%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP112^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n011.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 08:55:39 AM UTC 2026
% Result : Theorem 2.96s 2.00s
% Output : Refutation 2.96s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP112^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.14/0.33 % Computer : n011.cluster.edu
% 0.14/0.33 % Model : x86_64 x86_64
% 0.14/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.33 % Memory : 8042.1875MB
% 0.14/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.33 % CPULimit : 300
% 0.14/0.33 % WCLimit : 300
% 0.14/0.33 % DateTime : Tue Feb 24 08:26:17 EST 2026
% 0.14/0.33 % CPUTime :
% 0.14/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.31/0.46 ---- Original DTF file ---
% 0.31/0.46 thf(spec,logic,$$dhol).
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 % File : ITP112^2 : TPTP v9.2.1. Released v7.5.0.
% 0.31/0.46 % Domain : Interactive Theorem Proving
% 0.31/0.46 % Problem : Sledgehammer Lower_Semicontinuous problem prob_385__6250846_1
% 0.31/0.46 % Version : Especial.
% 0.31/0.46 % English :
% 0.31/0.46
% 0.31/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.31/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.31/0.46 % Source : [Des21]
% 0.31/0.46 % Names : Lower_Semicontinuous/prob_385__6250846_1 [Des21]
% 0.31/0.46
% 0.31/0.46 % Status : Theorem
% 0.31/0.46 % Rating : 0.67 v8.1.0, 0.50 v7.5.0
% 0.31/0.46 % Syntax : Number of formulae : 446 ( 181 unt; 64 typ; 0 def)
% 0.31/0.46 % Number of atoms : 1062 ( 375 equ; 0 cnn)
% 0.31/0.46 % Maximal formula atoms : 81 ( 2 avg)
% 0.31/0.46 % Number of connectives : 4309 ( 97 ~; 20 |; 46 &;3617 @)
% 0.31/0.46 % ( 0 <=>; 529 =>; 0 <=; 0 <~>)
% 0.31/0.46 % Maximal formula depth : 34 ( 6 avg)
% 0.31/0.46 % Number of types : 6 ( 5 usr)
% 0.31/0.46 % Number of type conns : 219 ( 219 >; 0 *; 0 +; 0 <<)
% 0.31/0.46 % Number of symbols : 60 ( 59 usr; 2 con; 0-6 aty)
% 0.31/0.46 % Number of variables : 1126 ( 100 ^; 929 !; 39 ?;1126 :)
% 0.31/0.46 % ( 58 !>; 0 ?*; 0 @-; 0 @+)
% 0.31/0.46 % SPC : TH1_THM_EQU_NAR
% 0.31/0.46
% 0.31/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:26:15.949
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 % Could-be-implicit typings (7)
% 0.31/0.46 thf(ty_t_Extended__Real_Oereal,type,
% 0.31/0.46 extended_ereal: $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_Filter_Ofilter,type,
% 0.31/0.46 filter: $tType > $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_Real_Oreal,type,
% 0.31/0.46 real: $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_Set_Oset,type,
% 0.31/0.46 set: $tType > $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_Nat_Onat,type,
% 0.31/0.46 nat: $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_Int_Oint,type,
% 0.31/0.46 int: $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_tf_a,type,
% 0.31/0.46 a: $tType ).
% 0.31/0.46
% 0.31/0.46 % Explicit typings (57)
% 0.31/0.46 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.31/0.46 finite_finite:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_HOL_Otype,type,
% 0.31/0.46 type:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Countable_Ocountable,type,
% 0.31/0.46 countable:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Ouminus,type,
% 0.31/0.46 uminus:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.31/0.46 ord:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.31/0.46 order:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.31/0.46 group_add:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.31/0.46 linorder:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.31/0.46 preorder:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Oab__group__add,type,
% 0.31/0.46 ab_group_add:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.31/0.46 semigroup_add:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.31/0.46 comm_monoid_add:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.31/0.46 ab_semigroup_add:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 0.31/0.46 boolean_algebra:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.31/0.46 cancel_semigroup_add:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.31/0.46 ordered_ab_group_add:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Topological__Spaces_Ot2__space,type,
% 0.31/0.46 topological_t2_space:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Limits_Otopological__group__add,type,
% 0.31/0.46 topolo799126099up_add:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Limits_Otopological__monoid__add,type,
% 0.31/0.46 topolo1314133330id_add:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.31/0.46 ordere779506340up_add:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
% 0.31/0.46 real_V2090557954_space:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Topological__Spaces_Ouniform__space,type,
% 0.31/0.46 topolo47006728_space:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.31/0.46 linord1659791738miring:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Topological__Spaces_Oorder__topology,type,
% 0.31/0.46 topolo259154727pology:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 0.31/0.46 comple187826305attice:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
% 0.31/0.46 comple1035589618norder:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Topological__Spaces_Olinorder__topology,type,
% 0.31/0.46 topolo2117631714pology:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Topological__Spaces_Otopological__space,type,
% 0.31/0.46 topolo503727757_space:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.31/0.46 canoni770627133id_add:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
% 0.31/0.46 real_V55928688vector:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.31/0.46 ordere236663937imp_le:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Topological__Spaces_Ofirst__countable__topology,type,
% 0.31/0.46 topolo2135403230pology:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
% 0.31/0.46 extend1396239628finity:
% 0.31/0.46 !>[A: $tType] : A ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Extended__Real_Oereal_Ocase__ereal,type,
% 0.31/0.46 extended_case_ereal:
% 0.31/0.46 !>[A: $tType] : ( ( real > A ) > A > A > extended_ereal > A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Extended__Real_Oereal_Oereal,type,
% 0.31/0.46 extended_ereal2: real > extended_ereal ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Filter_Oat__bot,type,
% 0.31/0.46 at_bot:
% 0.31/0.46 !>[A: $tType] : ( filter @ A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Filter_Oat__top,type,
% 0.31/0.46 at_top:
% 0.31/0.46 !>[A: $tType] : ( filter @ A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Filter_Ofilterlim,type,
% 0.31/0.46 filterlim:
% 0.31/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Fun_Ocomp,type,
% 0.31/0.46 comp:
% 0.31/0.46 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.31/0.46 plus_plus:
% 0.31/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.31/0.46 uminus_uminus:
% 0.31/0.46 !>[A: $tType] : ( A > A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_HOL_OEx1,type,
% 0.31/0.46 ex1:
% 0.31/0.46 !>[A: $tType] : ( ( A > $o ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_HOL_OUniq,type,
% 0.31/0.46 uniq:
% 0.31/0.46 !>[A: $tType] : ( ( A > $o ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Liminf__Limsup_OLiminf,type,
% 0.31/0.46 liminf_Liminf:
% 0.31/0.46 !>[A: $tType,B: $tType] : ( ( filter @ A ) > ( A > B ) > B ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc__at,type,
% 0.31/0.46 lower_582600101lsc_at:
% 0.31/0.46 !>[A: $tType,B: $tType] : ( A > ( A > B ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Ousc__at,type,
% 0.31/0.46 lower_708585518usc_at:
% 0.31/0.46 !>[A: $tType,B: $tType] : ( A > ( A > B ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 0.31/0.46 semiring_1_of_nat:
% 0.31/0.46 !>[A: $tType] : ( nat > A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.31/0.46 ord_less_eq:
% 0.31/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Set_OCollect,type,
% 0.31/0.46 collect:
% 0.31/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Topological__Spaces_Omonoseq,type,
% 0.31/0.46 topological_monoseq:
% 0.31/0.46 !>[A: $tType] : ( ( nat > A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
% 0.31/0.46 topolo1920029354e_nhds:
% 0.31/0.46 !>[A: $tType] : ( A > ( filter @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocauchy__filter,type,
% 0.31/0.46 topolo1334162427filter:
% 0.31/0.46 !>[A: $tType] : ( ( filter @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_member,type,
% 0.31/0.46 member:
% 0.31/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_v_A____,type,
% 0.31/0.46 a2: extended_ereal ).
% 0.31/0.46
% 0.31/0.46 thf(sy_v_f,type,
% 0.31/0.46 f: a > extended_ereal ).
% 0.31/0.46
% 0.31/0.46 thf(sy_v_x0,type,
% 0.31/0.46 x0: a ).
% 0.31/0.46
% 0.31/0.46 thf(sy_v_x____,type,
% 0.31/0.46 x: nat > a ).
% 0.31/0.46
% 0.31/0.46 % Relevant facts (256)
% 0.31/0.46 thf(fact_0_x__def_I1_J,axiom,
% 0.31/0.46 filterlim @ nat @ a @ x @ ( topolo1920029354e_nhds @ a @ x0 ) @ ( at_top @ nat ) ).
% 0.31/0.46
% 0.31/0.46 % x_def(1)
% 0.31/0.46 thf(fact_1_x__def_I2_J,axiom,
% 0.31/0.46 filterlim @ nat @ extended_ereal @ ( comp @ a @ extended_ereal @ nat @ f @ x ) @ ( topolo1920029354e_nhds @ extended_ereal @ a2 ) @ ( at_top @ nat ) ).
% 0.31/0.46
% 0.31/0.46 % x_def(2)
% 0.31/0.46 thf(fact_2__092_060open_062_092_060And_062F_O_A_I_If_A_092_060circ_062_Ax_J_A_092_060longlongrightarrow_062_AA_J_AF_A_092_060Longrightarrow_062_A_I_I_092_060lambda_062xa_O_A_N_A_If_A_092_060circ_062_Ax_J_Axa_J_A_092_060longlongrightarrow_062_A_N_AA_J_AF_092_060close_062,axiom,
% 0.31/0.46 ! [F: filter @ nat] :
% 0.31/0.46 ( ( filterlim @ nat @ extended_ereal @ ( comp @ a @ extended_ereal @ nat @ f @ x ) @ ( topolo1920029354e_nhds @ extended_ereal @ a2 ) @ F )
% 0.31/0.46 => ( filterlim @ nat @ extended_ereal
% 0.31/0.46 @ ^ [X: nat] : ( uminus_uminus @ extended_ereal @ ( comp @ a @ extended_ereal @ nat @ f @ x @ X ) )
% 0.31/0.46 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ a2 ) )
% 0.31/0.46 @ F ) ) ).
% 0.31/0.46
% 0.31/0.46 % \<open>\<And>F. ((f \<circ> x) \<longlongrightarrow> A) F \<Longrightarrow> ((\<lambda>xa. - (f \<circ> x) xa) \<longlongrightarrow> - A) F\<close>
% 0.31/0.46 thf(fact_3_lsc,axiom,
% 0.31/0.46 ( lower_582600101lsc_at @ a @ extended_ereal @ x0
% 0.31/0.46 @ ^ [X: a] : ( uminus_uminus @ extended_ereal @ ( f @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lsc
% 0.31/0.46 thf(fact_4_tendsto__uminus__ereal,axiom,
% 0.31/0.46 ! [A: $tType,F2: A > extended_ereal,X2: extended_ereal,F: filter @ A] :
% 0.31/0.46 ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ X2 ) @ F )
% 0.31/0.46 => ( filterlim @ A @ extended_ereal
% 0.31/0.46 @ ^ [X: A] : ( uminus_uminus @ extended_ereal @ ( F2 @ X ) )
% 0.31/0.46 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ X2 ) )
% 0.31/0.46 @ F ) ) ).
% 0.31/0.46
% 0.31/0.46 % tendsto_uminus_ereal
% 0.31/0.46 thf(fact_5_tendsto__const,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( topolo503727757_space @ A )
% 0.31/0.46 => ! [K: A,F: filter @ B] :
% 0.31/0.46 ( filterlim @ B @ A
% 0.31/0.46 @ ^ [X: B] : K
% 0.31/0.46 @ ( topolo1920029354e_nhds @ A @ K )
% 0.31/0.46 @ F ) ) ).
% 0.31/0.46
% 0.31/0.46 % tendsto_const
% 0.31/0.46 thf(fact_6_ereal__Lim__uminus,axiom,
% 0.31/0.46 ! [A: $tType,F2: A > extended_ereal,F0: extended_ereal,Net: filter @ A] :
% 0.31/0.46 ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ F0 ) @ Net )
% 0.31/0.46 = ( filterlim @ A @ extended_ereal
% 0.31/0.46 @ ^ [X: A] : ( uminus_uminus @ extended_ereal @ ( F2 @ X ) )
% 0.31/0.46 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ F0 ) )
% 0.31/0.46 @ Net ) ) ).
% 0.31/0.46
% 0.31/0.46 % ereal_Lim_uminus
% 0.31/0.46 thf(fact_7_LIMSEQ__const__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( topological_t2_space @ A )
% 0.31/0.46 => ! [K: A,L: A] :
% 0.31/0.46 ( ( filterlim @ nat @ A
% 0.31/0.46 @ ^ [N: nat] : K
% 0.31/0.46 @ ( topolo1920029354e_nhds @ A @ L )
% 0.31/0.46 @ ( at_top @ nat ) )
% 0.31/0.46 = ( K = L ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % LIMSEQ_const_iff
% 0.31/0.46 thf(fact_8_tendsto__minus__cancel,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( topolo799126099up_add @ A )
% 0.31/0.46 => ! [F2: B > A,A2: A,F: filter @ B] :
% 0.31/0.46 ( ( filterlim @ B @ A
% 0.31/0.46 @ ^ [X: B] : ( uminus_uminus @ A @ ( F2 @ X ) )
% 0.31/0.46 @ ( topolo1920029354e_nhds @ A @ ( uminus_uminus @ A @ A2 ) )
% 0.31/0.46 @ F )
% 0.31/0.46 => ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ A2 ) @ F ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % tendsto_minus_cancel
% 0.31/0.46 thf(fact_9_tendsto__minus__cancel__left,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( topolo799126099up_add @ B )
% 0.31/0.46 => ! [F2: A > B,Y: B,F: filter @ A] :
% 0.31/0.46 ( ( filterlim @ A @ B @ F2 @ ( topolo1920029354e_nhds @ B @ ( uminus_uminus @ B @ Y ) ) @ F )
% 0.31/0.46 = ( filterlim @ A @ B
% 0.31/0.46 @ ^ [X: A] : ( uminus_uminus @ B @ ( F2 @ X ) )
% 0.31/0.46 @ ( topolo1920029354e_nhds @ B @ Y )
% 0.31/0.46 @ F ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % tendsto_minus_cancel_left
% 0.31/0.46 thf(fact_10_tendsto__minus,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( topolo799126099up_add @ A )
% 0.31/0.46 => ! [F2: B > A,A2: A,F: filter @ B] :
% 0.31/0.46 ( ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ A2 ) @ F )
% 0.31/0.46 => ( filterlim @ B @ A
% 0.31/0.46 @ ^ [X: B] : ( uminus_uminus @ A @ ( F2 @ X ) )
% 0.31/0.46 @ ( topolo1920029354e_nhds @ A @ ( uminus_uminus @ A @ A2 ) )
% 0.31/0.46 @ F ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % tendsto_minus
% 0.31/0.46 thf(fact_11_LIMSEQ__unique,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( topological_t2_space @ A )
% 0.31/0.46 => ! [X3: nat > A,A2: A,B2: A] :
% 0.31/0.46 ( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ B2 ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( A2 = B2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % LIMSEQ_unique
% 0.31/0.46 thf(fact_12_tendsto__uminus__nhds,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( topolo799126099up_add @ A )
% 0.31/0.46 => ! [A2: A] : ( filterlim @ A @ A @ ( uminus_uminus @ A ) @ ( topolo1920029354e_nhds @ A @ ( uminus_uminus @ A @ A2 ) ) @ ( topolo1920029354e_nhds @ A @ A2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % tendsto_uminus_nhds
% 0.31/0.46 thf(fact_13_ereal__uminus__eq__iff,axiom,
% 0.31/0.46 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.31/0.46 ( ( ( uminus_uminus @ extended_ereal @ A2 )
% 0.31/0.46 = ( uminus_uminus @ extended_ereal @ B2 ) )
% 0.31/0.46 = ( A2 = B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % ereal_uminus_eq_iff
% 0.31/0.46 thf(fact_14_ereal__uminus__uminus,axiom,
% 0.31/0.46 ! [A2: extended_ereal] :
% 0.31/0.46 ( ( uminus_uminus @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) )
% 0.31/0.46 = A2 ) ).
% 0.31/0.46
% 0.31/0.46 % ereal_uminus_uminus
% 0.31/0.46 thf(fact_15_verit__minus__simplify_I4_J,axiom,
% 0.31/0.46 ! [B: $tType] :
% 0.31/0.46 ( ( group_add @ B )
% 0.31/0.46 => ! [B2: B] :
% 0.31/0.46 ( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
% 0.31/0.46 = B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % verit_minus_simplify(4)
% 0.31/0.46 thf(fact_16_verit__negate__coefficient_I3_J,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ordered_ab_group_add @ A )
% 0.31/0.46 => ! [A2: A,B2: A] :
% 0.31/0.46 ( ( A2 = B2 )
% 0.31/0.46 => ( ( uminus_uminus @ A @ A2 )
% 0.31/0.46 = ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % verit_negate_coefficient(3)
% 0.31/0.46 thf(fact_17_ereal__uminus__eq__reorder,axiom,
% 0.31/0.46 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.31/0.46 ( ( ( uminus_uminus @ extended_ereal @ A2 )
% 0.31/0.46 = B2 )
% 0.31/0.46 = ( A2
% 0.31/0.46 = ( uminus_uminus @ extended_ereal @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ereal_uminus_eq_reorder
% 0.31/0.46 thf(fact_18_tendsto__eq__rhs,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( topolo503727757_space @ A )
% 0.31/0.46 => ! [F2: B > A,X2: A,F: filter @ B,Y: A] :
% 0.31/0.46 ( ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ X2 ) @ F )
% 0.31/0.46 => ( ( X2 = Y )
% 0.31/0.46 => ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ Y ) @ F ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % tendsto_eq_rhs
% 0.31/0.46 thf(fact_19_tendsto__cong__limit,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( topolo503727757_space @ B )
% 0.31/0.46 => ! [F2: A > B,L: B,F: filter @ A,K: B] :
% 0.31/0.46 ( ( filterlim @ A @ B @ F2 @ ( topolo1920029354e_nhds @ B @ L ) @ F )
% 0.31/0.46 => ( ( K = L )
% 0.31/0.46 => ( filterlim @ A @ B @ F2 @ ( topolo1920029354e_nhds @ B @ K ) @ F ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % tendsto_cong_limit
% 0.31/0.46 thf(fact_20_comp__apply,axiom,
% 0.31/0.46 ! [C: $tType,A: $tType,B: $tType] :
% 0.31/0.46 ( ( comp @ B @ A @ C )
% 0.31/0.46 = ( ^ [F3: B > A,G: C > B,X: C] : ( F3 @ ( G @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % comp_apply
% 0.31/0.46 thf(fact_21_compl__eq__compl__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( boolean_algebra @ A )
% 0.31/0.46 => ! [X2: A,Y: A] :
% 0.31/0.46 ( ( ( uminus_uminus @ A @ X2 )
% 0.31/0.46 = ( uminus_uminus @ A @ Y ) )
% 0.31/0.46 = ( X2 = Y ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % compl_eq_compl_iff
% 0.31/0.46 thf(fact_22_double__compl,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( boolean_algebra @ A )
% 0.31/0.46 => ! [X2: A] :
% 0.31/0.46 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X2 ) )
% 0.31/0.46 = X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % double_compl
% 0.31/0.46 thf(fact_23_neg__equal__iff__equal,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( group_add @ A )
% 0.31/0.46 => ! [A2: A,B2: A] :
% 0.31/0.46 ( ( ( uminus_uminus @ A @ A2 )
% 0.31/0.46 = ( uminus_uminus @ A @ B2 ) )
% 0.31/0.46 = ( A2 = B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % neg_equal_iff_equal
% 0.31/0.46 thf(fact_24_add_Oinverse__inverse,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( group_add @ A )
% 0.31/0.46 => ! [A2: A] :
% 0.31/0.46 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 0.31/0.46 = A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % add.inverse_inverse
% 0.31/0.46 thf(fact_25_uminus__apply,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( uminus @ B )
% 0.31/0.46 => ( ( uminus_uminus @ ( A > B ) )
% 0.31/0.46 = ( ^ [A3: A > B,X: A] : ( uminus_uminus @ B @ ( A3 @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % uminus_apply
% 0.31/0.46 thf(fact_26_lsc__sequentially__mem,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( real_V2090557954_space @ A )
% 0.31/0.46 => ! [X0: A,F2: A > extended_ereal,X2: nat > A,C2: nat > extended_ereal,C0: extended_ereal] :
% 0.31/0.46 ( ( lower_582600101lsc_at @ A @ extended_ereal @ X0 @ F2 )
% 0.31/0.46 => ( ( filterlim @ nat @ A @ X2 @ ( topolo1920029354e_nhds @ A @ X0 ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ( filterlim @ nat @ extended_ereal @ C2 @ ( topolo1920029354e_nhds @ extended_ereal @ C0 ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ! [N2: nat] : ( ord_less_eq @ extended_ereal @ ( F2 @ ( X2 @ N2 ) ) @ ( C2 @ N2 ) )
% 0.31/0.46 => ( ord_less_eq @ extended_ereal @ ( F2 @ X0 ) @ C0 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lsc_sequentially_mem
% 0.31/0.46 thf(fact_27_lsc__sequentially__gen,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( real_V2090557954_space @ A )
% 0.31/0.46 => ( ( lower_582600101lsc_at @ A @ extended_ereal )
% 0.31/0.46 = ( ^ [X02: A,F3: A > extended_ereal] :
% 0.31/0.46 ! [X: nat > A,C3: nat > extended_ereal,C02: extended_ereal] :
% 0.31/0.46 ( ( ( filterlim @ nat @ A @ X @ ( topolo1920029354e_nhds @ A @ X02 ) @ ( at_top @ nat ) )
% 0.31/0.46 & ( filterlim @ nat @ extended_ereal @ C3 @ ( topolo1920029354e_nhds @ extended_ereal @ C02 ) @ ( at_top @ nat ) )
% 0.31/0.46 & ! [N: nat] : ( ord_less_eq @ extended_ereal @ ( F3 @ ( X @ N ) ) @ ( C3 @ N ) ) )
% 0.31/0.46 => ( ord_less_eq @ extended_ereal @ ( F3 @ X02 ) @ C02 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lsc_sequentially_gen
% 0.31/0.46 thf(fact_28_lsc__at__mem,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( topolo259154727pology @ B )
% 0.31/0.46 & ( topolo503727757_space @ A ) )
% 0.31/0.46 => ! [X0: A,F2: A > B,X2: nat > A,A4: B] :
% 0.31/0.46 ( ( lower_582600101lsc_at @ A @ B @ X0 @ F2 )
% 0.31/0.46 => ( ( filterlim @ nat @ A @ X2 @ ( topolo1920029354e_nhds @ A @ X0 ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ( filterlim @ nat @ B @ ( comp @ A @ B @ nat @ F2 @ X2 ) @ ( topolo1920029354e_nhds @ B @ A4 ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ord_less_eq @ B @ ( F2 @ X0 ) @ A4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lsc_at_mem
% 0.31/0.46 thf(fact_29_lsc__at__def,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ( topolo503727757_space @ A )
% 0.31/0.46 & ( topolo259154727pology @ B ) )
% 0.31/0.46 => ( ( lower_582600101lsc_at @ A @ B )
% 0.31/0.46 = ( ^ [X02: A,F3: A > B] :
% 0.31/0.46 ! [X4: nat > A,L2: B] :
% 0.31/0.46 ( ( ( filterlim @ nat @ A @ X4 @ ( topolo1920029354e_nhds @ A @ X02 ) @ ( at_top @ nat ) )
% 0.31/0.46 & ( filterlim @ nat @ B @ ( comp @ A @ B @ nat @ F3 @ X4 ) @ ( topolo1920029354e_nhds @ B @ L2 ) @ ( at_top @ nat ) ) )
% 0.31/0.46 => ( ord_less_eq @ B @ ( F3 @ X02 ) @ L2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lsc_at_def
% 0.31/0.46 thf(fact_30_neg__le__iff__le,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ordered_ab_group_add @ A )
% 0.31/0.46 => ! [B2: A,A2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 0.31/0.46 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % neg_le_iff_le
% 0.31/0.46 thf(fact_31_compl__le__compl__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( boolean_algebra @ A )
% 0.31/0.46 => ! [X2: A,Y: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y ) )
% 0.31/0.46 = ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % compl_le_compl_iff
% 0.31/0.46 thf(fact_32_ereal__minus__le__minus,axiom,
% 0.31/0.46 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.31/0.46 ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) @ ( uminus_uminus @ extended_ereal @ B2 ) )
% 0.31/0.46 = ( ord_less_eq @ extended_ereal @ B2 @ A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % ereal_minus_le_minus
% 0.31/0.46 thf(fact_33_verit__la__disequality,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A )
% 0.31/0.46 => ! [A2: A,B2: A] :
% 0.31/0.46 ( ( A2 = B2 )
% 0.31/0.46 | ~ ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.46 | ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % verit_la_disequality
% 0.31/0.46 thf(fact_34_ereal__complete__Inf,axiom,
% 0.31/0.46 ! [S: set @ extended_ereal] :
% 0.31/0.46 ? [X5: extended_ereal] :
% 0.31/0.46 ( ! [Xa: extended_ereal] :
% 0.31/0.46 ( ( member @ extended_ereal @ Xa @ S )
% 0.31/0.46 => ( ord_less_eq @ extended_ereal @ X5 @ Xa ) )
% 0.31/0.46 & ! [Z: extended_ereal] :
% 0.31/0.46 ( ! [Xa2: extended_ereal] :
% 0.31/0.46 ( ( member @ extended_ereal @ Xa2 @ S )
% 0.31/0.46 => ( ord_less_eq @ extended_ereal @ Z @ Xa2 ) )
% 0.31/0.46 => ( ord_less_eq @ extended_ereal @ Z @ X5 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ereal_complete_Inf
% 0.31/0.46 thf(fact_35_ereal__complete__Sup,axiom,
% 0.31/0.46 ! [S: set @ extended_ereal] :
% 0.31/0.46 ? [X5: extended_ereal] :
% 0.31/0.46 ( ! [Xa: extended_ereal] :
% 0.31/0.46 ( ( member @ extended_ereal @ Xa @ S )
% 0.31/0.46 => ( ord_less_eq @ extended_ereal @ Xa @ X5 ) )
% 0.31/0.46 & ! [Z: extended_ereal] :
% 0.31/0.46 ( ! [Xa2: extended_ereal] :
% 0.31/0.46 ( ( member @ extended_ereal @ Xa2 @ S )
% 0.31/0.46 => ( ord_less_eq @ extended_ereal @ Xa2 @ Z ) )
% 0.31/0.46 => ( ord_less_eq @ extended_ereal @ X5 @ Z ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ereal_complete_Sup
% 0.31/0.46 thf(fact_36_compl__mono,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( boolean_algebra @ A )
% 0.31/0.46 => ! [X2: A,Y: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.31/0.46 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % compl_mono
% 0.31/0.46 thf(fact_37_compl__le__swap1,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( boolean_algebra @ A )
% 0.31/0.46 => ! [Y: A,X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X2 ) )
% 0.31/0.46 => ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % compl_le_swap1
% 0.31/0.46 thf(fact_38_compl__le__swap2,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( boolean_algebra @ A )
% 0.31/0.46 => ! [Y: A,X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X2 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ Y ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % compl_le_swap2
% 0.31/0.46 thf(fact_39_le__minus__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ordered_ab_group_add @ A )
% 0.31/0.46 => ! [A2: A,B2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 0.31/0.46 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_minus_iff
% 0.31/0.46 thf(fact_40_minus__le__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ordered_ab_group_add @ A )
% 0.31/0.46 => ! [A2: A,B2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 0.31/0.46 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % minus_le_iff
% 0.31/0.46 thf(fact_41_le__imp__neg__le,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ordered_ab_group_add @ A )
% 0.31/0.46 => ! [A2: A,B2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_imp_neg_le
% 0.31/0.46 thf(fact_42_ereal__uminus__le__reorder,axiom,
% 0.31/0.46 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.31/0.46 ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) @ B2 )
% 0.31/0.46 = ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ B2 ) @ A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % ereal_uminus_le_reorder
% 0.31/0.46 thf(fact_43_lim__mono,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( topolo2117631714pology @ A )
% 0.31/0.46 => ! [N3: nat,X3: nat > A,Y2: nat > A,X2: A,Y: A] :
% 0.31/0.46 ( ! [N2: nat] :
% 0.31/0.46 ( ( ord_less_eq @ nat @ N3 @ N2 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 0.31/0.46 => ( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ( filterlim @ nat @ A @ Y2 @ ( topolo1920029354e_nhds @ A @ Y ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lim_mono
% 0.31/0.46 thf(fact_44_LIMSEQ__le,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( topolo2117631714pology @ A )
% 0.31/0.46 => ! [X3: nat > A,X2: A,Y2: nat > A,Y: A] :
% 0.31/0.46 ( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ( filterlim @ nat @ A @ Y2 @ ( topolo1920029354e_nhds @ A @ Y ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ? [N4: nat] :
% 0.31/0.46 ! [N2: nat] :
% 0.31/0.46 ( ( ord_less_eq @ nat @ N4 @ N2 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 0.31/0.46 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % LIMSEQ_le
% 0.31/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.31/0.46 ! [A: $tType,A2: A,P: A > $o] :
% 0.31/0.46 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.31/0.46 = ( P @ A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % mem_Collect_eq
% 0.31/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.31/0.46 ! [A: $tType,A4: set @ A] :
% 0.31/0.46 ( ( collect @ A
% 0.31/0.46 @ ^ [X: A] : ( member @ A @ X @ A4 ) )
% 0.31/0.46 = A4 ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mem_eq
% 0.31/0.46 thf(fact_47_Collect__cong,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/0.46 ( ! [X5: A] :
% 0.31/0.46 ( ( P @ X5 )
% 0.31/0.46 = ( Q @ X5 ) )
% 0.31/0.46 => ( ( collect @ A @ P )
% 0.31/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_cong
% 0.31/0.46 thf(fact_48_ext,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
% 0.31/0.46 ( ! [X5: A] :
% 0.31/0.46 ( ( F2 @ X5 )
% 0.31/0.46 = ( G2 @ X5 ) )
% 0.31/0.46 => ( F2 = G2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % ext
% 0.31/0.46 thf(fact_49_Lim__bounded,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( topolo2117631714pology @ A )
% 0.31/0.46 => ! [F2: nat > A,L: A,M: nat,C4: A] :
% 0.31/0.46 ( ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ! [N2: nat] :
% 0.31/0.46 ( ( ord_less_eq @ nat @ M @ N2 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( F2 @ N2 ) @ C4 ) )
% 0.31/0.46 => ( ord_less_eq @ A @ L @ C4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Lim_bounded
% 0.31/0.46 thf(fact_50_Lim__bounded2,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( topolo2117631714pology @ A )
% 0.31/0.46 => ! [F2: nat > A,L: A,N3: nat,C4: A] :
% 0.31/0.46 ( ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ! [N2: nat] :
% 0.31/0.46 ( ( ord_less_eq @ nat @ N3 @ N2 )
% 0.31/0.46 => ( ord_less_eq @ A @ C4 @ ( F2 @ N2 ) ) )
% 0.31/0.46 => ( ord_less_eq @ A @ C4 @ L ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Lim_bounded2
% 0.31/0.46 thf(fact_51_LIMSEQ__le__const,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( topolo2117631714pology @ A )
% 0.31/0.46 => ! [X3: nat > A,X2: A,A2: A] :
% 0.31/0.46 ( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ? [N4: nat] :
% 0.31/0.46 ! [N2: nat] :
% 0.31/0.46 ( ( ord_less_eq @ nat @ N4 @ N2 )
% 0.31/0.46 => ( ord_less_eq @ A @ A2 @ ( X3 @ N2 ) ) )
% 0.31/0.46 => ( ord_less_eq @ A @ A2 @ X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % LIMSEQ_le_const
% 0.31/0.46 thf(fact_52_LIMSEQ__le__const2,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( topolo2117631714pology @ A )
% 0.31/0.46 => ! [X3: nat > A,X2: A,A2: A] :
% 0.31/0.46 ( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ? [N4: nat] :
% 0.31/0.46 ! [N2: nat] :
% 0.31/0.46 ( ( ord_less_eq @ nat @ N4 @ N2 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( X3 @ N2 ) @ A2 ) )
% 0.31/0.46 => ( ord_less_eq @ A @ X2 @ A2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % LIMSEQ_le_const2
% 0.31/0.46 thf(fact_53_tendsto__mono,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( topolo503727757_space @ A )
% 0.31/0.46 => ! [F: filter @ B,F4: filter @ B,F2: B > A,L: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( filter @ B ) @ F @ F4 )
% 0.31/0.46 => ( ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L ) @ F4 )
% 0.31/0.46 => ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L ) @ F ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % tendsto_mono
% 0.31/0.46 thf(fact_54_fun__Compl__def,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( uminus @ B )
% 0.31/0.46 => ( ( uminus_uminus @ ( A > B ) )
% 0.31/0.46 = ( ^ [A3: A > B,X: A] : ( uminus_uminus @ B @ ( A3 @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % fun_Compl_def
% 0.31/0.46 thf(fact_55_equation__minus__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( group_add @ A )
% 0.31/0.46 => ! [A2: A,B2: A] :
% 0.31/0.46 ( ( A2
% 0.31/0.46 = ( uminus_uminus @ A @ B2 ) )
% 0.31/0.46 = ( B2
% 0.31/0.46 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % equation_minus_iff
% 0.31/0.46 thf(fact_56_minus__equation__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( group_add @ A )
% 0.31/0.46 => ! [A2: A,B2: A] :
% 0.31/0.46 ( ( ( uminus_uminus @ A @ A2 )
% 0.31/0.46 = B2 )
% 0.31/0.46 = ( ( uminus_uminus @ A @ B2 )
% 0.31/0.46 = A2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % minus_equation_iff
% 0.31/0.46 thf(fact_57_comp__eq__dest__lhs,axiom,
% 0.31/0.46 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
% 0.31/0.46 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.31/0.46 = C2 )
% 0.31/0.46 => ( ( A2 @ ( B2 @ V ) )
% 0.31/0.46 = ( C2 @ V ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % comp_eq_dest_lhs
% 0.31/0.46 thf(fact_58_comp__eq__elim,axiom,
% 0.31/0.46 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D] :
% 0.31/0.46 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.31/0.46 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 0.31/0.46 => ! [V2: A] :
% 0.31/0.46 ( ( A2 @ ( B2 @ V2 ) )
% 0.31/0.46 = ( C2 @ ( D2 @ V2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % comp_eq_elim
% 0.31/0.46 thf(fact_59_comp__eq__dest,axiom,
% 0.31/0.46 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D,V: A] :
% 0.31/0.46 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.31/0.46 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 0.31/0.46 => ( ( A2 @ ( B2 @ V ) )
% 0.31/0.46 = ( C2 @ ( D2 @ V ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % comp_eq_dest
% 0.31/0.46 thf(fact_60_comp__assoc,axiom,
% 0.31/0.46 ! [B: $tType,D: $tType,C: $tType,A: $tType,F2: D > B,G2: C > D,H: A > C] :
% 0.31/0.46 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F2 @ G2 ) @ H )
% 0.31/0.46 = ( comp @ D @ B @ A @ F2 @ ( comp @ C @ D @ A @ G2 @ H ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % comp_assoc
% 0.31/0.46 thf(fact_61_comp__def,axiom,
% 0.31/0.46 ! [A: $tType,C: $tType,B: $tType] :
% 0.31/0.46 ( ( comp @ B @ C @ A )
% 0.31/0.46 = ( ^ [F3: B > C,G: A > B,X: A] : ( F3 @ ( G @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % comp_def
% 0.31/0.46 thf(fact_62_lsc__sequentially,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( real_V2090557954_space @ A )
% 0.31/0.46 => ( ( lower_582600101lsc_at @ A @ extended_ereal )
% 0.31/0.46 = ( ^ [X02: A,F3: A > extended_ereal] :
% 0.31/0.46 ! [X: nat > A,C3: extended_ereal] :
% 0.31/0.46 ( ( ( filterlim @ nat @ A @ X @ ( topolo1920029354e_nhds @ A @ X02 ) @ ( at_top @ nat ) )
% 0.31/0.46 & ! [N: nat] : ( ord_less_eq @ extended_ereal @ ( F3 @ ( X @ N ) ) @ C3 ) )
% 0.31/0.46 => ( ord_less_eq @ extended_ereal @ ( F3 @ X02 ) @ C3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lsc_sequentially
% 0.31/0.46 thf(fact_63_usc__at__def,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ( topolo503727757_space @ A )
% 0.31/0.46 & ( topolo259154727pology @ B ) )
% 0.31/0.46 => ( ( lower_708585518usc_at @ A @ B )
% 0.31/0.46 = ( ^ [X02: A,F3: A > B] :
% 0.31/0.46 ! [X4: nat > A,L2: B] :
% 0.31/0.46 ( ( ( filterlim @ nat @ A @ X4 @ ( topolo1920029354e_nhds @ A @ X02 ) @ ( at_top @ nat ) )
% 0.31/0.46 & ( filterlim @ nat @ B @ ( comp @ A @ B @ nat @ F3 @ X4 ) @ ( topolo1920029354e_nhds @ B @ L2 ) @ ( at_top @ nat ) ) )
% 0.31/0.46 => ( ord_less_eq @ B @ L2 @ ( F3 @ X02 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % usc_at_def
% 0.31/0.46 thf(fact_64_usc__at__mem,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ( topolo503727757_space @ A )
% 0.31/0.46 & ( topolo259154727pology @ B ) )
% 0.31/0.46 => ! [X0: A,F2: A > B,X2: nat > A,A4: B] :
% 0.31/0.46 ( ( lower_708585518usc_at @ A @ B @ X0 @ F2 )
% 0.31/0.46 => ( ( filterlim @ nat @ A @ X2 @ ( topolo1920029354e_nhds @ A @ X0 ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ( filterlim @ nat @ B @ ( comp @ A @ B @ nat @ F2 @ X2 ) @ ( topolo1920029354e_nhds @ B @ A4 ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ord_less_eq @ B @ A4 @ ( F2 @ X0 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % usc_at_mem
% 0.31/0.46 thf(fact_65_lim__decreasing__cl,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ( comple1035589618norder @ A )
% 0.31/0.46 & ( topolo2117631714pology @ A ) )
% 0.31/0.46 => ! [F2: nat > A] :
% 0.31/0.46 ( ! [N2: nat,M2: nat] :
% 0.31/0.46 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( F2 @ M2 ) ) )
% 0.31/0.46 => ~ ! [L3: A] :
% 0.31/0.46 ~ ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L3 ) @ ( at_top @ nat ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lim_decreasing_cl
% 0.31/0.46 thf(fact_66_lim__increasing__cl,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ( comple1035589618norder @ A )
% 0.31/0.46 & ( topolo2117631714pology @ A ) )
% 0.31/0.46 => ! [F2: nat > A] :
% 0.31/0.46 ( ! [N2: nat,M2: nat] :
% 0.31/0.46 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( F2 @ M2 ) @ ( F2 @ N2 ) ) )
% 0.31/0.46 => ~ ! [L3: A] :
% 0.31/0.46 ~ ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L3 ) @ ( at_top @ nat ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lim_increasing_cl
% 0.31/0.46 thf(fact_67_order__refl,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A )
% 0.31/0.46 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_refl
% 0.31/0.46 thf(fact_68_monoseq__le,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( topolo2117631714pology @ A )
% 0.31/0.46 => ! [A2: nat > A,X2: A] :
% 0.31/0.46 ( ( topological_monoseq @ A @ A2 )
% 0.31/0.46 => ( ( filterlim @ nat @ A @ A2 @ ( topolo1920029354e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ( ! [N5: nat] : ( ord_less_eq @ A @ ( A2 @ N5 ) @ X2 )
% 0.31/0.46 & ! [M3: nat,N5: nat] :
% 0.31/0.46 ( ( ord_less_eq @ nat @ M3 @ N5 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( A2 @ M3 ) @ ( A2 @ N5 ) ) ) )
% 0.31/0.46 | ( ! [N5: nat] : ( ord_less_eq @ A @ X2 @ ( A2 @ N5 ) )
% 0.31/0.46 & ! [M3: nat,N5: nat] :
% 0.31/0.46 ( ( ord_less_eq @ nat @ M3 @ N5 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( A2 @ N5 ) @ ( A2 @ M3 ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % monoseq_le
% 0.31/0.46 thf(fact_69_lsc__liminf,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( real_V2090557954_space @ A )
% 0.31/0.46 => ( ( lower_582600101lsc_at @ A @ extended_ereal )
% 0.31/0.46 = ( ^ [X02: A,F3: A > extended_ereal] :
% 0.31/0.46 ! [X: nat > A] :
% 0.31/0.46 ( ( filterlim @ nat @ A @ X @ ( topolo1920029354e_nhds @ A @ X02 ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ord_less_eq @ extended_ereal @ ( F3 @ X02 ) @ ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ ( comp @ A @ extended_ereal @ nat @ F3 @ X ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lsc_liminf
% 0.31/0.46 thf(fact_70_lsc__imp__liminf,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( real_V2090557954_space @ A )
% 0.31/0.46 => ! [X0: A,F2: A > extended_ereal,X2: nat > A] :
% 0.31/0.46 ( ( lower_582600101lsc_at @ A @ extended_ereal @ X0 @ F2 )
% 0.31/0.46 => ( ( filterlim @ nat @ A @ X2 @ ( topolo1920029354e_nhds @ A @ X0 ) @ ( at_top @ nat ) )
% 0.31/0.46 => ( ord_less_eq @ extended_ereal @ ( F2 @ X0 ) @ ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ ( comp @ A @ extended_ereal @ nat @ F2 @ X2 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lsc_imp_liminf
% 0.31/0.46 thf(fact_71_LIMSEQ__Uniq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( topological_t2_space @ A )
% 0.31/0.46 => ! [X3: nat > A] :
% 0.31/0.46 ( uniq @ A
% 0.31/0.46 @ ^ [L2: A] : ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ L2 ) @ ( at_top @ nat ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % LIMSEQ_Uniq
% 0.31/0.46 thf(fact_72_monoseq__def,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ( ( topological_monoseq @ A )
% 0.31/0.46 = ( ^ [X4: nat > A] :
% 0.31/0.46 ( ! [M4: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_eq @ nat @ M4 @ N )
% 0.31/0.46 => ( ord_less_eq @ A @ ( X4 @ M4 ) @ ( X4 @ N ) ) )
% 0.31/0.46 | ! [M4: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_eq @ nat @ M4 @ N )
% 0.31/0.46 => ( ord_less_eq @ A @ ( X4 @ N ) @ ( X4 @ M4 ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % monoseq_def
% 0.31/0.46 thf(fact_73_monoI2,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [X3: nat > A] :
% 0.31/0.46 ( ! [M2: nat,N2: nat] :
% 0.31/0.46 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( X3 @ N2 ) @ ( X3 @ M2 ) ) )
% 0.31/0.46 => ( topological_monoseq @ A @ X3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % monoI2
% 0.31/0.46 thf(fact_74_monoI1,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [X3: nat > A] :
% 0.31/0.46 ( ! [M2: nat,N2: nat] :
% 0.31/0.46 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( X3 @ M2 ) @ ( X3 @ N2 ) ) )
% 0.31/0.46 => ( topological_monoseq @ A @ X3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % monoI1
% 0.31/0.46 thf(fact_75_monoseq__minus,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ordered_ab_group_add @ A )
% 0.31/0.46 => ! [A2: nat > A] :
% 0.31/0.46 ( ( topological_monoseq @ A @ A2 )
% 0.31/0.46 => ( topological_monoseq @ A
% 0.31/0.46 @ ^ [N: nat] : ( uminus_uminus @ A @ ( A2 @ N ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % monoseq_minus
% 0.31/0.46 thf(fact_76_le__funD,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ord @ B )
% 0.31/0.46 => ! [F2: A > B,G2: A > B,X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 0.31/0.46 => ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_funD
% 0.31/0.46 thf(fact_77_le__funE,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ord @ B )
% 0.31/0.46 => ! [F2: A > B,G2: A > B,X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 0.31/0.46 => ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_funE
% 0.31/0.46 thf(fact_78_le__funI,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ord @ B )
% 0.31/0.46 => ! [F2: A > B,G2: A > B] :
% 0.31/0.46 ( ! [X5: A] : ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( G2 @ X5 ) )
% 0.31/0.46 => ( ord_less_eq @ ( A > B ) @ F2 @ G2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_funI
% 0.31/0.46 thf(fact_79_le__fun__def,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ord @ B )
% 0.31/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.31/0.46 = ( ^ [F3: A > B,G: A > B] :
% 0.31/0.46 ! [X: A] : ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_fun_def
% 0.31/0.46 thf(fact_80_order__subst1,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( order @ B )
% 0.31/0.46 & ( order @ A ) )
% 0.31/0.46 => ! [A2: A,F2: B > A,B2: B,C2: B] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) )
% 0.31/0.46 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.31/0.46 => ( ! [X5: B,Y3: B] :
% 0.31/0.46 ( ( ord_less_eq @ B @ X5 @ Y3 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_subst1
% 0.31/0.46 thf(fact_81_order__subst2,axiom,
% 0.31/0.46 ! [A: $tType,C: $tType] :
% 0.31/0.46 ( ( ( order @ C )
% 0.31/0.46 & ( order @ A ) )
% 0.31/0.46 => ! [A2: A,B2: A,F2: A > C,C2: C] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 )
% 0.31/0.46 => ( ! [X5: A,Y3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 0.31/0.46 => ( ord_less_eq @ C @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_subst2
% 0.31/0.46 thf(fact_82_ord__eq__le__subst,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( ord @ B )
% 0.31/0.46 & ( ord @ A ) )
% 0.31/0.46 => ! [A2: A,F2: B > A,B2: B,C2: B] :
% 0.31/0.46 ( ( A2
% 0.31/0.46 = ( F2 @ B2 ) )
% 0.31/0.46 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.31/0.46 => ( ! [X5: B,Y3: B] :
% 0.31/0.46 ( ( ord_less_eq @ B @ X5 @ Y3 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_le_subst
% 0.31/0.46 thf(fact_83_ord__le__eq__subst,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( ord @ B )
% 0.31/0.46 & ( ord @ A ) )
% 0.31/0.46 => ! [A2: A,B2: A,F2: A > B,C2: B] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.46 => ( ( ( F2 @ B2 )
% 0.31/0.46 = C2 )
% 0.31/0.46 => ( ! [X5: A,Y3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 0.31/0.46 => ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_le_eq_subst
% 0.31/0.46 thf(fact_84_eq__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.31/0.46 = ( ^ [X: A,Y5: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X @ Y5 )
% 0.31/0.46 & ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % eq_iff
% 0.31/0.46 thf(fact_85_antisym,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [X2: A,Y: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.31/0.46 => ( ( ord_less_eq @ A @ Y @ X2 )
% 0.31/0.46 => ( X2 = Y ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym
% 0.31/0.46 thf(fact_86_linear,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A )
% 0.31/0.46 => ! [X2: A,Y: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.31/0.46 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linear
% 0.31/0.46 thf(fact_87_eq__refl,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A )
% 0.31/0.46 => ! [X2: A,Y: A] :
% 0.31/0.46 ( ( X2 = Y )
% 0.31/0.46 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % eq_refl
% 0.31/0.46 thf(fact_88_le__cases,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A )
% 0.31/0.46 => ! [X2: A,Y: A] :
% 0.31/0.46 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 0.31/0.46 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_cases
% 0.31/0.46 thf(fact_89_order_Otrans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.31/0.46 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.trans
% 0.31/0.46 thf(fact_90_le__cases3,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A )
% 0.31/0.46 => ! [X2: A,Y: A,Z3: A] :
% 0.31/0.46 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 0.31/0.46 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ X2 @ Z3 ) )
% 0.31/0.46 => ( ( ( ord_less_eq @ A @ X2 @ Z3 )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 0.31/0.46 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 0.31/0.46 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ Z3 @ X2 ) )
% 0.31/0.46 => ~ ( ( ord_less_eq @ A @ Z3 @ X2 )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_cases3
% 0.31/0.46 thf(fact_91_antisym__conv,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [Y: A,X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ X2 @ Y )
% 0.31/0.46 = ( X2 = Y ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym_conv
% 0.31/0.46 thf(fact_92_order__class_Oorder_Oeq__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.31/0.46 = ( ^ [A5: A,B3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.31/0.46 & ( ord_less_eq @ A @ B3 @ A5 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_class.order.eq_iff
% 0.31/0.46 thf(fact_93_ord__eq__le__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord @ A )
% 0.31/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.46 ( ( A2 = B2 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.31/0.46 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_le_trans
% 0.31/0.46 thf(fact_94_ord__le__eq__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord @ A )
% 0.31/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.46 => ( ( B2 = C2 )
% 0.31/0.46 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_le_eq_trans
% 0.31/0.46 thf(fact_95_order__class_Oorder_Oantisym,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [A2: A,B2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.31/0.46 => ( A2 = B2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_class.order.antisym
% 0.31/0.46 thf(fact_96_order__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A )
% 0.31/0.46 => ! [X2: A,Y: A,Z3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.31/0.46 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.31/0.46 => ( ord_less_eq @ A @ X2 @ Z3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_trans
% 0.31/0.46 thf(fact_97_dual__order_Orefl,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.refl
% 0.31/0.46 thf(fact_98_linorder__wlog,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A )
% 0.31/0.46 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.31/0.46 ( ! [A6: A,B4: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A6 @ B4 )
% 0.31/0.46 => ( P @ A6 @ B4 ) )
% 0.31/0.46 => ( ! [A6: A,B4: A] :
% 0.31/0.46 ( ( P @ B4 @ A6 )
% 0.31/0.46 => ( P @ A6 @ B4 ) )
% 0.31/0.46 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linorder_wlog
% 0.31/0.46 thf(fact_99_dual__order_Otrans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [B2: A,A2: A,C2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.31/0.46 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.trans
% 0.31/0.46 thf(fact_100_dual__order_Oeq__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.31/0.46 = ( ^ [A5: A,B3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ B3 @ A5 )
% 0.31/0.46 & ( ord_less_eq @ A @ A5 @ B3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.eq_iff
% 0.31/0.46 thf(fact_101_dual__order_Oantisym,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [B2: A,A2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.46 => ( A2 = B2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.antisym
% 0.31/0.46 thf(fact_102_filterlim__mono,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,F2: A > B,F22: filter @ B,F1: filter @ A,F23: filter @ B,F12: filter @ A] :
% 0.31/0.46 ( ( filterlim @ A @ B @ F2 @ F22 @ F1 )
% 0.31/0.46 => ( ( ord_less_eq @ ( filter @ B ) @ F22 @ F23 )
% 0.31/0.46 => ( ( ord_less_eq @ ( filter @ A ) @ F12 @ F1 )
% 0.31/0.46 => ( filterlim @ A @ B @ F2 @ F23 @ F12 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % filterlim_mono
% 0.31/0.46 thf(fact_103_nhds__imp__cauchy__filter,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( topolo47006728_space @ A )
% 0.31/0.46 => ! [F: filter @ A,X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( filter @ A ) @ F @ ( topolo1920029354e_nhds @ A @ X2 ) )
% 0.31/0.46 => ( topolo1334162427filter @ A @ F ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nhds_imp_cauchy_filter
% 0.31/0.46 thf(fact_104_alt__ex1E_H,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o] :
% 0.31/0.46 ( ? [X6: A] :
% 0.31/0.46 ( ( P @ X6 )
% 0.31/0.46 & ! [Y3: A] :
% 0.31/0.46 ( ( P @ Y3 )
% 0.31/0.46 => ( Y3 = X6 ) ) )
% 0.31/0.46 => ~ ( ? [X_1: A] : ( P @ X_1 )
% 0.31/0.46 => ~ ( uniq @ A @ P ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % alt_ex1E'
% 0.31/0.46 thf(fact_105_ex1__iff__ex__Uniq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ex1 @ A )
% 0.31/0.46 = ( ^ [P2: A > $o] :
% 0.31/0.46 ( ? [X7: A] : ( P2 @ X7 )
% 0.31/0.46 & ( uniq @ A @ P2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ex1_iff_ex_Uniq
% 0.31/0.46 thf(fact_106_liminf__PInfty,axiom,
% 0.31/0.46 ! [X3: nat > extended_ereal] :
% 0.31/0.46 ( ( filterlim @ nat @ extended_ereal @ X3 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( at_top @ nat ) )
% 0.31/0.46 = ( ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ X3 )
% 0.31/0.46 = ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % liminf_PInfty
% 0.31/0.46 thf(fact_107_fun_Omap__ident,axiom,
% 0.31/0.46 ! [A: $tType,D: $tType,T: D > A] :
% 0.31/0.46 ( ( comp @ A @ A @ D
% 0.31/0.46 @ ^ [X: A] : X
% 0.31/0.46 @ T )
% 0.31/0.46 = T ) ).
