%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP149^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n001.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 08:55:57 AM UTC 2026
% Result : Theorem 3.13s 1.80s
% Output : Refutation 3.13s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP149^2 : TPTP v9.2.1. Released v7.5.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.35 % Computer : n001.cluster.edu
% 0.17/0.35 % Model : x86_64 x86_64
% 0.17/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.35 % Memory : 8042.1875MB
% 0.17/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.35 % CPULimit : 300
% 0.17/0.35 % WCLimit : 300
% 0.17/0.35 % DateTime : Tue Feb 24 08:28:50 EST 2026
% 0.17/0.35 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.31/0.47 ---- Original DTF file ---
% 0.31/0.47 thf(spec,logic,$$dhol).
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 % File : ITP149^2 : TPTP v9.2.1. Released v7.5.0.
% 0.31/0.47 % Domain : Interactive Theorem Proving
% 0.31/0.47 % Problem : Sledgehammer Poincare_Bendixson problem prob_755__19581512_1
% 0.31/0.47 % Version : Especial.
% 0.31/0.47 % English :
% 0.31/0.47
% 0.31/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.31/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.31/0.47 % Source : [Des21]
% 0.31/0.47 % Names : Poincare_Bendixson/prob_755__19581512_1 [Des21]
% 0.31/0.47
% 0.31/0.47 % Status : Theorem
% 0.31/0.47 % Rating : 0.33 v8.1.0, 0.50 v7.5.0
% 0.31/0.47 % Syntax : Number of formulae : 382 ( 86 unt; 62 typ; 0 def)
% 0.31/0.47 % Number of atoms : 869 ( 220 equ; 0 cnn)
% 0.31/0.47 % Maximal formula atoms : 9 ( 2 avg)
% 0.31/0.47 % Number of connectives : 3838 ( 59 ~; 8 |; 37 &;3373 @)
% 0.31/0.47 % ( 0 <=>; 361 =>; 0 <=; 0 <~>)
% 0.31/0.47 % Maximal formula depth : 22 ( 7 avg)
% 0.31/0.47 % Number of types : 3 ( 2 usr)
% 0.31/0.47 % Number of type conns : 146 ( 146 >; 0 *; 0 +; 0 <<)
% 0.31/0.47 % Number of symbols : 61 ( 60 usr; 8 con; 0-6 aty)
% 0.31/0.47 % Number of variables : 776 ( 39 ^; 680 !; 9 ?; 776 :)
% 0.31/0.47 % ( 48 !>; 0 ?*; 0 @-; 0 @+)
% 0.31/0.47 % SPC : TH1_THM_EQU_NAR
% 0.31/0.47
% 0.31/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:29:39.099
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 % Could-be-implicit typings (4)
% 0.31/0.47 thf(ty_t_Bounded__Linear__Function_Oblinfun,type,
% 0.31/0.47 bounde2145540817linfun: $tType > $tType > $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_t_Real_Oreal,type,
% 0.31/0.47 real: $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_t_Set_Oset,type,
% 0.31/0.47 set: $tType > $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_tf_a,type,
% 0.31/0.47 a: $tType ).
% 0.31/0.47
% 0.31/0.47 % Explicit typings (58)
% 0.31/0.47 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.31/0.47 finite_finite:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_HOL_Otype,type,
% 0.31/0.47 type:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Countable_Ocountable,type,
% 0.31/0.47 countable:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Ozero,type,
% 0.31/0.47 zero:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Ominus,type,
% 0.31/0.47 minus:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Ouminus,type,
% 0.31/0.47 uminus:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.31/0.47 ord:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.31/0.47 order:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Ogroup__add,type,
% 0.31/0.47 group_add:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.31/0.47 linorder:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.31/0.47 preorder:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Oab__group__add,type,
% 0.31/0.47 ab_group_add:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 0.31/0.47 comm_monoid_diff:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Inner__Product_Oreal__inner,type,
% 0.31/0.47 inner_real_inner:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 0.31/0.47 boolean_algebra:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.31/0.47 ordered_ab_group_add:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.31/0.47 cancel1352612707id_add:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.31/0.47 cancel146912293up_add:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.31/0.47 linord219039673up_add:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Euclidean__Space_Oeuclidean__space,type,
% 0.31/0.47 euclid925273238_space:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
% 0.31/0.47 real_V2090557954_space:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Topological__Spaces_Operfect__space,type,
% 0.31/0.47 topolo890362671_space:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.31/0.47 canoni770627133id_add:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
% 0.31/0.47 ordere890947078_space:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Executable__Euclidean__Space_Oexecutable__euclidean__space,type,
% 0.31/0.47 execut510477386_space:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Elementary__Metric__Spaces_Ocball,type,
% 0.31/0.47 elemen321786957_cball:
% 0.31/0.47 !>[A: $tType] : ( A > real > ( set @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Flow_Oauto__ll__on__open_Oexistence__ivl0,type,
% 0.31/0.47 auto_l1112008849e_ivl0:
% 0.31/0.47 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > ( set @ real ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.31/0.47 minus_minus:
% 0.31/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.31/0.47 uminus_uminus:
% 0.31/0.47 !>[A: $tType] : ( A > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.31/0.47 zero_zero:
% 0.31/0.47 !>[A: $tType] : A ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Initial__Value__Problem_Ointerval,type,
% 0.31/0.47 initia826609931terval: ( set @ real ) > $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Inner__Product_Oreal__inner__class_Oinner,type,
% 0.31/0.47 inner_780170721_inner:
% 0.31/0.47 !>[A: $tType] : ( A > A > real ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Invariance_Oauto__ll__on__open_Oinvariant,type,
% 0.31/0.47 auto_ll_on_invariant:
% 0.31/0.47 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > ( set @ A ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Line__Segment_Oclosed__segment,type,
% 0.31/0.47 line_closed_segment:
% 0.31/0.47 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Line__Segment_Oopen__segment,type,
% 0.31/0.47 line_open_segment:
% 0.31/0.47 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_ODE__Misc_Oauto__ll__on__open_Otrapped,type,
% 0.31/0.47 oDE_au1039603466rapped:
% 0.31/0.47 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > ( set @ A ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.31/0.47 ord_less:
% 0.31/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.31/0.47 ord_less_eq:
% 0.31/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.31/0.47 top_top:
% 0.31/0.47 !>[A: $tType] : A ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Oclosed__orbit,type,
% 0.31/0.47 period385816147_orbit:
% 0.31/0.47 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiod,type,
% 0.31/0.47 period1153813292period:
% 0.31/0.47 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > real ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiodic__orbit,type,
% 0.31/0.47 period862636932_orbit:
% 0.31/0.47 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Orot,type,
% 0.31/0.47 poinca1750768982en_rot:
% 0.31/0.47 !>[A: $tType] : ( A > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Otransversal__segment,type,
% 0.31/0.47 poinca272511729egment:
% 0.31/0.47 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > A > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Reachability__Analysis_Oc1__on__open__euclidean_Osection,type,
% 0.31/0.47 reacha1084862253ection:
% 0.31/0.47 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > ( A > real ) > ( A > ( bounde2145540817linfun @ A @ real ) ) > ( set @ A ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
% 0.31/0.47 real_V2000881966t_dist:
% 0.31/0.47 !>[A: $tType] : ( A > A > real ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Set_OCollect,type,
% 0.31/0.47 collect:
% 0.31/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
% 0.31/0.47 set_or331188842AtMost:
% 0.31/0.47 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_member,type,
% 0.31/0.47 member:
% 0.31/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_X,type,
% 0.31/0.47 x: set @ a ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_a,type,
% 0.31/0.47 a2: a ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_b,type,
% 0.31/0.47 b: a ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_d____,type,
% 0.31/0.47 d: real ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_f,type,
% 0.31/0.47 f: a > a ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_s____,type,
% 0.31/0.47 s: a ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_thesis,type,
% 0.31/0.47 thesis: $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_x____,type,
% 0.31/0.47 x2: a ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_z,type,
% 0.31/0.47 z: a ).
% 0.31/0.47
% 0.31/0.47 % Relevant facts (255)
% 0.31/0.47 thf(fact_0_d_I1_J,axiom,
% 0.31/0.47 ord_less @ real @ ( zero_zero @ real ) @ d ).
% 0.31/0.47
% 0.31/0.47 % d(1)
% 0.31/0.47 thf(fact_1__092_060open_062s_A_092_060in_062_Acball_Ax_Ad_092_060close_062,axiom,
% 0.31/0.47 member @ a @ s @ ( elemen321786957_cball @ a @ x2 @ d ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>s \<in> cball x d\<close>
% 0.31/0.47 thf(fact_2_seg_I1_J,axiom,
% 0.31/0.47 a2 != b ).
% 0.31/0.47
% 0.31/0.47 % seg(1)
% 0.31/0.47 thf(fact_3_nrm__dot,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( inner_780170721_inner @ a @ ( minus_minus @ a @ X @ Y ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) )
% 0.31/0.47 = ( zero_zero @ real ) ) ).
% 0.31/0.47
% 0.31/0.47 % nrm_dot
% 0.31/0.47 thf(fact_4__092_060open_062x_A_092_060in_062_AX_092_060close_062,axiom,
% 0.31/0.47 member @ a @ x2 @ x ).
% 0.31/0.47
% 0.31/0.47 % \<open>x \<in> X\<close>
% 0.31/0.47 thf(fact_5__092_060open_062_092_060forall_062x_092_060in_062cball_Ax_Ad_O_Af_As_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060le_062_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062,axiom,
% 0.31/0.47 ! [X2: a] :
% 0.31/0.47 ( ( member @ a @ X2 @ ( elemen321786957_cball @ a @ x2 @ d ) )
% 0.31/0.47 => ( ord_less_eq @ real @ ( inner_780170721_inner @ a @ ( f @ s ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) @ ( inner_780170721_inner @ a @ ( f @ X2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>\<forall>x\<in>cball x d. f s \<bullet> rot (a - b) \<le> f x \<bullet> rot (a - b)\<close>
% 0.31/0.47 thf(fact_6__092_060open_062f_Ax_A_092_060noteq_062_A_I0_058_058_Ha_J_092_060close_062,axiom,
% 0.31/0.47 ( ( f @ x2 )
% 0.31/0.47 != ( zero_zero @ a ) ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>f x \<noteq> (0::'a)\<close>
% 0.31/0.47 thf(fact_7_direction_I2_J,axiom,
% 0.31/0.47 ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ z ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % direction(2)
% 0.31/0.47 thf(fact_8_d_I2_J,axiom,
% 0.31/0.47 ! [Y2: a] :
% 0.31/0.47 ( ( member @ a @ Y2 @ ( elemen321786957_cball @ a @ x2 @ d ) )
% 0.31/0.47 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Y2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) )
% 0.31/0.47 & ( member @ a @ Y2 @ x )
% 0.31/0.47 & ( ( f @ Y2 )
% 0.31/0.47 != ( zero_zero @ a ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % d(2)
% 0.31/0.47 thf(fact_9__092_060open_0620_A_060_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062,axiom,
% 0.31/0.47 ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ x2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>0 < f x \<bullet> rot (a - b)\<close>
% 0.31/0.47 thf(fact_10_dot__ortho,axiom,
% 0.31/0.47 ! [X: a] :
% 0.31/0.47 ( ( inner_780170721_inner @ a @ X @ ( poinca1750768982en_rot @ a @ X ) )
% 0.31/0.47 = ( zero_zero @ real ) ) ).
% 0.31/0.47
% 0.31/0.47 % dot_ortho
% 0.31/0.47 thf(fact_11__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062s_O_A_092_060lbrakk_062s_A_092_060in_062_Acball_Ax_Ad_059_A_092_060forall_062x_092_060in_062cball_Ax_Ad_O_Af_As_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060le_062_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.31/0.47 ~ ! [S: a] :
% 0.31/0.47 ( ( member @ a @ S @ ( elemen321786957_cball @ a @ x2 @ d ) )
% 0.31/0.47 => ~ ! [X2: a] :
% 0.31/0.47 ( ( member @ a @ X2 @ ( elemen321786957_cball @ a @ x2 @ d ) )
% 0.31/0.47 => ( ord_less_eq @ real @ ( inner_780170721_inner @ a @ ( f @ S ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) @ ( inner_780170721_inner @ a @ ( f @ X2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>\<And>thesis. (\<And>s. \<lbrakk>s \<in> cball x d; \<forall>x\<in>cball x d. f s \<bullet> rot (a - b) \<le> f x \<bullet> rot (a - b)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.31/0.47 thf(fact_12__092_060open_062x_A_092_060in_062_A_123a_N_Nb_125_092_060close_062,axiom,
% 0.31/0.47 member @ a @ x2 @ ( line_closed_segment @ a @ a2 @ b ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>x \<in> {a--b}\<close>
% 0.31/0.47 thf(fact_13__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062d_O_A_092_060lbrakk_0620_A_060_Ad_059_A_092_060And_062y_O_Ay_A_092_060in_062_Acball_Ax_Ad_A_092_060Longrightarrow_062_A0_A_060_Af_Ay_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060and_062_Ay_A_092_060in_062_AX_A_092_060and_062_Af_Ay_A_092_060noteq_062_A_I0_058_058_Ha_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.31/0.47 ~ ! [D: real] :
% 0.31/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ D )
% 0.31/0.47 => ~ ! [Y3: a] :
% 0.31/0.47 ( ( member @ a @ Y3 @ ( elemen321786957_cball @ a @ x2 @ D ) )
% 0.31/0.47 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Y3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) )
% 0.31/0.47 & ( member @ a @ Y3 @ x )
% 0.31/0.47 & ( ( f @ Y3 )
% 0.31/0.47 != ( zero_zero @ a ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>\<And>thesis. (\<And>d. \<lbrakk>0 < d; \<And>y. y \<in> cball x d \<Longrightarrow> 0 < f y \<bullet> rot (a - b) \<and> y \<in> X \<and> f y \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.31/0.47 thf(fact_14_transversal,axiom,
% 0.31/0.47 poinca272511729egment @ a @ f @ x @ a2 @ b ).
% 0.31/0.47
% 0.31/0.47 % transversal
% 0.31/0.47 thf(fact_15_inner__gt__zero__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( inner_real_inner @ A )
% 0.31/0.47 => ! [X: A] :
% 0.31/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ A @ X @ X ) )
% 0.31/0.47 = ( X
% 0.31/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inner_gt_zero_iff
% 0.31/0.47 thf(fact_16_centre__in__cball,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( real_V2090557954_space @ A )
% 0.31/0.47 => ! [X: A,E: real] :
% 0.31/0.47 ( ( member @ A @ X @ ( elemen321786957_cball @ A @ X @ E ) )
% 0.31/0.47 = ( ord_less_eq @ real @ ( zero_zero @ real ) @ E ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % centre_in_cball
% 0.31/0.47 thf(fact_17_all__zero__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( inner_real_inner @ A )
% 0.31/0.47 => ! [X: A] :
% 0.31/0.47 ( ( ! [U: A] :
% 0.31/0.47 ( ( inner_780170721_inner @ A @ X @ U )
% 0.31/0.47 = ( zero_zero @ real ) ) )
% 0.31/0.47 = ( X
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % all_zero_iff
% 0.31/0.47 thf(fact_18_inner__zero__left,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( inner_real_inner @ A )
% 0.31/0.47 => ! [X: A] :
% 0.31/0.47 ( ( inner_780170721_inner @ A @ ( zero_zero @ A ) @ X )
% 0.31/0.47 = ( zero_zero @ real ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inner_zero_left
% 0.31/0.47 thf(fact_19_inner__zero__right,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( inner_real_inner @ A )
% 0.31/0.47 => ! [X: A] :
% 0.31/0.47 ( ( inner_780170721_inner @ A @ X @ ( zero_zero @ A ) )
% 0.31/0.47 = ( zero_zero @ real ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inner_zero_right
% 0.31/0.47 thf(fact_20_inner__eq__zero__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( inner_real_inner @ A )
% 0.31/0.47 => ! [X: A] :
% 0.31/0.47 ( ( ( inner_780170721_inner @ A @ X @ X )
% 0.31/0.47 = ( zero_zero @ real ) )
% 0.31/0.47 = ( X
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inner_eq_zero_iff
% 0.31/0.47 thf(fact_21_diff__gt__0__iff__gt,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordered_ab_group_add @ A )
% 0.31/0.47 => ! [A2: A,B: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 0.31/0.47 = ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_gt_0_iff_gt
% 0.31/0.47 thf(fact_22_diff__ge__0__iff__ge,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordered_ab_group_add @ A )
% 0.31/0.47 => ! [A2: A,B: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 0.31/0.47 = ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_ge_0_iff_ge
% 0.31/0.47 thf(fact_23_rot__0,axiom,
% 0.31/0.47 ( ( poinca1750768982en_rot @ a @ ( zero_zero @ a ) )
% 0.31/0.47 = ( zero_zero @ a ) ) ).
% 0.31/0.47
% 0.31/0.47 % rot_0
% 0.31/0.47 thf(fact_24_rot__eq__0__iff,axiom,
% 0.31/0.47 ! [X: a] :
% 0.31/0.47 ( ( ( poinca1750768982en_rot @ a @ X )
% 0.31/0.47 = ( zero_zero @ a ) )
% 0.31/0.47 = ( X
% 0.31/0.47 = ( zero_zero @ a ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % rot_eq_0_iff
% 0.31/0.47 thf(fact_25_seg_I3_J,axiom,
% 0.31/0.47 ! [Z: a] :
% 0.31/0.47 ( ( member @ a @ Z @ ( line_closed_segment @ a @ a2 @ b ) )
% 0.31/0.47 => ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % seg(3)
% 0.31/0.47 thf(fact_26_direction_I1_J,axiom,
% 0.31/0.47 member @ a @ z @ ( line_closed_segment @ a @ a2 @ b ) ).
% 0.31/0.47
% 0.31/0.47 % direction(1)
% 0.31/0.47 thf(fact_27_transversal__segment__reverse,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.31/0.47 => ( poinca272511729egment @ a @ f @ x @ Y @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % transversal_segment_reverse
% 0.31/0.47 thf(fact_28_transversal__segment__commute,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.31/0.47 = ( poinca272511729egment @ a @ f @ x @ Y @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % transversal_segment_commute
% 0.31/0.47 thf(fact_29_in__segment__inner__rot2,axiom,
% 0.31/0.47 ! [X: a,A2: a,B: a,Y: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( line_closed_segment @ a @ A2 @ B ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( line_closed_segment @ a @ A2 @ B ) )
% 0.31/0.47 => ( ( inner_780170721_inner @ a @ ( minus_minus @ a @ X @ Y ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ A2 @ B ) ) )
% 0.31/0.47 = ( zero_zero @ real ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % in_segment_inner_rot2
% 0.31/0.47 thf(fact_30_in__segment__inner__rot,axiom,
% 0.31/0.47 ! [X: a,A2: a,B: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( line_closed_segment @ a @ A2 @ B ) )
% 0.31/0.47 => ( ( inner_780170721_inner @ a @ ( minus_minus @ a @ X @ A2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ B @ A2 ) ) )
% 0.31/0.47 = ( zero_zero @ real ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % in_segment_inner_rot
% 0.31/0.47 thf(fact_31_seg_I2_J,axiom,
% 0.31/0.47 ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ a2 @ b ) @ x ).
% 0.31/0.47
% 0.31/0.47 % seg(2)
% 0.31/0.47 thf(fact_32_le__zero__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [N: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.31/0.47 = ( N
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_zero_eq
% 0.31/0.47 thf(fact_33_not__gr__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [N: A] :
% 0.31/0.47 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.31/0.47 = ( N
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_gr_zero
% 0.31/0.47 thf(fact_34_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel1352612707id_add @ A )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( minus_minus @ A @ A2 @ A2 )
% 0.31/0.47 = ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % cancel_comm_monoid_add_class.diff_cancel
% 0.31/0.47 thf(fact_35_diff__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel1352612707id_add @ A )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.31/0.47 = A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_zero
% 0.31/0.47 thf(fact_36_zero__diff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( comm_monoid_diff @ A )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 0.31/0.47 = ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_diff
% 0.31/0.47 thf(fact_37_diff__0__right,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.31/0.47 = A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_0_right
% 0.31/0.47 thf(fact_38_diff__self,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( minus_minus @ A @ A2 @ A2 )
% 0.31/0.47 = ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_self
% 0.31/0.47 thf(fact_39_subset__cball,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( real_V2090557954_space @ A )
% 0.31/0.47 => ! [D2: real,E: real,X: A] :
% 0.31/0.47 ( ( ord_less_eq @ real @ D2 @ E )
% 0.31/0.47 => ( ord_less_eq @ ( set @ A ) @ ( elemen321786957_cball @ A @ X @ D2 ) @ ( elemen321786957_cball @ A @ X @ E ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_cball
% 0.31/0.47 thf(fact_40_transversal__segment__def,axiom,
% 0.31/0.47 ! [A2: a,B: a] :
% 0.31/0.47 ( ( poinca272511729egment @ a @ f @ x @ A2 @ B )
% 0.31/0.47 = ( ( A2 != B )
% 0.31/0.47 & ( ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ A2 @ B ) @ x )
% 0.31/0.47 & ! [X3: a] :
% 0.31/0.47 ( ( member @ a @ X3 @ ( line_closed_segment @ a @ A2 @ B ) )
% 0.31/0.47 => ( ( inner_780170721_inner @ a @ ( f @ X3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ A2 @ B ) ) )
% 0.31/0.47 != ( zero_zero @ real ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % transversal_segment_def
% 0.31/0.47 thf(fact_41_transversal__segmentE,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.31/0.47 => ( ( ( X != Y )
% 0.31/0.47 => ( ( ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ X @ Y ) @ x )
% 0.31/0.47 => ~ ! [Z2: a] :
% 0.31/0.47 ( ( member @ a @ Z2 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.31/0.47 => ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) ) ) ) )
% 0.31/0.47 => ~ ( ( X != Y )
% 0.31/0.47 => ( ( ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ X @ Y ) @ x )
% 0.31/0.47 => ~ ! [Z2: a] :
% 0.31/0.47 ( ( member @ a @ Z2 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.31/0.47 => ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ Y @ X ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % transversal_segmentE
% 0.31/0.47 thf(fact_42_transversal__segment__posD_I2_J,axiom,
% 0.31/0.47 ! [X: a,Y: a,Z3: a] :
% 0.31/0.47 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.31/0.47 => ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.31/0.47 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) )
% 0.31/0.47 => ( ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ X @ Y ) @ x ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % transversal_segment_posD(2)
% 0.31/0.47 thf(fact_43_transversal__segment__negD_I2_J,axiom,
% 0.31/0.47 ! [X: a,Y: a,Z3: a] :
% 0.31/0.47 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.31/0.47 => ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.31/0.47 => ( ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) )
% 0.31/0.47 => ( ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ X @ Y ) @ x ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % transversal_segment_negD(2)
% 0.31/0.47 thf(fact_44_mem__Collect__eq,axiom,
% 0.31/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.31/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.31/0.47 = ( P @ A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % mem_Collect_eq
% 0.31/0.47 thf(fact_45_Collect__mem__eq,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A] :
% 0.31/0.47 ( ( collect @ A
% 0.31/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.31/0.47 = A3 ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_mem_eq
% 0.31/0.47 thf(fact_46_Collect__cong,axiom,
% 0.31/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/0.47 ( ! [X4: A] :
% 0.31/0.47 ( ( P @ X4 )
% 0.31/0.47 = ( Q @ X4 ) )
% 0.31/0.47 => ( ( collect @ A @ P )
% 0.31/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_cong
% 0.31/0.47 thf(fact_47_ext,axiom,
% 0.31/0.47 ! [B2: $tType,A: $tType,F: A > B2,G: A > B2] :
% 0.31/0.47 ( ! [X4: A] :
% 0.31/0.47 ( ( F @ X4 )
% 0.31/0.47 = ( G @ X4 ) )
% 0.31/0.47 => ( F = G ) ) ).
% 0.31/0.47
% 0.31/0.47 % ext
% 0.31/0.47 thf(fact_48_transversal__segment__sign__less,axiom,
% 0.31/0.47 ! [W: a,Y: a] :
% 0.31/0.47 ( ( poinca272511729egment @ a @ f @ x @ W @ Y )
% 0.31/0.47 => ( ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ W ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ W @ Y ) ) ) @ ( zero_zero @ real ) )
% 0.31/0.47 => ! [X2: a] :
% 0.31/0.47 ( ( member @ a @ X2 @ ( line_closed_segment @ a @ W @ Y ) )
% 0.31/0.47 => ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ X2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ W @ Y ) ) ) @ ( zero_zero @ real ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % transversal_segment_sign_less
% 0.31/0.47 thf(fact_49_transversal__segment__pos,axiom,
% 0.31/0.47 ! [X: a,Y: a,W: a] :
% 0.31/0.47 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.31/0.47 => ( ( member @ a @ W @ ( line_closed_segment @ a @ X @ Y ) )
% 0.31/0.47 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ W ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) )
% 0.31/0.47 => ! [X2: a] :
% 0.31/0.47 ( ( member @ a @ X2 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.31/0.47 => ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ X2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % transversal_segment_pos
% 0.31/0.47 thf(fact_50_transversal__segment__neg,axiom,
% 0.31/0.47 ! [X: a,Y: a,W: a] :
% 0.31/0.47 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.31/0.47 => ( ( member @ a @ W @ ( line_closed_segment @ a @ X @ Y ) )
% 0.31/0.47 => ( ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ W ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) )
% 0.31/0.47 => ! [X2: a] :
% 0.31/0.47 ( ( member @ a @ X2 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.31/0.47 => ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ X2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % transversal_segment_neg
% 0.31/0.47 thf(fact_51_transversal__segment__posD_I1_J,axiom,
% 0.31/0.47 ! [X: a,Y: a,Z3: a] :
% 0.31/0.47 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.31/0.47 => ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.31/0.47 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) )
% 0.31/0.47 => ( X != Y ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % transversal_segment_posD(1)
% 0.31/0.47 thf(fact_52_transversal__segment__posD_I3_J,axiom,
% 0.31/0.47 ! [X: a,Y: a,Za: a,Z3: a] :
% 0.31/0.47 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.31/0.47 => ( ( member @ a @ Za @ ( line_closed_segment @ a @ X @ Y ) )
% 0.31/0.47 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Za ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) )
% 0.31/0.47 => ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.31/0.47 => ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % transversal_segment_posD(3)
% 0.31/0.47 thf(fact_53_transversal__segment__negD_I1_J,axiom,
% 0.31/0.47 ! [X: a,Y: a,Z3: a] :
% 0.31/0.47 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.31/0.47 => ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.31/0.47 => ( ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) )
% 0.31/0.47 => ( X != Y ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % transversal_segment_negD(1)
% 0.31/0.47 thf(fact_54_transversal__segment__negD_I3_J,axiom,
% 0.31/0.47 ! [X: a,Y: a,Za: a,Z3: a] :
% 0.31/0.47 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.31/0.47 => ( ( member @ a @ Za @ ( line_closed_segment @ a @ X @ Y ) )
% 0.31/0.47 => ( ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ Za ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) )
% 0.31/0.47 => ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.31/0.47 => ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % transversal_segment_negD(3)
% 0.31/0.47 thf(fact_55__092_060open_062_092_060And_062z_O_Az_A_092_060in_062_A_123a_N_Nb_125_A_092_060Longrightarrow_062_A0_A_060_Af_Az_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062,axiom,
% 0.31/0.47 ! [Z3: a] :
% 0.31/0.47 ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ a2 @ b ) )
% 0.31/0.47 => ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>\<And>z. z \<in> {a--b} \<Longrightarrow> 0 < f z \<bullet> rot (a - b)\<close>
% 0.31/0.47 thf(fact_56_c1__on__open__R2_Otransversal__segment_Ocong,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( execut510477386_space @ A )
% 0.31/0.47 => ( ( poinca272511729egment @ A )
% 0.31/0.47 = ( poinca272511729egment @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % c1_on_open_R2.transversal_segment.cong
% 0.31/0.47 thf(fact_57_zero__reorient,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( zero @ A )
% 0.31/0.47 => ! [X: A] :
% 0.31/0.47 ( ( ( zero_zero @ A )
% 0.31/0.47 = X )
% 0.31/0.47 = ( X
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_reorient
% 0.31/0.47 thf(fact_58_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel146912293up_add @ A )
% 0.31/0.47 => ! [A2: A,C: A,B: A] :
% 0.31/0.47 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B )
% 0.31/0.47 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % cancel_ab_semigroup_add_class.diff_right_commute
% 0.31/0.47 thf(fact_59_diff__eq__diff__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A2: A,B: A,C: A,D2: A] :
% 0.31/0.47 ( ( ( minus_minus @ A @ A2 @ B )
% 0.31/0.47 = ( minus_minus @ A @ C @ D2 ) )
% 0.31/0.47 => ( ( A2 = B )
% 0.31/0.47 = ( C = D2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_eq_diff_eq
% 0.31/0.47 thf(fact_60_inner__commute,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( inner_real_inner @ A )
% 0.31/0.47 => ( ( inner_780170721_inner @ A )
% 0.31/0.47 = ( ^ [X3: A,Y4: A] : ( inner_780170721_inner @ A @ Y4 @ X3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inner_commute
% 0.31/0.47 thf(fact_61_zero__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_le
% 0.31/0.47 thf(fact_62_zero__less__iff__neq__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [N: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.31/0.47 = ( N
% 0.31/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_less_iff_neq_zero
% 0.31/0.47 thf(fact_63_gr__implies__not__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [M: A,N: A] :
% 0.31/0.47 ( ( ord_less @ A @ M @ N )
% 0.31/0.47 => ( N
% 0.31/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % gr_implies_not_zero
% 0.31/0.47 thf(fact_64_not__less__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [N: A] :
% 0.31/0.47 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_less_zero
% 0.31/0.47 thf(fact_65_gr__zeroI,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [N: A] :
% 0.31/0.47 ( ( N
% 0.31/0.47 != ( zero_zero @ A ) )
% 0.31/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % gr_zeroI
% 0.31/0.47 thf(fact_66_eq__iff__diff__eq__0,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.31/0.47 = ( ^ [A4: A,B3: A] :
% 0.31/0.47 ( ( minus_minus @ A @ A4 @ B3 )
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % eq_iff_diff_eq_0
% 0.31/0.47 thf(fact_67_diff__eq__diff__less__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordered_ab_group_add @ A )
% 0.31/0.47 => ! [A2: A,B: A,C: A,D2: A] :
% 0.31/0.47 ( ( ( minus_minus @ A @ A2 @ B )
% 0.31/0.47 = ( minus_minus @ A @ C @ D2 ) )
% 0.31/0.47 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.31/0.47 = ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_eq_diff_less_eq
% 0.31/0.47 thf(fact_68_diff__right__mono,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordered_ab_group_add @ A )
% 0.31/0.47 => ! [A2: A,B: A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.31/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_right_mono
% 0.31/0.47 thf(fact_69_diff__left__mono,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordered_ab_group_add @ A )
% 0.31/0.47 => ! [B: A,A2: A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ B @ A2 )
% 0.31/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_left_mono
% 0.31/0.47 thf(fact_70_diff__mono,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordered_ab_group_add @ A )
% 0.31/0.47 => ! [A2: A,B: A,D2: A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.31/0.47 => ( ( ord_less_eq @ A @ D2 @ C )
% 0.31/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_mono
% 0.31/0.47 thf(fact_71_diff__strict__right__mono,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordered_ab_group_add @ A )
% 0.31/0.47 => ! [A2: A,B: A,C: A] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ B )
% 0.31/0.47 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_strict_right_mono
% 0.31/0.47 thf(fact_72_diff__strict__left__mono,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ! [B: A,A2: A,C: A] :
% 0.31/0.48 ( ( ord_less @ A @ B @ A2 )
% 0.31/0.48 => ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % diff_strict_left_mono
% 0.31/0.48 thf(fact_73_diff__eq__diff__less,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ! [A2: A,B: A,C: A,D2: A] :
% 0.31/0.48 ( ( ( minus_minus @ A @ A2 @ B )
% 0.31/0.48 = ( minus_minus @ A @ C @ D2 ) )
% 0.31/0.48 => ( ( ord_less @ A @ A2 @ B )
% 0.31/0.48 = ( ord_less @ A @ C @ D2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % diff_eq_diff_less
% 0.31/0.48 thf(fact_74_diff__strict__mono,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ! [A2: A,B: A,D2: A,C: A] :
% 0.31/0.48 ( ( ord_less @ A @ A2 @ B )
% 0.31/0.48 => ( ( ord_less @ A @ D2 @ C )
% 0.31/0.48 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D2 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % diff_strict_mono
% 0.31/0.48 thf(fact_75_inner__diff__right,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( inner_real_inner @ A )
% 0.31/0.48 => ! [X: A,Y: A,Z3: A] :
% 0.31/0.48 ( ( inner_780170721_inner @ A @ X @ ( minus_minus @ A @ Y @ Z3 ) )
% 0.31/0.48 = ( minus_minus @ real @ ( inner_780170721_inner @ A @ X @ Y ) @ ( inner_780170721_inner @ A @ X @ Z3 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % inner_diff_right
% 0.31/0.48 thf(fact_76_inner__diff__left,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( inner_real_inner @ A )
% 0.31/0.48 => ! [X: A,Y: A,Z3: A] :
% 0.31/0.48 ( ( inner_780170721_inner @ A @ ( minus_minus @ A @ X @ Y ) @ Z3 )
% 0.31/0.48 = ( minus_minus @ real @ ( inner_780170721_inner @ A @ X @ Z3 ) @ ( inner_780170721_inner @ A @ Y @ Z3 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % inner_diff_left
% 0.31/0.48 thf(fact_77_mem__cball__leI,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( real_V2090557954_space @ A )
% 0.31/0.48 => ! [X: A,Y: A,E: real,F: real] :
% 0.31/0.48 ( ( member @ A @ X @ ( elemen321786957_cball @ A @ Y @ E ) )
% 0.31/0.48 => ( ( ord_less_eq @ real @ E @ F )
% 0.31/0.48 => ( member @ A @ X @ ( elemen321786957_cball @ A @ Y @ F ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % mem_cball_leI
% 0.31/0.48 thf(fact_78_le__iff__diff__le__0,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ( ( ord_less_eq @ A )
% 0.31/0.48 = ( ^ [A4: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % le_iff_diff_le_0
% 0.31/0.48 thf(fact_79_less__iff__diff__less__0,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ( ( ord_less @ A )
% 0.31/0.48 = ( ^ [A4: A,B3: A] : ( ord_less @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % less_iff_diff_less_0
% 0.31/0.48 thf(fact_80_inner__ge__zero,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( inner_real_inner @ A )
% 0.31/0.48 => ! [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ A @ X @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % inner_ge_zero
% 0.31/0.48 thf(fact_81_fixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( member @ a @ X @ x )
% 0.31/0.48 => ( ( ( f @ X )
% 0.31/0.48 = ( zero_zero @ a ) )
% 0.31/0.48 => ( ( period1153813292period @ a @ f @ x @ X )
% 0.31/0.48 = ( zero_zero @ real ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % fixed_point_imp_closed_orbit_period_zero(2)
% 0.31/0.48 thf(fact_82_that,axiom,
% 0.31/0.48 ! [D3: real,B4: real] :
% 0.31/0.48 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
% 0.31/0.48 => ( ( ord_less @ real @ ( zero_zero @ real ) @ B4 )
% 0.31/0.48 => ( ! [X4: a,Y6: a] :
% 0.31/0.48 ( ( member @ a @ X4 @ ( line_closed_segment @ a @ a2 @ b ) )
% 0.31/0.48 => ( ( ord_less_eq @ real @ ( real_V2000881966t_dist @ a @ X4 @ Y6 ) @ D3 )
% 0.31/0.48 => ( ord_less_eq @ real @ B4 @ ( inner_780170721_inner @ a @ ( f @ Y6 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ) ) )
% 0.31/0.48 => thesis ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % that
% 0.31/0.48 thf(fact_83_inner__nonneg__nonneg,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordere890947078_space @ A )
% 0.31/0.48 => ! [A2: A,B: A] :
% 0.31/0.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.31/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 0.31/0.48 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ A @ A2 @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % inner_nonneg_nonneg
% 0.31/0.48 thf(fact_84_in__closed__segment__iff__rot,axiom,
% 0.31/0.48 ! [A2: a,B: a,X: a] :
% 0.31/0.48 ( ( A2 != B )
% 0.31/0.48 => ( ( member @ a @ X @ ( line_closed_segment @ a @ A2 @ B ) )
% 0.31/0.48 = ( ( ( inner_780170721_inner @ a @ ( minus_minus @ a @ X @ A2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ B @ A2 ) ) )
% 0.31/0.48 = ( zero_zero @ real ) )
% 0.31/0.48 & ( member @ real @ ( inner_780170721_inner @ a @ X @ ( minus_minus @ a @ B @ A2 ) ) @ ( set_or331188842AtMost @ real @ ( inner_780170721_inner @ a @ A2 @ ( minus_minus @ a @ B @ A2 ) ) @ ( inner_780170721_inner @ a @ B @ ( minus_minus @ a @ B @ A2 ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % in_closed_segment_iff_rot
% 0.31/0.48 thf(fact_85_sectionD_I5_J,axiom,
% 0.31/0.48 ! [S2: a > real,Ds: a > ( bounde2145540817linfun @ a @ real ),S3: set @ a] :
% 0.31/0.48 ( ( reacha1084862253ection @ a @ f @ x @ S2 @ Ds @ S3 )
% 0.31/0.48 => ( ord_less_eq @ ( set @ a ) @ S3 @ x ) ) ).
