ one_one @ real)) & ($true = (member @ a @ (sK15 @ X1 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0)))) | (($true = (member @ a @ (sK16 @ X1 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0))) & ($true != (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ (sK16 @ X1 @ X0) @ (X1 @ (sK16 @ X1 @ X0))) @ (line_open_segment @ a @ a2 @ b)))) | ($true != (ord_less @ real @ zero_zero @ real @ X0)) | (thesisa = $true) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)))),
% 2.77/1.75 inference(skolemisation,[status(esa),new_symbols(skolem,[sK15,sK16])],[f986,f1044,f1043])).
% 2.77/1.75 thf(f1043,plain,(
% 2.77/1.75 ! [X0 : real,X1 : a > real] : (? [X2 : a] : (($true != (ord_less @ real @ (abs_abs @ real @ (X1 @ X2)) @ one_one @ real)) & ($true = (member @ a @ X2 @ (elemen1815086676c_ball @ a @ p @ X0)))) => (($true != (ord_less @ real @ (abs_abs @ real @ (X1 @ (sK15 @ X1 @ X0))) @ one_one @ real)) & ($true = (member @ a @ (sK15 @ X1 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0)))))),
% 2.77/1.75 introduced(definition,[],[choice_axiom])).
% 2.77/1.75 thf(f1044,plain,(
% 2.77/1.75 ! [X0 : real,X1 : a > real] : (? [X3 : a] : (($true = (member @ a @ X3 @ (elemen1815086676c_ball @ a @ p @ X0))) & ($true != (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ X3 @ (X1 @ X3)) @ (line_open_segment @ a @ a2 @ b)))) => (($true = (member @ a @ (sK16 @ X1 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0))) & ($true != (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ (sK16 @ X1 @ X0) @ (X1 @ (sK16 @ X1 @ X0))) @ (line_open_segment @ a @ a2 @ b)))))),
% 2.77/1.75 introduced(definition,[],[choice_axiom])).
% 2.77/1.75 thf(f986,plain,(
% 2.77/1.75 ! [X0 : real,X1 : a > real] : ((zero_zero @ real != (X1 @ p)) | ? [X2 : a] : (($true != (ord_less @ real @ (abs_abs @ real @ (X1 @ X2)) @ one_one @ real)) & ($true = (member @ a @ X2 @ (elemen1815086676c_ball @ a @ p @ X0)))) | ? [X3 : a] : (($true = (member @ a @ X3 @ (elemen1815086676c_ball @ a @ p @ X0))) & ($true != (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ X3 @ (X1 @ X3)) @ (line_open_segment @ a @ a2 @ b)))) | ($true != (ord_less @ real @ zero_zero @ real @ X0)) | (thesisa = $true) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)))),
% 2.77/1.75 inference(flattening,[status(thm)],[f985])).
% 2.77/1.75 thf(f985,plain,(
% 2.77/1.75 ! [X0 : real,X1 : a > real] : ((((((thesisa = $true) | ? [X3 : a] : (($true = (member @ a @ X3 @ (elemen1815086676c_ball @ a @ p @ X0))) & ($true != (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ X3 @ (X1 @ X3)) @ (line_open_segment @ a @ a2 @ b))))) | ? [X2 : a] : (($true != (ord_less @ real @ (abs_abs @ real @ (X1 @ X2)) @ one_one @ real)) & ($true = (member @ a @ X2 @ (elemen1815086676c_ball @ a @ p @ X0))))) | (zero_zero @ real != (X1 @ p))) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1))) | ($true != (ord_less @ real @ zero_zero @ real @ X0)))),
% 2.77/1.75 inference(ennf_transformation,[status(thm)],[f582])).
% 2.77/1.75 thf(f582,plain,(
% 2.77/1.75 ! [X0 : real,X1 : a > real] : (($true = (ord_less @ real @ zero_zero @ real @ X0)) => (($true = (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)) => ((zero_zero @ real = (X1 @ p)) => (! [X2 : a] : (($true = (member @ a @ X2 @ (elemen1815086676c_ball @ a @ p @ X0))) => ($true = (ord_less @ real @ (abs_abs @ real @ (X1 @ X2)) @ one_one @ real))) => (! [X3 : a] : (($true = (member @ a @ X3 @ (elemen1815086676c_ball @ a @ p @ X0))) => ($true = (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ X3 @ (X1 @ X3)) @ (line_open_segment @ a @ a2 @ b)))) => (thesisa = $true))))))),
% 2.77/1.75 inference(fool_elimination,[status(thm)],[f581])).