% 0.31/0.46
% 0.31/0.46 % fun.map_ident
% 0.31/0.46 thf(fact_108_K__record__comp,axiom,
% 0.31/0.46 ! [C: $tType,B: $tType,A: $tType,C2: B,F2: A > C] :
% 0.31/0.46 ( ( comp @ C @ B @ A
% 0.31/0.46 @ ^ [X: C] : C2
% 0.31/0.46 @ F2 )
% 0.31/0.46 = ( ^ [X: A] : C2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % K_record_comp
% 0.31/0.46 thf(fact_109_ereal__infty__less__eq_I1_J,axiom,
% 0.31/0.46 ! [X2: extended_ereal] :
% 0.31/0.46 ( ( ord_less_eq @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ X2 )
% 0.31/0.46 = ( X2
% 0.31/0.46 = ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ereal_infty_less_eq(1)
% 0.31/0.46 thf(fact_110_ereal__infty__less__eq_I2_J,axiom,
% 0.31/0.46 ! [X2: extended_ereal] :
% 0.31/0.46 ( ( ord_less_eq @ extended_ereal @ X2 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.46 = ( X2
% 0.31/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ereal_infty_less_eq(2)
% 0.31/0.46 thf(fact_111_MInfty__neq__PInfty_I1_J,axiom,
% 0.31/0.46 ( ( extend1396239628finity @ extended_ereal )
% 0.31/0.46 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % MInfty_neq_PInfty(1)
% 0.31/0.46 thf(fact_112_ereal__less__eq_I1_J,axiom,
% 0.31/0.46 ! [X2: extended_ereal] : ( ord_less_eq @ extended_ereal @ X2 @ ( extend1396239628finity @ extended_ereal ) ) ).
% 0.31/0.46
% 0.31/0.46 % ereal_less_eq(1)
% 0.31/0.46 thf(fact_113_ereal__infty__less__eq2_I1_J,axiom,
% 0.31/0.46 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.31/0.46 ( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
% 0.31/0.46 => ( ( A2
% 0.31/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.46 => ( B2
% 0.31/0.46 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ereal_infty_less_eq2(1)
% 0.31/0.46 thf(fact_114_neq__PInf__trans,axiom,
% 0.31/0.46 ! [Y: extended_ereal,X2: extended_ereal] :
% 0.31/0.46 ( ( Y
% 0.31/0.46 != ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.46 => ( ( ord_less_eq @ extended_ereal @ X2 @ Y )
% 0.31/0.46 => ( X2
% 0.31/0.46 != ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % neq_PInf_trans
% 0.31/0.46 thf(fact_115_ereal__infty__less__eq2_I2_J,axiom,
% 0.31/0.46 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.31/0.46 ( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
% 0.31/0.46 => ( ( B2
% 0.31/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.46 => ( A2
% 0.31/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ereal_infty_less_eq2(2)
% 0.31/0.46 thf(fact_116_ereal__less__eq_I2_J,axiom,
% 0.31/0.46 ! [X2: extended_ereal] : ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ X2 ) ).
% 0.31/0.46
% 0.31/0.46 % ereal_less_eq(2)
% 0.31/0.46 thf(fact_117_lsc__at__MInfty,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( topolo503727757_space @ A )
% 0.31/0.46 => ! [F2: A > extended_ereal,X0: A] :
% 0.31/0.46 ( ( ( F2 @ X0 )
% 0.31/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.46 => ( lower_582600101lsc_at @ A @ extended_ereal @ X0 @ F2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lsc_at_MInfty
% 0.31/0.46 thf(fact_118_fun_Omap__comp,axiom,
% 0.31/0.46 ! [B: $tType,C: $tType,A: $tType,D: $tType,G2: B > C,F2: A > B,V: D > A] :
% 0.31/0.46 ( ( comp @ B @ C @ D @ G2 @ ( comp @ A @ B @ D @ F2 @ V ) )
% 0.31/0.46 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G2 @ F2 ) @ V ) ) ).
% 0.31/0.46
% 0.31/0.46 % fun.map_comp
% 0.31/0.46 thf(fact_119_Uniq__def,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( uniq @ A )
% 0.31/0.46 = ( ^ [P2: A > $o] :
% 0.31/0.46 ! [X: A,Y5: A] :
% 0.31/0.46 ( ( P2 @ X )
% 0.31/0.46 => ( ( P2 @ Y5 )
% 0.31/0.46 => ( Y5 = X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Uniq_def
% 0.31/0.46 thf(fact_120_Uniq__I,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o] :
% 0.31/0.46 ( ! [X5: A,Y3: A] :
% 0.31/0.46 ( ( P @ X5 )
% 0.31/0.46 => ( ( P @ Y3 )
% 0.31/0.46 => ( Y3 = X5 ) ) )
% 0.31/0.46 => ( uniq @ A @ P ) ) ).
% 0.31/0.46
% 0.31/0.46 % Uniq_I
% 0.31/0.46 thf(fact_121_Uniq__D,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,A2: A,B2: A] :
% 0.31/0.46 ( ( uniq @ A @ P )
% 0.31/0.46 => ( ( P @ A2 )
% 0.31/0.46 => ( ( P @ B2 )
% 0.31/0.46 => ( A2 = B2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Uniq_D
% 0.31/0.46 thf(fact_122_filterlim__ident,axiom,
% 0.31/0.46 ! [A: $tType,F: filter @ A] :
% 0.31/0.46 ( filterlim @ A @ A
% 0.31/0.46 @ ^ [X: A] : X
% 0.31/0.46 @ F
% 0.31/0.46 @ F ) ).
% 0.31/0.46
% 0.31/0.46 % filterlim_ident
% 0.31/0.46 thf(fact_123_filterlim__compose,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,C: $tType,G2: A > B,F32: filter @ B,F22: filter @ A,F2: C > A,F1: filter @ C] :
% 0.31/0.46 ( ( filterlim @ A @ B @ G2 @ F32 @ F22 )
% 0.31/0.46 => ( ( filterlim @ C @ A @ F2 @ F22 @ F1 )
% 0.31/0.46 => ( filterlim @ C @ B
% 0.31/0.46 @ ^ [X: C] : ( G2 @ ( F2 @ X ) )
% 0.31/0.46 @ F32
% 0.31/0.46 @ F1 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % filterlim_compose
% 0.31/0.46 thf(fact_124_Lim__MInfty,axiom,
% 0.31/0.46 ! [F2: nat > extended_ereal] :
% 0.31/0.46 ( ( filterlim @ nat @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) @ ( at_top @ nat ) )
% 0.31/0.46 = ( ! [B5: real] :
% 0.31/0.46 ? [N6: nat] :
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( ord_less_eq @ nat @ N6 @ N )
% 0.31/0.46 => ( ord_less_eq @ extended_ereal @ ( F2 @ N ) @ ( extended_ereal2 @ B5 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Lim_MInfty
% 0.31/0.46 thf(fact_125_MInfty__cases,axiom,
% 0.31/0.46 ! [A: $tType,F2: real > A,Y: A,Z3: A] :
% 0.31/0.47 ( ( extended_case_ereal @ A @ F2 @ Y @ Z3 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 = Z3 ) ).
% 0.31/0.47
% 0.31/0.47 % MInfty_cases
% 0.31/0.47 thf(fact_126_Lim__bounded__MInfty,axiom,
% 0.31/0.47 ! [F2: nat > extended_ereal,L: extended_ereal,B6: real] :
% 0.31/0.47 ( ( filterlim @ nat @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ L ) @ ( at_top @ nat ) )
% 0.31/0.47 => ( ! [N2: nat] : ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ B6 ) @ ( F2 @ N2 ) )
% 0.31/0.47 => ( L
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Lim_bounded_MInfty
% 0.31/0.47 thf(fact_127_Lim__PInfty,axiom,
% 0.31/0.47 ! [F2: nat > extended_ereal] :
% 0.31/0.47 ( ( filterlim @ nat @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( at_top @ nat ) )
% 0.31/0.47 = ( ! [B5: real] :
% 0.31/0.47 ? [N6: nat] :
% 0.31/0.47 ! [N: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ N6 @ N )
% 0.31/0.47 => ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ B5 ) @ ( F2 @ N ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Lim_PInfty
% 0.31/0.47 thf(fact_128_Lim__bounded__PInfty2,axiom,
% 0.31/0.47 ! [F2: nat > extended_ereal,L: extended_ereal,N3: nat,B6: real] :
% 0.31/0.47 ( ( filterlim @ nat @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ L ) @ ( at_top @ nat ) )
% 0.31/0.47 => ( ! [N2: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ N3 @ N2 )
% 0.31/0.47 => ( ord_less_eq @ extended_ereal @ ( F2 @ N2 ) @ ( extended_ereal2 @ B6 ) ) )
% 0.31/0.47 => ( L
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Lim_bounded_PInfty2
% 0.31/0.47 thf(fact_129_ereal_Oinject,axiom,
% 0.31/0.47 ! [X1: real,Y1: real] :
% 0.31/0.47 ( ( ( extended_ereal2 @ X1 )
% 0.31/0.47 = ( extended_ereal2 @ Y1 ) )
% 0.31/0.47 = ( X1 = Y1 ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal.inject
% 0.31/0.47 thf(fact_130_ereal__cong,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ( X2 = Y )
% 0.31/0.47 => ( ( extended_ereal2 @ X2 )
% 0.31/0.47 = ( extended_ereal2 @ Y ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_cong
% 0.31/0.47 thf(fact_131_ereal__less__eq_I3_J,axiom,
% 0.31/0.47 ! [R: real,P3: real] :
% 0.31/0.47 ( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P3 ) )
% 0.31/0.47 = ( ord_less_eq @ real @ R @ P3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_less_eq(3)
% 0.31/0.47 thf(fact_132_PInfty__cases,axiom,
% 0.31/0.47 ! [A: $tType,F2: real > A,Y: A,Z3: A] :
% 0.31/0.47 ( ( extended_case_ereal @ A @ F2 @ Y @ Z3 @ ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 = Y ) ).
% 0.31/0.47
% 0.31/0.47 % PInfty_cases
% 0.31/0.47 thf(fact_133_tendsto__ereal,axiom,
% 0.31/0.47 ! [A: $tType,F2: A > real,X2: real,F: filter @ A] :
% 0.31/0.47 ( ( filterlim @ A @ real @ F2 @ ( topolo1920029354e_nhds @ real @ X2 ) @ F )
% 0.31/0.47 => ( filterlim @ A @ extended_ereal
% 0.31/0.47 @ ^ [X: A] : ( extended_ereal2 @ ( F2 @ X ) )
% 0.31/0.47 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extended_ereal2 @ X2 ) )
% 0.31/0.47 @ F ) ) ).
% 0.31/0.47
% 0.31/0.47 % tendsto_ereal
% 0.31/0.47 thf(fact_134_lim__ereal,axiom,
% 0.31/0.47 ! [A: $tType,F2: A > real,X2: real,Net: filter @ A] :
% 0.31/0.47 ( ( filterlim @ A @ extended_ereal
% 0.31/0.47 @ ^ [N: A] : ( extended_ereal2 @ ( F2 @ N ) )
% 0.31/0.47 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extended_ereal2 @ X2 ) )
% 0.31/0.47 @ Net )
% 0.31/0.47 = ( filterlim @ A @ real @ F2 @ ( topolo1920029354e_nhds @ real @ X2 ) @ Net ) ) ).
% 0.31/0.47
% 0.31/0.47 % lim_ereal
% 0.31/0.47 thf(fact_135_PInfty__neq__ereal_I1_J,axiom,
% 0.31/0.47 ! [R: real] :
% 0.31/0.47 ( ( extended_ereal2 @ R )
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) ).
% 0.31/0.47
% 0.31/0.47 % PInfty_neq_ereal(1)
% 0.31/0.47 thf(fact_136_ereal_Ocase__distrib,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,H: A > B,F13: real > A,F24: A,F33: A,Ereal: extended_ereal] :
% 0.31/0.47 ( ( H @ ( extended_case_ereal @ A @ F13 @ F24 @ F33 @ Ereal ) )
% 0.31/0.47 = ( extended_case_ereal @ B
% 0.31/0.47 @ ^ [X: real] : ( H @ ( F13 @ X ) )
% 0.31/0.47 @ ( H @ F24 )
% 0.31/0.47 @ ( H @ F33 )
% 0.31/0.47 @ Ereal ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal.case_distrib
% 0.31/0.47 thf(fact_137_uminus__ereal_Osimps_I1_J,axiom,
% 0.31/0.47 ! [R: real] :
% 0.31/0.47 ( ( uminus_uminus @ extended_ereal @ ( extended_ereal2 @ R ) )
% 0.31/0.47 = ( extended_ereal2 @ ( uminus_uminus @ real @ R ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % uminus_ereal.simps(1)
% 0.31/0.47 thf(fact_138_ereal__le__le,axiom,
% 0.31/0.47 ! [Y: real,A2: extended_ereal,X2: real] :
% 0.31/0.47 ( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ Y ) @ A2 )
% 0.31/0.47 => ( ( ord_less_eq @ real @ X2 @ Y )
% 0.31/0.47 => ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ X2 ) @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_le_le
% 0.31/0.47 thf(fact_139_le__ereal__le,axiom,
% 0.31/0.47 ! [A2: extended_ereal,X2: real,Y: real] :
% 0.31/0.47 ( ( ord_less_eq @ extended_ereal @ A2 @ ( extended_ereal2 @ X2 ) )
% 0.31/0.47 => ( ( ord_less_eq @ real @ X2 @ Y )
% 0.31/0.47 => ( ord_less_eq @ extended_ereal @ A2 @ ( extended_ereal2 @ Y ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_ereal_le
% 0.31/0.47 thf(fact_140_ereal__le__real,axiom,
% 0.31/0.47 ! [X2: extended_ereal,Y: extended_ereal] :
% 0.31/0.47 ( ! [Z4: real] :
% 0.31/0.47 ( ( ord_less_eq @ extended_ereal @ X2 @ ( extended_ereal2 @ Z4 ) )
% 0.31/0.47 => ( ord_less_eq @ extended_ereal @ Y @ ( extended_ereal2 @ Z4 ) ) )
% 0.31/0.47 => ( ord_less_eq @ extended_ereal @ Y @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_le_real
% 0.31/0.47 thf(fact_141_ereal_Osimps_I8_J,axiom,
% 0.31/0.47 ! [A: $tType,F13: real > A,F24: A,F33: A,X1: real] :
% 0.31/0.47 ( ( extended_case_ereal @ A @ F13 @ F24 @ F33 @ ( extended_ereal2 @ X1 ) )
% 0.31/0.47 = ( F13 @ X1 ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal.simps(8)
% 0.31/0.47 thf(fact_142_ereal__semiline__unique,axiom,
% 0.31/0.47 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.31/0.47 ( ( ( collect @ real
% 0.31/0.47 @ ^ [Y5: real] : ( ord_less_eq @ extended_ereal @ A2 @ ( extended_ereal2 @ Y5 ) ) )
% 0.31/0.47 = ( collect @ real
% 0.31/0.47 @ ^ [Y5: real] : ( ord_less_eq @ extended_ereal @ B2 @ ( extended_ereal2 @ Y5 ) ) ) )
% 0.31/0.47 = ( A2 = B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_semiline_unique
% 0.31/0.47 thf(fact_143_real__of__ereal_Oinduct,axiom,
% 0.31/0.47 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.31/0.47 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ( P @ ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( P @ A0 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % real_of_ereal.induct
% 0.31/0.47 thf(fact_144_real__of__ereal_Ocases,axiom,
% 0.31/0.47 ! [X2: extended_ereal] :
% 0.31/0.47 ( ! [R2: real] :
% 0.31/0.47 ( X2
% 0.31/0.47 != ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ( X2
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % real_of_ereal.cases
% 0.31/0.47 thf(fact_145_times__ereal_Oinduct,axiom,
% 0.31/0.47 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.31/0.47 ( ! [R2: real,P4: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P4 ) )
% 0.31/0.47 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( ! [R2: real] : ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ! [R2: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % times_ereal.induct
% 0.31/0.47 thf(fact_146_plus__ereal_Oinduct,axiom,
% 0.31/0.47 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.31/0.47 ( ! [R2: real,P4: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P4 ) )
% 0.31/0.47 => ( ! [X_1: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_1 )
% 0.31/0.47 => ( ! [A6: extended_ereal] : ( P @ A6 @ ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ! [P4: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ P4 ) )
% 0.31/0.47 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % plus_ereal.induct
% 0.31/0.47 thf(fact_147_less__ereal_Oinduct,axiom,
% 0.31/0.47 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.31/0.47 ( ! [X5: real,Y3: real] : ( P @ ( extended_ereal2 @ X5 ) @ ( extended_ereal2 @ Y3 ) )
% 0.31/0.47 => ( ! [X_1: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_1 )
% 0.31/0.47 => ( ! [A6: extended_ereal] : ( P @ A6 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ! [X5: real] : ( P @ ( extended_ereal2 @ X5 ) @ ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( ! [R2: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_ereal.induct
% 0.31/0.47 thf(fact_148_abs__ereal_Oinduct,axiom,
% 0.31/0.47 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.31/0.47 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( P @ ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( P @ A0 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % abs_ereal.induct
% 0.31/0.47 thf(fact_149_ereal__all__split,axiom,
% 0.31/0.47 ( ( ^ [P5: extended_ereal > $o] :
% 0.31/0.47 ! [X7: extended_ereal] : ( P5 @ X7 ) )
% 0.31/0.47 = ( ^ [P2: extended_ereal > $o] :
% 0.31/0.47 ( ( P2 @ ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 & ! [X: real] : ( P2 @ ( extended_ereal2 @ X ) )
% 0.31/0.47 & ( P2 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_all_split
% 0.31/0.47 thf(fact_150_abs__ereal_Ocases,axiom,
% 0.31/0.47 ! [X2: extended_ereal] :
% 0.31/0.47 ( ! [R2: real] :
% 0.31/0.47 ( X2
% 0.31/0.47 != ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ( X2
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( X2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % abs_ereal.cases
% 0.31/0.47 thf(fact_151_ereal__ex__split,axiom,
% 0.31/0.47 ( ( ^ [P5: extended_ereal > $o] :
% 0.31/0.47 ? [X7: extended_ereal] : ( P5 @ X7 ) )
% 0.31/0.47 = ( ^ [P2: extended_ereal > $o] :
% 0.31/0.47 ( ( P2 @ ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 | ? [X: real] : ( P2 @ ( extended_ereal2 @ X ) )
% 0.31/0.47 | ( P2 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_ex_split
% 0.31/0.47 thf(fact_152_ereal3__cases,axiom,
% 0.31/0.47 ! [X2: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
% 0.31/0.47 ( ( ? [R2: real] :
% 0.31/0.47 ( X2
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ? [Ra: real] :
% 0.31/0.47 ( Xa3
% 0.31/0.47 = ( extended_ereal2 @ Ra ) )
% 0.31/0.47 => ! [Rb: real] :
% 0.31/0.47 ( Xb
% 0.31/0.47 != ( extended_ereal2 @ Rb ) ) ) )
% 0.31/0.47 => ( ( ? [R2: real] :
% 0.31/0.47 ( X2
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ? [Ra: real] :
% 0.31/0.47 ( Xa3
% 0.31/0.47 = ( extended_ereal2 @ Ra ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.31/0.47 => ( ( ? [R2: real] :
% 0.31/0.47 ( X2
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ? [Ra: real] :
% 0.31/0.47 ( Xa3
% 0.31/0.47 = ( extended_ereal2 @ Ra ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.31/0.47 => ( ( ? [R2: real] :
% 0.31/0.47 ( X2
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ! [Ra: real] :
% 0.31/0.47 ( Xb
% 0.31/0.47 != ( extended_ereal2 @ Ra ) ) ) )
% 0.31/0.47 => ( ( ? [R2: real] :
% 0.31/0.47 ( X2
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.31/0.47 => ( ( ? [R2: real] :
% 0.31/0.47 ( X2
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.31/0.47 => ( ( ? [R2: real] :
% 0.31/0.47 ( X2
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ! [Ra: real] :
% 0.31/0.47 ( Xb
% 0.31/0.47 != ( extended_ereal2 @ Ra ) ) ) )
% 0.31/0.47 => ( ( ? [R2: real] :
% 0.31/0.47 ( X2
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.31/0.47 => ( ( ? [R2: real] :
% 0.31/0.47 ( X2
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( ? [R2: real] :
% 0.31/0.47 ( Xa3
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ! [Ra: real] :
% 0.31/0.47 ( Xb
% 0.31/0.47 != ( extended_ereal2 @ Ra ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( ? [R2: real] :
% 0.31/0.47 ( Xa3
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( ? [R2: real] :
% 0.31/0.47 ( Xa3
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ! [R2: real] :
% 0.31/0.47 ( Xb
% 0.31/0.47 != ( extended_ereal2 @ R2 ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ! [R2: real] :
% 0.31/0.47 ( Xb
% 0.31/0.47 != ( extended_ereal2 @ R2 ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ? [R2: real] :
% 0.31/0.47 ( Xa3
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ! [Ra: real] :
% 0.31/0.47 ( Xb
% 0.31/0.47 != ( extended_ereal2 @ Ra ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ? [R2: real] :
% 0.31/0.47 ( Xa3
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ? [R2: real] :
% 0.31/0.47 ( Xa3
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ! [R2: real] :
% 0.31/0.47 ( Xb
% 0.31/0.47 != ( extended_ereal2 @ R2 ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ! [R2: real] :
% 0.31/0.47 ( Xb
% 0.31/0.47 != ( extended_ereal2 @ R2 ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.31/0.47 => ~ ( ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( Xb
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal3_cases
% 0.31/0.47 thf(fact_153_ereal2__cases,axiom,
% 0.31/0.47 ! [X2: extended_ereal,Xa3: extended_ereal] :
% 0.31/0.47 ( ( ? [R2: real] :
% 0.31/0.47 ( X2
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ! [Ra: real] :
% 0.31/0.47 ( Xa3
% 0.31/0.47 != ( extended_ereal2 @ Ra ) ) )
% 0.31/0.47 => ( ( ? [R2: real] :
% 0.31/0.47 ( X2
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( Xa3
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( ? [R2: real] :
% 0.31/0.47 ( X2
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( Xa3
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ! [R2: real] :
% 0.31/0.47 ( Xa3
% 0.31/0.47 != ( extended_ereal2 @ R2 ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( Xa3
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( Xa3
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ! [R2: real] :
% 0.31/0.47 ( Xa3
% 0.31/0.47 != ( extended_ereal2 @ R2 ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( Xa3
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ~ ( ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( Xa3
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal2_cases
% 0.31/0.47 thf(fact_154_ereal__cases,axiom,
% 0.31/0.47 ! [X2: extended_ereal] :
% 0.31/0.47 ( ! [R2: real] :
% 0.31/0.47 ( X2
% 0.31/0.47 != ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ( X2
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_cases
% 0.31/0.47 thf(fact_155_MInfty__neq__ereal_I1_J,axiom,
% 0.31/0.47 ! [R: real] :
% 0.31/0.47 ( ( extended_ereal2 @ R )
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % MInfty_neq_ereal(1)
% 0.31/0.47 thf(fact_156_ereal__top,axiom,
% 0.31/0.47 ! [X2: extended_ereal] :
% 0.31/0.47 ( ! [B7: real] : ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ B7 ) @ X2 )
% 0.31/0.47 => ( X2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_top
% 0.31/0.47 thf(fact_157_ereal__tendsto__simps2_I1_J,axiom,
% 0.31/0.47 ! [A: $tType,F2: A > real,A2: real,F: filter @ A] :
% 0.31/0.47 ( ( filterlim @ A @ extended_ereal @ ( comp @ real @ extended_ereal @ A @ extended_ereal2 @ F2 ) @ ( topolo1920029354e_nhds @ extended_ereal @ ( extended_ereal2 @ A2 ) ) @ F )
% 0.31/0.47 = ( filterlim @ A @ real @ F2 @ ( topolo1920029354e_nhds @ real @ A2 ) @ F ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_tendsto_simps2(1)
% 0.31/0.47 thf(fact_158_ereal__bot,axiom,
% 0.31/0.47 ! [X2: extended_ereal] :
% 0.31/0.47 ( ! [B7: real] : ( ord_less_eq @ extended_ereal @ X2 @ ( extended_ereal2 @ B7 ) )
% 0.31/0.47 => ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_bot
% 0.31/0.47 thf(fact_159_Lim__bounded__PInfty,axiom,
% 0.31/0.47 ! [F2: nat > extended_ereal,L: extended_ereal,B6: real] :
% 0.31/0.47 ( ( filterlim @ nat @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ L ) @ ( at_top @ nat ) )
% 0.31/0.47 => ( ! [N2: nat] : ( ord_less_eq @ extended_ereal @ ( F2 @ N2 ) @ ( extended_ereal2 @ B6 ) )
% 0.31/0.47 => ( L
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Lim_bounded_PInfty
% 0.31/0.47 thf(fact_160_ereal__tendsto__simps2_I3_J,axiom,
% 0.31/0.47 ! [A: $tType,F2: A > real,F: filter @ A] :
% 0.31/0.47 ( ( filterlim @ A @ extended_ereal @ ( comp @ real @ extended_ereal @ A @ extended_ereal2 @ F2 ) @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) @ F )
% 0.31/0.47 = ( filterlim @ A @ real @ F2 @ ( at_bot @ real ) @ F ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_tendsto_simps2(3)
% 0.31/0.47 thf(fact_161_tendsto__cadd__ereal,axiom,
% 0.31/0.47 ! [A: $tType,Y: extended_ereal,X2: extended_ereal,F2: A > extended_ereal,F: filter @ A] :
% 0.31/0.47 ( ( Y
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( X2
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ X2 ) @ F )
% 0.31/0.47 => ( filterlim @ A @ extended_ereal
% 0.31/0.47 @ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F2 @ X ) @ Y )
% 0.31/0.47 @ ( topolo1920029354e_nhds @ extended_ereal @ ( plus_plus @ extended_ereal @ X2 @ Y ) )
% 0.31/0.47 @ F ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % tendsto_cadd_ereal
% 0.31/0.47 thf(fact_162_ereal__minus__real__tendsto__MInf,axiom,
% 0.31/0.47 ( filterlim @ nat @ extended_ereal
% 0.31/0.47 @ ^ [X: nat] : ( extended_ereal2 @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ X ) ) )
% 0.31/0.47 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 @ ( at_top @ nat ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_minus_real_tendsto_MInf
% 0.31/0.47 thf(fact_163_tendsto__add__ereal__nonneg,axiom,
% 0.31/0.47 ! [A: $tType,X2: extended_ereal,Y: extended_ereal,F2: A > extended_ereal,F: filter @ A,G2: A > extended_ereal] :
% 0.31/0.47 ( ( X2
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( Y
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ X2 ) @ F )
% 0.31/0.47 => ( ( filterlim @ A @ extended_ereal @ G2 @ ( topolo1920029354e_nhds @ extended_ereal @ Y ) @ F )
% 0.31/0.47 => ( filterlim @ A @ extended_ereal
% 0.31/0.47 @ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F2 @ X ) @ ( G2 @ X ) )
% 0.31/0.47 @ ( topolo1920029354e_nhds @ extended_ereal @ ( plus_plus @ extended_ereal @ X2 @ Y ) )
% 0.31/0.47 @ F ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % tendsto_add_ereal_nonneg
% 0.31/0.47 thf(fact_164_add__left__cancel,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel_semigroup_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.31/0.47 = ( plus_plus @ A @ A2 @ C2 ) )
% 0.31/0.47 = ( B2 = C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_left_cancel
% 0.31/0.47 thf(fact_165_add__right__cancel,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel_semigroup_add @ A )
% 0.31/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.31/0.47 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.31/0.47 = ( plus_plus @ A @ C2 @ A2 ) )
% 0.31/0.47 = ( B2 = C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_right_cancel
% 0.31/0.47 thf(fact_166_add__le__cancel__left,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere236663937imp_le @ A )
% 0.31/0.47 => ! [C2: A,A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.31/0.47 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_le_cancel_left
% 0.31/0.47 thf(fact_167_add__le__cancel__right,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere236663937imp_le @ A )
% 0.31/0.47 => ! [A2: A,C2: A,B2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.31/0.47 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_le_cancel_right
% 0.31/0.47 thf(fact_168_minus__add__distrib,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ab_group_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.31/0.47 = ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minus_add_distrib
% 0.31/0.47 thf(fact_169_minus__add__cancel,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.31/0.47 = B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % minus_add_cancel
% 0.31/0.47 thf(fact_170_add__minus__cancel,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
% 0.31/0.47 = B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_minus_cancel
% 0.31/0.47 thf(fact_171_ereal__PInfty__eq__plus,axiom,
% 0.31/0.47 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.31/0.47 ( ( ( extend1396239628finity @ extended_ereal )
% 0.31/0.47 = ( plus_plus @ extended_ereal @ A2 @ B2 ) )
% 0.31/0.47 = ( ( A2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 | ( B2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_PInfty_eq_plus
% 0.31/0.47 thf(fact_172_ereal__plus__eq__PInfty,axiom,
% 0.31/0.47 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.31/0.47 ( ( ( plus_plus @ extended_ereal @ A2 @ B2 )
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 = ( ( A2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 | ( B2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_plus_eq_PInfty
% 0.31/0.47 thf(fact_173_ereal__MInfty__eq__plus,axiom,
% 0.31/0.47 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.31/0.47 ( ( ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 = ( plus_plus @ extended_ereal @ A2 @ B2 ) )
% 0.31/0.47 = ( ( ( A2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 & ( B2
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 | ( ( B2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 & ( A2
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_MInfty_eq_plus
% 0.31/0.47 thf(fact_174_ereal__plus__eq__MInfty,axiom,
% 0.31/0.47 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.31/0.47 ( ( ( plus_plus @ extended_ereal @ A2 @ B2 )
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 = ( ( ( A2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 | ( B2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.31/0.47 & ( A2
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 & ( B2
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_plus_eq_MInfty
% 0.31/0.47 thf(fact_175_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ab_semigroup_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 0.31/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ab_semigroup_add_class.add_ac(1)
% 0.31/0.47 thf(fact_176_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere779506340up_add @ A )
% 0.31/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.31/0.47 ( ( ( I = J )
% 0.31/0.47 & ( K = L ) )
% 0.31/0.47 => ( ( plus_plus @ A @ I @ K )
% 0.31/0.47 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_mono_thms_linordered_semiring(4)
% 0.31/0.47 thf(fact_177_group__cancel_Oadd1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( comm_monoid_add @ A )
% 0.31/0.47 => ! [A4: A,K: A,A2: A,B2: A] :
% 0.31/0.47 ( ( A4
% 0.31/0.47 = ( plus_plus @ A @ K @ A2 ) )
% 0.31/0.47 => ( ( plus_plus @ A @ A4 @ B2 )
% 0.31/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % group_cancel.add1
% 0.31/0.47 thf(fact_178_group__cancel_Oadd2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( comm_monoid_add @ A )
% 0.31/0.47 => ! [B6: A,K: A,B2: A,A2: A] :
% 0.31/0.47 ( ( B6
% 0.31/0.47 = ( plus_plus @ A @ K @ B2 ) )
% 0.31/0.47 => ( ( plus_plus @ A @ A2 @ B6 )
% 0.31/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % group_cancel.add2
% 0.31/0.47 thf(fact_179_add_Oassoc,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semigroup_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 0.31/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.assoc
% 0.31/0.47 thf(fact_180_add_Oleft__cancel,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.31/0.47 = ( plus_plus @ A @ A2 @ C2 ) )
% 0.31/0.47 = ( B2 = C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.left_cancel
% 0.31/0.47 thf(fact_181_add_Oright__cancel,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.31/0.47 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.31/0.47 = ( plus_plus @ A @ C2 @ A2 ) )
% 0.31/0.47 = ( B2 = C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.right_cancel
% 0.31/0.47 thf(fact_182_add_Ocommute,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ab_semigroup_add @ A )
% 0.31/0.47 => ( ( plus_plus @ A )
% 0.31/0.47 = ( ^ [A5: A,B3: A] : ( plus_plus @ A @ B3 @ A5 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.commute
% 0.31/0.47 thf(fact_183_add_Oleft__commute,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ab_semigroup_add @ A )
% 0.31/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.31/0.47 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
% 0.31/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.left_commute
% 0.31/0.47 thf(fact_184_add__left__imp__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel_semigroup_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.31/0.47 = ( plus_plus @ A @ A2 @ C2 ) )
% 0.31/0.47 => ( B2 = C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_left_imp_eq
% 0.31/0.47 thf(fact_185_add__right__imp__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel_semigroup_add @ A )
% 0.31/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.31/0.47 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.31/0.47 = ( plus_plus @ A @ C2 @ A2 ) )
% 0.31/0.47 => ( B2 = C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_right_imp_eq
% 0.31/0.47 thf(fact_186_add_Oinverse__distrib__swap,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.31/0.47 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.inverse_distrib_swap
% 0.31/0.47 thf(fact_187_group__cancel_Oneg1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ab_group_add @ A )
% 0.31/0.47 => ! [A4: A,K: A,A2: A] :
% 0.31/0.47 ( ( A4
% 0.31/0.47 = ( plus_plus @ A @ K @ A2 ) )
% 0.31/0.47 => ( ( uminus_uminus @ A @ A4 )
% 0.31/0.47 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % group_cancel.neg1
% 0.31/0.47 thf(fact_188_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere779506340up_add @ A )
% 0.31/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.31/0.47 ( ( ( ord_less_eq @ A @ I @ J )
% 0.31/0.47 & ( K = L ) )
% 0.31/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_mono_thms_linordered_semiring(3)
% 0.31/0.47 thf(fact_189_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere779506340up_add @ A )
% 0.31/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.31/0.47 ( ( ( I = J )
% 0.31/0.47 & ( ord_less_eq @ A @ K @ L ) )
% 0.31/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_mono_thms_linordered_semiring(2)
% 0.31/0.47 thf(fact_190_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere779506340up_add @ A )
% 0.31/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.31/0.47 ( ( ( ord_less_eq @ A @ I @ J )
% 0.31/0.47 & ( ord_less_eq @ A @ K @ L ) )
% 0.31/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_mono_thms_linordered_semiring(1)
% 0.31/0.47 thf(fact_191_add__mono,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere779506340up_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A,D2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 0.31/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_mono
% 0.31/0.47 thf(fact_192_add__left__mono,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere779506340up_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_left_mono
% 0.31/0.47 thf(fact_193_less__eqE,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.47 => ~ ! [C5: A] :
% 0.31/0.47 ( B2
% 0.31/0.47 != ( plus_plus @ A @ A2 @ C5 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_eqE
% 0.31/0.47 thf(fact_194_add__right__mono,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere779506340up_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_right_mono
% 0.31/0.47 thf(fact_195_le__iff__add,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ( ( ord_less_eq @ A )
% 0.31/0.47 = ( ^ [A5: A,B3: A] :
% 0.31/0.47 ? [C3: A] :
% 0.31/0.47 ( B3
% 0.31/0.47 = ( plus_plus @ A @ A5 @ C3 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_iff_add
% 0.31/0.47 thf(fact_196_add__le__imp__le__left,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere236663937imp_le @ A )
% 0.31/0.47 => ! [C2: A,A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.31/0.47 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_le_imp_le_left
% 0.31/0.47 thf(fact_197_add__le__imp__le__right,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere236663937imp_le @ A )
% 0.31/0.47 => ! [A2: A,C2: A,B2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.31/0.47 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_le_imp_le_right
% 0.31/0.47 thf(fact_198_plus__ereal_Osimps_I2_J,axiom,
% 0.31/0.47 ! [A2: extended_ereal] :
% 0.31/0.47 ( ( plus_plus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ A2 )
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) ) ).
% 0.31/0.47
% 0.31/0.47 % plus_ereal.simps(2)
% 0.31/0.47 thf(fact_199_plus__ereal_Osimps_I3_J,axiom,
% 0.31/0.47 ! [A2: extended_ereal] :
% 0.31/0.47 ( ( plus_plus @ extended_ereal @ A2 @ ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) ) ).
% 0.31/0.47
% 0.31/0.47 % plus_ereal.simps(3)
% 0.31/0.47 thf(fact_200_plus__ereal_Osimps_I6_J,axiom,
% 0.31/0.47 ( ( plus_plus @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % plus_ereal.simps(6)
% 0.31/0.47 thf(fact_201_ereal__add__cancel__left,axiom,
% 0.31/0.47 ! [A2: extended_ereal,B2: extended_ereal,C2: extended_ereal] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( ( plus_plus @ extended_ereal @ A2 @ B2 )
% 0.31/0.47 = ( plus_plus @ extended_ereal @ A2 @ C2 ) )
% 0.31/0.47 = ( ( A2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 | ( B2 = C2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_add_cancel_left
% 0.31/0.47 thf(fact_202_ereal__add__cancel__right,axiom,
% 0.31/0.47 ! [A2: extended_ereal,B2: extended_ereal,C2: extended_ereal] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( ( plus_plus @ extended_ereal @ B2 @ A2 )
% 0.31/0.47 = ( plus_plus @ extended_ereal @ C2 @ A2 ) )
% 0.31/0.47 = ( ( A2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 | ( B2 = C2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_add_cancel_right
% 0.31/0.47 thf(fact_203_tendsto__add,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( topolo1314133330id_add @ A )
% 0.31/0.47 => ! [F2: B > A,A2: A,F: filter @ B,G2: B > A,B2: A] :
% 0.31/0.47 ( ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ A2 ) @ F )
% 0.31/0.47 => ( ( filterlim @ B @ A @ G2 @ ( topolo1920029354e_nhds @ A @ B2 ) @ F )
% 0.31/0.47 => ( filterlim @ B @ A
% 0.31/0.47 @ ^ [X: B] : ( plus_plus @ A @ ( F2 @ X ) @ ( G2 @ X ) )
% 0.31/0.47 @ ( topolo1920029354e_nhds @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.31/0.47 @ F ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % tendsto_add
% 0.31/0.47 thf(fact_204_tendsto__add__const__iff,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( real_V55928688vector @ A )
% 0.31/0.47 => ! [C2: A,F2: B > A,D2: A,F: filter @ B] :
% 0.31/0.47 ( ( filterlim @ B @ A
% 0.31/0.47 @ ^ [X: B] : ( plus_plus @ A @ C2 @ ( F2 @ X ) )
% 0.31/0.47 @ ( topolo1920029354e_nhds @ A @ ( plus_plus @ A @ C2 @ D2 ) )
% 0.31/0.47 @ F )
% 0.31/0.47 = ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ D2 ) @ F ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % tendsto_add_const_iff
% 0.31/0.47 thf(fact_205_plus__ereal_Osimps_I5_J,axiom,
% 0.31/0.47 ! [P3: real] :
% 0.31/0.47 ( ( plus_plus @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ P3 ) )
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % plus_ereal.simps(5)
% 0.31/0.47 thf(fact_206_plus__ereal_Osimps_I4_J,axiom,
% 0.31/0.47 ! [R: real] :
% 0.31/0.47 ( ( plus_plus @ extended_ereal @ ( extended_ereal2 @ R ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % plus_ereal.simps(4)
% 0.31/0.47 thf(fact_207_ereal__add__le__add__iff,axiom,
% 0.31/0.47 ! [C2: extended_ereal,A2: extended_ereal,B2: extended_ereal] :
% 0.31/0.47 ( ( ord_less_eq @ extended_ereal @ ( plus_plus @ extended_ereal @ C2 @ A2 ) @ ( plus_plus @ extended_ereal @ C2 @ B2 ) )
% 0.31/0.47 = ( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
% 0.31/0.47 | ( C2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 | ( ( C2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 & ( A2
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 & ( B2
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_add_le_add_iff
% 0.31/0.47 thf(fact_208_ereal__add__le__add__iff2,axiom,
% 0.31/0.47 ! [A2: extended_ereal,C2: extended_ereal,B2: extended_ereal] :
% 0.31/0.47 ( ( ord_less_eq @ extended_ereal @ ( plus_plus @ extended_ereal @ A2 @ C2 ) @ ( plus_plus @ extended_ereal @ B2 @ C2 ) )
% 0.31/0.47 = ( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
% 0.31/0.47 | ( C2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 | ( ( C2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 & ( A2
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 & ( B2
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_add_le_add_iff2
% 0.31/0.47 thf(fact_209_tendsto__add__ereal__MInf,axiom,
% 0.31/0.47 ! [A: $tType,Y: extended_ereal,F2: A > extended_ereal,F: filter @ A,G2: A > extended_ereal] :
% 0.31/0.47 ( ( Y
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) @ F )
% 0.31/0.47 => ( ( filterlim @ A @ extended_ereal @ G2 @ ( topolo1920029354e_nhds @ extended_ereal @ Y ) @ F )
% 0.31/0.47 => ( filterlim @ A @ extended_ereal
% 0.31/0.47 @ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F2 @ X ) @ ( G2 @ X ) )
% 0.31/0.47 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 @ F ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % tendsto_add_ereal_MInf
% 0.31/0.47 thf(fact_210_tendsto__add__ereal__PInf,axiom,
% 0.31/0.47 ! [A: $tType,Y: extended_ereal,F2: A > extended_ereal,F: filter @ A,G2: A > extended_ereal] :
% 0.31/0.47 ( ( Y
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ F )
% 0.31/0.47 => ( ( filterlim @ A @ extended_ereal @ G2 @ ( topolo1920029354e_nhds @ extended_ereal @ Y ) @ F )
% 0.31/0.47 => ( filterlim @ A @ extended_ereal
% 0.31/0.47 @ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F2 @ X ) @ ( G2 @ X ) )
% 0.31/0.47 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 @ F ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % tendsto_add_ereal_PInf
% 0.31/0.47 thf(fact_211_tendsto__add__ereal__general,axiom,
% 0.31/0.47 ! [A: $tType,X2: extended_ereal,Y: extended_ereal,F2: A > extended_ereal,F: filter @ A,G2: A > extended_ereal] :
% 0.31/0.47 ( ~ ( ( ( X2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 & ( Y
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.31/0.47 | ( ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 & ( Y
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.31/0.47 => ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ X2 ) @ F )
% 0.31/0.47 => ( ( filterlim @ A @ extended_ereal @ G2 @ ( topolo1920029354e_nhds @ extended_ereal @ Y ) @ F )
% 0.31/0.47 => ( filterlim @ A @ extended_ereal
% 0.31/0.47 @ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F2 @ X ) @ ( G2 @ X ) )
% 0.31/0.47 @ ( topolo1920029354e_nhds @ extended_ereal @ ( plus_plus @ extended_ereal @ X2 @ Y ) )
% 0.31/0.47 @ F ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % tendsto_add_ereal_general
% 0.31/0.47 thf(fact_212_tendsto__at__botI__sequentially,axiom,
% 0.31/0.47 ! [B: $tType] :
% 0.31/0.47 ( ( topolo2135403230pology @ B )
% 0.31/0.47 => ! [F2: real > B,Y: B] :
% 0.31/0.47 ( ! [X8: nat > real] :
% 0.31/0.47 ( ( filterlim @ nat @ real @ X8 @ ( at_bot @ real ) @ ( at_top @ nat ) )
% 0.31/0.47 => ( filterlim @ nat @ B
% 0.31/0.47 @ ^ [N: nat] : ( F2 @ ( X8 @ N ) )
% 0.31/0.47 @ ( topolo1920029354e_nhds @ B @ Y )
% 0.31/0.47 @ ( at_top @ nat ) ) )
% 0.31/0.47 => ( filterlim @ real @ B @ F2 @ ( topolo1920029354e_nhds @ B @ Y ) @ ( at_bot @ real ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % tendsto_at_botI_sequentially
% 0.31/0.47 thf(fact_213_id__nat__ereal__tendsto__PInf,axiom,
% 0.31/0.47 ( filterlim @ nat @ extended_ereal
% 0.31/0.47 @ ^ [X: nat] : ( extended_ereal2 @ ( semiring_1_of_nat @ real @ X ) )
% 0.31/0.47 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 @ ( at_top @ nat ) ) ).
% 0.31/0.47
% 0.31/0.47 % id_nat_ereal_tendsto_PInf
% 0.31/0.47 thf(fact_214_ereal__liminf__add__mono,axiom,
% 0.31/0.47 ! [U: nat > extended_ereal,V: nat > extended_ereal] :
% 0.31/0.47 ( ~ ( ( ( ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ U )
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 & ( ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ V )
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.31/0.47 | ( ( ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ U )
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 & ( ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ V )
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.31/0.47 => ( ord_less_eq @ extended_ereal @ ( plus_plus @ extended_ereal @ ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ U ) @ ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ V ) )
% 0.31/0.47 @ ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat )
% 0.31/0.47 @ ^ [N: nat] : ( plus_plus @ extended_ereal @ ( U @ N ) @ ( V @ N ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_liminf_add_mono
% 0.31/0.47 thf(fact_215_of__nat__le__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord1659791738miring @ A )
% 0.31/0.47 => ! [M5: nat,N7: nat] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M5 ) @ ( semiring_1_of_nat @ A @ N7 ) )
% 0.31/0.47 = ( ord_less_eq @ nat @ M5 @ N7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % of_nat_le_iff
% 0.31/0.47 thf(fact_216_of__nat__mono,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord1659791738miring @ A )
% 0.31/0.47 => ! [I: nat,J: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ I @ J )
% 0.31/0.47 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % of_nat_mono
% 0.31/0.47 thf(fact_217_ereal__tendsto__simps2_I2_J,axiom,
% 0.31/0.47 ! [A: $tType,F2: A > real,F: filter @ A] :
% 0.31/0.47 ( ( filterlim @ A @ extended_ereal @ ( comp @ real @ extended_ereal @ A @ extended_ereal2 @ F2 ) @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ F )
% 0.31/0.47 = ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F ) ) ).
% 0.31/0.47
% 0.31/0.47 % ereal_tendsto_simps2(2)
% 0.31/0.47 thf(fact_218_nat__add__left__cancel__le,axiom,
% 0.31/0.47 ! [K: nat,M5: nat,N7: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M5 ) @ ( plus_plus @ nat @ K @ N7 ) )
% 0.31/0.47 = ( ord_less_eq @ nat @ M5 @ N7 ) ) ).
% 0.31/0.47
% 0.31/0.47 % nat_add_left_cancel_le
% 0.31/0.47 thf(fact_219_filterlim__at__top__add__at__top,axiom,
% 0.31/0.47 ! [A: $tType,F2: A > real,F: filter @ A,G2: A > real] :
% 0.31/0.47 ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F )
% 0.31/0.47 => ( ( filterlim @ A @ real @ G2 @ ( at_top @ real ) @ F )
% 0.31/0.47 => ( filterlim @ A @ real
% 0.31/0.47 @ ^ [X: A] : ( plus_plus @ real @ ( F2 @ X ) @ ( G2 @ X ) )
% 0.31/0.47 @ ( at_top @ real )
% 0.31/0.47 @ F ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % filterlim_at_top_add_at_top
% 0.31/0.47 thf(fact_220_filterlim__tendsto__add__at__top,axiom,
% 0.31/0.47 ! [A: $tType,F2: A > real,C2: real,F: filter @ A,G2: A > real] :
% 0.31/0.47 ( ( filterlim @ A @ real @ F2 @ ( topolo1920029354e_nhds @ real @ C2 ) @ F )
% 0.31/0.47 => ( ( filterlim @ A @ real @ G2 @ ( at_top @ real ) @ F )
% 0.31/0.47 => ( filterlim @ A @ real
% 0.31/0.47 @ ^ [X: A] : ( plus_plus @ real @ ( F2 @ X ) @ ( G2 @ X ) )
% 0.31/0.47 @ ( at_top @ real )
% 0.31/0.47 @ F ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % filterlim_tendsto_add_at_top
% 0.31/0.47 thf(fact_221_nat__int__comparison_I3_J,axiom,
% 0.31/0.47 ( ( ord_less_eq @ nat )
% 0.31/0.47 = ( ^ [A5: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % nat_int_comparison(3)
% 0.31/0.47 thf(fact_222_add__leE,axiom,
% 0.31/0.47 ! [M5: nat,K: nat,N7: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M5 @ K ) @ N7 )
% 0.31/0.47 => ~ ( ( ord_less_eq @ nat @ M5 @ N7 )
% 0.31/0.47 => ~ ( ord_less_eq @ nat @ K @ N7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_leE
% 0.31/0.47 thf(fact_223_le__add1,axiom,
% 0.31/0.47 ! [N7: nat,M5: nat] : ( ord_less_eq @ nat @ N7 @ ( plus_plus @ nat @ N7 @ M5 ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_add1
% 0.31/0.47 thf(fact_224_le__add2,axiom,
% 0.31/0.47 ! [N7: nat,M5: nat] : ( ord_less_eq @ nat @ N7 @ ( plus_plus @ nat @ M5 @ N7 ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_add2
% 0.31/0.47 thf(fact_225_add__leD1,axiom,
% 0.31/0.47 ! [M5: nat,K: nat,N7: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M5 @ K ) @ N7 )
% 0.31/0.47 => ( ord_less_eq @ nat @ M5 @ N7 ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_leD1
% 0.31/0.47 thf(fact_226_add__leD2,axiom,
% 0.31/0.47 ! [M5: nat,K: nat,N7: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M5 @ K ) @ N7 )
% 0.31/0.47 => ( ord_less_eq @ nat @ K @ N7 ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_leD2
% 0.31/0.47 thf(fact_227_le__Suc__ex,axiom,
% 0.31/0.47 ! [K: nat,L: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ K @ L )
% 0.31/0.47 => ? [N2: nat] :
% 0.31/0.47 ( L
% 0.31/0.47 = ( plus_plus @ nat @ K @ N2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_Suc_ex
% 0.31/0.47 thf(fact_228_add__le__mono,axiom,
% 0.31/0.47 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ I @ J )
% 0.31/0.47 => ( ( ord_less_eq @ nat @ K @ L )
% 0.31/0.47 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_le_mono
% 0.31/0.47 thf(fact_229_add__le__mono1,axiom,
% 0.31/0.47 ! [I: nat,J: nat,K: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ I @ J )
% 0.31/0.47 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_le_mono1
% 0.31/0.47 thf(fact_230_trans__le__add1,axiom,
% 0.31/0.47 ! [I: nat,J: nat,M5: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ I @ J )
% 0.31/0.47 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M5 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % trans_le_add1
% 0.31/0.47 thf(fact_231_trans__le__add2,axiom,
% 0.31/0.47 ! [I: nat,J: nat,M5: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ I @ J )
% 0.31/0.47 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M5 @ J ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % trans_le_add2
% 0.31/0.47 thf(fact_232_nat__le__iff__add,axiom,
% 0.31/0.47 ( ( ord_less_eq @ nat )
% 0.31/0.47 = ( ^ [M4: nat,N: nat] :
% 0.31/0.47 ? [K2: nat] :
% 0.31/0.47 ( N
% 0.31/0.47 = ( plus_plus @ nat @ M4 @ K2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % nat_le_iff_add
% 0.31/0.47 thf(fact_233_nat__leq__as__int,axiom,
% 0.31/0.47 ( ( ord_less_eq @ nat )
% 0.31/0.47 = ( ^ [A5: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % nat_leq_as_int
% 0.31/0.47 thf(fact_234_plus__ereal_Osimps_I1_J,axiom,
% 0.31/0.47 ! [R: real,P3: real] :
% 0.31/0.47 ( ( plus_plus @ extended_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P3 ) )
% 0.31/0.47 = ( extended_ereal2 @ ( plus_plus @ real @ R @ P3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % plus_ereal.simps(1)
% 0.31/0.47 thf(fact_235_filterlim__at__bot__mirror,axiom,
% 0.31/0.47 ! [A: $tType,F2: real > A,F: filter @ A] :
% 0.31/0.47 ( ( filterlim @ real @ A @ F2 @ F @ ( at_bot @ real ) )
% 0.31/0.47 = ( filterlim @ real @ A
% 0.31/0.47 @ ^ [X: real] : ( F2 @ ( uminus_uminus @ real @ X ) )
% 0.31/0.47 @ F
% 0.31/0.47 @ ( at_top @ real ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % filterlim_at_bot_mirror
% 0.31/0.47 thf(fact_236_filterlim__at__top__mirror,axiom,
% 0.31/0.47 ! [A: $tType,F2: real > A,F: filter @ A] :
% 0.31/0.47 ( ( filterlim @ real @ A @ F2 @ F @ ( at_top @ real ) )
% 0.31/0.47 = ( filterlim @ real @ A
% 0.31/0.47 @ ^ [X: real] : ( F2 @ ( uminus_uminus @ real @ X ) )
% 0.31/0.47 @ F
% 0.31/0.47 @ ( at_bot @ real ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % filterlim_at_top_mirror
% 0.31/0.47 thf(fact_237_filterlim__uminus__at__bot__at__top,axiom,
% 0.31/0.47 filterlim @ real @ real @ ( uminus_uminus @ real ) @ ( at_bot @ real ) @ ( at_top @ real ) ).