% 0.31/0.48
% 0.31/0.48 % sectionD(5)
% 0.31/0.48 thf(fact_86_transversal__segment__exists,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( member @ a @ X @ x )
% 0.31/0.48 => ( ( ( f @ X )
% 0.31/0.48 != ( zero_zero @ a ) )
% 0.31/0.48 => ~ ! [A5: a,B5: a] :
% 0.31/0.48 ( ( member @ a @ X @ ( line_open_segment @ a @ A5 @ B5 ) )
% 0.31/0.48 => ~ ( poinca272511729egment @ a @ f @ x @ A5 @ B5 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % transversal_segment_exists
% 0.31/0.48 thf(fact_87_invariant__iff__compl__invariant,axiom,
% 0.31/0.48 ! [M2: set @ a] :
% 0.31/0.48 ( ( auto_ll_on_invariant @ a @ f @ x @ M2 )
% 0.31/0.48 = ( auto_ll_on_invariant @ a @ f @ x @ ( minus_minus @ ( set @ a ) @ x @ M2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % invariant_iff_compl_invariant
% 0.31/0.48 thf(fact_88_cball__eq__cball__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( euclid925273238_space @ A )
% 0.31/0.48 => ! [X: A,D2: real,Y: A,E: real] :
% 0.31/0.48 ( ( ( elemen321786957_cball @ A @ X @ D2 )
% 0.31/0.48 = ( elemen321786957_cball @ A @ Y @ E ) )
% 0.31/0.48 = ( ( ( ord_less @ real @ D2 @ ( zero_zero @ real ) )
% 0.31/0.48 & ( ord_less @ real @ E @ ( zero_zero @ real ) ) )
% 0.31/0.48 | ( ( X = Y )
% 0.31/0.48 & ( D2 = E ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % cball_eq_cball_iff
% 0.31/0.48 thf(fact_89_Bolzano,axiom,
% 0.31/0.48 ! [A2: real,B: real,P: real > real > $o] :
% 0.31/0.48 ( ( ord_less_eq @ real @ A2 @ B )
% 0.31/0.48 => ( ! [A5: real,B5: real,C2: real] :
% 0.31/0.48 ( ( P @ A5 @ B5 )
% 0.31/0.48 => ( ( P @ B5 @ C2 )
% 0.31/0.48 => ( ( ord_less_eq @ real @ A5 @ B5 )
% 0.31/0.48 => ( ( ord_less_eq @ real @ B5 @ C2 )
% 0.31/0.48 => ( P @ A5 @ C2 ) ) ) ) )
% 0.31/0.48 => ( ! [X4: real] :
% 0.31/0.48 ( ( ord_less_eq @ real @ A2 @ X4 )
% 0.31/0.48 => ( ( ord_less_eq @ real @ X4 @ B )
% 0.31/0.48 => ? [D4: real] :
% 0.31/0.48 ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
% 0.31/0.48 & ! [A5: real,B5: real] :
% 0.31/0.48 ( ( ( ord_less_eq @ real @ A5 @ X4 )
% 0.31/0.48 & ( ord_less_eq @ real @ X4 @ B5 )
% 0.31/0.48 & ( ord_less @ real @ ( minus_minus @ real @ B5 @ A5 ) @ D4 ) )
% 0.31/0.48 => ( P @ A5 @ B5 ) ) ) ) )
% 0.31/0.48 => ( P @ A2 @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % Bolzano
% 0.31/0.48 thf(fact_90_mem__cball,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( real_V2090557954_space @ A )
% 0.31/0.48 => ! [Y: A,X: A,E: real] :
% 0.31/0.48 ( ( member @ A @ Y @ ( elemen321786957_cball @ A @ X @ E ) )
% 0.31/0.48 = ( ord_less_eq @ real @ ( real_V2000881966t_dist @ A @ X @ Y ) @ E ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % mem_cball
% 0.31/0.48 thf(fact_91_perfect__choose__dist,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ( real_V2090557954_space @ A )
% 0.31/0.48 & ( topolo890362671_space @ A ) )
% 0.31/0.48 => ! [R: real,X: A] :
% 0.31/0.48 ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
% 0.31/0.48 => ? [A5: A] :
% 0.31/0.48 ( ( A5 != X )
% 0.31/0.48 & ( ord_less @ real @ ( real_V2000881966t_dist @ A @ A5 @ X ) @ R ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % perfect_choose_dist
% 0.31/0.48 thf(fact_92_interval__inner__leI_I2_J,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordere890947078_space @ A )
% 0.31/0.48 => ! [X: A,A2: A,B: A,I: A] :
% 0.31/0.48 ( ( member @ A @ X @ ( set_or331188842AtMost @ A @ A2 @ B ) )
% 0.31/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ I )
% 0.31/0.48 => ( ord_less_eq @ real @ ( inner_780170721_inner @ A @ X @ I ) @ ( inner_780170721_inner @ A @ B @ I ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % interval_inner_leI(2)
% 0.31/0.48 thf(fact_93_interval__inner__leI_I1_J,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordere890947078_space @ A )
% 0.31/0.48 => ! [X: A,A2: A,B: A,I: A] :
% 0.31/0.48 ( ( member @ A @ X @ ( set_or331188842AtMost @ A @ A2 @ B ) )
% 0.31/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ I )
% 0.31/0.48 => ( ord_less_eq @ real @ ( inner_780170721_inner @ A @ A2 @ I ) @ ( inner_780170721_inner @ A @ X @ I ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % interval_inner_leI(1)
% 0.31/0.48 thf(fact_94_eucl__less__le__not__le,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordere890947078_space @ A )
% 0.31/0.48 => ( ( ord_less @ A )
% 0.31/0.48 = ( ^ [X3: A,Y4: A] :
% 0.31/0.48 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.31/0.48 & ~ ( ord_less_eq @ A @ Y4 @ X3 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % eucl_less_le_not_le
% 0.31/0.48 thf(fact_95_invariant__def,axiom,
% 0.31/0.48 ! [M2: set @ a] :
% 0.31/0.48 ( ( auto_ll_on_invariant @ a @ f @ x @ M2 )
% 0.31/0.48 = ( ! [X3: a] :
% 0.31/0.48 ( ( member @ a @ X3 @ M2 )
% 0.31/0.48 => ( oDE_au1039603466rapped @ a @ f @ x @ X3 @ M2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % invariant_def
% 0.31/0.48 thf(fact_96_periodic__orbit__period_I1_J,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( period862636932_orbit @ a @ f @ x @ X )
% 0.31/0.48 => ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % periodic_orbit_period(1)
% 0.31/0.48 thf(fact_97_zero__less__dist__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( real_V2090557954_space @ A )
% 0.31/0.48 => ! [X: A,Y: A] :
% 0.31/0.48 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X @ Y ) )
% 0.31/0.48 = ( X != Y ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % zero_less_dist_iff
% 0.31/0.48 thf(fact_98_dist__le__zero__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( real_V2090557954_space @ A )
% 0.31/0.48 => ! [X: A,Y: A] :
% 0.31/0.48 ( ( ord_less_eq @ real @ ( real_V2000881966t_dist @ A @ X @ Y ) @ ( zero_zero @ real ) )
% 0.31/0.48 = ( X = Y ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dist_le_zero_iff
% 0.31/0.48 thf(fact_99_atLeastatMost__subset__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( preorder @ A )
% 0.31/0.48 => ! [A2: A,B: A,C: A,D2: A] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A2 @ B ) @ ( set_or331188842AtMost @ A @ C @ D2 ) )
% 0.31/0.48 = ( ~ ( ord_less_eq @ A @ A2 @ B )
% 0.31/0.48 | ( ( ord_less_eq @ A @ C @ A2 )
% 0.31/0.48 & ( ord_less_eq @ A @ B @ D2 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % atLeastatMost_subset_iff
% 0.31/0.48 thf(fact_100_dist__self,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( real_V2090557954_space @ A )
% 0.31/0.48 => ! [X: A] :
% 0.31/0.48 ( ( real_V2000881966t_dist @ A @ X @ X )
% 0.31/0.48 = ( zero_zero @ real ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dist_self
% 0.31/0.48 thf(fact_101_atLeastAtMost__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ord @ A )
% 0.31/0.48 => ! [I: A,L: A,U2: A] :
% 0.31/0.48 ( ( member @ A @ I @ ( set_or331188842AtMost @ A @ L @ U2 ) )
% 0.31/0.48 = ( ( ord_less_eq @ A @ L @ I )
% 0.31/0.48 & ( ord_less_eq @ A @ I @ U2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % atLeastAtMost_iff
% 0.31/0.48 thf(fact_102_Icc__eq__Icc,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( order @ A )
% 0.31/0.48 => ! [L: A,H: A,L2: A,H2: A] :
% 0.31/0.48 ( ( ( set_or331188842AtMost @ A @ L @ H )
% 0.31/0.48 = ( set_or331188842AtMost @ A @ L2 @ H2 ) )
% 0.31/0.48 = ( ( ( L = L2 )
% 0.31/0.48 & ( H = H2 ) )
% 0.31/0.48 | ( ~ ( ord_less_eq @ A @ L @ H )
% 0.31/0.48 & ~ ( ord_less_eq @ A @ L2 @ H2 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % Icc_eq_Icc
% 0.31/0.48 thf(fact_103_dist__eq__0__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( real_V2090557954_space @ A )
% 0.31/0.48 => ! [X: A,Y: A] :
% 0.31/0.48 ( ( ( real_V2000881966t_dist @ A @ X @ Y )
% 0.31/0.48 = ( zero_zero @ real ) )
% 0.31/0.48 = ( X = Y ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dist_eq_0_iff
% 0.31/0.48 thf(fact_104_dist__commute,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( real_V2090557954_space @ A )
% 0.31/0.48 => ( ( real_V2000881966t_dist @ A )
% 0.31/0.48 = ( ^ [X3: A,Y4: A] : ( real_V2000881966t_dist @ A @ Y4 @ X3 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dist_commute
% 0.31/0.48 thf(fact_105_dist__commute__lessI,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( real_V2090557954_space @ A )
% 0.31/0.48 => ! [Y: A,X: A,E: real] :
% 0.31/0.48 ( ( ord_less @ real @ ( real_V2000881966t_dist @ A @ Y @ X ) @ E )
% 0.31/0.48 => ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X @ Y ) @ E ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dist_commute_lessI
% 0.31/0.48 thf(fact_106_atLeastatMost__psubset__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( preorder @ A )
% 0.31/0.48 => ! [A2: A,B: A,C: A,D2: A] :
% 0.31/0.48 ( ( ord_less @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A2 @ B ) @ ( set_or331188842AtMost @ A @ C @ D2 ) )
% 0.31/0.48 = ( ( ~ ( ord_less_eq @ A @ A2 @ B )
% 0.31/0.48 | ( ( ord_less_eq @ A @ C @ A2 )
% 0.31/0.48 & ( ord_less_eq @ A @ B @ D2 )
% 0.31/0.48 & ( ( ord_less @ A @ C @ A2 )
% 0.31/0.48 | ( ord_less @ A @ B @ D2 ) ) ) )
% 0.31/0.48 & ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % atLeastatMost_psubset_iff
% 0.31/0.48 thf(fact_107_zero__le__dist,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( real_V2090557954_space @ A )
% 0.31/0.48 => ! [X: A,Y: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X @ Y ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % zero_le_dist
% 0.31/0.48 thf(fact_108_dist__not__less__zero,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( real_V2090557954_space @ A )
% 0.31/0.48 => ! [X: A,Y: A] :
% 0.31/0.48 ~ ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X @ Y ) @ ( zero_zero @ real ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dist_not_less_zero
% 0.31/0.48 thf(fact_109_dist__pos__lt,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( real_V2090557954_space @ A )
% 0.31/0.48 => ! [X: A,Y: A] :
% 0.31/0.48 ( ( X != Y )
% 0.31/0.48 => ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X @ Y ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dist_pos_lt
% 0.31/0.48 thf(fact_110_rev_Operiodic__orbit__period_I1_J,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.31/0.48 => ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.periodic_orbit_period(1)
% 0.31/0.48 thf(fact_111_periodic__orbit__def,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( period862636932_orbit @ a @ f @ x @ X )
% 0.31/0.48 = ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.31/0.48 & ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % periodic_orbit_def
% 0.31/0.48 thf(fact_112_rev_Ofixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( member @ a @ X @ x )
% 0.31/0.48 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 0.31/0.48 = ( zero_zero @ a ) )
% 0.31/0.48 => ( ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.31/0.48 = ( zero_zero @ real ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.fixed_point_imp_closed_orbit_period_zero(2)
% 0.31/0.48 thf(fact_113_closed__orbit__period__zero__fixed__point,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.31/0.48 => ( ( ( period1153813292period @ a @ f @ x @ X )
% 0.31/0.48 = ( zero_zero @ real ) )
% 0.31/0.48 => ( ( f @ X )
% 0.31/0.48 = ( zero_zero @ a ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % closed_orbit_period_zero_fixed_point
% 0.31/0.48 thf(fact_114_closed__orbit__period__nonneg,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.31/0.48 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % closed_orbit_period_nonneg
% 0.31/0.48 thf(fact_115_closed__orbit__in__domain,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.31/0.48 => ( member @ a @ X @ x ) ) ).
% 0.31/0.48
% 0.31/0.48 % closed_orbit_in_domain
% 0.31/0.48 thf(fact_116_add_Oinverse__inverse,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( group_add @ A )
% 0.31/0.48 => ! [A2: A] :
% 0.31/0.48 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 0.31/0.48 = A2 ) ) ).
% 0.31/0.48
% 0.31/0.48 % add.inverse_inverse
% 0.31/0.48 thf(fact_117_neg__equal__iff__equal,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( group_add @ A )
% 0.31/0.48 => ! [A2: A,B: A] :
% 0.31/0.48 ( ( ( uminus_uminus @ A @ A2 )
% 0.31/0.48 = ( uminus_uminus @ A @ B ) )
% 0.31/0.48 = ( A2 = B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % neg_equal_iff_equal
% 0.31/0.48 thf(fact_118_rev_Oclosed__orbit__eq__rev,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.31/0.48 = ( period385816147_orbit @ a @ f @ x @ X ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.closed_orbit_eq_rev
% 0.31/0.48 thf(fact_119_rev_Oclosed__orbit__in__domain,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.31/0.48 => ( member @ a @ X @ x ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.closed_orbit_in_domain
% 0.31/0.48 thf(fact_120_rev_Oinvariant__eq__rev,axiom,
% 0.31/0.48 ! [M2: set @ a] :
% 0.31/0.48 ( ( auto_ll_on_invariant @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ M2 )
% 0.31/0.48 = ( auto_ll_on_invariant @ a @ f @ x @ M2 ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.invariant_eq_rev
% 0.31/0.48 thf(fact_121_fixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( member @ a @ X @ x )
% 0.31/0.48 => ( ( ( f @ X )
% 0.31/0.48 = ( zero_zero @ a ) )
% 0.31/0.48 => ( period385816147_orbit @ a @ f @ x @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % fixed_point_imp_closed_orbit_period_zero(1)
% 0.31/0.48 thf(fact_122_rev_Ofixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( member @ a @ X @ x )
% 0.31/0.48 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 0.31/0.48 = ( zero_zero @ a ) )
% 0.31/0.48 => ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.fixed_point_imp_closed_orbit_period_zero(1)
% 0.31/0.48 thf(fact_123_rev_OsectionD_I5_J,axiom,
% 0.31/0.48 ! [S2: a > real,Ds: a > ( bounde2145540817linfun @ a @ real ),S3: set @ a] :
% 0.31/0.48 ( ( reacha1084862253ection @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ S2 @ Ds @ S3 )
% 0.31/0.48 => ( ord_less_eq @ ( set @ a ) @ S3 @ x ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.sectionD(5)
% 0.31/0.48 thf(fact_124_rev_Oinvariant__iff__compl__invariant,axiom,
% 0.31/0.48 ! [M2: set @ a] :
% 0.31/0.48 ( ( auto_ll_on_invariant @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ M2 )
% 0.31/0.48 = ( auto_ll_on_invariant @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( minus_minus @ ( set @ a ) @ x @ M2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.invariant_iff_compl_invariant
% 0.31/0.48 thf(fact_125_closed__orbit__periodic,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.31/0.48 => ( ( ( f @ X )
% 0.31/0.48 != ( zero_zero @ a ) )
% 0.31/0.48 => ( period862636932_orbit @ a @ f @ x @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % closed_orbit_periodic
% 0.31/0.48 thf(fact_126_rev_Oinvariant__def,axiom,
% 0.31/0.48 ! [M2: set @ a] :
% 0.31/0.48 ( ( auto_ll_on_invariant @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ M2 )
% 0.31/0.48 = ( ! [X3: a] :
% 0.31/0.48 ( ( member @ a @ X3 @ M2 )
% 0.31/0.48 => ( oDE_au1039603466rapped @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X3 @ M2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.invariant_def
% 0.31/0.48 thf(fact_127_rev_Oclosed__orbit__periodic,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.31/0.48 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 0.31/0.48 != ( zero_zero @ a ) )
% 0.31/0.48 => ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.closed_orbit_periodic
% 0.31/0.48 thf(fact_128_add_Oinverse__neutral,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( group_add @ A )
% 0.31/0.48 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 0.31/0.48 = ( zero_zero @ A ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % add.inverse_neutral
% 0.31/0.48 thf(fact_129_neg__0__equal__iff__equal,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( group_add @ A )
% 0.31/0.48 => ! [A2: A] :
% 0.31/0.48 ( ( ( zero_zero @ A )
% 0.31/0.48 = ( uminus_uminus @ A @ A2 ) )
% 0.31/0.48 = ( ( zero_zero @ A )
% 0.31/0.48 = A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % neg_0_equal_iff_equal
% 0.31/0.48 thf(fact_130_neg__equal__0__iff__equal,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( group_add @ A )
% 0.31/0.48 => ! [A2: A] :
% 0.31/0.48 ( ( ( uminus_uminus @ A @ A2 )
% 0.31/0.48 = ( zero_zero @ A ) )
% 0.31/0.48 = ( A2
% 0.31/0.48 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % neg_equal_0_iff_equal
% 0.31/0.48 thf(fact_131_equal__neg__zero,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( linord219039673up_add @ A )
% 0.31/0.48 => ! [A2: A] :
% 0.31/0.48 ( ( A2
% 0.31/0.48 = ( uminus_uminus @ A @ A2 ) )
% 0.31/0.48 = ( A2
% 0.31/0.48 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % equal_neg_zero
% 0.31/0.48 thf(fact_132_neg__equal__zero,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( linord219039673up_add @ A )
% 0.31/0.48 => ! [A2: A] :
% 0.31/0.48 ( ( ( uminus_uminus @ A @ A2 )
% 0.31/0.48 = A2 )
% 0.31/0.48 = ( A2
% 0.31/0.48 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % neg_equal_zero
% 0.31/0.48 thf(fact_133_neg__le__iff__le,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ! [B: A,A2: A] :
% 0.31/0.48 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B ) @ ( uminus_uminus @ A @ A2 ) )
% 0.31/0.48 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % neg_le_iff_le
% 0.31/0.48 thf(fact_134_neg__less__iff__less,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ! [B: A,A2: A] :
% 0.31/0.48 ( ( ord_less @ A @ ( uminus_uminus @ A @ B ) @ ( uminus_uminus @ A @ A2 ) )
% 0.31/0.48 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % neg_less_iff_less
% 0.31/0.48 thf(fact_135_minus__diff__eq,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( group_add @ A )
% 0.31/0.48 => ! [A2: A,B: A] :
% 0.31/0.48 ( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B ) )
% 0.31/0.48 = ( minus_minus @ A @ B @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % minus_diff_eq
% 0.31/0.48 thf(fact_136_inner__minus__left,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( inner_real_inner @ A )
% 0.31/0.48 => ! [X: A,Y: A] :
% 0.31/0.48 ( ( inner_780170721_inner @ A @ ( uminus_uminus @ A @ X ) @ Y )
% 0.31/0.48 = ( uminus_uminus @ real @ ( inner_780170721_inner @ A @ X @ Y ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % inner_minus_left
% 0.31/0.48 thf(fact_137_inner__minus__right,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( inner_real_inner @ A )
% 0.31/0.48 => ! [X: A,Y: A] :
% 0.31/0.48 ( ( inner_780170721_inner @ A @ X @ ( uminus_uminus @ A @ Y ) )
% 0.31/0.48 = ( uminus_uminus @ real @ ( inner_780170721_inner @ A @ X @ Y ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % inner_minus_right
% 0.31/0.48 thf(fact_138_rev_Oclosed__orbit__period__nonneg,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.31/0.48 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.closed_orbit_period_nonneg
% 0.31/0.48 thf(fact_139_rev_Oclosed__orbit__period__zero__fixed__point,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.31/0.48 => ( ( ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.31/0.48 = ( zero_zero @ real ) )
% 0.31/0.48 => ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 0.31/0.48 = ( zero_zero @ a ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.closed_orbit_period_zero_fixed_point
% 0.31/0.48 thf(fact_140_rev_Operiodic__orbit__def,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.31/0.48 = ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.31/0.48 & ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.periodic_orbit_def
% 0.31/0.48 thf(fact_141_neg__0__le__iff__le,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ! [A2: A] :
% 0.31/0.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 0.31/0.48 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % neg_0_le_iff_le
% 0.31/0.48 thf(fact_142_neg__le__0__iff__le,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ! [A2: A] :
% 0.31/0.48 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.31/0.48 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % neg_le_0_iff_le
% 0.31/0.48 thf(fact_143_less__eq__neg__nonpos,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( linord219039673up_add @ A )
% 0.31/0.48 => ! [A2: A] :
% 0.31/0.48 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 0.31/0.48 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % less_eq_neg_nonpos
% 0.31/0.48 thf(fact_144_neg__less__eq__nonneg,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( linord219039673up_add @ A )
% 0.31/0.48 => ! [A2: A] :
% 0.31/0.48 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 0.31/0.48 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % neg_less_eq_nonneg
% 0.31/0.48 thf(fact_145_less__neg__neg,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( linord219039673up_add @ A )
% 0.31/0.48 => ! [A2: A] :
% 0.31/0.48 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 0.31/0.48 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % less_neg_neg
% 0.31/0.48 thf(fact_146_neg__less__pos,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( linord219039673up_add @ A )
% 0.31/0.48 => ! [A2: A] :
% 0.31/0.48 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 0.31/0.48 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % neg_less_pos
% 0.31/0.48 thf(fact_147_neg__0__less__iff__less,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ! [A2: A] :
% 0.31/0.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 0.31/0.48 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % neg_0_less_iff_less
% 0.31/0.48 thf(fact_148_neg__less__0__iff__less,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ! [A2: A] :
% 0.31/0.48 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.31/0.48 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % neg_less_0_iff_less
% 0.31/0.48 thf(fact_149_diff__0,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( group_add @ A )
% 0.31/0.48 => ! [A2: A] :
% 0.31/0.48 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 0.31/0.48 = ( uminus_uminus @ A @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % diff_0
% 0.31/0.48 thf(fact_150_le__imp__neg__le,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ! [A2: A,B: A] :
% 0.31/0.48 ( ( ord_less_eq @ A @ A2 @ B )
% 0.31/0.48 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % le_imp_neg_le
% 0.31/0.48 thf(fact_151_minus__le__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ! [A2: A,B: A] :
% 0.31/0.48 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B )
% 0.31/0.48 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B ) @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % minus_le_iff
% 0.31/0.48 thf(fact_152_le__minus__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ! [A2: A,B: A] :
% 0.31/0.48 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B ) )
% 0.31/0.48 = ( ord_less_eq @ A @ B @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % le_minus_iff
% 0.31/0.48 thf(fact_153_minus__less__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ! [A2: A,B: A] :
% 0.31/0.48 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B )
% 0.31/0.48 = ( ord_less @ A @ ( uminus_uminus @ A @ B ) @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % minus_less_iff
% 0.31/0.48 thf(fact_154_less__minus__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ! [A2: A,B: A] :
% 0.31/0.48 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B ) )
% 0.31/0.48 = ( ord_less @ A @ B @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % less_minus_iff
% 0.31/0.48 thf(fact_155_minus__diff__commute,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ab_group_add @ A )
% 0.31/0.48 => ! [B: A,A2: A] :
% 0.31/0.48 ( ( minus_minus @ A @ ( uminus_uminus @ A @ B ) @ A2 )
% 0.31/0.48 = ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % minus_diff_commute
% 0.31/0.48 thf(fact_156_equation__minus__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( group_add @ A )
% 0.31/0.48 => ! [A2: A,B: A] :
% 0.31/0.48 ( ( A2
% 0.31/0.48 = ( uminus_uminus @ A @ B ) )
% 0.31/0.48 = ( B
% 0.31/0.48 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % equation_minus_iff
% 0.31/0.48 thf(fact_157_minus__equation__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( group_add @ A )
% 0.31/0.48 => ! [A2: A,B: A] :
% 0.31/0.48 ( ( ( uminus_uminus @ A @ A2 )
% 0.31/0.48 = B )
% 0.31/0.48 = ( ( uminus_uminus @ A @ B )
% 0.31/0.48 = A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % minus_equation_iff
% 0.31/0.48 thf(fact_158_verit__minus__simplify_I3_J,axiom,
% 0.31/0.48 ! [B2: $tType] :
% 0.31/0.48 ( ( group_add @ B2 )
% 0.31/0.48 => ! [B: B2] :
% 0.31/0.48 ( ( minus_minus @ B2 @ ( zero_zero @ B2 ) @ B )
% 0.31/0.48 = ( uminus_uminus @ B2 @ B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % verit_minus_simplify(3)
% 0.31/0.48 thf(fact_159_psubsetI,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.31/0.48 => ( ( A3 != B6 )
% 0.31/0.48 => ( ord_less @ ( set @ A ) @ A3 @ B6 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % psubsetI
% 0.31/0.48 thf(fact_160_uminus__apply,axiom,
% 0.31/0.48 ! [B2: $tType,A: $tType] :
% 0.31/0.48 ( ( uminus @ B2 )
% 0.31/0.48 => ( ( uminus_uminus @ ( A > B2 ) )
% 0.31/0.48 = ( ^ [A6: A > B2,X3: A] : ( uminus_uminus @ B2 @ ( A6 @ X3 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % uminus_apply
% 0.31/0.48 thf(fact_161_compl__le__compl__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( boolean_algebra @ A )
% 0.31/0.48 => ! [X: A,Y: A] :
% 0.31/0.48 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
% 0.31/0.48 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % compl_le_compl_iff
% 0.31/0.48 thf(fact_162_rot__diff__commute,axiom,
% 0.31/0.48 ! [B: a,A2: a] :
% 0.31/0.48 ( ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ B @ A2 ) )
% 0.31/0.48 = ( uminus_uminus @ a @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ A2 @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rot_diff_commute
% 0.31/0.48 thf(fact_163_nrm__reverse,axiom,
% 0.31/0.48 ! [A2: a,X: a,Y: a] :
% 0.31/0.48 ( ( inner_780170721_inner @ a @ A2 @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) )
% 0.31/0.48 = ( inner_780170721_inner @ a @ ( uminus_uminus @ a @ A2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ Y @ X ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % nrm_reverse
% 0.31/0.48 thf(fact_164_verit__minus__simplify_I4_J,axiom,
% 0.31/0.48 ! [B2: $tType] :
% 0.31/0.48 ( ( group_add @ B2 )
% 0.31/0.48 => ! [B: B2] :
% 0.31/0.48 ( ( uminus_uminus @ B2 @ ( uminus_uminus @ B2 @ B ) )
% 0.31/0.48 = B ) ) ).
% 0.31/0.48
% 0.31/0.48 % verit_minus_simplify(4)
% 0.31/0.48 thf(fact_165_double__compl,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( boolean_algebra @ A )
% 0.31/0.48 => ! [X: A] :
% 0.31/0.48 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
% 0.31/0.48 = X ) ) ).
% 0.31/0.48
% 0.31/0.48 % double_compl
% 0.31/0.48 thf(fact_166_compl__eq__compl__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( boolean_algebra @ A )
% 0.31/0.48 => ! [X: A,Y: A] :
% 0.31/0.48 ( ( ( uminus_uminus @ A @ X )
% 0.31/0.48 = ( uminus_uminus @ A @ Y ) )
% 0.31/0.48 = ( X = Y ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % compl_eq_compl_iff
% 0.31/0.48 thf(fact_167_minus__apply,axiom,
% 0.31/0.48 ! [B2: $tType,A: $tType] :
% 0.31/0.48 ( ( minus @ B2 )
% 0.31/0.48 => ( ( minus_minus @ ( A > B2 ) )
% 0.31/0.48 = ( ^ [A6: A > B2,B7: A > B2,X3: A] : ( minus_minus @ B2 @ ( A6 @ X3 ) @ ( B7 @ X3 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % minus_apply
% 0.31/0.48 thf(fact_168_Compl__subset__Compl__iff,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) )
% 0.31/0.48 = ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ).
% 0.31/0.48
% 0.31/0.48 % Compl_subset_Compl_iff
% 0.31/0.48 thf(fact_169_Compl__anti__mono,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.31/0.48 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % Compl_anti_mono
% 0.31/0.48 thf(fact_170_subset__antisym,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.31/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
% 0.31/0.48 => ( A3 = B6 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_antisym
% 0.31/0.48 thf(fact_171_subsetI,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ! [X4: A] :
% 0.31/0.48 ( ( member @ A @ X4 @ A3 )
% 0.31/0.48 => ( member @ A @ X4 @ B6 ) )
% 0.31/0.48 => ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% 0.31/0.48
% 0.31/0.48 % subsetI
% 0.31/0.48 thf(fact_172_Diff__idemp,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ B6 )
% 0.31/0.48 = ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ).
% 0.31/0.48
% 0.31/0.48 % Diff_idemp
% 0.31/0.48 thf(fact_173_Diff__iff,axiom,
% 0.31/0.48 ! [A: $tType,C: A,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
% 0.31/0.48 = ( ( member @ A @ C @ A3 )
% 0.31/0.48 & ~ ( member @ A @ C @ B6 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % Diff_iff
% 0.31/0.48 thf(fact_174_DiffI,axiom,
% 0.31/0.48 ! [A: $tType,C: A,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( member @ A @ C @ A3 )
% 0.31/0.48 => ( ~ ( member @ A @ C @ B6 )
% 0.31/0.48 => ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % DiffI
% 0.31/0.48 thf(fact_175_rot__rot,axiom,
% 0.31/0.48 ! [V: a] :
% 0.31/0.48 ( ( poinca1750768982en_rot @ a @ ( poinca1750768982en_rot @ a @ V ) )
% 0.31/0.48 = ( uminus_uminus @ a @ V ) ) ).
% 0.31/0.48
% 0.31/0.48 % rot_rot
% 0.31/0.48 thf(fact_176_verit__la__disequality,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( linorder @ A )
% 0.31/0.48 => ! [A2: A,B: A] :
% 0.31/0.48 ( ( A2 = B )
% 0.31/0.48 | ~ ( ord_less_eq @ A @ A2 @ B )
% 0.31/0.48 | ~ ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % verit_la_disequality
% 0.31/0.48 thf(fact_177_verit__comp__simplify1_I1_J,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( order @ A )
% 0.31/0.48 => ! [A2: A] :
% 0.31/0.48 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.31/0.48
% 0.31/0.48 % verit_comp_simplify1(1)
% 0.31/0.48 thf(fact_178_verit__negate__coefficient_I3_J,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ! [A2: A,B: A] :
% 0.31/0.48 ( ( A2 = B )
% 0.31/0.48 => ( ( uminus_uminus @ A @ A2 )
% 0.31/0.48 = ( uminus_uminus @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % verit_negate_coefficient(3)
% 0.31/0.48 thf(fact_179_fun__diff__def,axiom,
% 0.31/0.48 ! [B2: $tType,A: $tType] :
% 0.31/0.48 ( ( minus @ B2 )
% 0.31/0.48 => ( ( minus_minus @ ( A > B2 ) )
% 0.31/0.48 = ( ^ [A6: A > B2,B7: A > B2,X3: A] : ( minus_minus @ B2 @ ( A6 @ X3 ) @ ( B7 @ X3 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % fun_diff_def
% 0.31/0.48 thf(fact_180_Collect__mono__iff,axiom,
% 0.31/0.48 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.31/0.48 = ( ! [X3: A] :
% 0.31/0.48 ( ( P @ X3 )
% 0.31/0.48 => ( Q @ X3 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % Collect_mono_iff
% 0.31/0.48 thf(fact_181_set__eq__subset,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ^ [Y5: set @ A,Z4: set @ A] : ( Y5 = Z4 ) )
% 0.31/0.48 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.31/0.48 & ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % set_eq_subset
% 0.31/0.48 thf(fact_182_subset__trans,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A,C3: set @ A] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.31/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C3 )
% 0.31/0.48 => ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_trans
% 0.31/0.48 thf(fact_183_Collect__mono,axiom,
% 0.31/0.48 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/0.48 ( ! [X4: A] :
% 0.31/0.48 ( ( P @ X4 )
% 0.31/0.48 => ( Q @ X4 ) )
% 0.31/0.48 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % Collect_mono
% 0.31/0.48 thf(fact_184_subset__refl,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 0.31/0.48
% 0.31/0.48 % subset_refl
% 0.31/0.48 thf(fact_185_subset__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) )
% 0.31/0.48 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.31/0.48 ! [T: A] :
% 0.31/0.48 ( ( member @ A @ T @ A6 )
% 0.31/0.48 => ( member @ A @ T @ B7 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_iff
% 0.31/0.48 thf(fact_186_equalityD2,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( A3 = B6 )
% 0.31/0.48 => ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ).
% 0.31/0.48
% 0.31/0.48 % equalityD2
% 0.31/0.48 thf(fact_187_equalityD1,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( A3 = B6 )
% 0.31/0.48 => ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% 0.31/0.48
% 0.31/0.48 % equalityD1
% 0.31/0.48 thf(fact_188_subset__eq,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) )
% 0.31/0.48 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.31/0.48 ! [X3: A] :
% 0.31/0.48 ( ( member @ A @ X3 @ A6 )
% 0.31/0.48 => ( member @ A @ X3 @ B7 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_eq
% 0.31/0.48 thf(fact_189_equalityE,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( A3 = B6 )
% 0.31/0.48 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.31/0.48 => ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % equalityE
% 0.31/0.48 thf(fact_190_subsetD,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A,C: A] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.31/0.48 => ( ( member @ A @ C @ A3 )
% 0.31/0.48 => ( member @ A @ C @ B6 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subsetD
% 0.31/0.48 thf(fact_191_in__mono,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A,X: A] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.31/0.48 => ( ( member @ A @ X @ A3 )
% 0.31/0.48 => ( member @ A @ X @ B6 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % in_mono
% 0.31/0.48 thf(fact_192_DiffD2,axiom,
% 0.31/0.48 ! [A: $tType,C: A,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
% 0.31/0.48 => ~ ( member @ A @ C @ B6 ) ) ).
% 0.31/0.48
% 0.31/0.48 % DiffD2
% 0.31/0.48 thf(fact_193_DiffD1,axiom,
% 0.31/0.48 ! [A: $tType,C: A,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
% 0.31/0.48 => ( member @ A @ C @ A3 ) ) ).
% 0.31/0.48
% 0.31/0.48 % DiffD1
% 0.31/0.48 thf(fact_194_DiffE,axiom,
% 0.31/0.48 ! [A: $tType,C: A,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
% 0.31/0.48 => ~ ( ( member @ A @ C @ A3 )
% 0.31/0.48 => ( member @ A @ C @ B6 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % DiffE
% 0.31/0.48 thf(fact_195_psubset__trans,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A,C3: set @ A] :
% 0.31/0.48 ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
% 0.31/0.48 => ( ( ord_less @ ( set @ A ) @ B6 @ C3 )
% 0.31/0.48 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % psubset_trans
% 0.31/0.48 thf(fact_196_psubsetD,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A,C: A] :
% 0.31/0.48 ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
% 0.31/0.48 => ( ( member @ A @ C @ A3 )
% 0.31/0.48 => ( member @ A @ C @ B6 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % psubsetD
% 0.31/0.48 thf(fact_197_verit__comp__simplify1_I3_J,axiom,
% 0.31/0.48 ! [B2: $tType] :
% 0.31/0.48 ( ( linorder @ B2 )
% 0.31/0.48 => ! [B8: B2,A7: B2] :
% 0.31/0.48 ( ( ~ ( ord_less_eq @ B2 @ B8 @ A7 ) )
% 0.31/0.48 = ( ord_less @ B2 @ A7 @ B8 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % verit_comp_simplify1(3)
% 0.31/0.48 thf(fact_198_compl__le__swap2,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( boolean_algebra @ A )
% 0.31/0.48 => ! [Y: A,X: A] :
% 0.31/0.48 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
% 0.31/0.48 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % compl_le_swap2
% 0.31/0.48 thf(fact_199_compl__le__swap1,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( boolean_algebra @ A )
% 0.31/0.48 => ! [Y: A,X: A] :
% 0.31/0.48 ( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
% 0.31/0.48 => ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % compl_le_swap1
% 0.31/0.48 thf(fact_200_compl__mono,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( boolean_algebra @ A )
% 0.31/0.48 => ! [X: A,Y: A] :
% 0.31/0.48 ( ( ord_less_eq @ A @ X @ Y )
% 0.31/0.48 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % compl_mono
% 0.31/0.48 thf(fact_201_verit__negate__coefficient_I2_J,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ordered_ab_group_add @ A )
% 0.31/0.48 => ! [A2: A,B: A] :
% 0.31/0.48 ( ( ord_less @ A @ A2 @ B )
% 0.31/0.48 => ( ord_less @ A @ ( uminus_uminus @ A @ B ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % verit_negate_coefficient(2)
% 0.31/0.48 thf(fact_202_compl__less__swap1,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( boolean_algebra @ A )
% 0.31/0.48 => ! [Y: A,X: A] :
% 0.31/0.48 ( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X ) )
% 0.31/0.48 => ( ord_less @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % compl_less_swap1
% 0.31/0.48 thf(fact_203_compl__less__swap2,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( boolean_algebra @ A )
% 0.31/0.48 => ! [Y: A,X: A] :
% 0.31/0.48 ( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X )
% 0.31/0.48 => ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % compl_less_swap2
% 0.31/0.48 thf(fact_204_compl__less__compl__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( boolean_algebra @ A )
% 0.31/0.48 => ! [X: A,Y: A] :
% 0.31/0.48 ( ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
% 0.31/0.48 = ( ord_less @ A @ Y @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % compl_less_compl_iff
% 0.31/0.48 thf(fact_205_fun__Compl__def,axiom,
% 0.31/0.48 ! [B2: $tType,A: $tType] :
% 0.31/0.48 ( ( uminus @ B2 )
% 0.31/0.48 => ( ( uminus_uminus @ ( A > B2 ) )
% 0.31/0.48 = ( ^ [A6: A > B2,X3: A] : ( uminus_uminus @ B2 @ ( A6 @ X3 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % fun_Compl_def
% 0.31/0.48 thf(fact_206_double__diff,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A,C3: set @ A] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.31/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C3 )
% 0.31/0.48 => ( ( minus_minus @ ( set @ A ) @ B6 @ ( minus_minus @ ( set @ A ) @ C3 @ A3 ) )
% 0.31/0.48 = A3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % double_diff
% 0.31/0.48 thf(fact_207_Diff__subset,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ A3 ) ).
