%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP110^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n031.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 08:55:38 AM UTC 2026
% Result : Theorem 2.52s 1.70s
% Output : Refutation 2.52s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP110^2 : TPTP v9.2.1. Released v7.5.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.34 % Computer : n031.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Tue Feb 24 08:27:03 EST 2026
% 0.18/0.34 % CPUTime :
% 0.18/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.32/0.46 ---- Original DTF file ---
% 0.32/0.46 thf(spec,logic,$$dhol).
% 0.32/0.46 %------------------------------------------------------------------------------
% 0.32/0.46 % File : ITP110^2 : TPTP v9.2.1. Released v7.5.0.
% 0.32/0.46 % Domain : Interactive Theorem Proving
% 0.32/0.46 % Problem : Sledgehammer Lower_Semicontinuous problem prob_1321__6263368_1
% 0.32/0.46 % Version : Especial.
% 0.32/0.46 % English :
% 0.32/0.46
% 0.32/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.32/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.32/0.46 % Source : [Des21]
% 0.32/0.46 % Names : Lower_Semicontinuous/prob_1321__6263368_1 [Des21]
% 0.32/0.46
% 0.32/0.46 % Status : Theorem
% 0.32/0.46 % Rating : 0.67 v8.1.0, 0.50 v7.5.0
% 0.32/0.46 % Syntax : Number of formulae : 404 ( 131 unt; 62 typ; 0 def)
% 0.32/0.46 % Number of atoms : 969 ( 384 equ; 0 cnn)
% 0.32/0.46 % Maximal formula atoms : 81 ( 2 avg)
% 0.32/0.46 % Number of connectives : 3379 ( 92 ~; 16 |; 48 &;2734 @)
% 0.32/0.46 % ( 0 <=>; 489 =>; 0 <=; 0 <~>)
% 0.32/0.46 % Maximal formula depth : 34 ( 6 avg)
% 0.32/0.46 % Number of types : 4 ( 3 usr)
% 0.32/0.46 % Number of type conns : 125 ( 125 >; 0 *; 0 +; 0 <<)
% 0.32/0.46 % Number of symbols : 60 ( 59 usr; 2 con; 0-5 aty)
% 0.32/0.46 % Number of variables : 881 ( 37 ^; 757 !; 35 ?; 881 :)
% 0.32/0.46 % ( 52 !>; 0 ?*; 0 @-; 0 @+)
% 0.32/0.46 % SPC : TH1_THM_EQU_NAR
% 0.32/0.46
% 0.32/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:29:37.519
% 0.32/0.46 %------------------------------------------------------------------------------
% 0.32/0.46 % Could-be-implicit typings (5)
% 0.32/0.46 thf(ty_t_Extended__Real_Oereal,type,
% 0.32/0.46 extended_ereal: $tType ).
% 0.32/0.46
% 0.32/0.46 thf(ty_t_Product__Type_Oprod,type,
% 0.32/0.46 product_prod: $tType > $tType > $tType ).
% 0.32/0.46
% 0.32/0.46 thf(ty_t_Real_Oreal,type,
% 0.32/0.46 real: $tType ).
% 0.32/0.46
% 0.32/0.46 thf(ty_t_Set_Oset,type,
% 0.32/0.46 set: $tType > $tType ).
% 0.32/0.46
% 0.32/0.46 thf(ty_tf_a,type,
% 0.32/0.46 a: $tType ).
% 0.32/0.46
% 0.32/0.46 % Explicit typings (57)
% 0.32/0.46 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
% 0.32/0.46 ordere890947078_space:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_HOL_Otype,type,
% 0.32/0.46 type:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Rings_Oring,type,
% 0.32/0.46 ring:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Groups_Ozero,type,
% 0.32/0.46 zero:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Fields_Ofield,type,
% 0.32/0.46 field:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Groups_Ouminus,type,
% 0.32/0.46 uminus:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.32/0.46 ord:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Orderings_Otop,type,
% 0.32/0.46 top:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.32/0.46 order:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Rings_Omult__zero,type,
% 0.32/0.46 mult_zero:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.32/0.46 group_add:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Orderings_Ono__bot,type,
% 0.32/0.46 no_bot:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.32/0.46 linorder:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.32/0.46 preorder:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Enum_Ofinite__lattice,type,
% 0.32/0.46 finite_lattice:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Orderings_Oorder__top,type,
% 0.32/0.46 order_top:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Fields_Odivision__ring,type,
% 0.32/0.46 division_ring:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Parity_Osemiring__bits,type,
% 0.32/0.46 semiring_bits:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Rings_Osemidom__divide,type,
% 0.32/0.46 semidom_divide:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Fields_Olinordered__field,type,
% 0.32/0.46 linordered_field:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 0.32/0.46 boolean_algebra:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.32/0.46 ordered_ab_group_add:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.32/0.46 linord219039673up_add:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
% 0.32/0.46 real_V1076094709vector:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Euclidean__Space_Oeuclidean__space,type,
% 0.32/0.46 euclid925273238_space:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
% 0.32/0.46 real_V2090557954_space:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 0.32/0.46 semiri1193490041visors:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.32/0.46 linord1659791738miring:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 0.32/0.46 comple187826305attice:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.32/0.46 canoni770627133id_add:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
% 0.32/0.46 semiri1923998003cancel:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
% 0.32/0.46 euclid24285859cancel:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
% 0.32/0.46 condit378418413attice:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
% 0.32/0.46 complete_Sup_Sup:
% 0.32/0.46 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Convex_Oconvex,type,
% 0.32/0.46 convex:
% 0.32/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
% 0.32/0.46 extend1396239628finity:
% 0.32/0.46 !>[A: $tType] : A ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Extended__Real_Oereal_OMInfty,type,
% 0.32/0.46 extended_MInfty: extended_ereal ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Extended__Real_Oereal_OPInfty,type,
% 0.32/0.46 extended_PInfty: extended_ereal ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Extended__Real_Oereal_Oereal,type,
% 0.32/0.46 extended_ereal2: real > extended_ereal ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Extended__Real_Oereal_Orec__ereal,type,
% 0.32/0.46 extended_rec_ereal:
% 0.32/0.46 !>[A: $tType] : ( ( real > A ) > A > A > extended_ereal > A ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Extended__Real_Oreal__of__ereal,type,
% 0.32/0.46 extend1716541707_ereal: extended_ereal > real ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 0.32/0.46 times_times:
% 0.32/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.32/0.46 uminus_uminus:
% 0.32/0.46 !>[A: $tType] : ( A > A ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.32/0.46 zero_zero:
% 0.32/0.46 !>[A: $tType] : A ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_OEpigraph,type,
% 0.32/0.46 lower_261390618igraph:
% 0.32/0.46 !>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > ( set @ ( product_prod @ A @ real ) ) ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Oaffine__on,type,
% 0.32/0.46 lower_500881736ine_on:
% 0.32/0.46 !>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > $o ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Oconcave__on,type,
% 0.32/0.46 lower_708069572ave_on:
% 0.32/0.46 !>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > $o ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Oconvex__on,type,
% 0.32/0.46 lower_673667120vex_on:
% 0.32/0.46 !>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > $o ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Odomain,type,
% 0.32/0.46 lower_272802190domain:
% 0.32/0.46 !>[A: $tType] : ( ( A > extended_ereal ) > ( set @ A ) ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Ofinite__on,type,
% 0.32/0.46 lower_728871928ite_on:
% 0.32/0.46 !>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > $o ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.32/0.46 ord_less_eq:
% 0.32/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 0.32/0.46 order_Greatest:
% 0.32/0.46 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.32/0.46 top_top:
% 0.32/0.46 !>[A: $tType] : A ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 0.32/0.46 divide_divide:
% 0.32/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Set_OCollect,type,
% 0.32/0.46 collect:
% 0.32/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_member,type,
% 0.32/0.46 member:
% 0.32/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_v_f,type,
% 0.32/0.46 f: a > extended_ereal ).
% 0.32/0.46
% 0.32/0.46 % Relevant facts (255)
% 0.32/0.46 thf(fact_0__092_060open_062convex_A_Idomain_Af_J_A_092_060Longrightarrow_062_ALower__Semicontinuous__Mirabelle__quczrylfpw_Oconvex__on_AUNIV_Af_092_060close_062,axiom,
% 0.32/0.46 ( ( convex @ a @ ( lower_272802190domain @ a @ f ) )
% 0.32/0.46 => ( lower_673667120vex_on @ a @ ( top_top @ ( set @ a ) ) @ f ) ) ).
% 0.32/0.46
% 0.32/0.46 % \<open>convex (domain f) \<Longrightarrow> Lower_Semicontinuous_Mirabelle_quczrylfpw.convex_on UNIV f\<close>
% 0.32/0.46 thf(fact_1_convex__domain,axiom,
% 0.32/0.46 ! [A: $tType] :
% 0.32/0.46 ( ( euclid925273238_space @ A )
% 0.32/0.46 => ! [F: A > extended_ereal] :
% 0.32/0.46 ( ( lower_673667120vex_on @ A @ ( top_top @ ( set @ A ) ) @ F )
% 0.32/0.46 => ( convex @ A @ ( lower_272802190domain @ A @ F ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % convex_domain
% 0.32/0.46 thf(fact_2_convex__on__domain,axiom,
% 0.32/0.46 ! [A: $tType] :
% 0.32/0.46 ( ( real_V1076094709vector @ A )
% 0.32/0.46 => ! [F: A > extended_ereal] :
% 0.32/0.46 ( ( lower_673667120vex_on @ A @ ( lower_272802190domain @ A @ F ) @ F )
% 0.32/0.46 = ( lower_673667120vex_on @ A @ ( top_top @ ( set @ A ) ) @ F ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % convex_on_domain
% 0.32/0.46 thf(fact_3_convex__on__domain2,axiom,
% 0.32/0.46 ! [A: $tType] :
% 0.32/0.46 ( ( real_V1076094709vector @ A )
% 0.32/0.46 => ! [F: A > extended_ereal] :
% 0.32/0.46 ( ( lower_673667120vex_on @ A @ ( lower_272802190domain @ A @ F ) @ F )
% 0.32/0.46 = ( ! [S: set @ A] : ( lower_673667120vex_on @ A @ S @ F ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % convex_on_domain2
% 0.32/0.46 thf(fact_4_convex__on__ereal__univ,axiom,
% 0.32/0.46 ! [A: $tType] :
% 0.32/0.46 ( ( real_V1076094709vector @ A )
% 0.32/0.46 => ! [F: A > extended_ereal] :
% 0.32/0.46 ( ( lower_673667120vex_on @ A @ ( top_top @ ( set @ A ) ) @ F )
% 0.32/0.46 = ( ! [S: set @ A] : ( lower_673667120vex_on @ A @ S @ F ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % convex_on_ereal_univ
% 0.32/0.46 thf(fact_5_convex__UNIV,axiom,
% 0.32/0.46 ! [A: $tType] :
% 0.32/0.46 ( ( real_V1076094709vector @ A )
% 0.32/0.46 => ( convex @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % convex_UNIV
% 0.32/0.46 thf(fact_6_UNIV__I,axiom,
% 0.32/0.46 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % UNIV_I
% 0.32/0.46 thf(fact_7_iso__tuple__UNIV__I,axiom,
% 0.32/0.46 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % iso_tuple_UNIV_I
% 0.32/0.46 thf(fact_8_top__apply,axiom,
% 0.32/0.46 ! [C: $tType,D: $tType] :
% 0.32/0.46 ( ( top @ C )
% 0.32/0.46 => ( ( top_top @ ( D > C ) )
% 0.32/0.46 = ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % top_apply
% 0.32/0.46 thf(fact_9_assms,axiom,
% 0.32/0.46 ! [X3: a] :
% 0.32/0.46 ( ( ( f @ X3 )
% 0.32/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.46 | ( ( f @ X3 )
% 0.32/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % assms
% 0.32/0.46 thf(fact_10_UNIV__eq__I,axiom,
% 0.32/0.46 ! [A: $tType,A2: set @ A] :
% 0.32/0.46 ( ! [X4: A] : ( member @ A @ X4 @ A2 )
% 0.32/0.46 => ( ( top_top @ ( set @ A ) )
% 0.32/0.46 = A2 ) ) ).