% 0.31/0.47
% 0.31/0.47 % filterlim_uminus_at_bot_at_top
% 0.31/0.47 thf(fact_238_filterlim__uminus__at__top__at__bot,axiom,
% 0.31/0.47 filterlim @ real @ real @ ( uminus_uminus @ real ) @ ( at_top @ real ) @ ( at_bot @ real ) ).
% 0.31/0.47
% 0.31/0.47 % filterlim_uminus_at_top_at_bot
% 0.31/0.47 thf(fact_239_liminf__shift__k,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( comple187826305attice @ A )
% 0.31/0.47 => ! [U: nat > A,K: nat] :
% 0.31/0.47 ( ( liminf_Liminf @ nat @ A @ ( at_top @ nat )
% 0.31/0.47 @ ^ [N: nat] : ( U @ ( plus_plus @ nat @ N @ K ) ) )
% 0.31/0.47 = ( liminf_Liminf @ nat @ A @ ( at_top @ nat ) @ U ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % liminf_shift_k
% 0.31/0.47 thf(fact_240_Nat_Oex__has__greatest__nat,axiom,
% 0.31/0.47 ! [P: nat > $o,K: nat,B2: nat] :
% 0.31/0.47 ( ( P @ K )
% 0.31/0.47 => ( ! [Y3: nat] :
% 0.31/0.47 ( ( P @ Y3 )
% 0.31/0.47 => ( ord_less_eq @ nat @ Y3 @ B2 ) )
% 0.31/0.47 => ? [X5: nat] :
% 0.31/0.47 ( ( P @ X5 )
% 0.31/0.47 & ! [Y6: nat] :
% 0.31/0.47 ( ( P @ Y6 )
% 0.31/0.47 => ( ord_less_eq @ nat @ Y6 @ X5 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Nat.ex_has_greatest_nat
% 0.31/0.47 thf(fact_241_nat__le__linear,axiom,
% 0.31/0.47 ! [M5: nat,N7: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ M5 @ N7 )
% 0.31/0.47 | ( ord_less_eq @ nat @ N7 @ M5 ) ) ).
% 0.31/0.47
% 0.31/0.47 % nat_le_linear
% 0.31/0.47 thf(fact_242_le__antisym,axiom,
% 0.31/0.47 ! [M5: nat,N7: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ M5 @ N7 )
% 0.31/0.47 => ( ( ord_less_eq @ nat @ N7 @ M5 )
% 0.31/0.47 => ( M5 = N7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_antisym
% 0.31/0.47 thf(fact_243_eq__imp__le,axiom,
% 0.31/0.47 ! [M5: nat,N7: nat] :
% 0.31/0.47 ( ( M5 = N7 )
% 0.31/0.47 => ( ord_less_eq @ nat @ M5 @ N7 ) ) ).
% 0.31/0.47
% 0.31/0.47 % eq_imp_le
% 0.31/0.47 thf(fact_244_le__trans,axiom,
% 0.31/0.47 ! [I: nat,J: nat,K: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ I @ J )
% 0.31/0.47 => ( ( ord_less_eq @ nat @ J @ K )
% 0.31/0.47 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_trans
% 0.31/0.47 thf(fact_245_le__refl,axiom,
% 0.31/0.47 ! [N7: nat] : ( ord_less_eq @ nat @ N7 @ N7 ) ).
% 0.31/0.47
% 0.31/0.47 % le_refl
% 0.31/0.47 thf(fact_246_filterlim__uminus__at__bot,axiom,
% 0.31/0.47 ! [A: $tType,F2: A > real,F: filter @ A] :
% 0.31/0.47 ( ( filterlim @ A @ real @ F2 @ ( at_bot @ real ) @ F )
% 0.31/0.47 = ( filterlim @ A @ real
% 0.31/0.47 @ ^ [X: A] : ( uminus_uminus @ real @ ( F2 @ X ) )
% 0.31/0.47 @ ( at_top @ real )
% 0.31/0.47 @ F ) ) ).
% 0.31/0.47
% 0.31/0.47 % filterlim_uminus_at_bot
% 0.31/0.47 thf(fact_247_filterlim__uminus__at__top,axiom,
% 0.31/0.47 ! [A: $tType,F2: A > real,F: filter @ A] :
% 0.31/0.47 ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F )
% 0.31/0.47 = ( filterlim @ A @ real
% 0.31/0.47 @ ^ [X: A] : ( uminus_uminus @ real @ ( F2 @ X ) )
% 0.31/0.47 @ ( at_bot @ real )
% 0.31/0.47 @ F ) ) ).
% 0.31/0.47
% 0.31/0.47 % filterlim_uminus_at_top
% 0.31/0.47 thf(fact_248_LIMSEQ__ignore__initial__segment,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( topolo503727757_space @ A )
% 0.31/0.47 => ! [F2: nat > A,A2: A,K: nat] :
% 0.31/0.47 ( ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
% 0.31/0.47 => ( filterlim @ nat @ A
% 0.31/0.47 @ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) )
% 0.31/0.47 @ ( topolo1920029354e_nhds @ A @ A2 )
% 0.31/0.47 @ ( at_top @ nat ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % LIMSEQ_ignore_initial_segment
% 0.31/0.47 thf(fact_249_LIMSEQ__offset,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( topolo503727757_space @ A )
% 0.31/0.47 => ! [F2: nat > A,K: nat,A2: A] :
% 0.31/0.47 ( ( filterlim @ nat @ A
% 0.31/0.47 @ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) )
% 0.31/0.47 @ ( topolo1920029354e_nhds @ A @ A2 )
% 0.31/0.47 @ ( at_top @ nat ) )
% 0.31/0.47 => ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % LIMSEQ_offset
% 0.31/0.47 thf(fact_250_plus__ereal_Oelims,axiom,
% 0.31/0.47 ! [X2: extended_ereal,Xa3: extended_ereal,Y: extended_ereal] :
% 0.31/0.47 ( ( ( plus_plus @ extended_ereal @ X2 @ Xa3 )
% 0.31/0.47 = Y )
% 0.31/0.47 => ( ! [R2: real] :
% 0.31/0.47 ( ( X2
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ! [P4: real] :
% 0.31/0.47 ( ( Xa3
% 0.31/0.47 = ( extended_ereal2 @ P4 ) )
% 0.31/0.47 => ( Y
% 0.31/0.47 != ( extended_ereal2 @ ( plus_plus @ real @ R2 @ P4 ) ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( Y
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( ( Xa3
% 0.31/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 => ( Y
% 0.31/0.47 != ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( ? [R2: real] :
% 0.31/0.47 ( X2
% 0.31/0.47 = ( extended_ereal2 @ R2 ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( Y
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.31/0.47 => ( ( ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ? [P4: real] :
% 0.31/0.47 ( Xa3
% 0.31/0.47 = ( extended_ereal2 @ P4 ) )
% 0.31/0.47 => ( Y
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.31/0.47 => ~ ( ( X2
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( ( Xa3
% 0.31/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.31/0.47 => ( Y
% 0.31/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % plus_ereal.elims
% 0.31/0.47 thf(fact_251_tendsto__PInfty__eq__at__top,axiom,
% 0.31/0.47 ! [A: $tType,F2: A > real,F: filter @ A] :
% 0.31/0.47 ( ( filterlim @ A @ extended_ereal
% 0.31/0.47 @ ^ [Z5: A] : ( extended_ereal2 @ ( F2 @ Z5 ) )
% 0.31/0.47 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
% 0.31/0.47 @ F )
% 0.31/0.47 = ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F ) ) ).
% 0.31/0.47
% 0.31/0.47 % tendsto_PInfty_eq_at_top
% 0.31/0.47 thf(fact_252_tendsto__at__topI__sequentially,axiom,
% 0.31/0.47 ! [B: $tType] :
% 0.31/0.47 ( ( topolo2135403230pology @ B )
% 0.31/0.47 => ! [F2: real > B,Y: B] :
% 0.31/0.47 ( ! [X8: nat > real] :
% 0.31/0.47 ( ( filterlim @ nat @ real @ X8 @ ( at_top @ real ) @ ( at_top @ nat ) )
% 0.31/0.47 => ( filterlim @ nat @ B
% 0.31/0.47 @ ^ [N: nat] : ( F2 @ ( X8 @ N ) )
% 0.31/0.47 @ ( topolo1920029354e_nhds @ B @ Y )
% 0.31/0.47 @ ( at_top @ nat ) ) )
% 0.31/0.47 => ( filterlim @ real @ B @ F2 @ ( topolo1920029354e_nhds @ B @ Y ) @ ( at_top @ real ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % tendsto_at_topI_sequentially
% 0.31/0.47 thf(fact_253_filterlim__real__sequentially,axiom,
% 0.31/0.47 filterlim @ nat @ real @ ( semiring_1_of_nat @ real ) @ ( at_top @ real ) @ ( at_top @ nat ) ).
% 0.31/0.47
% 0.31/0.47 % filterlim_real_sequentially
% 0.31/0.47 thf(fact_254_int__ops_I5_J,axiom,
% 0.31/0.47 ! [A2: nat,B2: nat] :
% 0.31/0.47 ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A2 @ B2 ) )
% 0.31/0.47 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % int_ops(5)
% 0.31/0.47 thf(fact_255_nat__int__comparison_I1_J,axiom,
% 0.31/0.47 ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
% 0.31/0.47 = ( ^ [A5: nat,B3: nat] :
% 0.31/0.47 ( ( semiring_1_of_nat @ int @ A5 )
% 0.31/0.47 = ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % nat_int_comparison(1)
% 0.31/0.47
% 0.31/0.47 % Subclasses (5)
% 0.31/0.47 thf(subcl_Real__Vector__Spaces_Ometric__space___HOL_Otype,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( real_V2090557954_space @ A )
% 0.31/0.47 => ( type @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Ot2__space,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( real_V2090557954_space @ A )
% 0.31/0.47 => ( topological_t2_space @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Ouniform__space,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( real_V2090557954_space @ A )
% 0.31/0.47 => ( topolo47006728_space @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Otopological__space,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( real_V2090557954_space @ A )
% 0.31/0.47 => ( topolo503727757_space @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Ofirst__countable__topology,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( real_V2090557954_space @ A )
% 0.31/0.47 => ( topolo2135403230pology @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 % Type constructors (119)
% 0.31/0.47 thf(tcon_HOL_Obool___Countable_Ocountable,axiom,
% 0.31/0.47 countable @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Countable_Ocountable_1,axiom,
% 0.31/0.47 ! [A7: $tType] :
% 0.31/0.47 ( ( finite_finite @ A7 )
% 0.31/0.47 => ( countable @ ( set @ A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Countable_Ocountable_2,axiom,
% 0.31/0.47 countable @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Countable_Ocountable_3,axiom,
% 0.31/0.47 countable @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Countable_Ocountable_4,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A7 )
% 0.31/0.47 & ( countable @ A8 ) )
% 0.31/0.47 => ( countable @ ( A7 > A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A7 )
% 0.31/0.47 & ( finite_finite @ A8 ) )
% 0.31/0.47 => ( finite_finite @ ( A7 > A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
% 0.31/0.47 ! [A7: $tType] :
% 0.31/0.47 ( ( finite_finite @ A7 )
% 0.31/0.47 => ( finite_finite @ ( set @ A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Finite__Set_Ofinite_6,axiom,
% 0.31/0.47 finite_finite @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Topological__Spaces_Ofirst__countable__topology,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( ( countable @ A7 )
% 0.31/0.47 & ( topolo2135403230pology @ A8 ) )
% 0.31/0.47 => ( topolo2135403230pology @ ( A7 > A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( topolo503727757_space @ A8 )
% 0.31/0.47 => ( topolo503727757_space @ ( A7 > A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( comple187826305attice @ A8 )
% 0.31/0.47 => ( comple187826305attice @ ( A7 > A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( boolean_algebra @ A8 )
% 0.31/0.47 => ( boolean_algebra @ ( A7 > A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( preorder @ A8 )
% 0.31/0.47 => ( preorder @ ( A7 > A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( order @ A8 )
% 0.31/0.47 => ( order @ ( A7 > A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( ord @ A8 )
% 0.31/0.47 => ( ord @ ( A7 > A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Groups_Ouminus,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( uminus @ A8 )
% 0.31/0.47 => ( uminus @ ( A7 > A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.31/0.47 ordere236663937imp_le @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Topological__Spaces_Otopological__space_7,axiom,
% 0.31/0.47 topolo503727757_space @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
% 0.31/0.47 topolo2117631714pology @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
% 0.31/0.47 topolo259154727pology @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 0.31/0.47 linord1659791738miring @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
% 0.31/0.47 ordere779506340up_add @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom,
% 0.31/0.47 topolo1314133330id_add @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
% 0.31/0.47 topological_t2_space @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 0.31/0.47 ordered_ab_group_add @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
% 0.31/0.47 cancel_semigroup_add @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
% 0.31/0.47 ab_semigroup_add @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
% 0.31/0.47 comm_monoid_add @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
% 0.31/0.47 semigroup_add @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
% 0.31/0.47 ab_group_add @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Orderings_Opreorder_8,axiom,
% 0.31/0.47 preorder @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.31/0.47 linorder @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 0.31/0.47 group_add @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Orderings_Oorder_9,axiom,
% 0.31/0.47 order @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Orderings_Oord_10,axiom,
% 0.31/0.47 ord @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Int_Oint___Groups_Ouminus_11,axiom,
% 0.31/0.47 uminus @ int ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Topological__Spaces_Ofirst__countable__topology_12,axiom,
% 0.31/0.47 topolo2135403230pology @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_13,axiom,
% 0.31/0.47 ordere236663937imp_le @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.31/0.47 canoni770627133id_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_14,axiom,
% 0.31/0.47 topolo503727757_space @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_15,axiom,
% 0.31/0.47 topolo2117631714pology @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_16,axiom,
% 0.31/0.47 topolo259154727pology @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_17,axiom,
% 0.31/0.47 linord1659791738miring @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_18,axiom,
% 0.31/0.47 ordere779506340up_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_19,axiom,
% 0.31/0.47 topolo1314133330id_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_20,axiom,
% 0.31/0.47 topological_t2_space @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_21,axiom,
% 0.31/0.47 cancel_semigroup_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_22,axiom,
% 0.31/0.47 ab_semigroup_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_23,axiom,
% 0.31/0.47 comm_monoid_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Osemigroup__add_24,axiom,
% 0.31/0.47 semigroup_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_25,axiom,
% 0.31/0.47 preorder @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Olinorder_26,axiom,
% 0.31/0.47 linorder @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_27,axiom,
% 0.31/0.47 order @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Oord_28,axiom,
% 0.31/0.47 ord @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_29,axiom,
% 0.31/0.47 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_30,axiom,
% 0.31/0.47 ! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Groups_Oab__semigroup__add_31,axiom,
% 0.31/0.47 ! [A7: $tType] :
% 0.31/0.47 ( ( ab_semigroup_add @ A7 )
% 0.31/0.47 => ( ab_semigroup_add @ ( set @ A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Groups_Ocomm__monoid__add_32,axiom,
% 0.31/0.47 ! [A7: $tType] :
% 0.31/0.47 ( ( comm_monoid_add @ A7 )
% 0.31/0.47 => ( comm_monoid_add @ ( set @ A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Groups_Osemigroup__add_33,axiom,
% 0.31/0.47 ! [A7: $tType] :
% 0.31/0.47 ( ( semigroup_add @ A7 )
% 0.31/0.47 => ( semigroup_add @ ( set @ A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_34,axiom,
% 0.31/0.47 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Oorder_35,axiom,
% 0.31/0.47 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Oord_36,axiom,
% 0.31/0.47 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Groups_Ouminus_37,axiom,
% 0.31/0.47 ! [A7: $tType] : ( uminus @ ( set @ A7 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_38,axiom,
% 0.31/0.47 topolo503727757_space @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_39,axiom,
% 0.31/0.47 topolo2117631714pology @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_40,axiom,
% 0.31/0.47 comple187826305attice @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_41,axiom,
% 0.31/0.47 topolo259154727pology @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_42,axiom,
% 0.31/0.47 topological_t2_space @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_43,axiom,
% 0.31/0.47 boolean_algebra @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_44,axiom,
% 0.31/0.47 preorder @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder_45,axiom,
% 0.31/0.47 linorder @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_46,axiom,
% 0.31/0.47 order @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Oord_47,axiom,
% 0.31/0.47 ord @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Groups_Ouminus_48,axiom,
% 0.31/0.47 uminus @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Topological__Spaces_Ofirst__countable__topology_49,axiom,
% 0.31/0.47 topolo2135403230pology @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_50,axiom,
% 0.31/0.47 ordere236663937imp_le @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
% 0.31/0.47 real_V55928688vector @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_51,axiom,
% 0.31/0.47 topolo503727757_space @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_52,axiom,
% 0.31/0.47 topolo2117631714pology @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_53,axiom,
% 0.31/0.47 topolo259154727pology @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_54,axiom,
% 0.31/0.47 linord1659791738miring @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
% 0.31/0.47 topolo47006728_space @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
% 0.31/0.47 real_V2090557954_space @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_55,axiom,
% 0.31/0.47 ordere779506340up_add @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Limits_Otopological__monoid__add_56,axiom,
% 0.31/0.47 topolo1314133330id_add @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
% 0.31/0.47 topolo799126099up_add @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_57,axiom,
% 0.31/0.47 topological_t2_space @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_58,axiom,
% 0.31/0.47 ordered_ab_group_add @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_59,axiom,
% 0.31/0.47 cancel_semigroup_add @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_60,axiom,
% 0.31/0.47 ab_semigroup_add @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_61,axiom,
% 0.31/0.47 comm_monoid_add @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Groups_Osemigroup__add_62,axiom,
% 0.31/0.47 semigroup_add @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Groups_Oab__group__add_63,axiom,
% 0.31/0.47 ab_group_add @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Orderings_Opreorder_64,axiom,
% 0.31/0.47 preorder @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Orderings_Olinorder_65,axiom,
% 0.31/0.47 linorder @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Groups_Ogroup__add_66,axiom,
% 0.31/0.47 group_add @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Orderings_Oorder_67,axiom,
% 0.31/0.47 order @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Orderings_Oord_68,axiom,
% 0.31/0.47 ord @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Groups_Ouminus_69,axiom,
% 0.31/0.47 uminus @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_70,axiom,
% 0.31/0.47 ! [A7: $tType] : ( comple187826305attice @ ( filter @ A7 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Filter_Ofilter___Orderings_Opreorder_71,axiom,
% 0.31/0.47 ! [A7: $tType] : ( preorder @ ( filter @ A7 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Filter_Ofilter___Orderings_Oorder_72,axiom,
% 0.31/0.47 ! [A7: $tType] : ( order @ ( filter @ A7 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Filter_Ofilter___Orderings_Oord_73,axiom,
% 0.31/0.47 ! [A7: $tType] : ( ord @ ( filter @ A7 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Ofirst__countable__topology_74,axiom,
% 0.31/0.47 topolo2135403230pology @ extended_ereal ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Otopological__space_75,axiom,
% 0.31/0.47 topolo503727757_space @ extended_ereal ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Olinorder__topology_76,axiom,
% 0.31/0.47 topolo2117631714pology @ extended_ereal ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Real_Oereal___Complete__Lattices_Ocomplete__linorder,axiom,
% 0.31/0.47 comple1035589618norder @ extended_ereal ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Real_Oereal___Complete__Lattices_Ocomplete__lattice_77,axiom,
% 0.31/0.47 comple187826305attice @ extended_ereal ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Oorder__topology_78,axiom,
% 0.31/0.47 topolo259154727pology @ extended_ereal ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Real_Oereal___Groups_Oordered__ab__semigroup__add_79,axiom,
% 0.31/0.47 ordere779506340up_add @ extended_ereal ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Ot2__space_80,axiom,
% 0.31/0.47 topological_t2_space @ extended_ereal ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Real_Oereal___Groups_Oab__semigroup__add_81,axiom,
% 0.31/0.47 ab_semigroup_add @ extended_ereal ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Real_Oereal___Groups_Ocomm__monoid__add_82,axiom,
% 0.31/0.47 comm_monoid_add @ extended_ereal ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Real_Oereal___Groups_Osemigroup__add_83,axiom,
% 0.31/0.47 semigroup_add @ extended_ereal ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Real_Oereal___Orderings_Opreorder_84,axiom,
% 0.31/0.47 preorder @ extended_ereal ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Real_Oereal___Orderings_Olinorder_85,axiom,
% 0.31/0.47 linorder @ extended_ereal ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Real_Oereal___Orderings_Oorder_86,axiom,
% 0.31/0.47 order @ extended_ereal ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Real_Oereal___Orderings_Oord_87,axiom,
% 0.31/0.47 ord @ extended_ereal ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Real_Oereal___Groups_Ouminus_88,axiom,
% 0.31/0.47 uminus @ extended_ereal ).
% 0.31/0.47
% 0.31/0.47 % Free types (1)
% 0.31/0.47 thf(tfree_0,hypothesis,
% 0.31/0.47 real_V2090557954_space @ a ).
% 0.31/0.47
% 0.31/0.47 % Conjectures (1)
% 0.31/0.47 thf(conj_0,conjecture,
% 0.31/0.47 ( filterlim @ nat @ extended_ereal
% 0.31/0.47 @ ^ [I2: nat] : ( uminus_uminus @ extended_ereal @ ( f @ ( x @ I2 ) ) )
% 0.31/0.47 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ a2 ) )
% 0.31/0.47 @ ( at_top @ nat ) ) ).
% 0.31/0.47
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 ------------------------
% 2.88/1.53 ---- Embedded in THF ---
% 2.88/1.53 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.88/1.53 %%% Generated on Tue Feb 24 08:26:18 EST 2026
% 2.88/1.53 %%% using '$$dhol' embedding, version 1.3.0.
% 2.88/1.53 %%% Logic specification used:
% 2.88/1.53 %%% thf(spec, logic, $$dhol).
% 2.88/1.53
% 2.88/1.53 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.WRkRsD2zqq/DTF2DTF29324.p
% 2.88/1.53 thf(ty_t_Extended__Real_Oereal, type, extended_ereal: $tType).
% 2.88/1.53 thf(ty_t_Filter_Ofilter, type, filter: ($tType > $tType)).
% 2.88/1.53 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.88/1.53 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.88/1.53 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.88/1.53 thf(ty_t_Int_Oint, type, int: $tType).
% 2.88/1.53 thf(ty_tf_a, type, a: $tType).
% 2.88/1.53 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Countable_Ocountable, type, countable: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Groups_Ouminus, type, uminus: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Groups_Oab__group__add, type, ab_group_add: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Lattices_Oboolean__algebra, type, boolean_algebra: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Topological__Spaces_Ot2__space, type, topological_t2_space: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Limits_Otopological__group__add, type, topolo799126099up_add: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Limits_Otopological__monoid__add, type, topolo1314133330id_add: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Real__Vector__Spaces_Ometric__space, type, real_V2090557954_space: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Topological__Spaces_Ouniform__space, type, topolo47006728_space: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Topological__Spaces_Oorder__topology, type, topolo259154727pology: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple187826305attice: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Complete__Lattices_Ocomplete__linorder, type, comple1035589618norder: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Topological__Spaces_Olinorder__topology, type, topolo2117631714pology: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Topological__Spaces_Otopological__space, type, topolo503727757_space: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector, type, real_V55928688vector: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_cl_Topological__Spaces_Ofirst__countable__topology, type, topolo2135403230pology: (!> [A:$tType]: ($o))).
% 2.88/1.53 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity, type, extend1396239628finity: (!> [A:$tType]: (A))).
% 2.88/1.53 thf(sy_c_Extended__Real_Oereal_Ocase__ereal, type, extended_case_ereal: (!> [A:$tType]: (((real > A) > (A > (A > (extended_ereal > A))))))).
% 2.88/1.53 thf(sy_c_Extended__Real_Oereal_Oereal, type, extended_ereal2: (real > extended_ereal)).
% 2.88/1.53 thf(sy_c_Filter_Oat__bot, type, at_bot: (!> [A:$tType]: ((filter @ A)))).
% 2.88/1.53 thf(sy_c_Filter_Oat__top, type, at_top: (!> [A:$tType]: ((filter @ A)))).
% 2.88/1.53 thf(sy_c_Filter_Ofilterlim, type, filterlim: (!> [A:$tType,B:$tType]: (((A > B) > ((filter @ B) > ((filter @ A) > $o)))))).
% 2.88/1.53 thf(sy_c_Fun_Ocomp, type, comp: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > ((A > B) > (A > C)))))).
% 2.88/1.53 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.88/1.53 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 2.88/1.53 thf(sy_c_HOL_OEx1, type, ex1: (!> [A:$tType]: (((A > $o) > $o)))).
% 2.88/1.53 thf(sy_c_HOL_OUniq, type, uniq: (!> [A:$tType]: (((A > $o) > $o)))).
% 2.88/1.53 thf(sy_c_Liminf__Limsup_OLiminf, type, liminf_Liminf: (!> [A:$tType,B:$tType]: (((filter @ A) > ((A > B) > B))))).
% 2.88/1.53 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc__at, type, lower_582600101lsc_at: (!> [A:$tType,B:$tType]: ((A > ((A > B) > $o))))).
% 2.88/1.53 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Ousc__at, type, lower_708585518usc_at: (!> [A:$tType,B:$tType]: ((A > ((A > B) > $o))))).
% 2.88/1.53 thf(sy_c_Nat_Osemiring__1__class_Oof__nat, type, semiring_1_of_nat: (!> [A:$tType]: ((nat > A)))).
% 2.88/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.88/1.53 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.88/1.53 thf(sy_c_Topological__Spaces_Omonoseq, type, topological_monoseq: (!> [A:$tType]: (((nat > A) > $o)))).
% 2.88/1.53 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds, type, topolo1920029354e_nhds: (!> [A:$tType]: ((A > (filter @ A))))).
% 2.88/1.53 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocauchy__filter, type, topolo1334162427filter: (!> [A:$tType]: (((filter @ A) > $o)))).
% 2.88/1.53 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.88/1.53 thf(sy_v_A____, type, a2: extended_ereal).
% 2.88/1.53 thf(sy_v_f, type, f: (a > extended_ereal)).
% 2.88/1.53 thf(sy_v_x0, type, x0: a).
% 2.88/1.53 thf(sy_v_x____, type, x: (nat > a)).
% 2.88/1.53 thf(fact_0_x__def_I1_J, axiom, (((((filterlim @ nat) @ a) @ x) @ ((topolo1920029354e_nhds @ a) @ x0)) @ (at_top @ nat))).
% 2.88/1.53 thf(fact_1_x__def_I2_J, axiom, (((((filterlim @ nat) @ extended_ereal) @ (((((comp @ a) @ extended_ereal) @ nat) @ f) @ x)) @ ((topolo1920029354e_nhds @ extended_ereal) @ a2)) @ (at_top @ nat))).
% 2.88/1.53 thf(fact_2__092_060open_062_092_060And_062F_O_A_I_If_A_092_060circ_062_Ax_J_A_092_060longlongrightarrow_062_AA_J_AF_A_092_060Longrightarrow_062_A_I_I_092_060lambda_062xa_O_A_N_A_If_A_092_060circ_062_Ax_J_Axa_J_A_092_060longlongrightarrow_062_A_N_AA_J_AF_092_060close_062, axiom, (! [F:(filter @ nat)]: (((((((filterlim @ nat) @ extended_ereal) @ (((((comp @ a) @ extended_ereal) @ nat) @ f) @ x)) @ ((topolo1920029354e_nhds @ extended_ereal) @ a2)) @ F) => (((((filterlim @ nat) @ extended_ereal) @ (^ [X:nat]: (((uminus_uminus @ extended_ereal) @ ((((((comp @ a) @ extended_ereal) @ nat) @ f) @ x) @ X))))) @ ((topolo1920029354e_nhds @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ a2))) @ F))))).
% 2.88/1.53 thf(fact_3_lsc, axiom, ((((lower_582600101lsc_at @ a) @ extended_ereal) @ x0) @ (^ [X:a]: (((uminus_uminus @ extended_ereal) @ (f @ X)))))).
% 2.88/1.53 thf(fact_4_tendsto__uminus__ereal, axiom, (! [A:$tType,F2:(A > extended_ereal),X2:extended_ereal,F:(filter @ A)]: (((((((filterlim @ A) @ extended_ereal) @ F2) @ ((topolo1920029354e_nhds @ extended_ereal) @ X2)) @ F) => (((((filterlim @ A) @ extended_ereal) @ (^ [X:A]: (((uminus_uminus @ extended_ereal) @ (F2 @ X))))) @ ((topolo1920029354e_nhds @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ X2))) @ F))))).
% 2.88/1.53 thf(fact_5_tendsto__const, axiom, (! [A:$tType,B:$tType]: (((topolo503727757_space @ A) => (! [K:A,F:(filter @ B)]: ((((((filterlim @ B) @ A) @ (^ [X:B]: (K))) @ ((topolo1920029354e_nhds @ A) @ K)) @ F))))))).
% 2.88/1.53 thf(fact_6_ereal__Lim__uminus, axiom, (! [A:$tType,F2:(A > extended_ereal),F0:extended_ereal,Net:(filter @ A)]: (((((((filterlim @ A) @ extended_ereal) @ F2) @ ((topolo1920029354e_nhds @ extended_ereal) @ F0)) @ Net) = (((((filterlim @ A) @ extended_ereal) @ (^ [X:A]: (((uminus_uminus @ extended_ereal) @ (F2 @ X))))) @ ((topolo1920029354e_nhds @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ F0))) @ Net))))).
% 2.88/1.53 thf(fact_7_LIMSEQ__const__iff, axiom, (! [A:$tType]: (((topological_t2_space @ A) => (! [K:A,L:A]: (((((((filterlim @ nat) @ A) @ (^ [N:nat]: (K))) @ ((topolo1920029354e_nhds @ A) @ L)) @ (at_top @ nat)) = (K = L)))))))).
% 2.88/1.53 thf(fact_8_tendsto__minus__cancel, axiom, (! [A:$tType,B:$tType]: (((topolo799126099up_add @ A) => (! [F2:(B > A),A2:A,F:(filter @ B)]: (((((((filterlim @ B) @ A) @ (^ [X:B]: (((uminus_uminus @ A) @ (F2 @ X))))) @ ((topolo1920029354e_nhds @ A) @ ((uminus_uminus @ A) @ A2))) @ F) => (((((filterlim @ B) @ A) @ F2) @ ((topolo1920029354e_nhds @ A) @ A2)) @ F)))))))).
% 2.88/1.53 thf(fact_9_tendsto__minus__cancel__left, axiom, (! [B:$tType,A:$tType]: (((topolo799126099up_add @ B) => (! [F2:(A > B),Y:B,F:(filter @ A)]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo1920029354e_nhds @ B) @ ((uminus_uminus @ B) @ Y))) @ F) = (((((filterlim @ A) @ B) @ (^ [X:A]: (((uminus_uminus @ B) @ (F2 @ X))))) @ ((topolo1920029354e_nhds @ B) @ Y)) @ F)))))))).
% 2.88/1.53 thf(fact_10_tendsto__minus, axiom, (! [A:$tType,B:$tType]: (((topolo799126099up_add @ A) => (! [F2:(B > A),A2:A,F:(filter @ B)]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo1920029354e_nhds @ A) @ A2)) @ F) => (((((filterlim @ B) @ A) @ (^ [X:B]: (((uminus_uminus @ A) @ (F2 @ X))))) @ ((topolo1920029354e_nhds @ A) @ ((uminus_uminus @ A) @ A2))) @ F)))))))).
% 2.88/1.53 thf(fact_11_LIMSEQ__unique, axiom, (! [A:$tType]: (((topological_t2_space @ A) => (! [X3:(nat > A),A2:A,B2:A]: (((((((filterlim @ nat) @ A) @ X3) @ ((topolo1920029354e_nhds @ A) @ A2)) @ (at_top @ nat)) => ((((((filterlim @ nat) @ A) @ X3) @ ((topolo1920029354e_nhds @ A) @ B2)) @ (at_top @ nat)) => (A2 = B2))))))))).
% 2.88/1.53 thf(fact_12_tendsto__uminus__nhds, axiom, (! [A:$tType]: (((topolo799126099up_add @ A) => (! [A2:A]: ((((((filterlim @ A) @ A) @ (uminus_uminus @ A)) @ ((topolo1920029354e_nhds @ A) @ ((uminus_uminus @ A) @ A2))) @ ((topolo1920029354e_nhds @ A) @ A2)))))))).
% 2.88/1.53 thf(fact_13_ereal__uminus__eq__iff, axiom, (! [A2:extended_ereal,B2:extended_ereal]: (((((uminus_uminus @ extended_ereal) @ A2) = ((uminus_uminus @ extended_ereal) @ B2)) = (A2 = B2))))).
% 2.88/1.53 thf(fact_14_ereal__uminus__uminus, axiom, (! [A2:extended_ereal]: ((((uminus_uminus @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ A2)) = A2)))).
% 2.88/1.53 thf(fact_15_verit__minus__simplify_I4_J, axiom, (! [B:$tType]: (((group_add @ B) => (! [B2:B]: ((((uminus_uminus @ B) @ ((uminus_uminus @ B) @ B2)) = B2))))))).
% 2.88/1.53 thf(fact_16_verit__negate__coefficient_I3_J, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((A2 = B2) => (((uminus_uminus @ A) @ A2) = ((uminus_uminus @ A) @ B2))))))))).
% 2.88/1.53 thf(fact_17_ereal__uminus__eq__reorder, axiom, (! [A2:extended_ereal,B2:extended_ereal]: (((((uminus_uminus @ extended_ereal) @ A2) = B2) = (A2 = ((uminus_uminus @ extended_ereal) @ B2)))))).
% 2.88/1.53 thf(fact_18_tendsto__eq__rhs, axiom, (! [A:$tType,B:$tType]: (((topolo503727757_space @ A) => (! [F2:(B > A),X2:A,F:(filter @ B),Y:A]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo1920029354e_nhds @ A) @ X2)) @ F) => ((X2 = Y) => (((((filterlim @ B) @ A) @ F2) @ ((topolo1920029354e_nhds @ A) @ Y)) @ F))))))))).
% 2.88/1.53 thf(fact_19_tendsto__cong__limit, axiom, (! [B:$tType,A:$tType]: (((topolo503727757_space @ B) => (! [F2:(A > B),L:B,F:(filter @ A),K:B]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo1920029354e_nhds @ B) @ L)) @ F) => ((K = L) => (((((filterlim @ A) @ B) @ F2) @ ((topolo1920029354e_nhds @ B) @ K)) @ F))))))))).
% 2.88/1.53 thf(fact_20_comp__apply, axiom, (! [C:$tType,A:$tType,B:$tType]: (((((comp @ B) @ A) @ C) = (^ [F3:(B > A),G:(C > B),X:C]: ((F3 @ (G @ X)))))))).
% 2.88/1.53 thf(fact_21_compl__eq__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X2:A,Y:A]: (((((uminus_uminus @ A) @ X2) = ((uminus_uminus @ A) @ Y)) = (X2 = Y)))))))).
% 2.88/1.53 thf(fact_22_double__compl, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X2:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ X2)) = X2))))))).
% 2.88/1.53 thf(fact_23_neg__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((uminus_uminus @ A) @ A2) = ((uminus_uminus @ A) @ B2)) = (A2 = B2)))))))).
% 2.88/1.53 thf(fact_24_add_Oinverse__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ A2)) = A2))))))).
% 2.88/1.53 thf(fact_25_uminus__apply, axiom, (! [B:$tType,A:$tType]: (((uminus @ B) => ((uminus_uminus @ (A > B)) = (^ [A3:(A > B),X:A]: (((uminus_uminus @ B) @ (A3 @ X))))))))).
% 2.88/1.53 thf(fact_26_lsc__sequentially__mem, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X0:A,F2:(A > extended_ereal),X2:(nat > A),C2:(nat > extended_ereal),C0:extended_ereal]: ((((((lower_582600101lsc_at @ A) @ extended_ereal) @ X0) @ F2) => ((((((filterlim @ nat) @ A) @ X2) @ ((topolo1920029354e_nhds @ A) @ X0)) @ (at_top @ nat)) => ((((((filterlim @ nat) @ extended_ereal) @ C2) @ ((topolo1920029354e_nhds @ extended_ereal) @ C0)) @ (at_top @ nat)) => ((! [N2:nat]: ((((ord_less_eq @ extended_ereal) @ (F2 @ (X2 @ N2))) @ (C2 @ N2)))) => (((ord_less_eq @ extended_ereal) @ (F2 @ X0)) @ C0))))))))))).
% 2.88/1.53 thf(fact_27_lsc__sequentially__gen, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (((lower_582600101lsc_at @ A) @ extended_ereal) = (^ [X02:A,F3:(A > extended_ereal)]: ((! [X:(nat > A),C3:(nat > extended_ereal),C02:extended_ereal]: ((((((((filterlim @ nat) @ A) @ X) @ ((topolo1920029354e_nhds @ A) @ X02)) @ (at_top @ nat)) & ((((((filterlim @ nat) @ extended_ereal) @ C3) @ ((topolo1920029354e_nhds @ extended_ereal) @ C02)) @ (at_top @ nat)) & (! [N:nat]: ((((ord_less_eq @ extended_ereal) @ (F3 @ (X @ N))) @ (C3 @ N)))))) => (((ord_less_eq @ extended_ereal) @ (F3 @ X02)) @ C02))))))))))).
% 2.88/1.53 thf(fact_28_lsc__at__mem, axiom, (! [A:$tType,B:$tType]: ((((topolo259154727pology @ B) & (topolo503727757_space @ A)) => (! [X0:A,F2:(A > B),X2:(nat > A),A4:B]: ((((((lower_582600101lsc_at @ A) @ B) @ X0) @ F2) => ((((((filterlim @ nat) @ A) @ X2) @ ((topolo1920029354e_nhds @ A) @ X0)) @ (at_top @ nat)) => ((((((filterlim @ nat) @ B) @ (((((comp @ A) @ B) @ nat) @ F2) @ X2)) @ ((topolo1920029354e_nhds @ B) @ A4)) @ (at_top @ nat)) => (((ord_less_eq @ B) @ (F2 @ X0)) @ A4)))))))))).
% 2.88/1.53 thf(fact_29_lsc__at__def, axiom, (! [B:$tType,A:$tType]: ((((topolo503727757_space @ A) & (topolo259154727pology @ B)) => (((lower_582600101lsc_at @ A) @ B) = (^ [X02:A,F3:(A > B)]: ((! [X4:(nat > A),L2:B]: ((((((((filterlim @ nat) @ A) @ X4) @ ((topolo1920029354e_nhds @ A) @ X02)) @ (at_top @ nat)) & (((((filterlim @ nat) @ B) @ (((((comp @ A) @ B) @ nat) @ F3) @ X4)) @ ((topolo1920029354e_nhds @ B) @ L2)) @ (at_top @ nat))) => (((ord_less_eq @ B) @ (F3 @ X02)) @ L2))))))))))).
% 2.88/1.53 thf(fact_30_neg__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.88/1.53 thf(fact_31_compl__le__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X2)) @ ((uminus_uminus @ A) @ Y)) = (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.88/1.53 thf(fact_32_ereal__minus__le__minus, axiom, (! [A2:extended_ereal,B2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ A2)) @ ((uminus_uminus @ extended_ereal) @ B2)) = (((ord_less_eq @ extended_ereal) @ B2) @ A2))))).
% 2.88/1.53 thf(fact_33_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((A2 = B2) | ((~ ((((ord_less_eq @ A) @ A2) @ B2))) | (~ ((((ord_less_eq @ A) @ B2) @ A2))))))))))).
% 2.88/1.53 thf(fact_34_ereal__complete__Inf, axiom, (! [S:(set @ extended_ereal)]: ((? [X5:extended_ereal]: (((! [Xa:extended_ereal]: (((((member @ extended_ereal) @ Xa) @ S) => (((ord_less_eq @ extended_ereal) @ X5) @ Xa)))) & (! [Z:extended_ereal]: (((! [Xa2:extended_ereal]: (((((member @ extended_ereal) @ Xa2) @ S) => (((ord_less_eq @ extended_ereal) @ Z) @ Xa2)))) => (((ord_less_eq @ extended_ereal) @ Z) @ X5)))))))))).
% 2.88/1.53 thf(fact_35_ereal__complete__Sup, axiom, (! [S:(set @ extended_ereal)]: ((? [X5:extended_ereal]: (((! [Xa:extended_ereal]: (((((member @ extended_ereal) @ Xa) @ S) => (((ord_less_eq @ extended_ereal) @ Xa) @ X5)))) & (! [Z:extended_ereal]: (((! [Xa2:extended_ereal]: (((((member @ extended_ereal) @ Xa2) @ S) => (((ord_less_eq @ extended_ereal) @ Xa2) @ Z)))) => (((ord_less_eq @ extended_ereal) @ X5) @ Z)))))))))).
% 2.88/1.53 thf(fact_36_compl__mono, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y)) @ ((uminus_uminus @ A) @ X2))))))))).
% 2.88/1.53 thf(fact_37_compl__le__swap1, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ ((uminus_uminus @ A) @ X2)) => (((ord_less_eq @ A) @ X2) @ ((uminus_uminus @ A) @ Y))))))))).
% 2.88/1.53 thf(fact_38_compl__le__swap2, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y)) @ X2) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X2)) @ Y)))))))).
% 2.88/1.53 thf(fact_39_le__minus__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ B2)) = (((ord_less_eq @ A) @ B2) @ ((uminus_uminus @ A) @ A2))))))))).
% 2.88/1.53 thf(fact_40_minus__le__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ A2)))))))).
% 2.88/1.53 thf(fact_41_le__imp__neg__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2))))))))).
% 2.88/1.53 thf(fact_42_ereal__uminus__le__reorder, axiom, (! [A2:extended_ereal,B2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ A2)) @ B2) = (((ord_less_eq @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ B2)) @ A2))))).
% 2.88/1.53 thf(fact_43_lim__mono, axiom, (! [A:$tType]: (((topolo2117631714pology @ A) => (! [N3:nat,X3:(nat > A),Y2:(nat > A),X2:A,Y:A]: (((! [N2:nat]: (((((ord_less_eq @ nat) @ N3) @ N2) => (((ord_less_eq @ A) @ (X3 @ N2)) @ (Y2 @ N2))))) => ((((((filterlim @ nat) @ A) @ X3) @ ((topolo1920029354e_nhds @ A) @ X2)) @ (at_top @ nat)) => ((((((filterlim @ nat) @ A) @ Y2) @ ((topolo1920029354e_nhds @ A) @ Y)) @ (at_top @ nat)) => (((ord_less_eq @ A) @ X2) @ Y)))))))))).
% 2.88/1.53 thf(fact_44_LIMSEQ__le, axiom, (! [A:$tType]: (((topolo2117631714pology @ A) => (! [X3:(nat > A),X2:A,Y2:(nat > A),Y:A]: (((((((filterlim @ nat) @ A) @ X3) @ ((topolo1920029354e_nhds @ A) @ X2)) @ (at_top @ nat)) => ((((((filterlim @ nat) @ A) @ Y2) @ ((topolo1920029354e_nhds @ A) @ Y)) @ (at_top @ nat)) => ((? [N4:nat]: ((! [N2:nat]: (((((ord_less_eq @ nat) @ N4) @ N2) => (((ord_less_eq @ A) @ (X3 @ N2)) @ (Y2 @ N2))))))) => (((ord_less_eq @ A) @ X2) @ Y)))))))))).
% 2.88/1.53 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.88/1.53 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A4)))) = A4)))).
% 2.88/1.53 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X5:A]: (((P @ X5) = (Q @ X5)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.88/1.53 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F2:(A > B),G2:(A > B)]: (((! [X5:A]: (((F2 @ X5) = (G2 @ X5)))) => (F2 = G2))))).
% 2.88/1.53 thf(fact_49_Lim__bounded, axiom, (! [A:$tType]: (((topolo2117631714pology @ A) => (! [F2:(nat > A),L:A,M:nat,C4:A]: (((((((filterlim @ nat) @ A) @ F2) @ ((topolo1920029354e_nhds @ A) @ L)) @ (at_top @ nat)) => ((! [N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => (((ord_less_eq @ A) @ (F2 @ N2)) @ C4)))) => (((ord_less_eq @ A) @ L) @ C4))))))))).
% 2.88/1.53 thf(fact_50_Lim__bounded2, axiom, (! [A:$tType]: (((topolo2117631714pology @ A) => (! [F2:(nat > A),L:A,N3:nat,C4:A]: (((((((filterlim @ nat) @ A) @ F2) @ ((topolo1920029354e_nhds @ A) @ L)) @ (at_top @ nat)) => ((! [N2:nat]: (((((ord_less_eq @ nat) @ N3) @ N2) => (((ord_less_eq @ A) @ C4) @ (F2 @ N2))))) => (((ord_less_eq @ A) @ C4) @ L))))))))).
% 2.88/1.53 thf(fact_51_LIMSEQ__le__const, axiom, (! [A:$tType]: (((topolo2117631714pology @ A) => (! [X3:(nat > A),X2:A,A2:A]: (((((((filterlim @ nat) @ A) @ X3) @ ((topolo1920029354e_nhds @ A) @ X2)) @ (at_top @ nat)) => ((? [N4:nat]: ((! [N2:nat]: (((((ord_less_eq @ nat) @ N4) @ N2) => (((ord_less_eq @ A) @ A2) @ (X3 @ N2))))))) => (((ord_less_eq @ A) @ A2) @ X2))))))))).
% 2.88/1.53 thf(fact_52_LIMSEQ__le__const2, axiom, (! [A:$tType]: (((topolo2117631714pology @ A) => (! [X3:(nat > A),X2:A,A2:A]: (((((((filterlim @ nat) @ A) @ X3) @ ((topolo1920029354e_nhds @ A) @ X2)) @ (at_top @ nat)) => ((? [N4:nat]: ((! [N2:nat]: (((((ord_less_eq @ nat) @ N4) @ N2) => (((ord_less_eq @ A) @ (X3 @ N2)) @ A2)))))) => (((ord_less_eq @ A) @ X2) @ A2))))))))).
% 2.88/1.53 thf(fact_53_tendsto__mono, axiom, (! [A:$tType,B:$tType]: (((topolo503727757_space @ A) => (! [F:(filter @ B),F4:(filter @ B),F2:(B > A),L:A]: (((((ord_less_eq @ (filter @ B)) @ F) @ F4) => ((((((filterlim @ B) @ A) @ F2) @ ((topolo1920029354e_nhds @ A) @ L)) @ F4) => (((((filterlim @ B) @ A) @ F2) @ ((topolo1920029354e_nhds @ A) @ L)) @ F))))))))).
% 2.88/1.53 thf(fact_54_fun__Compl__def, axiom, (! [B:$tType,A:$tType]: (((uminus @ B) => ((uminus_uminus @ (A > B)) = (^ [A3:(A > B),X:A]: (((uminus_uminus @ B) @ (A3 @ X))))))))).
% 2.88/1.53 thf(fact_55_equation__minus__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((A2 = ((uminus_uminus @ A) @ B2)) = (B2 = ((uminus_uminus @ A) @ A2))))))))).
% 2.88/1.53 thf(fact_56_minus__equation__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((uminus_uminus @ A) @ A2) = B2) = (((uminus_uminus @ A) @ B2) = A2)))))))).
% 2.88/1.53 thf(fact_57_comp__eq__dest__lhs, axiom, (! [C:$tType,B:$tType,A:$tType,A2:(C > B),B2:(A > C),C2:(A > B),V:A]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = C2) => ((A2 @ (B2 @ V)) = (C2 @ V)))))).
% 2.88/1.53 thf(fact_58_comp__eq__elim, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A2:(C > B),B2:(A > C),C2:(D > B),D2:(A > D)]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = (((((comp @ D) @ B) @ A) @ C2) @ D2)) => (! [V2:A]: (((A2 @ (B2 @ V2)) = (C2 @ (D2 @ V2))))))))).
% 2.88/1.53 thf(fact_59_comp__eq__dest, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A2:(C > B),B2:(A > C),C2:(D > B),D2:(A > D),V:A]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = (((((comp @ D) @ B) @ A) @ C2) @ D2)) => ((A2 @ (B2 @ V)) = (C2 @ (D2 @ V))))))).
% 2.88/1.53 thf(fact_60_comp__assoc, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F2:(D > B),G2:(C > D),H:(A > C)]: (((((((comp @ C) @ B) @ A) @ (((((comp @ D) @ B) @ C) @ F2) @ G2)) @ H) = (((((comp @ D) @ B) @ A) @ F2) @ (((((comp @ C) @ D) @ A) @ G2) @ H)))))).
% 2.88/1.53 thf(fact_61_comp__def, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((comp @ B) @ C) @ A) = (^ [F3:(B > C),G:(A > B),X:A]: ((F3 @ (G @ X)))))))).
% 2.88/1.53 thf(fact_62_lsc__sequentially, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (((lower_582600101lsc_at @ A) @ extended_ereal) = (^ [X02:A,F3:(A > extended_ereal)]: ((! [X:(nat > A),C3:extended_ereal]: ((((((((filterlim @ nat) @ A) @ X) @ ((topolo1920029354e_nhds @ A) @ X02)) @ (at_top @ nat)) & (! [N:nat]: ((((ord_less_eq @ extended_ereal) @ (F3 @ (X @ N))) @ C3)))) => (((ord_less_eq @ extended_ereal) @ (F3 @ X02)) @ C3))))))))))).