% 2.77/1.75 thf(f581,plain,(
% 2.77/1.75 ! [X0 : real,X1 : a > real] : ((ord_less @ real @ zero_zero @ real @ X0) => ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1) => ((zero_zero @ real = (X1 @ p)) => (! [X2 : a] : ((member @ a @ X2 @ (elemen1815086676c_ball @ a @ p @ X0)) => (ord_less @ real @ (abs_abs @ real @ (X1 @ X2)) @ one_one @ real)) => (! [X3 : a] : ((member @ a @ X3 @ (elemen1815086676c_ball @ a @ p @ X0)) => (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ X3 @ (X1 @ X3)) @ (line_open_segment @ a @ a2 @ b))) => thesisa)))))),
% 2.77/1.75 inference(rectify,[status(thm)],[f319])).
% 2.77/1.75 thf(f319,axiom,(
% 2.77/1.75 ! [X57 : real,X58 : a > real] : ((ord_less @ real @ zero_zero @ real @ X57) => ((topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X57) @ X58) => ((zero_zero @ real = (X58 @ p)) => (! [X11 : a] : ((member @ a @ X11 @ (elemen1815086676c_ball @ a @ p @ X57)) => (ord_less @ real @ (abs_abs @ real @ (X58 @ X11)) @ one_one @ real)) => (! [X11 : a] : ((member @ a @ X11 @ (elemen1815086676c_ball @ a @ p @ X57)) => (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ X11 @ (X58 @ X11)) @ (line_open_segment @ a @ a2 @ b))) => thesisa)))))),
% 2.77/1.75 file('/export/starexec/sandbox2/tmp/tmp.SqIxTx6pAT/DTF2THF_24163.p',conj_0)).
% 2.77/1.75 thf(f1142,plain,(
% 2.77/1.75 spl17_1 | spl17_4),
% 2.77/1.75 inference(avatar_split_clause,[status(thm)],[f1117,f1140,f1128])).
% 2.77/1.75 thf(f1117,plain,(
% 2.77/1.75 ( ! [X0 : real,X1 : a > real] : (($true = (member @ a @ (sK16 @ X1 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0))) | ($true != (ord_less @ real @ (abs_abs @ real @ (X1 @ (sK15 @ X1 @ X0))) @ one_one @ real)) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)) | (thesisa = $true) | (zero_zero @ real != (X1 @ p)) | ($true != (ord_less @ real @ zero_zero @ real @ X0))) )),
% 2.77/1.75 inference(cnf_transformation,[status(thm)],[f1045])).
% 2.77/1.75 thf(f1138,plain,(
% 2.77/1.75 spl17_1 | spl17_3),
% 2.77/1.75 inference(avatar_split_clause,[status(thm)],[f1116,f1136,f1128])).
% 2.77/1.75 thf(f1116,plain,(
% 2.77/1.75 ( ! [X0 : real,X1 : a > real] : ((zero_zero @ real != (X1 @ p)) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1)) | ($true != (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ (sK16 @ X1 @ X0) @ (X1 @ (sK16 @ X1 @ X0))) @ (line_open_segment @ a @ a2 @ b))) | ($true != (ord_less @ real @ zero_zero @ real @ X0)) | ($true != (ord_less @ real @ (abs_abs @ real @ (X1 @ (sK15 @ X1 @ X0))) @ one_one @ real)) | (thesisa = $true)) )),
% 2.77/1.75 inference(cnf_transformation,[status(thm)],[f1045])).
% 2.77/1.75 thf(f1134,plain,(
% 2.77/1.75 spl17_1 | spl17_2),
% 2.77/1.75 inference(avatar_split_clause,[status(thm)],[f1114,f1132,f1128])).
% 2.77/1.75 thf(f1114,plain,(
% 2.77/1.75 ( ! [X0 : real,X1 : a > real] : ((thesisa = $true) | ($true = (member @ a @ (sK15 @ X1 @ X0) @ (elemen1815086676c_ball @ a @ p @ X0))) | ($true != (ord_less @ real @ zero_zero @ real @ X0)) | ($true != (member @ a @ (auto_ll_on_flow0 @ a @ f @ x @ (sK16 @ X1 @ X0) @ (X1 @ (sK16 @ X1 @ X0))) @ (line_open_segment @ a @ a2 @ b))) | (zero_zero @ real != (X1 @ p)) | ($true != (topolo2071040574ous_on @ a @ real @ (elemen1815086676c_ball @ a @ p @ X0) @ X1))) )),
% 2.77/1.75 inference(cnf_transformation,[status(thm)],[f1045])).
% 2.77/1.75 % SZS output end Proof for DTF2THF_24163
% 2.77/1.75 % (24376)------------------------------
% 2.77/1.75 % (24376)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.75 % (24376)Termination reason: Refutation
% 2.77/1.75
% 2.77/1.75 % (24376)Memory used [KB]: 6780
% 2.77/1.75 % (24376)Time elapsed: 0.091 s
% 2.77/1.75 % (24376)Instructions burned: 176 (million)
% 2.77/1.75 % (24376)------------------------------
% 2.77/1.75 % (24376)------------------------------
% 2.77/1.75 % (24369)Success in time 0.108 s
%------------------------------------------------------------------------------