% 0.32/0.46
% 0.32/0.46 % UNIV_eq_I
% 0.32/0.46 thf(fact_11_UNIV__witness,axiom,
% 0.32/0.46 ! [A: $tType] :
% 0.32/0.46 ? [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % UNIV_witness
% 0.32/0.46 thf(fact_12_convex__Epigraph,axiom,
% 0.32/0.46 ! [A: $tType] :
% 0.32/0.46 ( ( real_V1076094709vector @ A )
% 0.32/0.46 => ! [S2: set @ A,F: A > extended_ereal] :
% 0.32/0.46 ( ( convex @ A @ S2 )
% 0.32/0.46 => ( ( convex @ ( product_prod @ A @ real ) @ ( lower_261390618igraph @ A @ S2 @ F ) )
% 0.32/0.46 = ( lower_673667120vex_on @ A @ S2 @ F ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % convex_Epigraph
% 0.32/0.46 thf(fact_13_convex__EpigraphI,axiom,
% 0.32/0.46 ! [A: $tType] :
% 0.32/0.46 ( ( real_V1076094709vector @ A )
% 0.32/0.46 => ! [S3: set @ A,F: A > extended_ereal] :
% 0.32/0.46 ( ( lower_673667120vex_on @ A @ S3 @ F )
% 0.32/0.46 => ( ( convex @ A @ S3 )
% 0.32/0.46 => ( convex @ ( product_prod @ A @ real ) @ ( lower_261390618igraph @ A @ S3 @ F ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % convex_EpigraphI
% 0.32/0.46 thf(fact_14_affine__on__def,axiom,
% 0.32/0.46 ! [A: $tType] :
% 0.32/0.46 ( ( real_V1076094709vector @ A )
% 0.32/0.46 => ( ( lower_500881736ine_on @ A )
% 0.32/0.46 = ( ^ [S: set @ A,F2: A > extended_ereal] :
% 0.32/0.46 ( ( lower_673667120vex_on @ A @ S @ F2 )
% 0.32/0.46 & ( lower_708069572ave_on @ A @ S @ F2 )
% 0.32/0.46 & ( lower_728871928ite_on @ A @ S @ F2 ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % affine_on_def
% 0.32/0.46 thf(fact_15_Sup__UNIV,axiom,
% 0.32/0.46 ! [A: $tType] :
% 0.32/0.46 ( ( comple187826305attice @ A )
% 0.32/0.46 => ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
% 0.32/0.46 = ( top_top @ A ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % Sup_UNIV
% 0.32/0.46 thf(fact_16_convex__on__ereal__subset,axiom,
% 0.32/0.46 ! [A: $tType] :
% 0.32/0.46 ( ( real_V1076094709vector @ A )
% 0.32/0.46 => ! [T: set @ A,F: A > extended_ereal,S3: set @ A] :
% 0.32/0.46 ( ( lower_673667120vex_on @ A @ T @ F )
% 0.32/0.46 => ( ( ord_less_eq @ ( set @ A ) @ S3 @ T )
% 0.32/0.46 => ( lower_673667120vex_on @ A @ S3 @ F ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % convex_on_ereal_subset
% 0.32/0.46 thf(fact_17_order__refl,axiom,
% 0.32/0.46 ! [A: $tType] :
% 0.32/0.46 ( ( preorder @ A )
% 0.32/0.46 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_refl
% 0.32/0.46 thf(fact_18_subsetI,axiom,
% 0.32/0.46 ! [A: $tType,A2: set @ A,B: set @ A] :
% 0.32/0.46 ( ! [X4: A] :
% 0.32/0.46 ( ( member @ A @ X4 @ A2 )
% 0.32/0.46 => ( member @ A @ X4 @ B ) )
% 0.32/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B ) ) ).
% 0.32/0.46
% 0.32/0.46 % subsetI
% 0.32/0.46 thf(fact_19_subset__antisym,axiom,
% 0.32/0.46 ! [A: $tType,A2: set @ A,B: set @ A] :
% 0.32/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
% 0.32/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B @ A2 )
% 0.32/0.46 => ( A2 = B ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % subset_antisym
% 0.32/0.46 thf(fact_20_in__mono,axiom,
% 0.32/0.46 ! [A: $tType,A2: set @ A,B: set @ A,X: A] :
% 0.32/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
% 0.32/0.46 => ( ( member @ A @ X @ A2 )
% 0.32/0.46 => ( member @ A @ X @ B ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % in_mono
% 0.32/0.46 thf(fact_21_subsetD,axiom,
% 0.32/0.46 ! [A: $tType,A2: set @ A,B: set @ A,C2: A] :
% 0.32/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
% 0.32/0.46 => ( ( member @ A @ C2 @ A2 )
% 0.32/0.46 => ( member @ A @ C2 @ B ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % subsetD
% 0.32/0.46 thf(fact_22_equalityE,axiom,
% 0.32/0.46 ! [A: $tType,A2: set @ A,B: set @ A] :
% 0.32/0.46 ( ( A2 = B )
% 0.32/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
% 0.32/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ B @ A2 ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % equalityE
% 0.32/0.46 thf(fact_23_subset__eq,axiom,
% 0.32/0.46 ! [A: $tType] :
% 0.32/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/0.46 = ( ^ [A3: set @ A,B2: set @ A] :
% 0.32/0.46 ! [X2: A] :
% 0.32/0.46 ( ( member @ A @ X2 @ A3 )
% 0.32/0.46 => ( member @ A @ X2 @ B2 ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % subset_eq
% 0.32/0.46 thf(fact_24_equalityD1,axiom,
% 0.32/0.46 ! [A: $tType,A2: set @ A,B: set @ A] :
% 0.32/0.46 ( ( A2 = B )
% 0.32/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B ) ) ).
% 0.32/0.46
% 0.32/0.46 % equalityD1
% 0.32/0.46 thf(fact_25_equalityD2,axiom,
% 0.32/0.46 ! [A: $tType,A2: set @ A,B: set @ A] :
% 0.32/0.46 ( ( A2 = B )
% 0.32/0.46 => ( ord_less_eq @ ( set @ A ) @ B @ A2 ) ) ).
% 0.32/0.46
% 0.32/0.46 % equalityD2
% 0.32/0.46 thf(fact_26_subset__iff,axiom,
% 0.32/0.46 ! [A: $tType] :
% 0.32/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/0.46 = ( ^ [A3: set @ A,B2: set @ A] :
% 0.32/0.46 ! [T2: A] :
% 0.32/0.46 ( ( member @ A @ T2 @ A3 )
% 0.32/0.46 => ( member @ A @ T2 @ B2 ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % subset_iff
% 0.32/0.46 thf(fact_27_subset__refl,axiom,
% 0.32/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.32/0.46
% 0.32/0.46 % subset_refl
% 0.32/0.46 thf(fact_28_Collect__mono,axiom,
% 0.32/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/0.46 ( ! [X4: A] :
% 0.32/0.46 ( ( P @ X4 )
% 0.32/0.46 => ( Q @ X4 ) )
% 0.32/0.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % Collect_mono
% 0.32/0.46 thf(fact_29_subset__trans,axiom,
% 0.32/0.46 ! [A: $tType,A2: set @ A,B: set @ A,C3: set @ A] :
% 0.32/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
% 0.32/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B @ C3 )
% 0.32/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % subset_trans
% 0.32/0.46 thf(fact_30_le__funD,axiom,
% 0.32/0.46 ! [B3: $tType,A: $tType] :
% 0.32/0.46 ( ( ord @ B3 )
% 0.32/0.46 => ! [F: A > B3,G: A > B3,X: A] :
% 0.32/0.46 ( ( ord_less_eq @ ( A > B3 ) @ F @ G )
% 0.32/0.46 => ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % le_funD
% 0.32/0.46 thf(fact_31_le__funE,axiom,
% 0.32/0.46 ! [B3: $tType,A: $tType] :
% 0.32/0.46 ( ( ord @ B3 )
% 0.32/0.46 => ! [F: A > B3,G: A > B3,X: A] :
% 0.32/0.46 ( ( ord_less_eq @ ( A > B3 ) @ F @ G )
% 0.32/0.46 => ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % le_funE
% 0.32/0.46 thf(fact_32_le__funI,axiom,
% 0.32/0.46 ! [B3: $tType,A: $tType] :
% 0.32/0.46 ( ( ord @ B3 )
% 0.32/0.46 => ! [F: A > B3,G: A > B3] :
% 0.32/0.46 ( ! [X4: A] : ( ord_less_eq @ B3 @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.32/0.46 => ( ord_less_eq @ ( A > B3 ) @ F @ G ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % le_funI
% 0.32/0.46 thf(fact_33_set__eq__subset,axiom,
% 0.32/0.46 ! [A: $tType] :
% 0.32/0.46 ( ( ^ [Y: set @ A,Z: set @ A] : ( Y = Z ) )
% 0.32/0.46 = ( ^ [A3: set @ A,B2: set @ A] :
% 0.32/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.32/0.46 & ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % set_eq_subset
% 0.32/0.46 thf(fact_34_le__fun__def,axiom,
% 0.32/0.46 ! [B3: $tType,A: $tType] :
% 0.32/0.46 ( ( ord @ B3 )
% 0.32/0.46 => ( ( ord_less_eq @ ( A > B3 ) )
% 0.32/0.46 = ( ^ [F2: A > B3,G2: A > B3] :
% 0.32/0.46 ! [X2: A] : ( ord_less_eq @ B3 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % le_fun_def
% 0.32/0.46 thf(fact_35_Collect__mono__iff,axiom,
% 0.32/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.32/0.46 = ( ! [X2: A] :
% 0.32/0.46 ( ( P @ X2 )
% 0.32/0.46 => ( Q @ X2 ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % Collect_mono_iff
% 0.32/0.46 thf(fact_36_order__subst1,axiom,
% 0.32/0.46 ! [A: $tType,B3: $tType] :
% 0.32/0.46 ( ( ( order @ B3 )
% 0.32/0.46 & ( order @ A ) )
% 0.32/0.46 => ! [A4: A,F: B3 > A,B4: B3,C2: B3] :
% 0.32/0.46 ( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
% 0.32/0.46 => ( ( ord_less_eq @ B3 @ B4 @ C2 )
% 0.32/0.46 => ( ! [X4: B3,Y2: B3] :
% 0.32/0.46 ( ( ord_less_eq @ B3 @ X4 @ Y2 )
% 0.32/0.46 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.32/0.46 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst1
% 0.32/0.46 thf(fact_37_order__subst2,axiom,
% 0.32/0.46 ! [A: $tType,C: $tType] :
% 0.32/0.46 ( ( ( order @ C )
% 0.32/0.46 & ( order @ A ) )
% 0.32/0.46 => ! [A4: A,B4: A,F: A > C,C2: C] :
% 0.32/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.32/0.46 => ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
% 0.32/0.46 => ( ! [X4: A,Y2: A] :
% 0.32/0.46 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.32/0.46 => ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.32/0.46 => ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst2
% 0.32/0.46 thf(fact_38_ord__eq__le__subst,axiom,
% 0.32/0.46 ! [A: $tType,B3: $tType] :
% 0.32/0.46 ( ( ( ord @ B3 )
% 0.32/0.46 & ( ord @ A ) )
% 0.32/0.46 => ! [A4: A,F: B3 > A,B4: B3,C2: B3] :
% 0.32/0.46 ( ( A4
% 0.32/0.46 = ( F @ B4 ) )
% 0.32/0.46 => ( ( ord_less_eq @ B3 @ B4 @ C2 )
% 0.32/0.46 => ( ! [X4: B3,Y2: B3] :
% 0.32/0.46 ( ( ord_less_eq @ B3 @ X4 @ Y2 )
% 0.32/0.46 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.32/0.46 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % ord_eq_le_subst
% 0.32/0.46 thf(fact_39_ord__le__eq__subst,axiom,
% 0.32/0.46 ! [A: $tType,B3: $tType] :
% 0.32/0.46 ( ( ( ord @ B3 )
% 0.32/0.46 & ( ord @ A ) )
% 0.32/0.46 => ! [A4: A,B4: A,F: A > B3,C2: B3] :
% 0.32/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.32/0.46 => ( ( ( F @ B4 )
% 0.32/0.46 = C2 )
% 0.32/0.46 => ( ! [X4: A,Y2: A] :
% 0.32/0.46 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.32/0.46 => ( ord_less_eq @ B3 @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.32/0.46 => ( ord_less_eq @ B3 @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % ord_le_eq_subst
% 0.32/0.46 thf(fact_40_Union__mono,axiom,
% 0.32/0.46 ! [A: $tType,A2: set @ ( set @ A ),B: set @ ( set @ A )] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ B )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Union_mono
% 0.32/0.47 thf(fact_41_eq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.32/0.47 = ( ^ [X2: A,Y3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.32/0.47 & ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_iff
% 0.32/0.47 thf(fact_42_Union__least,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ ( set @ A ),C3: set @ A] :
% 0.32/0.47 ( ! [X5: set @ A] :
% 0.32/0.47 ( ( member @ ( set @ A ) @ X5 @ A2 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ X5 @ C3 ) )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ C3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % Union_least
% 0.32/0.47 thf(fact_43_Union__upper,axiom,
% 0.32/0.47 ! [A: $tType,B: set @ A,A2: set @ ( set @ A )] :
% 0.32/0.47 ( ( member @ ( set @ A ) @ B @ A2 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ B @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Union_upper
% 0.32/0.47 thf(fact_44_antisym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X: A,Y4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Y4 @ X )
% 0.32/0.47 => ( X = Y4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym
% 0.32/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.32/0.47 ! [A: $tType,A4: A,P: A > $o] :
% 0.32/0.47 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 0.32/0.47 = ( P @ A4 ) ) ).
% 0.32/0.47
% 0.32/0.47 % mem_Collect_eq
% 0.32/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A] :
% 0.32/0.47 ( ( collect @ A
% 0.32/0.47 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 0.32/0.47 = A2 ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mem_eq
% 0.32/0.47 thf(fact_47_Collect__cong,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/0.47 ( ! [X4: A] :
% 0.32/0.47 ( ( P @ X4 )
% 0.32/0.47 = ( Q @ X4 ) )
% 0.32/0.47 => ( ( collect @ A @ P )
% 0.32/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_cong
% 0.32/0.47 thf(fact_48_ext,axiom,
% 0.32/0.47 ! [B3: $tType,A: $tType,F: A > B3,G: A > B3] :
% 0.32/0.47 ( ! [X4: A] :
% 0.32/0.47 ( ( F @ X4 )
% 0.32/0.47 = ( G @ X4 ) )
% 0.32/0.47 => ( F = G ) ) ).