% 0.31/0.48
% 0.31/0.48 % Diff_subset
% 0.31/0.48 thf(fact_208_Diff__mono,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,C3: set @ A,D5: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
% 0.31/0.48 => ( ( ord_less_eq @ ( set @ A ) @ D5 @ B6 )
% 0.31/0.48 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ ( minus_minus @ ( set @ A ) @ C3 @ D5 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % Diff_mono
% 0.31/0.48 thf(fact_209_subset__iff__psubset__eq,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) )
% 0.31/0.48 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.31/0.48 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 0.31/0.48 | ( A6 = B7 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_iff_psubset_eq
% 0.31/0.48 thf(fact_210_subset__psubset__trans,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A,C3: set @ A] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.31/0.48 => ( ( ord_less @ ( set @ A ) @ B6 @ C3 )
% 0.31/0.48 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_psubset_trans
% 0.31/0.48 thf(fact_211_subset__not__subset__eq,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ord_less @ ( set @ A ) )
% 0.31/0.48 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.31/0.48 & ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_not_subset_eq
% 0.31/0.48 thf(fact_212_psubset__subset__trans,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A,C3: set @ A] :
% 0.31/0.48 ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
% 0.31/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C3 )
% 0.31/0.48 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % psubset_subset_trans
% 0.31/0.48 thf(fact_213_psubset__imp__subset,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
% 0.31/0.48 => ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% 0.31/0.48
% 0.31/0.48 % psubset_imp_subset
% 0.31/0.48 thf(fact_214_psubset__eq,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ord_less @ ( set @ A ) )
% 0.31/0.48 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.31/0.48 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.31/0.48 & ( A6 != B7 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % psubset_eq
% 0.31/0.48 thf(fact_215_psubsetE,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
% 0.31/0.48 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.31/0.48 => ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % psubsetE
% 0.31/0.48 thf(fact_216_psubset__imp__ex__mem,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
% 0.31/0.48 => ? [B5: A] : ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B6 @ A3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % psubset_imp_ex_mem
% 0.31/0.48 thf(fact_217_rev_Oopen__existence__ivl_H,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( member @ a @ X @ x )
% 0.31/0.48 => ~ ! [A5: real] :
% 0.31/0.48 ( ( ord_less @ real @ ( zero_zero @ real ) @ A5 )
% 0.31/0.48 => ~ ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ ( uminus_uminus @ real @ A5 ) @ A5 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.open_existence_ivl'
% 0.31/0.48 thf(fact_218_open__existence__ivl_H,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( member @ a @ X @ x )
% 0.31/0.48 => ~ ! [A5: real] :
% 0.31/0.48 ( ( ord_less @ real @ ( zero_zero @ real ) @ A5 )
% 0.31/0.48 => ~ ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ ( uminus_uminus @ real @ A5 ) @ A5 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % open_existence_ivl'
% 0.31/0.48 thf(fact_219_rev_Oopen__existence__ivl0,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( member @ a @ X @ x )
% 0.31/0.48 => ? [A5: real,B5: real] :
% 0.31/0.48 ( ( ord_less @ real @ A5 @ ( zero_zero @ real ) )
% 0.31/0.48 & ( ord_less @ real @ ( zero_zero @ real ) @ B5 )
% 0.31/0.48 & ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ A5 @ B5 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.open_existence_ivl0
% 0.31/0.48 thf(fact_220_existence__ivl0__cong,axiom,
% 0.31/0.48 ! [B2: $tType,Y7: set @ a,G: a > a,X0: a] :
% 0.31/0.48 ( ( x = Y7 )
% 0.31/0.48 => ( ! [X4: a,T2: B2] :
% 0.31/0.48 ( ( member @ a @ X4 @ Y7 )
% 0.31/0.48 => ( ( f @ X4 )
% 0.31/0.48 = ( G @ X4 ) ) )
% 0.31/0.48 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
% 0.31/0.48 = ( auto_l1112008849e_ivl0 @ a @ G @ Y7 @ X0 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % existence_ivl0_cong
% 0.31/0.48 thf(fact_221_existence__ivl__zero,axiom,
% 0.31/0.48 ! [X0: a] :
% 0.31/0.48 ( ( member @ a @ X0 @ x )
% 0.31/0.48 => ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % existence_ivl_zero
% 0.31/0.48 thf(fact_222_rev_Oexistence__ivl0__cong,axiom,
% 0.31/0.48 ! [B2: $tType,Y7: set @ a,G: a > a,X0: a] :
% 0.31/0.48 ( ( x = Y7 )
% 0.31/0.48 => ( ! [X4: a,T2: B2] :
% 0.31/0.48 ( ( member @ a @ X4 @ Y7 )
% 0.31/0.48 => ( ( uminus_uminus @ ( a > a ) @ f @ X4 )
% 0.31/0.48 = ( G @ X4 ) ) )
% 0.31/0.48 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
% 0.31/0.48 = ( auto_l1112008849e_ivl0 @ a @ G @ Y7 @ X0 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.existence_ivl0_cong
% 0.31/0.48 thf(fact_223_general_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.31/0.48 ! [T3: real,T0: real,X0: a] :
% 0.31/0.48 ( ( member @ real @ ( minus_minus @ real @ T3 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.31/0.48 => ( member @ a @ X0 @ x ) ) ).
% 0.31/0.48
% 0.31/0.48 % general.mem_existence_ivl_iv_defined(2)
% 0.31/0.48 thf(fact_224_Compl__eq__Compl__iff,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.31/0.48 ( ( ( uminus_uminus @ ( set @ A ) @ A3 )
% 0.31/0.48 = ( uminus_uminus @ ( set @ A ) @ B6 ) )
% 0.31/0.48 = ( A3 = B6 ) ) ).
% 0.31/0.48
% 0.31/0.48 % Compl_eq_Compl_iff
% 0.31/0.48 thf(fact_225_Compl__iff,axiom,
% 0.31/0.48 ! [A: $tType,C: A,A3: set @ A] :
% 0.31/0.48 ( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
% 0.31/0.48 = ( ~ ( member @ A @ C @ A3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % Compl_iff
% 0.31/0.48 thf(fact_226_ComplI,axiom,
% 0.31/0.48 ! [A: $tType,C: A,A3: set @ A] :
% 0.31/0.48 ( ~ ( member @ A @ C @ A3 )
% 0.31/0.48 => ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ComplI
% 0.31/0.48 thf(fact_227_rev_Oexistence__ivl__zero,axiom,
% 0.31/0.48 ! [X0: a] :
% 0.31/0.48 ( ( member @ a @ X0 @ x )
% 0.31/0.48 => ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.existence_ivl_zero
% 0.31/0.48 thf(fact_228_rev_Ogeneral_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.31/0.48 ! [T3: real,T0: real,X0: a] :
% 0.31/0.48 ( ( member @ real @ ( minus_minus @ real @ T3 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.31/0.48 => ( member @ a @ X0 @ x ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.general.mem_existence_ivl_iv_defined(2)
% 0.31/0.48 thf(fact_229_local_Oivl__subset__existence__ivl,axiom,
% 0.31/0.48 ! [T3: real,X0: a] :
% 0.31/0.48 ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.31/0.48 => ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ T3 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % local.ivl_subset_existence_ivl
% 0.31/0.48 thf(fact_230_local_Oivl__subset__existence__ivl_H,axiom,
% 0.31/0.48 ! [T3: real,X0: a] :
% 0.31/0.48 ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.31/0.48 => ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ T3 @ ( zero_zero @ real ) ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % local.ivl_subset_existence_ivl'
% 0.31/0.48 thf(fact_231_open__existence__ivl0,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( member @ a @ X @ x )
% 0.31/0.48 => ? [A5: real,B5: real] :
% 0.31/0.48 ( ( ord_less @ real @ A5 @ ( zero_zero @ real ) )
% 0.31/0.48 & ( ord_less @ real @ ( zero_zero @ real ) @ B5 )
% 0.31/0.48 & ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ A5 @ B5 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % open_existence_ivl0
% 0.31/0.48 thf(fact_232_local_Orev_Oivl__subset__existence__ivl,axiom,
% 0.31/0.48 ! [T3: real,X0: a] :
% 0.31/0.48 ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.31/0.48 => ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ T3 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % local.rev.ivl_subset_existence_ivl
% 0.31/0.48 thf(fact_233_local_Orev_Oivl__subset__existence__ivl_H,axiom,
% 0.31/0.48 ! [T3: real,X0: a] :
% 0.31/0.48 ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.31/0.48 => ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ T3 @ ( zero_zero @ real ) ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % local.rev.ivl_subset_existence_ivl'
% 0.31/0.48 thf(fact_234_local_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.31/0.48 ! [T3: real,X0: a] :
% 0.31/0.48 ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.31/0.48 => ( member @ a @ X0 @ x ) ) ).
% 0.31/0.48
% 0.31/0.48 % local.mem_existence_ivl_iv_defined(2)
% 0.31/0.48 thf(fact_235_local_Orev_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.31/0.48 ! [T3: real,X0: a] :
% 0.31/0.48 ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.31/0.48 => ( member @ a @ X0 @ x ) ) ).
% 0.31/0.48
% 0.31/0.48 % local.rev.mem_existence_ivl_iv_defined(2)
% 0.31/0.48 thf(fact_236_double__complement,axiom,
% 0.31/0.48 ! [A: $tType,A3: set @ A] :
% 0.31/0.48 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
% 0.31/0.48 = A3 ) ).
% 0.31/0.48
% 0.31/0.48 % double_complement
% 0.31/0.48 thf(fact_237_ComplD,axiom,
% 0.31/0.48 ! [A: $tType,C: A,A3: set @ A] :
% 0.31/0.48 ( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
% 0.31/0.48 => ~ ( member @ A @ C @ A3 ) ) ).
% 0.31/0.48
% 0.31/0.48 % ComplD
% 0.31/0.48 thf(fact_238_rev_Omvar_Ointerval__axioms,axiom,
% 0.31/0.48 ! [X0: a] : ( initia826609931terval @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.mvar.interval_axioms
% 0.31/0.48 thf(fact_239_mvar_Ointerval__axioms,axiom,
% 0.31/0.48 ! [X0: a] : ( initia826609931terval @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ).
% 0.31/0.48
% 0.31/0.48 % mvar.interval_axioms
% 0.31/0.48 thf(fact_240_rev_Oclosed__orbit__global__existence,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.31/0.48 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.31/0.48 = ( top_top @ ( set @ real ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.closed_orbit_global_existence
% 0.31/0.48 thf(fact_241_rev_Ofixpoint__sol_I1_J,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( member @ a @ X @ x )
% 0.31/0.48 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 0.31/0.48 = ( zero_zero @ a ) )
% 0.31/0.48 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.31/0.48 = ( top_top @ ( set @ real ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev.fixpoint_sol(1)
% 0.31/0.48 thf(fact_242_interval__axioms,axiom,
% 0.31/0.48 initia826609931terval @ ( top_top @ ( set @ real ) ) ).
% 0.31/0.48
% 0.31/0.48 % interval_axioms
% 0.31/0.48 thf(fact_243_UNIV__I,axiom,
% 0.31/0.48 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % UNIV_I
% 0.31/0.48 thf(fact_244_local_Omem__existence__ivl__subset,axiom,
% 0.31/0.48 ! [T3: real,X0: a] :
% 0.31/0.48 ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.31/0.48 => ( member @ real @ T3 @ ( top_top @ ( set @ real ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % local.mem_existence_ivl_subset
% 0.31/0.48 thf(fact_245_local_Oexistence__ivl__initial__time,axiom,
% 0.31/0.48 ! [X0: a] :
% 0.31/0.48 ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.31/0.48 => ( ( member @ a @ X0 @ x )
% 0.31/0.48 => ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % local.existence_ivl_initial_time
% 0.31/0.48 thf(fact_246_local_Orev_Omem__existence__ivl__subset,axiom,
% 0.31/0.48 ! [T3: real,X0: a] :
% 0.31/0.48 ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.31/0.48 => ( member @ real @ T3 @ ( top_top @ ( set @ real ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % local.rev.mem_existence_ivl_subset
% 0.31/0.48 thf(fact_247_general_Omem__existence__ivl__iv__defined_I1_J,axiom,
% 0.31/0.48 ! [T3: real,T0: real,X0: a] :
% 0.31/0.48 ( ( member @ real @ ( minus_minus @ real @ T3 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.31/0.48 => ( member @ real @ T0 @ ( top_top @ ( set @ real ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % general.mem_existence_ivl_iv_defined(1)
% 0.31/0.48 thf(fact_248_general_Oexistence__ivl__initial__time,axiom,
% 0.31/0.48 ! [T0: real,X0: a] :
% 0.31/0.48 ( ( member @ real @ T0 @ ( top_top @ ( set @ real ) ) )
% 0.31/0.48 => ( ( member @ a @ X0 @ x )
% 0.31/0.48 => ( member @ real @ ( minus_minus @ real @ T0 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % general.existence_ivl_initial_time
% 0.31/0.48 thf(fact_249_mem__existence__ivl__shift__autonomous2,axiom,
% 0.31/0.48 ! [T3: real,S2: real,X: a] :
% 0.31/0.48 ( ( member @ real @ ( minus_minus @ real @ T3 @ S2 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
% 0.31/0.48 => ( ( member @ a @ X @ x )
% 0.31/0.48 => ( ! [S: real,T2: real,X4: a] :
% 0.31/0.48 ( ( member @ a @ X4 @ x )
% 0.31/0.48 => ( ( f @ X4 )
% 0.31/0.48 = ( f @ X4 ) ) )
% 0.31/0.48 => ( ( ( top_top @ ( set @ real ) )
% 0.31/0.48 = ( top_top @ ( set @ real ) ) )
% 0.31/0.48 => ( member @ real @ ( minus_minus @ real @ T3 @ S2 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % mem_existence_ivl_shift_autonomous2
% 0.31/0.48 thf(fact_250_general_Omem__existence__ivl__subset,axiom,
% 0.31/0.48 ! [T3: real,T0: real,X0: a] :
% 0.31/0.48 ( ( member @ real @ ( minus_minus @ real @ T3 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.31/0.48 => ( member @ real @ T3 @ ( top_top @ ( set @ real ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % general.mem_existence_ivl_subset
% 0.31/0.48 thf(fact_251_local_Oexistence__ivl__subset,axiom,
% 0.31/0.48 ! [X0: a] : ( ord_less_eq @ ( set @ real ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) @ ( top_top @ ( set @ real ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % local.existence_ivl_subset
% 0.31/0.48 thf(fact_252_fixpoint__sol_I1_J,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( member @ a @ X @ x )
% 0.31/0.48 => ( ( ( f @ X )
% 0.31/0.48 = ( zero_zero @ a ) )
% 0.31/0.48 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X )
% 0.31/0.48 = ( top_top @ ( set @ real ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % fixpoint_sol(1)
% 0.31/0.48 thf(fact_253_closed__orbit__global__existence,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.31/0.48 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X )
% 0.31/0.48 = ( top_top @ ( set @ real ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % closed_orbit_global_existence
% 0.31/0.48 thf(fact_254_local_Orev_Oexistence__ivl__initial__time,axiom,
% 0.31/0.48 ! [X0: a] :
% 0.31/0.48 ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.31/0.48 => ( ( member @ a @ X0 @ x )
% 0.31/0.48 => ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % local.rev.existence_ivl_initial_time
% 0.31/0.48
% 0.31/0.48 % Subclasses (17)
% 0.31/0.48 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___HOL_Otype,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( execut510477386_space @ A )
% 0.31/0.48 => ( type @ A ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ozero,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( execut510477386_space @ A )
% 0.31/0.48 => ( zero @ A ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ominus,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( execut510477386_space @ A )
% 0.31/0.48 => ( minus @ A ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ouminus,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( execut510477386_space @ A )
% 0.31/0.48 => ( uminus @ A ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Orderings_Oord,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( execut510477386_space @ A )
% 0.31/0.48 => ( ord @ A ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Orderings_Oorder,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( execut510477386_space @ A )
% 0.31/0.48 => ( order @ A ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ogroup__add,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( execut510477386_space @ A )
% 0.31/0.48 => ( group_add @ A ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Orderings_Opreorder,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( execut510477386_space @ A )
% 0.31/0.48 => ( preorder @ A ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oab__group__add,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( execut510477386_space @ A )
% 0.31/0.48 => ( ab_group_add @ A ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Inner__Product_Oreal__inner,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( execut510477386_space @ A )
% 0.31/0.48 => ( inner_real_inner @ A ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__group__add,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( execut510477386_space @ A )
% 0.31/0.48 => ( ordered_ab_group_add @ A ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocancel__comm__monoid__add,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( execut510477386_space @ A )
% 0.31/0.48 => ( cancel1352612707id_add @ A ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( execut510477386_space @ A )
% 0.31/0.48 => ( cancel146912293up_add @ A ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Euclidean__Space_Oeuclidean__space,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( execut510477386_space @ A )
% 0.31/0.48 => ( euclid925273238_space @ A ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Real__Vector__Spaces_Ometric__space,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( execut510477386_space @ A )
% 0.31/0.48 => ( real_V2090557954_space @ A ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Topological__Spaces_Operfect__space,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( execut510477386_space @ A )
% 0.31/0.48 => ( topolo890362671_space @ A ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( execut510477386_space @ A )
% 0.31/0.48 => ( ordere890947078_space @ A ) ) ).
% 0.31/0.48
% 0.31/0.48 % Type constructors (46)
% 0.31/0.48 thf(tcon_HOL_Obool___Countable_Ocountable,axiom,
% 0.31/0.48 countable @ $o ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Set_Oset___Countable_Ocountable_1,axiom,
% 0.31/0.48 ! [A8: $tType] :
% 0.31/0.48 ( ( finite_finite @ A8 )
% 0.31/0.48 => ( countable @ ( set @ A8 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_fun___Countable_Ocountable_2,axiom,
% 0.31/0.48 ! [A8: $tType,A9: $tType] :
% 0.31/0.48 ( ( ( finite_finite @ A8 )
% 0.31/0.48 & ( countable @ A9 ) )
% 0.31/0.48 => ( countable @ ( A8 > A9 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.31/0.48 ! [A8: $tType,A9: $tType] :
% 0.31/0.48 ( ( ( finite_finite @ A8 )
% 0.31/0.48 & ( finite_finite @ A9 ) )
% 0.31/0.48 => ( finite_finite @ ( A8 > A9 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Set_Oset___Finite__Set_Ofinite_3,axiom,
% 0.31/0.48 ! [A8: $tType] :
% 0.31/0.48 ( ( finite_finite @ A8 )
% 0.31/0.48 => ( finite_finite @ ( set @ A8 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_HOL_Obool___Finite__Set_Ofinite_4,axiom,
% 0.31/0.48 finite_finite @ $o ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_fun___Real__Vector__Spaces_Ometric__space,axiom,
% 0.31/0.48 ! [A8: $tType,A9: $tType] :
% 0.31/0.48 ( ( ( countable @ A8 )
% 0.31/0.48 & ( real_V2090557954_space @ A9 ) )
% 0.31/0.48 => ( real_V2090557954_space @ ( A8 > A9 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 0.31/0.48 ! [A8: $tType,A9: $tType] :
% 0.31/0.48 ( ( boolean_algebra @ A9 )
% 0.31/0.48 => ( boolean_algebra @ ( A8 > A9 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.31/0.48 ! [A8: $tType,A9: $tType] :
% 0.31/0.48 ( ( preorder @ A9 )
% 0.31/0.48 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.31/0.48 ! [A8: $tType,A9: $tType] :
% 0.31/0.48 ( ( order @ A9 )
% 0.31/0.48 => ( order @ ( A8 > A9 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_fun___Orderings_Oord,axiom,
% 0.31/0.48 ! [A8: $tType,A9: $tType] :
% 0.31/0.48 ( ( ord @ A9 )
% 0.31/0.48 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_fun___Groups_Ouminus,axiom,
% 0.31/0.48 ! [A8: $tType,A9: $tType] :
% 0.31/0.48 ( ( uminus @ A9 )
% 0.31/0.48 => ( uminus @ ( A8 > A9 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_fun___Groups_Ominus,axiom,
% 0.31/0.48 ! [A8: $tType,A9: $tType] :
% 0.31/0.48 ( ( minus @ A9 )
% 0.31/0.48 => ( minus @ ( A8 > A9 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_5,axiom,
% 0.31/0.48 ! [A8: $tType] : ( boolean_algebra @ ( set @ A8 ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Set_Oset___Orderings_Opreorder_6,axiom,
% 0.31/0.48 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Set_Oset___Orderings_Oorder_7,axiom,
% 0.31/0.48 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 0.31/0.48 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Set_Oset___Groups_Ouminus_9,axiom,
% 0.31/0.48 ! [A8: $tType] : ( uminus @ ( set @ A8 ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Set_Oset___Groups_Ominus_10,axiom,
% 0.31/0.48 ! [A8: $tType] : ( minus @ ( set @ A8 ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Set_Oset___Groups_Ozero,axiom,
% 0.31/0.48 ! [A8: $tType] :
% 0.31/0.48 ( ( zero @ A8 )
% 0.31/0.48 => ( zero @ ( set @ A8 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_11,axiom,
% 0.31/0.48 boolean_algebra @ $o ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
% 0.31/0.48 preorder @ $o ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.31/0.48 linorder @ $o ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_HOL_Obool___Orderings_Oorder_13,axiom,
% 0.31/0.48 order @ $o ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_HOL_Obool___Orderings_Oord_14,axiom,
% 0.31/0.48 ord @ $o ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_HOL_Obool___Groups_Ouminus_15,axiom,
% 0.31/0.48 uminus @ $o ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_HOL_Obool___Groups_Ominus_16,axiom,
% 0.31/0.48 minus @ $o ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Executable__Euclidean__Space_Oexecutable__euclidean__space,axiom,
% 0.31/0.48 execut510477386_space @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
% 0.31/0.48 ordere890947078_space @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
% 0.31/0.48 topolo890362671_space @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space_17,axiom,
% 0.31/0.48 real_V2090557954_space @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space,axiom,
% 0.31/0.48 euclid925273238_space @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
% 0.31/0.48 linord219039673up_add @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.31/0.48 cancel146912293up_add @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add,axiom,
% 0.31/0.48 cancel1352612707id_add @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
% 0.31/0.48 ordered_ab_group_add @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Inner__Product_Oreal__inner,axiom,
% 0.31/0.48 inner_real_inner @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Groups_Oab__group__add,axiom,
% 0.31/0.48 ab_group_add @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Orderings_Opreorder_18,axiom,
% 0.31/0.48 preorder @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Orderings_Olinorder_19,axiom,
% 0.31/0.48 linorder @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 0.31/0.48 group_add @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Orderings_Oorder_20,axiom,
% 0.31/0.48 order @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Orderings_Oord_21,axiom,
% 0.31/0.48 ord @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Groups_Ouminus_22,axiom,
% 0.31/0.48 uminus @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Groups_Ominus_23,axiom,
% 0.31/0.48 minus @ real ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Real_Oreal___Groups_Ozero_24,axiom,
% 0.31/0.48 zero @ real ).
% 0.31/0.48
% 0.31/0.48 % Free types (1)
% 0.31/0.48 thf(tfree_0,hypothesis,
% 0.31/0.48 execut510477386_space @ a ).
% 0.31/0.48
% 0.31/0.48 % Conjectures (1)
% 0.31/0.48 thf(conj_0,conjecture,
% 0.31/0.48 ? [D4: real] :
% 0.31/0.48 ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
% 0.31/0.48 & ? [B9: real] :
% 0.31/0.48 ( ( ord_less @ real @ ( zero_zero @ real ) @ B9 )
% 0.31/0.48 & ! [X4: a] :
% 0.31/0.48 ( ( member @ a @ X4 @ ( elemen321786957_cball @ a @ x2 @ D4 ) )
% 0.31/0.48 => ( ord_less_eq @ real @ B9 @ ( inner_780170721_inner @ a @ ( f @ X4 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 %------------------------------------------------------------------------------
% 0.31/0.48 ------------------------
% 0.32/1.60 ---- Embedded in THF ---
% 0.32/1.60 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.32/1.60 %%% Generated on Tue Feb 24 08:28:51 EST 2026
% 0.32/1.60 %%% using '$$dhol' embedding, version 1.3.0.
% 0.32/1.60 %%% Logic specification used:
% 0.32/1.60 %%% thf(spec, logic, $$dhol).
% 0.32/1.60
% 0.32/1.60 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.laqx631JN2/DTF2DTF21273.p
% 0.32/1.60 thf(ty_t_Bounded__Linear__Function_Oblinfun, type, bounde2145540817linfun: ($tType > ($tType > $tType))).
% 0.32/1.60 thf(ty_t_Real_Oreal, type, real: $tType).
% 0.32/1.60 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 0.32/1.60 thf(ty_tf_a, type, a: $tType).
% 0.32/1.60 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Countable_Ocountable, type, countable: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Groups_Ouminus, type, uminus: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Groups_Oab__group__add, type, ab_group_add: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Groups_Ocomm__monoid__diff, type, comm_monoid_diff: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Inner__Product_Oreal__inner, type, inner_real_inner: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Lattices_Oboolean__algebra, type, boolean_algebra: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Euclidean__Space_Oeuclidean__space, type, euclid925273238_space: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Real__Vector__Spaces_Ometric__space, type, real_V2090557954_space: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Topological__Spaces_Operfect__space, type, topolo890362671_space: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space, type, ordere890947078_space: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_cl_Executable__Euclidean__Space_Oexecutable__euclidean__space, type, execut510477386_space: (!> [A:$tType]: ($o))).
% 0.32/1.60 thf(sy_c_Elementary__Metric__Spaces_Ocball, type, elemen321786957_cball: (!> [A:$tType]: ((A > (real > (set @ A)))))).
% 0.32/1.60 thf(sy_c_Flow_Oauto__ll__on__open_Oexistence__ivl0, type, auto_l1112008849e_ivl0: (!> [A:$tType]: (((A > A) > ((set @ A) > (A > (set @ real))))))).
% 0.32/1.60 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 0.32/1.60 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 0.32/1.60 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 0.32/1.60 thf(sy_c_Initial__Value__Problem_Ointerval, type, initia826609931terval: ((set @ real) > $o)).
% 0.32/1.60 thf(sy_c_Inner__Product_Oreal__inner__class_Oinner, type, inner_780170721_inner: (!> [A:$tType]: ((A > (A > real))))).
% 0.32/1.60 thf(sy_c_Invariance_Oauto__ll__on__open_Oinvariant, type, auto_ll_on_invariant: (!> [A:$tType]: (((A > A) > ((set @ A) > ((set @ A) > $o)))))).
% 0.32/1.60 thf(sy_c_Line__Segment_Oclosed__segment, type, line_closed_segment: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 0.32/1.60 thf(sy_c_Line__Segment_Oopen__segment, type, line_open_segment: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 0.32/1.60 thf(sy_c_ODE__Misc_Oauto__ll__on__open_Otrapped, type, oDE_au1039603466rapped: (!> [A:$tType]: (((A > A) > ((set @ A) > (A > ((set @ A) > $o))))))).
% 0.32/1.60 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 0.32/1.60 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 0.32/1.60 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 0.32/1.60 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Oclosed__orbit, type, period385816147_orbit: (!> [A:$tType]: (((A > A) > ((set @ A) > (A > $o)))))).
% 0.32/1.60 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiod, type, period1153813292period: (!> [A:$tType]: (((A > A) > ((set @ A) > (A > real)))))).
% 0.32/1.60 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiodic__orbit, type, period862636932_orbit: (!> [A:$tType]: (((A > A) > ((set @ A) > (A > $o)))))).
% 0.32/1.60 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Orot, type, poinca1750768982en_rot: (!> [A:$tType]: ((A > A)))).
% 0.32/1.60 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Otransversal__segment, type, poinca272511729egment: (!> [A:$tType]: (((A > A) > ((set @ A) > (A > (A > $o))))))).
% 0.32/1.60 thf(sy_c_Reachability__Analysis_Oc1__on__open__euclidean_Osection, type, reacha1084862253ection: (!> [A:$tType]: (((A > A) > ((set @ A) > ((A > real) > ((A > ((bounde2145540817linfun @ A) @ real)) > ((set @ A) > $o)))))))).
% 0.32/1.60 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist, type, real_V2000881966t_dist: (!> [A:$tType]: ((A > (A > real))))).
% 0.32/1.60 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 0.32/1.60 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost, type, set_or331188842AtMost: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 0.32/1.60 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 0.32/1.60 thf(sy_v_X, type, x: (set @ a)).
% 0.32/1.60 thf(sy_v_a, type, a2: a).
% 0.32/1.60 thf(sy_v_b, type, b: a).
% 0.32/1.60 thf(sy_v_d____, type, d: real).
% 0.32/1.60 thf(sy_v_f, type, f: (a > a)).
% 0.32/1.60 thf(sy_v_s____, type, s: a).
% 0.32/1.60 thf(sy_v_thesis, type, thesis: $o).
% 0.32/1.60 thf(sy_v_x____, type, x2: a).
% 0.32/1.60 thf(sy_v_z, type, z: a).
% 0.32/1.60 thf(fact_0_d_I1_J, axiom, (((ord_less @ real) @ (zero_zero @ real)) @ d)).
% 0.32/1.60 thf(fact_1__092_060open_062s_A_092_060in_062_Acball_Ax_Ad_092_060close_062, axiom, (((member @ a) @ s) @ (((elemen321786957_cball @ a) @ x2) @ d))).
% 0.32/1.60 thf(fact_2_seg_I1_J, axiom, (a2 != b)).
% 0.32/1.60 thf(fact_3_nrm__dot, axiom, (! [X:a,Y:a]: (((((inner_780170721_inner @ a) @ (((minus_minus @ a) @ X) @ Y)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ X) @ Y))) = (zero_zero @ real))))).
% 0.32/1.60 thf(fact_4__092_060open_062x_A_092_060in_062_AX_092_060close_062, axiom, (((member @ a) @ x2) @ x)).
% 0.32/1.60 thf(fact_5__092_060open_062_092_060forall_062x_092_060in_062cball_Ax_Ad_O_Af_As_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060le_062_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062, axiom, (! [X2:a]: (((((member @ a) @ X2) @ (((elemen321786957_cball @ a) @ x2) @ d)) => (((ord_less_eq @ real) @ (((inner_780170721_inner @ a) @ (f @ s)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ a2) @ b)))) @ (((inner_780170721_inner @ a) @ (f @ X2)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ a2) @ b)))))))).
% 0.32/1.60 thf(fact_6__092_060open_062f_Ax_A_092_060noteq_062_A_I0_058_058_Ha_J_092_060close_062, axiom, ((f @ x2) != (zero_zero @ a))).
% 0.32/1.60 thf(fact_7_direction_I2_J, axiom, (((ord_less @ real) @ (zero_zero @ real)) @ (((inner_780170721_inner @ a) @ (f @ z)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ a2) @ b))))).
% 0.32/1.60 thf(fact_8_d_I2_J, axiom, (! [Y2:a]: (((((member @ a) @ Y2) @ (((elemen321786957_cball @ a) @ x2) @ d)) => ((((ord_less @ real) @ (zero_zero @ real)) @ (((inner_780170721_inner @ a) @ (f @ Y2)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ a2) @ b)))) & ((((member @ a) @ Y2) @ x) & ((f @ Y2) != (zero_zero @ a)))))))).
% 0.32/1.60 thf(fact_9__092_060open_0620_A_060_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062, axiom, (((ord_less @ real) @ (zero_zero @ real)) @ (((inner_780170721_inner @ a) @ (f @ x2)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ a2) @ b))))).
% 0.32/1.60 thf(fact_10_dot__ortho, axiom, (! [X:a]: (((((inner_780170721_inner @ a) @ X) @ ((poinca1750768982en_rot @ a) @ X)) = (zero_zero @ real))))).
% 0.32/1.60 thf(fact_11__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062s_O_A_092_060lbrakk_062s_A_092_060in_062_Acball_Ax_Ad_059_A_092_060forall_062x_092_060in_062cball_Ax_Ad_O_Af_As_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060le_062_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [S:a]: (((((member @ a) @ S) @ (((elemen321786957_cball @ a) @ x2) @ d)) => (~ ((! [X2:a]: (((((member @ a) @ X2) @ (((elemen321786957_cball @ a) @ x2) @ d)) => (((ord_less_eq @ real) @ (((inner_780170721_inner @ a) @ (f @ S)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ a2) @ b)))) @ (((inner_780170721_inner @ a) @ (f @ X2)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ a2) @ b))))))))))))))).
% 0.32/1.60 thf(fact_12__092_060open_062x_A_092_060in_062_A_123a_N_Nb_125_092_060close_062, axiom, (((member @ a) @ x2) @ (((line_closed_segment @ a) @ a2) @ b))).
% 0.32/1.60 thf(fact_13__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062d_O_A_092_060lbrakk_0620_A_060_Ad_059_A_092_060And_062y_O_Ay_A_092_060in_062_Acball_Ax_Ad_A_092_060Longrightarrow_062_A0_A_060_Af_Ay_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060and_062_Ay_A_092_060in_062_AX_A_092_060and_062_Af_Ay_A_092_060noteq_062_A_I0_058_058_Ha_J_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) => (~ ((! [Y3:a]: (((((member @ a) @ Y3) @ (((elemen321786957_cball @ a) @ x2) @ D)) => ((((ord_less @ real) @ (zero_zero @ real)) @ (((inner_780170721_inner @ a) @ (f @ Y3)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ a2) @ b)))) & ((((member @ a) @ Y3) @ x) & ((f @ Y3) != (zero_zero @ a))))))))))))))).
% 0.32/1.60 thf(fact_14_transversal, axiom, (((((poinca272511729egment @ a) @ f) @ x) @ a2) @ b)).
% 0.32/1.60 thf(fact_15_inner__gt__zero__iff, axiom, (! [A:$tType]: (((inner_real_inner @ A) => (! [X:A]: (((((ord_less @ real) @ (zero_zero @ real)) @ (((inner_780170721_inner @ A) @ X) @ X)) = (X != (zero_zero @ A))))))))).
% 0.32/1.60 thf(fact_16_centre__in__cball, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X:A,E:real]: (((((member @ A) @ X) @ (((elemen321786957_cball @ A) @ X) @ E)) = (((ord_less_eq @ real) @ (zero_zero @ real)) @ E)))))))).
% 0.32/1.60 thf(fact_17_all__zero__iff, axiom, (! [A:$tType]: (((inner_real_inner @ A) => (! [X:A]: (((! [U:A]: (((((inner_780170721_inner @ A) @ X) @ U) = (zero_zero @ real)))) = (X = (zero_zero @ A))))))))).
% 0.32/1.60 thf(fact_18_inner__zero__left, axiom, (! [A:$tType]: (((inner_real_inner @ A) => (! [X:A]: (((((inner_780170721_inner @ A) @ (zero_zero @ A)) @ X) = (zero_zero @ real)))))))).
% 0.32/1.60 thf(fact_19_inner__zero__right, axiom, (! [A:$tType]: (((inner_real_inner @ A) => (! [X:A]: (((((inner_780170721_inner @ A) @ X) @ (zero_zero @ A)) = (zero_zero @ real)))))))).
% 0.32/1.60 thf(fact_20_inner__eq__zero__iff, axiom, (! [A:$tType]: (((inner_real_inner @ A) => (! [X:A]: ((((((inner_780170721_inner @ A) @ X) @ X) = (zero_zero @ real)) = (X = (zero_zero @ A))))))))).
% 0.32/1.60 thf(fact_21_diff__gt__0__iff__gt, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A2) @ B)) = (((ord_less @ A) @ B) @ A2)))))))).
% 0.32/1.60 thf(fact_22_diff__ge__0__iff__ge, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A2) @ B)) = (((ord_less_eq @ A) @ B) @ A2)))))))).
% 0.32/1.60 thf(fact_23_rot__0, axiom, (((poinca1750768982en_rot @ a) @ (zero_zero @ a)) = (zero_zero @ a))).
% 0.32/1.60 thf(fact_24_rot__eq__0__iff, axiom, (! [X:a]: (((((poinca1750768982en_rot @ a) @ X) = (zero_zero @ a)) = (X = (zero_zero @ a)))))).
% 0.32/1.60 thf(fact_25_seg_I3_J, axiom, (! [Z:a]: (((((member @ a) @ Z) @ (((line_closed_segment @ a) @ a2) @ b)) => (((ord_less @ real) @ (zero_zero @ real)) @ (((inner_780170721_inner @ a) @ (f @ Z)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ a2) @ b)))))))).
% 0.32/1.60 thf(fact_26_direction_I1_J, axiom, (((member @ a) @ z) @ (((line_closed_segment @ a) @ a2) @ b))).
% 0.32/1.60 thf(fact_27_transversal__segment__reverse, axiom, (! [X:a,Y:a]: (((((((poinca272511729egment @ a) @ f) @ x) @ X) @ Y) => (((((poinca272511729egment @ a) @ f) @ x) @ Y) @ X))))).
% 0.32/1.60 thf(fact_28_transversal__segment__commute, axiom, (! [X:a,Y:a]: (((((((poinca272511729egment @ a) @ f) @ x) @ X) @ Y) = (((((poinca272511729egment @ a) @ f) @ x) @ Y) @ X))))).
% 0.32/1.60 thf(fact_29_in__segment__inner__rot2, axiom, (! [X:a,A2:a,B:a,Y:a]: (((((member @ a) @ X) @ (((line_closed_segment @ a) @ A2) @ B)) => ((((member @ a) @ Y) @ (((line_closed_segment @ a) @ A2) @ B)) => ((((inner_780170721_inner @ a) @ (((minus_minus @ a) @ X) @ Y)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ A2) @ B))) = (zero_zero @ real))))))).