% 2.88/1.53 thf(fact_63_usc__at__def, axiom, (! [B:$tType,A:$tType]: ((((topolo503727757_space @ A) & (topolo259154727pology @ B)) => (((lower_708585518usc_at @ A) @ B) = (^ [X02:A,F3:(A > B)]: ((! [X4:(nat > A),L2:B]: ((((((((filterlim @ nat) @ A) @ X4) @ ((topolo1920029354e_nhds @ A) @ X02)) @ (at_top @ nat)) & (((((filterlim @ nat) @ B) @ (((((comp @ A) @ B) @ nat) @ F3) @ X4)) @ ((topolo1920029354e_nhds @ B) @ L2)) @ (at_top @ nat))) => (((ord_less_eq @ B) @ L2) @ (F3 @ X02)))))))))))).
% 2.88/1.53 thf(fact_64_usc__at__mem, axiom, (! [B:$tType,A:$tType]: ((((topolo503727757_space @ A) & (topolo259154727pology @ B)) => (! [X0:A,F2:(A > B),X2:(nat > A),A4:B]: ((((((lower_708585518usc_at @ A) @ B) @ X0) @ F2) => ((((((filterlim @ nat) @ A) @ X2) @ ((topolo1920029354e_nhds @ A) @ X0)) @ (at_top @ nat)) => ((((((filterlim @ nat) @ B) @ (((((comp @ A) @ B) @ nat) @ F2) @ X2)) @ ((topolo1920029354e_nhds @ B) @ A4)) @ (at_top @ nat)) => (((ord_less_eq @ B) @ A4) @ (F2 @ X0))))))))))).
% 2.88/1.53 thf(fact_65_lim__decreasing__cl, axiom, (! [A:$tType]: ((((comple1035589618norder @ A) & (topolo2117631714pology @ A)) => (! [F2:(nat > A)]: (((! [N2:nat,M2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) => (((ord_less_eq @ A) @ (F2 @ N2)) @ (F2 @ M2))))) => (~ ((! [L3:A]: ((~ ((((((filterlim @ nat) @ A) @ F2) @ ((topolo1920029354e_nhds @ A) @ L3)) @ (at_top @ nat))))))))))))))).
% 2.88/1.53 thf(fact_66_lim__increasing__cl, axiom, (! [A:$tType]: ((((comple1035589618norder @ A) & (topolo2117631714pology @ A)) => (! [F2:(nat > A)]: (((! [N2:nat,M2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) => (((ord_less_eq @ A) @ (F2 @ M2)) @ (F2 @ N2))))) => (~ ((! [L3:A]: ((~ ((((((filterlim @ nat) @ A) @ F2) @ ((topolo1920029354e_nhds @ A) @ L3)) @ (at_top @ nat))))))))))))))).
% 2.88/1.53 thf(fact_67_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ X2) @ X2))))))).
% 2.88/1.53 thf(fact_68_monoseq__le, axiom, (! [A:$tType]: (((topolo2117631714pology @ A) => (! [A2:(nat > A),X2:A]: ((((topological_monoseq @ A) @ A2) => ((((((filterlim @ nat) @ A) @ A2) @ ((topolo1920029354e_nhds @ A) @ X2)) @ (at_top @ nat)) => (((! [N5:nat]: ((((ord_less_eq @ A) @ (A2 @ N5)) @ X2))) & (! [M3:nat,N5:nat]: (((((ord_less_eq @ nat) @ M3) @ N5) => (((ord_less_eq @ A) @ (A2 @ M3)) @ (A2 @ N5)))))) | ((! [N5:nat]: ((((ord_less_eq @ A) @ X2) @ (A2 @ N5)))) & (! [M3:nat,N5:nat]: (((((ord_less_eq @ nat) @ M3) @ N5) => (((ord_less_eq @ A) @ (A2 @ N5)) @ (A2 @ M3))))))))))))))).
% 2.88/1.53 thf(fact_69_lsc__liminf, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (((lower_582600101lsc_at @ A) @ extended_ereal) = (^ [X02:A,F3:(A > extended_ereal)]: ((! [X:(nat > A)]: (((((((filterlim @ nat) @ A) @ X) @ ((topolo1920029354e_nhds @ A) @ X02)) @ (at_top @ nat)) => (((ord_less_eq @ extended_ereal) @ (F3 @ X02)) @ ((((liminf_Liminf @ nat) @ extended_ereal) @ (at_top @ nat)) @ (((((comp @ A) @ extended_ereal) @ nat) @ F3) @ X))))))))))))).
% 2.88/1.53 thf(fact_70_lsc__imp__liminf, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X0:A,F2:(A > extended_ereal),X2:(nat > A)]: ((((((lower_582600101lsc_at @ A) @ extended_ereal) @ X0) @ F2) => ((((((filterlim @ nat) @ A) @ X2) @ ((topolo1920029354e_nhds @ A) @ X0)) @ (at_top @ nat)) => (((ord_less_eq @ extended_ereal) @ (F2 @ X0)) @ ((((liminf_Liminf @ nat) @ extended_ereal) @ (at_top @ nat)) @ (((((comp @ A) @ extended_ereal) @ nat) @ F2) @ X2))))))))))).
% 2.88/1.53 thf(fact_71_LIMSEQ__Uniq, axiom, (! [A:$tType]: (((topological_t2_space @ A) => (! [X3:(nat > A)]: (((uniq @ A) @ (^ [L2:A]: ((((((filterlim @ nat) @ A) @ X3) @ ((topolo1920029354e_nhds @ A) @ L2)) @ (at_top @ nat))))))))))).
% 2.88/1.53 thf(fact_72_monoseq__def, axiom, (! [A:$tType]: (((order @ A) => ((topological_monoseq @ A) = (^ [X4:(nat > A)]: (((! [M4:nat,N:nat]: (((((ord_less_eq @ nat) @ M4) @ N) => (((ord_less_eq @ A) @ (X4 @ M4)) @ (X4 @ N))))) | (! [M4:nat,N:nat]: (((((ord_less_eq @ nat) @ M4) @ N) => (((ord_less_eq @ A) @ (X4 @ N)) @ (X4 @ M4))))))))))))).
% 2.88/1.53 thf(fact_73_monoI2, axiom, (! [A:$tType]: (((order @ A) => (! [X3:(nat > A)]: (((! [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) => (((ord_less_eq @ A) @ (X3 @ N2)) @ (X3 @ M2))))) => ((topological_monoseq @ A) @ X3)))))))).
% 2.88/1.53 thf(fact_74_monoI1, axiom, (! [A:$tType]: (((order @ A) => (! [X3:(nat > A)]: (((! [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) => (((ord_less_eq @ A) @ (X3 @ M2)) @ (X3 @ N2))))) => ((topological_monoseq @ A) @ X3)))))))).
% 2.88/1.53 thf(fact_75_monoseq__minus, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:(nat > A)]: ((((topological_monoseq @ A) @ A2) => ((topological_monoseq @ A) @ (^ [N:nat]: (((uminus_uminus @ A) @ (A2 @ N)))))))))))).
% 2.88/1.53 thf(fact_76_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G2:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F2) @ G2) => (((ord_less_eq @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 2.88/1.53 thf(fact_77_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G2:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F2) @ G2) => (((ord_less_eq @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 2.88/1.53 thf(fact_78_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G2:(A > B)]: (((! [X5:A]: ((((ord_less_eq @ B) @ (F2 @ X5)) @ (G2 @ X5)))) => (((ord_less_eq @ (A > B)) @ F2) @ G2)))))))).
% 2.88/1.53 thf(fact_79_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F3:(A > B),G:(A > B)]: ((! [X:A]: ((((ord_less_eq @ B) @ (F3 @ X)) @ (G @ X))))))))))).
% 2.88/1.53 thf(fact_80_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F2:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F2 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X5:B,Y3:B]: (((((ord_less_eq @ B) @ X5) @ Y3) => (((ord_less_eq @ A) @ (F2 @ X5)) @ (F2 @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F2 @ C2))))))))))).
% 2.88/1.53 thf(fact_81_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F2:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F2 @ B2)) @ C2) => ((! [X5:A,Y3:A]: (((((ord_less_eq @ A) @ X5) @ Y3) => (((ord_less_eq @ C) @ (F2 @ X5)) @ (F2 @ Y3))))) => (((ord_less_eq @ C) @ (F2 @ A2)) @ C2)))))))))).
% 2.88/1.53 thf(fact_82_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F2:(B > A),B2:B,C2:B]: (((A2 = (F2 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X5:B,Y3:B]: (((((ord_less_eq @ B) @ X5) @ Y3) => (((ord_less_eq @ A) @ (F2 @ X5)) @ (F2 @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F2 @ C2))))))))))).
% 2.88/1.53 thf(fact_83_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F2:(A > B),C2:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F2 @ B2) = C2) => ((! [X5:A,Y3:A]: (((((ord_less_eq @ A) @ X5) @ Y3) => (((ord_less_eq @ B) @ (F2 @ X5)) @ (F2 @ Y3))))) => (((ord_less_eq @ B) @ (F2 @ A2)) @ C2)))))))))).
% 2.88/1.53 thf(fact_84_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [X:A,Y5:A]: (((((ord_less_eq @ A) @ X) @ Y5) & (((ord_less_eq @ A) @ Y5) @ X))))))))).
% 2.88/1.53 thf(fact_85_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ X2) => (X2 = Y))))))))).
% 2.88/1.53 thf(fact_86_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) | (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.88/1.53 thf(fact_87_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((X2 = Y) => (((ord_less_eq @ A) @ X2) @ Y)))))))).
% 2.88/1.53 thf(fact_88_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y))) => (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.88/1.53 thf(fact_89_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.88/1.53 thf(fact_90_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A,Z3:A]: ((((((ord_less_eq @ A) @ X2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z3)))) => (((((ord_less_eq @ A) @ Y) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Z3)))) => (((((ord_less_eq @ A) @ X2) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ Y)))) => (((((ord_less_eq @ A) @ Z3) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X2)))) => (((((ord_less_eq @ A) @ Y) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ X2)))) => (~ (((((ord_less_eq @ A) @ Z3) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Y))))))))))))))))).
% 2.88/1.53 thf(fact_91_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((ord_less_eq @ A) @ X2) @ Y) = (X2 = Y))))))))).
% 2.88/1.53 thf(fact_92_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [A5:A,B3:A]: (((((ord_less_eq @ A) @ A5) @ B3) & (((ord_less_eq @ A) @ B3) @ A5))))))))).
% 2.88/1.53 thf(fact_93_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.88/1.53 thf(fact_94_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.88/1.53 thf(fact_95_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.88/1.53 thf(fact_96_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,Z3:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z3) => (((ord_less_eq @ A) @ X2) @ Z3))))))))).
% 2.88/1.53 thf(fact_97_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.88/1.53 thf(fact_98_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A6:A,B4:A]: (((((ord_less_eq @ A) @ A6) @ B4) => ((P @ A6) @ B4)))) => ((! [A6:A,B4:A]: ((((P @ B4) @ A6) => ((P @ A6) @ B4)))) => ((P @ A2) @ B2))))))))).
% 2.88/1.53 thf(fact_99_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ A2))))))))).
% 2.88/1.53 thf(fact_100_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [A5:A,B3:A]: (((((ord_less_eq @ A) @ B3) @ A5) & (((ord_less_eq @ A) @ A5) @ B3))))))))).
% 2.88/1.53 thf(fact_101_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.88/1.53 thf(fact_102_filterlim__mono, axiom, (! [B:$tType,A:$tType,F2:(A > B),F22:(filter @ B),F1:(filter @ A),F23:(filter @ B),F12:(filter @ A)]: (((((((filterlim @ A) @ B) @ F2) @ F22) @ F1) => ((((ord_less_eq @ (filter @ B)) @ F22) @ F23) => ((((ord_less_eq @ (filter @ A)) @ F12) @ F1) => (((((filterlim @ A) @ B) @ F2) @ F23) @ F12))))))).
% 2.88/1.53 thf(fact_103_nhds__imp__cauchy__filter, axiom, (! [A:$tType]: (((topolo47006728_space @ A) => (! [F:(filter @ A),X2:A]: (((((ord_less_eq @ (filter @ A)) @ F) @ ((topolo1920029354e_nhds @ A) @ X2)) => ((topolo1334162427filter @ A) @ F)))))))).
% 2.88/1.53 thf(fact_104_alt__ex1E_H, axiom, (! [A:$tType,P:(A > $o)]: (((? [X6:A]: (((P @ X6) & (! [Y3:A]: (((P @ Y3) => (Y3 = X6))))))) => (~ (((? [X_1:A]: ((P @ X_1))) => (~ (((uniq @ A) @ P)))))))))).
% 2.88/1.53 thf(fact_105_ex1__iff__ex__Uniq, axiom, (! [A:$tType]: (((ex1 @ A) = (^ [P2:(A > $o)]: (((? [X7:A]: ((P2 @ X7))) & ((uniq @ A) @ P2)))))))).
% 2.88/1.53 thf(fact_106_liminf__PInfty, axiom, (! [X3:(nat > extended_ereal)]: (((((((filterlim @ nat) @ extended_ereal) @ X3) @ ((topolo1920029354e_nhds @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (at_top @ nat)) = (((((liminf_Liminf @ nat) @ extended_ereal) @ (at_top @ nat)) @ X3) = (extend1396239628finity @ extended_ereal)))))).
% 2.88/1.53 thf(fact_107_fun_Omap__ident, axiom, (! [A:$tType,D:$tType,T:(D > A)]: (((((((comp @ A) @ A) @ D) @ (^ [X:A]: (X))) @ T) = T)))).
% 2.88/1.53 thf(fact_108_K__record__comp, axiom, (! [C:$tType,B:$tType,A:$tType,C2:B,F2:(A > C)]: (((((((comp @ C) @ B) @ A) @ (^ [X:C]: (C2))) @ F2) = (^ [X:A]: (C2)))))).
% 2.88/1.53 thf(fact_109_ereal__infty__less__eq_I1_J, axiom, (! [X2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ (extend1396239628finity @ extended_ereal)) @ X2) = (X2 = (extend1396239628finity @ extended_ereal)))))).
% 2.88/1.53 thf(fact_110_ereal__infty__less__eq_I2_J, axiom, (! [X2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ X2) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) = (X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))).
% 2.88/1.53 thf(fact_111_MInfty__neq__PInfty_I1_J, axiom, ((extend1396239628finity @ extended_ereal) != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))).
% 2.88/1.53 thf(fact_112_ereal__less__eq_I1_J, axiom, (! [X2:extended_ereal]: ((((ord_less_eq @ extended_ereal) @ X2) @ (extend1396239628finity @ extended_ereal))))).
% 2.88/1.53 thf(fact_113_ereal__infty__less__eq2_I1_J, axiom, (! [A2:extended_ereal,B2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ A2) @ B2) => ((A2 = (extend1396239628finity @ extended_ereal)) => (B2 = (extend1396239628finity @ extended_ereal))))))).
% 2.88/1.53 thf(fact_114_neq__PInf__trans, axiom, (! [Y:extended_ereal,X2:extended_ereal]: (((Y != (extend1396239628finity @ extended_ereal)) => ((((ord_less_eq @ extended_ereal) @ X2) @ Y) => (X2 != (extend1396239628finity @ extended_ereal))))))).
% 2.88/1.53 thf(fact_115_ereal__infty__less__eq2_I2_J, axiom, (! [A2:extended_ereal,B2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ A2) @ B2) => ((B2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (A2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))).
% 2.88/1.53 thf(fact_116_ereal__less__eq_I2_J, axiom, (! [X2:extended_ereal]: ((((ord_less_eq @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ X2)))).
% 2.88/1.53 thf(fact_117_lsc__at__MInfty, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [F2:(A > extended_ereal),X0:A]: ((((F2 @ X0) = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((((lower_582600101lsc_at @ A) @ extended_ereal) @ X0) @ F2)))))))).
% 2.88/1.53 thf(fact_118_fun_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,D:$tType,G2:(B > C),F2:(A > B),V:(D > A)]: (((((((comp @ B) @ C) @ D) @ G2) @ (((((comp @ A) @ B) @ D) @ F2) @ V)) = (((((comp @ A) @ C) @ D) @ (((((comp @ B) @ C) @ A) @ G2) @ F2)) @ V))))).
% 2.88/1.53 thf(fact_119_Uniq__def, axiom, (! [A:$tType]: (((uniq @ A) = (^ [P2:(A > $o)]: ((! [X:A,Y5:A]: (((P2 @ X) => ((P2 @ Y5) => (Y5 = X))))))))))).
% 2.88/1.53 thf(fact_120_Uniq__I, axiom, (! [A:$tType,P:(A > $o)]: (((! [X5:A,Y3:A]: (((P @ X5) => ((P @ Y3) => (Y3 = X5))))) => ((uniq @ A) @ P))))).
% 2.88/1.53 thf(fact_121_Uniq__D, axiom, (! [A:$tType,P:(A > $o),A2:A,B2:A]: ((((uniq @ A) @ P) => ((P @ A2) => ((P @ B2) => (A2 = B2))))))).
% 2.88/1.53 thf(fact_122_filterlim__ident, axiom, (! [A:$tType,F:(filter @ A)]: ((((((filterlim @ A) @ A) @ (^ [X:A]: (X))) @ F) @ F)))).
% 2.88/1.53 thf(fact_123_filterlim__compose, axiom, (! [B:$tType,A:$tType,C:$tType,G2:(A > B),F32:(filter @ B),F22:(filter @ A),F2:(C > A),F1:(filter @ C)]: (((((((filterlim @ A) @ B) @ G2) @ F32) @ F22) => ((((((filterlim @ C) @ A) @ F2) @ F22) @ F1) => (((((filterlim @ C) @ B) @ (^ [X:C]: ((G2 @ (F2 @ X))))) @ F32) @ F1)))))).
% 2.88/1.53 thf(fact_124_Lim__MInfty, axiom, (! [F2:(nat > extended_ereal)]: (((((((filterlim @ nat) @ extended_ereal) @ F2) @ ((topolo1920029354e_nhds @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))) @ (at_top @ nat)) = (! [B5:real]: ((? [N6:nat]: ((! [N:nat]: (((((ord_less_eq @ nat) @ N6) @ N) => (((ord_less_eq @ extended_ereal) @ (F2 @ N)) @ (extended_ereal2 @ B5))))))))))))).
% 2.88/1.53 thf(fact_125_MInfty__cases, axiom, (! [A:$tType,F2:(real > A),Y:A,Z3:A]: (((((((extended_case_ereal @ A) @ F2) @ Y) @ Z3) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) = Z3)))).
% 2.88/1.53 thf(fact_126_Lim__bounded__MInfty, axiom, (! [F2:(nat > extended_ereal),L:extended_ereal,B6:real]: (((((((filterlim @ nat) @ extended_ereal) @ F2) @ ((topolo1920029354e_nhds @ extended_ereal) @ L)) @ (at_top @ nat)) => ((! [N2:nat]: ((((ord_less_eq @ extended_ereal) @ (extended_ereal2 @ B6)) @ (F2 @ N2)))) => (L != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))).
% 2.88/1.53 thf(fact_127_Lim__PInfty, axiom, (! [F2:(nat > extended_ereal)]: (((((((filterlim @ nat) @ extended_ereal) @ F2) @ ((topolo1920029354e_nhds @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (at_top @ nat)) = (! [B5:real]: ((? [N6:nat]: ((! [N:nat]: (((((ord_less_eq @ nat) @ N6) @ N) => (((ord_less_eq @ extended_ereal) @ (extended_ereal2 @ B5)) @ (F2 @ N))))))))))))).
% 2.88/1.53 thf(fact_128_Lim__bounded__PInfty2, axiom, (! [F2:(nat > extended_ereal),L:extended_ereal,N3:nat,B6:real]: (((((((filterlim @ nat) @ extended_ereal) @ F2) @ ((topolo1920029354e_nhds @ extended_ereal) @ L)) @ (at_top @ nat)) => ((! [N2:nat]: (((((ord_less_eq @ nat) @ N3) @ N2) => (((ord_less_eq @ extended_ereal) @ (F2 @ N2)) @ (extended_ereal2 @ B6))))) => (L != (extend1396239628finity @ extended_ereal))))))).
% 2.88/1.53 thf(fact_129_ereal_Oinject, axiom, (! [X1:real,Y1:real]: ((((extended_ereal2 @ X1) = (extended_ereal2 @ Y1)) = (X1 = Y1))))).
% 2.88/1.53 thf(fact_130_ereal__cong, axiom, (! [X2:real,Y:real]: (((X2 = Y) => ((extended_ereal2 @ X2) = (extended_ereal2 @ Y)))))).
% 2.88/1.53 thf(fact_131_ereal__less__eq_I3_J, axiom, (! [R:real,P3:real]: (((((ord_less_eq @ extended_ereal) @ (extended_ereal2 @ R)) @ (extended_ereal2 @ P3)) = (((ord_less_eq @ real) @ R) @ P3))))).
% 2.88/1.53 thf(fact_132_PInfty__cases, axiom, (! [A:$tType,F2:(real > A),Y:A,Z3:A]: (((((((extended_case_ereal @ A) @ F2) @ Y) @ Z3) @ (extend1396239628finity @ extended_ereal)) = Y)))).
% 2.88/1.53 thf(fact_133_tendsto__ereal, axiom, (! [A:$tType,F2:(A > real),X2:real,F:(filter @ A)]: (((((((filterlim @ A) @ real) @ F2) @ ((topolo1920029354e_nhds @ real) @ X2)) @ F) => (((((filterlim @ A) @ extended_ereal) @ (^ [X:A]: ((extended_ereal2 @ (F2 @ X))))) @ ((topolo1920029354e_nhds @ extended_ereal) @ (extended_ereal2 @ X2))) @ F))))).
% 2.88/1.53 thf(fact_134_lim__ereal, axiom, (! [A:$tType,F2:(A > real),X2:real,Net:(filter @ A)]: (((((((filterlim @ A) @ extended_ereal) @ (^ [N:A]: ((extended_ereal2 @ (F2 @ N))))) @ ((topolo1920029354e_nhds @ extended_ereal) @ (extended_ereal2 @ X2))) @ Net) = (((((filterlim @ A) @ real) @ F2) @ ((topolo1920029354e_nhds @ real) @ X2)) @ Net))))).
% 2.88/1.53 thf(fact_135_PInfty__neq__ereal_I1_J, axiom, (! [R:real]: (((extended_ereal2 @ R) != (extend1396239628finity @ extended_ereal))))).
% 2.88/1.53 thf(fact_136_ereal_Ocase__distrib, axiom, (! [A:$tType,B:$tType,H:(A > B),F13:(real > A),F24:A,F33:A,Ereal:extended_ereal]: (((H @ (((((extended_case_ereal @ A) @ F13) @ F24) @ F33) @ Ereal)) = (((((extended_case_ereal @ B) @ (^ [X:real]: ((H @ (F13 @ X))))) @ (H @ F24)) @ (H @ F33)) @ Ereal))))).
% 2.88/1.53 thf(fact_137_uminus__ereal_Osimps_I1_J, axiom, (! [R:real]: ((((uminus_uminus @ extended_ereal) @ (extended_ereal2 @ R)) = (extended_ereal2 @ ((uminus_uminus @ real) @ R)))))).
% 2.88/1.53 thf(fact_138_ereal__le__le, axiom, (! [Y:real,A2:extended_ereal,X2:real]: (((((ord_less_eq @ extended_ereal) @ (extended_ereal2 @ Y)) @ A2) => ((((ord_less_eq @ real) @ X2) @ Y) => (((ord_less_eq @ extended_ereal) @ (extended_ereal2 @ X2)) @ A2)))))).
% 2.88/1.53 thf(fact_139_le__ereal__le, axiom, (! [A2:extended_ereal,X2:real,Y:real]: (((((ord_less_eq @ extended_ereal) @ A2) @ (extended_ereal2 @ X2)) => ((((ord_less_eq @ real) @ X2) @ Y) => (((ord_less_eq @ extended_ereal) @ A2) @ (extended_ereal2 @ Y))))))).
% 2.88/1.53 thf(fact_140_ereal__le__real, axiom, (! [X2:extended_ereal,Y:extended_ereal]: (((! [Z4:real]: (((((ord_less_eq @ extended_ereal) @ X2) @ (extended_ereal2 @ Z4)) => (((ord_less_eq @ extended_ereal) @ Y) @ (extended_ereal2 @ Z4))))) => (((ord_less_eq @ extended_ereal) @ Y) @ X2))))).
% 2.88/1.53 thf(fact_141_ereal_Osimps_I8_J, axiom, (! [A:$tType,F13:(real > A),F24:A,F33:A,X1:real]: (((((((extended_case_ereal @ A) @ F13) @ F24) @ F33) @ (extended_ereal2 @ X1)) = (F13 @ X1))))).
% 2.88/1.53 thf(fact_142_ereal__semiline__unique, axiom, (! [A2:extended_ereal,B2:extended_ereal]: (((((collect @ real) @ (^ [Y5:real]: ((((ord_less_eq @ extended_ereal) @ A2) @ (extended_ereal2 @ Y5))))) = ((collect @ real) @ (^ [Y5:real]: ((((ord_less_eq @ extended_ereal) @ B2) @ (extended_ereal2 @ Y5)))))) = (A2 = B2))))).
% 2.88/1.53 thf(fact_143_real__of__ereal_Oinduct, axiom, (! [P:(extended_ereal > $o),A0:extended_ereal]: (((! [R2:real]: ((P @ (extended_ereal2 @ R2)))) => ((P @ (extend1396239628finity @ extended_ereal)) => ((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (P @ A0))))))).
% 2.88/1.53 thf(fact_144_real__of__ereal_Ocases, axiom, (! [X2:extended_ereal]: (((! [R2:real]: ((X2 != (extended_ereal2 @ R2)))) => ((X2 != (extend1396239628finity @ extended_ereal)) => (X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))).
% 2.88/1.53 thf(fact_145_times__ereal_Oinduct, axiom, (! [P:(extended_ereal > (extended_ereal > $o)),A0:extended_ereal,A1:extended_ereal]: (((! [R2:real,P4:real]: (((P @ (extended_ereal2 @ R2)) @ (extended_ereal2 @ P4)))) => ((! [R2:real]: (((P @ (extended_ereal2 @ R2)) @ (extend1396239628finity @ extended_ereal)))) => ((! [R2:real]: (((P @ (extend1396239628finity @ extended_ereal)) @ (extended_ereal2 @ R2)))) => ((! [R2:real]: (((P @ (extended_ereal2 @ R2)) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => ((! [R2:real]: (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (extended_ereal2 @ R2)))) => (((P @ (extend1396239628finity @ extended_ereal)) @ (extend1396239628finity @ extended_ereal)) => (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (extend1396239628finity @ extended_ereal)) => (((P @ (extend1396239628finity @ extended_ereal)) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((P @ A0) @ A1))))))))))))).
% 2.88/1.53 thf(fact_146_plus__ereal_Oinduct, axiom, (! [P:(extended_ereal > (extended_ereal > $o)),A0:extended_ereal,A1:extended_ereal]: (((! [R2:real,P4:real]: (((P @ (extended_ereal2 @ R2)) @ (extended_ereal2 @ P4)))) => ((! [X_1:extended_ereal]: (((P @ (extend1396239628finity @ extended_ereal)) @ X_1))) => ((! [A6:extended_ereal]: (((P @ A6) @ (extend1396239628finity @ extended_ereal)))) => ((! [R2:real]: (((P @ (extended_ereal2 @ R2)) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => ((! [P4:real]: (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (extended_ereal2 @ P4)))) => (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((P @ A0) @ A1)))))))))).
% 2.88/1.53 thf(fact_147_less__ereal_Oinduct, axiom, (! [P:(extended_ereal > (extended_ereal > $o)),A0:extended_ereal,A1:extended_ereal]: (((! [X5:real,Y3:real]: (((P @ (extended_ereal2 @ X5)) @ (extended_ereal2 @ Y3)))) => ((! [X_1:extended_ereal]: (((P @ (extend1396239628finity @ extended_ereal)) @ X_1))) => ((! [A6:extended_ereal]: (((P @ A6) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => ((! [X5:real]: (((P @ (extended_ereal2 @ X5)) @ (extend1396239628finity @ extended_ereal)))) => ((! [R2:real]: (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (extended_ereal2 @ R2)))) => (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (extend1396239628finity @ extended_ereal)) => ((P @ A0) @ A1)))))))))).
% 2.88/1.53 thf(fact_148_abs__ereal_Oinduct, axiom, (! [P:(extended_ereal > $o),A0:extended_ereal]: (((! [R2:real]: ((P @ (extended_ereal2 @ R2)))) => ((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((P @ (extend1396239628finity @ extended_ereal)) => (P @ A0))))))).
% 2.88/1.53 thf(fact_149_ereal__all__split, axiom, ((^ [P5:(extended_ereal > $o)]: ((! [X7:extended_ereal]: ((P5 @ X7))))) = (^ [P2:(extended_ereal > $o)]: (((P2 @ (extend1396239628finity @ extended_ereal)) & ((! [X:real]: ((P2 @ (extended_ereal2 @ X)))) & (P2 @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))))).
% 2.88/1.53 thf(fact_150_abs__ereal_Ocases, axiom, (! [X2:extended_ereal]: (((! [R2:real]: ((X2 != (extended_ereal2 @ R2)))) => ((X2 != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (X2 = (extend1396239628finity @ extended_ereal))))))).
% 2.88/1.53 thf(fact_151_ereal__ex__split, axiom, ((^ [P5:(extended_ereal > $o)]: ((? [X7:extended_ereal]: ((P5 @ X7))))) = (^ [P2:(extended_ereal > $o)]: (((P2 @ (extend1396239628finity @ extended_ereal)) | ((? [X:real]: ((P2 @ (extended_ereal2 @ X)))) | (P2 @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))))).
% 2.88/1.53 thf(fact_152_ereal3__cases, axiom, (! [X2:extended_ereal,Xa3:extended_ereal,Xb:extended_ereal]: ((((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((? [Ra:real]: ((Xa3 = (extended_ereal2 @ Ra)))) => (! [Rb:real]: ((Xb != (extended_ereal2 @ Rb)))))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((? [Ra:real]: ((Xa3 = (extended_ereal2 @ Ra)))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((? [Ra:real]: ((Xa3 = (extended_ereal2 @ Ra)))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (! [Ra:real]: ((Xb != (extended_ereal2 @ Ra)))))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (! [Ra:real]: ((Xb != (extended_ereal2 @ Ra)))))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((? [R2:real]: ((Xa3 = (extended_ereal2 @ R2)))) => (! [Ra:real]: ((Xb != (extended_ereal2 @ Ra)))))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((? [R2:real]: ((Xa3 = (extended_ereal2 @ R2)))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((? [R2:real]: ((Xa3 = (extended_ereal2 @ R2)))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (! [R2:real]: ((Xb != (extended_ereal2 @ R2)))))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (! [R2:real]: ((Xb != (extended_ereal2 @ R2)))))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((? [R2:real]: ((Xa3 = (extended_ereal2 @ R2)))) => (! [Ra:real]: ((Xb != (extended_ereal2 @ Ra)))))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((? [R2:real]: ((Xa3 = (extended_ereal2 @ R2)))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((? [R2:real]: ((Xa3 = (extended_ereal2 @ R2)))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (! [R2:real]: ((Xb != (extended_ereal2 @ R2)))))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (! [R2:real]: ((Xb != (extended_ereal2 @ R2)))))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (~ (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))))))))))))))))))))))))))))))).
% 2.88/1.53 thf(fact_153_ereal2__cases, axiom, (! [X2:extended_ereal,Xa3:extended_ereal]: ((((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => (! [Ra:real]: ((Xa3 != (extended_ereal2 @ Ra))))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => (Xa3 != (extend1396239628finity @ extended_ereal))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => (Xa3 != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))) => (((X2 = (extend1396239628finity @ extended_ereal)) => (! [R2:real]: ((Xa3 != (extended_ereal2 @ R2))))) => (((X2 = (extend1396239628finity @ extended_ereal)) => (Xa3 != (extend1396239628finity @ extended_ereal))) => (((X2 = (extend1396239628finity @ extended_ereal)) => (Xa3 != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (! [R2:real]: ((Xa3 != (extended_ereal2 @ R2))))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xa3 != (extend1396239628finity @ extended_ereal))) => (~ (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xa3 != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))))))))))))).
% 2.88/1.53 thf(fact_154_ereal__cases, axiom, (! [X2:extended_ereal]: (((! [R2:real]: ((X2 != (extended_ereal2 @ R2)))) => ((X2 != (extend1396239628finity @ extended_ereal)) => (X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))).
% 2.88/1.53 thf(fact_155_MInfty__neq__ereal_I1_J, axiom, (! [R:real]: (((extended_ereal2 @ R) != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))).
% 2.88/1.53 thf(fact_156_ereal__top, axiom, (! [X2:extended_ereal]: (((! [B7:real]: ((((ord_less_eq @ extended_ereal) @ (extended_ereal2 @ B7)) @ X2))) => (X2 = (extend1396239628finity @ extended_ereal)))))).
% 2.88/1.53 thf(fact_157_ereal__tendsto__simps2_I1_J, axiom, (! [A:$tType,F2:(A > real),A2:real,F:(filter @ A)]: (((((((filterlim @ A) @ extended_ereal) @ (((((comp @ real) @ extended_ereal) @ A) @ extended_ereal2) @ F2)) @ ((topolo1920029354e_nhds @ extended_ereal) @ (extended_ereal2 @ A2))) @ F) = (((((filterlim @ A) @ real) @ F2) @ ((topolo1920029354e_nhds @ real) @ A2)) @ F))))).
% 2.88/1.53 thf(fact_158_ereal__bot, axiom, (! [X2:extended_ereal]: (((! [B7:real]: ((((ord_less_eq @ extended_ereal) @ X2) @ (extended_ereal2 @ B7)))) => (X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))).
% 2.88/1.53 thf(fact_159_Lim__bounded__PInfty, axiom, (! [F2:(nat > extended_ereal),L:extended_ereal,B6:real]: (((((((filterlim @ nat) @ extended_ereal) @ F2) @ ((topolo1920029354e_nhds @ extended_ereal) @ L)) @ (at_top @ nat)) => ((! [N2:nat]: ((((ord_less_eq @ extended_ereal) @ (F2 @ N2)) @ (extended_ereal2 @ B6)))) => (L != (extend1396239628finity @ extended_ereal))))))).
% 2.88/1.53 thf(fact_160_ereal__tendsto__simps2_I3_J, axiom, (! [A:$tType,F2:(A > real),F:(filter @ A)]: (((((((filterlim @ A) @ extended_ereal) @ (((((comp @ real) @ extended_ereal) @ A) @ extended_ereal2) @ F2)) @ ((topolo1920029354e_nhds @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))) @ F) = (((((filterlim @ A) @ real) @ F2) @ (at_bot @ real)) @ F))))).
% 2.88/1.53 thf(fact_161_tendsto__cadd__ereal, axiom, (! [A:$tType,Y:extended_ereal,X2:extended_ereal,F2:(A > extended_ereal),F:(filter @ A)]: (((Y != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((X2 != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((((((filterlim @ A) @ extended_ereal) @ F2) @ ((topolo1920029354e_nhds @ extended_ereal) @ X2)) @ F) => (((((filterlim @ A) @ extended_ereal) @ (^ [X:A]: ((((plus_plus @ extended_ereal) @ (F2 @ X)) @ Y)))) @ ((topolo1920029354e_nhds @ extended_ereal) @ (((plus_plus @ extended_ereal) @ X2) @ Y))) @ F))))))).
% 2.88/1.53 thf(fact_162_ereal__minus__real__tendsto__MInf, axiom, (((((filterlim @ nat) @ extended_ereal) @ (^ [X:nat]: ((extended_ereal2 @ ((uminus_uminus @ real) @ ((semiring_1_of_nat @ real) @ X)))))) @ ((topolo1920029354e_nhds @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))) @ (at_top @ nat))).
% 2.88/1.53 thf(fact_163_tendsto__add__ereal__nonneg, axiom, (! [A:$tType,X2:extended_ereal,Y:extended_ereal,F2:(A > extended_ereal),F:(filter @ A),G2:(A > extended_ereal)]: (((X2 != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Y != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((((((filterlim @ A) @ extended_ereal) @ F2) @ ((topolo1920029354e_nhds @ extended_ereal) @ X2)) @ F) => ((((((filterlim @ A) @ extended_ereal) @ G2) @ ((topolo1920029354e_nhds @ extended_ereal) @ Y)) @ F) => (((((filterlim @ A) @ extended_ereal) @ (^ [X:A]: ((((plus_plus @ extended_ereal) @ (F2 @ X)) @ (G2 @ X))))) @ ((topolo1920029354e_nhds @ extended_ereal) @ (((plus_plus @ extended_ereal) @ X2) @ Y))) @ F)))))))).
% 2.88/1.53 thf(fact_164_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C2)) = (B2 = C2)))))))).
% 2.88/1.53 thf(fact_165_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A2:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C2) @ A2)) = (B2 = C2)))))))).
% 2.88/1.53 thf(fact_166_add__le__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.88/1.53 thf(fact_167_add__le__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.88/1.53 thf(fact_168_minus__add__distrib, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,B2:A]: ((((uminus_uminus @ A) @ (((plus_plus @ A) @ A2) @ B2)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ ((uminus_uminus @ A) @ B2))))))))).
% 2.88/1.53 thf(fact_169_minus__add__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ (((plus_plus @ A) @ A2) @ B2)) = B2))))))).
% 2.88/1.53 thf(fact_170_add__minus__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ B2)) = B2))))))).
% 2.88/1.53 thf(fact_171_ereal__PInfty__eq__plus, axiom, (! [A2:extended_ereal,B2:extended_ereal]: ((((extend1396239628finity @ extended_ereal) = (((plus_plus @ extended_ereal) @ A2) @ B2)) = ((A2 = (extend1396239628finity @ extended_ereal)) | (B2 = (extend1396239628finity @ extended_ereal))))))).
% 2.88/1.53 thf(fact_172_ereal__plus__eq__PInfty, axiom, (! [A2:extended_ereal,B2:extended_ereal]: ((((((plus_plus @ extended_ereal) @ A2) @ B2) = (extend1396239628finity @ extended_ereal)) = ((A2 = (extend1396239628finity @ extended_ereal)) | (B2 = (extend1396239628finity @ extended_ereal))))))).
% 2.88/1.53 thf(fact_173_ereal__MInfty__eq__plus, axiom, (! [A2:extended_ereal,B2:extended_ereal]: (((((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)) = (((plus_plus @ extended_ereal) @ A2) @ B2)) = (((A2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) & (B2 != (extend1396239628finity @ extended_ereal))) | ((B2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) & (A2 != (extend1396239628finity @ extended_ereal)))))))).
% 2.88/1.53 thf(fact_174_ereal__plus__eq__MInfty, axiom, (! [A2:extended_ereal,B2:extended_ereal]: ((((((plus_plus @ extended_ereal) @ A2) @ B2) = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) = (((A2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) | (B2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))) & ((A2 != (extend1396239628finity @ extended_ereal)) & (B2 != (extend1396239628finity @ extended_ereal)))))))).
% 2.88/1.53 thf(fact_175_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.88/1.53 thf(fact_176_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (K = L)) => ((((plus_plus @ A) @ I) @ K) = (((plus_plus @ A) @ J) @ L))))))))).
% 2.88/1.53 thf(fact_177_group__cancel_Oadd1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A4:A,K:A,A2:A,B2:A]: (((A4 = (((plus_plus @ A) @ K) @ A2)) => ((((plus_plus @ A) @ A4) @ B2) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.88/1.53 thf(fact_178_group__cancel_Oadd2, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [B6:A,K:A,B2:A,A2:A]: (((B6 = (((plus_plus @ A) @ K) @ B2)) => ((((plus_plus @ A) @ A2) @ B6) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.88/1.53 thf(fact_179_add_Oassoc, axiom, (! [A:$tType]: (((semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.88/1.53 thf(fact_180_add_Oleft__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C2)) = (B2 = C2)))))))).
% 2.88/1.53 thf(fact_181_add_Oright__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [B2:A,A2:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C2) @ A2)) = (B2 = C2)))))))).
% 2.88/1.53 thf(fact_182_add_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((plus_plus @ A) = (^ [A5:A,B3:A]: ((((plus_plus @ A) @ B3) @ A5)))))))).
% 2.88/1.53 thf(fact_183_add_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [B2:A,A2:A,C2:A]: (((((plus_plus @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C2)) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.88/1.53 thf(fact_184_add__left__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C2)) => (B2 = C2)))))))).
% 2.88/1.53 thf(fact_185_add__right__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A2:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C2) @ A2)) => (B2 = C2)))))))).
% 2.88/1.53 thf(fact_186_add_Oinverse__distrib__swap, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: ((((uminus_uminus @ A) @ (((plus_plus @ A) @ A2) @ B2)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2))))))))).
% 2.88/1.53 thf(fact_187_group__cancel_Oneg1, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A4:A,K:A,A2:A]: (((A4 = (((plus_plus @ A) @ K) @ A2)) => (((uminus_uminus @ A) @ A4) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ K)) @ ((uminus_uminus @ A) @ A2)))))))))).
% 2.88/1.53 thf(fact_188_add__mono__thms__linordered__semiring_I3_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (K = L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.88/1.53 thf(fact_189_add__mono__thms__linordered__semiring_I2_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.88/1.53 thf(fact_190_add__mono__thms__linordered__semiring_I1_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.88/1.53 thf(fact_191_add__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ D2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ D2)))))))))).
% 2.88/1.53 thf(fact_192_add__left__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2))))))))).
% 2.88/1.53 thf(fact_193_less__eqE, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (~ ((! [C5:A]: ((B2 != (((plus_plus @ A) @ A2) @ C5))))))))))))).
% 2.88/1.53 thf(fact_194_add__right__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.88/1.53 thf(fact_195_le__iff__add, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => ((ord_less_eq @ A) = (^ [A5:A,B3:A]: ((? [C3:A]: ((B3 = (((plus_plus @ A) @ A5) @ C3))))))))))).
% 2.88/1.53 thf(fact_196_add__le__imp__le__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2)) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.88/1.53 thf(fact_197_add__le__imp__le__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.88/1.53 thf(fact_198_plus__ereal_Osimps_I2_J, axiom, (! [A2:extended_ereal]: (((((plus_plus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)) @ A2) = (extend1396239628finity @ extended_ereal))))).
% 2.88/1.53 thf(fact_199_plus__ereal_Osimps_I3_J, axiom, (! [A2:extended_ereal]: (((((plus_plus @ extended_ereal) @ A2) @ (extend1396239628finity @ extended_ereal)) = (extend1396239628finity @ extended_ereal))))).
% 2.88/1.53 thf(fact_200_plus__ereal_Osimps_I6_J, axiom, ((((plus_plus @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))).
% 2.88/1.53 thf(fact_201_ereal__add__cancel__left, axiom, (! [A2:extended_ereal,B2:extended_ereal,C2:extended_ereal]: (((A2 != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (((((plus_plus @ extended_ereal) @ A2) @ B2) = (((plus_plus @ extended_ereal) @ A2) @ C2)) = ((A2 = (extend1396239628finity @ extended_ereal)) | (B2 = C2))))))).
% 2.88/1.53 thf(fact_202_ereal__add__cancel__right, axiom, (! [A2:extended_ereal,B2:extended_ereal,C2:extended_ereal]: (((A2 != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (((((plus_plus @ extended_ereal) @ B2) @ A2) = (((plus_plus @ extended_ereal) @ C2) @ A2)) = ((A2 = (extend1396239628finity @ extended_ereal)) | (B2 = C2))))))).
% 2.88/1.53 thf(fact_203_tendsto__add, axiom, (! [A:$tType,B:$tType]: (((topolo1314133330id_add @ A) => (! [F2:(B > A),A2:A,F:(filter @ B),G2:(B > A),B2:A]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo1920029354e_nhds @ A) @ A2)) @ F) => ((((((filterlim @ B) @ A) @ G2) @ ((topolo1920029354e_nhds @ A) @ B2)) @ F) => (((((filterlim @ B) @ A) @ (^ [X:B]: ((((plus_plus @ A) @ (F2 @ X)) @ (G2 @ X))))) @ ((topolo1920029354e_nhds @ A) @ (((plus_plus @ A) @ A2) @ B2))) @ F))))))))).
% 2.88/1.53 thf(fact_204_tendsto__add__const__iff, axiom, (! [A:$tType,B:$tType]: (((real_V55928688vector @ A) => (! [C2:A,F2:(B > A),D2:A,F:(filter @ B)]: (((((((filterlim @ B) @ A) @ (^ [X:B]: ((((plus_plus @ A) @ C2) @ (F2 @ X))))) @ ((topolo1920029354e_nhds @ A) @ (((plus_plus @ A) @ C2) @ D2))) @ F) = (((((filterlim @ B) @ A) @ F2) @ ((topolo1920029354e_nhds @ A) @ D2)) @ F)))))))).
% 2.88/1.53 thf(fact_205_plus__ereal_Osimps_I5_J, axiom, (! [P3:real]: (((((plus_plus @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (extended_ereal2 @ P3)) = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))).
% 2.88/1.53 thf(fact_206_plus__ereal_Osimps_I4_J, axiom, (! [R:real]: (((((plus_plus @ extended_ereal) @ (extended_ereal2 @ R)) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))).
% 2.88/1.53 thf(fact_207_ereal__add__le__add__iff, axiom, (! [C2:extended_ereal,A2:extended_ereal,B2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ (((plus_plus @ extended_ereal) @ C2) @ A2)) @ (((plus_plus @ extended_ereal) @ C2) @ B2)) = ((((ord_less_eq @ extended_ereal) @ A2) @ B2) | ((C2 = (extend1396239628finity @ extended_ereal)) | ((C2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) & ((A2 != (extend1396239628finity @ extended_ereal)) & (B2 != (extend1396239628finity @ extended_ereal)))))))))).
% 2.88/1.53 thf(fact_208_ereal__add__le__add__iff2, axiom, (! [A2:extended_ereal,C2:extended_ereal,B2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ (((plus_plus @ extended_ereal) @ A2) @ C2)) @ (((plus_plus @ extended_ereal) @ B2) @ C2)) = ((((ord_less_eq @ extended_ereal) @ A2) @ B2) | ((C2 = (extend1396239628finity @ extended_ereal)) | ((C2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) & ((A2 != (extend1396239628finity @ extended_ereal)) & (B2 != (extend1396239628finity @ extended_ereal)))))))))).
% 2.88/1.53 thf(fact_209_tendsto__add__ereal__MInf, axiom, (! [A:$tType,Y:extended_ereal,F2:(A > extended_ereal),F:(filter @ A),G2:(A > extended_ereal)]: (((Y != (extend1396239628finity @ extended_ereal)) => ((((((filterlim @ A) @ extended_ereal) @ F2) @ ((topolo1920029354e_nhds @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))) @ F) => ((((((filterlim @ A) @ extended_ereal) @ G2) @ ((topolo1920029354e_nhds @ extended_ereal) @ Y)) @ F) => (((((filterlim @ A) @ extended_ereal) @ (^ [X:A]: ((((plus_plus @ extended_ereal) @ (F2 @ X)) @ (G2 @ X))))) @ ((topolo1920029354e_nhds @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))) @ F))))))).
% 2.88/1.53 thf(fact_210_tendsto__add__ereal__PInf, axiom, (! [A:$tType,Y:extended_ereal,F2:(A > extended_ereal),F:(filter @ A),G2:(A > extended_ereal)]: (((Y != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((((((filterlim @ A) @ extended_ereal) @ F2) @ ((topolo1920029354e_nhds @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ F) => ((((((filterlim @ A) @ extended_ereal) @ G2) @ ((topolo1920029354e_nhds @ extended_ereal) @ Y)) @ F) => (((((filterlim @ A) @ extended_ereal) @ (^ [X:A]: ((((plus_plus @ extended_ereal) @ (F2 @ X)) @ (G2 @ X))))) @ ((topolo1920029354e_nhds @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ F))))))).
% 2.88/1.53 thf(fact_211_tendsto__add__ereal__general, axiom, (! [A:$tType,X2:extended_ereal,Y:extended_ereal,F2:(A > extended_ereal),F:(filter @ A),G2:(A > extended_ereal)]: (((~ ((((X2 = (extend1396239628finity @ extended_ereal)) & (Y = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))) | ((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) & (Y = (extend1396239628finity @ extended_ereal)))))) => ((((((filterlim @ A) @ extended_ereal) @ F2) @ ((topolo1920029354e_nhds @ extended_ereal) @ X2)) @ F) => ((((((filterlim @ A) @ extended_ereal) @ G2) @ ((topolo1920029354e_nhds @ extended_ereal) @ Y)) @ F) => (((((filterlim @ A) @ extended_ereal) @ (^ [X:A]: ((((plus_plus @ extended_ereal) @ (F2 @ X)) @ (G2 @ X))))) @ ((topolo1920029354e_nhds @ extended_ereal) @ (((plus_plus @ extended_ereal) @ X2) @ Y))) @ F))))))).
% 2.88/1.53 thf(fact_212_tendsto__at__botI__sequentially, axiom, (! [B:$tType]: (((topolo2135403230pology @ B) => (! [F2:(real > B),Y:B]: (((! [X8:(nat > real)]: (((((((filterlim @ nat) @ real) @ X8) @ (at_bot @ real)) @ (at_top @ nat)) => (((((filterlim @ nat) @ B) @ (^ [N:nat]: ((F2 @ (X8 @ N))))) @ ((topolo1920029354e_nhds @ B) @ Y)) @ (at_top @ nat))))) => (((((filterlim @ real) @ B) @ F2) @ ((topolo1920029354e_nhds @ B) @ Y)) @ (at_bot @ real))))))))).
% 2.88/1.53 thf(fact_213_id__nat__ereal__tendsto__PInf, axiom, (((((filterlim @ nat) @ extended_ereal) @ (^ [X:nat]: ((extended_ereal2 @ ((semiring_1_of_nat @ real) @ X))))) @ ((topolo1920029354e_nhds @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (at_top @ nat))).
% 2.88/1.53 thf(fact_214_ereal__liminf__add__mono, axiom, (! [U:(nat > extended_ereal),V:(nat > extended_ereal)]: (((~ ((((((((liminf_Liminf @ nat) @ extended_ereal) @ (at_top @ nat)) @ U) = (extend1396239628finity @ extended_ereal)) & (((((liminf_Liminf @ nat) @ extended_ereal) @ (at_top @ nat)) @ V) = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))) | ((((((liminf_Liminf @ nat) @ extended_ereal) @ (at_top @ nat)) @ U) = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) & (((((liminf_Liminf @ nat) @ extended_ereal) @ (at_top @ nat)) @ V) = (extend1396239628finity @ extended_ereal)))))) => (((ord_less_eq @ extended_ereal) @ (((plus_plus @ extended_ereal) @ ((((liminf_Liminf @ nat) @ extended_ereal) @ (at_top @ nat)) @ U)) @ ((((liminf_Liminf @ nat) @ extended_ereal) @ (at_top @ nat)) @ V))) @ ((((liminf_Liminf @ nat) @ extended_ereal) @ (at_top @ nat)) @ (^ [N:nat]: ((((plus_plus @ extended_ereal) @ (U @ N)) @ (V @ N)))))))))).
% 2.88/1.53 thf(fact_215_of__nat__le__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M5:nat,N7:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M5)) @ ((semiring_1_of_nat @ A) @ N7)) = (((ord_less_eq @ nat) @ M5) @ N7)))))))).
% 2.88/1.53 thf(fact_216_of__nat__mono, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [I:nat,J:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ I)) @ ((semiring_1_of_nat @ A) @ J))))))))).
% 2.88/1.53 thf(fact_217_ereal__tendsto__simps2_I2_J, axiom, (! [A:$tType,F2:(A > real),F:(filter @ A)]: (((((((filterlim @ A) @ extended_ereal) @ (((((comp @ real) @ extended_ereal) @ A) @ extended_ereal2) @ F2)) @ ((topolo1920029354e_nhds @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ F) = (((((filterlim @ A) @ real) @ F2) @ (at_top @ real)) @ F))))).