% 0.32/0.47
% 0.32/0.47 % ext
% 0.32/0.47 thf(fact_49_Union__subsetI,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ ( set @ A ),B: set @ ( set @ A )] :
% 0.32/0.47 ( ! [X4: set @ A] :
% 0.32/0.47 ( ( member @ ( set @ A ) @ X4 @ A2 )
% 0.32/0.47 => ? [Y5: set @ A] :
% 0.32/0.47 ( ( member @ ( set @ A ) @ Y5 @ B )
% 0.32/0.47 & ( ord_less_eq @ ( set @ A ) @ X4 @ Y5 ) ) )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Union_subsetI
% 0.32/0.47 thf(fact_50_linear,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.32/0.47 | ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linear
% 0.32/0.47 thf(fact_51_eq__refl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y4: A] :
% 0.32/0.47 ( ( X = Y4 )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_refl
% 0.32/0.47 thf(fact_52_le__cases,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y4: A] :
% 0.32/0.47 ( ~ ( ord_less_eq @ A @ X @ Y4 )
% 0.32/0.47 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases
% 0.32/0.47 thf(fact_53_order_Otrans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A4: A,B4: A,C2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 0.32/0.47 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.trans
% 0.32/0.47 thf(fact_54_le__cases3,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y4: A,Z2: A] :
% 0.32/0.47 ( ( ( ord_less_eq @ A @ X @ Y4 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Y4 @ Z2 ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ Y4 @ X )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Z2 @ Y4 ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ Z2 @ Y4 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Y4 @ X ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 0.32/0.47 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases3
% 0.32/0.47 thf(fact_55_antisym__conv,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [Y4: A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ Y4 @ X )
% 0.32/0.47 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.32/0.47 = ( X = Y4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv
% 0.32/0.47 thf(fact_56_order__class_Oorder_Oeq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.32/0.47 = ( ^ [A5: A,B5: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.32/0.47 & ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.eq_iff
% 0.32/0.47 thf(fact_57_ord__eq__le__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A )
% 0.32/0.47 => ! [A4: A,B4: A,C2: A] :
% 0.32/0.47 ( ( A4 = B4 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 0.32/0.47 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_trans
% 0.32/0.47 thf(fact_58_ord__le__eq__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A )
% 0.32/0.47 => ! [A4: A,B4: A,C2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.32/0.47 => ( ( B4 = C2 )
% 0.32/0.47 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_trans
% 0.32/0.47 thf(fact_59_order__class_Oorder_Oantisym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.32/0.47 => ( A4 = B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.antisym
% 0.32/0.47 thf(fact_60_order__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y4: A,Z2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_trans
% 0.32/0.47 thf(fact_61_dual__order_Orefl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.refl
% 0.32/0.47 thf(fact_62_linorder__wlog,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [P: A > A > $o,A4: A,B4: A] :
% 0.32/0.47 ( ! [A6: A,B6: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A6 @ B6 )
% 0.32/0.47 => ( P @ A6 @ B6 ) )
% 0.32/0.47 => ( ! [A6: A,B6: A] :
% 0.32/0.47 ( ( P @ B6 @ A6 )
% 0.32/0.47 => ( P @ A6 @ B6 ) )
% 0.32/0.47 => ( P @ A4 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_wlog
% 0.32/0.47 thf(fact_63_dual__order_Otrans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B4: A,A4: A,C2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ C2 @ B4 )
% 0.32/0.47 => ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.trans
% 0.32/0.47 thf(fact_64_dual__order_Oeq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.32/0.47 = ( ^ [A5: A,B5: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B5 @ A5 )
% 0.32/0.47 & ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.eq_iff
% 0.32/0.47 thf(fact_65_dual__order_Oantisym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B4: A,A4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.32/0.47 => ( A4 = B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.antisym
% 0.32/0.47 thf(fact_66_Sup__eqI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comple187826305attice @ A )
% 0.32/0.47 => ! [A2: set @ A,X: A] :
% 0.32/0.47 ( ! [Y2: A] :
% 0.32/0.47 ( ( member @ A @ Y2 @ A2 )
% 0.32/0.47 => ( ord_less_eq @ A @ Y2 @ X ) )
% 0.32/0.47 => ( ! [Y2: A] :
% 0.32/0.47 ( ! [Z3: A] :
% 0.32/0.47 ( ( member @ A @ Z3 @ A2 )
% 0.32/0.47 => ( ord_less_eq @ A @ Z3 @ Y2 ) )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ Y2 ) )
% 0.32/0.47 => ( ( complete_Sup_Sup @ A @ A2 )
% 0.32/0.47 = X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Sup_eqI
% 0.32/0.47 thf(fact_67_Sup__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comple187826305attice @ A )
% 0.32/0.47 => ! [A2: set @ A,B: set @ A] :
% 0.32/0.47 ( ! [A6: A] :
% 0.32/0.47 ( ( member @ A @ A6 @ A2 )
% 0.32/0.47 => ? [X3: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ B )
% 0.32/0.47 & ( ord_less_eq @ A @ A6 @ X3 ) ) )
% 0.32/0.47 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Sup_mono
% 0.32/0.47 thf(fact_68_Sup__least,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comple187826305attice @ A )
% 0.32/0.47 => ! [A2: set @ A,Z2: A] :
% 0.32/0.47 ( ! [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ A2 )
% 0.32/0.47 => ( ord_less_eq @ A @ X4 @ Z2 ) )
% 0.32/0.47 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ Z2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Sup_least
% 0.32/0.47 thf(fact_69_Sup__upper,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comple187826305attice @ A )
% 0.32/0.47 => ! [X: A,A2: set @ A] :
% 0.32/0.47 ( ( member @ A @ X @ A2 )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Sup_upper
% 0.32/0.47 thf(fact_70_Sup__le__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comple187826305attice @ A )
% 0.32/0.47 => ! [A2: set @ A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ B4 )
% 0.32/0.47 = ( ! [X2: A] :
% 0.32/0.47 ( ( member @ A @ X2 @ A2 )
% 0.32/0.47 => ( ord_less_eq @ A @ X2 @ B4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Sup_le_iff
% 0.32/0.47 thf(fact_71_Sup__upper2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comple187826305attice @ A )
% 0.32/0.47 => ! [U: A,A2: set @ A,V: A] :
% 0.32/0.47 ( ( member @ A @ U @ A2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ V @ U )
% 0.32/0.47 => ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Sup_upper2
% 0.32/0.47 thf(fact_72_finite__on__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( real_V1076094709vector @ A )
% 0.32/0.47 => ( ( lower_728871928ite_on @ A )
% 0.32/0.47 = ( ^ [S: set @ A,F2: A > extended_ereal] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( member @ A @ X2 @ S )
% 0.32/0.47 => ( ( ( F2 @ X2 )
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 & ( ( F2 @ X2 )
% 0.32/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_on_def
% 0.32/0.47 thf(fact_73_Sup__subset__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comple187826305attice @ A )
% 0.32/0.47 => ! [A2: set @ A,B: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
% 0.32/0.47 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Sup_subset_mono
% 0.32/0.47 thf(fact_74_top__set__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( top_top @ ( set @ A ) )
% 0.32/0.47 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % top_set_def
% 0.32/0.47 thf(fact_75_Union__UNIV,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( complete_Sup_Sup @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
% 0.32/0.47 = ( top_top @ ( set @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Union_UNIV
% 0.32/0.47 thf(fact_76_top_Oextremum__uniqueI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order_top @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
% 0.32/0.47 => ( A4
% 0.32/0.47 = ( top_top @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % top.extremum_uniqueI
% 0.32/0.47 thf(fact_77_top_Oextremum__unique,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order_top @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
% 0.32/0.47 = ( A4
% 0.32/0.47 = ( top_top @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % top.extremum_unique
% 0.32/0.47 thf(fact_78_top__greatest,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order_top @ A )
% 0.32/0.47 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( top_top @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % top_greatest
% 0.32/0.47 thf(fact_79_subset__UNIV,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_UNIV
% 0.32/0.47 thf(fact_80_epigraph__invertible,axiom,
% 0.32/0.47 ! [A: $tType,F: A > extended_ereal,G: A > extended_ereal] :
% 0.32/0.47 ( ( ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ F )
% 0.32/0.47 = ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ G ) )
% 0.32/0.47 => ( F = G ) ) ).
% 0.32/0.47
% 0.32/0.47 % epigraph_invertible
% 0.32/0.47 thf(fact_81_neg__le__iff__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [B4: A,A4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A4 ) )
% 0.32/0.47 = ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % neg_le_iff_le
% 0.32/0.47 thf(fact_82_compl__le__compl__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( boolean_algebra @ A )
% 0.32/0.47 => ! [X: A,Y4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y4 ) )
% 0.32/0.47 = ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % compl_le_compl_iff
% 0.32/0.47 thf(fact_83_top__finite__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( finite_lattice @ A )
% 0.32/0.47 => ( ( top_top @ A )
% 0.32/0.47 = ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % top_finite_def
% 0.32/0.47 thf(fact_84_ereal__uminus__eq__iff,axiom,
% 0.32/0.47 ! [A4: extended_ereal,B4: extended_ereal] :
% 0.32/0.47 ( ( ( uminus_uminus @ extended_ereal @ A4 )
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ B4 ) )
% 0.32/0.47 = ( A4 = B4 ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_uminus_eq_iff
% 0.32/0.47 thf(fact_85_ereal__uminus__uminus,axiom,
% 0.32/0.47 ! [A4: extended_ereal] :
% 0.32/0.47 ( ( uminus_uminus @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A4 ) )
% 0.32/0.47 = A4 ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_uminus_uminus
% 0.32/0.47 thf(fact_86_verit__minus__simplify_I4_J,axiom,
% 0.32/0.47 ! [B3: $tType] :
% 0.32/0.47 ( ( group_add @ B3 )
% 0.32/0.47 => ! [B4: B3] :
% 0.32/0.47 ( ( uminus_uminus @ B3 @ ( uminus_uminus @ B3 @ B4 ) )
% 0.32/0.47 = B4 ) ) ).
% 0.32/0.47
% 0.32/0.47 % verit_minus_simplify(4)
% 0.32/0.47 thf(fact_87_uminus__apply,axiom,
% 0.32/0.47 ! [B3: $tType,A: $tType] :
% 0.32/0.47 ( ( uminus @ B3 )
% 0.32/0.47 => ( ( uminus_uminus @ ( A > B3 ) )
% 0.32/0.47 = ( ^ [A3: A > B3,X2: A] : ( uminus_uminus @ B3 @ ( A3 @ X2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % uminus_apply
% 0.32/0.47 thf(fact_88_add_Oinverse__inverse,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( group_add @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A4 ) )
% 0.32/0.47 = A4 ) ) ).
% 0.32/0.47
% 0.32/0.47 % add.inverse_inverse
% 0.32/0.47 thf(fact_89_neg__equal__iff__equal,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( group_add @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( ( uminus_uminus @ A @ A4 )
% 0.32/0.47 = ( uminus_uminus @ A @ B4 ) )
% 0.32/0.47 = ( A4 = B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % neg_equal_iff_equal
% 0.32/0.47 thf(fact_90_double__compl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( boolean_algebra @ A )
% 0.32/0.47 => ! [X: A] :
% 0.32/0.47 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
% 0.32/0.47 = X ) ) ).