% 0.32/1.60 thf(fact_30_in__segment__inner__rot, axiom, (! [X:a,A2:a,B:a]: (((((member @ a) @ X) @ (((line_closed_segment @ a) @ A2) @ B)) => ((((inner_780170721_inner @ a) @ (((minus_minus @ a) @ X) @ A2)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ B) @ A2))) = (zero_zero @ real)))))).
% 0.32/1.60 thf(fact_31_seg_I2_J, axiom, (((ord_less_eq @ (set @ a)) @ (((line_closed_segment @ a) @ a2) @ b)) @ x)).
% 0.32/1.60 thf(fact_32_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 0.32/1.60 thf(fact_33_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 0.32/1.60 thf(fact_34_cancel__comm__monoid__add__class_Odiff__cancel, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 0.32/1.60 thf(fact_35_diff__zero, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 0.32/1.60 thf(fact_36_zero__diff, axiom, (! [A:$tType]: (((comm_monoid_diff @ A) => (! [A2:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 0.32/1.60 thf(fact_37_diff__0__right, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 0.32/1.60 thf(fact_38_diff__self, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 0.32/1.60 thf(fact_39_subset__cball, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [D2:real,E:real,X:A]: (((((ord_less_eq @ real) @ D2) @ E) => (((ord_less_eq @ (set @ A)) @ (((elemen321786957_cball @ A) @ X) @ D2)) @ (((elemen321786957_cball @ A) @ X) @ E))))))))).
% 0.32/1.60 thf(fact_40_transversal__segment__def, axiom, (! [A2:a,B:a]: (((((((poinca272511729egment @ a) @ f) @ x) @ A2) @ B) = ((A2 != B) & ((((ord_less_eq @ (set @ a)) @ (((line_closed_segment @ a) @ A2) @ B)) @ x) & (! [X3:a]: (((((member @ a) @ X3) @ (((line_closed_segment @ a) @ A2) @ B)) => ((((inner_780170721_inner @ a) @ (f @ X3)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ A2) @ B))) != (zero_zero @ real))))))))))).
% 0.32/1.60 thf(fact_41_transversal__segmentE, axiom, (! [X:a,Y:a]: (((((((poinca272511729egment @ a) @ f) @ x) @ X) @ Y) => (((X != Y) => ((((ord_less_eq @ (set @ a)) @ (((line_closed_segment @ a) @ X) @ Y)) @ x) => (~ ((! [Z2:a]: (((((member @ a) @ Z2) @ (((line_closed_segment @ a) @ X) @ Y)) => (((ord_less @ real) @ (zero_zero @ real)) @ (((inner_780170721_inner @ a) @ (f @ Z2)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ X) @ Y))))))))))) => (~ (((X != Y) => ((((ord_less_eq @ (set @ a)) @ (((line_closed_segment @ a) @ X) @ Y)) @ x) => (~ ((! [Z2:a]: (((((member @ a) @ Z2) @ (((line_closed_segment @ a) @ X) @ Y)) => (((ord_less @ real) @ (zero_zero @ real)) @ (((inner_780170721_inner @ a) @ (f @ Z2)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ Y) @ X)))))))))))))))))).
% 0.32/1.60 thf(fact_42_transversal__segment__posD_I2_J, axiom, (! [X:a,Y:a,Z3:a]: (((((((poinca272511729egment @ a) @ f) @ x) @ X) @ Y) => ((((member @ a) @ Z3) @ (((line_closed_segment @ a) @ X) @ Y)) => ((((ord_less @ real) @ (zero_zero @ real)) @ (((inner_780170721_inner @ a) @ (f @ Z3)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ X) @ Y)))) => (((ord_less_eq @ (set @ a)) @ (((line_closed_segment @ a) @ X) @ Y)) @ x))))))).
% 0.32/1.60 thf(fact_43_transversal__segment__negD_I2_J, axiom, (! [X:a,Y:a,Z3:a]: (((((((poinca272511729egment @ a) @ f) @ x) @ X) @ Y) => ((((member @ a) @ Z3) @ (((line_closed_segment @ a) @ X) @ Y)) => ((((ord_less @ real) @ (((inner_780170721_inner @ a) @ (f @ Z3)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ X) @ Y)))) @ (zero_zero @ real)) => (((ord_less_eq @ (set @ a)) @ (((line_closed_segment @ a) @ X) @ Y)) @ x))))))).
% 0.32/1.60 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 0.32/1.60 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A3)))) = A3)))).
% 0.32/1.60 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) = (Q @ X4)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 0.32/1.60 thf(fact_47_ext, axiom, (! [B2:$tType,A:$tType,F:(A > B2),G:(A > B2)]: (((! [X4:A]: (((F @ X4) = (G @ X4)))) => (F = G))))).
% 0.32/1.60 thf(fact_48_transversal__segment__sign__less, axiom, (! [W:a,Y:a]: (((((((poinca272511729egment @ a) @ f) @ x) @ W) @ Y) => ((((ord_less @ real) @ (((inner_780170721_inner @ a) @ (f @ W)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ W) @ Y)))) @ (zero_zero @ real)) => (! [X2:a]: (((((member @ a) @ X2) @ (((line_closed_segment @ a) @ W) @ Y)) => (((ord_less @ real) @ (((inner_780170721_inner @ a) @ (f @ X2)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ W) @ Y)))) @ (zero_zero @ real)))))))))).
% 0.32/1.60 thf(fact_49_transversal__segment__pos, axiom, (! [X:a,Y:a,W:a]: (((((((poinca272511729egment @ a) @ f) @ x) @ X) @ Y) => ((((member @ a) @ W) @ (((line_closed_segment @ a) @ X) @ Y)) => ((((ord_less @ real) @ (zero_zero @ real)) @ (((inner_780170721_inner @ a) @ (f @ W)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ X) @ Y)))) => (! [X2:a]: (((((member @ a) @ X2) @ (((line_closed_segment @ a) @ X) @ Y)) => (((ord_less @ real) @ (zero_zero @ real)) @ (((inner_780170721_inner @ a) @ (f @ X2)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ X) @ Y))))))))))))).
% 0.32/1.60 thf(fact_50_transversal__segment__neg, axiom, (! [X:a,Y:a,W:a]: (((((((poinca272511729egment @ a) @ f) @ x) @ X) @ Y) => ((((member @ a) @ W) @ (((line_closed_segment @ a) @ X) @ Y)) => ((((ord_less @ real) @ (((inner_780170721_inner @ a) @ (f @ W)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ X) @ Y)))) @ (zero_zero @ real)) => (! [X2:a]: (((((member @ a) @ X2) @ (((line_closed_segment @ a) @ X) @ Y)) => (((ord_less @ real) @ (((inner_780170721_inner @ a) @ (f @ X2)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ X) @ Y)))) @ (zero_zero @ real))))))))))).
% 0.32/1.60 thf(fact_51_transversal__segment__posD_I1_J, axiom, (! [X:a,Y:a,Z3:a]: (((((((poinca272511729egment @ a) @ f) @ x) @ X) @ Y) => ((((member @ a) @ Z3) @ (((line_closed_segment @ a) @ X) @ Y)) => ((((ord_less @ real) @ (zero_zero @ real)) @ (((inner_780170721_inner @ a) @ (f @ Z3)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ X) @ Y)))) => (X != Y))))))).
% 0.32/1.60 thf(fact_52_transversal__segment__posD_I3_J, axiom, (! [X:a,Y:a,Za:a,Z3:a]: (((((((poinca272511729egment @ a) @ f) @ x) @ X) @ Y) => ((((member @ a) @ Za) @ (((line_closed_segment @ a) @ X) @ Y)) => ((((ord_less @ real) @ (zero_zero @ real)) @ (((inner_780170721_inner @ a) @ (f @ Za)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ X) @ Y)))) => ((((member @ a) @ Z3) @ (((line_closed_segment @ a) @ X) @ Y)) => (((ord_less @ real) @ (zero_zero @ real)) @ (((inner_780170721_inner @ a) @ (f @ Z3)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ X) @ Y))))))))))).
% 0.32/1.60 thf(fact_53_transversal__segment__negD_I1_J, axiom, (! [X:a,Y:a,Z3:a]: (((((((poinca272511729egment @ a) @ f) @ x) @ X) @ Y) => ((((member @ a) @ Z3) @ (((line_closed_segment @ a) @ X) @ Y)) => ((((ord_less @ real) @ (((inner_780170721_inner @ a) @ (f @ Z3)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ X) @ Y)))) @ (zero_zero @ real)) => (X != Y))))))).
% 0.32/1.60 thf(fact_54_transversal__segment__negD_I3_J, axiom, (! [X:a,Y:a,Za:a,Z3:a]: (((((((poinca272511729egment @ a) @ f) @ x) @ X) @ Y) => ((((member @ a) @ Za) @ (((line_closed_segment @ a) @ X) @ Y)) => ((((ord_less @ real) @ (((inner_780170721_inner @ a) @ (f @ Za)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ X) @ Y)))) @ (zero_zero @ real)) => ((((member @ a) @ Z3) @ (((line_closed_segment @ a) @ X) @ Y)) => (((ord_less @ real) @ (((inner_780170721_inner @ a) @ (f @ Z3)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ X) @ Y)))) @ (zero_zero @ real))))))))).
% 0.32/1.60 thf(fact_55__092_060open_062_092_060And_062z_O_Az_A_092_060in_062_A_123a_N_Nb_125_A_092_060Longrightarrow_062_A0_A_060_Af_Az_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062, axiom, (! [Z3:a]: (((((member @ a) @ Z3) @ (((line_closed_segment @ a) @ a2) @ b)) => (((ord_less @ real) @ (zero_zero @ real)) @ (((inner_780170721_inner @ a) @ (f @ Z3)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ a2) @ b)))))))).
% 0.32/1.60 thf(fact_56_c1__on__open__R2_Otransversal__segment_Ocong, axiom, (! [A:$tType]: (((execut510477386_space @ A) => ((poinca272511729egment @ A) = (poinca272511729egment @ A)))))).
% 0.32/1.60 thf(fact_57_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 0.32/1.60 thf(fact_58_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,C:A,B:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C)) @ B) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B)) @ C)))))))).
% 0.32/1.60 thf(fact_59_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B:A,C:A,D2:A]: ((((((minus_minus @ A) @ A2) @ B) = (((minus_minus @ A) @ C) @ D2)) => ((A2 = B) = (C = D2))))))))).
% 0.32/1.60 thf(fact_60_inner__commute, axiom, (! [A:$tType]: (((inner_real_inner @ A) => ((inner_780170721_inner @ A) = (^ [X3:A,Y4:A]: ((((inner_780170721_inner @ A) @ Y4) @ X3)))))))).
% 0.32/1.60 thf(fact_61_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 0.32/1.60 thf(fact_62_zero__less__iff__neq__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 0.32/1.60 thf(fact_63_gr__implies__not__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 0.32/1.60 thf(fact_64_not__less__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 0.32/1.60 thf(fact_65_gr__zeroI, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 0.32/1.60 thf(fact_66_eq__iff__diff__eq__0, axiom, (! [A:$tType]: (((group_add @ A) => ((^ [Y5:A,Z4:A]: ((Y5 = Z4))) = (^ [A4:A,B3:A]: (((((minus_minus @ A) @ A4) @ B3) = (zero_zero @ A))))))))).
% 0.32/1.60 thf(fact_67_diff__eq__diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,C:A,D2:A]: ((((((minus_minus @ A) @ A2) @ B) = (((minus_minus @ A) @ C) @ D2)) => ((((ord_less_eq @ A) @ A2) @ B) = (((ord_less_eq @ A) @ C) @ D2))))))))).
% 0.32/1.60 thf(fact_68_diff__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B) @ C))))))))).
% 0.32/1.60 thf(fact_69_diff__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B) @ A2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C) @ A2)) @ (((minus_minus @ A) @ C) @ B))))))))).
% 0.32/1.60 thf(fact_70_diff__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,D2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ D2) @ C) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B) @ D2)))))))))).
% 0.32/1.60 thf(fact_71_diff__strict__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => (((ord_less @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B) @ C))))))))).
% 0.32/1.60 thf(fact_72_diff__strict__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B:A,A2:A,C:A]: (((((ord_less @ A) @ B) @ A2) => (((ord_less @ A) @ (((minus_minus @ A) @ C) @ A2)) @ (((minus_minus @ A) @ C) @ B))))))))).
% 0.32/1.60 thf(fact_73_diff__eq__diff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,C:A,D2:A]: ((((((minus_minus @ A) @ A2) @ B) = (((minus_minus @ A) @ C) @ D2)) => ((((ord_less @ A) @ A2) @ B) = (((ord_less @ A) @ C) @ D2))))))))).
% 0.32/1.60 thf(fact_74_diff__strict__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,D2:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ A) @ D2) @ C) => (((ord_less @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B) @ D2)))))))))).
% 0.32/1.60 thf(fact_75_inner__diff__right, axiom, (! [A:$tType]: (((inner_real_inner @ A) => (! [X:A,Y:A,Z3:A]: (((((inner_780170721_inner @ A) @ X) @ (((minus_minus @ A) @ Y) @ Z3)) = (((minus_minus @ real) @ (((inner_780170721_inner @ A) @ X) @ Y)) @ (((inner_780170721_inner @ A) @ X) @ Z3))))))))).
% 0.32/1.60 thf(fact_76_inner__diff__left, axiom, (! [A:$tType]: (((inner_real_inner @ A) => (! [X:A,Y:A,Z3:A]: (((((inner_780170721_inner @ A) @ (((minus_minus @ A) @ X) @ Y)) @ Z3) = (((minus_minus @ real) @ (((inner_780170721_inner @ A) @ X) @ Z3)) @ (((inner_780170721_inner @ A) @ Y) @ Z3))))))))).
% 0.32/1.60 thf(fact_77_mem__cball__leI, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X:A,Y:A,E:real,F:real]: (((((member @ A) @ X) @ (((elemen321786957_cball @ A) @ Y) @ E)) => ((((ord_less_eq @ real) @ E) @ F) => (((member @ A) @ X) @ (((elemen321786957_cball @ A) @ Y) @ F)))))))))).
% 0.32/1.60 thf(fact_78_le__iff__diff__le__0, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => ((ord_less_eq @ A) = (^ [A4:A,B3:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ A4) @ B3)) @ (zero_zero @ A))))))))).
% 0.32/1.60 thf(fact_79_less__iff__diff__less__0, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => ((ord_less @ A) = (^ [A4:A,B3:A]: ((((ord_less @ A) @ (((minus_minus @ A) @ A4) @ B3)) @ (zero_zero @ A))))))))).
% 0.32/1.60 thf(fact_80_inner__ge__zero, axiom, (! [A:$tType]: (((inner_real_inner @ A) => (! [X:A]: ((((ord_less_eq @ real) @ (zero_zero @ real)) @ (((inner_780170721_inner @ A) @ X) @ X)))))))).
% 0.32/1.60 thf(fact_81_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))))))).
% 0.32/1.60 thf(fact_82_that, axiom, (! [D3:real,B4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D3) => ((((ord_less @ real) @ (zero_zero @ real)) @ B4) => ((! [X4:a,Y6:a]: (((((member @ a) @ X4) @ (((line_closed_segment @ a) @ a2) @ b)) => ((((ord_less_eq @ real) @ (((real_V2000881966t_dist @ a) @ X4) @ Y6)) @ D3) => (((ord_less_eq @ real) @ B4) @ (((inner_780170721_inner @ a) @ (f @ Y6)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ a2) @ b)))))))) => thesis)))))).
% 0.32/1.60 thf(fact_83_inner__nonneg__nonneg, axiom, (! [A:$tType]: (((ordere890947078_space @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B) => (((ord_less_eq @ real) @ (zero_zero @ real)) @ (((inner_780170721_inner @ A) @ A2) @ B)))))))))).
% 0.32/1.60 thf(fact_84_in__closed__segment__iff__rot, axiom, (! [A2:a,B:a,X:a]: (((A2 != B) => ((((member @ a) @ X) @ (((line_closed_segment @ a) @ A2) @ B)) = (((((inner_780170721_inner @ a) @ (((minus_minus @ a) @ X) @ A2)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ B) @ A2))) = (zero_zero @ real)) & (((member @ real) @ (((inner_780170721_inner @ a) @ X) @ (((minus_minus @ a) @ B) @ A2))) @ (((set_or331188842AtMost @ real) @ (((inner_780170721_inner @ a) @ A2) @ (((minus_minus @ a) @ B) @ A2))) @ (((inner_780170721_inner @ a) @ B) @ (((minus_minus @ a) @ B) @ A2)))))))))).
% 0.32/1.60 thf(fact_85_sectionD_I5_J, axiom, (! [S2:(a > real),Ds:(a > ((bounde2145540817linfun @ a) @ real)),S3:(set @ a)]: ((((((((reacha1084862253ection @ a) @ f) @ x) @ S2) @ Ds) @ S3) => (((ord_less_eq @ (set @ a)) @ S3) @ x))))).
% 0.32/1.60 thf(fact_86_transversal__segment__exists, axiom, (! [X:a]: (((((member @ a) @ X) @ x) => (((f @ X) != (zero_zero @ a)) => (~ ((! [A5:a,B5:a]: (((((member @ a) @ X) @ (((line_open_segment @ a) @ A5) @ B5)) => (~ ((((((poinca272511729egment @ a) @ f) @ x) @ A5) @ B5))))))))))))).
% 0.32/1.60 thf(fact_87_invariant__iff__compl__invariant, axiom, (! [M2:(set @ a)]: ((((((auto_ll_on_invariant @ a) @ f) @ x) @ M2) = ((((auto_ll_on_invariant @ a) @ f) @ x) @ (((minus_minus @ (set @ a)) @ x) @ M2)))))).
% 0.32/1.60 thf(fact_88_cball__eq__cball__iff, axiom, (! [A:$tType]: (((euclid925273238_space @ A) => (! [X:A,D2:real,Y:A,E:real]: ((((((elemen321786957_cball @ A) @ X) @ D2) = (((elemen321786957_cball @ A) @ Y) @ E)) = (((((ord_less @ real) @ D2) @ (zero_zero @ real)) & (((ord_less @ real) @ E) @ (zero_zero @ real))) | ((X = Y) & (D2 = E)))))))))).
% 0.32/1.60 thf(fact_89_Bolzano, axiom, (! [A2:real,B:real,P:(real > (real > $o))]: (((((ord_less_eq @ real) @ A2) @ B) => ((! [A5:real,B5:real,C2:real]: ((((P @ A5) @ B5) => (((P @ B5) @ C2) => ((((ord_less_eq @ real) @ A5) @ B5) => ((((ord_less_eq @ real) @ B5) @ C2) => ((P @ A5) @ C2))))))) => ((! [X4:real]: (((((ord_less_eq @ real) @ A2) @ X4) => ((((ord_less_eq @ real) @ X4) @ B) => (? [D4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D4) & (! [A5:real,B5:real]: ((((((ord_less_eq @ real) @ A5) @ X4) & ((((ord_less_eq @ real) @ X4) @ B5) & (((ord_less @ real) @ (((minus_minus @ real) @ B5) @ A5)) @ D4))) => ((P @ A5) @ B5))))))))))) => ((P @ A2) @ B))))))).
% 0.32/1.60 thf(fact_90_mem__cball, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [Y:A,X:A,E:real]: (((((member @ A) @ Y) @ (((elemen321786957_cball @ A) @ X) @ E)) = (((ord_less_eq @ real) @ (((real_V2000881966t_dist @ A) @ X) @ Y)) @ E)))))))).
% 0.32/1.60 thf(fact_91_perfect__choose__dist, axiom, (! [A:$tType]: ((((real_V2090557954_space @ A) & (topolo890362671_space @ A)) => (! [R:real,X:A]: (((((ord_less @ real) @ (zero_zero @ real)) @ R) => (? [A5:A]: (((A5 != X) & (((ord_less @ real) @ (((real_V2000881966t_dist @ A) @ A5) @ X)) @ R))))))))))).
% 0.32/1.60 thf(fact_92_interval__inner__leI_I2_J, axiom, (! [A:$tType]: (((ordere890947078_space @ A) => (! [X:A,A2:A,B:A,I:A]: (((((member @ A) @ X) @ (((set_or331188842AtMost @ A) @ A2) @ B)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ I) => (((ord_less_eq @ real) @ (((inner_780170721_inner @ A) @ X) @ I)) @ (((inner_780170721_inner @ A) @ B) @ I)))))))))).
% 0.32/1.60 thf(fact_93_interval__inner__leI_I1_J, axiom, (! [A:$tType]: (((ordere890947078_space @ A) => (! [X:A,A2:A,B:A,I:A]: (((((member @ A) @ X) @ (((set_or331188842AtMost @ A) @ A2) @ B)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ I) => (((ord_less_eq @ real) @ (((inner_780170721_inner @ A) @ A2) @ I)) @ (((inner_780170721_inner @ A) @ X) @ I)))))))))).
% 0.32/1.60 thf(fact_94_eucl__less__le__not__le, axiom, (! [A:$tType]: (((ordere890947078_space @ A) => ((ord_less @ A) = (^ [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) & (~ ((((ord_less_eq @ A) @ Y4) @ X3))))))))))).
% 0.32/1.60 thf(fact_95_invariant__def, axiom, (! [M2:(set @ a)]: ((((((auto_ll_on_invariant @ a) @ f) @ x) @ M2) = (! [X3:a]: (((((member @ a) @ X3) @ M2) => (((((oDE_au1039603466rapped @ a) @ f) @ x) @ X3) @ M2)))))))).
% 0.32/1.60 thf(fact_96_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)))))).
% 0.32/1.60 thf(fact_97_zero__less__dist__iff, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X:A,Y:A]: (((((ord_less @ real) @ (zero_zero @ real)) @ (((real_V2000881966t_dist @ A) @ X) @ Y)) = (X != Y)))))))).
% 0.32/1.60 thf(fact_98_dist__le__zero__iff, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ real) @ (((real_V2000881966t_dist @ A) @ X) @ Y)) @ (zero_zero @ real)) = (X = Y)))))))).
% 0.32/1.60 thf(fact_99_atLeastatMost__subset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B:A,C:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or331188842AtMost @ A) @ A2) @ B)) @ (((set_or331188842AtMost @ A) @ C) @ D2)) = ((~ ((((ord_less_eq @ A) @ A2) @ B))) | ((((ord_less_eq @ A) @ C) @ A2) & (((ord_less_eq @ A) @ B) @ D2)))))))))).
% 0.32/1.60 thf(fact_100_dist__self, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X:A]: (((((real_V2000881966t_dist @ A) @ X) @ X) = (zero_zero @ real)))))))).
% 0.32/1.60 thf(fact_101_atLeastAtMost__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,L:A,U2:A]: (((((member @ A) @ I) @ (((set_or331188842AtMost @ A) @ L) @ U2)) = ((((ord_less_eq @ A) @ L) @ I) & (((ord_less_eq @ A) @ I) @ U2))))))))).
% 0.32/1.60 thf(fact_102_Icc__eq__Icc, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,H:A,L2:A,H2:A]: ((((((set_or331188842AtMost @ A) @ L) @ H) = (((set_or331188842AtMost @ A) @ L2) @ H2)) = (((L = L2) & (H = H2)) | ((~ ((((ord_less_eq @ A) @ L) @ H))) & (~ ((((ord_less_eq @ A) @ L2) @ H2)))))))))))).
% 0.32/1.60 thf(fact_103_dist__eq__0__iff, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X:A,Y:A]: ((((((real_V2000881966t_dist @ A) @ X) @ Y) = (zero_zero @ real)) = (X = Y)))))))).
% 0.32/1.60 thf(fact_104_dist__commute, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => ((real_V2000881966t_dist @ A) = (^ [X3:A,Y4:A]: ((((real_V2000881966t_dist @ A) @ Y4) @ X3)))))))).
% 0.32/1.60 thf(fact_105_dist__commute__lessI, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [Y:A,X:A,E:real]: (((((ord_less @ real) @ (((real_V2000881966t_dist @ A) @ Y) @ X)) @ E) => (((ord_less @ real) @ (((real_V2000881966t_dist @ A) @ X) @ Y)) @ E)))))))).
% 0.32/1.60 thf(fact_106_atLeastatMost__psubset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B:A,C:A,D2:A]: (((((ord_less @ (set @ A)) @ (((set_or331188842AtMost @ A) @ A2) @ B)) @ (((set_or331188842AtMost @ A) @ C) @ D2)) = (((~ ((((ord_less_eq @ A) @ A2) @ B))) | ((((ord_less_eq @ A) @ C) @ A2) & ((((ord_less_eq @ A) @ B) @ D2) & ((((ord_less @ A) @ C) @ A2) | (((ord_less @ A) @ B) @ D2))))) & (((ord_less_eq @ A) @ C) @ D2))))))))).
% 0.32/1.60 thf(fact_107_zero__le__dist, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X:A,Y:A]: ((((ord_less_eq @ real) @ (zero_zero @ real)) @ (((real_V2000881966t_dist @ A) @ X) @ Y)))))))).
% 0.32/1.60 thf(fact_108_dist__not__less__zero, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X:A,Y:A]: ((~ ((((ord_less @ real) @ (((real_V2000881966t_dist @ A) @ X) @ Y)) @ (zero_zero @ real)))))))))).
% 0.32/1.60 thf(fact_109_dist__pos__lt, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X:A,Y:A]: (((X != Y) => (((ord_less @ real) @ (zero_zero @ real)) @ (((real_V2000881966t_dist @ A) @ X) @ Y))))))))).
% 0.32/1.60 thf(fact_110_rev_Operiodic__orbit__period_I1_J, axiom, (! [X:a]: ((((((period862636932_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) => (((ord_less @ real) @ (zero_zero @ real)) @ ((((period1153813292period @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X)))))).
% 0.32/1.60 thf(fact_111_periodic__orbit__def, axiom, (! [X:a]: ((((((period862636932_orbit @ a) @ f) @ x) @ X) = (((((period385816147_orbit @ a) @ f) @ x) @ X) & (((ord_less @ real) @ (zero_zero @ real)) @ ((((period1153813292period @ a) @ f) @ x) @ X))))))).
% 0.32/1.60 thf(fact_112_rev_Ofixed__point__imp__closed__orbit__period__zero_I2_J, axiom, (! [X:a]: (((((member @ a) @ X) @ x) => (((((uminus_uminus @ (a > a)) @ f) @ X) = (zero_zero @ a)) => (((((period1153813292period @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) = (zero_zero @ real))))))).
% 0.32/1.60 thf(fact_113_closed__orbit__period__zero__fixed__point, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ f) @ x) @ X) => ((((((period1153813292period @ a) @ f) @ x) @ X) = (zero_zero @ real)) => ((f @ X) = (zero_zero @ a))))))).
% 0.32/1.60 thf(fact_114_closed__orbit__period__nonneg, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ f) @ x) @ X) => (((ord_less_eq @ real) @ (zero_zero @ real)) @ ((((period1153813292period @ a) @ f) @ x) @ X)))))).
% 0.32/1.60 thf(fact_115_closed__orbit__in__domain, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ f) @ x) @ X) => (((member @ a) @ X) @ x))))).
% 0.32/1.60 thf(fact_116_add_Oinverse__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ A2)) = A2))))))).
% 0.32/1.60 thf(fact_117_neg__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B:A]: (((((uminus_uminus @ A) @ A2) = ((uminus_uminus @ A) @ B)) = (A2 = B)))))))).
% 0.32/1.60 thf(fact_118_rev_Oclosed__orbit__eq__rev, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) = ((((period385816147_orbit @ a) @ f) @ x) @ X))))).
% 0.32/1.60 thf(fact_119_rev_Oclosed__orbit__in__domain, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) => (((member @ a) @ X) @ x))))).
% 0.32/1.60 thf(fact_120_rev_Oinvariant__eq__rev, axiom, (! [M2:(set @ a)]: ((((((auto_ll_on_invariant @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ M2) = ((((auto_ll_on_invariant @ a) @ f) @ x) @ M2))))).
% 0.32/1.60 thf(fact_121_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)))))).
% 0.32/1.60 thf(fact_122_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)))))).
% 0.32/1.60 thf(fact_123_rev_OsectionD_I5_J, axiom, (! [S2:(a > real),Ds:(a > ((bounde2145540817linfun @ a) @ real)),S3:(set @ a)]: ((((((((reacha1084862253ection @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ S2) @ Ds) @ S3) => (((ord_less_eq @ (set @ a)) @ S3) @ x))))).
% 0.32/1.60 thf(fact_124_rev_Oinvariant__iff__compl__invariant, axiom, (! [M2:(set @ a)]: ((((((auto_ll_on_invariant @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ M2) = ((((auto_ll_on_invariant @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ (((minus_minus @ (set @ a)) @ x) @ M2)))))).
% 0.32/1.60 thf(fact_125_closed__orbit__periodic, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ f) @ x) @ X) => (((f @ X) != (zero_zero @ a)) => ((((period862636932_orbit @ a) @ f) @ x) @ X)))))).
% 0.32/1.60 thf(fact_126_rev_Oinvariant__def, axiom, (! [M2:(set @ a)]: ((((((auto_ll_on_invariant @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ M2) = (! [X3:a]: (((((member @ a) @ X3) @ M2) => (((((oDE_au1039603466rapped @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X3) @ M2)))))))).
% 0.32/1.60 thf(fact_127_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)))))).
% 0.32/1.60 thf(fact_128_add_Oinverse__neutral, axiom, (! [A:$tType]: (((group_add @ A) => (((uminus_uminus @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 0.32/1.60 thf(fact_129_neg__0__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: ((((zero_zero @ A) = ((uminus_uminus @ A) @ A2)) = ((zero_zero @ A) = A2)))))))).
% 0.32/1.60 thf(fact_130_neg__equal__0__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((uminus_uminus @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 0.32/1.60 thf(fact_131_equal__neg__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((A2 = ((uminus_uminus @ A) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 0.32/1.60 thf(fact_132_neg__equal__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((uminus_uminus @ A) @ A2) = A2) = (A2 = (zero_zero @ A))))))))).
% 0.32/1.60 thf(fact_133_neg__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ B)))))))).
% 0.32/1.60 thf(fact_134_neg__less__iff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B:A,A2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ B)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less @ A) @ A2) @ B)))))))).
% 0.32/1.60 thf(fact_135_minus__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B:A]: ((((uminus_uminus @ A) @ (((minus_minus @ A) @ A2) @ B)) = (((minus_minus @ A) @ B) @ A2)))))))).
% 0.32/1.60 thf(fact_136_inner__minus__left, axiom, (! [A:$tType]: (((inner_real_inner @ A) => (! [X:A,Y:A]: (((((inner_780170721_inner @ A) @ ((uminus_uminus @ A) @ X)) @ Y) = ((uminus_uminus @ real) @ (((inner_780170721_inner @ A) @ X) @ Y))))))))).
% 0.32/1.60 thf(fact_137_inner__minus__right, axiom, (! [A:$tType]: (((inner_real_inner @ A) => (! [X:A,Y:A]: (((((inner_780170721_inner @ A) @ X) @ ((uminus_uminus @ A) @ Y)) = ((uminus_uminus @ real) @ (((inner_780170721_inner @ A) @ X) @ Y))))))))).
% 0.32/1.60 thf(fact_138_rev_Oclosed__orbit__period__nonneg, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) => (((ord_less_eq @ real) @ (zero_zero @ real)) @ ((((period1153813292period @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X)))))).
% 0.32/1.60 thf(fact_139_rev_Oclosed__orbit__period__zero__fixed__point, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) => ((((((period1153813292period @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) = (zero_zero @ real)) => ((((uminus_uminus @ (a > a)) @ f) @ X) = (zero_zero @ a))))))).
% 0.32/1.60 thf(fact_140_rev_Operiodic__orbit__def, axiom, (! [X:a]: ((((((period862636932_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) = (((((period385816147_orbit @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X) & (((ord_less @ real) @ (zero_zero @ real)) @ ((((period1153813292period @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X))))))).
% 0.32/1.60 thf(fact_141_neg__0__le__iff__le, axiom, (! [A:$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))))))))).
% 0.32/1.60 thf(fact_142_neg__le__0__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 0.32/1.60 thf(fact_143_less__eq__neg__nonpos, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 0.32/1.60 thf(fact_144_neg__less__eq__nonneg, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 0.32/1.60 thf(fact_145_less__neg__neg, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less @ A) @ A2) @ ((uminus_uminus @ A) @ A2)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 0.32/1.60 thf(fact_146_neg__less__pos, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 0.32/1.60 thf(fact_147_neg__0__less__iff__less, axiom, (! [A:$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))))))))).
% 0.32/1.60 thf(fact_148_neg__less__0__iff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 0.32/1.60 thf(fact_149_diff__0, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A2) = ((uminus_uminus @ A) @ A2)))))))).
% 0.32/1.60 thf(fact_150_le__imp__neg__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B)) @ ((uminus_uminus @ A) @ A2))))))))).
% 0.32/1.60 thf(fact_151_minus__le__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ B) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B)) @ A2)))))))).
% 0.32/1.60 thf(fact_152_le__minus__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ B)) = (((ord_less_eq @ A) @ B) @ ((uminus_uminus @ A) @ A2))))))))).
% 0.32/1.60 thf(fact_153_minus__less__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A2)) @ B) = (((ord_less @ A) @ ((uminus_uminus @ A) @ B)) @ A2)))))))).
% 0.32/1.60 thf(fact_154_less__minus__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ ((uminus_uminus @ A) @ B)) = (((ord_less @ A) @ B) @ ((uminus_uminus @ A) @ A2))))))))).
% 0.32/1.60 thf(fact_155_minus__diff__commute, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [B:A,A2:A]: (((((minus_minus @ A) @ ((uminus_uminus @ A) @ B)) @ A2) = (((minus_minus @ A) @ ((uminus_uminus @ A) @ A2)) @ B)))))))).
% 0.32/1.60 thf(fact_156_equation__minus__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B:A]: (((A2 = ((uminus_uminus @ A) @ B)) = (B = ((uminus_uminus @ A) @ A2))))))))).
% 0.32/1.60 thf(fact_157_minus__equation__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B:A]: (((((uminus_uminus @ A) @ A2) = B) = (((uminus_uminus @ A) @ B) = A2)))))))).
% 0.32/1.60 thf(fact_158_verit__minus__simplify_I3_J, axiom, (! [B2:$tType]: (((group_add @ B2) => (! [B:B2]: (((((minus_minus @ B2) @ (zero_zero @ B2)) @ B) = ((uminus_uminus @ B2) @ B)))))))).
% 0.32/1.60 thf(fact_159_psubsetI, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B6) => ((A3 != B6) => (((ord_less @ (set @ A)) @ A3) @ B6)))))).
% 0.32/1.60 thf(fact_160_uminus__apply, axiom, (! [B2:$tType,A:$tType]: (((uminus @ B2) => ((uminus_uminus @ (A > B2)) = (^ [A6:(A > B2),X3:A]: (((uminus_uminus @ B2) @ (A6 @ X3))))))))).
% 0.32/1.60 thf(fact_161_compl__le__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X)) @ ((uminus_uminus @ A) @ Y)) = (((ord_less_eq @ A) @ Y) @ X)))))))).
% 0.32/1.60 thf(fact_162_rot__diff__commute, axiom, (! [B:a,A2:a]: ((((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ B) @ A2)) = ((uminus_uminus @ a) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ A2) @ B))))))).
% 0.32/1.60 thf(fact_163_nrm__reverse, axiom, (! [A2:a,X:a,Y:a]: (((((inner_780170721_inner @ a) @ A2) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ X) @ Y))) = (((inner_780170721_inner @ a) @ ((uminus_uminus @ a) @ A2)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ Y) @ X))))))).
% 0.32/1.60 thf(fact_164_verit__minus__simplify_I4_J, axiom, (! [B2:$tType]: (((group_add @ B2) => (! [B:B2]: ((((uminus_uminus @ B2) @ ((uminus_uminus @ B2) @ B)) = B))))))).
% 0.32/1.60 thf(fact_165_double__compl, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ X)) = X))))))).
% 0.32/1.60 thf(fact_166_compl__eq__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X:A,Y:A]: (((((uminus_uminus @ A) @ X) = ((uminus_uminus @ A) @ Y)) = (X = Y)))))))).
% 0.32/1.60 thf(fact_167_minus__apply, axiom, (! [B2:$tType,A:$tType]: (((minus @ B2) => ((minus_minus @ (A > B2)) = (^ [A6:(A > B2),B7:(A > B2),X3:A]: ((((minus_minus @ B2) @ (A6 @ X3)) @ (B7 @ X3))))))))).
% 0.32/1.60 thf(fact_168_Compl__subset__Compl__iff, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A3)) @ ((uminus_uminus @ (set @ A)) @ B6)) = (((ord_less_eq @ (set @ A)) @ B6) @ A3))))).
% 0.32/1.60 thf(fact_169_Compl__anti__mono, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B6) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ B6)) @ ((uminus_uminus @ (set @ A)) @ A3)))))).
% 0.32/1.60 thf(fact_170_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ A3) => (A3 = B6)))))).
% 0.32/1.60 thf(fact_171_subsetI, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((! [X4:A]: (((((member @ A) @ X4) @ A3) => (((member @ A) @ X4) @ B6)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B6))))).
% 0.32/1.60 thf(fact_172_Diff__idemp, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B6)) @ B6) = (((minus_minus @ (set @ A)) @ A3) @ B6))))).
% 0.32/1.60 thf(fact_173_Diff__iff, axiom, (! [A:$tType,C:A,A3:(set @ A),B6:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A3) @ B6)) = ((((member @ A) @ C) @ A3) & (~ ((((member @ A) @ C) @ B6)))))))).
% 0.32/1.60 thf(fact_174_DiffI, axiom, (! [A:$tType,C:A,A3:(set @ A),B6:(set @ A)]: (((((member @ A) @ C) @ A3) => ((~ ((((member @ A) @ C) @ B6))) => (((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A3) @ B6))))))).
% 0.32/1.60 thf(fact_175_rot__rot, axiom, (! [V:a]: ((((poinca1750768982en_rot @ a) @ ((poinca1750768982en_rot @ a) @ V)) = ((uminus_uminus @ a) @ V))))).
% 0.32/1.60 thf(fact_176_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B:A]: (((A2 = B) | ((~ ((((ord_less_eq @ A) @ A2) @ B))) | (~ ((((ord_less_eq @ A) @ B) @ A2))))))))))).