% 2.88/1.53 thf(fact_218_nat__add__left__cancel__le, axiom, (! [K:nat,M5:nat,N7:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ K) @ M5)) @ (((plus_plus @ nat) @ K) @ N7)) = (((ord_less_eq @ nat) @ M5) @ N7))))).
% 2.88/1.53 thf(fact_219_filterlim__at__top__add__at__top, axiom, (! [A:$tType,F2:(A > real),F:(filter @ A),G2:(A > real)]: (((((((filterlim @ A) @ real) @ F2) @ (at_top @ real)) @ F) => ((((((filterlim @ A) @ real) @ G2) @ (at_top @ real)) @ F) => (((((filterlim @ A) @ real) @ (^ [X:A]: ((((plus_plus @ real) @ (F2 @ X)) @ (G2 @ X))))) @ (at_top @ real)) @ F)))))).
% 2.88/1.53 thf(fact_220_filterlim__tendsto__add__at__top, axiom, (! [A:$tType,F2:(A > real),C2:real,F:(filter @ A),G2:(A > real)]: (((((((filterlim @ A) @ real) @ F2) @ ((topolo1920029354e_nhds @ real) @ C2)) @ F) => ((((((filterlim @ A) @ real) @ G2) @ (at_top @ real)) @ F) => (((((filterlim @ A) @ real) @ (^ [X:A]: ((((plus_plus @ real) @ (F2 @ X)) @ (G2 @ X))))) @ (at_top @ real)) @ F)))))).
% 2.88/1.53 thf(fact_221_nat__int__comparison_I3_J, axiom, ((ord_less_eq @ nat) = (^ [A5:nat,B3:nat]: ((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ A5)) @ ((semiring_1_of_nat @ int) @ B3)))))).
% 2.88/1.53 thf(fact_222_add__leE, axiom, (! [M5:nat,K:nat,N7:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M5) @ K)) @ N7) => (~ (((((ord_less_eq @ nat) @ M5) @ N7) => (~ ((((ord_less_eq @ nat) @ K) @ N7)))))))))).
% 2.88/1.53 thf(fact_223_le__add1, axiom, (! [N7:nat,M5:nat]: ((((ord_less_eq @ nat) @ N7) @ (((plus_plus @ nat) @ N7) @ M5))))).
% 2.88/1.53 thf(fact_224_le__add2, axiom, (! [N7:nat,M5:nat]: ((((ord_less_eq @ nat) @ N7) @ (((plus_plus @ nat) @ M5) @ N7))))).
% 2.88/1.53 thf(fact_225_add__leD1, axiom, (! [M5:nat,K:nat,N7:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M5) @ K)) @ N7) => (((ord_less_eq @ nat) @ M5) @ N7))))).
% 2.88/1.53 thf(fact_226_add__leD2, axiom, (! [M5:nat,K:nat,N7:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M5) @ K)) @ N7) => (((ord_less_eq @ nat) @ K) @ N7))))).
% 2.88/1.53 thf(fact_227_le__Suc__ex, axiom, (! [K:nat,L:nat]: (((((ord_less_eq @ nat) @ K) @ L) => (? [N2:nat]: ((L = (((plus_plus @ nat) @ K) @ N2)))))))).
% 2.88/1.53 thf(fact_228_add__le__mono, axiom, (! [I:nat,J:nat,K:nat,L:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ K) @ L) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ L))))))).
% 2.88/1.53 thf(fact_229_add__le__mono1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ K)))))).
% 2.88/1.53 thf(fact_230_trans__le__add1, axiom, (! [I:nat,J:nat,M5:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ I) @ (((plus_plus @ nat) @ J) @ M5)))))).
% 2.88/1.53 thf(fact_231_trans__le__add2, axiom, (! [I:nat,J:nat,M5:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ I) @ (((plus_plus @ nat) @ M5) @ J)))))).
% 2.88/1.53 thf(fact_232_nat__le__iff__add, axiom, ((ord_less_eq @ nat) = (^ [M4:nat,N:nat]: ((? [K2:nat]: ((N = (((plus_plus @ nat) @ M4) @ K2)))))))).
% 2.88/1.53 thf(fact_233_nat__leq__as__int, axiom, ((ord_less_eq @ nat) = (^ [A5:nat,B3:nat]: ((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ A5)) @ ((semiring_1_of_nat @ int) @ B3)))))).
% 2.88/1.53 thf(fact_234_plus__ereal_Osimps_I1_J, axiom, (! [R:real,P3:real]: (((((plus_plus @ extended_ereal) @ (extended_ereal2 @ R)) @ (extended_ereal2 @ P3)) = (extended_ereal2 @ (((plus_plus @ real) @ R) @ P3)))))).
% 2.88/1.53 thf(fact_235_filterlim__at__bot__mirror, axiom, (! [A:$tType,F2:(real > A),F:(filter @ A)]: (((((((filterlim @ real) @ A) @ F2) @ F) @ (at_bot @ real)) = (((((filterlim @ real) @ A) @ (^ [X:real]: ((F2 @ ((uminus_uminus @ real) @ X))))) @ F) @ (at_top @ real)))))).
% 2.88/1.53 thf(fact_236_filterlim__at__top__mirror, axiom, (! [A:$tType,F2:(real > A),F:(filter @ A)]: (((((((filterlim @ real) @ A) @ F2) @ F) @ (at_top @ real)) = (((((filterlim @ real) @ A) @ (^ [X:real]: ((F2 @ ((uminus_uminus @ real) @ X))))) @ F) @ (at_bot @ real)))))).
% 2.88/1.53 thf(fact_237_filterlim__uminus__at__bot__at__top, axiom, (((((filterlim @ real) @ real) @ (uminus_uminus @ real)) @ (at_bot @ real)) @ (at_top @ real))).
% 2.88/1.53 thf(fact_238_filterlim__uminus__at__top__at__bot, axiom, (((((filterlim @ real) @ real) @ (uminus_uminus @ real)) @ (at_top @ real)) @ (at_bot @ real))).
% 2.88/1.53 thf(fact_239_liminf__shift__k, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [U:(nat > A),K:nat]: ((((((liminf_Liminf @ nat) @ A) @ (at_top @ nat)) @ (^ [N:nat]: ((U @ (((plus_plus @ nat) @ N) @ K))))) = ((((liminf_Liminf @ nat) @ A) @ (at_top @ nat)) @ U)))))))).
% 2.88/1.53 thf(fact_240_Nat_Oex__has__greatest__nat, axiom, (! [P:(nat > $o),K:nat,B2:nat]: (((P @ K) => ((! [Y3:nat]: (((P @ Y3) => (((ord_less_eq @ nat) @ Y3) @ B2)))) => (? [X5:nat]: (((P @ X5) & (! [Y6:nat]: (((P @ Y6) => (((ord_less_eq @ nat) @ Y6) @ X5)))))))))))).
% 2.88/1.53 thf(fact_241_nat__le__linear, axiom, (! [M5:nat,N7:nat]: (((((ord_less_eq @ nat) @ M5) @ N7) | (((ord_less_eq @ nat) @ N7) @ M5))))).
% 2.88/1.53 thf(fact_242_le__antisym, axiom, (! [M5:nat,N7:nat]: (((((ord_less_eq @ nat) @ M5) @ N7) => ((((ord_less_eq @ nat) @ N7) @ M5) => (M5 = N7)))))).
% 2.88/1.53 thf(fact_243_eq__imp__le, axiom, (! [M5:nat,N7:nat]: (((M5 = N7) => (((ord_less_eq @ nat) @ M5) @ N7))))).
% 2.88/1.53 thf(fact_244_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))).
% 2.88/1.53 thf(fact_245_le__refl, axiom, (! [N7:nat]: ((((ord_less_eq @ nat) @ N7) @ N7)))).
% 2.88/1.53 thf(fact_246_filterlim__uminus__at__bot, axiom, (! [A:$tType,F2:(A > real),F:(filter @ A)]: (((((((filterlim @ A) @ real) @ F2) @ (at_bot @ real)) @ F) = (((((filterlim @ A) @ real) @ (^ [X:A]: (((uminus_uminus @ real) @ (F2 @ X))))) @ (at_top @ real)) @ F))))).
% 2.88/1.53 thf(fact_247_filterlim__uminus__at__top, axiom, (! [A:$tType,F2:(A > real),F:(filter @ A)]: (((((((filterlim @ A) @ real) @ F2) @ (at_top @ real)) @ F) = (((((filterlim @ A) @ real) @ (^ [X:A]: (((uminus_uminus @ real) @ (F2 @ X))))) @ (at_bot @ real)) @ F))))).
% 2.88/1.53 thf(fact_248_LIMSEQ__ignore__initial__segment, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [F2:(nat > A),A2:A,K:nat]: (((((((filterlim @ nat) @ A) @ F2) @ ((topolo1920029354e_nhds @ A) @ A2)) @ (at_top @ nat)) => (((((filterlim @ nat) @ A) @ (^ [N:nat]: ((F2 @ (((plus_plus @ nat) @ N) @ K))))) @ ((topolo1920029354e_nhds @ A) @ A2)) @ (at_top @ nat))))))))).
% 2.88/1.53 thf(fact_249_LIMSEQ__offset, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [F2:(nat > A),K:nat,A2:A]: (((((((filterlim @ nat) @ A) @ (^ [N:nat]: ((F2 @ (((plus_plus @ nat) @ N) @ K))))) @ ((topolo1920029354e_nhds @ A) @ A2)) @ (at_top @ nat)) => (((((filterlim @ nat) @ A) @ F2) @ ((topolo1920029354e_nhds @ A) @ A2)) @ (at_top @ nat))))))))).
% 2.88/1.53 thf(fact_250_plus__ereal_Oelims, axiom, (! [X2:extended_ereal,Xa3:extended_ereal,Y:extended_ereal]: ((((((plus_plus @ extended_ereal) @ X2) @ Xa3) = Y) => ((! [R2:real]: (((X2 = (extended_ereal2 @ R2)) => (! [P4:real]: (((Xa3 = (extended_ereal2 @ P4)) => (Y != (extended_ereal2 @ (((plus_plus @ real) @ R2) @ P4))))))))) => (((X2 = (extend1396239628finity @ extended_ereal)) => (Y != (extend1396239628finity @ extended_ereal))) => (((Xa3 = (extend1396239628finity @ extended_ereal)) => (Y != (extend1396239628finity @ extended_ereal))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Y != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((? [P4:real]: ((Xa3 = (extended_ereal2 @ P4)))) => (Y != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (~ (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Y != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))))))))))).
% 2.88/1.53 thf(fact_251_tendsto__PInfty__eq__at__top, axiom, (! [A:$tType,F2:(A > real),F:(filter @ A)]: (((((((filterlim @ A) @ extended_ereal) @ (^ [Z5:A]: ((extended_ereal2 @ (F2 @ Z5))))) @ ((topolo1920029354e_nhds @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ F) = (((((filterlim @ A) @ real) @ F2) @ (at_top @ real)) @ F))))).
% 2.88/1.53 thf(fact_252_tendsto__at__topI__sequentially, axiom, (! [B:$tType]: (((topolo2135403230pology @ B) => (! [F2:(real > B),Y:B]: (((! [X8:(nat > real)]: (((((((filterlim @ nat) @ real) @ X8) @ (at_top @ real)) @ (at_top @ nat)) => (((((filterlim @ nat) @ B) @ (^ [N:nat]: ((F2 @ (X8 @ N))))) @ ((topolo1920029354e_nhds @ B) @ Y)) @ (at_top @ nat))))) => (((((filterlim @ real) @ B) @ F2) @ ((topolo1920029354e_nhds @ B) @ Y)) @ (at_top @ real))))))))).
% 2.88/1.53 thf(fact_253_filterlim__real__sequentially, axiom, (((((filterlim @ nat) @ real) @ (semiring_1_of_nat @ real)) @ (at_top @ real)) @ (at_top @ nat))).
% 2.88/1.53 thf(fact_254_int__ops_I5_J, axiom, (! [A2:nat,B2:nat]: ((((semiring_1_of_nat @ int) @ (((plus_plus @ nat) @ A2) @ B2)) = (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ A2)) @ ((semiring_1_of_nat @ int) @ B2)))))).
% 2.88/1.53 thf(fact_255_nat__int__comparison_I1_J, axiom, ((^ [Y4:nat,Z2:nat]: ((Y4 = Z2))) = (^ [A5:nat,B3:nat]: ((((semiring_1_of_nat @ int) @ A5) = ((semiring_1_of_nat @ int) @ B3)))))).
% 2.88/1.53 thf(subcl_Real__Vector__Spaces_Ometric__space___HOL_Otype, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (type @ A))))).
% 2.88/1.53 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Ot2__space, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (topological_t2_space @ A))))).
% 2.88/1.53 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Ouniform__space, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (topolo47006728_space @ A))))).
% 2.88/1.53 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Otopological__space, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (topolo503727757_space @ A))))).
% 2.88/1.53 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Ofirst__countable__topology, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (topolo2135403230pology @ A))))).
% 2.88/1.53 thf(tcon_HOL_Obool___Countable_Ocountable, axiom, (countable @ $o)).
% 2.88/1.53 thf(tcon_Set_Oset___Countable_Ocountable_1, axiom, (! [A7:$tType]: (((finite_finite @ A7) => (countable @ (set @ A7)))))).
% 2.88/1.53 thf(tcon_Nat_Onat___Countable_Ocountable_2, axiom, (countable @ nat)).
% 2.88/1.53 thf(tcon_Int_Oint___Countable_Ocountable_3, axiom, (countable @ int)).
% 2.88/1.53 thf(tcon_fun___Countable_Ocountable_4, axiom, (! [A7:$tType,A8:$tType]: ((((finite_finite @ A7) & (countable @ A8)) => (countable @ (A7 > A8)))))).
% 2.88/1.53 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A7:$tType,A8:$tType]: ((((finite_finite @ A7) & (finite_finite @ A8)) => (finite_finite @ (A7 > A8)))))).
% 2.88/1.53 thf(tcon_Set_Oset___Finite__Set_Ofinite_5, axiom, (! [A7:$tType]: (((finite_finite @ A7) => (finite_finite @ (set @ A7)))))).
% 2.88/1.53 thf(tcon_HOL_Obool___Finite__Set_Ofinite_6, axiom, (finite_finite @ $o)).
% 2.88/1.53 thf(tcon_fun___Topological__Spaces_Ofirst__countable__topology, axiom, (! [A7:$tType,A8:$tType]: ((((countable @ A7) & (topolo2135403230pology @ A8)) => (topolo2135403230pology @ (A7 > A8)))))).
% 2.88/1.53 thf(tcon_fun___Topological__Spaces_Otopological__space, axiom, (! [A7:$tType,A8:$tType]: (((topolo503727757_space @ A8) => (topolo503727757_space @ (A7 > A8)))))).
% 2.88/1.53 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A7:$tType,A8:$tType]: (((comple187826305attice @ A8) => (comple187826305attice @ (A7 > A8)))))).
% 2.88/1.53 thf(tcon_fun___Lattices_Oboolean__algebra, axiom, (! [A7:$tType,A8:$tType]: (((boolean_algebra @ A8) => (boolean_algebra @ (A7 > A8)))))).
% 2.88/1.53 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: (((preorder @ A8) => (preorder @ (A7 > A8)))))).
% 2.88/1.53 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: (((order @ A8) => (order @ (A7 > A8)))))).
% 2.88/1.53 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: (((ord @ A8) => (ord @ (A7 > A8)))))).
% 2.88/1.53 thf(tcon_fun___Groups_Ouminus, axiom, (! [A7:$tType,A8:$tType]: (((uminus @ A8) => (uminus @ (A7 > A8)))))).
% 2.88/1.53 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le, axiom, (ordere236663937imp_le @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Topological__Spaces_Otopological__space_7, axiom, (topolo503727757_space @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology, axiom, (topolo2117631714pology @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Topological__Spaces_Oorder__topology, axiom, (topolo259154727pology @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add, axiom, (ordere779506340up_add @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Limits_Otopological__monoid__add, axiom, (topolo1314133330id_add @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Topological__Spaces_Ot2__space, axiom, (topological_t2_space @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add, axiom, (ordered_ab_group_add @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add, axiom, (cancel_semigroup_add @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Groups_Oab__semigroup__add, axiom, (ab_semigroup_add @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add, axiom, (comm_monoid_add @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Groups_Osemigroup__add, axiom, (semigroup_add @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Groups_Oab__group__add, axiom, (ab_group_add @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Orderings_Opreorder_8, axiom, (preorder @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, (linorder @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Groups_Ogroup__add, axiom, (group_add @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Orderings_Oorder_9, axiom, (order @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Orderings_Oord_10, axiom, (ord @ int)).
% 2.88/1.53 thf(tcon_Int_Oint___Groups_Ouminus_11, axiom, (uminus @ int)).
% 2.88/1.53 thf(tcon_Nat_Onat___Topological__Spaces_Ofirst__countable__topology_12, axiom, (topolo2135403230pology @ nat)).
% 2.88/1.53 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_13, axiom, (ordere236663937imp_le @ nat)).
% 2.88/1.53 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.88/1.53 thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_14, axiom, (topolo503727757_space @ nat)).
% 2.88/1.53 thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_15, axiom, (topolo2117631714pology @ nat)).
% 2.88/1.53 thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_16, axiom, (topolo259154727pology @ nat)).
% 2.88/1.53 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_17, axiom, (linord1659791738miring @ nat)).
% 2.88/1.53 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_18, axiom, (ordere779506340up_add @ nat)).
% 2.88/1.53 thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_19, axiom, (topolo1314133330id_add @ nat)).
% 2.88/1.53 thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_20, axiom, (topological_t2_space @ nat)).
% 2.88/1.53 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_21, axiom, (cancel_semigroup_add @ nat)).
% 2.88/1.53 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_22, axiom, (ab_semigroup_add @ nat)).
% 2.88/1.53 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_23, axiom, (comm_monoid_add @ nat)).
% 2.88/1.53 thf(tcon_Nat_Onat___Groups_Osemigroup__add_24, axiom, (semigroup_add @ nat)).
% 2.88/1.53 thf(tcon_Nat_Onat___Orderings_Opreorder_25, axiom, (preorder @ nat)).
% 2.88/1.53 thf(tcon_Nat_Onat___Orderings_Olinorder_26, axiom, (linorder @ nat)).
% 2.88/1.53 thf(tcon_Nat_Onat___Orderings_Oorder_27, axiom, (order @ nat)).
% 2.88/1.53 thf(tcon_Nat_Onat___Orderings_Oord_28, axiom, (ord @ nat)).
% 2.88/1.53 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_29, axiom, (! [A7:$tType]: ((comple187826305attice @ (set @ A7))))).
% 2.88/1.53 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_30, axiom, (! [A7:$tType]: ((boolean_algebra @ (set @ A7))))).
% 2.88/1.53 thf(tcon_Set_Oset___Groups_Oab__semigroup__add_31, axiom, (! [A7:$tType]: (((ab_semigroup_add @ A7) => (ab_semigroup_add @ (set @ A7)))))).
% 2.88/1.53 thf(tcon_Set_Oset___Groups_Ocomm__monoid__add_32, axiom, (! [A7:$tType]: (((comm_monoid_add @ A7) => (comm_monoid_add @ (set @ A7)))))).
% 2.88/1.53 thf(tcon_Set_Oset___Groups_Osemigroup__add_33, axiom, (! [A7:$tType]: (((semigroup_add @ A7) => (semigroup_add @ (set @ A7)))))).
% 2.88/1.53 thf(tcon_Set_Oset___Orderings_Opreorder_34, axiom, (! [A7:$tType]: ((preorder @ (set @ A7))))).
% 2.88/1.53 thf(tcon_Set_Oset___Orderings_Oorder_35, axiom, (! [A7:$tType]: ((order @ (set @ A7))))).
% 2.88/1.53 thf(tcon_Set_Oset___Orderings_Oord_36, axiom, (! [A7:$tType]: ((ord @ (set @ A7))))).
% 2.88/1.53 thf(tcon_Set_Oset___Groups_Ouminus_37, axiom, (! [A7:$tType]: ((uminus @ (set @ A7))))).
% 2.88/1.53 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_38, axiom, (topolo503727757_space @ $o)).
% 2.88/1.53 thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_39, axiom, (topolo2117631714pology @ $o)).
% 2.88/1.53 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_40, axiom, (comple187826305attice @ $o)).
% 2.88/1.53 thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_41, axiom, (topolo259154727pology @ $o)).
% 2.88/1.53 thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_42, axiom, (topological_t2_space @ $o)).
% 2.88/1.53 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_43, axiom, (boolean_algebra @ $o)).
% 2.88/1.53 thf(tcon_HOL_Obool___Orderings_Opreorder_44, axiom, (preorder @ $o)).
% 2.88/1.53 thf(tcon_HOL_Obool___Orderings_Olinorder_45, axiom, (linorder @ $o)).
% 2.88/1.53 thf(tcon_HOL_Obool___Orderings_Oorder_46, axiom, (order @ $o)).
% 2.88/1.53 thf(tcon_HOL_Obool___Orderings_Oord_47, axiom, (ord @ $o)).
% 2.88/1.53 thf(tcon_HOL_Obool___Groups_Ouminus_48, axiom, (uminus @ $o)).
% 2.88/1.53 thf(tcon_Real_Oreal___Topological__Spaces_Ofirst__countable__topology_49, axiom, (topolo2135403230pology @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_50, axiom, (ordere236663937imp_le @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector, axiom, (real_V55928688vector @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_51, axiom, (topolo503727757_space @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_52, axiom, (topolo2117631714pology @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_53, axiom, (topolo259154727pology @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_54, axiom, (linord1659791738miring @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Topological__Spaces_Ouniform__space, axiom, (topolo47006728_space @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space, axiom, (real_V2090557954_space @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_55, axiom, (ordere779506340up_add @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Limits_Otopological__monoid__add_56, axiom, (topolo1314133330id_add @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Limits_Otopological__group__add, axiom, (topolo799126099up_add @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_57, axiom, (topological_t2_space @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_58, axiom, (ordered_ab_group_add @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_59, axiom, (cancel_semigroup_add @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_60, axiom, (ab_semigroup_add @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_61, axiom, (comm_monoid_add @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Groups_Osemigroup__add_62, axiom, (semigroup_add @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Groups_Oab__group__add_63, axiom, (ab_group_add @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Orderings_Opreorder_64, axiom, (preorder @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Orderings_Olinorder_65, axiom, (linorder @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Groups_Ogroup__add_66, axiom, (group_add @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Orderings_Oorder_67, axiom, (order @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Orderings_Oord_68, axiom, (ord @ real)).
% 2.88/1.53 thf(tcon_Real_Oreal___Groups_Ouminus_69, axiom, (uminus @ real)).
% 2.88/1.53 thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_70, axiom, (! [A7:$tType]: ((comple187826305attice @ (filter @ A7))))).
% 2.88/1.53 thf(tcon_Filter_Ofilter___Orderings_Opreorder_71, axiom, (! [A7:$tType]: ((preorder @ (filter @ A7))))).
% 2.88/1.53 thf(tcon_Filter_Ofilter___Orderings_Oorder_72, axiom, (! [A7:$tType]: ((order @ (filter @ A7))))).
% 2.88/1.53 thf(tcon_Filter_Ofilter___Orderings_Oord_73, axiom, (! [A7:$tType]: ((ord @ (filter @ A7))))).
% 2.88/1.53 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Ofirst__countable__topology_74, axiom, (topolo2135403230pology @ extended_ereal)).
% 2.88/1.53 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Otopological__space_75, axiom, (topolo503727757_space @ extended_ereal)).
% 2.88/1.53 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Olinorder__topology_76, axiom, (topolo2117631714pology @ extended_ereal)).
% 2.88/1.53 thf(tcon_Extended__Real_Oereal___Complete__Lattices_Ocomplete__linorder, axiom, (comple1035589618norder @ extended_ereal)).
% 2.88/1.53 thf(tcon_Extended__Real_Oereal___Complete__Lattices_Ocomplete__lattice_77, axiom, (comple187826305attice @ extended_ereal)).
% 2.88/1.53 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Oorder__topology_78, axiom, (topolo259154727pology @ extended_ereal)).
% 2.88/1.53 thf(tcon_Extended__Real_Oereal___Groups_Oordered__ab__semigroup__add_79, axiom, (ordere779506340up_add @ extended_ereal)).
% 2.88/1.53 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Ot2__space_80, axiom, (topological_t2_space @ extended_ereal)).
% 2.88/1.53 thf(tcon_Extended__Real_Oereal___Groups_Oab__semigroup__add_81, axiom, (ab_semigroup_add @ extended_ereal)).
% 2.88/1.53 thf(tcon_Extended__Real_Oereal___Groups_Ocomm__monoid__add_82, axiom, (comm_monoid_add @ extended_ereal)).
% 2.88/1.53 thf(tcon_Extended__Real_Oereal___Groups_Osemigroup__add_83, axiom, (semigroup_add @ extended_ereal)).
% 2.88/1.53 thf(tcon_Extended__Real_Oereal___Orderings_Opreorder_84, axiom, (preorder @ extended_ereal)).
% 2.88/1.53 thf(tcon_Extended__Real_Oereal___Orderings_Olinorder_85, axiom, (linorder @ extended_ereal)).
% 2.88/1.53 thf(tcon_Extended__Real_Oereal___Orderings_Oorder_86, axiom, (order @ extended_ereal)).
% 2.88/1.53 thf(tcon_Extended__Real_Oereal___Orderings_Oord_87, axiom, (ord @ extended_ereal)).
% 2.88/1.53 thf(tcon_Extended__Real_Oereal___Groups_Ouminus_88, axiom, (uminus @ extended_ereal)).
% 2.88/1.53 thf(tfree_0, hypothesis, (real_V2090557954_space @ a)).
% 2.88/1.53 thf(conj_0, conjecture, (((((filterlim @ nat) @ extended_ereal) @ (^ [I2:nat]: (((uminus_uminus @ extended_ereal) @ (f @ (x @ I2)))))) @ ((topolo1920029354e_nhds @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ a2))) @ (at_top @ nat))).
% 2.88/1.53 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.WRkRsD2zqq/DTF2DTF29324.p
% 2.88/1.53 ------------------------
% 2.88/1.56 ---- Cleaned THF ---
% 2.88/1.56 thf(ty_t_Extended__Real_Oereal,type,
% 2.88/1.56 extended_ereal: $tType ).
% 2.88/1.56
% 2.88/1.56 thf(ty_t_Filter_Ofilter,type,
% 2.88/1.56 filter: $tType > $tType ).
% 2.88/1.56
% 2.88/1.56 thf(ty_t_Real_Oreal,type,
% 2.88/1.56 real: $tType ).
% 2.88/1.56
% 2.88/1.56 thf(ty_t_Set_Oset,type,
% 2.88/1.56 set: $tType > $tType ).
% 2.88/1.56
% 2.88/1.56 thf(ty_t_Nat_Onat,type,
% 2.88/1.56 nat: $tType ).
% 2.88/1.56
% 2.88/1.56 thf(ty_t_Int_Oint,type,
% 2.88/1.56 int: $tType ).
% 2.88/1.56
% 2.88/1.56 thf(ty_tf_a,type,
% 2.88/1.56 a: $tType ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.88/1.56 finite_finite:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_HOL_Otype,type,
% 2.88/1.56 type:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Countable_Ocountable,type,
% 2.88/1.56 countable:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Groups_Ouminus,type,
% 2.88/1.56 uminus:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Orderings_Oord,type,
% 2.88/1.56 ord:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Orderings_Oorder,type,
% 2.88/1.56 order:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Groups_Ogroup__add,type,
% 2.88/1.56 group_add:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Orderings_Olinorder,type,
% 2.88/1.56 linorder:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Orderings_Opreorder,type,
% 2.88/1.56 preorder:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Groups_Oab__group__add,type,
% 2.88/1.56 ab_group_add:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Groups_Osemigroup__add,type,
% 2.88/1.56 semigroup_add:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 2.88/1.56 comm_monoid_add:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 2.88/1.56 ab_semigroup_add:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 2.88/1.56 boolean_algebra:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 2.88/1.56 cancel_semigroup_add:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.88/1.56 ordered_ab_group_add:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Topological__Spaces_Ot2__space,type,
% 2.88/1.56 topological_t2_space:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Limits_Otopological__group__add,type,
% 2.88/1.56 topolo799126099up_add:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Limits_Otopological__monoid__add,type,
% 2.88/1.56 topolo1314133330id_add:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 2.88/1.56 ordere779506340up_add:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
% 2.88/1.56 real_V2090557954_space:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Topological__Spaces_Ouniform__space,type,
% 2.88/1.56 topolo47006728_space:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.88/1.56 linord1659791738miring:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Topological__Spaces_Oorder__topology,type,
% 2.88/1.56 topolo259154727pology:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 2.88/1.56 comple187826305attice:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
% 2.88/1.56 comple1035589618norder:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Topological__Spaces_Olinorder__topology,type,
% 2.88/1.56 topolo2117631714pology:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Topological__Spaces_Otopological__space,type,
% 2.88/1.56 topolo503727757_space:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.88/1.56 canoni770627133id_add:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
% 2.88/1.56 real_V55928688vector:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 2.88/1.56 ordere236663937imp_le:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_cl_Topological__Spaces_Ofirst__countable__topology,type,
% 2.88/1.56 topolo2135403230pology:
% 2.88/1.56 !>[A: $tType] : $o ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
% 2.88/1.56 extend1396239628finity:
% 2.88/1.56 !>[A: $tType] : A ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Extended__Real_Oereal_Ocase__ereal,type,
% 2.88/1.56 extended_case_ereal:
% 2.88/1.56 !>[A: $tType] : ( ( real > A ) > A > A > extended_ereal > A ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Extended__Real_Oereal_Oereal,type,
% 2.88/1.56 extended_ereal2: real > extended_ereal ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Filter_Oat__bot,type,
% 2.88/1.56 at_bot:
% 2.88/1.56 !>[A: $tType] : ( filter @ A ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Filter_Oat__top,type,
% 2.88/1.56 at_top:
% 2.88/1.56 !>[A: $tType] : ( filter @ A ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Filter_Ofilterlim,type,
% 2.88/1.56 filterlim:
% 2.88/1.56 !>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Fun_Ocomp,type,
% 2.88/1.56 comp:
% 2.88/1.56 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.88/1.56 plus_plus:
% 2.88/1.56 !>[A: $tType] : ( A > A > A ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 2.88/1.56 uminus_uminus:
% 2.88/1.56 !>[A: $tType] : ( A > A ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_HOL_OEx1,type,
% 2.88/1.56 ex1:
% 2.88/1.56 !>[A: $tType] : ( ( A > $o ) > $o ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_HOL_OUniq,type,
% 2.88/1.56 uniq:
% 2.88/1.56 !>[A: $tType] : ( ( A > $o ) > $o ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Liminf__Limsup_OLiminf,type,
% 2.88/1.56 liminf_Liminf:
% 2.88/1.56 !>[A: $tType,B: $tType] : ( ( filter @ A ) > ( A > B ) > B ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc__at,type,
% 2.88/1.56 lower_582600101lsc_at:
% 2.88/1.56 !>[A: $tType,B: $tType] : ( A > ( A > B ) > $o ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Ousc__at,type,
% 2.88/1.56 lower_708585518usc_at:
% 2.88/1.56 !>[A: $tType,B: $tType] : ( A > ( A > B ) > $o ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 2.88/1.56 semiring_1_of_nat:
% 2.88/1.56 !>[A: $tType] : ( nat > A ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.88/1.56 ord_less_eq:
% 2.88/1.56 !>[A: $tType] : ( A > A > $o ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Set_OCollect,type,
% 2.88/1.56 collect:
% 2.88/1.56 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Topological__Spaces_Omonoseq,type,
% 2.88/1.56 topological_monoseq:
% 2.88/1.56 !>[A: $tType] : ( ( nat > A ) > $o ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
% 2.88/1.56 topolo1920029354e_nhds:
% 2.88/1.56 !>[A: $tType] : ( A > ( filter @ A ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocauchy__filter,type,
% 2.88/1.56 topolo1334162427filter:
% 2.88/1.56 !>[A: $tType] : ( ( filter @ A ) > $o ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_c_member,type,
% 2.88/1.56 member:
% 2.88/1.56 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.88/1.56
% 2.88/1.56 thf(sy_v_A____,type,
% 2.88/1.56 a2: extended_ereal ).
% 2.88/1.56
% 2.88/1.56 thf(sy_v_f,type,
% 2.88/1.56 f: a > extended_ereal ).
% 2.88/1.56
% 2.88/1.56 thf(sy_v_x0,type,
% 2.88/1.56 x0: a ).
% 2.88/1.56
% 2.88/1.56 thf(sy_v_x____,type,
% 2.88/1.56 x: nat > a ).
% 2.88/1.56
% 2.88/1.56 thf(fact_0_x__def_I1_J,axiom,
% 2.88/1.56 filterlim @ nat @ a @ x @ ( topolo1920029354e_nhds @ a @ x0 ) @ ( at_top @ nat ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_1_x__def_I2_J,axiom,
% 2.88/1.56 filterlim @ nat @ extended_ereal @ ( comp @ a @ extended_ereal @ nat @ f @ x ) @ ( topolo1920029354e_nhds @ extended_ereal @ a2 ) @ ( at_top @ nat ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_2__092_060open_062_092_060And_062F_O_A_I_If_A_092_060circ_062_Ax_J_A_092_060longlongrightarrow_062_AA_J_AF_A_092_060Longrightarrow_062_A_I_I_092_060lambda_062xa_O_A_N_A_If_A_092_060circ_062_Ax_J_Axa_J_A_092_060longlongrightarrow_062_A_N_AA_J_AF_092_060close_062,axiom,
% 2.88/1.56 ! [F: filter @ nat] :
% 2.88/1.56 ( ( filterlim @ nat @ extended_ereal @ ( comp @ a @ extended_ereal @ nat @ f @ x ) @ ( topolo1920029354e_nhds @ extended_ereal @ a2 ) @ F )
% 2.88/1.56 => ( filterlim @ nat @ extended_ereal
% 2.88/1.56 @ ^ [X: nat] : ( uminus_uminus @ extended_ereal @ ( comp @ a @ extended_ereal @ nat @ f @ x @ X ) )
% 2.88/1.56 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ a2 ) )
% 2.88/1.56 @ F ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_3_lsc,axiom,
% 2.88/1.56 ( lower_582600101lsc_at @ a @ extended_ereal @ x0
% 2.88/1.56 @ ^ [X: a] : ( uminus_uminus @ extended_ereal @ ( f @ X ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_4_tendsto__uminus__ereal,axiom,
% 2.88/1.56 ! [A: $tType,F2: A > extended_ereal,X2: extended_ereal,F: filter @ A] :
% 2.88/1.56 ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ X2 ) @ F )
% 2.88/1.56 => ( filterlim @ A @ extended_ereal
% 2.88/1.56 @ ^ [X: A] : ( uminus_uminus @ extended_ereal @ ( F2 @ X ) )
% 2.88/1.56 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ X2 ) )
% 2.88/1.56 @ F ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_5_tendsto__const,axiom,
% 2.88/1.56 ! [A: $tType,B: $tType] :
% 2.88/1.56 ( ( topolo503727757_space @ A )
% 2.88/1.56 => ! [K: A,F: filter @ B] :
% 2.88/1.56 ( filterlim @ B @ A
% 2.88/1.56 @ ^ [X: B] : K
% 2.88/1.56 @ ( topolo1920029354e_nhds @ A @ K )
% 2.88/1.56 @ F ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_6_ereal__Lim__uminus,axiom,
% 2.88/1.56 ! [A: $tType,F2: A > extended_ereal,F0: extended_ereal,Net: filter @ A] :
% 2.88/1.56 ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ F0 ) @ Net )
% 2.88/1.56 = ( filterlim @ A @ extended_ereal
% 2.88/1.56 @ ^ [X: A] : ( uminus_uminus @ extended_ereal @ ( F2 @ X ) )
% 2.88/1.56 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ F0 ) )
% 2.88/1.56 @ Net ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_7_LIMSEQ__const__iff,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( topological_t2_space @ A )
% 2.88/1.56 => ! [K: A,L: A] :
% 2.88/1.56 ( ( filterlim @ nat @ A
% 2.88/1.56 @ ^ [N: nat] : K
% 2.88/1.56 @ ( topolo1920029354e_nhds @ A @ L )
% 2.88/1.56 @ ( at_top @ nat ) )
% 2.88/1.56 = ( K = L ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_8_tendsto__minus__cancel,axiom,
% 2.88/1.56 ! [A: $tType,B: $tType] :
% 2.88/1.56 ( ( topolo799126099up_add @ A )
% 2.88/1.56 => ! [F2: B > A,A2: A,F: filter @ B] :
% 2.88/1.56 ( ( filterlim @ B @ A
% 2.88/1.56 @ ^ [X: B] : ( uminus_uminus @ A @ ( F2 @ X ) )
% 2.88/1.56 @ ( topolo1920029354e_nhds @ A @ ( uminus_uminus @ A @ A2 ) )
% 2.88/1.56 @ F )
% 2.88/1.56 => ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ A2 ) @ F ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_9_tendsto__minus__cancel__left,axiom,
% 2.88/1.56 ! [B: $tType,A: $tType] :
% 2.88/1.56 ( ( topolo799126099up_add @ B )
% 2.88/1.56 => ! [F2: A > B,Y: B,F: filter @ A] :
% 2.88/1.56 ( ( filterlim @ A @ B @ F2 @ ( topolo1920029354e_nhds @ B @ ( uminus_uminus @ B @ Y ) ) @ F )
% 2.88/1.56 = ( filterlim @ A @ B
% 2.88/1.56 @ ^ [X: A] : ( uminus_uminus @ B @ ( F2 @ X ) )
% 2.88/1.56 @ ( topolo1920029354e_nhds @ B @ Y )
% 2.88/1.56 @ F ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_10_tendsto__minus,axiom,
% 2.88/1.56 ! [A: $tType,B: $tType] :
% 2.88/1.56 ( ( topolo799126099up_add @ A )
% 2.88/1.56 => ! [F2: B > A,A2: A,F: filter @ B] :
% 2.88/1.56 ( ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ A2 ) @ F )
% 2.88/1.56 => ( filterlim @ B @ A
% 2.88/1.56 @ ^ [X: B] : ( uminus_uminus @ A @ ( F2 @ X ) )
% 2.88/1.56 @ ( topolo1920029354e_nhds @ A @ ( uminus_uminus @ A @ A2 ) )
% 2.88/1.56 @ F ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_11_LIMSEQ__unique,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( topological_t2_space @ A )
% 2.88/1.56 => ! [X3: nat > A,A2: A,B2: A] :
% 2.88/1.56 ( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ B2 ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( A2 = B2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_12_tendsto__uminus__nhds,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( topolo799126099up_add @ A )
% 2.88/1.56 => ! [A2: A] : ( filterlim @ A @ A @ ( uminus_uminus @ A ) @ ( topolo1920029354e_nhds @ A @ ( uminus_uminus @ A @ A2 ) ) @ ( topolo1920029354e_nhds @ A @ A2 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_13_ereal__uminus__eq__iff,axiom,
% 2.88/1.56 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.88/1.56 ( ( ( uminus_uminus @ extended_ereal @ A2 )
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ B2 ) )
% 2.88/1.56 = ( A2 = B2 ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_14_ereal__uminus__uminus,axiom,
% 2.88/1.56 ! [A2: extended_ereal] :
% 2.88/1.56 ( ( uminus_uminus @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) )
% 2.88/1.56 = A2 ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_15_verit__minus__simplify_I4_J,axiom,
% 2.88/1.56 ! [B: $tType] :
% 2.88/1.56 ( ( group_add @ B )
% 2.88/1.56 => ! [B2: B] :
% 2.88/1.56 ( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
% 2.88/1.56 = B2 ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_16_verit__negate__coefficient_I3_J,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( ordered_ab_group_add @ A )
% 2.88/1.56 => ! [A2: A,B2: A] :
% 2.88/1.56 ( ( A2 = B2 )
% 2.88/1.56 => ( ( uminus_uminus @ A @ A2 )
% 2.88/1.56 = ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_17_ereal__uminus__eq__reorder,axiom,
% 2.88/1.56 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.88/1.56 ( ( ( uminus_uminus @ extended_ereal @ A2 )
% 2.88/1.56 = B2 )
% 2.88/1.56 = ( A2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ B2 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_18_tendsto__eq__rhs,axiom,
% 2.88/1.56 ! [A: $tType,B: $tType] :
% 2.88/1.56 ( ( topolo503727757_space @ A )
% 2.88/1.56 => ! [F2: B > A,X2: A,F: filter @ B,Y: A] :
% 2.88/1.56 ( ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ X2 ) @ F )
% 2.88/1.56 => ( ( X2 = Y )
% 2.88/1.56 => ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ Y ) @ F ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_19_tendsto__cong__limit,axiom,
% 2.88/1.56 ! [B: $tType,A: $tType] :
% 2.88/1.56 ( ( topolo503727757_space @ B )
% 2.88/1.56 => ! [F2: A > B,L: B,F: filter @ A,K: B] :
% 2.88/1.56 ( ( filterlim @ A @ B @ F2 @ ( topolo1920029354e_nhds @ B @ L ) @ F )
% 2.88/1.56 => ( ( K = L )
% 2.88/1.56 => ( filterlim @ A @ B @ F2 @ ( topolo1920029354e_nhds @ B @ K ) @ F ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_20_comp__apply,axiom,
% 2.88/1.56 ! [C: $tType,A: $tType,B: $tType] :
% 2.88/1.56 ( ( comp @ B @ A @ C )
% 2.88/1.56 = ( ^ [F3: B > A,G: C > B,X: C] : ( F3 @ ( G @ X ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_21_compl__eq__compl__iff,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( boolean_algebra @ A )
% 2.88/1.56 => ! [X2: A,Y: A] :
% 2.88/1.56 ( ( ( uminus_uminus @ A @ X2 )
% 2.88/1.56 = ( uminus_uminus @ A @ Y ) )
% 2.88/1.56 = ( X2 = Y ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_22_double__compl,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( boolean_algebra @ A )
% 2.88/1.56 => ! [X2: A] :
% 2.88/1.56 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X2 ) )
% 2.88/1.56 = X2 ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_23_neg__equal__iff__equal,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( group_add @ A )
% 2.88/1.56 => ! [A2: A,B2: A] :
% 2.88/1.56 ( ( ( uminus_uminus @ A @ A2 )
% 2.88/1.56 = ( uminus_uminus @ A @ B2 ) )
% 2.88/1.56 = ( A2 = B2 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_24_add_Oinverse__inverse,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( group_add @ A )
% 2.88/1.56 => ! [A2: A] :
% 2.88/1.56 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 2.88/1.56 = A2 ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_25_uminus__apply,axiom,
% 2.88/1.56 ! [B: $tType,A: $tType] :
% 2.88/1.56 ( ( uminus @ B )
% 2.88/1.56 => ( ( uminus_uminus @ ( A > B ) )
% 2.88/1.56 = ( ^ [A3: A > B,X: A] : ( uminus_uminus @ B @ ( A3 @ X ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_26_lsc__sequentially__mem,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( real_V2090557954_space @ A )
% 2.88/1.56 => ! [X0: A,F2: A > extended_ereal,X2: nat > A,C2: nat > extended_ereal,C0: extended_ereal] :
% 2.88/1.56 ( ( lower_582600101lsc_at @ A @ extended_ereal @ X0 @ F2 )
% 2.88/1.56 => ( ( filterlim @ nat @ A @ X2 @ ( topolo1920029354e_nhds @ A @ X0 ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ( filterlim @ nat @ extended_ereal @ C2 @ ( topolo1920029354e_nhds @ extended_ereal @ C0 ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ! [N2: nat] : ( ord_less_eq @ extended_ereal @ ( F2 @ ( X2 @ N2 ) ) @ ( C2 @ N2 ) )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ ( F2 @ X0 ) @ C0 ) ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_27_lsc__sequentially__gen,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( real_V2090557954_space @ A )
% 2.88/1.56 => ( ( lower_582600101lsc_at @ A @ extended_ereal )
% 2.88/1.56 = ( ^ [X02: A,F3: A > extended_ereal] :
% 2.88/1.56 ! [X: nat > A,C3: nat > extended_ereal,C02: extended_ereal] :
% 2.88/1.56 ( ( ( filterlim @ nat @ A @ X @ ( topolo1920029354e_nhds @ A @ X02 ) @ ( at_top @ nat ) )
% 2.88/1.56 & ( filterlim @ nat @ extended_ereal @ C3 @ ( topolo1920029354e_nhds @ extended_ereal @ C02 ) @ ( at_top @ nat ) )
% 2.88/1.56 & ! [N: nat] : ( ord_less_eq @ extended_ereal @ ( F3 @ ( X @ N ) ) @ ( C3 @ N ) ) )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ ( F3 @ X02 ) @ C02 ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_28_lsc__at__mem,axiom,
% 2.88/1.56 ! [A: $tType,B: $tType] :
% 2.88/1.56 ( ( ( topolo259154727pology @ B )
% 2.88/1.56 & ( topolo503727757_space @ A ) )
% 2.88/1.56 => ! [X0: A,F2: A > B,X2: nat > A,A4: B] :
% 2.88/1.56 ( ( lower_582600101lsc_at @ A @ B @ X0 @ F2 )
% 2.88/1.56 => ( ( filterlim @ nat @ A @ X2 @ ( topolo1920029354e_nhds @ A @ X0 ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ( filterlim @ nat @ B @ ( comp @ A @ B @ nat @ F2 @ X2 ) @ ( topolo1920029354e_nhds @ B @ A4 ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ord_less_eq @ B @ ( F2 @ X0 ) @ A4 ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_29_lsc__at__def,axiom,
% 2.88/1.56 ! [B: $tType,A: $tType] :
% 2.88/1.56 ( ( ( topolo503727757_space @ A )
% 2.88/1.56 & ( topolo259154727pology @ B ) )
% 2.88/1.56 => ( ( lower_582600101lsc_at @ A @ B )
% 2.88/1.56 = ( ^ [X02: A,F3: A > B] :
% 2.88/1.56 ! [X4: nat > A,L2: B] :
% 2.88/1.56 ( ( ( filterlim @ nat @ A @ X4 @ ( topolo1920029354e_nhds @ A @ X02 ) @ ( at_top @ nat ) )
% 2.88/1.56 & ( filterlim @ nat @ B @ ( comp @ A @ B @ nat @ F3 @ X4 ) @ ( topolo1920029354e_nhds @ B @ L2 ) @ ( at_top @ nat ) ) )
% 2.88/1.56 => ( ord_less_eq @ B @ ( F3 @ X02 ) @ L2 ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_30_neg__le__iff__le,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( ordered_ab_group_add @ A )
% 2.88/1.56 => ! [B2: A,A2: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 2.88/1.56 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_31_compl__le__compl__iff,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( boolean_algebra @ A )
% 2.88/1.56 => ! [X2: A,Y: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y ) )
% 2.88/1.56 = ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_32_ereal__minus__le__minus,axiom,
% 2.88/1.56 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.88/1.56 ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) @ ( uminus_uminus @ extended_ereal @ B2 ) )
% 2.88/1.56 = ( ord_less_eq @ extended_ereal @ B2 @ A2 ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_33_verit__la__disequality,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( linorder @ A )
% 2.88/1.56 => ! [A2: A,B2: A] :
% 2.88/1.56 ( ( A2 = B2 )
% 2.88/1.56 | ~ ( ord_less_eq @ A @ A2 @ B2 )
% 2.88/1.56 | ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_34_ereal__complete__Inf,axiom,
% 2.88/1.56 ! [S: set @ extended_ereal] :
% 2.88/1.56 ? [X5: extended_ereal] :
% 2.88/1.56 ( ! [Xa: extended_ereal] :
% 2.88/1.56 ( ( member @ extended_ereal @ Xa @ S )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ X5 @ Xa ) )
% 2.88/1.56 & ! [Z: extended_ereal] :
% 2.88/1.56 ( ! [Xa2: extended_ereal] :
% 2.88/1.56 ( ( member @ extended_ereal @ Xa2 @ S )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ Z @ Xa2 ) )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ Z @ X5 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_35_ereal__complete__Sup,axiom,
% 2.88/1.56 ! [S: set @ extended_ereal] :
% 2.88/1.56 ? [X5: extended_ereal] :
% 2.88/1.56 ( ! [Xa: extended_ereal] :
% 2.88/1.56 ( ( member @ extended_ereal @ Xa @ S )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ Xa @ X5 ) )
% 2.88/1.56 & ! [Z: extended_ereal] :
% 2.88/1.56 ( ! [Xa2: extended_ereal] :
% 2.88/1.56 ( ( member @ extended_ereal @ Xa2 @ S )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ Xa2 @ Z ) )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ X5 @ Z ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_36_compl__mono,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( boolean_algebra @ A )
% 2.88/1.56 => ! [X2: A,Y: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.88/1.56 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_37_compl__le__swap1,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( boolean_algebra @ A )
% 2.88/1.56 => ! [Y: A,X2: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X2 ) )
% 2.88/1.56 => ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_38_compl__le__swap2,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( boolean_algebra @ A )
% 2.88/1.56 => ! [Y: A,X2: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X2 )
% 2.88/1.56 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ Y ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_39_le__minus__iff,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( ordered_ab_group_add @ A )
% 2.88/1.56 => ! [A2: A,B2: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 2.88/1.56 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_40_minus__le__iff,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( ordered_ab_group_add @ A )
% 2.88/1.56 => ! [A2: A,B2: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 2.88/1.56 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_41_le__imp__neg__le,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( ordered_ab_group_add @ A )
% 2.88/1.56 => ! [A2: A,B2: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.88/1.56 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_42_ereal__uminus__le__reorder,axiom,
% 2.88/1.56 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.88/1.56 ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) @ B2 )
% 2.88/1.56 = ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ B2 ) @ A2 ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_43_lim__mono,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( topolo2117631714pology @ A )
% 2.88/1.56 => ! [N3: nat,X3: nat > A,Y2: nat > A,X2: A,Y: A] :
% 2.88/1.56 ( ! [N2: nat] :
% 2.88/1.56 ( ( ord_less_eq @ nat @ N3 @ N2 )
% 2.88/1.56 => ( ord_less_eq @ A @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 2.88/1.56 => ( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ( filterlim @ nat @ A @ Y2 @ ( topolo1920029354e_nhds @ A @ Y ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_44_LIMSEQ__le,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( topolo2117631714pology @ A )
% 2.88/1.56 => ! [X3: nat > A,X2: A,Y2: nat > A,Y: A] :
% 2.88/1.56 ( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ( filterlim @ nat @ A @ Y2 @ ( topolo1920029354e_nhds @ A @ Y ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ? [N4: nat] :
% 2.88/1.56 ! [N2: nat] :
% 2.88/1.56 ( ( ord_less_eq @ nat @ N4 @ N2 )
% 2.88/1.56 => ( ord_less_eq @ A @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 2.88/1.56 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_45_mem__Collect__eq,axiom,
% 2.88/1.56 ! [A: $tType,A2: A,P: A > $o] :
% 2.88/1.56 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.88/1.56 = ( P @ A2 ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_46_Collect__mem__eq,axiom,
% 2.88/1.56 ! [A: $tType,A4: set @ A] :
% 2.88/1.56 ( ( collect @ A
% 2.88/1.56 @ ^ [X: A] : ( member @ A @ X @ A4 ) )
% 2.88/1.56 = A4 ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_47_Collect__cong,axiom,
% 2.88/1.56 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.88/1.56 ( ! [X5: A] :
% 2.88/1.56 ( ( P @ X5 )
% 2.88/1.56 = ( Q @ X5 ) )
% 2.88/1.56 => ( ( collect @ A @ P )
% 2.88/1.56 = ( collect @ A @ Q ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_48_ext,axiom,
% 2.88/1.56 ! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
% 2.88/1.56 ( ! [X5: A] :
% 2.88/1.56 ( ( F2 @ X5 )
% 2.88/1.56 = ( G2 @ X5 ) )
% 2.88/1.56 => ( F2 = G2 ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_49_Lim__bounded,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( topolo2117631714pology @ A )
% 2.88/1.56 => ! [F2: nat > A,L: A,M: nat,C4: A] :
% 2.88/1.56 ( ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ! [N2: nat] :
% 2.88/1.56 ( ( ord_less_eq @ nat @ M @ N2 )
% 2.88/1.56 => ( ord_less_eq @ A @ ( F2 @ N2 ) @ C4 ) )
% 2.88/1.56 => ( ord_less_eq @ A @ L @ C4 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_50_Lim__bounded2,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( topolo2117631714pology @ A )
% 2.88/1.56 => ! [F2: nat > A,L: A,N3: nat,C4: A] :
% 2.88/1.56 ( ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ! [N2: nat] :
% 2.88/1.56 ( ( ord_less_eq @ nat @ N3 @ N2 )
% 2.88/1.56 => ( ord_less_eq @ A @ C4 @ ( F2 @ N2 ) ) )
% 2.88/1.56 => ( ord_less_eq @ A @ C4 @ L ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_51_LIMSEQ__le__const,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( topolo2117631714pology @ A )
% 2.88/1.56 => ! [X3: nat > A,X2: A,A2: A] :
% 2.88/1.56 ( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ? [N4: nat] :
% 2.88/1.56 ! [N2: nat] :
% 2.88/1.56 ( ( ord_less_eq @ nat @ N4 @ N2 )
% 2.88/1.56 => ( ord_less_eq @ A @ A2 @ ( X3 @ N2 ) ) )
% 2.88/1.56 => ( ord_less_eq @ A @ A2 @ X2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_52_LIMSEQ__le__const2,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( topolo2117631714pology @ A )
% 2.88/1.56 => ! [X3: nat > A,X2: A,A2: A] :
% 2.88/1.56 ( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ? [N4: nat] :
% 2.88/1.56 ! [N2: nat] :
% 2.88/1.56 ( ( ord_less_eq @ nat @ N4 @ N2 )
% 2.88/1.56 => ( ord_less_eq @ A @ ( X3 @ N2 ) @ A2 ) )