% 0.32/0.47
% 0.32/0.47 % double_compl
% 0.32/0.47 thf(fact_91_compl__eq__compl__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( boolean_algebra @ A )
% 0.32/0.47 => ! [X: A,Y4: A] :
% 0.32/0.47 ( ( ( uminus_uminus @ A @ X )
% 0.32/0.47 = ( uminus_uminus @ A @ Y4 ) )
% 0.32/0.47 = ( X = Y4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % compl_eq_compl_iff
% 0.32/0.47 thf(fact_92_Compl__subset__Compl__iff,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B ) )
% 0.32/0.47 = ( ord_less_eq @ ( set @ A ) @ B @ A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % Compl_subset_Compl_iff
% 0.32/0.47 thf(fact_93_Compl__anti__mono,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Compl_anti_mono
% 0.32/0.47 thf(fact_94_ereal__infty__less__eq_I1_J,axiom,
% 0.32/0.47 ! [X: extended_ereal] :
% 0.32/0.47 ( ( ord_less_eq @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ X )
% 0.32/0.47 = ( X
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_infty_less_eq(1)
% 0.32/0.47 thf(fact_95_UN__ball__bex__simps_I3_J,axiom,
% 0.32/0.47 ! [D: $tType,A2: set @ ( set @ D ),P: D > $o] :
% 0.32/0.47 ( ( ? [X2: D] :
% 0.32/0.47 ( ( member @ D @ X2 @ ( complete_Sup_Sup @ ( set @ D ) @ A2 ) )
% 0.32/0.47 & ( P @ X2 ) ) )
% 0.32/0.47 = ( ? [X2: set @ D] :
% 0.32/0.47 ( ( member @ ( set @ D ) @ X2 @ A2 )
% 0.32/0.47 & ? [Y3: D] :
% 0.32/0.47 ( ( member @ D @ Y3 @ X2 )
% 0.32/0.47 & ( P @ Y3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % UN_ball_bex_simps(3)
% 0.32/0.47 thf(fact_96_UN__ball__bex__simps_I1_J,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ ( set @ A ),P: A > $o] :
% 0.32/0.47 ( ( ! [X2: A] :
% 0.32/0.47 ( ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
% 0.32/0.47 => ( P @ X2 ) ) )
% 0.32/0.47 = ( ! [X2: set @ A] :
% 0.32/0.47 ( ( member @ ( set @ A ) @ X2 @ A2 )
% 0.32/0.47 => ! [Y3: A] :
% 0.32/0.47 ( ( member @ A @ Y3 @ X2 )
% 0.32/0.47 => ( P @ Y3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % UN_ball_bex_simps(1)
% 0.32/0.47 thf(fact_97_UnionI,axiom,
% 0.32/0.47 ! [A: $tType,X6: set @ A,C3: set @ ( set @ A ),A2: A] :
% 0.32/0.47 ( ( member @ ( set @ A ) @ X6 @ C3 )
% 0.32/0.47 => ( ( member @ A @ A2 @ X6 )
% 0.32/0.47 => ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % UnionI
% 0.32/0.47 thf(fact_98_Union__iff,axiom,
% 0.32/0.47 ! [A: $tType,A2: A,C3: set @ ( set @ A )] :
% 0.32/0.47 ( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
% 0.32/0.47 = ( ? [X2: set @ A] :
% 0.32/0.47 ( ( member @ ( set @ A ) @ X2 @ C3 )
% 0.32/0.47 & ( member @ A @ A2 @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Union_iff
% 0.32/0.47 thf(fact_99_ereal__minus__le__minus,axiom,
% 0.32/0.47 ! [A4: extended_ereal,B4: extended_ereal] :
% 0.32/0.47 ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A4 ) @ ( uminus_uminus @ extended_ereal @ B4 ) )
% 0.32/0.47 = ( ord_less_eq @ extended_ereal @ B4 @ A4 ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_minus_le_minus
% 0.32/0.47 thf(fact_100_top1I,axiom,
% 0.32/0.47 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 0.32/0.47
% 0.32/0.47 % top1I
% 0.32/0.47 thf(fact_101_ereal__infty__less__eq_I2_J,axiom,
% 0.32/0.47 ! [X: extended_ereal] :
% 0.32/0.47 ( ( ord_less_eq @ extended_ereal @ X @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 = ( X
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_infty_less_eq(2)
% 0.32/0.47 thf(fact_102_UnionE,axiom,
% 0.32/0.47 ! [A: $tType,A2: A,C3: set @ ( set @ A )] :
% 0.32/0.47 ( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
% 0.32/0.47 => ~ ! [X5: set @ A] :
% 0.32/0.47 ( ( member @ A @ A2 @ X5 )
% 0.32/0.47 => ~ ( member @ ( set @ A ) @ X5 @ C3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % UnionE
% 0.32/0.47 thf(fact_103_neq__PInf__trans,axiom,
% 0.32/0.47 ! [Y4: extended_ereal,X: extended_ereal] :
% 0.32/0.47 ( ( Y4
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( ( ord_less_eq @ extended_ereal @ X @ Y4 )
% 0.32/0.47 => ( X
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % neq_PInf_trans
% 0.32/0.47 thf(fact_104_ereal__infty__less__eq2_I1_J,axiom,
% 0.32/0.47 ! [A4: extended_ereal,B4: extended_ereal] :
% 0.32/0.47 ( ( ord_less_eq @ extended_ereal @ A4 @ B4 )
% 0.32/0.47 => ( ( A4
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( B4
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_infty_less_eq2(1)
% 0.32/0.47 thf(fact_105_ereal__less__eq_I1_J,axiom,
% 0.32/0.47 ! [X: extended_ereal] : ( ord_less_eq @ extended_ereal @ X @ ( extend1396239628finity @ extended_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_less_eq(1)
% 0.32/0.47 thf(fact_106_ereal__uminus__le__reorder,axiom,
% 0.32/0.47 ! [A4: extended_ereal,B4: extended_ereal] :
% 0.32/0.47 ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A4 ) @ B4 )
% 0.32/0.47 = ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ B4 ) @ A4 ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_uminus_le_reorder
% 0.32/0.47 thf(fact_107_ereal__infty__less__eq2_I2_J,axiom,
% 0.32/0.47 ! [A4: extended_ereal,B4: extended_ereal] :
% 0.32/0.47 ( ( ord_less_eq @ extended_ereal @ A4 @ B4 )
% 0.32/0.47 => ( ( B4
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( A4
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_infty_less_eq2(2)
% 0.32/0.47 thf(fact_108_ereal__less__eq_I2_J,axiom,
% 0.32/0.47 ! [X: extended_ereal] : ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ X ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_less_eq(2)
% 0.32/0.47 thf(fact_109_epigraph__subset__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( real_V2090557954_space @ A )
% 0.32/0.47 => ! [F: A > extended_ereal,G: A > extended_ereal] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ real ) ) @ ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ F ) @ ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ G ) )
% 0.32/0.47 = ( ! [X2: A] : ( ord_less_eq @ extended_ereal @ ( G @ X2 ) @ ( F @ X2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % epigraph_subset_iff
% 0.32/0.47 thf(fact_110_verit__la__disequality,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( A4 = B4 )
% 0.32/0.47 | ~ ( ord_less_eq @ A @ A4 @ B4 )
% 0.32/0.47 | ~ ( ord_less_eq @ A @ B4 @ A4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % verit_la_disequality
% 0.32/0.47 thf(fact_111_minus__equation__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( group_add @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( ( uminus_uminus @ A @ A4 )
% 0.32/0.47 = B4 )
% 0.32/0.47 = ( ( uminus_uminus @ A @ B4 )
% 0.32/0.47 = A4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minus_equation_iff
% 0.32/0.47 thf(fact_112_equation__minus__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( group_add @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( A4
% 0.32/0.47 = ( uminus_uminus @ A @ B4 ) )
% 0.32/0.47 = ( B4
% 0.32/0.47 = ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % equation_minus_iff
% 0.32/0.47 thf(fact_113_fun__Compl__def,axiom,
% 0.32/0.47 ! [B3: $tType,A: $tType] :
% 0.32/0.47 ( ( uminus @ B3 )
% 0.32/0.47 => ( ( uminus_uminus @ ( A > B3 ) )
% 0.32/0.47 = ( ^ [A3: A > B3,X2: A] : ( uminus_uminus @ B3 @ ( A3 @ X2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % fun_Compl_def
% 0.32/0.47 thf(fact_114_verit__negate__coefficient_I3_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( A4 = B4 )
% 0.32/0.47 => ( ( uminus_uminus @ A @ A4 )
% 0.32/0.47 = ( uminus_uminus @ A @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % verit_negate_coefficient(3)
% 0.32/0.47 thf(fact_115_top__ereal__def,axiom,
% 0.32/0.47 ( ( top_top @ extended_ereal )
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % top_ereal_def
% 0.32/0.47 thf(fact_116_Sup__eq__PInfty,axiom,
% 0.32/0.47 ! [S2: set @ extended_ereal] :
% 0.32/0.47 ( ( member @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ S2 )
% 0.32/0.47 => ( ( complete_Sup_Sup @ extended_ereal @ S2 )
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Sup_eq_PInfty
% 0.32/0.47 thf(fact_117_ereal__uminus__eq__reorder,axiom,
% 0.32/0.47 ! [A4: extended_ereal,B4: extended_ereal] :
% 0.32/0.47 ( ( ( uminus_uminus @ extended_ereal @ A4 )
% 0.32/0.47 = B4 )
% 0.32/0.47 = ( A4
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_uminus_eq_reorder
% 0.32/0.47 thf(fact_118_le__imp__neg__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_imp_neg_le
% 0.32/0.47 thf(fact_119_minus__le__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ B4 )
% 0.32/0.47 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ A4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minus_le_iff
% 0.32/0.47 thf(fact_120_le__minus__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ B4 ) )
% 0.32/0.47 = ( ord_less_eq @ A @ B4 @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_minus_iff
% 0.32/0.47 thf(fact_121_compl__le__swap2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( boolean_algebra @ A )
% 0.32/0.47 => ! [Y4: A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y4 ) @ X )
% 0.32/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % compl_le_swap2
% 0.32/0.47 thf(fact_122_compl__le__swap1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( boolean_algebra @ A )
% 0.32/0.47 => ! [Y4: A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ Y4 @ ( uminus_uminus @ A @ X ) )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % compl_le_swap1
% 0.32/0.47 thf(fact_123_compl__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( boolean_algebra @ A )
% 0.32/0.47 => ! [X: A,Y4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y4 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % compl_mono
% 0.32/0.47 thf(fact_124_MInfty__neq__PInfty_I1_J,axiom,
% 0.32/0.47 ( ( extend1396239628finity @ extended_ereal )
% 0.32/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % MInfty_neq_PInfty(1)
% 0.32/0.47 thf(fact_125_top__empty__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( top_top @ ( A > $o ) )
% 0.32/0.47 = ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % top_empty_eq
% 0.32/0.47 thf(fact_126_cSup__eq__maximum,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( condit378418413attice @ A )
% 0.32/0.47 => ! [Z2: A,X6: set @ A] :
% 0.32/0.47 ( ( member @ A @ Z2 @ X6 )
% 0.32/0.47 => ( ! [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ X6 )
% 0.32/0.47 => ( ord_less_eq @ A @ X4 @ Z2 ) )
% 0.32/0.47 => ( ( complete_Sup_Sup @ A @ X6 )
% 0.32/0.47 = Z2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % cSup_eq_maximum
% 0.32/0.47 thf(fact_127_cSup__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ( condit378418413attice @ A )
% 0.32/0.47 & ( no_bot @ A ) )
% 0.32/0.47 => ! [X6: set @ A,A4: A] :
% 0.32/0.47 ( ! [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ X6 )
% 0.32/0.47 => ( ord_less_eq @ A @ X4 @ A4 ) )
% 0.32/0.47 => ( ! [Y2: A] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ X6 )
% 0.32/0.47 => ( ord_less_eq @ A @ X3 @ Y2 ) )
% 0.32/0.47 => ( ord_less_eq @ A @ A4 @ Y2 ) )
% 0.32/0.47 => ( ( complete_Sup_Sup @ A @ X6 )
% 0.32/0.47 = A4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % cSup_eq
% 0.32/0.47 thf(fact_128_top__conj_I2_J,axiom,
% 0.32/0.47 ! [A: $tType,P: $o,X: A] :
% 0.32/0.47 ( ( P
% 0.32/0.47 & ( top_top @ ( A > $o ) @ X ) )
% 0.32/0.47 = P ) ).
% 0.32/0.47
% 0.32/0.47 % top_conj(2)
% 0.32/0.47 thf(fact_129_top__conj_I1_J,axiom,
% 0.32/0.47 ! [A: $tType,X: A,P: $o] :
% 0.32/0.47 ( ( ( top_top @ ( A > $o ) @ X )
% 0.32/0.47 & P )
% 0.32/0.47 = P ) ).
% 0.32/0.47
% 0.32/0.47 % top_conj(1)
% 0.32/0.47 thf(fact_130_MInfty__eq__minfinity,axiom,
% 0.32/0.47 ( extended_MInfty
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % MInfty_eq_minfinity
% 0.32/0.47 thf(fact_131_ComplI,axiom,
% 0.32/0.47 ! [A: $tType,C2: A,A2: set @ A] :
% 0.32/0.47 ( ~ ( member @ A @ C2 @ A2 )
% 0.32/0.47 => ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ComplI
% 0.32/0.47 thf(fact_132_Compl__iff,axiom,
% 0.32/0.47 ! [A: $tType,C2: A,A2: set @ A] :
% 0.32/0.47 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.32/0.47 = ( ~ ( member @ A @ C2 @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Compl_iff
% 0.32/0.47 thf(fact_133_Compl__eq__Compl__iff,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B: set @ A] :
% 0.32/0.47 ( ( ( uminus_uminus @ ( set @ A ) @ A2 )
% 0.32/0.47 = ( uminus_uminus @ ( set @ A ) @ B ) )
% 0.32/0.47 = ( A2 = B ) ) ).
% 0.32/0.47
% 0.32/0.47 % Compl_eq_Compl_iff
% 0.32/0.47 thf(fact_134_ComplD,axiom,
% 0.32/0.47 ! [A: $tType,C2: A,A2: set @ A] :
% 0.32/0.47 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.32/0.47 => ~ ( member @ A @ C2 @ A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % ComplD
% 0.32/0.47 thf(fact_135_double__complement,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A] :
% 0.32/0.47 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.32/0.47 = A2 ) ).
% 0.32/0.47
% 0.32/0.47 % double_complement
% 0.32/0.47 thf(fact_136_ereal__complete__Inf,axiom,
% 0.32/0.47 ! [S2: set @ extended_ereal] :
% 0.32/0.47 ? [X4: extended_ereal] :
% 0.32/0.47 ( ! [Xa: extended_ereal] :
% 0.32/0.47 ( ( member @ extended_ereal @ Xa @ S2 )
% 0.32/0.47 => ( ord_less_eq @ extended_ereal @ X4 @ Xa ) )
% 0.32/0.47 & ! [Z3: extended_ereal] :
% 0.32/0.47 ( ! [Xa2: extended_ereal] :
% 0.32/0.47 ( ( member @ extended_ereal @ Xa2 @ S2 )
% 0.32/0.47 => ( ord_less_eq @ extended_ereal @ Z3 @ Xa2 ) )
% 0.32/0.47 => ( ord_less_eq @ extended_ereal @ Z3 @ X4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_complete_Inf
% 0.32/0.47 thf(fact_137_ereal__complete__Sup,axiom,
% 0.32/0.47 ! [S2: set @ extended_ereal] :
% 0.32/0.47 ? [X4: extended_ereal] :
% 0.32/0.47 ( ! [Xa: extended_ereal] :
% 0.32/0.47 ( ( member @ extended_ereal @ Xa @ S2 )
% 0.32/0.47 => ( ord_less_eq @ extended_ereal @ Xa @ X4 ) )
% 0.32/0.47 & ! [Z3: extended_ereal] :
% 0.32/0.47 ( ! [Xa2: extended_ereal] :
% 0.32/0.47 ( ( member @ extended_ereal @ Xa2 @ S2 )
% 0.32/0.47 => ( ord_less_eq @ extended_ereal @ Xa2 @ Z3 ) )
% 0.32/0.47 => ( ord_less_eq @ extended_ereal @ X4 @ Z3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_complete_Sup
% 0.32/0.47 thf(fact_138_uminus__ereal_Osimps_I3_J,axiom,
% 0.32/0.47 ( ( uminus_uminus @ extended_ereal @ extended_MInfty )
% 0.32/0.47 = extended_PInfty ) ).
% 0.32/0.47
% 0.32/0.47 % uminus_ereal.simps(3)
% 0.32/0.47 thf(fact_139_uminus__ereal_Osimps_I2_J,axiom,
% 0.32/0.47 ( ( uminus_uminus @ extended_ereal @ extended_PInfty )
% 0.32/0.47 = extended_MInfty ) ).
% 0.32/0.47
% 0.32/0.47 % uminus_ereal.simps(2)
% 0.32/0.47 thf(fact_140_exists__diff,axiom,
% 0.32/0.47 ! [A: $tType,P: ( set @ A ) > $o] :
% 0.32/0.47 ( ( ? [S: set @ A] : ( P @ ( uminus_uminus @ ( set @ A ) @ S ) ) )
% 0.32/0.47 = ( ? [X7: set @ A] : ( P @ X7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % exists_diff
% 0.32/0.47 thf(fact_141_GreatestI2__order,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [P: A > $o,X: A,Q: A > $o] :
% 0.32/0.47 ( ( P @ X )
% 0.32/0.47 => ( ! [Y2: A] :
% 0.32/0.47 ( ( P @ Y2 )
% 0.32/0.47 => ( ord_less_eq @ A @ Y2 @ X ) )
% 0.32/0.47 => ( ! [X4: A] :
% 0.32/0.47 ( ( P @ X4 )
% 0.32/0.47 => ( ! [Y5: A] :
% 0.32/0.47 ( ( P @ Y5 )
% 0.32/0.47 => ( ord_less_eq @ A @ Y5 @ X4 ) )
% 0.32/0.47 => ( Q @ X4 ) ) )
% 0.32/0.47 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % GreatestI2_order
% 0.32/0.47 thf(fact_142_infinity__ereal__def,axiom,
% 0.32/0.47 ( ( extend1396239628finity @ extended_ereal )
% 0.32/0.47 = extended_PInfty ) ).