% 0.32/1.60 thf(fact_177_verit__comp__simplify1_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 0.32/1.60 thf(fact_178_verit__negate__coefficient_I3_J, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A]: (((A2 = B) => (((uminus_uminus @ A) @ A2) = ((uminus_uminus @ A) @ B))))))))).
% 0.32/1.60 thf(fact_179_fun__diff__def, axiom, (! [B2:$tType,A:$tType]: (((minus @ B2) => ((minus_minus @ (A > B2)) = (^ [A6:(A > B2),B7:(A > B2),X3:A]: ((((minus_minus @ B2) @ (A6 @ X3)) @ (B7 @ X3))))))))).
% 0.32/1.60 thf(fact_180_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X3:A]: (((P @ X3) => (Q @ X3)))))))).
% 0.32/1.60 thf(fact_181_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y5:(set @ A),Z4:(set @ A)]: ((Y5 = Z4))) = (^ [A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) & (((ord_less_eq @ (set @ A)) @ B7) @ A6)))))))).
% 0.32/1.60 thf(fact_182_subset__trans, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ C3) => (((ord_less_eq @ (set @ A)) @ A3) @ C3)))))).
% 0.32/1.60 thf(fact_183_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) => (Q @ X4)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 0.32/1.60 thf(fact_184_subset__refl, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ A3)))).
% 0.32/1.60 thf(fact_185_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: ((! [T:A]: (((((member @ A) @ T) @ A6) => (((member @ A) @ T) @ B7)))))))))).
% 0.32/1.60 thf(fact_186_equalityD2, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((A3 = B6) => (((ord_less_eq @ (set @ A)) @ B6) @ A3))))).
% 0.32/1.60 thf(fact_187_equalityD1, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((A3 = B6) => (((ord_less_eq @ (set @ A)) @ A3) @ B6))))).
% 0.32/1.60 thf(fact_188_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((member @ A) @ X3) @ B7)))))))))).
% 0.32/1.60 thf(fact_189_equalityE, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((A3 = B6) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B6) => (~ ((((ord_less_eq @ (set @ A)) @ B6) @ A3)))))))))).
% 0.32/1.60 thf(fact_190_subsetD, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B6) => ((((member @ A) @ C) @ A3) => (((member @ A) @ C) @ B6)))))).
% 0.32/1.60 thf(fact_191_in__mono, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B6) => ((((member @ A) @ X) @ A3) => (((member @ A) @ X) @ B6)))))).
% 0.32/1.60 thf(fact_192_DiffD2, axiom, (! [A:$tType,C:A,A3:(set @ A),B6:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A3) @ B6)) => (~ ((((member @ A) @ C) @ B6))))))).
% 0.32/1.60 thf(fact_193_DiffD1, axiom, (! [A:$tType,C:A,A3:(set @ A),B6:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A3) @ B6)) => (((member @ A) @ C) @ A3))))).
% 0.32/1.60 thf(fact_194_DiffE, axiom, (! [A:$tType,C:A,A3:(set @ A),B6:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A3) @ B6)) => (~ (((((member @ A) @ C) @ A3) => (((member @ A) @ C) @ B6)))))))).
% 0.32/1.60 thf(fact_195_psubset__trans, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B6) => ((((ord_less @ (set @ A)) @ B6) @ C3) => (((ord_less @ (set @ A)) @ A3) @ C3)))))).
% 0.32/1.60 thf(fact_196_psubsetD, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A),C:A]: (((((ord_less @ (set @ A)) @ A3) @ B6) => ((((member @ A) @ C) @ A3) => (((member @ A) @ C) @ B6)))))).
% 0.32/1.60 thf(fact_197_verit__comp__simplify1_I3_J, axiom, (! [B2:$tType]: (((linorder @ B2) => (! [B8:B2,A7:B2]: (((~ ((((ord_less_eq @ B2) @ B8) @ A7))) = (((ord_less @ B2) @ A7) @ B8)))))))).
% 0.32/1.60 thf(fact_198_compl__le__swap2, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y)) @ X) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X)) @ Y)))))))).
% 0.32/1.60 thf(fact_199_compl__le__swap1, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ ((uminus_uminus @ A) @ X)) => (((ord_less_eq @ A) @ X) @ ((uminus_uminus @ A) @ Y))))))))).
% 0.32/1.60 thf(fact_200_compl__mono, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y)) @ ((uminus_uminus @ A) @ X))))))))).
% 0.32/1.60 thf(fact_201_verit__negate__coefficient_I2_J, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (((ord_less @ A) @ ((uminus_uminus @ A) @ B)) @ ((uminus_uminus @ A) @ A2))))))))).
% 0.32/1.60 thf(fact_202_compl__less__swap1, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y:A,X:A]: (((((ord_less @ A) @ Y) @ ((uminus_uminus @ A) @ X)) => (((ord_less @ A) @ X) @ ((uminus_uminus @ A) @ Y))))))))).
% 0.32/1.60 thf(fact_203_compl__less__swap2, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y:A,X:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ Y)) @ X) => (((ord_less @ A) @ ((uminus_uminus @ A) @ X)) @ Y)))))))).
% 0.32/1.60 thf(fact_204_compl__less__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ X)) @ ((uminus_uminus @ A) @ Y)) = (((ord_less @ A) @ Y) @ X)))))))).
% 0.32/1.60 thf(fact_205_fun__Compl__def, axiom, (! [B2:$tType,A:$tType]: (((uminus @ B2) => ((uminus_uminus @ (A > B2)) = (^ [A6:(A > B2),X3:A]: (((uminus_uminus @ B2) @ (A6 @ X3))))))))).
% 0.32/1.60 thf(fact_206_double__diff, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ C3) => ((((minus_minus @ (set @ A)) @ B6) @ (((minus_minus @ (set @ A)) @ C3) @ A3)) = A3)))))).
% 0.32/1.60 thf(fact_207_Diff__subset, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B6)) @ A3)))).
% 0.32/1.60 thf(fact_208_Diff__mono, axiom, (! [A:$tType,A3:(set @ A),C3:(set @ A),D5:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ C3) => ((((ord_less_eq @ (set @ A)) @ D5) @ B6) => (((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B6)) @ (((minus_minus @ (set @ A)) @ C3) @ D5))))))).
% 0.32/1.60 thf(fact_209_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: (((((ord_less @ (set @ A)) @ A6) @ B7) | (A6 = B7)))))))).
% 0.32/1.60 thf(fact_210_subset__psubset__trans, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B6) => ((((ord_less @ (set @ A)) @ B6) @ C3) => (((ord_less @ (set @ A)) @ A3) @ C3)))))).
% 0.32/1.60 thf(fact_211_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) & (~ ((((ord_less_eq @ (set @ A)) @ B7) @ A6)))))))))).
% 0.32/1.60 thf(fact_212_psubset__subset__trans, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ C3) => (((ord_less @ (set @ A)) @ A3) @ C3)))))).
% 0.32/1.60 thf(fact_213_psubset__imp__subset, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B6) => (((ord_less_eq @ (set @ A)) @ A3) @ B6))))).
% 0.32/1.60 thf(fact_214_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) & (A6 != B7)))))))).
% 0.32/1.60 thf(fact_215_psubsetE, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B6) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B6) => (((ord_less_eq @ (set @ A)) @ B6) @ A3)))))))).
% 0.32/1.60 thf(fact_216_psubset__imp__ex__mem, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B6) => (? [B5:A]: ((((member @ A) @ B5) @ (((minus_minus @ (set @ A)) @ B6) @ A3)))))))).
% 0.32/1.60 thf(fact_217_rev_Oopen__existence__ivl_H, axiom, (! [X:a]: (((((member @ a) @ X) @ x) => (~ ((! [A5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ A5) => (~ ((((ord_less_eq @ (set @ real)) @ (((set_or331188842AtMost @ real) @ ((uminus_uminus @ real) @ A5)) @ A5)) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X))))))))))))).
% 0.32/1.60 thf(fact_218_open__existence__ivl_H, axiom, (! [X:a]: (((((member @ a) @ X) @ x) => (~ ((! [A5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ A5) => (~ ((((ord_less_eq @ (set @ real)) @ (((set_or331188842AtMost @ real) @ ((uminus_uminus @ real) @ A5)) @ A5)) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X))))))))))))).
% 0.32/1.60 thf(fact_219_rev_Oopen__existence__ivl0, axiom, (! [X:a]: (((((member @ a) @ X) @ x) => (? [A5:real,B5:real]: (((((ord_less @ real) @ A5) @ (zero_zero @ real)) & ((((ord_less @ real) @ (zero_zero @ real)) @ B5) & (((ord_less_eq @ (set @ real)) @ (((set_or331188842AtMost @ real) @ A5) @ B5)) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X)))))))))).
% 0.32/1.60 thf(fact_220_existence__ivl0__cong, axiom, (! [B2:$tType,Y7:(set @ a),G:(a > a),X0:a]: (((x = Y7) => ((! [X4:a,T2:B2]: (((((member @ a) @ X4) @ Y7) => ((f @ X4) = (G @ X4))))) => (((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0) = ((((auto_l1112008849e_ivl0 @ a) @ G) @ Y7) @ X0))))))).
% 0.32/1.60 thf(fact_221_existence__ivl__zero, axiom, (! [X0:a]: (((((member @ a) @ X0) @ x) => (((member @ real) @ (zero_zero @ real)) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)))))).
% 0.32/1.60 thf(fact_222_rev_Oexistence__ivl0__cong, axiom, (! [B2:$tType,Y7:(set @ a),G:(a > a),X0:a]: (((x = Y7) => ((! [X4:a,T2:B2]: (((((member @ a) @ X4) @ Y7) => ((((uminus_uminus @ (a > a)) @ f) @ X4) = (G @ X4))))) => (((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0) = ((((auto_l1112008849e_ivl0 @ a) @ G) @ Y7) @ X0))))))).
% 0.32/1.60 thf(fact_223_general_Omem__existence__ivl__iv__defined_I2_J, axiom, (! [T3:real,T0:real,X0:a]: (((((member @ real) @ (((minus_minus @ real) @ T3) @ T0)) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) => (((member @ a) @ X0) @ x))))).
% 0.32/1.60 thf(fact_224_Compl__eq__Compl__iff, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((((uminus_uminus @ (set @ A)) @ A3) = ((uminus_uminus @ (set @ A)) @ B6)) = (A3 = B6))))).
% 0.32/1.60 thf(fact_225_Compl__iff, axiom, (! [A:$tType,C:A,A3:(set @ A)]: (((((member @ A) @ C) @ ((uminus_uminus @ (set @ A)) @ A3)) = (~ ((((member @ A) @ C) @ A3))))))).
% 0.32/1.60 thf(fact_226_ComplI, axiom, (! [A:$tType,C:A,A3:(set @ A)]: (((~ ((((member @ A) @ C) @ A3))) => (((member @ A) @ C) @ ((uminus_uminus @ (set @ A)) @ A3)))))).
% 0.32/1.60 thf(fact_227_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)))))).
% 0.32/1.60 thf(fact_228_rev_Ogeneral_Omem__existence__ivl__iv__defined_I2_J, axiom, (! [T3:real,T0:real,X0:a]: (((((member @ real) @ (((minus_minus @ real) @ T3) @ T0)) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)) => (((member @ a) @ X0) @ x))))).
% 0.32/1.60 thf(fact_229_local_Oivl__subset__existence__ivl, axiom, (! [T3:real,X0:a]: (((((member @ real) @ T3) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) => (((ord_less_eq @ (set @ real)) @ (((set_or331188842AtMost @ real) @ (zero_zero @ real)) @ T3)) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)))))).
% 0.32/1.60 thf(fact_230_local_Oivl__subset__existence__ivl_H, axiom, (! [T3:real,X0:a]: (((((member @ real) @ T3) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) => (((ord_less_eq @ (set @ real)) @ (((set_or331188842AtMost @ real) @ T3) @ (zero_zero @ real))) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)))))).
% 0.32/1.60 thf(fact_231_open__existence__ivl0, axiom, (! [X:a]: (((((member @ a) @ X) @ x) => (? [A5:real,B5:real]: (((((ord_less @ real) @ A5) @ (zero_zero @ real)) & ((((ord_less @ real) @ (zero_zero @ real)) @ B5) & (((ord_less_eq @ (set @ real)) @ (((set_or331188842AtMost @ real) @ A5) @ B5)) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X)))))))))).
% 0.32/1.60 thf(fact_232_local_Orev_Oivl__subset__existence__ivl, axiom, (! [T3:real,X0:a]: (((((member @ real) @ T3) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)) => (((ord_less_eq @ (set @ real)) @ (((set_or331188842AtMost @ real) @ (zero_zero @ real)) @ T3)) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)))))).
% 0.32/1.60 thf(fact_233_local_Orev_Oivl__subset__existence__ivl_H, axiom, (! [T3:real,X0:a]: (((((member @ real) @ T3) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)) => (((ord_less_eq @ (set @ real)) @ (((set_or331188842AtMost @ real) @ T3) @ (zero_zero @ real))) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)))))).
% 0.32/1.60 thf(fact_234_local_Omem__existence__ivl__iv__defined_I2_J, axiom, (! [T3:real,X0:a]: (((((member @ real) @ T3) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) => (((member @ a) @ X0) @ x))))).
% 0.32/1.60 thf(fact_235_local_Orev_Omem__existence__ivl__iv__defined_I2_J, axiom, (! [T3:real,X0:a]: (((((member @ real) @ T3) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)) => (((member @ a) @ X0) @ x))))).
% 0.32/1.60 thf(fact_236_double__complement, axiom, (! [A:$tType,A3:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A3)) = A3)))).
% 0.32/1.60 thf(fact_237_ComplD, axiom, (! [A:$tType,C:A,A3:(set @ A)]: (((((member @ A) @ C) @ ((uminus_uminus @ (set @ A)) @ A3)) => (~ ((((member @ A) @ C) @ A3))))))).
% 0.32/1.60 thf(fact_238_rev_Omvar_Ointerval__axioms, axiom, (! [X0:a]: ((initia826609931terval @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0))))).
% 0.32/1.60 thf(fact_239_mvar_Ointerval__axioms, axiom, (! [X0:a]: ((initia826609931terval @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0))))).
% 0.32/1.60 thf(fact_240_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))))))).
% 0.32/1.60 thf(fact_241_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)))))))).
% 0.32/1.60 thf(fact_242_interval__axioms, axiom, (initia826609931terval @ (top_top @ (set @ real)))).
% 0.32/1.60 thf(fact_243_UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 0.32/1.60 thf(fact_244_local_Omem__existence__ivl__subset, axiom, (! [T3:real,X0:a]: (((((member @ real) @ T3) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) => (((member @ real) @ T3) @ (top_top @ (set @ real))))))).
% 0.32/1.60 thf(fact_245_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))))))).
% 0.32/1.60 thf(fact_246_local_Orev_Omem__existence__ivl__subset, axiom, (! [T3:real,X0:a]: (((((member @ real) @ T3) @ ((((auto_l1112008849e_ivl0 @ a) @ ((uminus_uminus @ (a > a)) @ f)) @ x) @ X0)) => (((member @ real) @ T3) @ (top_top @ (set @ real))))))).
% 0.32/1.60 thf(fact_247_general_Omem__existence__ivl__iv__defined_I1_J, axiom, (! [T3:real,T0:real,X0:a]: (((((member @ real) @ (((minus_minus @ real) @ T3) @ T0)) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) => (((member @ real) @ T0) @ (top_top @ (set @ real))))))).
% 0.32/1.60 thf(fact_248_general_Oexistence__ivl__initial__time, axiom, (! [T0:real,X0:a]: (((((member @ real) @ T0) @ (top_top @ (set @ real))) => ((((member @ a) @ X0) @ x) => (((member @ real) @ (((minus_minus @ real) @ T0) @ T0)) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0))))))).
% 0.32/1.60 thf(fact_249_mem__existence__ivl__shift__autonomous2, axiom, (! [T3:real,S2:real,X:a]: (((((member @ real) @ (((minus_minus @ real) @ T3) @ S2)) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X)) => ((((member @ a) @ X) @ x) => ((! [S:real,T2:real,X4:a]: (((((member @ a) @ X4) @ x) => ((f @ X4) = (f @ X4))))) => (((top_top @ (set @ real)) = (top_top @ (set @ real))) => (((member @ real) @ (((minus_minus @ real) @ T3) @ S2)) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X))))))))).
% 0.32/1.60 thf(fact_250_general_Omem__existence__ivl__subset, axiom, (! [T3:real,T0:real,X0:a]: (((((member @ real) @ (((minus_minus @ real) @ T3) @ T0)) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) => (((member @ real) @ T3) @ (top_top @ (set @ real))))))).
% 0.32/1.60 thf(fact_251_local_Oexistence__ivl__subset, axiom, (! [X0:a]: ((((ord_less_eq @ (set @ real)) @ ((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X0)) @ (top_top @ (set @ real)))))).
% 0.32/1.60 thf(fact_252_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)))))))).
% 0.32/1.60 thf(fact_253_closed__orbit__global__existence, axiom, (! [X:a]: ((((((period385816147_orbit @ a) @ f) @ x) @ X) => (((((auto_l1112008849e_ivl0 @ a) @ f) @ x) @ X) = (top_top @ (set @ real))))))).
% 0.32/1.60 thf(fact_254_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))))))).
% 0.32/1.60 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___HOL_Otype, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (type @ A))))).
% 0.32/1.60 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ozero, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (zero @ A))))).
% 0.32/1.60 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ominus, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (minus @ A))))).
% 0.32/1.60 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ouminus, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (uminus @ A))))).
% 0.32/1.60 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Orderings_Oord, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (ord @ A))))).
% 0.32/1.60 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Orderings_Oorder, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (order @ A))))).
% 0.32/1.60 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ogroup__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (group_add @ A))))).
% 0.32/1.60 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Orderings_Opreorder, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (preorder @ A))))).
% 0.32/1.60 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oab__group__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (ab_group_add @ A))))).
% 0.32/1.60 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Inner__Product_Oreal__inner, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (inner_real_inner @ A))))).
% 0.32/1.60 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__group__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (ordered_ab_group_add @ A))))).
% 0.32/1.60 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocancel__comm__monoid__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (cancel1352612707id_add @ A))))).
% 0.32/1.60 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocancel__ab__semigroup__add, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (cancel146912293up_add @ A))))).
% 0.32/1.60 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Euclidean__Space_Oeuclidean__space, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (euclid925273238_space @ A))))).
% 0.32/1.60 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Real__Vector__Spaces_Ometric__space, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (real_V2090557954_space @ A))))).
% 0.32/1.60 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Topological__Spaces_Operfect__space, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (topolo890362671_space @ A))))).
% 0.32/1.60 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Ordered__Euclidean__Space_Oordered__euclidean__space, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (ordere890947078_space @ A))))).
% 0.32/1.60 thf(tcon_HOL_Obool___Countable_Ocountable, axiom, (countable @ $o)).
% 0.32/1.60 thf(tcon_Set_Oset___Countable_Ocountable_1, axiom, (! [A8:$tType]: (((finite_finite @ A8) => (countable @ (set @ A8)))))).
% 0.32/1.60 thf(tcon_fun___Countable_Ocountable_2, axiom, (! [A8:$tType,A9:$tType]: ((((finite_finite @ A8) & (countable @ A9)) => (countable @ (A8 > A9)))))).
% 0.32/1.60 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A8:$tType,A9:$tType]: ((((finite_finite @ A8) & (finite_finite @ A9)) => (finite_finite @ (A8 > A9)))))).
% 0.32/1.60 thf(tcon_Set_Oset___Finite__Set_Ofinite_3, axiom, (! [A8:$tType]: (((finite_finite @ A8) => (finite_finite @ (set @ A8)))))).
% 0.32/1.60 thf(tcon_HOL_Obool___Finite__Set_Ofinite_4, axiom, (finite_finite @ $o)).
% 0.32/1.60 thf(tcon_fun___Real__Vector__Spaces_Ometric__space, axiom, (! [A8:$tType,A9:$tType]: ((((countable @ A8) & (real_V2090557954_space @ A9)) => (real_V2090557954_space @ (A8 > A9)))))).
% 0.32/1.60 thf(tcon_fun___Lattices_Oboolean__algebra, axiom, (! [A8:$tType,A9:$tType]: (((boolean_algebra @ A9) => (boolean_algebra @ (A8 > A9)))))).
% 0.32/1.60 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 0.32/1.60 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 0.32/1.60 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 0.32/1.60 thf(tcon_fun___Groups_Ouminus, axiom, (! [A8:$tType,A9:$tType]: (((uminus @ A9) => (uminus @ (A8 > A9)))))).
% 0.32/1.60 thf(tcon_fun___Groups_Ominus, axiom, (! [A8:$tType,A9:$tType]: (((minus @ A9) => (minus @ (A8 > A9)))))).
% 0.32/1.60 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_5, axiom, (! [A8:$tType]: ((boolean_algebra @ (set @ A8))))).
% 0.32/1.60 thf(tcon_Set_Oset___Orderings_Opreorder_6, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 0.32/1.60 thf(tcon_Set_Oset___Orderings_Oorder_7, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 0.32/1.60 thf(tcon_Set_Oset___Orderings_Oord_8, axiom, (! [A8:$tType]: ((ord @ (set @ A8))))).
% 0.32/1.60 thf(tcon_Set_Oset___Groups_Ouminus_9, axiom, (! [A8:$tType]: ((uminus @ (set @ A8))))).
% 0.32/1.60 thf(tcon_Set_Oset___Groups_Ominus_10, axiom, (! [A8:$tType]: ((minus @ (set @ A8))))).
% 0.32/1.60 thf(tcon_Set_Oset___Groups_Ozero, axiom, (! [A8:$tType]: (((zero @ A8) => (zero @ (set @ A8)))))).
% 0.32/1.60 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_11, axiom, (boolean_algebra @ $o)).
% 0.32/1.60 thf(tcon_HOL_Obool___Orderings_Opreorder_12, axiom, (preorder @ $o)).
% 0.32/1.60 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 0.32/1.60 thf(tcon_HOL_Obool___Orderings_Oorder_13, axiom, (order @ $o)).
% 0.32/1.60 thf(tcon_HOL_Obool___Orderings_Oord_14, axiom, (ord @ $o)).
% 0.32/1.60 thf(tcon_HOL_Obool___Groups_Ouminus_15, axiom, (uminus @ $o)).
% 0.32/1.60 thf(tcon_HOL_Obool___Groups_Ominus_16, axiom, (minus @ $o)).
% 0.32/1.60 thf(tcon_Real_Oreal___Executable__Euclidean__Space_Oexecutable__euclidean__space, axiom, (execut510477386_space @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space, axiom, (ordere890947078_space @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Topological__Spaces_Operfect__space, axiom, (topolo890362671_space @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space_17, axiom, (real_V2090557954_space @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space, axiom, (euclid925273238_space @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add, axiom, (linord219039673up_add @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add, axiom, (cancel146912293up_add @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add, axiom, (cancel1352612707id_add @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add, axiom, (ordered_ab_group_add @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Inner__Product_Oreal__inner, axiom, (inner_real_inner @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Groups_Oab__group__add, axiom, (ab_group_add @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Orderings_Opreorder_18, axiom, (preorder @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Orderings_Olinorder_19, axiom, (linorder @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Groups_Ogroup__add, axiom, (group_add @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Orderings_Oorder_20, axiom, (order @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Orderings_Oord_21, axiom, (ord @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Groups_Ouminus_22, axiom, (uminus @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Groups_Ominus_23, axiom, (minus @ real)).
% 0.32/1.60 thf(tcon_Real_Oreal___Groups_Ozero_24, axiom, (zero @ real)).
% 0.32/1.60 thf(tfree_0, hypothesis, (execut510477386_space @ a)).
% 0.32/1.60 thf(conj_0, conjecture, (? [D4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D4) & (? [B9:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ B9) & (! [X4:a]: (((((member @ a) @ X4) @ (((elemen321786957_cball @ a) @ x2) @ D4)) => (((ord_less_eq @ real) @ B9) @ (((inner_780170721_inner @ a) @ (f @ X4)) @ ((poinca1750768982en_rot @ a) @ (((minus_minus @ a) @ a2) @ b)))))))))))))).
% 0.32/1.60 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.laqx631JN2/DTF2DTF21273.p
% 0.32/1.60 ------------------------
% 0.32/1.63 ---- Cleaned THF ---
% 0.32/1.63 thf(ty_t_Bounded__Linear__Function_Oblinfun,type,
% 0.32/1.63 bounde2145540817linfun: $tType > $tType > $tType ).
% 0.32/1.63
% 0.32/1.63 thf(ty_t_Real_Oreal,type,
% 0.32/1.63 real: $tType ).
% 0.32/1.63
% 0.32/1.63 thf(ty_t_Set_Oset,type,
% 0.32/1.63 set: $tType > $tType ).
% 0.32/1.63
% 0.32/1.63 thf(ty_tf_a,type,
% 0.32/1.63 a: $tType ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.32/1.63 finite_finite:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_HOL_Otype,type,
% 0.32/1.63 type:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Countable_Ocountable,type,
% 0.32/1.63 countable:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Groups_Ozero,type,
% 0.32/1.63 zero:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Groups_Ominus,type,
% 0.32/1.63 minus:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Groups_Ouminus,type,
% 0.32/1.63 uminus:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Orderings_Oord,type,
% 0.32/1.63 ord:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Orderings_Oorder,type,
% 0.32/1.63 order:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Groups_Ogroup__add,type,
% 0.32/1.63 group_add:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Orderings_Olinorder,type,
% 0.32/1.63 linorder:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Orderings_Opreorder,type,
% 0.32/1.63 preorder:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Groups_Oab__group__add,type,
% 0.32/1.63 ab_group_add:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 0.32/1.63 comm_monoid_diff:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Inner__Product_Oreal__inner,type,
% 0.32/1.63 inner_real_inner:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 0.32/1.63 boolean_algebra:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.32/1.63 ordered_ab_group_add:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.32/1.63 cancel1352612707id_add:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.32/1.63 cancel146912293up_add:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.32/1.63 linord219039673up_add:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Euclidean__Space_Oeuclidean__space,type,
% 0.32/1.63 euclid925273238_space:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
% 0.32/1.63 real_V2090557954_space:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Topological__Spaces_Operfect__space,type,
% 0.32/1.63 topolo890362671_space:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.32/1.63 canoni770627133id_add:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
% 0.32/1.63 ordere890947078_space:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_cl_Executable__Euclidean__Space_Oexecutable__euclidean__space,type,
% 0.32/1.63 execut510477386_space:
% 0.32/1.63 !>[A: $tType] : $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Elementary__Metric__Spaces_Ocball,type,
% 0.32/1.63 elemen321786957_cball:
% 0.32/1.63 !>[A: $tType] : ( A > real > ( set @ A ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Flow_Oauto__ll__on__open_Oexistence__ivl0,type,
% 0.32/1.63 auto_l1112008849e_ivl0:
% 0.32/1.63 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > ( set @ real ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.32/1.63 minus_minus:
% 0.32/1.63 !>[A: $tType] : ( A > A > A ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.32/1.63 uminus_uminus:
% 0.32/1.63 !>[A: $tType] : ( A > A ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.32/1.63 zero_zero:
% 0.32/1.63 !>[A: $tType] : A ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Initial__Value__Problem_Ointerval,type,
% 0.32/1.63 initia826609931terval: ( set @ real ) > $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Inner__Product_Oreal__inner__class_Oinner,type,
% 0.32/1.63 inner_780170721_inner:
% 0.32/1.63 !>[A: $tType] : ( A > A > real ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Invariance_Oauto__ll__on__open_Oinvariant,type,
% 0.32/1.63 auto_ll_on_invariant:
% 0.32/1.63 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > ( set @ A ) > $o ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Line__Segment_Oclosed__segment,type,
% 0.32/1.63 line_closed_segment:
% 0.32/1.63 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Line__Segment_Oopen__segment,type,
% 0.32/1.63 line_open_segment:
% 0.32/1.63 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_ODE__Misc_Oauto__ll__on__open_Otrapped,type,
% 0.32/1.63 oDE_au1039603466rapped:
% 0.32/1.63 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > ( set @ A ) > $o ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.32/1.63 ord_less:
% 0.32/1.63 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.32/1.63 ord_less_eq:
% 0.32/1.63 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.32/1.63 top_top:
% 0.32/1.63 !>[A: $tType] : A ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Oclosed__orbit,type,
% 0.32/1.63 period385816147_orbit:
% 0.32/1.63 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > $o ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiod,type,
% 0.32/1.63 period1153813292period:
% 0.32/1.63 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > real ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiodic__orbit,type,
% 0.32/1.63 period862636932_orbit:
% 0.32/1.63 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > $o ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Orot,type,
% 0.32/1.63 poinca1750768982en_rot:
% 0.32/1.63 !>[A: $tType] : ( A > A ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Otransversal__segment,type,
% 0.32/1.63 poinca272511729egment:
% 0.32/1.63 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > A > $o ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Reachability__Analysis_Oc1__on__open__euclidean_Osection,type,
% 0.32/1.63 reacha1084862253ection:
% 0.32/1.63 !>[A: $tType] : ( ( A > A ) > ( set @ A ) > ( A > real ) > ( A > ( bounde2145540817linfun @ A @ real ) ) > ( set @ A ) > $o ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
% 0.32/1.63 real_V2000881966t_dist:
% 0.32/1.63 !>[A: $tType] : ( A > A > real ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Set_OCollect,type,
% 0.32/1.63 collect:
% 0.32/1.63 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
% 0.32/1.63 set_or331188842AtMost:
% 0.32/1.63 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_member,type,
% 0.32/1.63 member:
% 0.32/1.63 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.32/1.63
% 0.32/1.63 thf(sy_v_X,type,
% 0.32/1.63 x: set @ a ).
% 0.32/1.63
% 0.32/1.63 thf(sy_v_a,type,
% 0.32/1.63 a2: a ).
% 0.32/1.63
% 0.32/1.63 thf(sy_v_b,type,
% 0.32/1.63 b: a ).
% 0.32/1.63
% 0.32/1.63 thf(sy_v_d____,type,
% 0.32/1.63 d: real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_v_f,type,
% 0.32/1.63 f: a > a ).
% 0.32/1.63
% 0.32/1.63 thf(sy_v_s____,type,
% 0.32/1.63 s: a ).
% 0.32/1.63
% 0.32/1.63 thf(sy_v_thesis,type,
% 0.32/1.63 thesis: $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_v_x____,type,
% 0.32/1.63 x2: a ).
% 0.32/1.63
% 0.32/1.63 thf(sy_v_z,type,
% 0.32/1.63 z: a ).
% 0.32/1.63
% 0.32/1.63 thf(fact_0_d_I1_J,axiom,
% 0.32/1.63 ord_less @ real @ ( zero_zero @ real ) @ d ).
% 0.32/1.63
% 0.32/1.63 thf(fact_1__092_060open_062s_A_092_060in_062_Acball_Ax_Ad_092_060close_062,axiom,
% 0.32/1.63 member @ a @ s @ ( elemen321786957_cball @ a @ x2 @ d ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_2_seg_I1_J,axiom,
% 0.32/1.63 a2 != b ).
% 0.32/1.63
% 0.32/1.63 thf(fact_3_nrm__dot,axiom,
% 0.32/1.63 ! [X: a,Y: a] :
% 0.32/1.63 ( ( inner_780170721_inner @ a @ ( minus_minus @ a @ X @ Y ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) )
% 0.32/1.63 = ( zero_zero @ real ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_4__092_060open_062x_A_092_060in_062_AX_092_060close_062,axiom,
% 0.32/1.63 member @ a @ x2 @ x ).
% 0.32/1.63
% 0.32/1.63 thf(fact_5__092_060open_062_092_060forall_062x_092_060in_062cball_Ax_Ad_O_Af_As_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060le_062_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062,axiom,
% 0.32/1.63 ! [X2: a] :
% 0.32/1.63 ( ( member @ a @ X2 @ ( elemen321786957_cball @ a @ x2 @ d ) )
% 0.32/1.63 => ( ord_less_eq @ real @ ( inner_780170721_inner @ a @ ( f @ s ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) @ ( inner_780170721_inner @ a @ ( f @ X2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_6__092_060open_062f_Ax_A_092_060noteq_062_A_I0_058_058_Ha_J_092_060close_062,axiom,
% 0.32/1.63 ( ( f @ x2 )
% 0.32/1.63 != ( zero_zero @ a ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_7_direction_I2_J,axiom,
% 0.32/1.63 ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ z ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_8_d_I2_J,axiom,
% 0.32/1.63 ! [Y2: a] :
% 0.32/1.63 ( ( member @ a @ Y2 @ ( elemen321786957_cball @ a @ x2 @ d ) )
% 0.32/1.63 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Y2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) )
% 0.32/1.63 & ( member @ a @ Y2 @ x )
% 0.32/1.63 & ( ( f @ Y2 )
% 0.32/1.63 != ( zero_zero @ a ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_9__092_060open_0620_A_060_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062,axiom,
% 0.32/1.63 ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ x2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_10_dot__ortho,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( inner_780170721_inner @ a @ X @ ( poinca1750768982en_rot @ a @ X ) )
% 0.32/1.63 = ( zero_zero @ real ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_11__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062s_O_A_092_060lbrakk_062s_A_092_060in_062_Acball_Ax_Ad_059_A_092_060forall_062x_092_060in_062cball_Ax_Ad_O_Af_As_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060le_062_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.32/1.63 ~ ! [S: a] :
% 0.32/1.63 ( ( member @ a @ S @ ( elemen321786957_cball @ a @ x2 @ d ) )
% 0.32/1.63 => ~ ! [X2: a] :
% 0.32/1.63 ( ( member @ a @ X2 @ ( elemen321786957_cball @ a @ x2 @ d ) )
% 0.32/1.63 => ( ord_less_eq @ real @ ( inner_780170721_inner @ a @ ( f @ S ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) @ ( inner_780170721_inner @ a @ ( f @ X2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_12__092_060open_062x_A_092_060in_062_A_123a_N_Nb_125_092_060close_062,axiom,
% 0.32/1.63 member @ a @ x2 @ ( line_closed_segment @ a @ a2 @ b ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_13__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062d_O_A_092_060lbrakk_0620_A_060_Ad_059_A_092_060And_062y_O_Ay_A_092_060in_062_Acball_Ax_Ad_A_092_060Longrightarrow_062_A0_A_060_Af_Ay_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060and_062_Ay_A_092_060in_062_AX_A_092_060and_062_Af_Ay_A_092_060noteq_062_A_I0_058_058_Ha_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.32/1.63 ~ ! [D: real] :
% 0.32/1.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ D )
% 0.32/1.63 => ~ ! [Y3: a] :
% 0.32/1.63 ( ( member @ a @ Y3 @ ( elemen321786957_cball @ a @ x2 @ D ) )
% 0.32/1.63 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Y3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) )
% 0.32/1.63 & ( member @ a @ Y3 @ x )
% 0.32/1.63 & ( ( f @ Y3 )
% 0.32/1.63 != ( zero_zero @ a ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_14_transversal,axiom,
% 0.32/1.63 poinca272511729egment @ a @ f @ x @ a2 @ b ).
% 0.32/1.63
% 0.32/1.63 thf(fact_15_inner__gt__zero__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( inner_real_inner @ A )
% 0.32/1.63 => ! [X: A] :
% 0.32/1.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ A @ X @ X ) )
% 0.32/1.63 = ( X
% 0.32/1.63 != ( zero_zero @ A ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_16_centre__in__cball,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( real_V2090557954_space @ A )
% 0.32/1.63 => ! [X: A,E: real] :
% 0.32/1.63 ( ( member @ A @ X @ ( elemen321786957_cball @ A @ X @ E ) )
% 0.32/1.63 = ( ord_less_eq @ real @ ( zero_zero @ real ) @ E ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_17_all__zero__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( inner_real_inner @ A )
% 0.32/1.63 => ! [X: A] :
% 0.32/1.63 ( ( ! [U: A] :
% 0.32/1.63 ( ( inner_780170721_inner @ A @ X @ U )
% 0.32/1.63 = ( zero_zero @ real ) ) )
% 0.32/1.63 = ( X
% 0.32/1.63 = ( zero_zero @ A ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_18_inner__zero__left,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( inner_real_inner @ A )
% 0.32/1.63 => ! [X: A] :
% 0.32/1.63 ( ( inner_780170721_inner @ A @ ( zero_zero @ A ) @ X )
% 0.32/1.63 = ( zero_zero @ real ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_19_inner__zero__right,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( inner_real_inner @ A )
% 0.32/1.63 => ! [X: A] :
% 0.32/1.63 ( ( inner_780170721_inner @ A @ X @ ( zero_zero @ A ) )
% 0.32/1.63 = ( zero_zero @ real ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_20_inner__eq__zero__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( inner_real_inner @ A )
% 0.32/1.63 => ! [X: A] :
% 0.32/1.63 ( ( ( inner_780170721_inner @ A @ X @ X )
% 0.32/1.63 = ( zero_zero @ real ) )
% 0.32/1.63 = ( X
% 0.32/1.63 = ( zero_zero @ A ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_21_diff__gt__0__iff__gt,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A] :
% 0.32/1.63 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 0.32/1.63 = ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_22_diff__ge__0__iff__ge,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 0.32/1.63 = ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_23_rot__0,axiom,
% 0.32/1.63 ( ( poinca1750768982en_rot @ a @ ( zero_zero @ a ) )
% 0.32/1.63 = ( zero_zero @ a ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_24_rot__eq__0__iff,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( ( poinca1750768982en_rot @ a @ X )
% 0.32/1.63 = ( zero_zero @ a ) )
% 0.32/1.63 = ( X
% 0.32/1.63 = ( zero_zero @ a ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_25_seg_I3_J,axiom,
% 0.32/1.63 ! [Z: a] :
% 0.32/1.63 ( ( member @ a @ Z @ ( line_closed_segment @ a @ a2 @ b ) )
% 0.32/1.63 => ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_26_direction_I1_J,axiom,
% 0.32/1.63 member @ a @ z @ ( line_closed_segment @ a @ a2 @ b ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_27_transversal__segment__reverse,axiom,
% 0.32/1.63 ! [X: a,Y: a] :
% 0.32/1.63 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.32/1.63 => ( poinca272511729egment @ a @ f @ x @ Y @ X ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_28_transversal__segment__commute,axiom,
% 0.32/1.63 ! [X: a,Y: a] :
% 0.32/1.63 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.32/1.63 = ( poinca272511729egment @ a @ f @ x @ Y @ X ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_29_in__segment__inner__rot2,axiom,
% 0.32/1.63 ! [X: a,A2: a,B: a,Y: a] :
% 0.32/1.63 ( ( member @ a @ X @ ( line_closed_segment @ a @ A2 @ B ) )
% 0.32/1.63 => ( ( member @ a @ Y @ ( line_closed_segment @ a @ A2 @ B ) )
% 0.32/1.63 => ( ( inner_780170721_inner @ a @ ( minus_minus @ a @ X @ Y ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ A2 @ B ) ) )
% 0.32/1.63 = ( zero_zero @ real ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_30_in__segment__inner__rot,axiom,
% 0.32/1.63 ! [X: a,A2: a,B: a] :
% 0.32/1.63 ( ( member @ a @ X @ ( line_closed_segment @ a @ A2 @ B ) )
% 0.32/1.63 => ( ( inner_780170721_inner @ a @ ( minus_minus @ a @ X @ A2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ B @ A2 ) ) )
% 0.32/1.63 = ( zero_zero @ real ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_31_seg_I2_J,axiom,
% 0.32/1.63 ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ a2 @ b ) @ x ).