% 2.88/1.56 => ( ord_less_eq @ A @ X2 @ A2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_53_tendsto__mono,axiom,
% 2.88/1.56 ! [A: $tType,B: $tType] :
% 2.88/1.56 ( ( topolo503727757_space @ A )
% 2.88/1.56 => ! [F: filter @ B,F4: filter @ B,F2: B > A,L: A] :
% 2.88/1.56 ( ( ord_less_eq @ ( filter @ B ) @ F @ F4 )
% 2.88/1.56 => ( ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L ) @ F4 )
% 2.88/1.56 => ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L ) @ F ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_54_fun__Compl__def,axiom,
% 2.88/1.56 ! [B: $tType,A: $tType] :
% 2.88/1.56 ( ( uminus @ B )
% 2.88/1.56 => ( ( uminus_uminus @ ( A > B ) )
% 2.88/1.56 = ( ^ [A3: A > B,X: A] : ( uminus_uminus @ B @ ( A3 @ X ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_55_equation__minus__iff,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( group_add @ A )
% 2.88/1.56 => ! [A2: A,B2: A] :
% 2.88/1.56 ( ( A2
% 2.88/1.56 = ( uminus_uminus @ A @ B2 ) )
% 2.88/1.56 = ( B2
% 2.88/1.56 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_56_minus__equation__iff,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( group_add @ A )
% 2.88/1.56 => ! [A2: A,B2: A] :
% 2.88/1.56 ( ( ( uminus_uminus @ A @ A2 )
% 2.88/1.56 = B2 )
% 2.88/1.56 = ( ( uminus_uminus @ A @ B2 )
% 2.88/1.56 = A2 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_57_comp__eq__dest__lhs,axiom,
% 2.88/1.56 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
% 2.88/1.56 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 2.88/1.56 = C2 )
% 2.88/1.56 => ( ( A2 @ ( B2 @ V ) )
% 2.88/1.56 = ( C2 @ V ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_58_comp__eq__elim,axiom,
% 2.88/1.56 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D] :
% 2.88/1.56 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 2.88/1.56 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 2.88/1.56 => ! [V2: A] :
% 2.88/1.56 ( ( A2 @ ( B2 @ V2 ) )
% 2.88/1.56 = ( C2 @ ( D2 @ V2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_59_comp__eq__dest,axiom,
% 2.88/1.56 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D,V: A] :
% 2.88/1.56 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 2.88/1.56 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 2.88/1.56 => ( ( A2 @ ( B2 @ V ) )
% 2.88/1.56 = ( C2 @ ( D2 @ V ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_60_comp__assoc,axiom,
% 2.88/1.56 ! [B: $tType,D: $tType,C: $tType,A: $tType,F2: D > B,G2: C > D,H: A > C] :
% 2.88/1.56 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F2 @ G2 ) @ H )
% 2.88/1.56 = ( comp @ D @ B @ A @ F2 @ ( comp @ C @ D @ A @ G2 @ H ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_61_comp__def,axiom,
% 2.88/1.56 ! [A: $tType,C: $tType,B: $tType] :
% 2.88/1.56 ( ( comp @ B @ C @ A )
% 2.88/1.56 = ( ^ [F3: B > C,G: A > B,X: A] : ( F3 @ ( G @ X ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_62_lsc__sequentially,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( real_V2090557954_space @ A )
% 2.88/1.56 => ( ( lower_582600101lsc_at @ A @ extended_ereal )
% 2.88/1.56 = ( ^ [X02: A,F3: A > extended_ereal] :
% 2.88/1.56 ! [X: nat > A,C3: extended_ereal] :
% 2.88/1.56 ( ( ( filterlim @ nat @ A @ X @ ( topolo1920029354e_nhds @ A @ X02 ) @ ( at_top @ nat ) )
% 2.88/1.56 & ! [N: nat] : ( ord_less_eq @ extended_ereal @ ( F3 @ ( X @ N ) ) @ C3 ) )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ ( F3 @ X02 ) @ C3 ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_63_usc__at__def,axiom,
% 2.88/1.56 ! [B: $tType,A: $tType] :
% 2.88/1.56 ( ( ( topolo503727757_space @ A )
% 2.88/1.56 & ( topolo259154727pology @ B ) )
% 2.88/1.56 => ( ( lower_708585518usc_at @ A @ B )
% 2.88/1.56 = ( ^ [X02: A,F3: A > B] :
% 2.88/1.56 ! [X4: nat > A,L2: B] :
% 2.88/1.56 ( ( ( filterlim @ nat @ A @ X4 @ ( topolo1920029354e_nhds @ A @ X02 ) @ ( at_top @ nat ) )
% 2.88/1.56 & ( filterlim @ nat @ B @ ( comp @ A @ B @ nat @ F3 @ X4 ) @ ( topolo1920029354e_nhds @ B @ L2 ) @ ( at_top @ nat ) ) )
% 2.88/1.56 => ( ord_less_eq @ B @ L2 @ ( F3 @ X02 ) ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_64_usc__at__mem,axiom,
% 2.88/1.56 ! [B: $tType,A: $tType] :
% 2.88/1.56 ( ( ( topolo503727757_space @ A )
% 2.88/1.56 & ( topolo259154727pology @ B ) )
% 2.88/1.56 => ! [X0: A,F2: A > B,X2: nat > A,A4: B] :
% 2.88/1.56 ( ( lower_708585518usc_at @ A @ B @ X0 @ F2 )
% 2.88/1.56 => ( ( filterlim @ nat @ A @ X2 @ ( topolo1920029354e_nhds @ A @ X0 ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ( filterlim @ nat @ B @ ( comp @ A @ B @ nat @ F2 @ X2 ) @ ( topolo1920029354e_nhds @ B @ A4 ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ord_less_eq @ B @ A4 @ ( F2 @ X0 ) ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_65_lim__decreasing__cl,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( ( comple1035589618norder @ A )
% 2.88/1.56 & ( topolo2117631714pology @ A ) )
% 2.88/1.56 => ! [F2: nat > A] :
% 2.88/1.56 ( ! [N2: nat,M2: nat] :
% 2.88/1.56 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.88/1.56 => ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( F2 @ M2 ) ) )
% 2.88/1.56 => ~ ! [L3: A] :
% 2.88/1.56 ~ ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L3 ) @ ( at_top @ nat ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_66_lim__increasing__cl,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( ( comple1035589618norder @ A )
% 2.88/1.56 & ( topolo2117631714pology @ A ) )
% 2.88/1.56 => ! [F2: nat > A] :
% 2.88/1.56 ( ! [N2: nat,M2: nat] :
% 2.88/1.56 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.88/1.56 => ( ord_less_eq @ A @ ( F2 @ M2 ) @ ( F2 @ N2 ) ) )
% 2.88/1.56 => ~ ! [L3: A] :
% 2.88/1.56 ~ ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L3 ) @ ( at_top @ nat ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_67_order__refl,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( preorder @ A )
% 2.88/1.56 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_68_monoseq__le,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( topolo2117631714pology @ A )
% 2.88/1.56 => ! [A2: nat > A,X2: A] :
% 2.88/1.56 ( ( topological_monoseq @ A @ A2 )
% 2.88/1.56 => ( ( filterlim @ nat @ A @ A2 @ ( topolo1920029354e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ( ! [N5: nat] : ( ord_less_eq @ A @ ( A2 @ N5 ) @ X2 )
% 2.88/1.56 & ! [M3: nat,N5: nat] :
% 2.88/1.56 ( ( ord_less_eq @ nat @ M3 @ N5 )
% 2.88/1.56 => ( ord_less_eq @ A @ ( A2 @ M3 ) @ ( A2 @ N5 ) ) ) )
% 2.88/1.56 | ( ! [N5: nat] : ( ord_less_eq @ A @ X2 @ ( A2 @ N5 ) )
% 2.88/1.56 & ! [M3: nat,N5: nat] :
% 2.88/1.56 ( ( ord_less_eq @ nat @ M3 @ N5 )
% 2.88/1.56 => ( ord_less_eq @ A @ ( A2 @ N5 ) @ ( A2 @ M3 ) ) ) ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_69_lsc__liminf,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( real_V2090557954_space @ A )
% 2.88/1.56 => ( ( lower_582600101lsc_at @ A @ extended_ereal )
% 2.88/1.56 = ( ^ [X02: A,F3: A > extended_ereal] :
% 2.88/1.56 ! [X: nat > A] :
% 2.88/1.56 ( ( filterlim @ nat @ A @ X @ ( topolo1920029354e_nhds @ A @ X02 ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ ( F3 @ X02 ) @ ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ ( comp @ A @ extended_ereal @ nat @ F3 @ X ) ) ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_70_lsc__imp__liminf,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( real_V2090557954_space @ A )
% 2.88/1.56 => ! [X0: A,F2: A > extended_ereal,X2: nat > A] :
% 2.88/1.56 ( ( lower_582600101lsc_at @ A @ extended_ereal @ X0 @ F2 )
% 2.88/1.56 => ( ( filterlim @ nat @ A @ X2 @ ( topolo1920029354e_nhds @ A @ X0 ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ ( F2 @ X0 ) @ ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ ( comp @ A @ extended_ereal @ nat @ F2 @ X2 ) ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_71_LIMSEQ__Uniq,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( topological_t2_space @ A )
% 2.88/1.56 => ! [X3: nat > A] :
% 2.88/1.56 ( uniq @ A
% 2.88/1.56 @ ^ [L2: A] : ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ L2 ) @ ( at_top @ nat ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_72_monoseq__def,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( order @ A )
% 2.88/1.56 => ( ( topological_monoseq @ A )
% 2.88/1.56 = ( ^ [X4: nat > A] :
% 2.88/1.56 ( ! [M4: nat,N: nat] :
% 2.88/1.56 ( ( ord_less_eq @ nat @ M4 @ N )
% 2.88/1.56 => ( ord_less_eq @ A @ ( X4 @ M4 ) @ ( X4 @ N ) ) )
% 2.88/1.56 | ! [M4: nat,N: nat] :
% 2.88/1.56 ( ( ord_less_eq @ nat @ M4 @ N )
% 2.88/1.56 => ( ord_less_eq @ A @ ( X4 @ N ) @ ( X4 @ M4 ) ) ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_73_monoI2,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( order @ A )
% 2.88/1.56 => ! [X3: nat > A] :
% 2.88/1.56 ( ! [M2: nat,N2: nat] :
% 2.88/1.56 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.88/1.56 => ( ord_less_eq @ A @ ( X3 @ N2 ) @ ( X3 @ M2 ) ) )
% 2.88/1.56 => ( topological_monoseq @ A @ X3 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_74_monoI1,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( order @ A )
% 2.88/1.56 => ! [X3: nat > A] :
% 2.88/1.56 ( ! [M2: nat,N2: nat] :
% 2.88/1.56 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.88/1.56 => ( ord_less_eq @ A @ ( X3 @ M2 ) @ ( X3 @ N2 ) ) )
% 2.88/1.56 => ( topological_monoseq @ A @ X3 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_75_monoseq__minus,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( ordered_ab_group_add @ A )
% 2.88/1.56 => ! [A2: nat > A] :
% 2.88/1.56 ( ( topological_monoseq @ A @ A2 )
% 2.88/1.56 => ( topological_monoseq @ A
% 2.88/1.56 @ ^ [N: nat] : ( uminus_uminus @ A @ ( A2 @ N ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_76_le__funD,axiom,
% 2.88/1.56 ! [B: $tType,A: $tType] :
% 2.88/1.56 ( ( ord @ B )
% 2.88/1.56 => ! [F2: A > B,G2: A > B,X2: A] :
% 2.88/1.56 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 2.88/1.56 => ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_77_le__funE,axiom,
% 2.88/1.56 ! [B: $tType,A: $tType] :
% 2.88/1.56 ( ( ord @ B )
% 2.88/1.56 => ! [F2: A > B,G2: A > B,X2: A] :
% 2.88/1.56 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 2.88/1.56 => ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_78_le__funI,axiom,
% 2.88/1.56 ! [B: $tType,A: $tType] :
% 2.88/1.56 ( ( ord @ B )
% 2.88/1.56 => ! [F2: A > B,G2: A > B] :
% 2.88/1.56 ( ! [X5: A] : ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( G2 @ X5 ) )
% 2.88/1.56 => ( ord_less_eq @ ( A > B ) @ F2 @ G2 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_79_le__fun__def,axiom,
% 2.88/1.56 ! [B: $tType,A: $tType] :
% 2.88/1.56 ( ( ord @ B )
% 2.88/1.56 => ( ( ord_less_eq @ ( A > B ) )
% 2.88/1.56 = ( ^ [F3: A > B,G: A > B] :
% 2.88/1.56 ! [X: A] : ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_80_order__subst1,axiom,
% 2.88/1.56 ! [A: $tType,B: $tType] :
% 2.88/1.56 ( ( ( order @ B )
% 2.88/1.56 & ( order @ A ) )
% 2.88/1.56 => ! [A2: A,F2: B > A,B2: B,C2: B] :
% 2.88/1.56 ( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) )
% 2.88/1.56 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 2.88/1.56 => ( ! [X5: B,Y3: B] :
% 2.88/1.56 ( ( ord_less_eq @ B @ X5 @ Y3 )
% 2.88/1.56 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 2.88/1.56 => ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_81_order__subst2,axiom,
% 2.88/1.56 ! [A: $tType,C: $tType] :
% 2.88/1.56 ( ( ( order @ C )
% 2.88/1.56 & ( order @ A ) )
% 2.88/1.56 => ! [A2: A,B2: A,F2: A > C,C2: C] :
% 2.88/1.56 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.88/1.56 => ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 )
% 2.88/1.56 => ( ! [X5: A,Y3: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 2.88/1.56 => ( ord_less_eq @ C @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 2.88/1.56 => ( ord_less_eq @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_82_ord__eq__le__subst,axiom,
% 2.88/1.56 ! [A: $tType,B: $tType] :
% 2.88/1.56 ( ( ( ord @ B )
% 2.88/1.56 & ( ord @ A ) )
% 2.88/1.56 => ! [A2: A,F2: B > A,B2: B,C2: B] :
% 2.88/1.56 ( ( A2
% 2.88/1.56 = ( F2 @ B2 ) )
% 2.88/1.56 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 2.88/1.56 => ( ! [X5: B,Y3: B] :
% 2.88/1.56 ( ( ord_less_eq @ B @ X5 @ Y3 )
% 2.88/1.56 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 2.88/1.56 => ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_83_ord__le__eq__subst,axiom,
% 2.88/1.56 ! [A: $tType,B: $tType] :
% 2.88/1.56 ( ( ( ord @ B )
% 2.88/1.56 & ( ord @ A ) )
% 2.88/1.56 => ! [A2: A,B2: A,F2: A > B,C2: B] :
% 2.88/1.56 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.88/1.56 => ( ( ( F2 @ B2 )
% 2.88/1.56 = C2 )
% 2.88/1.56 => ( ! [X5: A,Y3: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 2.88/1.56 => ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 2.88/1.56 => ( ord_less_eq @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_84_eq__iff,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( order @ A )
% 2.88/1.56 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.88/1.56 = ( ^ [X: A,Y5: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ X @ Y5 )
% 2.88/1.56 & ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_85_antisym,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( order @ A )
% 2.88/1.56 => ! [X2: A,Y: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.88/1.56 => ( ( ord_less_eq @ A @ Y @ X2 )
% 2.88/1.56 => ( X2 = Y ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_86_linear,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( linorder @ A )
% 2.88/1.56 => ! [X2: A,Y: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.88/1.56 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_87_eq__refl,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( preorder @ A )
% 2.88/1.56 => ! [X2: A,Y: A] :
% 2.88/1.56 ( ( X2 = Y )
% 2.88/1.56 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_88_le__cases,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( linorder @ A )
% 2.88/1.56 => ! [X2: A,Y: A] :
% 2.88/1.56 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 2.88/1.56 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_89_order_Otrans,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( order @ A )
% 2.88/1.56 => ! [A2: A,B2: A,C2: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.88/1.56 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 2.88/1.56 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_90_le__cases3,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( linorder @ A )
% 2.88/1.56 => ! [X2: A,Y: A,Z3: A] :
% 2.88/1.56 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 2.88/1.56 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 2.88/1.56 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 2.88/1.56 => ~ ( ord_less_eq @ A @ X2 @ Z3 ) )
% 2.88/1.56 => ( ( ( ord_less_eq @ A @ X2 @ Z3 )
% 2.88/1.56 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 2.88/1.56 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 2.88/1.56 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 2.88/1.56 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 2.88/1.56 => ~ ( ord_less_eq @ A @ Z3 @ X2 ) )
% 2.88/1.56 => ~ ( ( ord_less_eq @ A @ Z3 @ X2 )
% 2.88/1.56 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_91_antisym__conv,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( order @ A )
% 2.88/1.56 => ! [Y: A,X2: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ Y @ X2 )
% 2.88/1.56 => ( ( ord_less_eq @ A @ X2 @ Y )
% 2.88/1.56 = ( X2 = Y ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_92_order__class_Oorder_Oeq__iff,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( order @ A )
% 2.88/1.56 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.88/1.56 = ( ^ [A5: A,B3: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ A5 @ B3 )
% 2.88/1.56 & ( ord_less_eq @ A @ B3 @ A5 ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_93_ord__eq__le__trans,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( ord @ A )
% 2.88/1.56 => ! [A2: A,B2: A,C2: A] :
% 2.88/1.56 ( ( A2 = B2 )
% 2.88/1.56 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 2.88/1.56 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_94_ord__le__eq__trans,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( ord @ A )
% 2.88/1.56 => ! [A2: A,B2: A,C2: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.88/1.56 => ( ( B2 = C2 )
% 2.88/1.56 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_95_order__class_Oorder_Oantisym,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( order @ A )
% 2.88/1.56 => ! [A2: A,B2: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.88/1.56 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.88/1.56 => ( A2 = B2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_96_order__trans,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( preorder @ A )
% 2.88/1.56 => ! [X2: A,Y: A,Z3: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.88/1.56 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 2.88/1.56 => ( ord_less_eq @ A @ X2 @ Z3 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_97_dual__order_Orefl,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( order @ A )
% 2.88/1.56 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_98_linorder__wlog,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( linorder @ A )
% 2.88/1.56 => ! [P: A > A > $o,A2: A,B2: A] :
% 2.88/1.56 ( ! [A6: A,B4: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ A6 @ B4 )
% 2.88/1.56 => ( P @ A6 @ B4 ) )
% 2.88/1.56 => ( ! [A6: A,B4: A] :
% 2.88/1.56 ( ( P @ B4 @ A6 )
% 2.88/1.56 => ( P @ A6 @ B4 ) )
% 2.88/1.56 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_99_dual__order_Otrans,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( order @ A )
% 2.88/1.56 => ! [B2: A,A2: A,C2: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.88/1.56 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 2.88/1.56 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_100_dual__order_Oeq__iff,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( order @ A )
% 2.88/1.56 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.88/1.56 = ( ^ [A5: A,B3: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ B3 @ A5 )
% 2.88/1.56 & ( ord_less_eq @ A @ A5 @ B3 ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_101_dual__order_Oantisym,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( order @ A )
% 2.88/1.56 => ! [B2: A,A2: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.88/1.56 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.88/1.56 => ( A2 = B2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_102_filterlim__mono,axiom,
% 2.88/1.56 ! [B: $tType,A: $tType,F2: A > B,F22: filter @ B,F1: filter @ A,F23: filter @ B,F12: filter @ A] :
% 2.88/1.56 ( ( filterlim @ A @ B @ F2 @ F22 @ F1 )
% 2.88/1.56 => ( ( ord_less_eq @ ( filter @ B ) @ F22 @ F23 )
% 2.88/1.56 => ( ( ord_less_eq @ ( filter @ A ) @ F12 @ F1 )
% 2.88/1.56 => ( filterlim @ A @ B @ F2 @ F23 @ F12 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_103_nhds__imp__cauchy__filter,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( topolo47006728_space @ A )
% 2.88/1.56 => ! [F: filter @ A,X2: A] :
% 2.88/1.56 ( ( ord_less_eq @ ( filter @ A ) @ F @ ( topolo1920029354e_nhds @ A @ X2 ) )
% 2.88/1.56 => ( topolo1334162427filter @ A @ F ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_104_alt__ex1E_H,axiom,
% 2.88/1.56 ! [A: $tType,P: A > $o] :
% 2.88/1.56 ( ? [X6: A] :
% 2.88/1.56 ( ( P @ X6 )
% 2.88/1.56 & ! [Y3: A] :
% 2.88/1.56 ( ( P @ Y3 )
% 2.88/1.56 => ( Y3 = X6 ) ) )
% 2.88/1.56 => ~ ( ? [X_1: A] : ( P @ X_1 )
% 2.88/1.56 => ~ ( uniq @ A @ P ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_105_ex1__iff__ex__Uniq,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( ex1 @ A )
% 2.88/1.56 = ( ^ [P2: A > $o] :
% 2.88/1.56 ( ? [X7: A] : ( P2 @ X7 )
% 2.88/1.56 & ( uniq @ A @ P2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_106_liminf__PInfty,axiom,
% 2.88/1.56 ! [X3: nat > extended_ereal] :
% 2.88/1.56 ( ( filterlim @ nat @ extended_ereal @ X3 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( at_top @ nat ) )
% 2.88/1.56 = ( ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ X3 )
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_107_fun_Omap__ident,axiom,
% 2.88/1.56 ! [A: $tType,D: $tType,T: D > A] :
% 2.88/1.56 ( ( comp @ A @ A @ D
% 2.88/1.56 @ ^ [X: A] : X
% 2.88/1.56 @ T )
% 2.88/1.56 = T ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_108_K__record__comp,axiom,
% 2.88/1.56 ! [C: $tType,B: $tType,A: $tType,C2: B,F2: A > C] :
% 2.88/1.56 ( ( comp @ C @ B @ A
% 2.88/1.56 @ ^ [X: C] : C2
% 2.88/1.56 @ F2 )
% 2.88/1.56 = ( ^ [X: A] : C2 ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_109_ereal__infty__less__eq_I1_J,axiom,
% 2.88/1.56 ! [X2: extended_ereal] :
% 2.88/1.56 ( ( ord_less_eq @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ X2 )
% 2.88/1.56 = ( X2
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_110_ereal__infty__less__eq_I2_J,axiom,
% 2.88/1.56 ! [X2: extended_ereal] :
% 2.88/1.56 ( ( ord_less_eq @ extended_ereal @ X2 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 = ( X2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_111_MInfty__neq__PInfty_I1_J,axiom,
% 2.88/1.56 ( ( extend1396239628finity @ extended_ereal )
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_112_ereal__less__eq_I1_J,axiom,
% 2.88/1.56 ! [X2: extended_ereal] : ( ord_less_eq @ extended_ereal @ X2 @ ( extend1396239628finity @ extended_ereal ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_113_ereal__infty__less__eq2_I1_J,axiom,
% 2.88/1.56 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.88/1.56 ( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
% 2.88/1.56 => ( ( A2
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( B2
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_114_neq__PInf__trans,axiom,
% 2.88/1.56 ! [Y: extended_ereal,X2: extended_ereal] :
% 2.88/1.56 ( ( Y
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( ( ord_less_eq @ extended_ereal @ X2 @ Y )
% 2.88/1.56 => ( X2
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_115_ereal__infty__less__eq2_I2_J,axiom,
% 2.88/1.56 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.88/1.56 ( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
% 2.88/1.56 => ( ( B2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( A2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_116_ereal__less__eq_I2_J,axiom,
% 2.88/1.56 ! [X2: extended_ereal] : ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ X2 ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_117_lsc__at__MInfty,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( topolo503727757_space @ A )
% 2.88/1.56 => ! [F2: A > extended_ereal,X0: A] :
% 2.88/1.56 ( ( ( F2 @ X0 )
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( lower_582600101lsc_at @ A @ extended_ereal @ X0 @ F2 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_118_fun_Omap__comp,axiom,
% 2.88/1.56 ! [B: $tType,C: $tType,A: $tType,D: $tType,G2: B > C,F2: A > B,V: D > A] :
% 2.88/1.56 ( ( comp @ B @ C @ D @ G2 @ ( comp @ A @ B @ D @ F2 @ V ) )
% 2.88/1.56 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G2 @ F2 ) @ V ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_119_Uniq__def,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( uniq @ A )
% 2.88/1.56 = ( ^ [P2: A > $o] :
% 2.88/1.56 ! [X: A,Y5: A] :
% 2.88/1.56 ( ( P2 @ X )
% 2.88/1.56 => ( ( P2 @ Y5 )
% 2.88/1.56 => ( Y5 = X ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_120_Uniq__I,axiom,
% 2.88/1.56 ! [A: $tType,P: A > $o] :
% 2.88/1.56 ( ! [X5: A,Y3: A] :
% 2.88/1.56 ( ( P @ X5 )
% 2.88/1.56 => ( ( P @ Y3 )
% 2.88/1.56 => ( Y3 = X5 ) ) )
% 2.88/1.56 => ( uniq @ A @ P ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_121_Uniq__D,axiom,
% 2.88/1.56 ! [A: $tType,P: A > $o,A2: A,B2: A] :
% 2.88/1.56 ( ( uniq @ A @ P )
% 2.88/1.56 => ( ( P @ A2 )
% 2.88/1.56 => ( ( P @ B2 )
% 2.88/1.56 => ( A2 = B2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_122_filterlim__ident,axiom,
% 2.88/1.56 ! [A: $tType,F: filter @ A] :
% 2.88/1.56 ( filterlim @ A @ A
% 2.88/1.56 @ ^ [X: A] : X
% 2.88/1.56 @ F
% 2.88/1.56 @ F ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_123_filterlim__compose,axiom,
% 2.88/1.56 ! [B: $tType,A: $tType,C: $tType,G2: A > B,F32: filter @ B,F22: filter @ A,F2: C > A,F1: filter @ C] :
% 2.88/1.56 ( ( filterlim @ A @ B @ G2 @ F32 @ F22 )
% 2.88/1.56 => ( ( filterlim @ C @ A @ F2 @ F22 @ F1 )
% 2.88/1.56 => ( filterlim @ C @ B
% 2.88/1.56 @ ^ [X: C] : ( G2 @ ( F2 @ X ) )
% 2.88/1.56 @ F32
% 2.88/1.56 @ F1 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_124_Lim__MInfty,axiom,
% 2.88/1.56 ! [F2: nat > extended_ereal] :
% 2.88/1.56 ( ( filterlim @ nat @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) @ ( at_top @ nat ) )
% 2.88/1.56 = ( ! [B5: real] :
% 2.88/1.56 ? [N6: nat] :
% 2.88/1.56 ! [N: nat] :
% 2.88/1.56 ( ( ord_less_eq @ nat @ N6 @ N )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ ( F2 @ N ) @ ( extended_ereal2 @ B5 ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_125_MInfty__cases,axiom,
% 2.88/1.56 ! [A: $tType,F2: real > A,Y: A,Z3: A] :
% 2.88/1.56 ( ( extended_case_ereal @ A @ F2 @ Y @ Z3 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 = Z3 ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_126_Lim__bounded__MInfty,axiom,
% 2.88/1.56 ! [F2: nat > extended_ereal,L: extended_ereal,B6: real] :
% 2.88/1.56 ( ( filterlim @ nat @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ L ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ! [N2: nat] : ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ B6 ) @ ( F2 @ N2 ) )
% 2.88/1.56 => ( L
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_127_Lim__PInfty,axiom,
% 2.88/1.56 ! [F2: nat > extended_ereal] :
% 2.88/1.56 ( ( filterlim @ nat @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( at_top @ nat ) )
% 2.88/1.56 = ( ! [B5: real] :
% 2.88/1.56 ? [N6: nat] :
% 2.88/1.56 ! [N: nat] :
% 2.88/1.56 ( ( ord_less_eq @ nat @ N6 @ N )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ B5 ) @ ( F2 @ N ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_128_Lim__bounded__PInfty2,axiom,
% 2.88/1.56 ! [F2: nat > extended_ereal,L: extended_ereal,N3: nat,B6: real] :
% 2.88/1.56 ( ( filterlim @ nat @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ L ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ! [N2: nat] :
% 2.88/1.56 ( ( ord_less_eq @ nat @ N3 @ N2 )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ ( F2 @ N2 ) @ ( extended_ereal2 @ B6 ) ) )
% 2.88/1.56 => ( L
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_129_ereal_Oinject,axiom,
% 2.88/1.56 ! [X1: real,Y1: real] :
% 2.88/1.56 ( ( ( extended_ereal2 @ X1 )
% 2.88/1.56 = ( extended_ereal2 @ Y1 ) )
% 2.88/1.56 = ( X1 = Y1 ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_130_ereal__cong,axiom,
% 2.88/1.56 ! [X2: real,Y: real] :
% 2.88/1.56 ( ( X2 = Y )
% 2.88/1.56 => ( ( extended_ereal2 @ X2 )
% 2.88/1.56 = ( extended_ereal2 @ Y ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_131_ereal__less__eq_I3_J,axiom,
% 2.88/1.56 ! [R: real,P3: real] :
% 2.88/1.56 ( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P3 ) )
% 2.88/1.56 = ( ord_less_eq @ real @ R @ P3 ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_132_PInfty__cases,axiom,
% 2.88/1.56 ! [A: $tType,F2: real > A,Y: A,Z3: A] :
% 2.88/1.56 ( ( extended_case_ereal @ A @ F2 @ Y @ Z3 @ ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 = Y ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_133_tendsto__ereal,axiom,
% 2.88/1.56 ! [A: $tType,F2: A > real,X2: real,F: filter @ A] :
% 2.88/1.56 ( ( filterlim @ A @ real @ F2 @ ( topolo1920029354e_nhds @ real @ X2 ) @ F )
% 2.88/1.56 => ( filterlim @ A @ extended_ereal
% 2.88/1.56 @ ^ [X: A] : ( extended_ereal2 @ ( F2 @ X ) )
% 2.88/1.56 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extended_ereal2 @ X2 ) )
% 2.88/1.56 @ F ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_134_lim__ereal,axiom,
% 2.88/1.56 ! [A: $tType,F2: A > real,X2: real,Net: filter @ A] :
% 2.88/1.56 ( ( filterlim @ A @ extended_ereal
% 2.88/1.56 @ ^ [N: A] : ( extended_ereal2 @ ( F2 @ N ) )
% 2.88/1.56 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extended_ereal2 @ X2 ) )
% 2.88/1.56 @ Net )
% 2.88/1.56 = ( filterlim @ A @ real @ F2 @ ( topolo1920029354e_nhds @ real @ X2 ) @ Net ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_135_PInfty__neq__ereal_I1_J,axiom,
% 2.88/1.56 ! [R: real] :
% 2.88/1.56 ( ( extended_ereal2 @ R )
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_136_ereal_Ocase__distrib,axiom,
% 2.88/1.56 ! [A: $tType,B: $tType,H: A > B,F13: real > A,F24: A,F33: A,Ereal: extended_ereal] :
% 2.88/1.56 ( ( H @ ( extended_case_ereal @ A @ F13 @ F24 @ F33 @ Ereal ) )
% 2.88/1.56 = ( extended_case_ereal @ B
% 2.88/1.56 @ ^ [X: real] : ( H @ ( F13 @ X ) )
% 2.88/1.56 @ ( H @ F24 )
% 2.88/1.56 @ ( H @ F33 )
% 2.88/1.56 @ Ereal ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_137_uminus__ereal_Osimps_I1_J,axiom,
% 2.88/1.56 ! [R: real] :
% 2.88/1.56 ( ( uminus_uminus @ extended_ereal @ ( extended_ereal2 @ R ) )
% 2.88/1.56 = ( extended_ereal2 @ ( uminus_uminus @ real @ R ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_138_ereal__le__le,axiom,
% 2.88/1.56 ! [Y: real,A2: extended_ereal,X2: real] :
% 2.88/1.56 ( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ Y ) @ A2 )
% 2.88/1.56 => ( ( ord_less_eq @ real @ X2 @ Y )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ X2 ) @ A2 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_139_le__ereal__le,axiom,
% 2.88/1.56 ! [A2: extended_ereal,X2: real,Y: real] :
% 2.88/1.56 ( ( ord_less_eq @ extended_ereal @ A2 @ ( extended_ereal2 @ X2 ) )
% 2.88/1.56 => ( ( ord_less_eq @ real @ X2 @ Y )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ A2 @ ( extended_ereal2 @ Y ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_140_ereal__le__real,axiom,
% 2.88/1.56 ! [X2: extended_ereal,Y: extended_ereal] :
% 2.88/1.56 ( ! [Z4: real] :
% 2.88/1.56 ( ( ord_less_eq @ extended_ereal @ X2 @ ( extended_ereal2 @ Z4 ) )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ Y @ ( extended_ereal2 @ Z4 ) ) )
% 2.88/1.56 => ( ord_less_eq @ extended_ereal @ Y @ X2 ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_141_ereal_Osimps_I8_J,axiom,
% 2.88/1.56 ! [A: $tType,F13: real > A,F24: A,F33: A,X1: real] :
% 2.88/1.56 ( ( extended_case_ereal @ A @ F13 @ F24 @ F33 @ ( extended_ereal2 @ X1 ) )
% 2.88/1.56 = ( F13 @ X1 ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_142_ereal__semiline__unique,axiom,
% 2.88/1.56 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.88/1.56 ( ( ( collect @ real
% 2.88/1.56 @ ^ [Y5: real] : ( ord_less_eq @ extended_ereal @ A2 @ ( extended_ereal2 @ Y5 ) ) )
% 2.88/1.56 = ( collect @ real
% 2.88/1.56 @ ^ [Y5: real] : ( ord_less_eq @ extended_ereal @ B2 @ ( extended_ereal2 @ Y5 ) ) ) )
% 2.88/1.56 = ( A2 = B2 ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_143_real__of__ereal_Oinduct,axiom,
% 2.88/1.56 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 2.88/1.56 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( ( P @ ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( P @ A0 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_144_real__of__ereal_Ocases,axiom,
% 2.88/1.56 ! [X2: extended_ereal] :
% 2.88/1.56 ( ! [R2: real] :
% 2.88/1.56 ( X2
% 2.88/1.56 != ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( ( X2
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( X2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_145_times__ereal_Oinduct,axiom,
% 2.88/1.56 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 2.88/1.56 ( ! [R2: real,P4: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P4 ) )
% 2.88/1.56 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( ! [R2: real] : ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ! [R2: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_146_plus__ereal_Oinduct,axiom,
% 2.88/1.56 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 2.88/1.56 ( ! [R2: real,P4: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P4 ) )
% 2.88/1.56 => ( ! [X_1: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_1 )
% 2.88/1.56 => ( ! [A6: extended_ereal] : ( P @ A6 @ ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ! [P4: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ P4 ) )
% 2.88/1.56 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_147_less__ereal_Oinduct,axiom,
% 2.88/1.56 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 2.88/1.56 ( ! [X5: real,Y3: real] : ( P @ ( extended_ereal2 @ X5 ) @ ( extended_ereal2 @ Y3 ) )
% 2.88/1.56 => ( ! [X_1: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_1 )
% 2.88/1.56 => ( ! [A6: extended_ereal] : ( P @ A6 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ! [X5: real] : ( P @ ( extended_ereal2 @ X5 ) @ ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( ! [R2: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_148_abs__ereal_Oinduct,axiom,
% 2.88/1.56 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 2.88/1.56 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ( P @ ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( P @ A0 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_149_ereal__all__split,axiom,
% 2.88/1.56 ( ( ^ [P5: extended_ereal > $o] :
% 2.88/1.56 ! [X7: extended_ereal] : ( P5 @ X7 ) )
% 2.88/1.56 = ( ^ [P2: extended_ereal > $o] :
% 2.88/1.56 ( ( P2 @ ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 & ! [X: real] : ( P2 @ ( extended_ereal2 @ X ) )
% 2.88/1.56 & ( P2 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_150_abs__ereal_Ocases,axiom,
% 2.88/1.56 ! [X2: extended_ereal] :
% 2.88/1.56 ( ! [R2: real] :
% 2.88/1.56 ( X2
% 2.88/1.56 != ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( ( X2
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( X2
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_151_ereal__ex__split,axiom,
% 2.88/1.56 ( ( ^ [P5: extended_ereal > $o] :
% 2.88/1.56 ? [X7: extended_ereal] : ( P5 @ X7 ) )
% 2.88/1.56 = ( ^ [P2: extended_ereal > $o] :
% 2.88/1.56 ( ( P2 @ ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 | ? [X: real] : ( P2 @ ( extended_ereal2 @ X ) )
% 2.88/1.56 | ( P2 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_152_ereal3__cases,axiom,
% 2.88/1.56 ! [X2: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
% 2.88/1.56 ( ( ? [R2: real] :
% 2.88/1.56 ( X2
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( ? [Ra: real] :
% 2.88/1.56 ( Xa3
% 2.88/1.56 = ( extended_ereal2 @ Ra ) )
% 2.88/1.56 => ! [Rb: real] :
% 2.88/1.56 ( Xb
% 2.88/1.56 != ( extended_ereal2 @ Rb ) ) ) )
% 2.88/1.56 => ( ( ? [R2: real] :
% 2.88/1.56 ( X2
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( ? [Ra: real] :
% 2.88/1.56 ( Xa3
% 2.88/1.56 = ( extended_ereal2 @ Ra ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.88/1.56 => ( ( ? [R2: real] :
% 2.88/1.56 ( X2
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( ? [Ra: real] :
% 2.88/1.56 ( Xa3
% 2.88/1.56 = ( extended_ereal2 @ Ra ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.88/1.56 => ( ( ? [R2: real] :
% 2.88/1.56 ( X2
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ! [Ra: real] :
% 2.88/1.56 ( Xb
% 2.88/1.56 != ( extended_ereal2 @ Ra ) ) ) )
% 2.88/1.56 => ( ( ? [R2: real] :
% 2.88/1.56 ( X2
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.88/1.56 => ( ( ? [R2: real] :
% 2.88/1.56 ( X2
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.88/1.56 => ( ( ? [R2: real] :
% 2.88/1.56 ( X2
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ! [Ra: real] :
% 2.88/1.56 ( Xb
% 2.88/1.56 != ( extended_ereal2 @ Ra ) ) ) )
% 2.88/1.56 => ( ( ? [R2: real] :
% 2.88/1.56 ( X2
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.88/1.56 => ( ( ? [R2: real] :
% 2.88/1.56 ( X2
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( ? [R2: real] :
% 2.88/1.56 ( Xa3
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ! [Ra: real] :
% 2.88/1.56 ( Xb
% 2.88/1.56 != ( extended_ereal2 @ Ra ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( ? [R2: real] :
% 2.88/1.56 ( Xa3
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( ? [R2: real] :
% 2.88/1.56 ( Xa3
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ! [R2: real] :
% 2.88/1.56 ( Xb
% 2.88/1.56 != ( extended_ereal2 @ R2 ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ! [R2: real] :
% 2.88/1.56 ( Xb
% 2.88/1.56 != ( extended_ereal2 @ R2 ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ? [R2: real] :
% 2.88/1.56 ( Xa3
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ! [Ra: real] :
% 2.88/1.56 ( Xb
% 2.88/1.56 != ( extended_ereal2 @ Ra ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ? [R2: real] :
% 2.88/1.56 ( Xa3
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ? [R2: real] :
% 2.88/1.56 ( Xa3
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ! [R2: real] :
% 2.88/1.56 ( Xb
% 2.88/1.56 != ( extended_ereal2 @ R2 ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ! [R2: real] :
% 2.88/1.56 ( Xb
% 2.88/1.56 != ( extended_ereal2 @ R2 ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.88/1.56 => ~ ( ( X2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ( Xa3
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( Xb
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_153_ereal2__cases,axiom,
% 2.88/1.56 ! [X2: extended_ereal,Xa3: extended_ereal] :
% 2.88/1.56 ( ( ? [R2: real] :
% 2.88/1.56 ( X2
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ! [Ra: real] :
% 2.88/1.56 ( Xa3
% 2.88/1.56 != ( extended_ereal2 @ Ra ) ) )
% 2.88/1.56 => ( ( ? [R2: real] :
% 2.88/1.56 ( X2
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( Xa3
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ( ? [R2: real] :
% 2.88/1.56 ( X2
% 2.88/1.56 = ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( Xa3
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ! [R2: real] :
% 2.88/1.56 ( Xa3
% 2.88/1.56 != ( extended_ereal2 @ R2 ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( Xa3
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( Xa3
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ! [R2: real] :
% 2.88/1.56 ( Xa3
% 2.88/1.56 != ( extended_ereal2 @ R2 ) ) )
% 2.88/1.56 => ( ( ( X2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( Xa3
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ~ ( ( X2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( Xa3
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_154_ereal__cases,axiom,
% 2.88/1.56 ! [X2: extended_ereal] :
% 2.88/1.56 ( ! [R2: real] :
% 2.88/1.56 ( X2
% 2.88/1.56 != ( extended_ereal2 @ R2 ) )
% 2.88/1.56 => ( ( X2
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.56 => ( X2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_155_MInfty__neq__ereal_I1_J,axiom,
% 2.88/1.56 ! [R: real] :
% 2.88/1.56 ( ( extended_ereal2 @ R )
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_156_ereal__top,axiom,
% 2.88/1.56 ! [X2: extended_ereal] :
% 2.88/1.56 ( ! [B7: real] : ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ B7 ) @ X2 )
% 2.88/1.56 => ( X2
% 2.88/1.56 = ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_157_ereal__tendsto__simps2_I1_J,axiom,
% 2.88/1.56 ! [A: $tType,F2: A > real,A2: real,F: filter @ A] :
% 2.88/1.56 ( ( filterlim @ A @ extended_ereal @ ( comp @ real @ extended_ereal @ A @ extended_ereal2 @ F2 ) @ ( topolo1920029354e_nhds @ extended_ereal @ ( extended_ereal2 @ A2 ) ) @ F )
% 2.88/1.56 = ( filterlim @ A @ real @ F2 @ ( topolo1920029354e_nhds @ real @ A2 ) @ F ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_158_ereal__bot,axiom,
% 2.88/1.56 ! [X2: extended_ereal] :
% 2.88/1.56 ( ! [B7: real] : ( ord_less_eq @ extended_ereal @ X2 @ ( extended_ereal2 @ B7 ) )
% 2.88/1.56 => ( X2
% 2.88/1.56 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_159_Lim__bounded__PInfty,axiom,
% 2.88/1.56 ! [F2: nat > extended_ereal,L: extended_ereal,B6: real] :
% 2.88/1.56 ( ( filterlim @ nat @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ L ) @ ( at_top @ nat ) )
% 2.88/1.56 => ( ! [N2: nat] : ( ord_less_eq @ extended_ereal @ ( F2 @ N2 ) @ ( extended_ereal2 @ B6 ) )
% 2.88/1.56 => ( L
% 2.88/1.56 != ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_160_ereal__tendsto__simps2_I3_J,axiom,
% 2.88/1.56 ! [A: $tType,F2: A > real,F: filter @ A] :
% 2.88/1.56 ( ( filterlim @ A @ extended_ereal @ ( comp @ real @ extended_ereal @ A @ extended_ereal2 @ F2 ) @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) @ F )
% 2.88/1.56 = ( filterlim @ A @ real @ F2 @ ( at_bot @ real ) @ F ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_161_tendsto__cadd__ereal,axiom,
% 2.88/1.56 ! [A: $tType,Y: extended_ereal,X2: extended_ereal,F2: A > extended_ereal,F: filter @ A] :
% 2.88/1.56 ( ( Y
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ( X2
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ X2 ) @ F )
% 2.88/1.56 => ( filterlim @ A @ extended_ereal
% 2.88/1.56 @ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F2 @ X ) @ Y )
% 2.88/1.56 @ ( topolo1920029354e_nhds @ extended_ereal @ ( plus_plus @ extended_ereal @ X2 @ Y ) )
% 2.88/1.56 @ F ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_162_ereal__minus__real__tendsto__MInf,axiom,
% 2.88/1.56 ( filterlim @ nat @ extended_ereal
% 2.88/1.56 @ ^ [X: nat] : ( extended_ereal2 @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ X ) ) )
% 2.88/1.56 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 @ ( at_top @ nat ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_163_tendsto__add__ereal__nonneg,axiom,
% 2.88/1.56 ! [A: $tType,X2: extended_ereal,Y: extended_ereal,F2: A > extended_ereal,F: filter @ A,G2: A > extended_ereal] :
% 2.88/1.56 ( ( X2
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ( Y
% 2.88/1.56 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.56 => ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ X2 ) @ F )
% 2.88/1.56 => ( ( filterlim @ A @ extended_ereal @ G2 @ ( topolo1920029354e_nhds @ extended_ereal @ Y ) @ F )
% 2.88/1.56 => ( filterlim @ A @ extended_ereal
% 2.88/1.56 @ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F2 @ X ) @ ( G2 @ X ) )
% 2.88/1.56 @ ( topolo1920029354e_nhds @ extended_ereal @ ( plus_plus @ extended_ereal @ X2 @ Y ) )
% 2.88/1.56 @ F ) ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_164_add__left__cancel,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( cancel_semigroup_add @ A )
% 2.88/1.56 => ! [A2: A,B2: A,C2: A] :
% 2.88/1.56 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.88/1.56 = ( plus_plus @ A @ A2 @ C2 ) )
% 2.88/1.56 = ( B2 = C2 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_165_add__right__cancel,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( cancel_semigroup_add @ A )
% 2.88/1.56 => ! [B2: A,A2: A,C2: A] :
% 2.88/1.56 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.88/1.56 = ( plus_plus @ A @ C2 @ A2 ) )
% 2.88/1.56 = ( B2 = C2 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_166_add__le__cancel__left,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( ordere236663937imp_le @ A )
% 2.88/1.56 => ! [C2: A,A2: A,B2: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 2.88/1.56 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_167_add__le__cancel__right,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( ordere236663937imp_le @ A )
% 2.88/1.56 => ! [A2: A,C2: A,B2: A] :
% 2.88/1.56 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 2.88/1.56 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_168_minus__add__distrib,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( ab_group_add @ A )
% 2.88/1.56 => ! [A2: A,B2: A] :
% 2.88/1.56 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.88/1.56 = ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 2.88/1.56
% 2.88/1.56 thf(fact_169_minus__add__cancel,axiom,
% 2.88/1.56 ! [A: $tType] :
% 2.88/1.56 ( ( group_add @ A )
% 2.88/1.56 => ! [A2: A,B2: A] :
% 2.88/1.57 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.88/1.57 = B2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_170_add__minus__cancel,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( group_add @ A )
% 2.88/1.57 => ! [A2: A,B2: A] :
% 2.88/1.57 ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
% 2.88/1.57 = B2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_171_ereal__PInfty__eq__plus,axiom,
% 2.88/1.57 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.88/1.57 ( ( ( extend1396239628finity @ extended_ereal )
% 2.88/1.57 = ( plus_plus @ extended_ereal @ A2 @ B2 ) )
% 2.88/1.57 = ( ( A2
% 2.88/1.57 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 | ( B2
% 2.88/1.57 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_172_ereal__plus__eq__PInfty,axiom,
% 2.88/1.57 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.88/1.57 ( ( ( plus_plus @ extended_ereal @ A2 @ B2 )
% 2.88/1.57 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 = ( ( A2
% 2.88/1.57 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 | ( B2
% 2.88/1.57 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_173_ereal__MInfty__eq__plus,axiom,
% 2.88/1.57 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.88/1.57 ( ( ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 = ( plus_plus @ extended_ereal @ A2 @ B2 ) )
% 2.88/1.57 = ( ( ( A2
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 & ( B2
% 2.88/1.57 != ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 | ( ( B2
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 & ( A2
% 2.88/1.57 != ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_174_ereal__plus__eq__MInfty,axiom,
% 2.88/1.57 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.88/1.57 ( ( ( plus_plus @ extended_ereal @ A2 @ B2 )
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 = ( ( ( A2
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 | ( B2
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.88/1.57 & ( A2
% 2.88/1.57 != ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 & ( B2
% 2.88/1.57 != ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_175_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ab_semigroup_add @ A )
% 2.88/1.57 => ! [A2: A,B2: A,C2: A] :
% 2.88/1.57 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 2.88/1.57 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_176_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ordere779506340up_add @ A )
% 2.88/1.57 => ! [I: A,J: A,K: A,L: A] :
% 2.88/1.57 ( ( ( I = J )
% 2.88/1.57 & ( K = L ) )
% 2.88/1.57 => ( ( plus_plus @ A @ I @ K )
% 2.88/1.57 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_177_group__cancel_Oadd1,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( comm_monoid_add @ A )
% 2.88/1.57 => ! [A4: A,K: A,A2: A,B2: A] :
% 2.88/1.57 ( ( A4
% 2.88/1.57 = ( plus_plus @ A @ K @ A2 ) )
% 2.88/1.57 => ( ( plus_plus @ A @ A4 @ B2 )
% 2.88/1.57 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_178_group__cancel_Oadd2,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( comm_monoid_add @ A )
% 2.88/1.57 => ! [B6: A,K: A,B2: A,A2: A] :
% 2.88/1.57 ( ( B6
% 2.88/1.57 = ( plus_plus @ A @ K @ B2 ) )
% 2.88/1.57 => ( ( plus_plus @ A @ A2 @ B6 )
% 2.88/1.57 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_179_add_Oassoc,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( semigroup_add @ A )
% 2.88/1.57 => ! [A2: A,B2: A,C2: A] :
% 2.88/1.57 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 2.88/1.57 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_180_add_Oleft__cancel,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( group_add @ A )
% 2.88/1.57 => ! [A2: A,B2: A,C2: A] :
% 2.88/1.57 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.88/1.57 = ( plus_plus @ A @ A2 @ C2 ) )
% 2.88/1.57 = ( B2 = C2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_181_add_Oright__cancel,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( group_add @ A )
% 2.88/1.57 => ! [B2: A,A2: A,C2: A] :
% 2.88/1.57 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.88/1.57 = ( plus_plus @ A @ C2 @ A2 ) )
% 2.88/1.57 = ( B2 = C2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_182_add_Ocommute,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ab_semigroup_add @ A )
% 2.88/1.57 => ( ( plus_plus @ A )
% 2.88/1.57 = ( ^ [A5: A,B3: A] : ( plus_plus @ A @ B3 @ A5 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_183_add_Oleft__commute,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ab_semigroup_add @ A )
% 2.88/1.57 => ! [B2: A,A2: A,C2: A] :
% 2.88/1.57 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
% 2.88/1.57 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_184_add__left__imp__eq,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( cancel_semigroup_add @ A )
% 2.88/1.57 => ! [A2: A,B2: A,C2: A] :
% 2.88/1.57 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.88/1.57 = ( plus_plus @ A @ A2 @ C2 ) )
% 2.88/1.57 => ( B2 = C2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_185_add__right__imp__eq,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( cancel_semigroup_add @ A )
% 2.88/1.57 => ! [B2: A,A2: A,C2: A] :
% 2.88/1.57 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.88/1.57 = ( plus_plus @ A @ C2 @ A2 ) )