% 0.32/0.47
% 0.32/0.47 % infinity_ereal_def
% 0.32/0.47 thf(fact_143_ereal_Odistinct_I5_J,axiom,
% 0.32/0.47 extended_PInfty != extended_MInfty ).
% 0.32/0.47
% 0.32/0.47 % ereal.distinct(5)
% 0.32/0.47 thf(fact_144_Greatest__equality,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [P: A > $o,X: A] :
% 0.32/0.47 ( ( P @ X )
% 0.32/0.47 => ( ! [Y2: A] :
% 0.32/0.47 ( ( P @ Y2 )
% 0.32/0.47 => ( ord_less_eq @ A @ Y2 @ X ) )
% 0.32/0.47 => ( ( order_Greatest @ A @ P )
% 0.32/0.47 = X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Greatest_equality
% 0.32/0.47 thf(fact_145_uminus__ereal_Oelims,axiom,
% 0.32/0.47 ! [X: extended_ereal,Y4: extended_ereal] :
% 0.32/0.47 ( ( ( uminus_uminus @ extended_ereal @ X )
% 0.32/0.47 = Y4 )
% 0.32/0.47 => ( ! [R: real] :
% 0.32/0.47 ( ( X
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( Y4
% 0.32/0.47 != ( extended_ereal2 @ ( uminus_uminus @ real @ R ) ) ) )
% 0.32/0.47 => ( ( ( X = extended_PInfty )
% 0.32/0.47 => ( Y4 != extended_MInfty ) )
% 0.32/0.47 => ~ ( ( X = extended_MInfty )
% 0.32/0.47 => ( Y4 != extended_PInfty ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % uminus_ereal.elims
% 0.32/0.47 thf(fact_146_not__MInfty__nonneg,axiom,
% 0.32/0.47 ! [X: extended_ereal] :
% 0.32/0.47 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ X )
% 0.32/0.47 => ( X
% 0.32/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_MInfty_nonneg
% 0.32/0.47 thf(fact_147_ereal_Osimps_I12_J,axiom,
% 0.32/0.47 ! [A: $tType,F1: real > A,F22: A,F3: A] :
% 0.32/0.47 ( ( extended_rec_ereal @ A @ F1 @ F22 @ F3 @ extended_PInfty )
% 0.32/0.47 = F22 ) ).
% 0.32/0.47
% 0.32/0.47 % ereal.simps(12)
% 0.32/0.47 thf(fact_148_ereal_Osimps_I13_J,axiom,
% 0.32/0.47 ! [A: $tType,F1: real > A,F22: A,F3: A] :
% 0.32/0.47 ( ( extended_rec_ereal @ A @ F1 @ F22 @ F3 @ extended_MInfty )
% 0.32/0.47 = F3 ) ).
% 0.32/0.47
% 0.32/0.47 % ereal.simps(13)
% 0.32/0.47 thf(fact_149_ereal_Oinject,axiom,
% 0.32/0.47 ! [X1: real,Y1: real] :
% 0.32/0.47 ( ( ( extended_ereal2 @ X1 )
% 0.32/0.47 = ( extended_ereal2 @ Y1 ) )
% 0.32/0.47 = ( X1 = Y1 ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal.inject
% 0.32/0.47 thf(fact_150_ereal__cong,axiom,
% 0.32/0.47 ! [X: real,Y4: real] :
% 0.32/0.47 ( ( X = Y4 )
% 0.32/0.47 => ( ( extended_ereal2 @ X )
% 0.32/0.47 = ( extended_ereal2 @ Y4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_cong
% 0.32/0.47 thf(fact_151_le__zero__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( canoni770627133id_add @ A )
% 0.32/0.47 => ! [N: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.32/0.47 = ( N
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_zero_eq
% 0.32/0.47 thf(fact_152_neg__equal__zero,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord219039673up_add @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( ( uminus_uminus @ A @ A4 )
% 0.32/0.47 = A4 )
% 0.32/0.47 = ( A4
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % neg_equal_zero
% 0.32/0.47 thf(fact_153_equal__neg__zero,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord219039673up_add @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( A4
% 0.32/0.47 = ( uminus_uminus @ A @ A4 ) )
% 0.32/0.47 = ( A4
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % equal_neg_zero
% 0.32/0.47 thf(fact_154_neg__equal__0__iff__equal,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( group_add @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( ( uminus_uminus @ A @ A4 )
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 = ( A4
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % neg_equal_0_iff_equal
% 0.32/0.47 thf(fact_155_neg__0__equal__iff__equal,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( group_add @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( ( zero_zero @ A )
% 0.32/0.47 = ( uminus_uminus @ A @ A4 ) )
% 0.32/0.47 = ( ( zero_zero @ A )
% 0.32/0.47 = A4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % neg_0_equal_iff_equal
% 0.32/0.47 thf(fact_156_add_Oinverse__neutral,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( group_add @ A )
% 0.32/0.47 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add.inverse_neutral
% 0.32/0.47 thf(fact_157_ereal__less__eq_I3_J,axiom,
% 0.32/0.47 ! [R2: real,P2: real] :
% 0.32/0.47 ( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
% 0.32/0.47 = ( ord_less_eq @ real @ R2 @ P2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_less_eq(3)
% 0.32/0.47 thf(fact_158_ereal__uminus__zero,axiom,
% 0.32/0.47 ( ( uminus_uminus @ extended_ereal @ ( zero_zero @ extended_ereal ) )
% 0.32/0.47 = ( zero_zero @ extended_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_uminus_zero
% 0.32/0.47 thf(fact_159_ereal__uminus__zero__iff,axiom,
% 0.32/0.47 ! [A4: extended_ereal] :
% 0.32/0.47 ( ( ( uminus_uminus @ extended_ereal @ A4 )
% 0.32/0.47 = ( zero_zero @ extended_ereal ) )
% 0.32/0.47 = ( A4
% 0.32/0.47 = ( zero_zero @ extended_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_uminus_zero_iff
% 0.32/0.47 thf(fact_160_neg__0__le__iff__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A4 ) )
% 0.32/0.47 = ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % neg_0_le_iff_le
% 0.32/0.47 thf(fact_161_neg__le__0__iff__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ ( zero_zero @ A ) )
% 0.32/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % neg_le_0_iff_le
% 0.32/0.47 thf(fact_162_less__eq__neg__nonpos,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord219039673up_add @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ A4 ) )
% 0.32/0.47 = ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_eq_neg_nonpos
% 0.32/0.47 thf(fact_163_neg__less__eq__nonneg,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord219039673up_add @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ A4 )
% 0.32/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % neg_less_eq_nonneg
% 0.32/0.47 thf(fact_164_ereal__uminus__le__0__iff,axiom,
% 0.32/0.47 ! [A4: extended_ereal] :
% 0.32/0.47 ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A4 ) @ ( zero_zero @ extended_ereal ) )
% 0.32/0.47 = ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A4 ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_uminus_le_0_iff
% 0.32/0.47 thf(fact_165_ereal__0__le__uminus__iff,axiom,
% 0.32/0.47 ! [A4: extended_ereal] :
% 0.32/0.47 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( uminus_uminus @ extended_ereal @ A4 ) )
% 0.32/0.47 = ( ord_less_eq @ extended_ereal @ A4 @ ( zero_zero @ extended_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_0_le_uminus_iff
% 0.32/0.47 thf(fact_166_ereal_Odistinct_I1_J,axiom,
% 0.32/0.47 ! [X1: real] :
% 0.32/0.47 ( ( extended_ereal2 @ X1 )
% 0.32/0.47 != extended_PInfty ) ).
% 0.32/0.47
% 0.32/0.47 % ereal.distinct(1)
% 0.32/0.47 thf(fact_167_zero__reorient,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ! [X: A] :
% 0.32/0.47 ( ( ( zero_zero @ A )
% 0.32/0.47 = X )
% 0.32/0.47 = ( X
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_reorient
% 0.32/0.47 thf(fact_168_ereal_Osimps_I11_J,axiom,
% 0.32/0.47 ! [A: $tType,F1: real > A,F22: A,F3: A,X1: real] :
% 0.32/0.47 ( ( extended_rec_ereal @ A @ F1 @ F22 @ F3 @ ( extended_ereal2 @ X1 ) )
% 0.32/0.47 = ( F1 @ X1 ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal.simps(11)
% 0.32/0.47 thf(fact_169_Infty__neq__0_I1_J,axiom,
% 0.32/0.47 ( ( extend1396239628finity @ extended_ereal )
% 0.32/0.47 != ( zero_zero @ extended_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % Infty_neq_0(1)
% 0.32/0.47 thf(fact_170_zero__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( canoni770627133id_add @ A )
% 0.32/0.47 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_le
% 0.32/0.47 thf(fact_171_uminus__ereal_Osimps_I1_J,axiom,
% 0.32/0.47 ! [R2: real] :
% 0.32/0.47 ( ( uminus_uminus @ extended_ereal @ ( extended_ereal2 @ R2 ) )
% 0.32/0.47 = ( extended_ereal2 @ ( uminus_uminus @ real @ R2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % uminus_ereal.simps(1)
% 0.32/0.47 thf(fact_172_PInfty__neq__ereal_I1_J,axiom,
% 0.32/0.47 ! [R2: real] :
% 0.32/0.47 ( ( extended_ereal2 @ R2 )
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % PInfty_neq_ereal(1)
% 0.32/0.47 thf(fact_173_ereal_Odistinct_I3_J,axiom,
% 0.32/0.47 ! [X1: real] :
% 0.32/0.47 ( ( extended_ereal2 @ X1 )
% 0.32/0.47 != extended_MInfty ) ).