% 0.32/1.63
% 0.32/1.63 thf(fact_32_le__zero__eq,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( canoni770627133id_add @ A )
% 0.32/1.63 => ! [N: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.32/1.63 = ( N
% 0.32/1.63 = ( zero_zero @ A ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_33_not__gr__zero,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( canoni770627133id_add @ A )
% 0.32/1.63 => ! [N: A] :
% 0.32/1.63 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.32/1.63 = ( N
% 0.32/1.63 = ( zero_zero @ A ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_34_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( cancel1352612707id_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( minus_minus @ A @ A2 @ A2 )
% 0.32/1.63 = ( zero_zero @ A ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_35_diff__zero,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( cancel1352612707id_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.32/1.63 = A2 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_36_zero__diff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( comm_monoid_diff @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 0.32/1.63 = ( zero_zero @ A ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_37_diff__0__right,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( group_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.32/1.63 = A2 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_38_diff__self,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( group_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( minus_minus @ A @ A2 @ A2 )
% 0.32/1.63 = ( zero_zero @ A ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_39_subset__cball,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( real_V2090557954_space @ A )
% 0.32/1.63 => ! [D2: real,E: real,X: A] :
% 0.32/1.63 ( ( ord_less_eq @ real @ D2 @ E )
% 0.32/1.63 => ( ord_less_eq @ ( set @ A ) @ ( elemen321786957_cball @ A @ X @ D2 ) @ ( elemen321786957_cball @ A @ X @ E ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_40_transversal__segment__def,axiom,
% 0.32/1.63 ! [A2: a,B: a] :
% 0.32/1.63 ( ( poinca272511729egment @ a @ f @ x @ A2 @ B )
% 0.32/1.63 = ( ( A2 != B )
% 0.32/1.63 & ( ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ A2 @ B ) @ x )
% 0.32/1.63 & ! [X3: a] :
% 0.32/1.63 ( ( member @ a @ X3 @ ( line_closed_segment @ a @ A2 @ B ) )
% 0.32/1.63 => ( ( inner_780170721_inner @ a @ ( f @ X3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ A2 @ B ) ) )
% 0.32/1.63 != ( zero_zero @ real ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_41_transversal__segmentE,axiom,
% 0.32/1.63 ! [X: a,Y: a] :
% 0.32/1.63 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.32/1.63 => ( ( ( X != Y )
% 0.32/1.63 => ( ( ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ X @ Y ) @ x )
% 0.32/1.63 => ~ ! [Z2: a] :
% 0.32/1.63 ( ( member @ a @ Z2 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.32/1.63 => ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) ) ) ) )
% 0.32/1.63 => ~ ( ( X != Y )
% 0.32/1.63 => ( ( ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ X @ Y ) @ x )
% 0.32/1.63 => ~ ! [Z2: a] :
% 0.32/1.63 ( ( member @ a @ Z2 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.32/1.63 => ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ Y @ X ) ) ) ) ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_42_transversal__segment__posD_I2_J,axiom,
% 0.32/1.63 ! [X: a,Y: a,Z3: a] :
% 0.32/1.63 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.32/1.63 => ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.32/1.63 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) )
% 0.32/1.63 => ( ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ X @ Y ) @ x ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_43_transversal__segment__negD_I2_J,axiom,
% 0.32/1.63 ! [X: a,Y: a,Z3: a] :
% 0.32/1.63 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.32/1.63 => ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.32/1.63 => ( ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) )
% 0.32/1.63 => ( ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ X @ Y ) @ x ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_44_mem__Collect__eq,axiom,
% 0.32/1.63 ! [A: $tType,A2: A,P: A > $o] :
% 0.32/1.63 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.32/1.63 = ( P @ A2 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_45_Collect__mem__eq,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A] :
% 0.32/1.63 ( ( collect @ A
% 0.32/1.63 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.32/1.63 = A3 ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_46_Collect__cong,axiom,
% 0.32/1.63 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/1.63 ( ! [X4: A] :
% 0.32/1.63 ( ( P @ X4 )
% 0.32/1.63 = ( Q @ X4 ) )
% 0.32/1.63 => ( ( collect @ A @ P )
% 0.32/1.63 = ( collect @ A @ Q ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_47_ext,axiom,
% 0.32/1.63 ! [B2: $tType,A: $tType,F: A > B2,G: A > B2] :
% 0.32/1.63 ( ! [X4: A] :
% 0.32/1.63 ( ( F @ X4 )
% 0.32/1.63 = ( G @ X4 ) )
% 0.32/1.63 => ( F = G ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_48_transversal__segment__sign__less,axiom,
% 0.32/1.63 ! [W: a,Y: a] :
% 0.32/1.63 ( ( poinca272511729egment @ a @ f @ x @ W @ Y )
% 0.32/1.63 => ( ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ W ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ W @ Y ) ) ) @ ( zero_zero @ real ) )
% 0.32/1.63 => ! [X2: a] :
% 0.32/1.63 ( ( member @ a @ X2 @ ( line_closed_segment @ a @ W @ Y ) )
% 0.32/1.63 => ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ X2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ W @ Y ) ) ) @ ( zero_zero @ real ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_49_transversal__segment__pos,axiom,
% 0.32/1.63 ! [X: a,Y: a,W: a] :
% 0.32/1.63 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.32/1.63 => ( ( member @ a @ W @ ( line_closed_segment @ a @ X @ Y ) )
% 0.32/1.63 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ W ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) )
% 0.32/1.63 => ! [X2: a] :
% 0.32/1.63 ( ( member @ a @ X2 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.32/1.63 => ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ X2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_50_transversal__segment__neg,axiom,
% 0.32/1.63 ! [X: a,Y: a,W: a] :
% 0.32/1.63 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.32/1.63 => ( ( member @ a @ W @ ( line_closed_segment @ a @ X @ Y ) )
% 0.32/1.63 => ( ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ W ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) )
% 0.32/1.63 => ! [X2: a] :
% 0.32/1.63 ( ( member @ a @ X2 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.32/1.63 => ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ X2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_51_transversal__segment__posD_I1_J,axiom,
% 0.32/1.63 ! [X: a,Y: a,Z3: a] :
% 0.32/1.63 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.32/1.63 => ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.32/1.63 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) )
% 0.32/1.63 => ( X != Y ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_52_transversal__segment__posD_I3_J,axiom,
% 0.32/1.63 ! [X: a,Y: a,Za: a,Z3: a] :
% 0.32/1.63 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.32/1.63 => ( ( member @ a @ Za @ ( line_closed_segment @ a @ X @ Y ) )
% 0.32/1.63 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Za ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) )
% 0.32/1.63 => ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.32/1.63 => ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_53_transversal__segment__negD_I1_J,axiom,
% 0.32/1.63 ! [X: a,Y: a,Z3: a] :
% 0.32/1.63 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.32/1.63 => ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.32/1.63 => ( ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) )
% 0.32/1.63 => ( X != Y ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_54_transversal__segment__negD_I3_J,axiom,
% 0.32/1.63 ! [X: a,Y: a,Za: a,Z3: a] :
% 0.32/1.63 ( ( poinca272511729egment @ a @ f @ x @ X @ Y )
% 0.32/1.63 => ( ( member @ a @ Za @ ( line_closed_segment @ a @ X @ Y ) )
% 0.32/1.63 => ( ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ Za ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) )
% 0.32/1.63 => ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
% 0.32/1.63 => ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_55__092_060open_062_092_060And_062z_O_Az_A_092_060in_062_A_123a_N_Nb_125_A_092_060Longrightarrow_062_A0_A_060_Af_Az_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062,axiom,
% 0.32/1.63 ! [Z3: a] :
% 0.32/1.63 ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ a2 @ b ) )
% 0.32/1.63 => ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_56_c1__on__open__R2_Otransversal__segment_Ocong,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( execut510477386_space @ A )
% 0.32/1.63 => ( ( poinca272511729egment @ A )
% 0.32/1.63 = ( poinca272511729egment @ A ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_57_zero__reorient,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( zero @ A )
% 0.32/1.63 => ! [X: A] :
% 0.32/1.63 ( ( ( zero_zero @ A )
% 0.32/1.63 = X )
% 0.32/1.63 = ( X
% 0.32/1.63 = ( zero_zero @ A ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_58_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( cancel146912293up_add @ A )
% 0.32/1.63 => ! [A2: A,C: A,B: A] :
% 0.32/1.63 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B )
% 0.32/1.63 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_59_diff__eq__diff__eq,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A,C: A,D2: A] :
% 0.32/1.63 ( ( ( minus_minus @ A @ A2 @ B )
% 0.32/1.63 = ( minus_minus @ A @ C @ D2 ) )
% 0.32/1.63 => ( ( A2 = B )
% 0.32/1.63 = ( C = D2 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_60_inner__commute,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( inner_real_inner @ A )
% 0.32/1.63 => ( ( inner_780170721_inner @ A )
% 0.32/1.63 = ( ^ [X3: A,Y4: A] : ( inner_780170721_inner @ A @ Y4 @ X3 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_61_zero__le,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( canoni770627133id_add @ A )
% 0.32/1.63 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_62_zero__less__iff__neq__zero,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( canoni770627133id_add @ A )
% 0.32/1.63 => ! [N: A] :
% 0.32/1.63 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.32/1.63 = ( N
% 0.32/1.63 != ( zero_zero @ A ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_63_gr__implies__not__zero,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( canoni770627133id_add @ A )
% 0.32/1.63 => ! [M: A,N: A] :
% 0.32/1.63 ( ( ord_less @ A @ M @ N )
% 0.32/1.63 => ( N
% 0.32/1.63 != ( zero_zero @ A ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_64_not__less__zero,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( canoni770627133id_add @ A )
% 0.32/1.63 => ! [N: A] :
% 0.32/1.63 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_65_gr__zeroI,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( canoni770627133id_add @ A )
% 0.32/1.63 => ! [N: A] :
% 0.32/1.63 ( ( N
% 0.32/1.63 != ( zero_zero @ A ) )
% 0.32/1.63 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_66_eq__iff__diff__eq__0,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( group_add @ A )
% 0.32/1.63 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.32/1.63 = ( ^ [A4: A,B3: A] :
% 0.32/1.63 ( ( minus_minus @ A @ A4 @ B3 )
% 0.32/1.63 = ( zero_zero @ A ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_67_diff__eq__diff__less__eq,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A,C: A,D2: A] :
% 0.32/1.63 ( ( ( minus_minus @ A @ A2 @ B )
% 0.32/1.63 = ( minus_minus @ A @ C @ D2 ) )
% 0.32/1.63 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.32/1.63 = ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_68_diff__right__mono,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A,C: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ A2 @ B )
% 0.32/1.63 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_69_diff__left__mono,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [B: A,A2: A,C: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ B @ A2 )
% 0.32/1.63 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_70_diff__mono,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A,D2: A,C: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ A2 @ B )
% 0.32/1.63 => ( ( ord_less_eq @ A @ D2 @ C )
% 0.32/1.63 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D2 ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_71_diff__strict__right__mono,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A,C: A] :
% 0.32/1.63 ( ( ord_less @ A @ A2 @ B )
% 0.32/1.63 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_72_diff__strict__left__mono,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [B: A,A2: A,C: A] :
% 0.32/1.63 ( ( ord_less @ A @ B @ A2 )
% 0.32/1.63 => ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_73_diff__eq__diff__less,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A,C: A,D2: A] :
% 0.32/1.63 ( ( ( minus_minus @ A @ A2 @ B )
% 0.32/1.63 = ( minus_minus @ A @ C @ D2 ) )
% 0.32/1.63 => ( ( ord_less @ A @ A2 @ B )
% 0.32/1.63 = ( ord_less @ A @ C @ D2 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_74_diff__strict__mono,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A,D2: A,C: A] :
% 0.32/1.63 ( ( ord_less @ A @ A2 @ B )
% 0.32/1.63 => ( ( ord_less @ A @ D2 @ C )
% 0.32/1.63 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D2 ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_75_inner__diff__right,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( inner_real_inner @ A )
% 0.32/1.63 => ! [X: A,Y: A,Z3: A] :
% 0.32/1.63 ( ( inner_780170721_inner @ A @ X @ ( minus_minus @ A @ Y @ Z3 ) )
% 0.32/1.63 = ( minus_minus @ real @ ( inner_780170721_inner @ A @ X @ Y ) @ ( inner_780170721_inner @ A @ X @ Z3 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_76_inner__diff__left,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( inner_real_inner @ A )
% 0.32/1.63 => ! [X: A,Y: A,Z3: A] :
% 0.32/1.63 ( ( inner_780170721_inner @ A @ ( minus_minus @ A @ X @ Y ) @ Z3 )
% 0.32/1.63 = ( minus_minus @ real @ ( inner_780170721_inner @ A @ X @ Z3 ) @ ( inner_780170721_inner @ A @ Y @ Z3 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_77_mem__cball__leI,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( real_V2090557954_space @ A )
% 0.32/1.63 => ! [X: A,Y: A,E: real,F: real] :
% 0.32/1.63 ( ( member @ A @ X @ ( elemen321786957_cball @ A @ Y @ E ) )
% 0.32/1.63 => ( ( ord_less_eq @ real @ E @ F )
% 0.32/1.63 => ( member @ A @ X @ ( elemen321786957_cball @ A @ Y @ F ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_78_le__iff__diff__le__0,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ( ( ord_less_eq @ A )
% 0.32/1.63 = ( ^ [A4: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_79_less__iff__diff__less__0,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ( ( ord_less @ A )
% 0.32/1.63 = ( ^ [A4: A,B3: A] : ( ord_less @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_80_inner__ge__zero,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( inner_real_inner @ A )
% 0.32/1.63 => ! [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ A @ X @ X ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_81_fixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( member @ a @ X @ x )
% 0.32/1.63 => ( ( ( f @ X )
% 0.32/1.63 = ( zero_zero @ a ) )
% 0.32/1.63 => ( ( period1153813292period @ a @ f @ x @ X )
% 0.32/1.63 = ( zero_zero @ real ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_82_that,axiom,
% 0.32/1.63 ! [D3: real,B4: real] :
% 0.32/1.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
% 0.32/1.63 => ( ( ord_less @ real @ ( zero_zero @ real ) @ B4 )
% 0.32/1.63 => ( ! [X4: a,Y6: a] :
% 0.32/1.63 ( ( member @ a @ X4 @ ( line_closed_segment @ a @ a2 @ b ) )
% 0.32/1.63 => ( ( ord_less_eq @ real @ ( real_V2000881966t_dist @ a @ X4 @ Y6 ) @ D3 )
% 0.32/1.63 => ( ord_less_eq @ real @ B4 @ ( inner_780170721_inner @ a @ ( f @ Y6 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ) ) )
% 0.32/1.63 => thesis ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_83_inner__nonneg__nonneg,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordere890947078_space @ A )
% 0.32/1.63 => ! [A2: A,B: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.32/1.63 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 0.32/1.63 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ A @ A2 @ B ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_84_in__closed__segment__iff__rot,axiom,
% 0.32/1.63 ! [A2: a,B: a,X: a] :
% 0.32/1.63 ( ( A2 != B )
% 0.32/1.63 => ( ( member @ a @ X @ ( line_closed_segment @ a @ A2 @ B ) )
% 0.32/1.63 = ( ( ( inner_780170721_inner @ a @ ( minus_minus @ a @ X @ A2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ B @ A2 ) ) )
% 0.32/1.63 = ( zero_zero @ real ) )
% 0.32/1.63 & ( member @ real @ ( inner_780170721_inner @ a @ X @ ( minus_minus @ a @ B @ A2 ) ) @ ( set_or331188842AtMost @ real @ ( inner_780170721_inner @ a @ A2 @ ( minus_minus @ a @ B @ A2 ) ) @ ( inner_780170721_inner @ a @ B @ ( minus_minus @ a @ B @ A2 ) ) ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_85_sectionD_I5_J,axiom,
% 0.32/1.63 ! [S2: a > real,Ds: a > ( bounde2145540817linfun @ a @ real ),S3: set @ a] :
% 0.32/1.63 ( ( reacha1084862253ection @ a @ f @ x @ S2 @ Ds @ S3 )
% 0.32/1.63 => ( ord_less_eq @ ( set @ a ) @ S3 @ x ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_86_transversal__segment__exists,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( member @ a @ X @ x )
% 0.32/1.63 => ( ( ( f @ X )
% 0.32/1.63 != ( zero_zero @ a ) )
% 0.32/1.63 => ~ ! [A5: a,B5: a] :
% 0.32/1.63 ( ( member @ a @ X @ ( line_open_segment @ a @ A5 @ B5 ) )
% 0.32/1.63 => ~ ( poinca272511729egment @ a @ f @ x @ A5 @ B5 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_87_invariant__iff__compl__invariant,axiom,
% 0.32/1.63 ! [M2: set @ a] :
% 0.32/1.63 ( ( auto_ll_on_invariant @ a @ f @ x @ M2 )
% 0.32/1.63 = ( auto_ll_on_invariant @ a @ f @ x @ ( minus_minus @ ( set @ a ) @ x @ M2 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_88_cball__eq__cball__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( euclid925273238_space @ A )
% 0.32/1.63 => ! [X: A,D2: real,Y: A,E: real] :
% 0.32/1.63 ( ( ( elemen321786957_cball @ A @ X @ D2 )
% 0.32/1.63 = ( elemen321786957_cball @ A @ Y @ E ) )
% 0.32/1.63 = ( ( ( ord_less @ real @ D2 @ ( zero_zero @ real ) )
% 0.32/1.63 & ( ord_less @ real @ E @ ( zero_zero @ real ) ) )
% 0.32/1.63 | ( ( X = Y )
% 0.32/1.63 & ( D2 = E ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_89_Bolzano,axiom,
% 0.32/1.63 ! [A2: real,B: real,P: real > real > $o] :
% 0.32/1.63 ( ( ord_less_eq @ real @ A2 @ B )
% 0.32/1.63 => ( ! [A5: real,B5: real,C2: real] :
% 0.32/1.63 ( ( P @ A5 @ B5 )
% 0.32/1.63 => ( ( P @ B5 @ C2 )
% 0.32/1.63 => ( ( ord_less_eq @ real @ A5 @ B5 )
% 0.32/1.63 => ( ( ord_less_eq @ real @ B5 @ C2 )
% 0.32/1.63 => ( P @ A5 @ C2 ) ) ) ) )
% 0.32/1.63 => ( ! [X4: real] :
% 0.32/1.63 ( ( ord_less_eq @ real @ A2 @ X4 )
% 0.32/1.63 => ( ( ord_less_eq @ real @ X4 @ B )
% 0.32/1.63 => ? [D4: real] :
% 0.32/1.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
% 0.32/1.63 & ! [A5: real,B5: real] :
% 0.32/1.63 ( ( ( ord_less_eq @ real @ A5 @ X4 )
% 0.32/1.63 & ( ord_less_eq @ real @ X4 @ B5 )
% 0.32/1.63 & ( ord_less @ real @ ( minus_minus @ real @ B5 @ A5 ) @ D4 ) )
% 0.32/1.63 => ( P @ A5 @ B5 ) ) ) ) )
% 0.32/1.63 => ( P @ A2 @ B ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_90_mem__cball,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( real_V2090557954_space @ A )
% 0.32/1.63 => ! [Y: A,X: A,E: real] :
% 0.32/1.63 ( ( member @ A @ Y @ ( elemen321786957_cball @ A @ X @ E ) )
% 0.32/1.63 = ( ord_less_eq @ real @ ( real_V2000881966t_dist @ A @ X @ Y ) @ E ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_91_perfect__choose__dist,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ( real_V2090557954_space @ A )
% 0.32/1.63 & ( topolo890362671_space @ A ) )
% 0.32/1.63 => ! [R: real,X: A] :
% 0.32/1.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
% 0.32/1.63 => ? [A5: A] :
% 0.32/1.63 ( ( A5 != X )
% 0.32/1.63 & ( ord_less @ real @ ( real_V2000881966t_dist @ A @ A5 @ X ) @ R ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_92_interval__inner__leI_I2_J,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordere890947078_space @ A )
% 0.32/1.63 => ! [X: A,A2: A,B: A,I: A] :
% 0.32/1.63 ( ( member @ A @ X @ ( set_or331188842AtMost @ A @ A2 @ B ) )
% 0.32/1.63 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ I )
% 0.32/1.63 => ( ord_less_eq @ real @ ( inner_780170721_inner @ A @ X @ I ) @ ( inner_780170721_inner @ A @ B @ I ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_93_interval__inner__leI_I1_J,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordere890947078_space @ A )
% 0.32/1.63 => ! [X: A,A2: A,B: A,I: A] :
% 0.32/1.63 ( ( member @ A @ X @ ( set_or331188842AtMost @ A @ A2 @ B ) )
% 0.32/1.63 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ I )
% 0.32/1.63 => ( ord_less_eq @ real @ ( inner_780170721_inner @ A @ A2 @ I ) @ ( inner_780170721_inner @ A @ X @ I ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_94_eucl__less__le__not__le,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordere890947078_space @ A )
% 0.32/1.63 => ( ( ord_less @ A )
% 0.32/1.63 = ( ^ [X3: A,Y4: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.32/1.63 & ~ ( ord_less_eq @ A @ Y4 @ X3 ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_95_invariant__def,axiom,
% 0.32/1.63 ! [M2: set @ a] :
% 0.32/1.63 ( ( auto_ll_on_invariant @ a @ f @ x @ M2 )
% 0.32/1.63 = ( ! [X3: a] :
% 0.32/1.63 ( ( member @ a @ X3 @ M2 )
% 0.32/1.63 => ( oDE_au1039603466rapped @ a @ f @ x @ X3 @ M2 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_96_periodic__orbit__period_I1_J,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( period862636932_orbit @ a @ f @ x @ X )
% 0.32/1.63 => ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_97_zero__less__dist__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( real_V2090557954_space @ A )
% 0.32/1.63 => ! [X: A,Y: A] :
% 0.32/1.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X @ Y ) )
% 0.32/1.63 = ( X != Y ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_98_dist__le__zero__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( real_V2090557954_space @ A )
% 0.32/1.63 => ! [X: A,Y: A] :
% 0.32/1.63 ( ( ord_less_eq @ real @ ( real_V2000881966t_dist @ A @ X @ Y ) @ ( zero_zero @ real ) )
% 0.32/1.63 = ( X = Y ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_99_atLeastatMost__subset__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( preorder @ A )
% 0.32/1.63 => ! [A2: A,B: A,C: A,D2: A] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A2 @ B ) @ ( set_or331188842AtMost @ A @ C @ D2 ) )
% 0.32/1.63 = ( ~ ( ord_less_eq @ A @ A2 @ B )
% 0.32/1.63 | ( ( ord_less_eq @ A @ C @ A2 )
% 0.32/1.63 & ( ord_less_eq @ A @ B @ D2 ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_100_dist__self,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( real_V2090557954_space @ A )
% 0.32/1.63 => ! [X: A] :
% 0.32/1.63 ( ( real_V2000881966t_dist @ A @ X @ X )
% 0.32/1.63 = ( zero_zero @ real ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_101_atLeastAtMost__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ord @ A )
% 0.32/1.63 => ! [I: A,L: A,U2: A] :
% 0.32/1.63 ( ( member @ A @ I @ ( set_or331188842AtMost @ A @ L @ U2 ) )
% 0.32/1.63 = ( ( ord_less_eq @ A @ L @ I )
% 0.32/1.63 & ( ord_less_eq @ A @ I @ U2 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_102_Icc__eq__Icc,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( order @ A )
% 0.32/1.63 => ! [L: A,H: A,L2: A,H2: A] :
% 0.32/1.63 ( ( ( set_or331188842AtMost @ A @ L @ H )
% 0.32/1.63 = ( set_or331188842AtMost @ A @ L2 @ H2 ) )
% 0.32/1.63 = ( ( ( L = L2 )
% 0.32/1.63 & ( H = H2 ) )
% 0.32/1.63 | ( ~ ( ord_less_eq @ A @ L @ H )
% 0.32/1.63 & ~ ( ord_less_eq @ A @ L2 @ H2 ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_103_dist__eq__0__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( real_V2090557954_space @ A )
% 0.32/1.63 => ! [X: A,Y: A] :
% 0.32/1.63 ( ( ( real_V2000881966t_dist @ A @ X @ Y )
% 0.32/1.63 = ( zero_zero @ real ) )
% 0.32/1.63 = ( X = Y ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_104_dist__commute,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( real_V2090557954_space @ A )
% 0.32/1.63 => ( ( real_V2000881966t_dist @ A )
% 0.32/1.63 = ( ^ [X3: A,Y4: A] : ( real_V2000881966t_dist @ A @ Y4 @ X3 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_105_dist__commute__lessI,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( real_V2090557954_space @ A )
% 0.32/1.63 => ! [Y: A,X: A,E: real] :
% 0.32/1.63 ( ( ord_less @ real @ ( real_V2000881966t_dist @ A @ Y @ X ) @ E )
% 0.32/1.63 => ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X @ Y ) @ E ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_106_atLeastatMost__psubset__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( preorder @ A )
% 0.32/1.63 => ! [A2: A,B: A,C: A,D2: A] :
% 0.32/1.63 ( ( ord_less @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A2 @ B ) @ ( set_or331188842AtMost @ A @ C @ D2 ) )
% 0.32/1.63 = ( ( ~ ( ord_less_eq @ A @ A2 @ B )
% 0.32/1.63 | ( ( ord_less_eq @ A @ C @ A2 )
% 0.32/1.63 & ( ord_less_eq @ A @ B @ D2 )
% 0.32/1.63 & ( ( ord_less @ A @ C @ A2 )
% 0.32/1.63 | ( ord_less @ A @ B @ D2 ) ) ) )
% 0.32/1.63 & ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_107_zero__le__dist,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( real_V2090557954_space @ A )
% 0.32/1.63 => ! [X: A,Y: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X @ Y ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_108_dist__not__less__zero,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( real_V2090557954_space @ A )
% 0.32/1.63 => ! [X: A,Y: A] :
% 0.32/1.63 ~ ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X @ Y ) @ ( zero_zero @ real ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_109_dist__pos__lt,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( real_V2090557954_space @ A )
% 0.32/1.63 => ! [X: A,Y: A] :
% 0.32/1.63 ( ( X != Y )
% 0.32/1.63 => ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X @ Y ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_110_rev_Operiodic__orbit__period_I1_J,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.32/1.63 => ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_111_periodic__orbit__def,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( period862636932_orbit @ a @ f @ x @ X )
% 0.32/1.63 = ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.32/1.63 & ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_112_rev_Ofixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( member @ a @ X @ x )
% 0.32/1.63 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 0.32/1.63 = ( zero_zero @ a ) )
% 0.32/1.63 => ( ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.32/1.63 = ( zero_zero @ real ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_113_closed__orbit__period__zero__fixed__point,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.32/1.63 => ( ( ( period1153813292period @ a @ f @ x @ X )
% 0.32/1.63 = ( zero_zero @ real ) )
% 0.32/1.63 => ( ( f @ X )
% 0.32/1.63 = ( zero_zero @ a ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_114_closed__orbit__period__nonneg,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.32/1.63 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_115_closed__orbit__in__domain,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.32/1.63 => ( member @ a @ X @ x ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_116_add_Oinverse__inverse,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( group_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 0.32/1.63 = A2 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_117_neg__equal__iff__equal,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A] :
% 0.32/1.63 ( ( ( uminus_uminus @ A @ A2 )
% 0.32/1.63 = ( uminus_uminus @ A @ B ) )
% 0.32/1.63 = ( A2 = B ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_118_rev_Oclosed__orbit__eq__rev,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.32/1.63 = ( period385816147_orbit @ a @ f @ x @ X ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_119_rev_Oclosed__orbit__in__domain,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.32/1.63 => ( member @ a @ X @ x ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_120_rev_Oinvariant__eq__rev,axiom,
% 0.32/1.63 ! [M2: set @ a] :
% 0.32/1.63 ( ( auto_ll_on_invariant @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ M2 )
% 0.32/1.63 = ( auto_ll_on_invariant @ a @ f @ x @ M2 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_121_fixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( member @ a @ X @ x )
% 0.32/1.63 => ( ( ( f @ X )
% 0.32/1.63 = ( zero_zero @ a ) )
% 0.32/1.63 => ( period385816147_orbit @ a @ f @ x @ X ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_122_rev_Ofixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( member @ a @ X @ x )
% 0.32/1.63 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 0.32/1.63 = ( zero_zero @ a ) )
% 0.32/1.63 => ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_123_rev_OsectionD_I5_J,axiom,
% 0.32/1.63 ! [S2: a > real,Ds: a > ( bounde2145540817linfun @ a @ real ),S3: set @ a] :
% 0.32/1.63 ( ( reacha1084862253ection @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ S2 @ Ds @ S3 )
% 0.32/1.63 => ( ord_less_eq @ ( set @ a ) @ S3 @ x ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_124_rev_Oinvariant__iff__compl__invariant,axiom,
% 0.32/1.63 ! [M2: set @ a] :
% 0.32/1.63 ( ( auto_ll_on_invariant @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ M2 )
% 0.32/1.63 = ( auto_ll_on_invariant @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( minus_minus @ ( set @ a ) @ x @ M2 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_125_closed__orbit__periodic,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.32/1.63 => ( ( ( f @ X )
% 0.32/1.63 != ( zero_zero @ a ) )
% 0.32/1.63 => ( period862636932_orbit @ a @ f @ x @ X ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_126_rev_Oinvariant__def,axiom,
% 0.32/1.63 ! [M2: set @ a] :
% 0.32/1.63 ( ( auto_ll_on_invariant @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ M2 )
% 0.32/1.63 = ( ! [X3: a] :
% 0.32/1.63 ( ( member @ a @ X3 @ M2 )
% 0.32/1.63 => ( oDE_au1039603466rapped @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X3 @ M2 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_127_rev_Oclosed__orbit__periodic,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.32/1.63 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 0.32/1.63 != ( zero_zero @ a ) )
% 0.32/1.63 => ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_128_add_Oinverse__neutral,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( group_add @ A )
% 0.32/1.63 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 0.32/1.63 = ( zero_zero @ A ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_129_neg__0__equal__iff__equal,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( group_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( ( zero_zero @ A )
% 0.32/1.63 = ( uminus_uminus @ A @ A2 ) )
% 0.32/1.63 = ( ( zero_zero @ A )
% 0.32/1.63 = A2 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_130_neg__equal__0__iff__equal,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( group_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( ( uminus_uminus @ A @ A2 )
% 0.32/1.63 = ( zero_zero @ A ) )
% 0.32/1.63 = ( A2
% 0.32/1.63 = ( zero_zero @ A ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_131_equal__neg__zero,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( linord219039673up_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( A2
% 0.32/1.63 = ( uminus_uminus @ A @ A2 ) )
% 0.32/1.63 = ( A2
% 0.32/1.63 = ( zero_zero @ A ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_132_neg__equal__zero,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( linord219039673up_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( ( uminus_uminus @ A @ A2 )
% 0.32/1.63 = A2 )
% 0.32/1.63 = ( A2
% 0.32/1.63 = ( zero_zero @ A ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_133_neg__le__iff__le,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [B: A,A2: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B ) @ ( uminus_uminus @ A @ A2 ) )
% 0.32/1.63 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_134_neg__less__iff__less,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [B: A,A2: A] :
% 0.32/1.63 ( ( ord_less @ A @ ( uminus_uminus @ A @ B ) @ ( uminus_uminus @ A @ A2 ) )
% 0.32/1.63 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_135_minus__diff__eq,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A] :
% 0.32/1.63 ( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B ) )
% 0.32/1.63 = ( minus_minus @ A @ B @ A2 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_136_inner__minus__left,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( inner_real_inner @ A )
% 0.32/1.63 => ! [X: A,Y: A] :
% 0.32/1.63 ( ( inner_780170721_inner @ A @ ( uminus_uminus @ A @ X ) @ Y )
% 0.32/1.63 = ( uminus_uminus @ real @ ( inner_780170721_inner @ A @ X @ Y ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_137_inner__minus__right,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( inner_real_inner @ A )
% 0.32/1.63 => ! [X: A,Y: A] :
% 0.32/1.63 ( ( inner_780170721_inner @ A @ X @ ( uminus_uminus @ A @ Y ) )
% 0.32/1.63 = ( uminus_uminus @ real @ ( inner_780170721_inner @ A @ X @ Y ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_138_rev_Oclosed__orbit__period__nonneg,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.32/1.63 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_139_rev_Oclosed__orbit__period__zero__fixed__point,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.32/1.63 => ( ( ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.32/1.63 = ( zero_zero @ real ) )
% 0.32/1.63 => ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 0.32/1.63 = ( zero_zero @ a ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_140_rev_Operiodic__orbit__def,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.32/1.63 = ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.32/1.63 & ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_141_neg__0__le__iff__le,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 0.32/1.63 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_142_neg__le__0__iff__le,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.32/1.63 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_143_less__eq__neg__nonpos,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( linord219039673up_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 0.32/1.63 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_144_neg__less__eq__nonneg,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( linord219039673up_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 0.32/1.63 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_145_less__neg__neg,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( linord219039673up_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 0.32/1.63 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_146_neg__less__pos,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( linord219039673up_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 0.32/1.63 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_147_neg__0__less__iff__less,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 0.32/1.63 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_148_neg__less__0__iff__less,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.32/1.63 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_149_diff__0,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( group_add @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 0.32/1.63 = ( uminus_uminus @ A @ A2 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_150_le__imp__neg__le,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ A2 @ B )
% 0.32/1.63 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_151_minus__le__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B )
% 0.32/1.63 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B ) @ A2 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_152_le__minus__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B ) )
% 0.32/1.63 = ( ord_less_eq @ A @ B @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_153_minus__less__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A] :
% 0.32/1.63 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B )
% 0.32/1.63 = ( ord_less @ A @ ( uminus_uminus @ A @ B ) @ A2 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_154_less__minus__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A] :
% 0.32/1.63 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B ) )
% 0.32/1.63 = ( ord_less @ A @ B @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_155_minus__diff__commute,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ab_group_add @ A )
% 0.32/1.63 => ! [B: A,A2: A] :
% 0.32/1.63 ( ( minus_minus @ A @ ( uminus_uminus @ A @ B ) @ A2 )
% 0.32/1.63 = ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ B ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_156_equation__minus__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A] :
% 0.32/1.63 ( ( A2
% 0.32/1.63 = ( uminus_uminus @ A @ B ) )
% 0.32/1.63 = ( B
% 0.32/1.63 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_157_minus__equation__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A] :
% 0.32/1.63 ( ( ( uminus_uminus @ A @ A2 )
% 0.32/1.63 = B )
% 0.32/1.63 = ( ( uminus_uminus @ A @ B )
% 0.32/1.63 = A2 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_158_verit__minus__simplify_I3_J,axiom,
% 0.32/1.63 ! [B2: $tType] :
% 0.32/1.63 ( ( group_add @ B2 )
% 0.32/1.63 => ! [B: B2] :
% 0.32/1.63 ( ( minus_minus @ B2 @ ( zero_zero @ B2 ) @ B )
% 0.32/1.63 = ( uminus_uminus @ B2 @ B ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_159_psubsetI,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.32/1.63 => ( ( A3 != B6 )
% 0.32/1.63 => ( ord_less @ ( set @ A ) @ A3 @ B6 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_160_uminus__apply,axiom,
% 0.32/1.63 ! [B2: $tType,A: $tType] :
% 0.32/1.63 ( ( uminus @ B2 )
% 0.32/1.63 => ( ( uminus_uminus @ ( A > B2 ) )
% 0.32/1.63 = ( ^ [A6: A > B2,X3: A] : ( uminus_uminus @ B2 @ ( A6 @ X3 ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_161_compl__le__compl__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( boolean_algebra @ A )
% 0.32/1.63 => ! [X: A,Y: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
% 0.32/1.63 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_162_rot__diff__commute,axiom,
% 0.32/1.63 ! [B: a,A2: a] :
% 0.32/1.63 ( ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ B @ A2 ) )
% 0.32/1.63 = ( uminus_uminus @ a @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ A2 @ B ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_163_nrm__reverse,axiom,
% 0.32/1.63 ! [A2: a,X: a,Y: a] :
% 0.32/1.63 ( ( inner_780170721_inner @ a @ A2 @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) )
% 0.32/1.63 = ( inner_780170721_inner @ a @ ( uminus_uminus @ a @ A2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ Y @ X ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_164_verit__minus__simplify_I4_J,axiom,
% 0.32/1.63 ! [B2: $tType] :
% 0.32/1.63 ( ( group_add @ B2 )
% 0.32/1.63 => ! [B: B2] :
% 0.32/1.63 ( ( uminus_uminus @ B2 @ ( uminus_uminus @ B2 @ B ) )
% 0.32/1.63 = B ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_165_double__compl,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( boolean_algebra @ A )