% 2.88/1.57 => ( B2 = C2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_186_add_Oinverse__distrib__swap,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( group_add @ A )
% 2.88/1.57 => ! [A2: A,B2: A] :
% 2.88/1.57 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.88/1.57 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_187_group__cancel_Oneg1,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ab_group_add @ A )
% 2.88/1.57 => ! [A4: A,K: A,A2: A] :
% 2.88/1.57 ( ( A4
% 2.88/1.57 = ( plus_plus @ A @ K @ A2 ) )
% 2.88/1.57 => ( ( uminus_uminus @ A @ A4 )
% 2.88/1.57 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_188_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ordere779506340up_add @ A )
% 2.88/1.57 => ! [I: A,J: A,K: A,L: A] :
% 2.88/1.57 ( ( ( ord_less_eq @ A @ I @ J )
% 2.88/1.57 & ( K = L ) )
% 2.88/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_189_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ordere779506340up_add @ A )
% 2.88/1.57 => ! [I: A,J: A,K: A,L: A] :
% 2.88/1.57 ( ( ( I = J )
% 2.88/1.57 & ( ord_less_eq @ A @ K @ L ) )
% 2.88/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_190_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ordere779506340up_add @ A )
% 2.88/1.57 => ! [I: A,J: A,K: A,L: A] :
% 2.88/1.57 ( ( ( ord_less_eq @ A @ I @ J )
% 2.88/1.57 & ( ord_less_eq @ A @ K @ L ) )
% 2.88/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_191_add__mono,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ordere779506340up_add @ A )
% 2.88/1.57 => ! [A2: A,B2: A,C2: A,D2: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.88/1.57 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 2.88/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_192_add__left__mono,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ordere779506340up_add @ A )
% 2.88/1.57 => ! [A2: A,B2: A,C2: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.88/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_193_less__eqE,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( canoni770627133id_add @ A )
% 2.88/1.57 => ! [A2: A,B2: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.88/1.57 => ~ ! [C5: A] :
% 2.88/1.57 ( B2
% 2.88/1.57 != ( plus_plus @ A @ A2 @ C5 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_194_add__right__mono,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ordere779506340up_add @ A )
% 2.88/1.57 => ! [A2: A,B2: A,C2: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.88/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_195_le__iff__add,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( canoni770627133id_add @ A )
% 2.88/1.57 => ( ( ord_less_eq @ A )
% 2.88/1.57 = ( ^ [A5: A,B3: A] :
% 2.88/1.57 ? [C3: A] :
% 2.88/1.57 ( B3
% 2.88/1.57 = ( plus_plus @ A @ A5 @ C3 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_196_add__le__imp__le__left,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ordere236663937imp_le @ A )
% 2.88/1.57 => ! [C2: A,A2: A,B2: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 2.88/1.57 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_197_add__le__imp__le__right,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ordere236663937imp_le @ A )
% 2.88/1.57 => ! [A2: A,C2: A,B2: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 2.88/1.57 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_198_plus__ereal_Osimps_I2_J,axiom,
% 2.88/1.57 ! [A2: extended_ereal] :
% 2.88/1.57 ( ( plus_plus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ A2 )
% 2.88/1.57 = ( extend1396239628finity @ extended_ereal ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_199_plus__ereal_Osimps_I3_J,axiom,
% 2.88/1.57 ! [A2: extended_ereal] :
% 2.88/1.57 ( ( plus_plus @ extended_ereal @ A2 @ ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 = ( extend1396239628finity @ extended_ereal ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_200_plus__ereal_Osimps_I6_J,axiom,
% 2.88/1.57 ( ( plus_plus @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_201_ereal__add__cancel__left,axiom,
% 2.88/1.57 ! [A2: extended_ereal,B2: extended_ereal,C2: extended_ereal] :
% 2.88/1.57 ( ( A2
% 2.88/1.57 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 => ( ( ( plus_plus @ extended_ereal @ A2 @ B2 )
% 2.88/1.57 = ( plus_plus @ extended_ereal @ A2 @ C2 ) )
% 2.88/1.57 = ( ( A2
% 2.88/1.57 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 | ( B2 = C2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_202_ereal__add__cancel__right,axiom,
% 2.88/1.57 ! [A2: extended_ereal,B2: extended_ereal,C2: extended_ereal] :
% 2.88/1.57 ( ( A2
% 2.88/1.57 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 => ( ( ( plus_plus @ extended_ereal @ B2 @ A2 )
% 2.88/1.57 = ( plus_plus @ extended_ereal @ C2 @ A2 ) )
% 2.88/1.57 = ( ( A2
% 2.88/1.57 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 | ( B2 = C2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_203_tendsto__add,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType] :
% 2.88/1.57 ( ( topolo1314133330id_add @ A )
% 2.88/1.57 => ! [F2: B > A,A2: A,F: filter @ B,G2: B > A,B2: A] :
% 2.88/1.57 ( ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ A2 ) @ F )
% 2.88/1.57 => ( ( filterlim @ B @ A @ G2 @ ( topolo1920029354e_nhds @ A @ B2 ) @ F )
% 2.88/1.57 => ( filterlim @ B @ A
% 2.88/1.57 @ ^ [X: B] : ( plus_plus @ A @ ( F2 @ X ) @ ( G2 @ X ) )
% 2.88/1.57 @ ( topolo1920029354e_nhds @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.88/1.57 @ F ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_204_tendsto__add__const__iff,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType] :
% 2.88/1.57 ( ( real_V55928688vector @ A )
% 2.88/1.57 => ! [C2: A,F2: B > A,D2: A,F: filter @ B] :
% 2.88/1.57 ( ( filterlim @ B @ A
% 2.88/1.57 @ ^ [X: B] : ( plus_plus @ A @ C2 @ ( F2 @ X ) )
% 2.88/1.57 @ ( topolo1920029354e_nhds @ A @ ( plus_plus @ A @ C2 @ D2 ) )
% 2.88/1.57 @ F )
% 2.88/1.57 = ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ D2 ) @ F ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_205_plus__ereal_Osimps_I5_J,axiom,
% 2.88/1.57 ! [P3: real] :
% 2.88/1.57 ( ( plus_plus @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ P3 ) )
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_206_plus__ereal_Osimps_I4_J,axiom,
% 2.88/1.57 ! [R: real] :
% 2.88/1.57 ( ( plus_plus @ extended_ereal @ ( extended_ereal2 @ R ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_207_ereal__add__le__add__iff,axiom,
% 2.88/1.57 ! [C2: extended_ereal,A2: extended_ereal,B2: extended_ereal] :
% 2.88/1.57 ( ( ord_less_eq @ extended_ereal @ ( plus_plus @ extended_ereal @ C2 @ A2 ) @ ( plus_plus @ extended_ereal @ C2 @ B2 ) )
% 2.88/1.57 = ( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
% 2.88/1.57 | ( C2
% 2.88/1.57 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 | ( ( C2
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 & ( A2
% 2.88/1.57 != ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 & ( B2
% 2.88/1.57 != ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_208_ereal__add__le__add__iff2,axiom,
% 2.88/1.57 ! [A2: extended_ereal,C2: extended_ereal,B2: extended_ereal] :
% 2.88/1.57 ( ( ord_less_eq @ extended_ereal @ ( plus_plus @ extended_ereal @ A2 @ C2 ) @ ( plus_plus @ extended_ereal @ B2 @ C2 ) )
% 2.88/1.57 = ( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
% 2.88/1.57 | ( C2
% 2.88/1.57 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 | ( ( C2
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 & ( A2
% 2.88/1.57 != ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 & ( B2
% 2.88/1.57 != ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_209_tendsto__add__ereal__MInf,axiom,
% 2.88/1.57 ! [A: $tType,Y: extended_ereal,F2: A > extended_ereal,F: filter @ A,G2: A > extended_ereal] :
% 2.88/1.57 ( ( Y
% 2.88/1.57 != ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 => ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) @ F )
% 2.88/1.57 => ( ( filterlim @ A @ extended_ereal @ G2 @ ( topolo1920029354e_nhds @ extended_ereal @ Y ) @ F )
% 2.88/1.57 => ( filterlim @ A @ extended_ereal
% 2.88/1.57 @ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F2 @ X ) @ ( G2 @ X ) )
% 2.88/1.57 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 @ F ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_210_tendsto__add__ereal__PInf,axiom,
% 2.88/1.57 ! [A: $tType,Y: extended_ereal,F2: A > extended_ereal,F: filter @ A,G2: A > extended_ereal] :
% 2.88/1.57 ( ( Y
% 2.88/1.57 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 => ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ F )
% 2.88/1.57 => ( ( filterlim @ A @ extended_ereal @ G2 @ ( topolo1920029354e_nhds @ extended_ereal @ Y ) @ F )
% 2.88/1.57 => ( filterlim @ A @ extended_ereal
% 2.88/1.57 @ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F2 @ X ) @ ( G2 @ X ) )
% 2.88/1.57 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 @ F ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_211_tendsto__add__ereal__general,axiom,
% 2.88/1.57 ! [A: $tType,X2: extended_ereal,Y: extended_ereal,F2: A > extended_ereal,F: filter @ A,G2: A > extended_ereal] :
% 2.88/1.57 ( ~ ( ( ( X2
% 2.88/1.57 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 & ( Y
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.88/1.57 | ( ( X2
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 & ( Y
% 2.88/1.57 = ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.88/1.57 => ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ X2 ) @ F )
% 2.88/1.57 => ( ( filterlim @ A @ extended_ereal @ G2 @ ( topolo1920029354e_nhds @ extended_ereal @ Y ) @ F )
% 2.88/1.57 => ( filterlim @ A @ extended_ereal
% 2.88/1.57 @ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F2 @ X ) @ ( G2 @ X ) )
% 2.88/1.57 @ ( topolo1920029354e_nhds @ extended_ereal @ ( plus_plus @ extended_ereal @ X2 @ Y ) )
% 2.88/1.57 @ F ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_212_tendsto__at__botI__sequentially,axiom,
% 2.88/1.57 ! [B: $tType] :
% 2.88/1.57 ( ( topolo2135403230pology @ B )
% 2.88/1.57 => ! [F2: real > B,Y: B] :
% 2.88/1.57 ( ! [X8: nat > real] :
% 2.88/1.57 ( ( filterlim @ nat @ real @ X8 @ ( at_bot @ real ) @ ( at_top @ nat ) )
% 2.88/1.57 => ( filterlim @ nat @ B
% 2.88/1.57 @ ^ [N: nat] : ( F2 @ ( X8 @ N ) )
% 2.88/1.57 @ ( topolo1920029354e_nhds @ B @ Y )
% 2.88/1.57 @ ( at_top @ nat ) ) )
% 2.88/1.57 => ( filterlim @ real @ B @ F2 @ ( topolo1920029354e_nhds @ B @ Y ) @ ( at_bot @ real ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_213_id__nat__ereal__tendsto__PInf,axiom,
% 2.88/1.57 ( filterlim @ nat @ extended_ereal
% 2.88/1.57 @ ^ [X: nat] : ( extended_ereal2 @ ( semiring_1_of_nat @ real @ X ) )
% 2.88/1.57 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 @ ( at_top @ nat ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_214_ereal__liminf__add__mono,axiom,
% 2.88/1.57 ! [U: nat > extended_ereal,V: nat > extended_ereal] :
% 2.88/1.57 ( ~ ( ( ( ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ U )
% 2.88/1.57 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 & ( ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ V )
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.88/1.57 | ( ( ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ U )
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 & ( ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ V )
% 2.88/1.57 = ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.88/1.57 => ( ord_less_eq @ extended_ereal @ ( plus_plus @ extended_ereal @ ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ U ) @ ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ V ) )
% 2.88/1.57 @ ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat )
% 2.88/1.57 @ ^ [N: nat] : ( plus_plus @ extended_ereal @ ( U @ N ) @ ( V @ N ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_215_of__nat__le__iff,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( linord1659791738miring @ A )
% 2.88/1.57 => ! [M5: nat,N7: nat] :
% 2.88/1.57 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M5 ) @ ( semiring_1_of_nat @ A @ N7 ) )
% 2.88/1.57 = ( ord_less_eq @ nat @ M5 @ N7 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_216_of__nat__mono,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( linord1659791738miring @ A )
% 2.88/1.57 => ! [I: nat,J: nat] :
% 2.88/1.57 ( ( ord_less_eq @ nat @ I @ J )
% 2.88/1.57 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_217_ereal__tendsto__simps2_I2_J,axiom,
% 2.88/1.57 ! [A: $tType,F2: A > real,F: filter @ A] :
% 2.88/1.57 ( ( filterlim @ A @ extended_ereal @ ( comp @ real @ extended_ereal @ A @ extended_ereal2 @ F2 ) @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ F )
% 2.88/1.57 = ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_218_nat__add__left__cancel__le,axiom,
% 2.88/1.57 ! [K: nat,M5: nat,N7: nat] :
% 2.88/1.57 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M5 ) @ ( plus_plus @ nat @ K @ N7 ) )
% 2.88/1.57 = ( ord_less_eq @ nat @ M5 @ N7 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_219_filterlim__at__top__add__at__top,axiom,
% 2.88/1.57 ! [A: $tType,F2: A > real,F: filter @ A,G2: A > real] :
% 2.88/1.57 ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F )
% 2.88/1.57 => ( ( filterlim @ A @ real @ G2 @ ( at_top @ real ) @ F )
% 2.88/1.57 => ( filterlim @ A @ real
% 2.88/1.57 @ ^ [X: A] : ( plus_plus @ real @ ( F2 @ X ) @ ( G2 @ X ) )
% 2.88/1.57 @ ( at_top @ real )
% 2.88/1.57 @ F ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_220_filterlim__tendsto__add__at__top,axiom,
% 2.88/1.57 ! [A: $tType,F2: A > real,C2: real,F: filter @ A,G2: A > real] :
% 2.88/1.57 ( ( filterlim @ A @ real @ F2 @ ( topolo1920029354e_nhds @ real @ C2 ) @ F )
% 2.88/1.57 => ( ( filterlim @ A @ real @ G2 @ ( at_top @ real ) @ F )
% 2.88/1.57 => ( filterlim @ A @ real
% 2.88/1.57 @ ^ [X: A] : ( plus_plus @ real @ ( F2 @ X ) @ ( G2 @ X ) )
% 2.88/1.57 @ ( at_top @ real )
% 2.88/1.57 @ F ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_221_nat__int__comparison_I3_J,axiom,
% 2.88/1.57 ( ( ord_less_eq @ nat )
% 2.88/1.57 = ( ^ [A5: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_222_add__leE,axiom,
% 2.88/1.57 ! [M5: nat,K: nat,N7: nat] :
% 2.88/1.57 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M5 @ K ) @ N7 )
% 2.88/1.57 => ~ ( ( ord_less_eq @ nat @ M5 @ N7 )
% 2.88/1.57 => ~ ( ord_less_eq @ nat @ K @ N7 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_223_le__add1,axiom,
% 2.88/1.57 ! [N7: nat,M5: nat] : ( ord_less_eq @ nat @ N7 @ ( plus_plus @ nat @ N7 @ M5 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_224_le__add2,axiom,
% 2.88/1.57 ! [N7: nat,M5: nat] : ( ord_less_eq @ nat @ N7 @ ( plus_plus @ nat @ M5 @ N7 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_225_add__leD1,axiom,
% 2.88/1.57 ! [M5: nat,K: nat,N7: nat] :
% 2.88/1.57 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M5 @ K ) @ N7 )
% 2.88/1.57 => ( ord_less_eq @ nat @ M5 @ N7 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_226_add__leD2,axiom,
% 2.88/1.57 ! [M5: nat,K: nat,N7: nat] :
% 2.88/1.57 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M5 @ K ) @ N7 )
% 2.88/1.57 => ( ord_less_eq @ nat @ K @ N7 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_227_le__Suc__ex,axiom,
% 2.88/1.57 ! [K: nat,L: nat] :
% 2.88/1.57 ( ( ord_less_eq @ nat @ K @ L )
% 2.88/1.57 => ? [N2: nat] :
% 2.88/1.57 ( L
% 2.88/1.57 = ( plus_plus @ nat @ K @ N2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_228_add__le__mono,axiom,
% 2.88/1.57 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.88/1.57 ( ( ord_less_eq @ nat @ I @ J )
% 2.88/1.57 => ( ( ord_less_eq @ nat @ K @ L )
% 2.88/1.57 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_229_add__le__mono1,axiom,
% 2.88/1.57 ! [I: nat,J: nat,K: nat] :
% 2.88/1.57 ( ( ord_less_eq @ nat @ I @ J )
% 2.88/1.57 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_230_trans__le__add1,axiom,
% 2.88/1.57 ! [I: nat,J: nat,M5: nat] :
% 2.88/1.57 ( ( ord_less_eq @ nat @ I @ J )
% 2.88/1.57 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M5 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_231_trans__le__add2,axiom,
% 2.88/1.57 ! [I: nat,J: nat,M5: nat] :
% 2.88/1.57 ( ( ord_less_eq @ nat @ I @ J )
% 2.88/1.57 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M5 @ J ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_232_nat__le__iff__add,axiom,
% 2.88/1.57 ( ( ord_less_eq @ nat )
% 2.88/1.57 = ( ^ [M4: nat,N: nat] :
% 2.88/1.57 ? [K2: nat] :
% 2.88/1.57 ( N
% 2.88/1.57 = ( plus_plus @ nat @ M4 @ K2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_233_nat__leq__as__int,axiom,
% 2.88/1.57 ( ( ord_less_eq @ nat )
% 2.88/1.57 = ( ^ [A5: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_234_plus__ereal_Osimps_I1_J,axiom,
% 2.88/1.57 ! [R: real,P3: real] :
% 2.88/1.57 ( ( plus_plus @ extended_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P3 ) )
% 2.88/1.57 = ( extended_ereal2 @ ( plus_plus @ real @ R @ P3 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_235_filterlim__at__bot__mirror,axiom,
% 2.88/1.57 ! [A: $tType,F2: real > A,F: filter @ A] :
% 2.88/1.57 ( ( filterlim @ real @ A @ F2 @ F @ ( at_bot @ real ) )
% 2.88/1.57 = ( filterlim @ real @ A
% 2.88/1.57 @ ^ [X: real] : ( F2 @ ( uminus_uminus @ real @ X ) )
% 2.88/1.57 @ F
% 2.88/1.57 @ ( at_top @ real ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_236_filterlim__at__top__mirror,axiom,
% 2.88/1.57 ! [A: $tType,F2: real > A,F: filter @ A] :
% 2.88/1.57 ( ( filterlim @ real @ A @ F2 @ F @ ( at_top @ real ) )
% 2.88/1.57 = ( filterlim @ real @ A
% 2.88/1.57 @ ^ [X: real] : ( F2 @ ( uminus_uminus @ real @ X ) )
% 2.88/1.57 @ F
% 2.88/1.57 @ ( at_bot @ real ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_237_filterlim__uminus__at__bot__at__top,axiom,
% 2.88/1.57 filterlim @ real @ real @ ( uminus_uminus @ real ) @ ( at_bot @ real ) @ ( at_top @ real ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_238_filterlim__uminus__at__top__at__bot,axiom,
% 2.88/1.57 filterlim @ real @ real @ ( uminus_uminus @ real ) @ ( at_top @ real ) @ ( at_bot @ real ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_239_liminf__shift__k,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( comple187826305attice @ A )
% 2.88/1.57 => ! [U: nat > A,K: nat] :
% 2.88/1.57 ( ( liminf_Liminf @ nat @ A @ ( at_top @ nat )
% 2.88/1.57 @ ^ [N: nat] : ( U @ ( plus_plus @ nat @ N @ K ) ) )
% 2.88/1.57 = ( liminf_Liminf @ nat @ A @ ( at_top @ nat ) @ U ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_240_Nat_Oex__has__greatest__nat,axiom,
% 2.88/1.57 ! [P: nat > $o,K: nat,B2: nat] :
% 2.88/1.57 ( ( P @ K )
% 2.88/1.57 => ( ! [Y3: nat] :
% 2.88/1.57 ( ( P @ Y3 )
% 2.88/1.57 => ( ord_less_eq @ nat @ Y3 @ B2 ) )
% 2.88/1.57 => ? [X5: nat] :
% 2.88/1.57 ( ( P @ X5 )
% 2.88/1.57 & ! [Y6: nat] :
% 2.88/1.57 ( ( P @ Y6 )
% 2.88/1.57 => ( ord_less_eq @ nat @ Y6 @ X5 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_241_nat__le__linear,axiom,
% 2.88/1.57 ! [M5: nat,N7: nat] :
% 2.88/1.57 ( ( ord_less_eq @ nat @ M5 @ N7 )
% 2.88/1.57 | ( ord_less_eq @ nat @ N7 @ M5 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_242_le__antisym,axiom,
% 2.88/1.57 ! [M5: nat,N7: nat] :
% 2.88/1.57 ( ( ord_less_eq @ nat @ M5 @ N7 )
% 2.88/1.57 => ( ( ord_less_eq @ nat @ N7 @ M5 )
% 2.88/1.57 => ( M5 = N7 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_243_eq__imp__le,axiom,
% 2.88/1.57 ! [M5: nat,N7: nat] :
% 2.88/1.57 ( ( M5 = N7 )
% 2.88/1.57 => ( ord_less_eq @ nat @ M5 @ N7 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_244_le__trans,axiom,
% 2.88/1.57 ! [I: nat,J: nat,K: nat] :
% 2.88/1.57 ( ( ord_less_eq @ nat @ I @ J )
% 2.88/1.57 => ( ( ord_less_eq @ nat @ J @ K )
% 2.88/1.57 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_245_le__refl,axiom,
% 2.88/1.57 ! [N7: nat] : ( ord_less_eq @ nat @ N7 @ N7 ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_246_filterlim__uminus__at__bot,axiom,
% 2.88/1.57 ! [A: $tType,F2: A > real,F: filter @ A] :
% 2.88/1.57 ( ( filterlim @ A @ real @ F2 @ ( at_bot @ real ) @ F )
% 2.88/1.57 = ( filterlim @ A @ real
% 2.88/1.57 @ ^ [X: A] : ( uminus_uminus @ real @ ( F2 @ X ) )
% 2.88/1.57 @ ( at_top @ real )
% 2.88/1.57 @ F ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_247_filterlim__uminus__at__top,axiom,
% 2.88/1.57 ! [A: $tType,F2: A > real,F: filter @ A] :
% 2.88/1.57 ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F )
% 2.88/1.57 = ( filterlim @ A @ real
% 2.88/1.57 @ ^ [X: A] : ( uminus_uminus @ real @ ( F2 @ X ) )
% 2.88/1.57 @ ( at_bot @ real )
% 2.88/1.57 @ F ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_248_LIMSEQ__ignore__initial__segment,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( topolo503727757_space @ A )
% 2.88/1.57 => ! [F2: nat > A,A2: A,K: nat] :
% 2.88/1.57 ( ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
% 2.88/1.57 => ( filterlim @ nat @ A
% 2.88/1.57 @ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) )
% 2.88/1.57 @ ( topolo1920029354e_nhds @ A @ A2 )
% 2.88/1.57 @ ( at_top @ nat ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_249_LIMSEQ__offset,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( topolo503727757_space @ A )
% 2.88/1.57 => ! [F2: nat > A,K: nat,A2: A] :
% 2.88/1.57 ( ( filterlim @ nat @ A
% 2.88/1.57 @ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) )
% 2.88/1.57 @ ( topolo1920029354e_nhds @ A @ A2 )
% 2.88/1.57 @ ( at_top @ nat ) )
% 2.88/1.57 => ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_250_plus__ereal_Oelims,axiom,
% 2.88/1.57 ! [X2: extended_ereal,Xa3: extended_ereal,Y: extended_ereal] :
% 2.88/1.57 ( ( ( plus_plus @ extended_ereal @ X2 @ Xa3 )
% 2.88/1.57 = Y )
% 2.88/1.57 => ( ! [R2: real] :
% 2.88/1.57 ( ( X2
% 2.88/1.57 = ( extended_ereal2 @ R2 ) )
% 2.88/1.57 => ! [P4: real] :
% 2.88/1.57 ( ( Xa3
% 2.88/1.57 = ( extended_ereal2 @ P4 ) )
% 2.88/1.57 => ( Y
% 2.88/1.57 != ( extended_ereal2 @ ( plus_plus @ real @ R2 @ P4 ) ) ) ) )
% 2.88/1.57 => ( ( ( X2
% 2.88/1.57 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 => ( Y
% 2.88/1.57 != ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 => ( ( ( Xa3
% 2.88/1.57 = ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 => ( Y
% 2.88/1.57 != ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 => ( ( ? [R2: real] :
% 2.88/1.57 ( X2
% 2.88/1.57 = ( extended_ereal2 @ R2 ) )
% 2.88/1.57 => ( ( Xa3
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 => ( Y
% 2.88/1.57 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.88/1.57 => ( ( ( X2
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 => ( ? [P4: real] :
% 2.88/1.57 ( Xa3
% 2.88/1.57 = ( extended_ereal2 @ P4 ) )
% 2.88/1.57 => ( Y
% 2.88/1.57 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.88/1.57 => ~ ( ( X2
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 => ( ( Xa3
% 2.88/1.57 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.88/1.57 => ( Y
% 2.88/1.57 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_251_tendsto__PInfty__eq__at__top,axiom,
% 2.88/1.57 ! [A: $tType,F2: A > real,F: filter @ A] :
% 2.88/1.57 ( ( filterlim @ A @ extended_ereal
% 2.88/1.57 @ ^ [Z5: A] : ( extended_ereal2 @ ( F2 @ Z5 ) )
% 2.88/1.57 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
% 2.88/1.57 @ F )
% 2.88/1.57 = ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_252_tendsto__at__topI__sequentially,axiom,
% 2.88/1.57 ! [B: $tType] :
% 2.88/1.57 ( ( topolo2135403230pology @ B )
% 2.88/1.57 => ! [F2: real > B,Y: B] :
% 2.88/1.57 ( ! [X8: nat > real] :
% 2.88/1.57 ( ( filterlim @ nat @ real @ X8 @ ( at_top @ real ) @ ( at_top @ nat ) )
% 2.88/1.57 => ( filterlim @ nat @ B
% 2.88/1.57 @ ^ [N: nat] : ( F2 @ ( X8 @ N ) )
% 2.88/1.57 @ ( topolo1920029354e_nhds @ B @ Y )
% 2.88/1.57 @ ( at_top @ nat ) ) )
% 2.88/1.57 => ( filterlim @ real @ B @ F2 @ ( topolo1920029354e_nhds @ B @ Y ) @ ( at_top @ real ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_253_filterlim__real__sequentially,axiom,
% 2.88/1.57 filterlim @ nat @ real @ ( semiring_1_of_nat @ real ) @ ( at_top @ real ) @ ( at_top @ nat ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_254_int__ops_I5_J,axiom,
% 2.88/1.57 ! [A2: nat,B2: nat] :
% 2.88/1.57 ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A2 @ B2 ) )
% 2.88/1.57 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_255_nat__int__comparison_I1_J,axiom,
% 2.88/1.57 ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
% 2.88/1.57 = ( ^ [A5: nat,B3: nat] :
% 2.88/1.57 ( ( semiring_1_of_nat @ int @ A5 )
% 2.88/1.57 = ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(subcl_Real__Vector__Spaces_Ometric__space___HOL_Otype,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( real_V2090557954_space @ A )
% 2.88/1.57 => ( type @ A ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Ot2__space,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( real_V2090557954_space @ A )
% 2.88/1.57 => ( topological_t2_space @ A ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Ouniform__space,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( real_V2090557954_space @ A )
% 2.88/1.57 => ( topolo47006728_space @ A ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Otopological__space,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( real_V2090557954_space @ A )
% 2.88/1.57 => ( topolo503727757_space @ A ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Ofirst__countable__topology,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( real_V2090557954_space @ A )
% 2.88/1.57 => ( topolo2135403230pology @ A ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Countable_Ocountable,axiom,
% 2.88/1.57 countable @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Countable_Ocountable_1,axiom,
% 2.88/1.57 ! [A7: $tType] :
% 2.88/1.57 ( ( finite_finite @ A7 )
% 2.88/1.57 => ( countable @ ( set @ A7 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Countable_Ocountable_2,axiom,
% 2.88/1.57 countable @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Countable_Ocountable_3,axiom,
% 2.88/1.57 countable @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Countable_Ocountable_4,axiom,
% 2.88/1.57 ! [A7: $tType,A8: $tType] :
% 2.88/1.57 ( ( ( finite_finite @ A7 )
% 2.88/1.57 & ( countable @ A8 ) )
% 2.88/1.57 => ( countable @ ( A7 > A8 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 2.88/1.57 ! [A7: $tType,A8: $tType] :
% 2.88/1.57 ( ( ( finite_finite @ A7 )
% 2.88/1.57 & ( finite_finite @ A8 ) )
% 2.88/1.57 => ( finite_finite @ ( A7 > A8 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
% 2.88/1.57 ! [A7: $tType] :
% 2.88/1.57 ( ( finite_finite @ A7 )
% 2.88/1.57 => ( finite_finite @ ( set @ A7 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Finite__Set_Ofinite_6,axiom,
% 2.88/1.57 finite_finite @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Topological__Spaces_Ofirst__countable__topology,axiom,
% 2.88/1.57 ! [A7: $tType,A8: $tType] :
% 2.88/1.57 ( ( ( countable @ A7 )
% 2.88/1.57 & ( topolo2135403230pology @ A8 ) )
% 2.88/1.57 => ( topolo2135403230pology @ ( A7 > A8 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
% 2.88/1.57 ! [A7: $tType,A8: $tType] :
% 2.88/1.57 ( ( topolo503727757_space @ A8 )
% 2.88/1.57 => ( topolo503727757_space @ ( A7 > A8 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 2.88/1.57 ! [A7: $tType,A8: $tType] :
% 2.88/1.57 ( ( comple187826305attice @ A8 )
% 2.88/1.57 => ( comple187826305attice @ ( A7 > A8 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 2.88/1.57 ! [A7: $tType,A8: $tType] :
% 2.88/1.57 ( ( boolean_algebra @ A8 )
% 2.88/1.57 => ( boolean_algebra @ ( A7 > A8 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.88/1.57 ! [A7: $tType,A8: $tType] :
% 2.88/1.57 ( ( preorder @ A8 )
% 2.88/1.57 => ( preorder @ ( A7 > A8 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.88/1.57 ! [A7: $tType,A8: $tType] :
% 2.88/1.57 ( ( order @ A8 )
% 2.88/1.57 => ( order @ ( A7 > A8 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Orderings_Oord,axiom,
% 2.88/1.57 ! [A7: $tType,A8: $tType] :
% 2.88/1.57 ( ( ord @ A8 )
% 2.88/1.57 => ( ord @ ( A7 > A8 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Groups_Ouminus,axiom,
% 2.88/1.57 ! [A7: $tType,A8: $tType] :
% 2.88/1.57 ( ( uminus @ A8 )
% 2.88/1.57 => ( uminus @ ( A7 > A8 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 2.88/1.57 ordere236663937imp_le @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Topological__Spaces_Otopological__space_7,axiom,
% 2.88/1.57 topolo503727757_space @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
% 2.88/1.57 topolo2117631714pology @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
% 2.88/1.57 topolo259154727pology @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 2.88/1.57 linord1659791738miring @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
% 2.88/1.57 ordere779506340up_add @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom,
% 2.88/1.57 topolo1314133330id_add @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
% 2.88/1.57 topological_t2_space @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 2.88/1.57 ordered_ab_group_add @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
% 2.88/1.57 cancel_semigroup_add @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
% 2.88/1.57 ab_semigroup_add @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
% 2.88/1.57 comm_monoid_add @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
% 2.88/1.57 semigroup_add @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
% 2.88/1.57 ab_group_add @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Orderings_Opreorder_8,axiom,
% 2.88/1.57 preorder @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 2.88/1.57 linorder @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 2.88/1.57 group_add @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Orderings_Oorder_9,axiom,
% 2.88/1.57 order @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Orderings_Oord_10,axiom,
% 2.88/1.57 ord @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Int_Oint___Groups_Ouminus_11,axiom,
% 2.88/1.57 uminus @ int ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Topological__Spaces_Ofirst__countable__topology_12,axiom,
% 2.88/1.57 topolo2135403230pology @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_13,axiom,
% 2.88/1.57 ordere236663937imp_le @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.88/1.57 canoni770627133id_add @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_14,axiom,
% 2.88/1.57 topolo503727757_space @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_15,axiom,
% 2.88/1.57 topolo2117631714pology @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_16,axiom,
% 2.88/1.57 topolo259154727pology @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_17,axiom,
% 2.88/1.57 linord1659791738miring @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_18,axiom,
% 2.88/1.57 ordere779506340up_add @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_19,axiom,
% 2.88/1.57 topolo1314133330id_add @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_20,axiom,
% 2.88/1.57 topological_t2_space @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_21,axiom,
% 2.88/1.57 cancel_semigroup_add @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_22,axiom,
% 2.88/1.57 ab_semigroup_add @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_23,axiom,
% 2.88/1.57 comm_monoid_add @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Groups_Osemigroup__add_24,axiom,
% 2.88/1.57 semigroup_add @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Orderings_Opreorder_25,axiom,
% 2.88/1.57 preorder @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Orderings_Olinorder_26,axiom,
% 2.88/1.57 linorder @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Orderings_Oorder_27,axiom,
% 2.88/1.57 order @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Nat_Onat___Orderings_Oord_28,axiom,
% 2.88/1.57 ord @ nat ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_29,axiom,
% 2.88/1.57 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_30,axiom,
% 2.88/1.57 ! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Groups_Oab__semigroup__add_31,axiom,
% 2.88/1.57 ! [A7: $tType] :
% 2.88/1.57 ( ( ab_semigroup_add @ A7 )
% 2.88/1.57 => ( ab_semigroup_add @ ( set @ A7 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Groups_Ocomm__monoid__add_32,axiom,
% 2.88/1.57 ! [A7: $tType] :
% 2.88/1.57 ( ( comm_monoid_add @ A7 )
% 2.88/1.57 => ( comm_monoid_add @ ( set @ A7 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Groups_Osemigroup__add_33,axiom,
% 2.88/1.57 ! [A7: $tType] :
% 2.88/1.57 ( ( semigroup_add @ A7 )
% 2.88/1.57 => ( semigroup_add @ ( set @ A7 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Orderings_Opreorder_34,axiom,
% 2.88/1.57 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Orderings_Oorder_35,axiom,
% 2.88/1.57 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Orderings_Oord_36,axiom,
% 2.88/1.57 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Groups_Ouminus_37,axiom,
% 2.88/1.57 ! [A7: $tType] : ( uminus @ ( set @ A7 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_38,axiom,
% 2.88/1.57 topolo503727757_space @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_39,axiom,
% 2.88/1.57 topolo2117631714pology @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_40,axiom,
% 2.88/1.57 comple187826305attice @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_41,axiom,
% 2.88/1.57 topolo259154727pology @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_42,axiom,
% 2.88/1.57 topological_t2_space @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_43,axiom,
% 2.88/1.57 boolean_algebra @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Orderings_Opreorder_44,axiom,
% 2.88/1.57 preorder @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Orderings_Olinorder_45,axiom,
% 2.88/1.57 linorder @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Orderings_Oorder_46,axiom,
% 2.88/1.57 order @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Orderings_Oord_47,axiom,
% 2.88/1.57 ord @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Groups_Ouminus_48,axiom,
% 2.88/1.57 uminus @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Topological__Spaces_Ofirst__countable__topology_49,axiom,
% 2.88/1.57 topolo2135403230pology @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_50,axiom,
% 2.88/1.57 ordere236663937imp_le @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
% 2.88/1.57 real_V55928688vector @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_51,axiom,
% 2.88/1.57 topolo503727757_space @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_52,axiom,
% 2.88/1.57 topolo2117631714pology @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_53,axiom,
% 2.88/1.57 topolo259154727pology @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_54,axiom,
% 2.88/1.57 linord1659791738miring @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
% 2.88/1.57 topolo47006728_space @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
% 2.88/1.57 real_V2090557954_space @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_55,axiom,
% 2.88/1.57 ordere779506340up_add @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Limits_Otopological__monoid__add_56,axiom,
% 2.88/1.57 topolo1314133330id_add @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
% 2.88/1.57 topolo799126099up_add @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_57,axiom,
% 2.88/1.57 topological_t2_space @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_58,axiom,
% 2.88/1.57 ordered_ab_group_add @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_59,axiom,
% 2.88/1.57 cancel_semigroup_add @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_60,axiom,
% 2.88/1.57 ab_semigroup_add @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_61,axiom,
% 2.88/1.57 comm_monoid_add @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Groups_Osemigroup__add_62,axiom,
% 2.88/1.57 semigroup_add @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Groups_Oab__group__add_63,axiom,
% 2.88/1.57 ab_group_add @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Orderings_Opreorder_64,axiom,
% 2.88/1.57 preorder @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Orderings_Olinorder_65,axiom,
% 2.88/1.57 linorder @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Groups_Ogroup__add_66,axiom,
% 2.88/1.57 group_add @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Orderings_Oorder_67,axiom,
% 2.88/1.57 order @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Orderings_Oord_68,axiom,
% 2.88/1.57 ord @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Real_Oreal___Groups_Ouminus_69,axiom,
% 2.88/1.57 uminus @ real ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_70,axiom,
% 2.88/1.57 ! [A7: $tType] : ( comple187826305attice @ ( filter @ A7 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Filter_Ofilter___Orderings_Opreorder_71,axiom,
% 2.88/1.57 ! [A7: $tType] : ( preorder @ ( filter @ A7 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Filter_Ofilter___Orderings_Oorder_72,axiom,
% 2.88/1.57 ! [A7: $tType] : ( order @ ( filter @ A7 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Filter_Ofilter___Orderings_Oord_73,axiom,
% 2.88/1.57 ! [A7: $tType] : ( ord @ ( filter @ A7 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Ofirst__countable__topology_74,axiom,
% 2.88/1.57 topolo2135403230pology @ extended_ereal ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Otopological__space_75,axiom,
% 2.88/1.57 topolo503727757_space @ extended_ereal ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Olinorder__topology_76,axiom,
% 2.88/1.57 topolo2117631714pology @ extended_ereal ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Extended__Real_Oereal___Complete__Lattices_Ocomplete__linorder,axiom,
% 2.88/1.57 comple1035589618norder @ extended_ereal ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Extended__Real_Oereal___Complete__Lattices_Ocomplete__lattice_77,axiom,
% 2.88/1.57 comple187826305attice @ extended_ereal ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Oorder__topology_78,axiom,
% 2.88/1.57 topolo259154727pology @ extended_ereal ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Extended__Real_Oereal___Groups_Oordered__ab__semigroup__add_79,axiom,
% 2.88/1.57 ordere779506340up_add @ extended_ereal ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Ot2__space_80,axiom,
% 2.88/1.57 topological_t2_space @ extended_ereal ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Extended__Real_Oereal___Groups_Oab__semigroup__add_81,axiom,
% 2.88/1.57 ab_semigroup_add @ extended_ereal ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Extended__Real_Oereal___Groups_Ocomm__monoid__add_82,axiom,
% 2.88/1.57 comm_monoid_add @ extended_ereal ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Extended__Real_Oereal___Groups_Osemigroup__add_83,axiom,
% 2.88/1.57 semigroup_add @ extended_ereal ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Extended__Real_Oereal___Orderings_Opreorder_84,axiom,
% 2.88/1.57 preorder @ extended_ereal ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Extended__Real_Oereal___Orderings_Olinorder_85,axiom,
% 2.88/1.57 linorder @ extended_ereal ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Extended__Real_Oereal___Orderings_Oorder_86,axiom,
% 2.88/1.57 order @ extended_ereal ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Extended__Real_Oereal___Orderings_Oord_87,axiom,
% 2.88/1.57 ord @ extended_ereal ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Extended__Real_Oereal___Groups_Ouminus_88,axiom,
% 2.88/1.57 uminus @ extended_ereal ).
% 2.88/1.57
% 2.88/1.57 thf(tfree_0,hypothesis,
% 2.88/1.57 real_V2090557954_space @ a ).
% 2.88/1.57
% 2.88/1.57 thf(conj_0,conjecture,
% 2.88/1.57 ( filterlim @ nat @ extended_ereal
% 2.88/1.57 @ ^ [I2: nat] : ( uminus_uminus @ extended_ereal @ ( f @ ( x @ I2 ) ) )
% 2.88/1.57 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ a2 ) )
% 2.88/1.57 @ ( at_top @ nat ) ) ).
% 2.88/1.57 ------------------------
% 2.96/1.60 % (29518)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_29324 for (2999ds/4Mi)
% 2.96/1.60 % (29519)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_29324 for (2999ds/27Mi)
% 2.96/1.60 % (29518)Instruction limit reached!
% 2.96/1.60 % (29518)------------------------------
% 2.96/1.60 % (29518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.60 % (29518)Termination reason: Unknown
% 2.96/1.60 % (29518)Termination phase: shuffling
% 2.96/1.60
% 2.96/1.60 % (29518)Memory used [KB]: 1663
% 2.96/1.60 % (29518)Time elapsed: 0.003 s
% 2.96/1.60 % (29518)Instructions burned: 6 (million)
% 2.96/1.60 % (29518)------------------------------
% 2.96/1.60 % (29518)------------------------------
% 2.96/1.61 % (29521)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_29324 for (2999ds/2Mi)
% 2.96/1.61 % (29517)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_29324 for (2999ds/183Mi)
% 2.96/1.61 % (29520)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_29324 for (2999ds/2Mi)
% 2.96/1.61 % (29522)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_29324 for (2999ds/275Mi)
% 2.96/1.61 % (29523)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_29324 for (2999ds/18Mi)
% 2.96/1.61 % (29521)Instruction limit reached!
% 2.96/1.61 % (29521)------------------------------
% 2.96/1.61 % (29521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.61 % (29521)Termination reason: Unknown
% 2.96/1.61 % (29521)Termination phase: shuffling
% 2.96/1.61
% 2.96/1.61 % (29521)Memory used [KB]: 1535
% 2.96/1.61 % (29521)Time elapsed: 0.004 s
% 2.96/1.61 % (29521)Instructions burned: 3 (million)
% 2.96/1.61 % (29521)------------------------------
% 2.96/1.61 % (29521)------------------------------
% 2.96/1.61 % (29520)Instruction limit reached!
% 2.96/1.61 % (29520)------------------------------
% 2.96/1.61 % (29520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.61 % (29520)Termination reason: Unknown
% 2.96/1.61 % (29520)Termination phase: shuffling
% 2.96/1.61
% 2.96/1.61 % (29520)Memory used [KB]: 1535
% 2.96/1.61 % (29520)Time elapsed: 0.004 s
% 2.96/1.61 % (29520)Instructions burned: 3 (million)
% 2.96/1.61 % (29520)------------------------------
% 2.96/1.61 % (29520)------------------------------
% 2.96/1.61 % (29519)Instruction limit reached!
% 2.96/1.61 % (29519)------------------------------
% 2.96/1.61 % (29519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.61 % (29519)Termination reason: Unknown
% 2.96/1.61 % (29519)Termination phase: Property scanning
% 2.96/1.61
% 2.96/1.61 % (29519)Memory used [KB]: 2046
% 2.96/1.61 % (29519)Time elapsed: 0.011 s
% 2.96/1.61 % (29519)Instructions burned: 28 (million)
% 2.96/1.61 % (29519)------------------------------
% 2.96/1.61 % (29519)------------------------------
% 2.96/1.61 % (29524)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_29324 for (2999ds/3Mi)
% 2.96/1.61 % (29524)Instruction limit reached!
% 2.96/1.61 % (29524)------------------------------
% 2.96/1.61 % (29524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.61 % (29524)Termination reason: Unknown
% 2.96/1.61 % (29524)Termination phase: shuffling
% 2.96/1.61
% 2.96/1.61 % (29524)Memory used [KB]: 1535
% 2.96/1.61 % (29524)Time elapsed: 0.002 s
% 2.96/1.61 % (29524)Instructions burned: 3 (million)
% 2.96/1.61 % (29524)------------------------------
% 2.96/1.61 % (29524)------------------------------
% 2.96/1.62 % (29523)Instruction limit reached!
% 2.96/1.62 % (29523)------------------------------
% 2.96/1.62 % (29523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.62 % (29523)Termination reason: Unknown
% 2.96/1.62 % (29523)Termination phase: shuffling
% 2.96/1.62
% 2.96/1.62 % (29523)Memory used [KB]: 1918
% 2.96/1.62 % (29523)Time elapsed: 0.013 s
% 2.96/1.62 % (29523)Instructions burned: 18 (million)
% 2.96/1.62 % (29523)------------------------------
% 2.96/1.62 % (29523)------------------------------
% 2.96/1.62 % (29527)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_29324 for (2999ds/3Mi)
% 2.96/1.62 % (29528)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_29324 for (2999ds/1041Mi)
% 2.96/1.62 % (29527)Instruction limit reached!
% 2.96/1.62 % (29527)------------------------------
% 2.96/1.62 % (29527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.62 % (29527)Termination reason: Unknown
% 2.96/1.62 % (29527)Termination phase: shuffling
% 2.96/1.62
% 2.96/1.62 % (29527)Memory used [KB]: 1535
% 2.96/1.62 % (29527)Time elapsed: 0.003 s
% 2.96/1.62 % (29527)Instructions burned: 4 (million)
% 2.96/1.62 % (29527)------------------------------
% 2.96/1.62 % (29527)------------------------------
% 2.96/1.63 % (29525)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_29324 for (2999ds/37Mi)
% 2.96/1.63 % (29526)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_29324 for (2999ds/15Mi)
% 2.96/1.63 % (29530)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_29324 for (2999ds/16Mi)
% 2.96/1.64 % (29526)Instruction limit reached!
% 2.96/1.64 % (29526)------------------------------
% 2.96/1.64 % (29526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.64 % (29526)Termination reason: Unknown
% 2.96/1.64 % (29526)Termination phase: shuffling
% 2.96/1.64
% 2.96/1.64 % (29526)Memory used [KB]: 1791
% 2.96/1.64 % (29526)Time elapsed: 0.012 s
% 2.96/1.64 % (29526)Instructions burned: 16 (million)
% 2.96/1.64 % (29526)------------------------------
% 2.96/1.64 % (29526)------------------------------
% 2.96/1.64 % (29529)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_29324 for (2999ds/7Mi)
% 2.96/1.64 % (29530)Instruction limit reached!
% 2.96/1.64 % (29530)------------------------------
% 2.96/1.64 % (29530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.64 % (29530)Termination reason: Unknown
% 2.96/1.64 % (29530)Termination phase: shuffling
% 2.96/1.64
% 2.96/1.64 % (29530)Memory used [KB]: 1791
% 2.96/1.64 % (29530)Time elapsed: 0.008 s
% 2.96/1.64 % (29530)Instructions burned: 17 (million)
% 2.96/1.64 % (29530)------------------------------
% 2.96/1.64 % (29530)------------------------------
% 2.96/1.64 % (29525)Instruction limit reached!
% 2.96/1.64 % (29525)------------------------------
% 2.96/1.64 % (29525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.64 % (29525)Termination reason: Unknown
% 2.96/1.64 % (29525)Termination phase: shuffling
% 2.96/1.64
% 2.96/1.64 % (29525)Memory used [KB]: 2302
% 2.96/1.64 % (29525)Time elapsed: 0.017 s
% 2.96/1.64 % (29525)Instructions burned: 39 (million)
% 2.96/1.64 % (29525)------------------------------
% 2.96/1.64 % (29525)------------------------------
% 2.96/1.64 % (29529)Instruction limit reached!
% 2.96/1.64 % (29529)------------------------------
% 2.96/1.64 % (29529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.64 % (29529)Termination reason: Unknown
% 2.96/1.64 % (29529)Termination phase: shuffling
% 2.96/1.64
% 2.96/1.64 % (29529)Memory used [KB]: 1663
% 2.96/1.64 % (29529)Time elapsed: 0.007 s
% 2.96/1.64 % (29529)Instructions burned: 8 (million)
% 2.96/1.64 % (29529)------------------------------
% 2.96/1.64 % (29529)------------------------------
% 2.96/1.65 % (29532)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_29324 for (2999ds/3Mi)
% 2.96/1.65 % (29532)Instruction limit reached!
% 2.96/1.65 % (29532)------------------------------
% 2.96/1.65 % (29532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.65 % (29532)Termination reason: Unknown
% 2.96/1.65 % (29532)Termination phase: shuffling
% 2.96/1.65
% 2.96/1.65 % (29532)Memory used [KB]: 1535
% 2.96/1.65 % (29532)Time elapsed: 0.003 s
% 2.96/1.65 % (29532)Instructions burned: 4 (million)
% 2.96/1.65 % (29532)------------------------------
% 2.96/1.65 % (29532)------------------------------
% 2.96/1.65 % (29533)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_29324 for (2999ds/7Mi)
% 2.96/1.65 % (29533)Instruction limit reached!
% 2.96/1.65 % (29533)------------------------------
% 2.96/1.65 % (29533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.65 % (29533)Termination reason: Unknown
% 2.96/1.65 % (29533)Termination phase: shuffling
% 2.96/1.65
% 2.96/1.65 % (29533)Memory used [KB]: 1535
% 2.96/1.65 % (29533)Time elapsed: 0.004 s
% 2.96/1.65 % (29533)Instructions burned: 7 (million)
% 2.96/1.65 % (29533)------------------------------
% 2.96/1.65 % (29533)------------------------------
% 2.96/1.66 % (29531)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_29324 for (2999ds/3Mi)
% 2.96/1.66 % (29531)Instruction limit reached!
% 2.96/1.66 % (29531)------------------------------
% 2.96/1.66 % (29531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.66 % (29531)Termination reason: Unknown
% 2.96/1.66 % (29531)Termination phase: shuffling
% 2.96/1.66
% 2.96/1.66 % (29531)Memory used [KB]: 1535
% 2.96/1.66 % (29531)Time elapsed: 0.003 s
% 2.96/1.66 % (29531)Instructions burned: 4 (million)
% 2.96/1.66 % (29531)------------------------------
% 2.96/1.66 % (29531)------------------------------
% 2.96/1.66 % (29534)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_29324 for (2999ds/3Mi)
% 2.96/1.66 % (29534)Instruction limit reached!