% 0.32/0.47
% 0.32/0.47 % ereal.distinct(3)
% 0.32/0.47 thf(fact_174_ereal__le__real,axiom,
% 0.32/0.47 ! [X: extended_ereal,Y4: extended_ereal] :
% 0.32/0.47 ( ! [Z4: real] :
% 0.32/0.47 ( ( ord_less_eq @ extended_ereal @ X @ ( extended_ereal2 @ Z4 ) )
% 0.32/0.47 => ( ord_less_eq @ extended_ereal @ Y4 @ ( extended_ereal2 @ Z4 ) ) )
% 0.32/0.47 => ( ord_less_eq @ extended_ereal @ Y4 @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_le_real
% 0.32/0.47 thf(fact_175_le__ereal__le,axiom,
% 0.32/0.47 ! [A4: extended_ereal,X: real,Y4: real] :
% 0.32/0.47 ( ( ord_less_eq @ extended_ereal @ A4 @ ( extended_ereal2 @ X ) )
% 0.32/0.47 => ( ( ord_less_eq @ real @ X @ Y4 )
% 0.32/0.47 => ( ord_less_eq @ extended_ereal @ A4 @ ( extended_ereal2 @ Y4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_ereal_le
% 0.32/0.47 thf(fact_176_ereal__le__le,axiom,
% 0.32/0.47 ! [Y4: real,A4: extended_ereal,X: real] :
% 0.32/0.47 ( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ Y4 ) @ A4 )
% 0.32/0.47 => ( ( ord_less_eq @ real @ X @ Y4 )
% 0.32/0.47 => ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ X ) @ A4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_le_le
% 0.32/0.47 thf(fact_177_MInfty__neq__ereal_I1_J,axiom,
% 0.32/0.47 ! [R2: real] :
% 0.32/0.47 ( ( extended_ereal2 @ R2 )
% 0.32/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % MInfty_neq_ereal(1)
% 0.32/0.47 thf(fact_178_ereal__cases,axiom,
% 0.32/0.47 ! [X: extended_ereal] :
% 0.32/0.47 ( ! [R: real] :
% 0.32/0.47 ( X
% 0.32/0.47 != ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ( X
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( X
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_cases
% 0.32/0.47 thf(fact_179_ereal2__cases,axiom,
% 0.32/0.47 ! [X: extended_ereal,Xa3: extended_ereal] :
% 0.32/0.47 ( ( ? [R: real] :
% 0.32/0.47 ( X
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ! [Ra: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 != ( extended_ereal2 @ Ra ) ) )
% 0.32/0.47 => ( ( ? [R: real] :
% 0.32/0.47 ( X
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( Xa3
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( ( ? [R: real] :
% 0.32/0.47 ( X
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( Xa3
% 0.32/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ! [R: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 != ( extended_ereal2 @ R ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( Xa3
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( Xa3
% 0.32/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ! [R: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 != ( extended_ereal2 @ R ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( Xa3
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ~ ( ( X
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( Xa3
% 0.32/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal2_cases
% 0.32/0.47 thf(fact_180_ereal3__cases,axiom,
% 0.32/0.47 ! [X: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
% 0.32/0.47 ( ( ? [R: real] :
% 0.32/0.47 ( X
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ? [Ra: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ Ra ) )
% 0.32/0.47 => ! [Rb: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ Rb ) ) ) )
% 0.32/0.47 => ( ( ? [R: real] :
% 0.32/0.47 ( X
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ? [Ra: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ Ra ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.32/0.47 => ( ( ? [R: real] :
% 0.32/0.47 ( X
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ? [Ra: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ Ra ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.32/0.47 => ( ( ? [R: real] :
% 0.32/0.47 ( X
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ! [Ra: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ Ra ) ) ) )
% 0.32/0.47 => ( ( ? [R: real] :
% 0.32/0.47 ( X
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.32/0.47 => ( ( ? [R: real] :
% 0.32/0.47 ( X
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.32/0.47 => ( ( ? [R: real] :
% 0.32/0.47 ( X
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ! [Ra: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ Ra ) ) ) )
% 0.32/0.47 => ( ( ? [R: real] :
% 0.32/0.47 ( X
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.32/0.47 => ( ( ? [R: real] :
% 0.32/0.47 ( X
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( ? [R: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ! [Ra: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ Ra ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( ? [R: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( ? [R: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ! [R: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ R ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ! [R: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ R ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( ? [R: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ! [Ra: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ Ra ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( ? [R: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( ? [R: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ! [R: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ R ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ! [R: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ R ) ) ) )
% 0.32/0.47 => ( ( ( X
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.32/0.47 => ~ ( ( X
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal3_cases
% 0.32/0.47 thf(fact_181_ereal__ex__split,axiom,
% 0.32/0.47 ( ( ^ [P3: extended_ereal > $o] :
% 0.32/0.47 ? [X7: extended_ereal] : ( P3 @ X7 ) )
% 0.32/0.47 = ( ^ [P4: extended_ereal > $o] :
% 0.32/0.47 ( ( P4 @ ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 | ? [X2: real] : ( P4 @ ( extended_ereal2 @ X2 ) )
% 0.32/0.47 | ( P4 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_ex_split
% 0.32/0.47 thf(fact_182_abs__ereal_Ocases,axiom,
% 0.32/0.47 ! [X: extended_ereal] :
% 0.32/0.47 ( ! [R: real] :
% 0.32/0.47 ( X
% 0.32/0.47 != ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ( X
% 0.32/0.47 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( X
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % abs_ereal.cases
% 0.32/0.47 thf(fact_183_ereal__all__split,axiom,
% 0.32/0.47 ( ( ^ [P3: extended_ereal > $o] :
% 0.32/0.47 ! [X7: extended_ereal] : ( P3 @ X7 ) )
% 0.32/0.47 = ( ^ [P4: extended_ereal > $o] :
% 0.32/0.47 ( ( P4 @ ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 & ! [X2: real] : ( P4 @ ( extended_ereal2 @ X2 ) )
% 0.32/0.47 & ( P4 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_all_split
% 0.32/0.47 thf(fact_184_abs__ereal_Oinduct,axiom,
% 0.32/0.47 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.32/0.47 ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( ( P @ ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( P @ A0 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % abs_ereal.induct
% 0.32/0.47 thf(fact_185_less__ereal_Oinduct,axiom,
% 0.32/0.47 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.32/0.47 ( ! [X4: real,Y2: real] : ( P @ ( extended_ereal2 @ X4 ) @ ( extended_ereal2 @ Y2 ) )
% 0.32/0.47 => ( ! [X_1: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_1 )
% 0.32/0.47 => ( ! [A6: extended_ereal] : ( P @ A6 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( ! [X4: real] : ( P @ ( extended_ereal2 @ X4 ) @ ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( ! [R: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_ereal.induct
% 0.32/0.47 thf(fact_186_plus__ereal_Oinduct,axiom,
% 0.32/0.47 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.32/0.47 ( ! [R: real,P5: real] : ( P @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P5 ) )
% 0.32/0.47 => ( ! [X_1: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_1 )
% 0.32/0.47 => ( ! [A6: extended_ereal] : ( P @ A6 @ ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( ! [P5: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ P5 ) )
% 0.32/0.47 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % plus_ereal.induct
% 0.32/0.47 thf(fact_187_times__ereal_Oinduct,axiom,
% 0.32/0.47 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.32/0.47 ( ! [R: real,P5: real] : ( P @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P5 ) )
% 0.32/0.47 => ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( ! [R: real] : ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( ! [R: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % times_ereal.induct
% 0.32/0.47 thf(fact_188_real__of__ereal_Ocases,axiom,
% 0.32/0.47 ! [X: extended_ereal] :
% 0.32/0.47 ( ! [R: real] :
% 0.32/0.47 ( X
% 0.32/0.47 != ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ( X
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( X
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % real_of_ereal.cases
% 0.32/0.47 thf(fact_189_real__of__ereal_Oinduct,axiom,
% 0.32/0.47 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.32/0.47 ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ( P @ ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 => ( P @ A0 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % real_of_ereal.induct
% 0.32/0.47 thf(fact_190_domain__Epigraph__aux,axiom,
% 0.32/0.47 ! [X: extended_ereal] :
% 0.32/0.47 ( ( X
% 0.32/0.47 != ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 => ? [R: real] : ( ord_less_eq @ extended_ereal @ X @ ( extended_ereal2 @ R ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % domain_Epigraph_aux
% 0.32/0.47 thf(fact_191_ereal__top,axiom,
% 0.32/0.47 ! [X: extended_ereal] :
% 0.32/0.47 ( ! [B7: real] : ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ B7 ) @ X )
% 0.32/0.47 => ( X
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_top
% 0.32/0.47 thf(fact_192_Infty__neq__0_I3_J,axiom,
% 0.32/0.47 ( ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 != ( zero_zero @ extended_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % Infty_neq_0(3)
% 0.32/0.47 thf(fact_193_ereal_Oinduct,axiom,
% 0.32/0.47 ! [P: extended_ereal > $o,Ereal: extended_ereal] :
% 0.32/0.47 ( ! [X4: real] : ( P @ ( extended_ereal2 @ X4 ) )
% 0.32/0.47 => ( ( P @ extended_PInfty )
% 0.32/0.47 => ( ( P @ extended_MInfty )
% 0.32/0.47 => ( P @ Ereal ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal.induct
% 0.32/0.47 thf(fact_194_ereal_Oexhaust,axiom,
% 0.32/0.47 ! [Y4: extended_ereal] :
% 0.32/0.47 ( ! [X12: real] :
% 0.32/0.47 ( Y4
% 0.32/0.47 != ( extended_ereal2 @ X12 ) )
% 0.32/0.47 => ( ( Y4 != extended_PInfty )
% 0.32/0.47 => ( Y4 = extended_MInfty ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal.exhaust
% 0.32/0.47 thf(fact_195_uminus__ereal_Ocases,axiom,
% 0.32/0.47 ! [X: extended_ereal] :
% 0.32/0.47 ( ! [R: real] :
% 0.32/0.47 ( X
% 0.32/0.47 != ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ( X != extended_PInfty )
% 0.32/0.47 => ( X = extended_MInfty ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % uminus_ereal.cases
% 0.32/0.47 thf(fact_196_uminus__ereal_Oinduct,axiom,
% 0.32/0.47 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.32/0.47 ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
% 0.32/0.47 => ( ( P @ extended_PInfty )
% 0.32/0.47 => ( ( P @ extended_MInfty )
% 0.32/0.47 => ( P @ A0 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % uminus_ereal.induct
% 0.32/0.47 thf(fact_197_ereal__bot,axiom,
% 0.32/0.47 ! [X: extended_ereal] :
% 0.32/0.47 ( ! [B7: real] : ( ord_less_eq @ extended_ereal @ X @ ( extended_ereal2 @ B7 ) )
% 0.32/0.47 => ( X
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_bot
% 0.32/0.47 thf(fact_198_le__numeral__extra_I3_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord1659791738miring @ A )
% 0.32/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_numeral_extra(3)
% 0.32/0.47 thf(fact_199_ereal__less__eq_I4_J,axiom,
% 0.32/0.47 ! [R2: real] :
% 0.32/0.47 ( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ R2 ) @ ( zero_zero @ extended_ereal ) )
% 0.32/0.47 = ( ord_less_eq @ real @ R2 @ ( zero_zero @ real ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_less_eq(4)
% 0.32/0.47 thf(fact_200_ereal__less__eq_I5_J,axiom,
% 0.32/0.47 ! [R2: real] :
% 0.32/0.47 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( extended_ereal2 @ R2 ) )
% 0.32/0.47 = ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_less_eq(5)
% 0.32/0.47 thf(fact_201_ereal__eq__0_I2_J,axiom,
% 0.32/0.47 ! [R2: real] :
% 0.32/0.47 ( ( ( zero_zero @ extended_ereal )
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 = ( R2
% 0.32/0.47 = ( zero_zero @ real ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_eq_0(2)
% 0.32/0.47 thf(fact_202_ereal__eq__0_I1_J,axiom,
% 0.32/0.47 ! [R2: real] :
% 0.32/0.47 ( ( ( extended_ereal2 @ R2 )
% 0.32/0.47 = ( zero_zero @ extended_ereal ) )
% 0.32/0.47 = ( R2
% 0.32/0.47 = ( zero_zero @ real ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_eq_0(1)
% 0.32/0.47 thf(fact_203_zero__ereal__def,axiom,
% 0.32/0.47 ( ( zero_zero @ extended_ereal )
% 0.32/0.47 = ( extended_ereal2 @ ( zero_zero @ real ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_ereal_def
% 0.32/0.47 thf(fact_204_real__eq__0__iff__le__ge__0,axiom,
% 0.32/0.47 ! [X: real] :
% 0.32/0.47 ( ( X
% 0.32/0.47 = ( zero_zero @ real ) )
% 0.32/0.47 = ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
% 0.32/0.47 & ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( uminus_uminus @ real @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % real_eq_0_iff_le_ge_0
% 0.32/0.47 thf(fact_205_ereal__divide__ereal,axiom,
% 0.32/0.47 ! [R2: real] :
% 0.32/0.47 ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 )
% 0.32/0.47 => ( ( divide_divide @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R2 ) )
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 & ( ~ ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 )
% 0.32/0.47 => ( ( divide_divide @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R2 ) )
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_divide_ereal
% 0.32/0.47 thf(fact_206_ereal__uminus__divide,axiom,
% 0.32/0.47 ! [X: extended_ereal,Y4: extended_ereal] :
% 0.32/0.47 ( ( divide_divide @ extended_ereal @ ( uminus_uminus @ extended_ereal @ X ) @ Y4 )
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( divide_divide @ extended_ereal @ X @ Y4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_uminus_divide
% 0.32/0.47 thf(fact_207_ereal__divide__zero__left,axiom,
% 0.32/0.47 ! [A4: extended_ereal] :
% 0.32/0.47 ( ( divide_divide @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A4 )
% 0.32/0.47 = ( zero_zero @ extended_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_divide_zero_left
% 0.32/0.47 thf(fact_208_ereal__divide__Infty_I1_J,axiom,
% 0.32/0.47 ! [X: extended_ereal] :
% 0.32/0.47 ( ( divide_divide @ extended_ereal @ X @ ( extend1396239628finity @ extended_ereal ) )
% 0.32/0.47 = ( zero_zero @ extended_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_divide_Infty(1)
% 0.32/0.47 thf(fact_209_ereal__divide__Infty_I2_J,axiom,
% 0.32/0.47 ! [X: extended_ereal] :
% 0.32/0.47 ( ( divide_divide @ extended_ereal @ X @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.32/0.47 = ( zero_zero @ extended_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_divide_Infty(2)
% 0.32/0.47 thf(fact_210_zero__le__divide__ereal,axiom,
% 0.32/0.47 ! [A4: extended_ereal,B4: extended_ereal] :
% 0.32/0.47 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A4 )
% 0.32/0.47 => ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ B4 )
% 0.32/0.47 => ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( divide_divide @ extended_ereal @ A4 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_le_divide_ereal
% 0.32/0.47 thf(fact_211_div__minus__minus,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( euclid24285859cancel @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( divide_divide @ A @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ A @ B4 ) )
% 0.32/0.47 = ( divide_divide @ A @ A4 @ B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % div_minus_minus
% 0.32/0.47 thf(fact_212_div__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semidom_divide @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A4 )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % div_0
% 0.32/0.47 thf(fact_213_div__by__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semidom_divide @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( divide_divide @ A @ A4 @ ( zero_zero @ A ) )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % div_by_0
% 0.32/0.47 thf(fact_214_div__minus__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( euclid24285859cancel @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( divide_divide @ A @ A4 @ ( uminus_uminus @ A @ B4 ) )
% 0.32/0.47 = ( divide_divide @ A @ ( uminus_uminus @ A @ A4 ) @ B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % div_minus_right