% 0.32/1.63 => ! [X: A] :
% 0.32/1.63 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
% 0.32/1.63 = X ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_166_compl__eq__compl__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( boolean_algebra @ A )
% 0.32/1.63 => ! [X: A,Y: A] :
% 0.32/1.63 ( ( ( uminus_uminus @ A @ X )
% 0.32/1.63 = ( uminus_uminus @ A @ Y ) )
% 0.32/1.63 = ( X = Y ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_167_minus__apply,axiom,
% 0.32/1.63 ! [B2: $tType,A: $tType] :
% 0.32/1.63 ( ( minus @ B2 )
% 0.32/1.63 => ( ( minus_minus @ ( A > B2 ) )
% 0.32/1.63 = ( ^ [A6: A > B2,B7: A > B2,X3: A] : ( minus_minus @ B2 @ ( A6 @ X3 ) @ ( B7 @ X3 ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_168_Compl__subset__Compl__iff,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) )
% 0.32/1.63 = ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_169_Compl__anti__mono,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.32/1.63 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_170_subset__antisym,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.32/1.63 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
% 0.32/1.63 => ( A3 = B6 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_171_subsetI,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ! [X4: A] :
% 0.32/1.63 ( ( member @ A @ X4 @ A3 )
% 0.32/1.63 => ( member @ A @ X4 @ B6 ) )
% 0.32/1.63 => ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_172_Diff__idemp,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ B6 )
% 0.32/1.63 = ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_173_Diff__iff,axiom,
% 0.32/1.63 ! [A: $tType,C: A,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
% 0.32/1.63 = ( ( member @ A @ C @ A3 )
% 0.32/1.63 & ~ ( member @ A @ C @ B6 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_174_DiffI,axiom,
% 0.32/1.63 ! [A: $tType,C: A,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( member @ A @ C @ A3 )
% 0.32/1.63 => ( ~ ( member @ A @ C @ B6 )
% 0.32/1.63 => ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_175_rot__rot,axiom,
% 0.32/1.63 ! [V: a] :
% 0.32/1.63 ( ( poinca1750768982en_rot @ a @ ( poinca1750768982en_rot @ a @ V ) )
% 0.32/1.63 = ( uminus_uminus @ a @ V ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_176_verit__la__disequality,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( linorder @ A )
% 0.32/1.63 => ! [A2: A,B: A] :
% 0.32/1.63 ( ( A2 = B )
% 0.32/1.63 | ~ ( ord_less_eq @ A @ A2 @ B )
% 0.32/1.63 | ~ ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_177_verit__comp__simplify1_I1_J,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( order @ A )
% 0.32/1.63 => ! [A2: A] :
% 0.32/1.63 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_178_verit__negate__coefficient_I3_J,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A] :
% 0.32/1.63 ( ( A2 = B )
% 0.32/1.63 => ( ( uminus_uminus @ A @ A2 )
% 0.32/1.63 = ( uminus_uminus @ A @ B ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_179_fun__diff__def,axiom,
% 0.32/1.63 ! [B2: $tType,A: $tType] :
% 0.32/1.63 ( ( minus @ B2 )
% 0.32/1.63 => ( ( minus_minus @ ( A > B2 ) )
% 0.32/1.63 = ( ^ [A6: A > B2,B7: A > B2,X3: A] : ( minus_minus @ B2 @ ( A6 @ X3 ) @ ( B7 @ X3 ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_180_Collect__mono__iff,axiom,
% 0.32/1.63 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.32/1.63 = ( ! [X3: A] :
% 0.32/1.63 ( ( P @ X3 )
% 0.32/1.63 => ( Q @ X3 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_181_set__eq__subset,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ^ [Y5: set @ A,Z4: set @ A] : ( Y5 = Z4 ) )
% 0.32/1.63 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.32/1.63 & ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_182_subset__trans,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A,C3: set @ A] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.32/1.63 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C3 )
% 0.32/1.63 => ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_183_Collect__mono,axiom,
% 0.32/1.63 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/1.63 ( ! [X4: A] :
% 0.32/1.63 ( ( P @ X4 )
% 0.32/1.63 => ( Q @ X4 ) )
% 0.32/1.63 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_184_subset__refl,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_185_subset__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/1.63 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.32/1.63 ! [T: A] :
% 0.32/1.63 ( ( member @ A @ T @ A6 )
% 0.32/1.63 => ( member @ A @ T @ B7 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_186_equalityD2,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( A3 = B6 )
% 0.32/1.63 => ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_187_equalityD1,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( A3 = B6 )
% 0.32/1.63 => ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_188_subset__eq,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/1.63 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.32/1.63 ! [X3: A] :
% 0.32/1.63 ( ( member @ A @ X3 @ A6 )
% 0.32/1.63 => ( member @ A @ X3 @ B7 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_189_equalityE,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( A3 = B6 )
% 0.32/1.63 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.32/1.63 => ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_190_subsetD,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A,C: A] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.32/1.63 => ( ( member @ A @ C @ A3 )
% 0.32/1.63 => ( member @ A @ C @ B6 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_191_in__mono,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A,X: A] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.32/1.63 => ( ( member @ A @ X @ A3 )
% 0.32/1.63 => ( member @ A @ X @ B6 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_192_DiffD2,axiom,
% 0.32/1.63 ! [A: $tType,C: A,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
% 0.32/1.63 => ~ ( member @ A @ C @ B6 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_193_DiffD1,axiom,
% 0.32/1.63 ! [A: $tType,C: A,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
% 0.32/1.63 => ( member @ A @ C @ A3 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_194_DiffE,axiom,
% 0.32/1.63 ! [A: $tType,C: A,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
% 0.32/1.63 => ~ ( ( member @ A @ C @ A3 )
% 0.32/1.63 => ( member @ A @ C @ B6 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_195_psubset__trans,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A,C3: set @ A] :
% 0.32/1.63 ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
% 0.32/1.63 => ( ( ord_less @ ( set @ A ) @ B6 @ C3 )
% 0.32/1.63 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_196_psubsetD,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A,C: A] :
% 0.32/1.63 ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
% 0.32/1.63 => ( ( member @ A @ C @ A3 )
% 0.32/1.63 => ( member @ A @ C @ B6 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_197_verit__comp__simplify1_I3_J,axiom,
% 0.32/1.63 ! [B2: $tType] :
% 0.32/1.63 ( ( linorder @ B2 )
% 0.32/1.63 => ! [B8: B2,A7: B2] :
% 0.32/1.63 ( ( ~ ( ord_less_eq @ B2 @ B8 @ A7 ) )
% 0.32/1.63 = ( ord_less @ B2 @ A7 @ B8 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_198_compl__le__swap2,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( boolean_algebra @ A )
% 0.32/1.63 => ! [Y: A,X: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
% 0.32/1.63 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_199_compl__le__swap1,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( boolean_algebra @ A )
% 0.32/1.63 => ! [Y: A,X: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
% 0.32/1.63 => ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_200_compl__mono,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( boolean_algebra @ A )
% 0.32/1.63 => ! [X: A,Y: A] :
% 0.32/1.63 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/1.63 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_201_verit__negate__coefficient_I2_J,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ordered_ab_group_add @ A )
% 0.32/1.63 => ! [A2: A,B: A] :
% 0.32/1.63 ( ( ord_less @ A @ A2 @ B )
% 0.32/1.63 => ( ord_less @ A @ ( uminus_uminus @ A @ B ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_202_compl__less__swap1,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( boolean_algebra @ A )
% 0.32/1.63 => ! [Y: A,X: A] :
% 0.32/1.63 ( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X ) )
% 0.32/1.63 => ( ord_less @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_203_compl__less__swap2,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( boolean_algebra @ A )
% 0.32/1.63 => ! [Y: A,X: A] :
% 0.32/1.63 ( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X )
% 0.32/1.63 => ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_204_compl__less__compl__iff,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( boolean_algebra @ A )
% 0.32/1.63 => ! [X: A,Y: A] :
% 0.32/1.63 ( ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
% 0.32/1.63 = ( ord_less @ A @ Y @ X ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_205_fun__Compl__def,axiom,
% 0.32/1.63 ! [B2: $tType,A: $tType] :
% 0.32/1.63 ( ( uminus @ B2 )
% 0.32/1.63 => ( ( uminus_uminus @ ( A > B2 ) )
% 0.32/1.63 = ( ^ [A6: A > B2,X3: A] : ( uminus_uminus @ B2 @ ( A6 @ X3 ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_206_double__diff,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A,C3: set @ A] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.32/1.63 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C3 )
% 0.32/1.63 => ( ( minus_minus @ ( set @ A ) @ B6 @ ( minus_minus @ ( set @ A ) @ C3 @ A3 ) )
% 0.32/1.63 = A3 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_207_Diff__subset,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ A3 ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_208_Diff__mono,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,C3: set @ A,D5: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
% 0.32/1.63 => ( ( ord_less_eq @ ( set @ A ) @ D5 @ B6 )
% 0.32/1.63 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ ( minus_minus @ ( set @ A ) @ C3 @ D5 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_209_subset__iff__psubset__eq,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/1.63 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.32/1.63 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 0.32/1.63 | ( A6 = B7 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_210_subset__psubset__trans,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A,C3: set @ A] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.32/1.63 => ( ( ord_less @ ( set @ A ) @ B6 @ C3 )
% 0.32/1.63 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_211_subset__not__subset__eq,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ord_less @ ( set @ A ) )
% 0.32/1.63 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.32/1.63 & ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_212_psubset__subset__trans,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A,C3: set @ A] :
% 0.32/1.63 ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
% 0.32/1.63 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C3 )
% 0.32/1.63 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_213_psubset__imp__subset,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
% 0.32/1.63 => ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_214_psubset__eq,axiom,
% 0.32/1.63 ! [A: $tType] :
% 0.32/1.63 ( ( ord_less @ ( set @ A ) )
% 0.32/1.63 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.32/1.63 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.32/1.63 & ( A6 != B7 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_215_psubsetE,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
% 0.32/1.63 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.32/1.63 => ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_216_psubset__imp__ex__mem,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
% 0.32/1.63 => ? [B5: A] : ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B6 @ A3 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_217_rev_Oopen__existence__ivl_H,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( member @ a @ X @ x )
% 0.32/1.63 => ~ ! [A5: real] :
% 0.32/1.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ A5 )
% 0.32/1.63 => ~ ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ ( uminus_uminus @ real @ A5 ) @ A5 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_218_open__existence__ivl_H,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( member @ a @ X @ x )
% 0.32/1.63 => ~ ! [A5: real] :
% 0.32/1.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ A5 )
% 0.32/1.63 => ~ ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ ( uminus_uminus @ real @ A5 ) @ A5 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_219_rev_Oopen__existence__ivl0,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( member @ a @ X @ x )
% 0.32/1.63 => ? [A5: real,B5: real] :
% 0.32/1.63 ( ( ord_less @ real @ A5 @ ( zero_zero @ real ) )
% 0.32/1.63 & ( ord_less @ real @ ( zero_zero @ real ) @ B5 )
% 0.32/1.63 & ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ A5 @ B5 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_220_existence__ivl0__cong,axiom,
% 0.32/1.63 ! [B2: $tType,Y7: set @ a,G: a > a,X0: a] :
% 0.32/1.63 ( ( x = Y7 )
% 0.32/1.63 => ( ! [X4: a,T2: B2] :
% 0.32/1.63 ( ( member @ a @ X4 @ Y7 )
% 0.32/1.63 => ( ( f @ X4 )
% 0.32/1.63 = ( G @ X4 ) ) )
% 0.32/1.63 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
% 0.32/1.63 = ( auto_l1112008849e_ivl0 @ a @ G @ Y7 @ X0 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_221_existence__ivl__zero,axiom,
% 0.32/1.63 ! [X0: a] :
% 0.32/1.63 ( ( member @ a @ X0 @ x )
% 0.32/1.63 => ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_222_rev_Oexistence__ivl0__cong,axiom,
% 0.32/1.63 ! [B2: $tType,Y7: set @ a,G: a > a,X0: a] :
% 0.32/1.63 ( ( x = Y7 )
% 0.32/1.63 => ( ! [X4: a,T2: B2] :
% 0.32/1.63 ( ( member @ a @ X4 @ Y7 )
% 0.32/1.63 => ( ( uminus_uminus @ ( a > a ) @ f @ X4 )
% 0.32/1.63 = ( G @ X4 ) ) )
% 0.32/1.63 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
% 0.32/1.63 = ( auto_l1112008849e_ivl0 @ a @ G @ Y7 @ X0 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_223_general_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.32/1.63 ! [T3: real,T0: real,X0: a] :
% 0.32/1.63 ( ( member @ real @ ( minus_minus @ real @ T3 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.32/1.63 => ( member @ a @ X0 @ x ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_224_Compl__eq__Compl__iff,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.32/1.63 ( ( ( uminus_uminus @ ( set @ A ) @ A3 )
% 0.32/1.63 = ( uminus_uminus @ ( set @ A ) @ B6 ) )
% 0.32/1.63 = ( A3 = B6 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_225_Compl__iff,axiom,
% 0.32/1.63 ! [A: $tType,C: A,A3: set @ A] :
% 0.32/1.63 ( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
% 0.32/1.63 = ( ~ ( member @ A @ C @ A3 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_226_ComplI,axiom,
% 0.32/1.63 ! [A: $tType,C: A,A3: set @ A] :
% 0.32/1.63 ( ~ ( member @ A @ C @ A3 )
% 0.32/1.63 => ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_227_rev_Oexistence__ivl__zero,axiom,
% 0.32/1.63 ! [X0: a] :
% 0.32/1.63 ( ( member @ a @ X0 @ x )
% 0.32/1.63 => ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_228_rev_Ogeneral_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.32/1.63 ! [T3: real,T0: real,X0: a] :
% 0.32/1.63 ( ( member @ real @ ( minus_minus @ real @ T3 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.32/1.63 => ( member @ a @ X0 @ x ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_229_local_Oivl__subset__existence__ivl,axiom,
% 0.32/1.63 ! [T3: real,X0: a] :
% 0.32/1.63 ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.32/1.63 => ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ T3 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_230_local_Oivl__subset__existence__ivl_H,axiom,
% 0.32/1.63 ! [T3: real,X0: a] :
% 0.32/1.63 ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.32/1.63 => ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ T3 @ ( zero_zero @ real ) ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_231_open__existence__ivl0,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( member @ a @ X @ x )
% 0.32/1.63 => ? [A5: real,B5: real] :
% 0.32/1.63 ( ( ord_less @ real @ A5 @ ( zero_zero @ real ) )
% 0.32/1.63 & ( ord_less @ real @ ( zero_zero @ real ) @ B5 )
% 0.32/1.63 & ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ A5 @ B5 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_232_local_Orev_Oivl__subset__existence__ivl,axiom,
% 0.32/1.63 ! [T3: real,X0: a] :
% 0.32/1.63 ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.32/1.63 => ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ T3 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_233_local_Orev_Oivl__subset__existence__ivl_H,axiom,
% 0.32/1.63 ! [T3: real,X0: a] :
% 0.32/1.63 ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.32/1.63 => ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ T3 @ ( zero_zero @ real ) ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_234_local_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.32/1.63 ! [T3: real,X0: a] :
% 0.32/1.63 ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.32/1.63 => ( member @ a @ X0 @ x ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_235_local_Orev_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.32/1.63 ! [T3: real,X0: a] :
% 0.32/1.63 ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.32/1.63 => ( member @ a @ X0 @ x ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_236_double__complement,axiom,
% 0.32/1.63 ! [A: $tType,A3: set @ A] :
% 0.32/1.63 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
% 0.32/1.63 = A3 ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_237_ComplD,axiom,
% 0.32/1.63 ! [A: $tType,C: A,A3: set @ A] :
% 0.32/1.63 ( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
% 0.32/1.63 => ~ ( member @ A @ C @ A3 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_238_rev_Omvar_Ointerval__axioms,axiom,
% 0.32/1.63 ! [X0: a] : ( initia826609931terval @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_239_mvar_Ointerval__axioms,axiom,
% 0.32/1.63 ! [X0: a] : ( initia826609931terval @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_240_rev_Oclosed__orbit__global__existence,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.32/1.63 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.32/1.63 = ( top_top @ ( set @ real ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_241_rev_Ofixpoint__sol_I1_J,axiom,
% 0.32/1.63 ! [X: a] :
% 0.32/1.63 ( ( member @ a @ X @ x )
% 0.32/1.63 => ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
% 0.32/1.63 = ( zero_zero @ a ) )
% 0.32/1.63 => ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
% 0.32/1.63 = ( top_top @ ( set @ real ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_242_interval__axioms,axiom,
% 0.32/1.63 initia826609931terval @ ( top_top @ ( set @ real ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_243_UNIV__I,axiom,
% 0.32/1.63 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_244_local_Omem__existence__ivl__subset,axiom,
% 0.32/1.63 ! [T3: real,X0: a] :
% 0.32/1.63 ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.32/1.63 => ( member @ real @ T3 @ ( top_top @ ( set @ real ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_245_local_Oexistence__ivl__initial__time,axiom,
% 0.32/1.63 ! [X0: a] :
% 0.32/1.63 ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.32/1.63 => ( ( member @ a @ X0 @ x )
% 0.32/1.63 => ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_246_local_Orev_Omem__existence__ivl__subset,axiom,
% 0.32/1.63 ! [T3: real,X0: a] :
% 0.32/1.63 ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
% 0.32/1.63 => ( member @ real @ T3 @ ( top_top @ ( set @ real ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_247_general_Omem__existence__ivl__iv__defined_I1_J,axiom,
% 0.32/1.63 ! [T3: real,T0: real,X0: a] :
% 0.32/1.63 ( ( member @ real @ ( minus_minus @ real @ T3 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.32/1.63 => ( member @ real @ T0 @ ( top_top @ ( set @ real ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_248_general_Oexistence__ivl__initial__time,axiom,
% 0.32/1.63 ! [T0: real,X0: a] :
% 0.32/1.63 ( ( member @ real @ T0 @ ( top_top @ ( set @ real ) ) )
% 0.32/1.63 => ( ( member @ a @ X0 @ x )
% 0.32/1.63 => ( member @ real @ ( minus_minus @ real @ T0 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_249_mem__existence__ivl__shift__autonomous2,axiom,
% 0.32/1.63 ! [T3: real,S2: real,X: a] :
% 0.32/1.63 ( ( member @ real @ ( minus_minus @ real @ T3 @ S2 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
% 0.32/1.63 => ( ( member @ a @ X @ x )
% 0.32/1.63 => ( ! [S: real,T2: real,X4: a] :
% 0.32/1.63 ( ( member @ a @ X4 @ x )
% 0.32/1.63 => ( ( f @ X4 )
% 0.32/1.63 = ( f @ X4 ) ) )
% 0.32/1.63 => ( ( ( top_top @ ( set @ real ) )
% 0.32/1.63 = ( top_top @ ( set @ real ) ) )
% 0.32/1.63 => ( member @ real @ ( minus_minus @ real @ T3 @ S2 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_250_general_Omem__existence__ivl__subset,axiom,
% 0.32/1.64 ! [T3: real,T0: real,X0: a] :
% 0.32/1.64 ( ( member @ real @ ( minus_minus @ real @ T3 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
% 0.32/1.64 => ( member @ real @ T3 @ ( top_top @ ( set @ real ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_251_local_Oexistence__ivl__subset,axiom,
% 0.32/1.64 ! [X0: a] : ( ord_less_eq @ ( set @ real ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) @ ( top_top @ ( set @ real ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_252_fixpoint__sol_I1_J,axiom,
% 0.32/1.64 ! [X: a] :
% 0.32/1.64 ( ( member @ a @ X @ x )
% 0.32/1.64 => ( ( ( f @ X )
% 0.32/1.64 = ( zero_zero @ a ) )
% 0.32/1.64 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X )
% 0.32/1.64 = ( top_top @ ( set @ real ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_253_closed__orbit__global__existence,axiom,
% 0.32/1.64 ! [X: a] :
% 0.32/1.64 ( ( period385816147_orbit @ a @ f @ x @ X )
% 0.32/1.64 => ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X )
% 0.32/1.64 = ( top_top @ ( set @ real ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_254_local_Orev_Oexistence__ivl__initial__time,axiom,
% 0.32/1.64 ! [X0: a] :
% 0.32/1.64 ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
% 0.32/1.64 => ( ( member @ a @ X0 @ x )
% 0.32/1.64 => ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___HOL_Otype,axiom,
% 0.32/1.64 ! [A: $tType] :
% 0.32/1.64 ( ( execut510477386_space @ A )
% 0.32/1.64 => ( type @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ozero,axiom,
% 0.32/1.64 ! [A: $tType] :
% 0.32/1.64 ( ( execut510477386_space @ A )
% 0.32/1.64 => ( zero @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ominus,axiom,
% 0.32/1.64 ! [A: $tType] :
% 0.32/1.64 ( ( execut510477386_space @ A )
% 0.32/1.64 => ( minus @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ouminus,axiom,
% 0.32/1.64 ! [A: $tType] :
% 0.32/1.64 ( ( execut510477386_space @ A )
% 0.32/1.64 => ( uminus @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Orderings_Oord,axiom,
% 0.32/1.64 ! [A: $tType] :
% 0.32/1.64 ( ( execut510477386_space @ A )
% 0.32/1.64 => ( ord @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Orderings_Oorder,axiom,
% 0.32/1.64 ! [A: $tType] :
% 0.32/1.64 ( ( execut510477386_space @ A )
% 0.32/1.64 => ( order @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ogroup__add,axiom,
% 0.32/1.64 ! [A: $tType] :
% 0.32/1.64 ( ( execut510477386_space @ A )
% 0.32/1.64 => ( group_add @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Orderings_Opreorder,axiom,
% 0.32/1.64 ! [A: $tType] :
% 0.32/1.64 ( ( execut510477386_space @ A )
% 0.32/1.64 => ( preorder @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oab__group__add,axiom,
% 0.32/1.64 ! [A: $tType] :
% 0.32/1.64 ( ( execut510477386_space @ A )
% 0.32/1.64 => ( ab_group_add @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Inner__Product_Oreal__inner,axiom,
% 0.32/1.64 ! [A: $tType] :
% 0.32/1.64 ( ( execut510477386_space @ A )
% 0.32/1.64 => ( inner_real_inner @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__group__add,axiom,
% 0.32/1.64 ! [A: $tType] :
% 0.32/1.64 ( ( execut510477386_space @ A )
% 0.32/1.64 => ( ordered_ab_group_add @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocancel__comm__monoid__add,axiom,
% 0.32/1.64 ! [A: $tType] :
% 0.32/1.64 ( ( execut510477386_space @ A )
% 0.32/1.64 => ( cancel1352612707id_add @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.32/1.64 ! [A: $tType] :
% 0.32/1.64 ( ( execut510477386_space @ A )
% 0.32/1.64 => ( cancel146912293up_add @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Euclidean__Space_Oeuclidean__space,axiom,
% 0.32/1.64 ! [A: $tType] :
% 0.32/1.64 ( ( execut510477386_space @ A )
% 0.32/1.64 => ( euclid925273238_space @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Real__Vector__Spaces_Ometric__space,axiom,
% 0.32/1.64 ! [A: $tType] :
% 0.32/1.64 ( ( execut510477386_space @ A )
% 0.32/1.64 => ( real_V2090557954_space @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Topological__Spaces_Operfect__space,axiom,
% 0.32/1.64 ! [A: $tType] :
% 0.32/1.64 ( ( execut510477386_space @ A )
% 0.32/1.64 => ( topolo890362671_space @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
% 0.32/1.64 ! [A: $tType] :
% 0.32/1.64 ( ( execut510477386_space @ A )
% 0.32/1.64 => ( ordere890947078_space @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_HOL_Obool___Countable_Ocountable,axiom,
% 0.32/1.64 countable @ $o ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Set_Oset___Countable_Ocountable_1,axiom,
% 0.32/1.64 ! [A8: $tType] :
% 0.32/1.64 ( ( finite_finite @ A8 )
% 0.32/1.64 => ( countable @ ( set @ A8 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_fun___Countable_Ocountable_2,axiom,
% 0.32/1.64 ! [A8: $tType,A9: $tType] :
% 0.32/1.64 ( ( ( finite_finite @ A8 )
% 0.32/1.64 & ( countable @ A9 ) )
% 0.32/1.64 => ( countable @ ( A8 > A9 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.32/1.64 ! [A8: $tType,A9: $tType] :
% 0.32/1.64 ( ( ( finite_finite @ A8 )
% 0.32/1.64 & ( finite_finite @ A9 ) )
% 0.32/1.64 => ( finite_finite @ ( A8 > A9 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Set_Oset___Finite__Set_Ofinite_3,axiom,
% 0.32/1.64 ! [A8: $tType] :
% 0.32/1.64 ( ( finite_finite @ A8 )
% 0.32/1.64 => ( finite_finite @ ( set @ A8 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_HOL_Obool___Finite__Set_Ofinite_4,axiom,
% 0.32/1.64 finite_finite @ $o ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_fun___Real__Vector__Spaces_Ometric__space,axiom,
% 0.32/1.64 ! [A8: $tType,A9: $tType] :
% 0.32/1.64 ( ( ( countable @ A8 )
% 0.32/1.64 & ( real_V2090557954_space @ A9 ) )
% 0.32/1.64 => ( real_V2090557954_space @ ( A8 > A9 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 0.32/1.64 ! [A8: $tType,A9: $tType] :
% 0.32/1.64 ( ( boolean_algebra @ A9 )
% 0.32/1.64 => ( boolean_algebra @ ( A8 > A9 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.32/1.64 ! [A8: $tType,A9: $tType] :
% 0.32/1.64 ( ( preorder @ A9 )
% 0.32/1.64 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.32/1.64 ! [A8: $tType,A9: $tType] :
% 0.32/1.64 ( ( order @ A9 )
% 0.32/1.64 => ( order @ ( A8 > A9 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_fun___Orderings_Oord,axiom,
% 0.32/1.64 ! [A8: $tType,A9: $tType] :
% 0.32/1.64 ( ( ord @ A9 )
% 0.32/1.64 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_fun___Groups_Ouminus,axiom,
% 0.32/1.64 ! [A8: $tType,A9: $tType] :
% 0.32/1.64 ( ( uminus @ A9 )
% 0.32/1.64 => ( uminus @ ( A8 > A9 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_fun___Groups_Ominus,axiom,
% 0.32/1.64 ! [A8: $tType,A9: $tType] :
% 0.32/1.64 ( ( minus @ A9 )
% 0.32/1.64 => ( minus @ ( A8 > A9 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_5,axiom,
% 0.32/1.64 ! [A8: $tType] : ( boolean_algebra @ ( set @ A8 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Set_Oset___Orderings_Opreorder_6,axiom,
% 0.32/1.64 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Set_Oset___Orderings_Oorder_7,axiom,
% 0.32/1.64 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 0.32/1.64 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Set_Oset___Groups_Ouminus_9,axiom,
% 0.32/1.64 ! [A8: $tType] : ( uminus @ ( set @ A8 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Set_Oset___Groups_Ominus_10,axiom,
% 0.32/1.64 ! [A8: $tType] : ( minus @ ( set @ A8 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Set_Oset___Groups_Ozero,axiom,
% 0.32/1.64 ! [A8: $tType] :
% 0.32/1.64 ( ( zero @ A8 )
% 0.32/1.64 => ( zero @ ( set @ A8 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_11,axiom,
% 0.32/1.64 boolean_algebra @ $o ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
% 0.32/1.64 preorder @ $o ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.32/1.64 linorder @ $o ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_HOL_Obool___Orderings_Oorder_13,axiom,
% 0.32/1.64 order @ $o ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_HOL_Obool___Orderings_Oord_14,axiom,
% 0.32/1.64 ord @ $o ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_HOL_Obool___Groups_Ouminus_15,axiom,
% 0.32/1.64 uminus @ $o ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_HOL_Obool___Groups_Ominus_16,axiom,
% 0.32/1.64 minus @ $o ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Executable__Euclidean__Space_Oexecutable__euclidean__space,axiom,
% 0.32/1.64 execut510477386_space @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
% 0.32/1.64 ordere890947078_space @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
% 0.32/1.64 topolo890362671_space @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space_17,axiom,
% 0.32/1.64 real_V2090557954_space @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space,axiom,
% 0.32/1.64 euclid925273238_space @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
% 0.32/1.64 linord219039673up_add @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.32/1.64 cancel146912293up_add @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add,axiom,
% 0.32/1.64 cancel1352612707id_add @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
% 0.32/1.64 ordered_ab_group_add @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Inner__Product_Oreal__inner,axiom,
% 0.32/1.64 inner_real_inner @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Groups_Oab__group__add,axiom,
% 0.32/1.64 ab_group_add @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Orderings_Opreorder_18,axiom,
% 0.32/1.64 preorder @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Orderings_Olinorder_19,axiom,
% 0.32/1.64 linorder @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 0.32/1.64 group_add @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Orderings_Oorder_20,axiom,
% 0.32/1.64 order @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Orderings_Oord_21,axiom,
% 0.32/1.64 ord @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Groups_Ouminus_22,axiom,
% 0.32/1.64 uminus @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Groups_Ominus_23,axiom,
% 0.32/1.64 minus @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tcon_Real_Oreal___Groups_Ozero_24,axiom,
% 0.32/1.64 zero @ real ).
% 0.32/1.64
% 0.32/1.64 thf(tfree_0,hypothesis,
% 0.32/1.64 execut510477386_space @ a ).
% 0.32/1.64
% 0.32/1.64 thf(conj_0,conjecture,
% 0.32/1.64 ? [D4: real] :
% 0.32/1.64 ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
% 0.32/1.64 & ? [B9: real] :
% 0.32/1.64 ( ( ord_less @ real @ ( zero_zero @ real ) @ B9 )
% 0.32/1.64 & ! [X4: a] :
% 0.32/1.64 ( ( member @ a @ X4 @ ( elemen321786957_cball @ a @ x2 @ D4 ) )
% 0.32/1.64 => ( ord_less_eq @ real @ B9 @ ( inner_780170721_inner @ a @ ( f @ X4 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ) ) ) ) ).
% 0.32/1.64 ------------------------
% 0.32/1.65 % (21485)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_21273 for (2999ds/183Mi)
% 0.32/1.66 % (21490)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_21273 for (2999ds/275Mi)
% 0.32/1.66 % (21491)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_21273 for (2999ds/18Mi)
% 0.32/1.66 % (21488)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_21273 for (2999ds/2Mi)
% 0.32/1.66 % (21486)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_21273 for (2999ds/4Mi)
% 0.32/1.66 % (21489)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_21273 for (2999ds/2Mi)
% 0.32/1.66 % (21487)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_21273 for (2999ds/27Mi)
% 0.32/1.66 % (21488)Instruction limit reached!
% 0.32/1.66 % (21488)------------------------------
% 0.32/1.66 % (21488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.66 % (21488)Termination reason: Unknown
% 0.32/1.66 % (21489)Instruction limit reached!
% 0.32/1.66 % (21489)------------------------------
% 0.32/1.66 % (21489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.66 % (21488)Termination phase: shuffling
% 0.32/1.66
% 0.32/1.66 % (21488)Memory used [KB]: 1407
% 0.32/1.66 % (21488)Time elapsed: 0.003 s
% 0.32/1.66 % (21488)Instructions burned: 2 (million)
% 0.32/1.66 % (21488)------------------------------
% 0.32/1.66 % (21488)------------------------------
% 0.32/1.66 % (21489)Termination reason: Unknown
% 0.32/1.66 % (21489)Termination phase: shuffling
% 0.32/1.66
% 0.32/1.66 % (21489)Memory used [KB]: 1407
% 0.32/1.66 % (21489)Time elapsed: 0.003 s
% 0.32/1.66 % (21489)Instructions burned: 2 (million)
% 0.32/1.66 % (21489)------------------------------
% 0.32/1.66 % (21489)------------------------------
% 0.32/1.66 % (21486)Instruction limit reached!
% 0.32/1.66 % (21486)------------------------------
% 0.32/1.66 % (21486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.66 % (21486)Termination reason: Unknown
% 0.32/1.66 % (21486)Termination phase: shuffling
% 0.32/1.66
% 0.32/1.66 % (21486)Memory used [KB]: 1407
% 0.32/1.66 % (21486)Time elapsed: 0.004 s
% 0.32/1.66 % (21486)Instructions burned: 4 (million)
% 0.32/1.66 % (21486)------------------------------
% 0.32/1.66 % (21486)------------------------------
% 0.32/1.67 % (21491)Instruction limit reached!
% 0.32/1.67 % (21491)------------------------------
% 0.32/1.67 % (21491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.67 % (21491)Termination reason: Unknown
% 0.32/1.67 % (21491)Termination phase: shuffling
% 0.32/1.67
% 0.32/1.67 % (21491)Memory used [KB]: 1663
% 0.32/1.67 % (21491)Time elapsed: 0.011 s
% 0.32/1.67 % (21491)Instructions burned: 18 (million)
% 0.32/1.67 % (21491)------------------------------
% 0.32/1.67 % (21491)------------------------------
% 0.32/1.68 % (21487)Instruction limit reached!
% 0.32/1.68 % (21487)------------------------------
% 0.32/1.68 % (21487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.68 % (21487)Termination reason: Unknown
% 0.32/1.68 % (21487)Termination phase: Property scanning
% 0.32/1.68
% 0.32/1.68 % (21487)Memory used [KB]: 1663
% 0.32/1.68 % (21487)Time elapsed: 0.016 s
% 0.32/1.68 % (21487)Instructions burned: 30 (million)
% 0.32/1.68 % (21487)------------------------------
% 0.32/1.68 % (21487)------------------------------
% 0.32/1.68 % (21497)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_21273 for (2999ds/3Mi)
% 0.32/1.68 % (21498)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_21273 for (2999ds/37Mi)
% 0.32/1.68 % (21499)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_21273 for (2999ds/15Mi)
% 0.32/1.68 % (21497)Instruction limit reached!
% 0.32/1.68 % (21497)------------------------------
% 0.32/1.68 % (21497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.68 % (21497)Termination reason: Unknown
% 0.32/1.68 % (21497)Termination phase: shuffling
% 0.32/1.68
% 0.32/1.68 % (21497)Memory used [KB]: 1407
% 0.32/1.68 % (21497)Time elapsed: 0.004 s
% 0.32/1.68 % (21497)Instructions burned: 4 (million)
% 0.32/1.68 % (21497)------------------------------
% 0.32/1.68 % (21497)------------------------------
% 0.32/1.69 % (21499)Instruction limit reached!
% 0.32/1.69 % (21499)------------------------------
% 0.32/1.69 % (21499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.69 % (21499)Termination reason: Unknown
% 0.32/1.69 % (21499)Termination phase: shuffling
% 0.32/1.69
% 0.32/1.69 % (21499)Memory used [KB]: 1535
% 0.32/1.69 % (21499)Time elapsed: 0.010 s
% 0.32/1.69 % (21499)Instructions burned: 15 (million)
% 0.32/1.69 % (21499)------------------------------
% 0.32/1.69 % (21499)------------------------------
% 0.32/1.69 % (21501)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_21273 for (2999ds/3Mi)
% 0.32/1.69 % (21501)Instruction limit reached!
% 0.32/1.69 % (21501)------------------------------
% 0.32/1.69 % (21501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.69 % (21501)Termination reason: Unknown
% 0.32/1.69 % (21501)Termination phase: shuffling
% 0.32/1.69
% 0.32/1.69 % (21501)Memory used [KB]: 1407
% 0.32/1.69 % (21501)Time elapsed: 0.004 s
% 0.32/1.69 % (21501)Instructions burned: 5 (million)
% 0.32/1.69 % (21501)------------------------------
% 0.32/1.69 % (21501)------------------------------
% 0.32/1.69 % (21504)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_21273 for (2999ds/1041Mi)
% 0.32/1.69 % (21498)Instruction limit reached!
% 0.32/1.69 % (21498)------------------------------
% 0.32/1.69 % (21498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.69 % (21498)Termination reason: Unknown
% 0.32/1.69 % (21498)Termination phase: Property scanning
% 0.32/1.69
% 0.32/1.69 % (21498)Memory used [KB]: 2046
% 0.32/1.69 % (21498)Time elapsed: 0.016 s
% 0.32/1.69 % (21498)Instructions burned: 38 (million)
% 0.32/1.69 % (21498)------------------------------
% 0.32/1.69 % (21498)------------------------------
% 3.13/1.70 % (21507)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_21273 for (2999ds/16Mi)
% 3.13/1.70 % (21506)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_21273 for (2999ds/7Mi)
% 3.13/1.70 % (21509)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_21273 for (2999ds/3Mi)
% 3.13/1.70 % (21506)Instruction limit reached!
% 3.13/1.70 % (21506)------------------------------
% 3.13/1.70 % (21506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.70 % (21506)Termination reason: Unknown
% 3.13/1.70 % (21506)Termination phase: shuffling
% 3.13/1.70
% 3.13/1.70 % (21506)Memory used [KB]: 1407
% 3.13/1.70 % (21506)Time elapsed: 0.004 s
% 3.13/1.70 % (21506)Instructions burned: 7 (million)
% 3.13/1.70 % (21506)------------------------------
% 3.13/1.70 % (21506)------------------------------
% 3.13/1.70 % (21509)Instruction limit reached!
% 3.13/1.70 % (21509)------------------------------
% 3.13/1.70 % (21509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.70 % (21509)Termination reason: Unknown
% 3.13/1.70 % (21509)Termination phase: shuffling
% 3.13/1.70
% 3.13/1.70 % (21509)Memory used [KB]: 1407
% 3.13/1.70 % (21509)Time elapsed: 0.003 s
% 3.13/1.70 % (21509)Instructions burned: 4 (million)
% 3.13/1.70 % (21509)------------------------------
% 3.13/1.70 % (21509)------------------------------
% 3.13/1.70 % (21507)Instruction limit reached!
% 3.13/1.70 % (21507)------------------------------
% 3.13/1.70 % (21507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.70 % (21507)Termination reason: Unknown
% 3.13/1.70 % (21507)Termination phase: shuffling
% 3.13/1.70
% 3.13/1.70 % (21507)Memory used [KB]: 1535
% 3.13/1.70 % (21507)Time elapsed: 0.007 s
% 3.13/1.70 % (21507)Instructions burned: 17 (million)
% 3.13/1.70 % (21507)------------------------------
% 3.13/1.70 % (21507)------------------------------
% 3.13/1.70 % (21510)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_21273 for (2999ds/3Mi)
% 3.13/1.71 % (21510)Instruction limit reached!