% 2.96/1.66 % (29534)------------------------------
% 2.96/1.66 % (29534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.66 % (29534)Termination reason: Unknown
% 2.96/1.66 % (29534)Termination phase: shuffling
% 2.96/1.66
% 2.96/1.66 % (29534)Memory used [KB]: 1535
% 2.96/1.66 % (29534)Time elapsed: 0.003 s
% 2.96/1.66 % (29534)Instructions burned: 4 (million)
% 2.96/1.66 % (29534)------------------------------
% 2.96/1.66 % (29534)------------------------------
% 2.96/1.66 % (29535)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_29324 for (2999ds/4Mi)
% 2.96/1.66 % (29535)Instruction limit reached!
% 2.96/1.66 % (29535)------------------------------
% 2.96/1.66 % (29535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.66 % (29535)Termination reason: Unknown
% 2.96/1.66 % (29535)Termination phase: shuffling
% 2.96/1.66
% 2.96/1.66 % (29535)Memory used [KB]: 1535
% 2.96/1.66 % (29535)Time elapsed: 0.003 s
% 2.96/1.66 % (29535)Instructions burned: 4 (million)
% 2.96/1.66 % (29535)------------------------------
% 2.96/1.66 % (29535)------------------------------
% 2.96/1.67 % (29536)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_29324 for (2999ds/18Mi)
% 2.96/1.67 % (29537)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_29324 for (2999ds/710Mi)
% 2.96/1.67 % (29536)Instruction limit reached!
% 2.96/1.67 % (29536)------------------------------
% 2.96/1.67 % (29536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.67 % (29536)Termination reason: Unknown
% 2.96/1.67 % (29536)Termination phase: shuffling
% 2.96/1.67
% 2.96/1.67 % (29536)Memory used [KB]: 1791
% 2.96/1.67 % (29536)Time elapsed: 0.008 s
% 2.96/1.67 % (29536)Instructions burned: 18 (million)
% 2.96/1.67 % (29536)------------------------------
% 2.96/1.67 % (29536)------------------------------
% 2.96/1.67 % (29538)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_29324 for (2999ds/6Mi)
% 2.96/1.67 % (29539)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_29324 for (2999ds/902Mi)
% 2.96/1.67 % (29538)Instruction limit reached!
% 2.96/1.67 % (29538)------------------------------
% 2.96/1.67 % (29538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.67 % (29538)Termination reason: Unknown
% 2.96/1.67 % (29538)Termination phase: shuffling
% 2.96/1.67
% 2.96/1.67 % (29538)Memory used [KB]: 1535
% 2.96/1.67 % (29538)Time elapsed: 0.004 s
% 2.96/1.67 % (29538)Instructions burned: 6 (million)
% 2.96/1.67 % (29538)------------------------------
% 2.96/1.67 % (29538)------------------------------
% 2.96/1.68 % (29540)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_29324 for (2999ds/21Mi)
% 2.96/1.68 % (29541)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_29324 for (2999ds/5Mi)
% 2.96/1.69 % (29541)Instruction limit reached!
% 2.96/1.69 % (29541)------------------------------
% 2.96/1.69 % (29541)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.69 % (29541)Termination reason: Unknown
% 2.96/1.69 % (29541)Termination phase: shuffling
% 2.96/1.69
% 2.96/1.69 % (29541)Memory used [KB]: 1663
% 2.96/1.69 % (29541)Time elapsed: 0.004 s
% 2.96/1.69 % (29541)Instructions burned: 7 (million)
% 2.96/1.69 % (29541)------------------------------
% 2.96/1.69 % (29541)------------------------------
% 2.96/1.69 % (29540)Instruction limit reached!
% 2.96/1.69 % (29540)------------------------------
% 2.96/1.69 % (29540)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.69 % (29540)Termination reason: Unknown
% 2.96/1.69 % (29540)Termination phase: shuffling
% 2.96/1.69
% 2.96/1.69 % (29540)Memory used [KB]: 1918
% 2.96/1.69 % (29540)Time elapsed: 0.010 s
% 2.96/1.69 % (29540)Instructions burned: 23 (million)
% 2.96/1.69 % (29540)------------------------------
% 2.96/1.69 % (29540)------------------------------
% 2.96/1.70 % (29517)Instruction limit reached!
% 2.96/1.70 % (29517)------------------------------
% 2.96/1.70 % (29517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.70 % (29517)Termination reason: Unknown
% 2.96/1.70 % (29517)Termination phase: Saturation
% 2.96/1.70
% 2.96/1.70 % (29517)Memory used [KB]: 7547
% 2.96/1.70 % (29517)Time elapsed: 0.088 s
% 2.96/1.70 % (29517)Instructions burned: 185 (million)
% 2.96/1.70 % (29517)------------------------------
% 2.96/1.70 % (29517)------------------------------
% 2.96/1.70 % (29542)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_29324 for (2998ds/6Mi)
% 2.96/1.70 % (29542)Instruction limit reached!
% 2.96/1.70 % (29542)------------------------------
% 2.96/1.70 % (29542)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.70 % (29542)Termination reason: Unknown
% 2.96/1.70 % (29542)Termination phase: shuffling
% 2.96/1.70
% 2.96/1.70 % (29542)Memory used [KB]: 1663
% 2.96/1.70 % (29542)Time elapsed: 0.003 s
% 2.96/1.70 % (29542)Instructions burned: 6 (million)
% 2.96/1.70 % (29542)------------------------------
% 2.96/1.70 % (29542)------------------------------
% 2.96/1.70 % (29543)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_29324 for (2998ds/377Mi)
% 2.96/1.71 % (29544)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_29324 for (2998ds/779Mi)
% 2.96/1.71 % (29545)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_29324 for (2998ds/19Mi)
% 2.96/1.72 % (29545)Instruction limit reached!
% 2.96/1.72 % (29545)------------------------------
% 2.96/1.72 % (29545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.72 % (29545)Termination reason: Unknown
% 2.96/1.72 % (29545)Termination phase: shuffling
% 2.96/1.72
% 2.96/1.72 % (29545)Memory used [KB]: 1918
% 2.96/1.72 % (29545)Time elapsed: 0.008 s
% 2.96/1.72 % (29545)Instructions burned: 21 (million)
% 2.96/1.72 % (29545)------------------------------
% 2.96/1.72 % (29545)------------------------------
% 2.96/1.73 % (29546)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_29324 for (2998ds/879Mi)
% 2.96/1.74 % (29522)Instruction limit reached!
% 2.96/1.74 % (29522)------------------------------
% 2.96/1.74 % (29522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.74 % (29522)Termination reason: Unknown
% 2.96/1.74 % (29522)Termination phase: Saturation
% 2.96/1.74
% 2.96/1.74 % (29522)Memory used [KB]: 6908
% 2.96/1.74 % (29522)Time elapsed: 0.136 s
% 2.96/1.74 % (29522)Instructions burned: 275 (million)
% 2.96/1.74 % (29522)------------------------------
% 2.96/1.74 % (29522)------------------------------
% 2.96/1.75 % (29547)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_29324 for (2998ds/17Mi)
% 2.96/1.76 % (29547)Instruction limit reached!
% 2.96/1.76 % (29547)------------------------------
% 2.96/1.76 % (29547)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.76 % (29547)Termination reason: Unknown
% 2.96/1.76 % (29547)Termination phase: shuffling
% 2.96/1.76
% 2.96/1.76 % (29547)Memory used [KB]: 1791
% 2.96/1.76 % (29547)Time elapsed: 0.008 s
% 2.96/1.76 % (29547)Instructions burned: 17 (million)
% 2.96/1.76 % (29547)------------------------------
% 2.96/1.76 % (29547)------------------------------
% 2.96/1.77 % (29548)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_29324 for (2998ds/3Mi)
% 2.96/1.77 % (29548)Instruction limit reached!
% 2.96/1.77 % (29548)------------------------------
% 2.96/1.77 % (29548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.77 % (29548)Termination reason: Unknown
% 2.96/1.77 % (29548)Termination phase: shuffling
% 2.96/1.77
% 2.96/1.77 % (29548)Memory used [KB]: 1535
% 2.96/1.77 % (29548)Time elapsed: 0.003 s
% 2.96/1.77 % (29548)Instructions burned: 4 (million)
% 2.96/1.77 % (29548)------------------------------
% 2.96/1.77 % (29548)------------------------------
% 2.96/1.78 % (29549)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_29324 for (2998ds/30Mi)
% 2.96/1.79 % (29549)Instruction limit reached!
% 2.96/1.79 % (29549)------------------------------
% 2.96/1.79 % (29549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.79 % (29549)Termination reason: Unknown
% 2.96/1.79 % (29549)Termination phase: Property scanning
% 2.96/1.79
% 2.96/1.79 % (29549)Memory used [KB]: 2046
% 2.96/1.79 % (29549)Time elapsed: 0.013 s
% 2.96/1.79 % (29549)Instructions burned: 32 (million)
% 2.96/1.79 % (29549)------------------------------
% 2.96/1.79 % (29549)------------------------------
% 2.96/1.80 % (29550)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_29324 for (2997ds/127Mi)
% 2.96/1.82 % (29543)Instruction limit reached!
% 2.96/1.82 % (29543)------------------------------
% 2.96/1.82 % (29543)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.82 % (29543)Termination reason: Unknown
% 2.96/1.82 % (29543)Termination phase: Saturation
% 2.96/1.82
% 2.96/1.82 % (29543)Memory used [KB]: 8571
% 2.96/1.82 % (29543)Time elapsed: 0.127 s
% 2.96/1.82 % (29543)Instructions burned: 378 (million)
% 2.96/1.82 % (29543)------------------------------
% 2.96/1.82 % (29543)------------------------------
% 2.96/1.84 % (29551)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_29324 for (2997ds/100Mi)
% 2.96/1.84 % (29550)Instruction limit reached!
% 2.96/1.84 % (29550)------------------------------
% 2.96/1.84 % (29550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.84 % (29550)Termination reason: Unknown
% 2.96/1.84 % (29550)Termination phase: Saturation
% 2.96/1.84
% 2.96/1.84 % (29550)Memory used [KB]: 7291
% 2.96/1.84 % (29550)Time elapsed: 0.043 s
% 2.96/1.84 % (29550)Instructions burned: 129 (million)
% 2.96/1.84 % (29550)------------------------------
% 2.96/1.84 % (29550)------------------------------
% 2.96/1.85 % (29552)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_29324 for (2997ds/3Mi)
% 2.96/1.86 % (29552)Instruction limit reached!
% 2.96/1.86 % (29552)------------------------------
% 2.96/1.86 % (29552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.86 % (29552)Termination reason: Unknown
% 2.96/1.86 % (29552)Termination phase: shuffling
% 2.96/1.86
% 2.96/1.86 % (29552)Memory used [KB]: 1535
% 2.96/1.86 % (29552)Time elapsed: 0.003 s
% 2.96/1.86 % (29552)Instructions burned: 4 (million)
% 2.96/1.86 % (29552)------------------------------
% 2.96/1.86 % (29552)------------------------------
% 2.96/1.86 % (29551)Instruction limit reached!
% 2.96/1.86 % (29551)------------------------------
% 2.96/1.86 % (29551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.86 % (29551)Termination reason: Unknown
% 2.96/1.86 % (29551)Termination phase: Property scanning
% 2.96/1.86
% 2.96/1.86 % (29551)Memory used [KB]: 2558
% 2.96/1.86 % (29551)Time elapsed: 0.029 s
% 2.96/1.86 % (29551)Instructions burned: 101 (million)
% 2.96/1.86 % (29551)------------------------------
% 2.96/1.86 % (29551)------------------------------
% 2.96/1.87 % (29553)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_29324 for (2997ds/20Mi)
% 2.96/1.87 % (29554)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_29324 for (2997ds/86Mi)
% 2.96/1.87 % (29553)Instruction limit reached!
% 2.96/1.87 % (29553)------------------------------
% 2.96/1.87 % (29553)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.87 % (29553)Termination reason: Unknown
% 2.96/1.87 % (29553)Termination phase: shuffling
% 2.96/1.87
% 2.96/1.87 % (29553)Memory used [KB]: 1918
% 2.96/1.87 % (29553)Time elapsed: 0.009 s
% 2.96/1.87 % (29553)Instructions burned: 21 (million)
% 2.96/1.87 % (29553)------------------------------
% 2.96/1.87 % (29553)------------------------------
% 2.96/1.88 % (29555)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_29324 for (2997ds/107Mi)
% 2.96/1.89 % (29537)Instruction limit reached!
% 2.96/1.89 % (29537)------------------------------
% 2.96/1.89 % (29537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.89 % (29537)Termination reason: Unknown
% 2.96/1.89 % (29537)Termination phase: Saturation
% 2.96/1.89
% 2.96/1.89 % (29537)Memory used [KB]: 7164
% 2.96/1.89 % (29537)Time elapsed: 0.224 s
% 2.96/1.89 % (29537)Instructions burned: 711 (million)
% 2.96/1.89 % (29537)------------------------------
% 2.96/1.89 % (29537)------------------------------
% 2.96/1.90 % (29554)Instruction limit reached!
% 2.96/1.90 % (29554)------------------------------
% 2.96/1.90 % (29554)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.90 % (29554)Termination reason: Unknown
% 2.96/1.90 % (29554)Termination phase: Property scanning
% 2.96/1.90
% 2.96/1.90 % (29554)Memory used [KB]: 2686
% 2.96/1.90 % (29554)Time elapsed: 0.027 s
% 2.96/1.90 % (29554)Instructions burned: 88 (million)
% 2.96/1.90 % (29554)------------------------------
% 2.96/1.90 % (29554)------------------------------
% 2.96/1.90 % (29556)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_29324 for (2996ds/39Mi)
% 2.96/1.91 % (29557)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_29324 for (2996ds/448Mi)
% 2.96/1.91 % (29556)Instruction limit reached!
% 2.96/1.91 % (29556)------------------------------
% 2.96/1.91 % (29556)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.91 % (29556)Termination reason: Unknown
% 2.96/1.91 % (29556)Termination phase: shuffling
% 2.96/1.91
% 2.96/1.91 % (29556)Memory used [KB]: 2302
% 2.96/1.91 % (29556)Time elapsed: 0.012 s
% 2.96/1.91 % (29556)Instructions burned: 39 (million)
% 2.96/1.91 % (29556)------------------------------
% 2.96/1.91 % (29556)------------------------------
% 2.96/1.92 % (29555)Instruction limit reached!
% 2.96/1.92 % (29555)------------------------------
% 2.96/1.92 % (29555)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.92 % (29555)Termination reason: Unknown
% 2.96/1.92 % (29555)Termination phase: Saturation
% 2.96/1.92
% 2.96/1.92 % (29555)Memory used [KB]: 7164
% 2.96/1.92 % (29555)Time elapsed: 0.036 s
% 2.96/1.92 % (29555)Instructions burned: 108 (million)
% 2.96/1.92 % (29555)------------------------------
% 2.96/1.92 % (29555)------------------------------
% 2.96/1.92 % (29558)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_29324 for (2996ds/46Mi)
% 2.96/1.93 % (29559)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_29324 for (2996ds/98Mi)
% 2.96/1.94 % (29558)Instruction limit reached!
% 2.96/1.94 % (29558)------------------------------
% 2.96/1.94 % (29558)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.94 % (29558)Termination reason: Unknown
% 2.96/1.94 % (29558)Termination phase: SInE selection
% 2.96/1.94
% 2.96/1.94 % (29558)Memory used [KB]: 2046
% 2.96/1.94 % (29558)Time elapsed: 0.015 s
% 2.96/1.94 % (29558)Instructions burned: 46 (million)
% 2.96/1.94 % (29558)------------------------------
% 2.96/1.94 % (29558)------------------------------
% 2.96/1.94 % (29560)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_29324 for (2996ds/507Mi)
% 2.96/1.95 % (29544)Instruction limit reached!
% 2.96/1.95 % (29544)------------------------------
% 2.96/1.95 % (29544)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.95 % (29544)Termination reason: Unknown
% 2.96/1.95 % (29544)Termination phase: Saturation
% 2.96/1.95
% 2.96/1.95 % (29544)Memory used [KB]: 8699
% 2.96/1.95 % (29544)Time elapsed: 0.241 s
% 2.96/1.95 % (29544)Instructions burned: 779 (million)
% 2.96/1.95 % (29544)------------------------------
% 2.96/1.95 % (29544)------------------------------
% 2.96/1.95 % (29539)Instruction limit reached!
% 2.96/1.95 % (29539)------------------------------
% 2.96/1.95 % (29539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.95 % (29539)Termination reason: Unknown
% 2.96/1.95 % (29539)Termination phase: Saturation
% 2.96/1.95
% 2.96/1.95 % (29539)Memory used [KB]: 9083
% 2.96/1.95 % (29539)Time elapsed: 0.274 s
% 2.96/1.95 % (29539)Instructions burned: 905 (million)
% 2.96/1.95 % (29539)------------------------------
% 2.96/1.95 % (29539)------------------------------
% 2.96/1.95 % (29528)Instruction limit reached!
% 2.96/1.95 % (29528)------------------------------
% 2.96/1.95 % (29528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.95 % (29528)Termination reason: Unknown
% 2.96/1.95 % (29528)Termination phase: Saturation
% 2.96/1.95
% 2.96/1.95 % (29528)Memory used [KB]: 10490
% 2.96/1.95 % (29528)Time elapsed: 0.328 s
% 2.96/1.95 % (29528)Instructions burned: 1043 (million)
% 2.96/1.95 % (29528)------------------------------
% 2.96/1.95 % (29528)------------------------------
% 2.96/1.96 % (29562)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_29324 for (2996ds/39Mi)
% 2.96/1.96 % (29561)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_29324 for (2996ds/149Mi)
% 2.96/1.96 % (29559)Instruction limit reached!
% 2.96/1.96 % (29559)------------------------------
% 2.96/1.96 % (29559)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.96 % (29559)Termination reason: Unknown
% 2.96/1.96 % (29559)Termination phase: Property scanning
% 2.96/1.96
% 2.96/1.96 % (29559)Memory used [KB]: 2558
% 2.96/1.96 % (29559)Time elapsed: 0.031 s
% 2.96/1.96 % (29559)Instructions burned: 100 (million)
% 2.96/1.96 % (29559)------------------------------
% 2.96/1.96 % (29559)------------------------------
% 2.96/1.96 % (29563)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_29324 for (2996ds/6Mi)
% 2.96/1.96 % (29563)Instruction limit reached!
% 2.96/1.96 % (29563)------------------------------
% 2.96/1.96 % (29563)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.96 % (29563)Termination reason: Unknown
% 2.96/1.96 % (29563)Termination phase: shuffling
% 2.96/1.96
% 2.96/1.96 % (29563)Memory used [KB]: 1663
% 2.96/1.96 % (29563)Time elapsed: 0.003 s
% 2.96/1.96 % (29563)Instructions burned: 7 (million)
% 2.96/1.96 % (29563)------------------------------
% 2.96/1.96 % (29563)------------------------------
% 2.96/1.97 % (29564)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_29324 for (2996ds/1087Mi)
% 2.96/1.97 % (29562)Instruction limit reached!
% 2.96/1.97 % (29562)------------------------------
% 2.96/1.97 % (29562)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.97 % (29562)Termination reason: Unknown
% 2.96/1.97 % (29562)Termination phase: Preprocessing 1
% 2.96/1.97
% 2.96/1.97 % (29562)Memory used [KB]: 2046
% 2.96/1.97 % (29562)Time elapsed: 0.014 s
% 2.96/1.97 % (29562)Instructions burned: 42 (million)
% 2.96/1.97 % (29562)------------------------------
% 2.96/1.97 % (29562)------------------------------
% 2.96/1.97 % (29565)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_29324 for (2996ds/47Mi)
% 2.96/1.98 % (29566)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_29324 for (2996ds/44Mi)
% 2.96/1.99 % (29565)Instruction limit reached!
% 2.96/1.99 % (29565)------------------------------
% 2.96/1.99 % (29565)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.99 % (29565)Termination reason: Unknown
% 2.96/1.99 % (29565)Termination phase: Preprocessing 3
% 2.96/1.99
% 2.96/1.99 % (29565)Memory used [KB]: 2302
% 2.96/1.99 % (29565)Time elapsed: 0.017 s
% 2.96/1.99 % (29565)Instructions burned: 49 (million)
% 2.96/1.99 % (29565)------------------------------
% 2.96/1.99 % (29565)------------------------------
% 2.96/1.99 % (29566)Instruction limit reached!
% 2.96/1.99 % (29566)------------------------------
% 2.96/1.99 % (29566)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.99 % (29566)Termination reason: Unknown
% 2.96/1.99 % (29566)Termination phase: shuffling
% 2.96/1.99
% 2.96/1.99 % (29566)Memory used [KB]: 2302
% 2.96/1.99 % (29566)Time elapsed: 0.014 s
% 2.96/1.99 % (29566)Instructions burned: 44 (million)
% 2.96/1.99 % (29566)------------------------------
% 2.96/1.99 % (29566)------------------------------
% 2.96/2.00 % (29560)First to succeed.
% 2.96/2.00 % (29560)Refutation found. Thanks to Tanya!
% 2.96/2.00 % SZS status Theorem for DTF2THF_29324
% 2.96/2.00 % SZS output start Proof for DTF2THF_29324
% 2.96/2.00 thf(type_def_5, type, real: $tType).
% 2.96/2.00 thf(type_def_7, type, extended_ereal: $tType).
% 2.96/2.00 thf(type_def_8, type, filter: ($tType) > $tType).
% 2.96/2.00 thf(type_def_9, type, nat: $tType).
% 2.96/2.00 thf(type_def_10, type, set: ($tType) > $tType).
% 2.96/2.00 thf(type_def_11, type, a: $tType).
% 2.96/2.00 thf(type_def_12, type, int: $tType).
% 2.96/2.00 thf(func_def_0, type, extended_ereal: $tType).
% 2.96/2.00 thf(func_def_1, type, filter: ($tType) > $tType).
% 2.96/2.00 thf(func_def_2, type, real: $tType).
% 2.96/2.00 thf(func_def_3, type, set: ($tType) > $tType).
% 2.96/2.00 thf(func_def_4, type, nat: $tType).
% 2.96/2.00 thf(func_def_5, type, int: $tType).
% 2.96/2.00 thf(func_def_6, type, a: $tType).
% 2.96/2.00 thf(func_def_7, type, finite_finite: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_8, type, type: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_9, type, countable: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_10, type, uminus: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_11, type, ord: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_12, type, order: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_13, type, group_add: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_14, type, linorder: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_15, type, preorder: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_16, type, ab_group_add: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_17, type, semigroup_add: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_18, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_19, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_20, type, boolean_algebra: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_21, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_22, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_23, type, topological_t2_space: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_24, type, topolo799126099up_add: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_25, type, topolo1314133330id_add: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_26, type, ordere779506340up_add: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_27, type, real_V2090557954_space: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_28, type, topolo47006728_space: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_29, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_30, type, topolo259154727pology: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_31, type, comple187826305attice: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_32, type, comple1035589618norder: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_33, type, topolo2117631714pology: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_34, type, topolo503727757_space: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_35, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_36, type, real_V55928688vector: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_37, type, ordere236663937imp_le: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_38, type, topolo2135403230pology: !>[X0: $tType]:($o)).
% 2.96/2.00 thf(func_def_39, type, extend1396239628finity: !>[X0: $tType]:(X0)).
% 2.96/2.00 thf(func_def_40, type, extended_case_ereal: !>[X0: $tType]:((real > X0) > X0 > X0 > extended_ereal > X0)).
% 2.96/2.00 thf(func_def_41, type, extended_ereal2: real > extended_ereal).
% 2.96/2.00 thf(func_def_42, type, at_bot: !>[X0: $tType]:(filter @ X0)).
% 2.96/2.00 thf(func_def_43, type, at_top: !>[X0: $tType]:(filter @ X0)).
% 2.96/2.00 thf(func_def_44, type, filterlim: !>[X0: $tType, X1: $tType]:((X0 > X1) > filter @ X1 > filter @ X0 > $o)).
% 2.96/2.00 thf(func_def_45, type, comp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X0) > X2 > X1)).
% 2.96/2.00 thf(func_def_46, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.96/2.00 thf(func_def_47, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 2.96/2.00 thf(func_def_48, type, ex1: !>[X0: $tType]:((X0 > $o) > $o)).
% 2.96/2.00 thf(func_def_49, type, uniq: !>[X0: $tType]:((X0 > $o) > $o)).
% 2.96/2.00 thf(func_def_50, type, liminf_Liminf: !>[X0: $tType, X1: $tType]:(filter @ X0 > (X0 > X1) > X1)).
% 2.96/2.00 thf(func_def_51, type, lower_582600101lsc_at: !>[X0: $tType, X1: $tType]:(X0 > (X0 > X1) > $o)).
% 2.96/2.00 thf(func_def_52, type, lower_708585518usc_at: !>[X0: $tType, X1: $tType]:(X0 > (X0 > X1) > $o)).
% 2.96/2.00 thf(func_def_53, type, semiring_1_of_nat: !>[X0: $tType]:(nat > X0)).
% 2.96/2.00 thf(func_def_54, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.96/2.00 thf(func_def_55, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.96/2.00 thf(func_def_56, type, topological_monoseq: !>[X0: $tType]:((nat > X0) > $o)).
% 2.96/2.00 thf(func_def_57, type, topolo1920029354e_nhds: !>[X0: $tType]:(X0 > filter @ X0)).
% 2.96/2.00 thf(func_def_58, type, topolo1334162427filter: !>[X0: $tType]:(filter @ X0 > $o)).
% 2.96/2.00 thf(func_def_59, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.96/2.00 thf(func_def_60, type, a2: extended_ereal).
% 2.96/2.00 thf(func_def_61, type, f: a > extended_ereal).
% 2.96/2.00 thf(func_def_62, type, x0: a).
% 2.96/2.00 thf(func_def_63, type, x: nat > a).
% 2.96/2.00 thf(func_def_80, type, sK0: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.96/2.00 thf(func_def_81, type, sK1: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.96/2.00 thf(func_def_82, type, sP2: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_83, type, sP3: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_84, type, sP4: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_85, type, sP5: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_86, type, sP6: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_87, type, sP7: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_88, type, sP8: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_89, type, sP9: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_90, type, sP10: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_91, type, sP11: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_92, type, sP12: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_93, type, sP13: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_94, type, sP14: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_95, type, sP15: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_96, type, sP16: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_97, type, sP17: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_98, type, sP18: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_99, type, sP19: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_100, type, sP20: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_101, type, sP21: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_102, type, sP22: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_103, type, sP23: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_104, type, sP24: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_105, type, sP25: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_106, type, sP26: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_107, type, sP27: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_108, type, sP28: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_109, type, sP29: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_110, type, sP30: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_111, type, sP31: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_112, type, sP32: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.96/2.00 thf(func_def_113, type, sP33: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.96/2.00 thf(func_def_114, type, sP34: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.96/2.00 thf(func_def_115, type, sP35: extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_116, type, sP36: extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_117, type, sP37: extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_118, type, sP38: extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_119, type, sP39: extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_120, type, sP40: extended_ereal > extended_ereal > $o).
% 2.96/2.00 thf(func_def_121, type, sK41: (extended_ereal > extended_ereal > $o) > real).
% 2.96/2.00 thf(func_def_122, type, sK42: (extended_ereal > extended_ereal > $o) > real).
% 2.96/2.00 thf(func_def_123, type, sK43: (extended_ereal > extended_ereal > $o) > real).
% 2.96/2.00 thf(func_def_124, type, sK44: (extended_ereal > extended_ereal > $o) > real).
% 2.96/2.00 thf(func_def_125, type, sK45: (extended_ereal > extended_ereal > $o) > real).
% 2.96/2.00 thf(func_def_126, type, sK46: (extended_ereal > extended_ereal > $o) > real).
% 2.96/2.00 thf(func_def_127, type, sK47: !>[X0: $tType]:(X0 > (real > X0) > nat > real)).
% 2.96/2.00 thf(func_def_128, type, sK48: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.96/2.00 thf(func_def_129, type, sK49: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.96/2.00 thf(func_def_130, type, sK50: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.96/2.00 thf(func_def_131, type, sK51: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.96/2.00 thf(func_def_132, type, sK52: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.96/2.00 thf(func_def_133, type, sK53: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.96/2.00 thf(func_def_134, type, sK54: extended_ereal > real).
% 2.96/2.00 thf(func_def_135, type, sK55: (extended_ereal > extended_ereal > $o) > real).
% 2.96/2.00 thf(func_def_136, type, sK56: (extended_ereal > extended_ereal > $o) > real).
% 2.96/2.00 thf(func_def_137, type, sK57: (extended_ereal > extended_ereal > $o) > real).
% 2.96/2.00 thf(func_def_138, type, sK58: (extended_ereal > extended_ereal > $o) > real).
% 2.96/2.00 thf(func_def_139, type, sK59: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 2.96/2.00 thf(func_def_140, type, sK60: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 2.96/2.00 thf(func_def_141, type, sK61: extended_ereal > extended_ereal > extended_ereal > real).
% 2.96/2.00 thf(func_def_142, type, sK62: extended_ereal > extended_ereal > extended_ereal > real).
% 2.96/2.00 thf(func_def_143, type, sK63: extended_ereal > real).
% 2.96/2.00 thf(func_def_144, type, sK64: extended_ereal > real).
% 2.96/2.00 thf(func_def_145, type, sK65: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.96/2.00 thf(func_def_146, type, sK66: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.96/2.00 thf(func_def_147, type, sK67: real > nat > (nat > extended_ereal) > nat).
% 2.96/2.00 thf(func_def_148, type, sK68: set @ extended_ereal > extended_ereal).
% 2.96/2.00 thf(func_def_149, type, sK69: extended_ereal > set @ extended_ereal > extended_ereal).
% 2.96/2.00 thf(func_def_150, type, sK70: extended_ereal > real).
% 2.96/2.00 thf(func_def_151, type, sK71: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.96/2.00 thf(func_def_152, type, sK72: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.96/2.00 thf(func_def_153, type, sK73: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.96/2.00 thf(func_def_154, type, sK74: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.96/2.00 thf(func_def_155, type, sK75: nat > (nat > $o) > nat).
% 2.96/2.00 thf(func_def_156, type, sK76: (nat > $o) > nat).
% 2.96/2.00 thf(func_def_157, type, sK77: set @ extended_ereal > extended_ereal).
% 2.96/2.00 thf(func_def_158, type, sK78: extended_ereal > set @ extended_ereal > extended_ereal).
% 2.96/2.00 thf(func_def_159, type, sK79: (extended_ereal > $o) > real).
% 2.96/2.00 thf(func_def_160, type, sK80: extended_ereal > real).
% 2.96/2.00 thf(func_def_161, type, sK81: extended_ereal > real).
% 2.96/2.00 thf(func_def_162, type, sK82: extended_ereal > real).
% 2.96/2.00 thf(func_def_163, type, sK83: extended_ereal > real).
% 2.96/2.00 thf(func_def_164, type, sK84: extended_ereal > real).
% 2.96/2.00 thf(func_def_165, type, sK85: extended_ereal > real).
% 2.96/2.00 thf(func_def_166, type, sK86: extended_ereal > real).
% 2.96/2.00 thf(func_def_167, type, sK87: extended_ereal > real).
% 2.96/2.00 thf(func_def_168, type, sK88: extended_ereal > real).
% 2.96/2.00 thf(func_def_169, type, sK89: extended_ereal > real).
% 2.96/2.00 thf(func_def_170, type, sK90: extended_ereal > real).
% 2.96/2.00 thf(func_def_171, type, sK91: extended_ereal > real).
% 2.96/2.00 thf(func_def_172, type, sK92: extended_ereal > real).
% 2.96/2.00 thf(func_def_173, type, sK93: extended_ereal > real).
% 2.96/2.00 thf(func_def_174, type, sK94: extended_ereal > real).
% 2.96/2.00 thf(func_def_175, type, sK95: extended_ereal > real).
% 2.96/2.00 thf(func_def_176, type, sK96: extended_ereal > real).
% 2.96/2.00 thf(func_def_177, type, sK97: extended_ereal > real).
% 2.96/2.00 thf(func_def_178, type, sK98: extended_ereal > real).
% 2.96/2.00 thf(func_def_179, type, sK99: extended_ereal > real).
% 2.96/2.00 thf(func_def_180, type, sK100: extended_ereal > real).
% 2.96/2.00 thf(func_def_181, type, sK101: extended_ereal > real).
% 2.96/2.00 thf(func_def_182, type, sK102: extended_ereal > real).
% 2.96/2.00 thf(func_def_183, type, sK103: extended_ereal > real).
% 2.96/2.00 thf(func_def_184, type, sK104: extended_ereal > real).
% 2.96/2.00 thf(func_def_185, type, sK105: extended_ereal > real).
% 2.96/2.00 thf(func_def_186, type, sK106: extended_ereal > real).
% 2.96/2.00 thf(func_def_187, type, sK107: nat > nat > nat).
% 2.96/2.00 thf(func_def_188, type, sK108: extended_ereal > real).
% 2.96/2.00 thf(func_def_189, type, sK109: !>[X0: $tType]:((nat > extended_ereal) > (nat > X0) > (X0 > extended_ereal) > nat)).
% 2.96/2.00 thf(func_def_190, type, sK110: !>[X0: $tType]:(nat > (nat > X0) > (nat > X0) > nat)).
% 2.96/2.00 thf(func_def_191, type, sK111: extended_ereal > extended_ereal > real).
% 2.96/2.00 thf(func_def_192, type, sK112: !>[X0: $tType]:(X0 > (real > X0) > nat > real)).
% 2.96/2.00 thf(func_def_193, type, sK113: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.96/2.00 thf(func_def_194, type, sK114: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.96/2.00 thf(func_def_195, type, sK115: !>[X0: $tType]:((nat > X0) > X0)).
% 2.96/2.00 thf(func_def_196, type, sK116: !>[X0: $tType]:((nat > X0) > nat)).
% 2.96/2.00 thf(func_def_197, type, sK117: !>[X0: $tType]:((nat > X0) > nat)).
% 2.96/2.00 thf(func_def_198, type, sK118: (extended_ereal > extended_ereal > $o) > real).
% 2.96/2.00 thf(func_def_199, type, sK119: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 2.96/2.00 thf(func_def_200, type, sK120: (extended_ereal > extended_ereal > $o) > real).
% 2.96/2.00 thf(func_def_201, type, sK121: (extended_ereal > extended_ereal > $o) > real).
% 2.96/2.00 thf(func_def_202, type, sK122: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 2.96/2.00 thf(func_def_203, type, sK123: (extended_ereal > extended_ereal > $o) > real).
% 2.96/2.00 thf(func_def_204, type, sK124: !>[X0: $tType]:((nat > X0) > (nat > X0) > nat > nat)).
% 2.96/2.00 thf(func_def_205, type, sK125: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.96/2.00 thf(func_def_206, type, sK126: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.96/2.00 thf(func_def_207, type, sK127: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.96/2.00 thf(func_def_208, type, sK128: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.96/2.00 thf(func_def_209, type, sK129: extended_ereal > real).
% 2.96/2.00 thf(func_def_210, type, sK130: !>[X0: $tType]:((nat > X0) > nat)).
% 2.96/2.00 thf(func_def_211, type, sK131: !>[X0: $tType]:((nat > X0) > nat)).
% 2.96/2.00 thf(func_def_212, type, sK132: extended_ereal > real).
% 2.96/2.00 thf(func_def_213, type, sK133: !>[X0: $tType]:(X0 > nat > (nat > X0) > nat)).
% 2.96/2.00 thf(func_def_214, type, sK134: !>[X0: $tType]:(nat > X0 > (nat > X0) > nat)).
% 2.96/2.00 thf(func_def_215, type, sK135: real > (nat > extended_ereal) > nat).
% 2.96/2.00 thf(func_def_216, type, sK136: !>[X0: $tType]:((nat > X0) > nat)).
% 2.96/2.00 thf(func_def_217, type, sK137: !>[X0: $tType]:((nat > X0) > nat)).
% 2.96/2.00 thf(func_def_218, type, sK138: !>[X0: $tType]:(nat > X0 > (nat > X0) > nat)).
% 2.96/2.00 thf(func_def_219, type, sK139: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.96/2.00 thf(func_def_220, type, sK140: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.96/2.00 thf(func_def_221, type, sK141: !>[X0: $tType]:(X0 > nat > (nat > X0) > nat)).
% 2.96/2.00 thf(func_def_222, type, sK142: (extended_ereal > $o) > real).
% 2.96/2.00 thf(func_def_223, type, sK143: real > (nat > extended_ereal) > nat).
% 2.96/2.00 thf(func_def_224, type, sK144: !>[X0: $tType]:((nat > X0) > X0)).
% 2.96/2.00 thf(func_def_225, type, sK145: !>[X0: $tType]:((nat > X0) > nat)).
% 2.96/2.00 thf(func_def_226, type, sK146: !>[X0: $tType]:((nat > X0) > nat)).
% 2.96/2.00 thf(func_def_227, type, sK147: extended_ereal > real).
% 2.96/2.00 thf(func_def_228, type, sK148: extended_ereal > real).
% 2.96/2.00 thf(func_def_229, type, sK149: extended_ereal > real).
% 2.96/2.00 thf(func_def_230, type, sK150: extended_ereal > real).
% 2.96/2.00 thf(func_def_231, type, sK151: extended_ereal > real).
% 2.96/2.00 thf(func_def_232, type, sK152: extended_ereal > real).
% 2.96/2.00 thf(func_def_234, type, ph154: !>[X0: $tType]:(X0)).
% 2.96/2.00 thf(f2399,plain,(
% 2.96/2.00 $false),
% 2.96/2.00 inference(subsumption_resolution,[status(thm)],[f2395,f2361])).
% 2.96/2.00 thf(f2361,plain,(
% 2.96/2.00 ($true = (filterlim @ nat @ extended_ereal @ (^[Y0 : nat]: (f @ (x @ Y0))) @ (topolo1920029354e_nhds @ extended_ereal @ a2) @ at_top @ nat))),
% 2.96/2.00 inference(beta_eta_normalization,[status(thm)],[f2319])).
% 2.96/2.00 thf(f2319,plain,(
% 2.96/2.00 ($true = (filterlim @ nat @ extended_ereal @ ((^[Y0 : a > extended_ereal]: ((^[Y1 : nat > a]: ((^[Y2 : nat]: (Y0 @ (Y1 @ Y2))))))) @ f @ x) @ (topolo1920029354e_nhds @ extended_ereal @ a2) @ at_top @ nat))),
% 2.96/2.00 inference(definition_unfolding,[status(thm)],[f2289,f2142])).
% 2.96/2.00 thf(f2142,plain,(
% 2.96/2.00 ( ! [X1 : $tType,X0 : $tType,X2 : $tType] : ((comp @ X2 @ X1 @ X0 = (^[Y0 : X2 > X1]: ((^[Y1 : X0 > X2]: ((^[Y2 : X0]: (Y0 @ (Y1 @ Y2))))))))) )),
% 2.96/2.00 inference(cnf_transformation,[status(thm)],[f970])).
% 2.96/2.00 thf(f970,plain,(
% 2.96/2.00 ! [X0 : $tType,X1 : $tType,X2 : $tType] : (comp @ X2 @ X1 @ X0 = (^[Y0 : X2 > X1]: ((^[Y1 : X0 > X2]: ((^[Y2 : X0]: (Y0 @ (Y1 @ Y2))))))))),
% 2.96/2.00 inference(fool_elimination,[status(thm)],[f969])).
% 2.96/2.00 thf(f969,plain,(
% 2.96/2.00 ! [X0 : $tType,X1 : $tType,X2 : $tType] : (comp @ X2 @ X1 @ X0 = (^[X3 : X2 > X1, X4 : X0 > X2, X5 : X0] : (X3 @ (X4 @ X5))))),
% 2.96/2.00 inference(rectify,[status(thm)],[f21])).
% 2.96/2.00 thf(f21,axiom,(
% 2.96/2.00 ! [X2 : $tType,X0 : $tType,X1 : $tType] : (comp @ X1 @ X0 @ X2 = (^[X16 : X1 > X0, X17 : X2 > X1, X4 : X2] : (X16 @ (X17 @ X4))))),
% 2.96/2.00 file('/export/starexec/sandbox2/tmp/tmp.WRkRsD2zqq/DTF2THF_29324.p',fact_20_comp__apply)).
% 2.96/2.00 thf(f2289,plain,(
% 2.96/2.00 ((filterlim @ nat @ extended_ereal @ (comp @ a @ extended_ereal @ nat @ f @ x) @ (topolo1920029354e_nhds @ extended_ereal @ a2) @ at_top @ nat) = $true)),
% 2.96/2.00 inference(cnf_transformation,[status(thm)],[f607])).
% 2.96/2.00 thf(f607,plain,(
% 2.96/2.00 ((filterlim @ nat @ extended_ereal @ (comp @ a @ extended_ereal @ nat @ f @ x) @ (topolo1920029354e_nhds @ extended_ereal @ a2) @ at_top @ nat) = $true)),
% 2.96/2.00 inference(fool_elimination,[status(thm)],[f606])).
% 2.96/2.00 thf(f606,plain,(
% 2.96/2.00 (filterlim @ nat @ extended_ereal @ (comp @ a @ extended_ereal @ nat @ f @ x) @ (topolo1920029354e_nhds @ extended_ereal @ a2) @ at_top @ nat)),
% 2.96/2.00 inference(rectify,[status(thm)],[f2])).
% 2.96/2.00 thf(f2,axiom,(
% 2.96/2.00 (filterlim @ nat @ extended_ereal @ (comp @ a @ extended_ereal @ nat @ f @ x) @ (topolo1920029354e_nhds @ extended_ereal @ a2) @ at_top @ nat)),
% 2.96/2.00 file('/export/starexec/sandbox2/tmp/tmp.WRkRsD2zqq/DTF2THF_29324.p',fact_1_x__def_I2_J)).
% 2.96/2.00 thf(f2395,plain,(
% 2.96/2.00 ($true != (filterlim @ nat @ extended_ereal @ (^[Y0 : nat]: (f @ (x @ Y0))) @ (topolo1920029354e_nhds @ extended_ereal @ a2) @ at_top @ nat))),
% 2.96/2.00 inference(trivial_inequality_removal,[status(thm)],[f2394])).
% 2.96/2.00 thf(f2394,plain,(
% 2.96/2.00 ($true != $true) | ($true != (filterlim @ nat @ extended_ereal @ (^[Y0 : nat]: (f @ (x @ Y0))) @ (topolo1920029354e_nhds @ extended_ereal @ a2) @ at_top @ nat))),
% 2.96/2.00 inference(superposition,[status(thm)],[f2123,f2380])).
% 2.96/2.00 thf(f2380,plain,(
% 2.96/2.00 ( ! [X0 : filter @ nat] : (($true = (filterlim @ nat @ extended_ereal @ (^[Y0 : nat]: (uminus_uminus @ extended_ereal @ (f @ (x @ Y0)))) @ (topolo1920029354e_nhds @ extended_ereal @ (uminus_uminus @ extended_ereal @ a2)) @ X0)) | ((filterlim @ nat @ extended_ereal @ (^[Y0 : nat]: (f @ (x @ Y0))) @ (topolo1920029354e_nhds @ extended_ereal @ a2) @ X0) != $true)) )),
% 2.96/2.00 inference(beta_eta_normalization,[status(thm)],[f2297])).
% 2.96/2.00 thf(f2297,plain,(
% 2.96/2.00 ( ! [X0 : filter @ nat] : (((filterlim @ nat @ extended_ereal @ ((^[Y0 : a > extended_ereal]: ((^[Y1 : nat > a]: ((^[Y2 : nat]: (Y0 @ (Y1 @ Y2))))))) @ f @ x) @ (topolo1920029354e_nhds @ extended_ereal @ a2) @ X0) != $true) | ($true = (filterlim @ nat @ extended_ereal @ (^[Y0 : nat]: (uminus_uminus @ extended_ereal @ ((^[Y1 : a > extended_ereal]: ((^[Y2 : nat > a]: ((^[Y3 : nat]: (Y1 @ (Y2 @ Y3))))))) @ f @ x @ Y0))) @ (topolo1920029354e_nhds @ extended_ereal @ (uminus_uminus @ extended_ereal @ a2)) @ X0))) )),
% 2.96/2.00 inference(definition_unfolding,[status(thm)],[f1840,f2142,f2142])).
% 2.96/2.00 thf(f1840,plain,(
% 2.96/2.00 ( ! [X0 : filter @ nat] : (((filterlim @ nat @ extended_ereal @ (^[Y0 : nat]: (uminus_uminus @ extended_ereal @ (comp @ a @ extended_ereal @ nat @ f @ x @ Y0))) @ (topolo1920029354e_nhds @ extended_ereal @ (uminus_uminus @ extended_ereal @ a2)) @ X0) = $true) | ((filterlim @ nat @ extended_ereal @ (comp @ a @ extended_ereal @ nat @ f @ x) @ (topolo1920029354e_nhds @ extended_ereal @ a2) @ X0) != $true)) )),
% 2.96/2.00 inference(cnf_transformation,[status(thm)],[f1223])).
% 2.96/2.00 thf(f1223,plain,(
% 2.96/2.00 ! [X0 : filter @ nat] : (((filterlim @ nat @ extended_ereal @ (^[Y0 : nat]: (uminus_uminus @ extended_ereal @ (comp @ a @ extended_ereal @ nat @ f @ x @ Y0))) @ (topolo1920029354e_nhds @ extended_ereal @ (uminus_uminus @ extended_ereal @ a2)) @ X0) = $true) | ((filterlim @ nat @ extended_ereal @ (comp @ a @ extended_ereal @ nat @ f @ x) @ (topolo1920029354e_nhds @ extended_ereal @ a2) @ X0) != $true))),
% 2.96/2.00 inference(ennf_transformation,[status(thm)],[f739])).
% 2.96/2.00 thf(f739,plain,(
% 2.96/2.00 ! [X0 : filter @ nat] : (((filterlim @ nat @ extended_ereal @ (comp @ a @ extended_ereal @ nat @ f @ x) @ (topolo1920029354e_nhds @ extended_ereal @ a2) @ X0) = $true) => ((filterlim @ nat @ extended_ereal @ (^[Y0 : nat]: (uminus_uminus @ extended_ereal @ (comp @ a @ extended_ereal @ nat @ f @ x @ Y0))) @ (topolo1920029354e_nhds @ extended_ereal @ (uminus_uminus @ extended_ereal @ a2)) @ X0) = $true))),
% 2.96/2.00 inference(fool_elimination,[status(thm)],[f738])).
% 2.96/2.00 thf(f738,plain,(
% 2.96/2.00 ! [X0 : filter @ nat] : ((filterlim @ nat @ extended_ereal @ (comp @ a @ extended_ereal @ nat @ f @ x) @ (topolo1920029354e_nhds @ extended_ereal @ a2) @ X0) => (filterlim @ nat @ extended_ereal @ (^[X1 : nat] : (uminus_uminus @ extended_ereal @ (comp @ a @ extended_ereal @ nat @ f @ x @ X1))) @ (topolo1920029354e_nhds @ extended_ereal @ (uminus_uminus @ extended_ereal @ a2)) @ X0))),
% 2.96/2.00 inference(rectify,[status(thm)],[f3])).
% 2.96/2.00 thf(f3,axiom,(
% 2.96/2.00 ! [X3 : filter @ nat] : ((filterlim @ nat @ extended_ereal @ (comp @ a @ extended_ereal @ nat @ f @ x) @ (topolo1920029354e_nhds @ extended_ereal @ a2) @ X3) => (filterlim @ nat @ extended_ereal @ (^[X4 : nat] : (uminus_uminus @ extended_ereal @ (comp @ a @ extended_ereal @ nat @ f @ x @ X4))) @ (topolo1920029354e_nhds @ extended_ereal @ (uminus_uminus @ extended_ereal @ a2)) @ X3))),
% 2.96/2.00 file('/export/starexec/sandbox2/tmp/tmp.WRkRsD2zqq/DTF2THF_29324.p',fact_2__092_060open_062_092_060And_062F_O_A_I_If_A_092_060circ_062_Ax_J_A_092_060longlongrightarrow_062_AA_J_AF_A_092_060Longrightarrow_062_A_I_I_092_060lambda_062xa_O_A_N_A_If_A_092_060circ_062_Ax_J_Axa_J_A_092_060longlongrightarrow_062_A_N_AA_J_AF_092_060close_062)).
% 2.96/2.00 thf(f2123,plain,(
% 2.96/2.00 ((filterlim @ nat @ extended_ereal @ (^[Y0 : nat]: (uminus_uminus @ extended_ereal @ (f @ (x @ Y0)))) @ (topolo1920029354e_nhds @ extended_ereal @ (uminus_uminus @ extended_ereal @ a2)) @ at_top @ nat) != $true)),
% 2.96/2.00 inference(cnf_transformation,[status(thm)],[f1166])).
% 2.96/2.00 thf(f1166,plain,(
% 2.96/2.00 ((filterlim @ nat @ extended_ereal @ (^[Y0 : nat]: (uminus_uminus @ extended_ereal @ (f @ (x @ Y0)))) @ (topolo1920029354e_nhds @ extended_ereal @ (uminus_uminus @ extended_ereal @ a2)) @ at_top @ nat) != $true)),
% 2.96/2.00 inference(flattening,[status(thm)],[f533])).
% 2.96/2.00 thf(f533,plain,(
% 2.96/2.00 ~((filterlim @ nat @ extended_ereal @ (^[Y0 : nat]: (uminus_uminus @ extended_ereal @ (f @ (x @ Y0)))) @ (topolo1920029354e_nhds @ extended_ereal @ (uminus_uminus @ extended_ereal @ a2)) @ at_top @ nat) = $true)),
% 2.96/2.00 inference(fool_elimination,[status(thm)],[f532])).
% 2.96/2.00 thf(f532,plain,(
% 2.96/2.00 ~(filterlim @ nat @ extended_ereal @ (^[X0 : nat] : (uminus_uminus @ extended_ereal @ (f @ (x @ X0)))) @ (topolo1920029354e_nhds @ extended_ereal @ (uminus_uminus @ extended_ereal @ a2)) @ at_top @ nat)),
% 2.96/2.00 inference(rectify,[status(thm)],[f383])).
% 2.96/2.00 thf(f383,negated_conjecture,(
% 2.96/2.00 ~(filterlim @ nat @ extended_ereal @ (^[X106 : nat] : (uminus_uminus @ extended_ereal @ (f @ (x @ X106)))) @ (topolo1920029354e_nhds @ extended_ereal @ (uminus_uminus @ extended_ereal @ a2)) @ at_top @ nat)),
% 2.96/2.00 inference(negated_conjecture,[status(cth)],[f382])).
% 2.96/2.00 thf(f382,conjecture,(
% 2.96/2.00 (filterlim @ nat @ extended_ereal @ (^[X106 : nat] : (uminus_uminus @ extended_ereal @ (f @ (x @ X106)))) @ (topolo1920029354e_nhds @ extended_ereal @ (uminus_uminus @ extended_ereal @ a2)) @ at_top @ nat)),
% 2.96/2.00 file('/export/starexec/sandbox2/tmp/tmp.WRkRsD2zqq/DTF2THF_29324.p',conj_0)).
% 2.96/2.00 % SZS output end Proof for DTF2THF_29324
% 2.96/2.00 % (29560)------------------------------
% 2.96/2.00 % (29560)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/2.00 % (29560)Termination reason: Refutation
% 2.96/2.00
% 2.96/2.00 % (29560)Memory used [KB]: 7931
% 2.96/2.00 % (29560)Time elapsed: 0.052 s
% 2.96/2.00 % (29560)Instructions burned: 161 (million)
% 2.96/2.00 % (29560)------------------------------
% 2.96/2.00 % (29560)------------------------------
% 2.96/2.00 % (29516)Success in time 0.426 s
%------------------------------------------------------------------------------