% 0.32/0.47 thf(fact_215_division__ring__divide__zero,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( division_ring @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( divide_divide @ A @ A4 @ ( zero_zero @ A ) )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % division_ring_divide_zero
% 0.32/0.47 thf(fact_216_bits__div__by__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semiring_bits @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( divide_divide @ A @ A4 @ ( zero_zero @ A ) )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % bits_div_by_0
% 0.32/0.47 thf(fact_217_divide__eq__0__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( ( divide_divide @ A @ A4 @ B4 )
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 = ( ( A4
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 | ( B4
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % divide_eq_0_iff
% 0.32/0.47 thf(fact_218_divide__cancel__left,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [C2: A,A4: A,B4: A] :
% 0.32/0.47 ( ( ( divide_divide @ A @ C2 @ A4 )
% 0.32/0.47 = ( divide_divide @ A @ C2 @ B4 ) )
% 0.32/0.47 = ( ( C2
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 | ( A4 = B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % divide_cancel_left
% 0.32/0.47 thf(fact_219_bits__div__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semiring_bits @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A4 )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % bits_div_0
% 0.32/0.47 thf(fact_220_divide__cancel__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [A4: A,C2: A,B4: A] :
% 0.32/0.47 ( ( ( divide_divide @ A @ A4 @ C2 )
% 0.32/0.47 = ( divide_divide @ A @ B4 @ C2 ) )
% 0.32/0.47 = ( ( C2
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 | ( A4 = B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % divide_cancel_right
% 0.32/0.47 thf(fact_221_minus__divide__left,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( division_ring @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ B4 ) )
% 0.32/0.47 = ( divide_divide @ A @ ( uminus_uminus @ A @ A4 ) @ B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minus_divide_left
% 0.32/0.47 thf(fact_222_minus__divide__divide,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( divide_divide @ A @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ A @ B4 ) )
% 0.32/0.47 = ( divide_divide @ A @ A4 @ B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minus_divide_divide
% 0.32/0.47 thf(fact_223_minus__divide__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ B4 ) )
% 0.32/0.47 = ( divide_divide @ A @ A4 @ ( uminus_uminus @ A @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minus_divide_right
% 0.32/0.47 thf(fact_224_divide__right__mono__neg,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [A4: A,B4: A,C2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.32/0.47 => ( ord_less_eq @ A @ ( divide_divide @ A @ B4 @ C2 ) @ ( divide_divide @ A @ A4 @ C2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % divide_right_mono_neg
% 0.32/0.47 thf(fact_225_divide__nonpos__nonpos,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [X: A,Y4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Y4 @ ( zero_zero @ A ) )
% 0.32/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % divide_nonpos_nonpos
% 0.32/0.47 thf(fact_226_divide__nonpos__nonneg,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [X: A,Y4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.32/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y4 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % divide_nonpos_nonneg
% 0.32/0.47 thf(fact_227_divide__nonneg__nonpos,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [X: A,Y4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Y4 @ ( zero_zero @ A ) )
% 0.32/0.47 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % divide_nonneg_nonpos
% 0.32/0.47 thf(fact_228_divide__nonneg__nonneg,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [X: A,Y4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.32/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y4 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % divide_nonneg_nonneg
% 0.32/0.47 thf(fact_229_zero__le__divide__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A4 @ B4 ) )
% 0.32/0.47 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 0.32/0.47 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) )
% 0.32/0.47 | ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
% 0.32/0.47 & ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_le_divide_iff
% 0.32/0.47 thf(fact_230_divide__right__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [A4: A,B4: A,C2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( divide_divide @ A @ A4 @ C2 ) @ ( divide_divide @ A @ B4 @ C2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % divide_right_mono
% 0.32/0.47 thf(fact_231_divide__le__0__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A4 @ B4 ) @ ( zero_zero @ A ) )
% 0.32/0.47 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 0.32/0.47 & ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) )
% 0.32/0.47 | ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
% 0.32/0.47 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % divide_le_0_iff
% 0.32/0.47 thf(fact_232_nonzero__minus__divide__divide,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( division_ring @ A )
% 0.32/0.47 => ! [B4: A,A4: A] :
% 0.32/0.47 ( ( B4
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( ( divide_divide @ A @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ A @ B4 ) )
% 0.32/0.47 = ( divide_divide @ A @ A4 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nonzero_minus_divide_divide
% 0.32/0.47 thf(fact_233_nonzero__minus__divide__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( division_ring @ A )
% 0.32/0.47 => ! [B4: A,A4: A] :
% 0.32/0.47 ( ( B4
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ B4 ) )
% 0.32/0.47 = ( divide_divide @ A @ A4 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nonzero_minus_divide_right
% 0.32/0.47 thf(fact_234_real__of__ereal__le__0,axiom,
% 0.32/0.47 ! [X: extended_ereal] :
% 0.32/0.47 ( ( ord_less_eq @ real @ ( extend1716541707_ereal @ X ) @ ( zero_zero @ real ) )
% 0.32/0.47 = ( ( ord_less_eq @ extended_ereal @ X @ ( zero_zero @ extended_ereal ) )
% 0.32/0.47 | ( X
% 0.32/0.47 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % real_of_ereal_le_0
% 0.32/0.47 thf(fact_235_ereal__divide,axiom,
% 0.32/0.47 ! [P2: real,R2: real] :
% 0.32/0.47 ( ( ( P2
% 0.32/0.47 = ( zero_zero @ real ) )
% 0.32/0.47 => ( ( divide_divide @ extended_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
% 0.32/0.47 = ( times_times @ extended_ereal @ ( extended_ereal2 @ R2 ) @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.32/0.47 & ( ( P2
% 0.32/0.47 != ( zero_zero @ real ) )
% 0.32/0.47 => ( ( divide_divide @ extended_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
% 0.32/0.47 = ( extended_ereal2 @ ( divide_divide @ real @ R2 @ P2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_divide
% 0.32/0.47 thf(fact_236_mult__zero__left,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( mult_zero @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( times_times @ A @ ( zero_zero @ A ) @ A4 )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % mult_zero_left
% 0.32/0.47 thf(fact_237_mult__zero__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( mult_zero @ A )
% 0.32/0.47 => ! [A4: A] :
% 0.32/0.47 ( ( times_times @ A @ A4 @ ( zero_zero @ A ) )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % mult_zero_right
% 0.32/0.47 thf(fact_238_mult__eq__0__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semiri1193490041visors @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( ( times_times @ A @ A4 @ B4 )
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 = ( ( A4
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 | ( B4
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % mult_eq_0_iff
% 0.32/0.47 thf(fact_239_mult__cancel__left,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semiri1923998003cancel @ A )
% 0.32/0.47 => ! [C2: A,A4: A,B4: A] :
% 0.32/0.47 ( ( ( times_times @ A @ C2 @ A4 )
% 0.32/0.47 = ( times_times @ A @ C2 @ B4 ) )
% 0.32/0.47 = ( ( C2
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 | ( A4 = B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % mult_cancel_left
% 0.32/0.47 thf(fact_240_mult__cancel__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semiri1923998003cancel @ A )
% 0.32/0.47 => ! [A4: A,C2: A,B4: A] :
% 0.32/0.47 ( ( ( times_times @ A @ A4 @ C2 )
% 0.32/0.47 = ( times_times @ A @ B4 @ C2 ) )
% 0.32/0.47 = ( ( C2
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 | ( A4 = B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % mult_cancel_right
% 0.32/0.47 thf(fact_241_mult__minus__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ring @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( times_times @ A @ A4 @ ( uminus_uminus @ A @ B4 ) )
% 0.32/0.47 = ( uminus_uminus @ A @ ( times_times @ A @ A4 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % mult_minus_right
% 0.32/0.47 thf(fact_242_minus__mult__minus,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ring @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( times_times @ A @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ A @ B4 ) )
% 0.32/0.47 = ( times_times @ A @ A4 @ B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minus_mult_minus
% 0.32/0.47 thf(fact_243_mult__minus__left,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ring @ A )
% 0.32/0.47 => ! [A4: A,B4: A] :
% 0.32/0.47 ( ( times_times @ A @ ( uminus_uminus @ A @ A4 ) @ B4 )
% 0.32/0.47 = ( uminus_uminus @ A @ ( times_times @ A @ A4 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % mult_minus_left
% 0.32/0.47 thf(fact_244_ereal__mult__minus__right,axiom,
% 0.32/0.47 ! [A4: extended_ereal,B4: extended_ereal] :
% 0.32/0.47 ( ( times_times @ extended_ereal @ A4 @ ( uminus_uminus @ extended_ereal @ B4 ) )
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( times_times @ extended_ereal @ A4 @ B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_mult_minus_right
% 0.32/0.47 thf(fact_245_ereal__mult__minus__left,axiom,
% 0.32/0.47 ! [A4: extended_ereal,B4: extended_ereal] :
% 0.32/0.47 ( ( times_times @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A4 ) @ B4 )
% 0.32/0.47 = ( uminus_uminus @ extended_ereal @ ( times_times @ extended_ereal @ A4 @ B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_mult_minus_left
% 0.32/0.47 thf(fact_246_ereal__mult__zero,axiom,
% 0.32/0.47 ! [A4: extended_ereal] :
% 0.32/0.47 ( ( times_times @ extended_ereal @ A4 @ ( zero_zero @ extended_ereal ) )
% 0.32/0.47 = ( zero_zero @ extended_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_mult_zero
% 0.32/0.47 thf(fact_247_ereal__zero__mult,axiom,
% 0.32/0.47 ! [A4: extended_ereal] :
% 0.32/0.47 ( ( times_times @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A4 )
% 0.32/0.47 = ( zero_zero @ extended_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_zero_mult
% 0.32/0.47 thf(fact_248_ereal__zero__times,axiom,
% 0.32/0.47 ! [A4: extended_ereal,B4: extended_ereal] :
% 0.32/0.47 ( ( ( times_times @ extended_ereal @ A4 @ B4 )
% 0.32/0.47 = ( zero_zero @ extended_ereal ) )
% 0.32/0.47 = ( ( A4
% 0.32/0.47 = ( zero_zero @ extended_ereal ) )
% 0.32/0.47 | ( B4
% 0.32/0.47 = ( zero_zero @ extended_ereal ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_zero_times
% 0.32/0.47 thf(fact_249_ereal__times__divide__eq__left,axiom,
% 0.32/0.47 ! [B4: extended_ereal,C2: extended_ereal,A4: extended_ereal] :
% 0.32/0.47 ( ( times_times @ extended_ereal @ ( divide_divide @ extended_ereal @ B4 @ C2 ) @ A4 )
% 0.32/0.47 = ( divide_divide @ extended_ereal @ ( times_times @ extended_ereal @ B4 @ A4 ) @ C2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_times_divide_eq_left
% 0.32/0.47 thf(fact_250_mult__divide__mult__cancel__left__if,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [C2: A,A4: A,B4: A] :
% 0.32/0.47 ( ( ( C2
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) )
% 0.32/0.47 = ( zero_zero @ A ) ) )
% 0.32/0.47 & ( ( C2
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) )
% 0.32/0.47 = ( divide_divide @ A @ A4 @ B4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % mult_divide_mult_cancel_left_if
% 0.32/0.47 thf(fact_251_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [C2: A,A4: A,B4: A] :
% 0.32/0.47 ( ( C2
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) )
% 0.32/0.47 = ( divide_divide @ A @ A4 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nonzero_mult_divide_mult_cancel_left
% 0.32/0.47 thf(fact_252_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [C2: A,A4: A,B4: A] :
% 0.32/0.47 ( ( C2
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ B4 @ C2 ) )
% 0.32/0.47 = ( divide_divide @ A @ A4 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nonzero_mult_divide_mult_cancel_left2
% 0.32/0.47 thf(fact_253_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [C2: A,A4: A,B4: A] :
% 0.32/0.47 ( ( C2
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ C2 ) )
% 0.32/0.47 = ( divide_divide @ A @ A4 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nonzero_mult_divide_mult_cancel_right
% 0.32/0.47 thf(fact_254_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [C2: A,A4: A,B4: A] :
% 0.32/0.47 ( ( C2
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ C2 @ B4 ) )
% 0.32/0.47 = ( divide_divide @ A @ A4 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nonzero_mult_divide_mult_cancel_right2
% 0.32/0.47
% 0.32/0.47 % Subclasses (6)
% 0.32/0.47 thf(subcl_Euclidean__Space_Oeuclidean__space___HOL_Otype,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( euclid925273238_space @ A )
% 0.32/0.47 => ( type @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subcl_Euclidean__Space_Oeuclidean__space___Groups_Ozero,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( euclid925273238_space @ A )
% 0.32/0.47 => ( zero @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subcl_Euclidean__Space_Oeuclidean__space___Groups_Ouminus,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( euclid925273238_space @ A )
% 0.32/0.47 => ( uminus @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subcl_Euclidean__Space_Oeuclidean__space___Groups_Ogroup__add,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( euclid925273238_space @ A )
% 0.32/0.47 => ( group_add @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subcl_Euclidean__Space_Oeuclidean__space___Real__Vector__Spaces_Oreal__vector,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( euclid925273238_space @ A )
% 0.32/0.47 => ( real_V1076094709vector @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subcl_Euclidean__Space_Oeuclidean__space___Real__Vector__Spaces_Ometric__space,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( euclid925273238_space @ A )
% 0.32/0.47 => ( real_V2090557954_space @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % Type constructors (79)
% 0.32/0.47 thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ( ordere890947078_space @ A7 )
% 0.32/0.47 & ( ordere890947078_space @ A8 ) )
% 0.32/0.47 => ( ordere890947078_space @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_1,axiom,
% 0.32/0.47 ordere890947078_space @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( comple187826305attice @ A8 )
% 0.32/0.47 => ( condit378418413attice @ ( A7 > A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( comple187826305attice @ A8 )
% 0.32/0.47 => ( comple187826305attice @ ( A7 > A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( boolean_algebra @ A8 )
% 0.32/0.47 => ( boolean_algebra @ ( A7 > A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( order_top @ A8 )
% 0.32/0.47 => ( order_top @ ( A7 > A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( preorder @ A8 )
% 0.32/0.47 => ( preorder @ ( A7 > A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( order @ A8 )
% 0.32/0.47 => ( order @ ( A7 > A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Otop,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( top @ A8 )
% 0.32/0.47 => ( top @ ( A7 > A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ord @ A8 )
% 0.32/0.47 => ( ord @ ( A7 > A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Groups_Ouminus,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( uminus @ A8 )
% 0.32/0.47 => ( uminus @ ( A7 > A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_2,axiom,
% 0.32/0.47 ! [A7: $tType] : ( condit378418413attice @ ( set @ A7 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_3,axiom,
% 0.32/0.47 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_4,axiom,
% 0.32/0.47 ! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Orderings_Oorder__top_5,axiom,