% 3.13/1.71 % (21510)------------------------------
% 3.13/1.71 % (21510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.71 % (21510)Termination reason: Unknown
% 3.13/1.71 % (21510)Termination phase: shuffling
% 3.13/1.71
% 3.13/1.71 % (21510)Memory used [KB]: 1407
% 3.13/1.71 % (21510)Time elapsed: 0.003 s
% 3.13/1.71 % (21510)Instructions burned: 4 (million)
% 3.13/1.71 % (21510)------------------------------
% 3.13/1.71 % (21510)------------------------------
% 3.13/1.71 % (21511)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_21273 for (2999ds/7Mi)
% 3.13/1.71 % (21512)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_21273 for (2999ds/3Mi)
% 3.13/1.71 % (21512)Instruction limit reached!
% 3.13/1.71 % (21512)------------------------------
% 3.13/1.71 % (21512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.71 % (21512)Termination reason: Unknown
% 3.13/1.71 % (21512)Termination phase: shuffling
% 3.13/1.71
% 3.13/1.71 % (21512)Memory used [KB]: 1407
% 3.13/1.71 % (21512)Time elapsed: 0.002 s
% 3.13/1.71 % (21512)Instructions burned: 4 (million)
% 3.13/1.71 % (21512)------------------------------
% 3.13/1.71 % (21512)------------------------------
% 3.13/1.71 % (21511)Instruction limit reached!
% 3.13/1.71 % (21511)------------------------------
% 3.13/1.71 % (21511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.71 % (21511)Termination reason: Unknown
% 3.13/1.71 % (21511)Termination phase: shuffling
% 3.13/1.71
% 3.13/1.71 % (21511)Memory used [KB]: 1407
% 3.13/1.71 % (21511)Time elapsed: 0.004 s
% 3.13/1.71 % (21511)Instructions burned: 9 (million)
% 3.13/1.71 % (21511)------------------------------
% 3.13/1.71 % (21511)------------------------------
% 3.13/1.71 % (21513)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_21273 for (2999ds/4Mi)
% 3.13/1.72 % (21513)Instruction limit reached!
% 3.13/1.72 % (21513)------------------------------
% 3.13/1.72 % (21513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.72 % (21513)Termination reason: Unknown
% 3.13/1.72 % (21513)Termination phase: shuffling
% 3.13/1.72
% 3.13/1.72 % (21513)Memory used [KB]: 1407
% 3.13/1.72 % (21513)Time elapsed: 0.003 s
% 3.13/1.72 % (21513)Instructions burned: 6 (million)
% 3.13/1.72 % (21513)------------------------------
% 3.13/1.72 % (21513)------------------------------
% 3.13/1.72 % (21514)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_21273 for (2999ds/18Mi)
% 3.13/1.72 % (21485)Instruction limit reached!
% 3.13/1.72 % (21485)------------------------------
% 3.13/1.72 % (21485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.72 % (21485)Termination reason: Unknown
% 3.13/1.72 % (21485)Termination phase: Saturation
% 3.13/1.72
% 3.13/1.72 % (21485)Memory used [KB]: 6908
% 3.13/1.72 % (21485)Time elapsed: 0.066 s
% 3.13/1.72 % (21485)Instructions burned: 184 (million)
% 3.13/1.72 % (21485)------------------------------
% 3.13/1.72 % (21485)------------------------------
% 3.13/1.72 % (21514)Instruction limit reached!
% 3.13/1.72 % (21514)------------------------------
% 3.13/1.72 % (21514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.72 % (21514)Termination reason: Unknown
% 3.13/1.72 % (21514)Termination phase: shuffling
% 3.13/1.72
% 3.13/1.72 % (21514)Memory used [KB]: 1663
% 3.13/1.72 % (21514)Time elapsed: 0.008 s
% 3.13/1.72 % (21514)Instructions burned: 19 (million)
% 3.13/1.72 % (21514)------------------------------
% 3.13/1.72 % (21514)------------------------------
% 3.13/1.72 % (21515)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_21273 for (2999ds/710Mi)
% 3.13/1.72 % (21516)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_21273 for (2999ds/6Mi)
% 3.13/1.73 % (21516)Instruction limit reached!
% 3.13/1.73 % (21516)------------------------------
% 3.13/1.73 % (21516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.73 % (21516)Termination reason: Unknown
% 3.13/1.73 % (21516)Termination phase: shuffling
% 3.13/1.73
% 3.13/1.73 % (21516)Memory used [KB]: 1407
% 3.13/1.73 % (21516)Time elapsed: 0.003 s
% 3.13/1.73 % (21516)Instructions burned: 6 (million)
% 3.13/1.73 % (21516)------------------------------
% 3.13/1.73 % (21516)------------------------------
% 3.13/1.73 % (21517)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_21273 for (2999ds/902Mi)
% 3.13/1.73 % (21518)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_21273 for (2999ds/21Mi)
% 3.13/1.73 % (21519)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_21273 for (2999ds/5Mi)
% 3.13/1.73 % (21519)Instruction limit reached!
% 3.13/1.73 % (21519)------------------------------
% 3.13/1.73 % (21519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.73 % (21519)Termination reason: Unknown
% 3.13/1.73 % (21519)Termination phase: shuffling
% 3.13/1.73
% 3.13/1.73 % (21519)Memory used [KB]: 1407
% 3.13/1.73 % (21519)Time elapsed: 0.003 s
% 3.13/1.73 % (21519)Instructions burned: 7 (million)
% 3.13/1.73 % (21519)------------------------------
% 3.13/1.73 % (21519)------------------------------
% 3.13/1.73 % (21520)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_21273 for (2999ds/6Mi)
% 3.13/1.74 % (21518)Instruction limit reached!
% 3.13/1.74 % (21518)------------------------------
% 3.13/1.74 % (21518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.74 % (21518)Termination reason: Unknown
% 3.13/1.74 % (21518)Termination phase: shuffling
% 3.13/1.74
% 3.13/1.74 % (21518)Memory used [KB]: 1663
% 3.13/1.74 % (21518)Time elapsed: 0.008 s
% 3.13/1.74 % (21518)Instructions burned: 21 (million)
% 3.13/1.74 % (21518)------------------------------
% 3.13/1.74 % (21518)------------------------------
% 3.13/1.74 % (21520)Instruction limit reached!
% 3.13/1.74 % (21520)------------------------------
% 3.13/1.74 % (21520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.74 % (21520)Termination reason: Unknown
% 3.13/1.74 % (21520)Termination phase: shuffling
% 3.13/1.74
% 3.13/1.74 % (21520)Memory used [KB]: 1407
% 3.13/1.74 % (21520)Time elapsed: 0.003 s
% 3.13/1.74 % (21520)Instructions burned: 7 (million)
% 3.13/1.74 % (21520)------------------------------
% 3.13/1.74 % (21520)------------------------------
% 3.13/1.74 % (21521)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_21273 for (2999ds/377Mi)
% 3.13/1.75 % (21522)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_21273 for (2999ds/779Mi)
% 3.13/1.75 % (21523)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_21273 for (2999ds/19Mi)
% 3.13/1.75 % (21490)Instruction limit reached!
% 3.13/1.75 % (21490)------------------------------
% 3.13/1.75 % (21490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.75 % (21490)Termination reason: Unknown
% 3.13/1.75 % (21490)Termination phase: Saturation
% 3.13/1.75
% 3.13/1.75 % (21490)Memory used [KB]: 6908
% 3.13/1.75 % (21490)Time elapsed: 0.091 s
% 3.13/1.75 % (21490)Instructions burned: 276 (million)
% 3.13/1.75 % (21490)------------------------------
% 3.13/1.75 % (21490)------------------------------
% 3.13/1.75 % (21523)Instruction limit reached!
% 3.13/1.75 % (21523)------------------------------
% 3.13/1.75 % (21523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.75 % (21523)Termination reason: Unknown
% 3.13/1.75 % (21523)Termination phase: shuffling
% 3.13/1.75
% 3.13/1.75 % (21523)Memory used [KB]: 1663
% 3.13/1.75 % (21523)Time elapsed: 0.007 s
% 3.13/1.75 % (21523)Instructions burned: 20 (million)
% 3.13/1.75 % (21523)------------------------------
% 3.13/1.75 % (21523)------------------------------
% 3.13/1.76 % (21524)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_21273 for (2998ds/879Mi)
% 3.13/1.76 % (21525)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_21273 for (2998ds/17Mi)
% 3.13/1.77 % (21525)Instruction limit reached!
% 3.13/1.77 % (21525)------------------------------
% 3.13/1.77 % (21525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.77 % (21525)Termination reason: Unknown
% 3.13/1.77 % (21525)Termination phase: shuffling
% 3.13/1.77
% 3.13/1.77 % (21525)Memory used [KB]: 1535
% 3.13/1.77 % (21525)Time elapsed: 0.007 s
% 3.13/1.77 % (21525)Instructions burned: 18 (million)
% 3.13/1.77 % (21525)------------------------------
% 3.13/1.77 % (21525)------------------------------
% 3.13/1.78 % (21526)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_21273 for (2998ds/3Mi)
% 3.13/1.78 % (21526)Instruction limit reached!
% 3.13/1.78 % (21526)------------------------------
% 3.13/1.78 % (21526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.78 % (21526)Termination reason: Unknown
% 3.13/1.78 % (21526)Termination phase: shuffling
% 3.13/1.78
% 3.13/1.78 % (21526)Memory used [KB]: 1407
% 3.13/1.78 % (21526)Time elapsed: 0.002 s
% 3.13/1.78 % (21526)Instructions burned: 3 (million)
% 3.13/1.78 % (21526)------------------------------
% 3.13/1.78 % (21526)------------------------------
% 3.13/1.79 % (21527)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_21273 for (2998ds/30Mi)
% 3.13/1.80 % (21517)First to succeed.
% 3.13/1.80 % (21517)Refutation found. Thanks to Tanya!
% 3.13/1.80 % SZS status Theorem for DTF2THF_21273
% 3.13/1.80 % SZS output start Proof for DTF2THF_21273
% 3.13/1.80 thf(type_def_5, type, real: $tType).
% 3.13/1.80 thf(type_def_6, type, set: ($tType) > $tType).
% 3.13/1.80 thf(type_def_8, type, bounde2145540817linfun: ($tType * $tType) > $tType).
% 3.13/1.80 thf(type_def_9, type, a: $tType).
% 3.13/1.80 thf(func_def_0, type, bounde2145540817linfun: ($tType * $tType) > $tType).
% 3.13/1.80 thf(func_def_1, type, real: $tType).
% 3.13/1.80 thf(func_def_2, type, set: ($tType) > $tType).
% 3.13/1.80 thf(func_def_3, type, a: $tType).
% 3.13/1.80 thf(func_def_4, type, finite_finite: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_5, type, type: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_6, type, countable: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_7, type, zero: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_8, type, minus: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_9, type, uminus: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_10, type, ord: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_11, type, order: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_12, type, group_add: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_13, type, linorder: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_14, type, preorder: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_15, type, ab_group_add: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_16, type, comm_monoid_diff: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_17, type, inner_real_inner: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_18, type, boolean_algebra: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_19, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_20, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_21, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_22, type, linord219039673up_add: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_23, type, euclid925273238_space: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_24, type, real_V2090557954_space: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_25, type, topolo890362671_space: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_26, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_27, type, ordere890947078_space: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_28, type, execut510477386_space: !>[X0: $tType]:($o)).
% 3.13/1.80 thf(func_def_29, type, elemen321786957_cball: !>[X0: $tType]:(X0 > real > set @ X0)).
% 3.13/1.80 thf(func_def_30, type, auto_l1112008849e_ivl0: !>[X0: $tType]:((X0 > X0) > set @ X0 > X0 > set @ real)).
% 3.13/1.80 thf(func_def_31, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.13/1.80 thf(func_def_32, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 3.13/1.80 thf(func_def_33, type, zero_zero: !>[X0: $tType]:(X0)).
% 3.13/1.80 thf(func_def_34, type, initia826609931terval: set @ real > $o).
% 3.13/1.80 thf(func_def_35, type, inner_780170721_inner: !>[X0: $tType]:(X0 > X0 > real)).
% 3.13/1.80 thf(func_def_36, type, auto_ll_on_invariant: !>[X0: $tType]:((X0 > X0) > set @ X0 > set @ X0 > $o)).
% 3.13/1.80 thf(func_def_37, type, line_closed_segment: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 3.13/1.80 thf(func_def_38, type, line_open_segment: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 3.13/1.80 thf(func_def_39, type, oDE_au1039603466rapped: !>[X0: $tType]:((X0 > X0) > set @ X0 > X0 > set @ X0 > $o)).
% 3.13/1.80 thf(func_def_40, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.13/1.80 thf(func_def_41, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.13/1.80 thf(func_def_42, type, top_top: !>[X0: $tType]:(X0)).
% 3.13/1.80 thf(func_def_43, type, period385816147_orbit: !>[X0: $tType]:((X0 > X0) > set @ X0 > X0 > $o)).
% 3.13/1.80 thf(func_def_44, type, period1153813292period: !>[X0: $tType]:((X0 > X0) > set @ X0 > X0 > real)).
% 3.13/1.80 thf(func_def_45, type, period862636932_orbit: !>[X0: $tType]:((X0 > X0) > set @ X0 > X0 > $o)).
% 3.13/1.80 thf(func_def_46, type, poinca1750768982en_rot: !>[X0: $tType]:(X0 > X0)).
% 3.13/1.80 thf(func_def_47, type, poinca272511729egment: !>[X0: $tType]:((X0 > X0) > set @ X0 > X0 > X0 > $o)).
% 3.13/1.80 thf(func_def_48, type, reacha1084862253ection: !>[X0: $tType]:((X0 > X0) > set @ X0 > (X0 > real) > (X0 > bounde2145540817linfun @ X0 @ real) > set @ X0 > $o)).
% 3.13/1.80 thf(func_def_49, type, real_V2000881966t_dist: !>[X0: $tType]:(X0 > X0 > real)).
% 3.13/1.80 thf(func_def_50, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.13/1.80 thf(func_def_51, type, set_or331188842AtMost: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 3.13/1.80 thf(func_def_52, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.13/1.80 thf(func_def_53, type, x: set @ a).
% 3.13/1.80 thf(func_def_54, type, a2: a).
% 3.13/1.80 thf(func_def_55, type, b: a).
% 3.13/1.80 thf(func_def_56, type, d: real).
% 3.13/1.80 thf(func_def_57, type, f: a > a).
% 3.13/1.80 thf(func_def_58, type, s: a).
% 3.13/1.80 thf(func_def_60, type, x2: a).
% 3.13/1.80 thf(func_def_61, type, z: a).
% 3.13/1.80 thf(func_def_75, type, sP0: a > a > $o).
% 3.13/1.80 thf(func_def_76, type, sP1: (real > real > $o) > real > real > $o).
% 3.13/1.80 thf(func_def_77, type, sK2: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.13/1.80 thf(func_def_78, type, sK3: a > real).
% 3.13/1.80 thf(func_def_79, type, sK4: a > real).
% 3.13/1.80 thf(func_def_80, type, sK5: a > real).
% 3.13/1.80 thf(func_def_81, type, sK6: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.13/1.80 thf(func_def_82, type, sK7: (a > a) > set @ a > a).
% 3.13/1.80 thf(func_def_83, type, sK8: !>[X0: $tType]:(X0 > real > X0)).
% 3.13/1.80 thf(func_def_84, type, sK9: set @ a > a).
% 3.13/1.80 thf(func_def_85, type, sK10: real).
% 3.13/1.80 thf(func_def_86, type, sK11: real > real > a).
% 3.13/1.80 thf(func_def_87, type, sK12: a > a > a).
% 3.13/1.80 thf(func_def_88, type, sK13: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 3.13/1.80 thf(func_def_89, type, sK14: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.13/1.80 thf(func_def_90, type, sK15: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 3.13/1.80 thf(func_def_91, type, sK16: real > real > a).
% 3.13/1.80 thf(func_def_92, type, sK17: real > real > a).
% 3.13/1.80 thf(func_def_93, type, sK18: a).
% 3.13/1.80 thf(func_def_94, type, sK19: !>[X0: $tType]:(X0 > X0)).
% 3.13/1.80 thf(func_def_95, type, sK20: a).
% 3.13/1.80 thf(func_def_96, type, sK21: set @ a > a).
% 3.13/1.80 thf(func_def_97, type, sK22: a > real).
% 3.13/1.80 thf(func_def_98, type, sK23: a > real).
% 3.13/1.80 thf(func_def_99, type, sK24: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.13/1.80 thf(func_def_100, type, sK25: (real > real > $o) > real > real > real).
% 3.13/1.80 thf(func_def_101, type, sK26: real > (real > real > $o) > real > real > real).
% 3.13/1.80 thf(func_def_102, type, sK27: real > (real > real > $o) > real > real > real).
% 3.13/1.80 thf(func_def_103, type, sK28: (real > real > $o) > real).
% 3.13/1.80 thf(func_def_104, type, sK29: (real > real > $o) > real).
% 3.13/1.80 thf(func_def_105, type, sK30: (real > real > $o) > real).
% 3.13/1.80 thf(func_def_106, type, sK31: a > a).
% 3.13/1.80 thf(func_def_107, type, sK32: a > a).
% 3.13/1.80 thf(func_def_108, type, sK33: (a > a) > set @ a > a).
% 3.13/1.80 thf(func_def_109, type, sK34: a > real).
% 3.13/1.80 thf(func_def_111, type, ph36: !>[X0: $tType]:(X0)).
% 3.13/1.80 thf(f2459,plain,(
% 3.13/1.80 $false),
% 3.13/1.80 inference(avatar_sat_refutation,[status(thm)],[f1930,f2397,f2455])).
% 3.13/1.80 thf(f2455,plain,(
% 3.13/1.80 ~spl35_13 | ~spl35_12),
% 3.13/1.80 inference(avatar_split_clause,[status(thm)],[f2454,f1924,f1927])).
% 3.13/1.80 thf(f1927,definition,(
% 3.13/1.80 spl35_13 <=> ($true = (ord_less @ real @ zero_zero @ real @ (inner_780170721_inner @ a @ (f @ sK20) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b)))))),
% 3.13/1.80 introduced(definition,[new_symbols(naming,[spl35_13])],[avatar_definition])).
% 3.13/1.80 thf(f1924,definition,(
% 3.13/1.80 spl35_12 <=> ! [X0 : real] : (($true != (ord_less @ real @ zero_zero @ real @ X0)) | ((member @ a @ (sK11 @ (inner_780170721_inner @ a @ (f @ sK20) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))) @ X0) @ (elemen321786957_cball @ a @ x2 @ d)) != $true))),
% 3.13/1.80 introduced(definition,[new_symbols(naming,[spl35_12])],[avatar_definition])).
% 3.13/1.80 thf(f2454,plain,(
% 3.13/1.80 ($true != (ord_less @ real @ zero_zero @ real @ (inner_780170721_inner @ a @ (f @ sK20) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))) | ~spl35_12),
% 3.13/1.80 inference(trivial_inequality_removal,[status(thm)],[f2453])).
% 3.13/1.80 thf(f2453,plain,(
% 3.13/1.80 ($true != (ord_less @ real @ zero_zero @ real @ (inner_780170721_inner @ a @ (f @ sK20) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))) | ($true != $true) | ~spl35_12),
% 3.13/1.80 inference(forward_demodulation,[status(thm)],[f2441,f1741])).
% 3.13/1.80 thf(f1741,plain,(
% 3.13/1.80 ((ord_less @ real @ zero_zero @ real @ d) = $true)),
% 3.13/1.80 inference(cnf_transformation,[status(thm)],[f910])).
% 3.13/1.80 thf(f910,plain,(
% 3.13/1.80 ((ord_less @ real @ zero_zero @ real @ d) = $true)),
% 3.13/1.80 inference(fool_elimination,[status(thm)],[f909])).
% 3.13/1.80 thf(f909,plain,(
% 3.13/1.80 (ord_less @ real @ zero_zero @ real @ d)),
% 3.13/1.80 inference(rectify,[status(thm)],[f1])).
% 3.13/1.80 thf(f1,axiom,(
% 3.13/1.80 (ord_less @ real @ zero_zero @ real @ d)),
% 3.13/1.80 file('/export/starexec/sandbox/tmp/tmp.laqx631JN2/DTF2THF_21273.p',fact_0_d_I1_J)).
% 3.13/1.80 thf(f2441,plain,(
% 3.13/1.80 ($true != (ord_less @ real @ zero_zero @ real @ (inner_780170721_inner @ a @ (f @ sK20) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))) | ((ord_less @ real @ zero_zero @ real @ d) != $true) | ~spl35_12),
% 3.13/1.80 inference(trivial_inequality_removal,[status(thm)],[f2440])).
% 3.13/1.80 thf(f2440,plain,(
% 3.13/1.80 ($true != (ord_less @ real @ zero_zero @ real @ (inner_780170721_inner @ a @ (f @ sK20) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))) | ($true != $true) | ((ord_less @ real @ zero_zero @ real @ d) != $true) | ~spl35_12),
% 3.13/1.80 inference(duplicate_literal_removal,[status(thm)],[f2433])).
% 3.13/1.80 thf(f2433,plain,(
% 3.13/1.80 ($true != $true) | ((ord_less @ real @ zero_zero @ real @ d) != $true) | ($true != (ord_less @ real @ zero_zero @ real @ (inner_780170721_inner @ a @ (f @ sK20) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))) | ((ord_less @ real @ zero_zero @ real @ d) != $true) | ~spl35_12),
% 3.13/1.80 inference(superposition,[status(thm)],[f1925,f1458])).
% 3.13/1.80 thf(f1458,plain,(
% 3.13/1.80 ( ! [X0 : real,X1 : real] : (($true = (member @ a @ (sK11 @ X1 @ X0) @ (elemen321786957_cball @ a @ x2 @ X0))) | ($true != (ord_less @ real @ zero_zero @ real @ X1)) | ($true != (ord_less @ real @ zero_zero @ real @ X0))) )),
% 3.13/1.80 inference(cnf_transformation,[status(thm)],[f1288])).
% 3.13/1.80 thf(f1288,plain,(
% 3.13/1.80 ! [X0 : real] : (! [X1 : real] : ((($true = (member @ a @ (sK11 @ X1 @ X0) @ (elemen321786957_cball @ a @ x2 @ X0))) & ($true != (ord_less_eq @ real @ X1 @ (inner_780170721_inner @ a @ (f @ (sK11 @ X1 @ X0)) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b)))))) | ($true != (ord_less @ real @ zero_zero @ real @ X1))) | ($true != (ord_less @ real @ zero_zero @ real @ X0)))),
% 3.13/1.80 inference(skolemisation,[status(esa),new_symbols(skolem,[sK11])],[f1074,f1287])).
% 3.13/1.80 thf(f1287,plain,(
% 3.13/1.80 ! [X0 : real,X1 : real] : (? [X2 : a] : (((member @ a @ X2 @ (elemen321786957_cball @ a @ x2 @ X0)) = $true) & ($true != (ord_less_eq @ real @ X1 @ (inner_780170721_inner @ a @ (f @ X2) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b)))))) => (($true = (member @ a @ (sK11 @ X1 @ X0) @ (elemen321786957_cball @ a @ x2 @ X0))) & ($true != (ord_less_eq @ real @ X1 @ (inner_780170721_inner @ a @ (f @ (sK11 @ X1 @ X0)) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b)))))))),
% 3.13/1.80 introduced(definition,[],[choice_axiom])).
% 3.13/1.80 thf(f1074,plain,(
% 3.13/1.80 ! [X0 : real] : (! [X1 : real] : (? [X2 : a] : (((member @ a @ X2 @ (elemen321786957_cball @ a @ x2 @ X0)) = $true) & ($true != (ord_less_eq @ real @ X1 @ (inner_780170721_inner @ a @ (f @ X2) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b)))))) | ($true != (ord_less @ real @ zero_zero @ real @ X1))) | ($true != (ord_less @ real @ zero_zero @ real @ X0)))),
% 3.13/1.80 inference(ennf_transformation,[status(thm)],[f378])).
% 3.13/1.80 thf(f378,plain,(
% 3.13/1.80 ~? [X0 : real] : (($true = (ord_less @ real @ zero_zero @ real @ X0)) & ? [X1 : real] : (($true = (ord_less @ real @ zero_zero @ real @ X1)) & ! [X2 : a] : (((member @ a @ X2 @ (elemen321786957_cball @ a @ x2 @ X0)) = $true) => ($true = (ord_less_eq @ real @ X1 @ (inner_780170721_inner @ a @ (f @ X2) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))))))),
% 3.13/1.80 inference(fool_elimination,[status(thm)],[f377])).
% 3.13/1.80 thf(f377,plain,(
% 3.13/1.80 ~? [X0 : real] : (? [X1 : real] : (! [X2 : a] : ((member @ a @ X2 @ (elemen321786957_cball @ a @ x2 @ X0)) => (ord_less_eq @ real @ X1 @ (inner_780170721_inner @ a @ (f @ X2) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))) & (ord_less @ real @ zero_zero @ real @ X1)) & (ord_less @ real @ zero_zero @ real @ X0))),
% 3.13/1.80 inference(rectify,[status(thm)],[f321])).
% 3.13/1.80 thf(f321,negated_conjecture,(
% 3.13/1.80 ~? [X44 : real] : (? [X68 : real] : (! [X21 : a] : ((member @ a @ X21 @ (elemen321786957_cball @ a @ x2 @ X44)) => (ord_less_eq @ real @ X68 @ (inner_780170721_inner @ a @ (f @ X21) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))) & (ord_less @ real @ zero_zero @ real @ X68)) & (ord_less @ real @ zero_zero @ real @ X44))),
% 3.13/1.80 inference(negated_conjecture,[status(cth)],[f320])).
% 3.13/1.80 thf(f320,conjecture,(
% 3.13/1.80 ? [X44 : real] : (? [X68 : real] : (! [X21 : a] : ((member @ a @ X21 @ (elemen321786957_cball @ a @ x2 @ X44)) => (ord_less_eq @ real @ X68 @ (inner_780170721_inner @ a @ (f @ X21) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))) & (ord_less @ real @ zero_zero @ real @ X68)) & (ord_less @ real @ zero_zero @ real @ X44))),
% 3.13/1.80 file('/export/starexec/sandbox/tmp/tmp.laqx631JN2/DTF2THF_21273.p',conj_0)).
% 3.13/1.80 thf(f1925,plain,(
% 3.13/1.80 ( ! [X0 : real] : (((member @ a @ (sK11 @ (inner_780170721_inner @ a @ (f @ sK20) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))) @ X0) @ (elemen321786957_cball @ a @ x2 @ d)) != $true) | ($true != (ord_less @ real @ zero_zero @ real @ X0))) ) | ~spl35_12),
% 3.13/1.80 inference(avatar_component_clause,[status(thm)],[f1924])).
% 3.13/1.80 thf(f2397,plain,(
% 3.13/1.80 spl35_13),
% 3.13/1.80 inference(avatar_contradiction_clause,[status(thm)],[f2396])).
% 3.13/1.80 thf(f2396,plain,(
% 3.13/1.80 $false | spl35_13),
% 3.13/1.80 inference(trivial_inequality_removal,[status(thm)],[f2393])).
% 3.13/1.80 thf(f2393,plain,(
% 3.13/1.80 ($true != $true) | spl35_13),
% 3.13/1.80 inference(superposition,[status(thm)],[f2000,f1552])).
% 3.13/1.80 thf(f1552,plain,(
% 3.13/1.80 ($true = (member @ a @ sK20 @ (elemen321786957_cball @ a @ x2 @ d)))),
% 3.13/1.80 inference(cnf_transformation,[status(thm)],[f1318])).
% 3.13/1.80 thf(f1318,plain,(
% 3.13/1.80 ! [X1 : a] : (($true = (ord_less_eq @ real @ (inner_780170721_inner @ a @ (f @ sK20) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))) @ (inner_780170721_inner @ a @ (f @ X1) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))) | ($true != (member @ a @ X1 @ (elemen321786957_cball @ a @ x2 @ d)))) & ($true = (member @ a @ sK20 @ (elemen321786957_cball @ a @ x2 @ d)))),
% 3.13/1.80 inference(skolemisation,[status(esa),new_symbols(skolem,[sK20])],[f984,f1317])).
% 3.13/1.80 thf(f1317,plain,(
% 3.13/1.80 ? [X0 : a] : (! [X1 : a] : (($true = (ord_less_eq @ real @ (inner_780170721_inner @ a @ (f @ X0) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))) @ (inner_780170721_inner @ a @ (f @ X1) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))) | ($true != (member @ a @ X1 @ (elemen321786957_cball @ a @ x2 @ d)))) & ($true = (member @ a @ X0 @ (elemen321786957_cball @ a @ x2 @ d)))) => (! [X1 : a] : (($true = (ord_less_eq @ real @ (inner_780170721_inner @ a @ (f @ sK20) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))) @ (inner_780170721_inner @ a @ (f @ X1) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))) | ($true != (member @ a @ X1 @ (elemen321786957_cball @ a @ x2 @ d)))) & ($true = (member @ a @ sK20 @ (elemen321786957_cball @ a @ x2 @ d))))),
% 3.13/1.80 introduced(definition,[],[choice_axiom])).
% 3.13/1.80 thf(f984,plain,(
% 3.13/1.80 ? [X0 : a] : (! [X1 : a] : (($true = (ord_less_eq @ real @ (inner_780170721_inner @ a @ (f @ X0) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))) @ (inner_780170721_inner @ a @ (f @ X1) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))) | ($true != (member @ a @ X1 @ (elemen321786957_cball @ a @ x2 @ d)))) & ($true = (member @ a @ X0 @ (elemen321786957_cball @ a @ x2 @ d))))),
% 3.13/1.80 inference(ennf_transformation,[status(thm)],[f868])).
% 3.13/1.80 thf(f868,plain,(
% 3.13/1.80 ~! [X0 : a] : (($true = (member @ a @ X0 @ (elemen321786957_cball @ a @ x2 @ d))) => ~! [X1 : a] : (($true = (member @ a @ X1 @ (elemen321786957_cball @ a @ x2 @ d))) => ($true = (ord_less_eq @ real @ (inner_780170721_inner @ a @ (f @ X0) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))) @ (inner_780170721_inner @ a @ (f @ X1) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b)))))))),
% 3.13/1.80 inference(fool_elimination,[status(thm)],[f867])).
% 3.13/1.80 thf(f867,plain,(
% 3.13/1.80 ~! [X0 : a] : ((member @ a @ X0 @ (elemen321786957_cball @ a @ x2 @ d)) => ~! [X1 : a] : ((member @ a @ X1 @ (elemen321786957_cball @ a @ x2 @ d)) => (ord_less_eq @ real @ (inner_780170721_inner @ a @ (f @ X0) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))) @ (inner_780170721_inner @ a @ (f @ X1) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))))),
% 3.13/1.80 inference(rectify,[status(thm)],[f12])).
% 3.13/1.80 thf(f12,axiom,(
% 3.13/1.80 ~! [X5 : a] : ((member @ a @ X5 @ (elemen321786957_cball @ a @ x2 @ d)) => ~! [X3 : a] : ((member @ a @ X3 @ (elemen321786957_cball @ a @ x2 @ d)) => (ord_less_eq @ real @ (inner_780170721_inner @ a @ (f @ X5) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))) @ (inner_780170721_inner @ a @ (f @ X3) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))))),
% 3.13/1.80 file('/export/starexec/sandbox/tmp/tmp.laqx631JN2/DTF2THF_21273.p',fact_11__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062s_O_A_092_060lbrakk_062s_A_092_060in_062_Acball_Ax_Ad_059_A_092_060forall_062x_092_060in_062cball_Ax_Ad_O_Af_As_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060le_062_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062)).
% 3.13/1.80 thf(f2000,plain,(
% 3.13/1.80 ($true != (member @ a @ sK20 @ (elemen321786957_cball @ a @ x2 @ d))) | spl35_13),
% 3.13/1.80 inference(trivial_inequality_removal,[status(thm)],[f1985])).
% 3.13/1.80 thf(f1985,plain,(
% 3.13/1.80 ($true != (member @ a @ sK20 @ (elemen321786957_cball @ a @ x2 @ d))) | ($true != $true) | spl35_13),
% 3.13/1.80 inference(superposition,[status(thm)],[f1929,f1583])).
% 3.13/1.80 thf(f1583,plain,(
% 3.13/1.80 ( ! [X0 : a] : (($true = (ord_less @ real @ zero_zero @ real @ (inner_780170721_inner @ a @ (f @ X0) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))) | ($true != (member @ a @ X0 @ (elemen321786957_cball @ a @ x2 @ d)))) )),
% 3.13/1.80 inference(cnf_transformation,[status(thm)],[f1202])).
% 3.13/1.80 thf(f1202,plain,(
% 3.13/1.80 ! [X0 : a] : (($true != (member @ a @ X0 @ (elemen321786957_cball @ a @ x2 @ d))) | (($true = (member @ a @ X0 @ x)) & (zero_zero @ a != (f @ X0)) & ($true = (ord_less @ real @ zero_zero @ real @ (inner_780170721_inner @ a @ (f @ X0) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b)))))))),
% 3.13/1.80 inference(ennf_transformation,[status(thm)],[f650])).
% 3.13/1.80 thf(f650,plain,(
% 3.13/1.80 ! [X0 : a] : (($true = (member @ a @ X0 @ (elemen321786957_cball @ a @ x2 @ d))) => (($true = (member @ a @ X0 @ x)) & (zero_zero @ a != (f @ X0)) & ($true = (ord_less @ real @ zero_zero @ real @ (inner_780170721_inner @ a @ (f @ X0) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b)))))))),
% 3.13/1.80 inference(fool_elimination,[status(thm)],[f649])).
% 3.13/1.80 thf(f649,plain,(
% 3.13/1.80 ! [X0 : a] : ((member @ a @ X0 @ (elemen321786957_cball @ a @ x2 @ d)) => ((member @ a @ X0 @ x) & (ord_less @ real @ zero_zero @ real @ (inner_780170721_inner @ a @ (f @ X0) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b)))) & (zero_zero @ a != (f @ X0))))),
% 3.13/1.80 inference(rectify,[status(thm)],[f9])).
% 3.13/1.80 thf(f9,axiom,(
% 3.13/1.80 ! [X4 : a] : ((member @ a @ X4 @ (elemen321786957_cball @ a @ x2 @ d)) => ((member @ a @ X4 @ x) & (ord_less @ real @ zero_zero @ real @ (inner_780170721_inner @ a @ (f @ X4) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b)))) & (zero_zero @ a != (f @ X4))))),
% 3.13/1.80 file('/export/starexec/sandbox/tmp/tmp.laqx631JN2/DTF2THF_21273.p',fact_8_d_I2_J)).
% 3.13/1.80 thf(f1929,plain,(
% 3.13/1.80 ($true != (ord_less @ real @ zero_zero @ real @ (inner_780170721_inner @ a @ (f @ sK20) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))) | spl35_13),
% 3.13/1.80 inference(avatar_component_clause,[status(thm)],[f1927])).
% 3.13/1.80 thf(f1930,plain,(
% 3.13/1.80 spl35_12 | ~spl35_13),
% 3.13/1.80 inference(avatar_split_clause,[status(thm)],[f1884,f1927,f1924])).
% 3.13/1.80 thf(f1884,plain,(
% 3.13/1.80 ( ! [X0 : real] : (($true != (ord_less @ real @ zero_zero @ real @ X0)) | ($true != (ord_less @ real @ zero_zero @ real @ (inner_780170721_inner @ a @ (f @ sK20) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))) | ((member @ a @ (sK11 @ (inner_780170721_inner @ a @ (f @ sK20) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))) @ X0) @ (elemen321786957_cball @ a @ x2 @ d)) != $true)) )),
% 3.13/1.80 inference(trivial_inequality_removal,[status(thm)],[f1872])).
% 3.13/1.80 thf(f1872,plain,(
% 3.13/1.80 ( ! [X0 : real] : (($true != $true) | ((member @ a @ (sK11 @ (inner_780170721_inner @ a @ (f @ sK20) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))) @ X0) @ (elemen321786957_cball @ a @ x2 @ d)) != $true) | ($true != (ord_less @ real @ zero_zero @ real @ X0)) | ($true != (ord_less @ real @ zero_zero @ real @ (inner_780170721_inner @ a @ (f @ sK20) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b)))))) )),
% 3.13/1.80 inference(superposition,[status(thm)],[f1457,f1553])).
% 3.13/1.80 thf(f1553,plain,(
% 3.13/1.80 ( ! [X1 : a] : (($true = (ord_less_eq @ real @ (inner_780170721_inner @ a @ (f @ sK20) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))) @ (inner_780170721_inner @ a @ (f @ X1) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))) | ($true != (member @ a @ X1 @ (elemen321786957_cball @ a @ x2 @ d)))) )),
% 3.13/1.80 inference(cnf_transformation,[status(thm)],[f1318])).
% 3.13/1.80 thf(f1457,plain,(
% 3.13/1.80 ( ! [X0 : real,X1 : real] : (($true != (ord_less_eq @ real @ X1 @ (inner_780170721_inner @ a @ (f @ (sK11 @ X1 @ X0)) @ (poinca1750768982en_rot @ a @ (minus_minus @ a @ a2 @ b))))) | ($true != (ord_less @ real @ zero_zero @ real @ X1)) | ($true != (ord_less @ real @ zero_zero @ real @ X0))) )),
% 3.13/1.80 inference(cnf_transformation,[status(thm)],[f1288])).
% 3.13/1.80 % SZS output end Proof for DTF2THF_21273
% 3.13/1.80 % (21517)------------------------------
% 3.13/1.80 % (21517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.80 % (21517)Termination reason: Refutation
% 3.13/1.80
% 3.13/1.80 % (21517)Memory used [KB]: 7291
% 3.13/1.80 % (21517)Time elapsed: 0.073 s
% 3.13/1.80 % (21517)Instructions burned: 228 (million)
% 3.13/1.80 % (21517)------------------------------
% 3.13/1.80 % (21517)------------------------------
% 3.13/1.80 % (21479)Success in time 0.153 s
%------------------------------------------------------------------------------