% 0.32/0.47 ! [A7: $tType] : ( order_top @ ( set @ A7 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_6,axiom,
% 0.32/0.47 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Orderings_Oorder_7,axiom,
% 0.32/0.47 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Orderings_Otop_8,axiom,
% 0.32/0.47 ! [A7: $tType] : ( top @ ( set @ A7 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
% 0.32/0.47 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Groups_Ouminus_10,axiom,
% 0.32/0.47 ! [A7: $tType] : ( uminus @ ( set @ A7 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Groups_Ozero,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( zero @ A7 )
% 0.32/0.47 => ( zero @ ( set @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11,axiom,
% 0.32/0.47 condit378418413attice @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_12,axiom,
% 0.32/0.47 comple187826305attice @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_13,axiom,
% 0.32/0.47 boolean_algebra @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Oorder__top_14,axiom,
% 0.32/0.47 order_top @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_15,axiom,
% 0.32/0.47 preorder @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.32/0.47 linorder @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
% 0.32/0.47 order @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Otop_17,axiom,
% 0.32/0.47 top @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Oord_18,axiom,
% 0.32/0.47 ord @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Groups_Ouminus_19,axiom,
% 0.32/0.47 uminus @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_20,axiom,
% 0.32/0.47 condit378418413attice @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel,axiom,
% 0.32/0.47 semiri1923998003cancel @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring,axiom,
% 0.32/0.47 linord1659791738miring @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors,axiom,
% 0.32/0.47 semiri1193490041visors @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
% 0.32/0.47 real_V2090557954_space @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space,axiom,
% 0.32/0.47 euclid925273238_space @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
% 0.32/0.47 real_V1076094709vector @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
% 0.32/0.47 linord219039673up_add @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
% 0.32/0.47 ordered_ab_group_add @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 0.32/0.47 linordered_field @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Osemidom__divide,axiom,
% 0.32/0.47 semidom_divide @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
% 0.32/0.47 division_ring @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Orderings_Opreorder_21,axiom,
% 0.32/0.47 preorder @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Orderings_Olinorder_22,axiom,
% 0.32/0.47 linorder @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Orderings_Ono__bot,axiom,
% 0.32/0.47 no_bot @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 0.32/0.47 group_add @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Omult__zero,axiom,
% 0.32/0.47 mult_zero @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Orderings_Oorder_23,axiom,
% 0.32/0.47 order @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Orderings_Oord_24,axiom,
% 0.32/0.47 ord @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Groups_Ouminus_25,axiom,
% 0.32/0.47 uminus @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Fields_Ofield,axiom,
% 0.32/0.47 field @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Groups_Ozero_26,axiom,
% 0.32/0.47 zero @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Oring,axiom,
% 0.32/0.47 ring @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Product__Type_Oprod___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_27,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ( condit378418413attice @ A7 )
% 0.32/0.47 & ( condit378418413attice @ A8 ) )
% 0.32/0.47 => ( condit378418413attice @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Product__Type_Oprod___Complete__Lattices_Ocomplete__lattice_28,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ( comple187826305attice @ A7 )
% 0.32/0.47 & ( comple187826305attice @ A8 ) )
% 0.32/0.47 => ( comple187826305attice @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Ometric__space_29,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ( real_V2090557954_space @ A7 )
% 0.32/0.47 & ( real_V2090557954_space @ A8 ) )
% 0.32/0.47 => ( real_V2090557954_space @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Product__Type_Oprod___Euclidean__Space_Oeuclidean__space_30,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ( euclid925273238_space @ A7 )
% 0.32/0.47 & ( euclid925273238_space @ A8 ) )
% 0.32/0.47 => ( euclid925273238_space @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__vector_31,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ( real_V1076094709vector @ A7 )
% 0.32/0.47 & ( real_V1076094709vector @ A8 ) )
% 0.32/0.47 => ( real_V1076094709vector @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__group__add_32,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ( ordere890947078_space @ A7 )
% 0.32/0.47 & ( ordere890947078_space @ A8 ) )
% 0.32/0.47 => ( ordered_ab_group_add @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Product__Type_Oprod___Lattices_Oboolean__algebra_33,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ( boolean_algebra @ A7 )
% 0.32/0.47 & ( boolean_algebra @ A8 ) )
% 0.32/0.47 => ( boolean_algebra @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Product__Type_Oprod___Orderings_Oorder__top_34,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ( order_top @ A7 )
% 0.32/0.47 & ( order_top @ A8 ) )
% 0.32/0.47 => ( order_top @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_35,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ( preorder @ A7 )
% 0.32/0.47 & ( preorder @ A8 ) )
% 0.32/0.47 => ( preorder @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Product__Type_Oprod___Groups_Ogroup__add_36,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ( group_add @ A7 )
% 0.32/0.47 & ( group_add @ A8 ) )
% 0.32/0.47 => ( group_add @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Product__Type_Oprod___Orderings_Oorder_37,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ( order @ A7 )
% 0.32/0.47 & ( order @ A8 ) )
% 0.32/0.47 => ( order @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Product__Type_Oprod___Orderings_Otop_38,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ( top @ A7 )
% 0.32/0.47 & ( top @ A8 ) )
% 0.32/0.47 => ( top @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Product__Type_Oprod___Orderings_Oord_39,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ( ord @ A7 )
% 0.32/0.47 & ( ord @ A8 ) )
% 0.32/0.47 => ( ord @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Product__Type_Oprod___Groups_Ouminus_40,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ( uminus @ A7 )
% 0.32/0.47 & ( uminus @ A8 ) )
% 0.32/0.47 => ( uminus @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Product__Type_Oprod___Groups_Ozero_41,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ( zero @ A7 )
% 0.32/0.47 & ( zero @ A8 ) )
% 0.32/0.47 => ( zero @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Extended__Real_Oereal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_42,axiom,
% 0.32/0.47 condit378418413attice @ extended_ereal ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Extended__Real_Oereal___Complete__Lattices_Ocomplete__lattice_43,axiom,
% 0.32/0.47 comple187826305attice @ extended_ereal ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Extended__Real_Oereal___Orderings_Oorder__top_44,axiom,
% 0.32/0.47 order_top @ extended_ereal ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Extended__Real_Oereal___Orderings_Opreorder_45,axiom,
% 0.32/0.47 preorder @ extended_ereal ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Extended__Real_Oereal___Orderings_Olinorder_46,axiom,
% 0.32/0.47 linorder @ extended_ereal ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Extended__Real_Oereal___Orderings_Oorder_47,axiom,
% 0.32/0.47 order @ extended_ereal ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Extended__Real_Oereal___Orderings_Otop_48,axiom,
% 0.32/0.47 top @ extended_ereal ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Extended__Real_Oereal___Orderings_Oord_49,axiom,
% 0.32/0.47 ord @ extended_ereal ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Extended__Real_Oereal___Groups_Ouminus_50,axiom,
% 0.32/0.47 uminus @ extended_ereal ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Extended__Real_Oereal___Groups_Ozero_51,axiom,
% 0.32/0.47 zero @ extended_ereal ).
% 0.32/0.47
% 0.32/0.47 % Free types (1)
% 0.32/0.47 thf(tfree_0,hypothesis,
% 0.32/0.47 euclid925273238_space @ a ).
% 0.32/0.47
% 0.32/0.47 % Conjectures (1)
% 0.32/0.47 thf(conj_0,conjecture,
% 0.32/0.47 ( ( lower_673667120vex_on @ a @ ( top_top @ ( set @ a ) ) @ f )
% 0.32/0.47 = ( convex @ a @ ( lower_272802190domain @ a @ f ) ) ) ).
% 0.32/0.47
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 ------------------------
% 2.31/1.57 ---- Embedded in THF ---
% 2.31/1.57 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.31/1.57 %%% Generated on Tue Feb 24 08:27:04 EST 2026
% 2.31/1.57 %%% using '$$dhol' embedding, version 1.3.0.
% 2.31/1.57 %%% Logic specification used:
% 2.31/1.57 %%% thf(spec, logic, $$dhol).
% 2.31/1.57
% 2.31/1.57 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.0x1kQQmmmY/DTF2DTF23220.p
% 2.31/1.57 thf(ty_t_Extended__Real_Oereal, type, extended_ereal: $tType).
% 2.31/1.57 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.31/1.57 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.31/1.57 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.31/1.57 thf(ty_tf_a, type, a: $tType).
% 2.31/1.57 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space, type, ordere890947078_space: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Rings_Oring, type, ring: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Fields_Ofield, type, field: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Groups_Ouminus, type, uminus: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Rings_Omult__zero, type, mult_zero: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Enum_Ofinite__lattice, type, finite_lattice: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Fields_Odivision__ring, type, division_ring: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Parity_Osemiring__bits, type, semiring_bits: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Rings_Osemidom__divide, type, semidom_divide: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Lattices_Oboolean__algebra, type, boolean_algebra: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Real__Vector__Spaces_Oreal__vector, type, real_V1076094709vector: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Euclidean__Space_Oeuclidean__space, type, euclid925273238_space: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Real__Vector__Spaces_Ometric__space, type, real_V2090557954_space: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Rings_Osemiring__no__zero__divisors, type, semiri1193490041visors: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple187826305attice: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel, type, semiri1923998003cancel: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel, type, euclid24285859cancel: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice, type, condit378418413attice: (!> [A:$tType]: ($o))).
% 2.31/1.57 thf(sy_c_Complete__Lattices_OSup__class_OSup, type, complete_Sup_Sup: (!> [A:$tType]: (((set @ A) > A)))).
% 2.31/1.57 thf(sy_c_Convex_Oconvex, type, convex: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.31/1.57 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity, type, extend1396239628finity: (!> [A:$tType]: (A))).
% 2.31/1.57 thf(sy_c_Extended__Real_Oereal_OMInfty, type, extended_MInfty: extended_ereal).
% 2.31/1.57 thf(sy_c_Extended__Real_Oereal_OPInfty, type, extended_PInfty: extended_ereal).
% 2.31/1.57 thf(sy_c_Extended__Real_Oereal_Oereal, type, extended_ereal2: (real > extended_ereal)).
% 2.31/1.57 thf(sy_c_Extended__Real_Oereal_Orec__ereal, type, extended_rec_ereal: (!> [A:$tType]: (((real > A) > (A > (A > (extended_ereal > A))))))).
% 2.31/1.57 thf(sy_c_Extended__Real_Oreal__of__ereal, type, extend1716541707_ereal: (extended_ereal > real)).
% 2.31/1.57 thf(sy_c_Groups_Otimes__class_Otimes, type, times_times: (!> [A:$tType]: ((A > (A > A))))).
% 2.31/1.57 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 2.31/1.57 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.31/1.57 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_OEpigraph, type, lower_261390618igraph: (!> [A:$tType]: (((set @ A) > ((A > extended_ereal) > (set @ ((product_prod @ A) @ real))))))).
% 2.31/1.57 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Oaffine__on, type, lower_500881736ine_on: (!> [A:$tType]: (((set @ A) > ((A > extended_ereal) > $o))))).
% 2.31/1.57 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Oconcave__on, type, lower_708069572ave_on: (!> [A:$tType]: (((set @ A) > ((A > extended_ereal) > $o))))).
% 2.31/1.57 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Oconvex__on, type, lower_673667120vex_on: (!> [A:$tType]: (((set @ A) > ((A > extended_ereal) > $o))))).
% 2.31/1.57 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Odomain, type, lower_272802190domain: (!> [A:$tType]: (((A > extended_ereal) > (set @ A))))).
% 2.31/1.57 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Ofinite__on, type, lower_728871928ite_on: (!> [A:$tType]: (((set @ A) > ((A > extended_ereal) > $o))))).
% 2.31/1.57 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.31/1.57 thf(sy_c_Orderings_Oorder__class_OGreatest, type, order_Greatest: (!> [A:$tType]: (((A > $o) > A)))).
% 2.31/1.57 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.31/1.57 thf(sy_c_Rings_Odivide__class_Odivide, type, divide_divide: (!> [A:$tType]: ((A > (A > A))))).
% 2.31/1.57 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.31/1.57 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.31/1.57 thf(sy_v_f, type, f: (a > extended_ereal)).
% 2.31/1.57 thf(fact_0__092_060open_062convex_A_Idomain_Af_J_A_092_060Longrightarrow_062_ALower__Semicontinuous__Mirabelle__quczrylfpw_Oconvex__on_AUNIV_Af_092_060close_062, axiom, (((convex @ a) @ ((lower_272802190domain @ a) @ f)) => (((lower_673667120vex_on @ a) @ (top_top @ (set @ a))) @ f))).
% 2.31/1.57 thf(fact_1_convex__domain, axiom, (! [A:$tType]: (((euclid925273238_space @ A) => (! [F:(A > extended_ereal)]: (((((lower_673667120vex_on @ A) @ (top_top @ (set @ A))) @ F) => ((convex @ A) @ ((lower_272802190domain @ A) @ F))))))))).
% 2.31/1.57 thf(fact_2_convex__on__domain, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [F:(A > extended_ereal)]: (((((lower_673667120vex_on @ A) @ ((lower_272802190domain @ A) @ F)) @ F) = (((lower_673667120vex_on @ A) @ (top_top @ (set @ A))) @ F)))))))).
% 2.31/1.57 thf(fact_3_convex__on__domain2, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [F:(A > extended_ereal)]: (((((lower_673667120vex_on @ A) @ ((lower_272802190domain @ A) @ F)) @ F) = (! [S:(set @ A)]: ((((lower_673667120vex_on @ A) @ S) @ F)))))))))).
% 2.31/1.57 thf(fact_4_convex__on__ereal__univ, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [F:(A > extended_ereal)]: (((((lower_673667120vex_on @ A) @ (top_top @ (set @ A))) @ F) = (! [S:(set @ A)]: ((((lower_673667120vex_on @ A) @ S) @ F)))))))))).
% 2.31/1.57 thf(fact_5_convex__UNIV, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => ((convex @ A) @ (top_top @ (set @ A))))))).
% 2.31/1.57 thf(fact_6_UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.31/1.57 thf(fact_7_iso__tuple__UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.31/1.57 thf(fact_8_top__apply, axiom, (! [C:$tType,D:$tType]: (((top @ C) => ((top_top @ (D > C)) = (^ [X2:D]: ((top_top @ C)))))))).
% 2.31/1.57 thf(fact_9_assms, axiom, (! [X3:a]: ((((f @ X3) = (extend1396239628finity @ extended_ereal)) | ((f @ X3) = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))).
% 2.31/1.57 thf(fact_10_UNIV__eq__I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X4:A]: ((((member @ A) @ X4) @ A2))) => ((top_top @ (set @ A)) = A2))))).
% 2.31/1.57 thf(fact_11_UNIV__witness, axiom, (! [A:$tType]: ((? [X4:A]: ((((member @ A) @ X4) @ (top_top @ (set @ A)))))))).
% 2.31/1.57 thf(fact_12_convex__Epigraph, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [S2:(set @ A),F:(A > extended_ereal)]: ((((convex @ A) @ S2) => (((convex @ ((product_prod @ A) @ real)) @ (((lower_261390618igraph @ A) @ S2) @ F)) = (((lower_673667120vex_on @ A) @ S2) @ F))))))))).
% 2.31/1.57 thf(fact_13_convex__EpigraphI, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [S3:(set @ A),F:(A > extended_ereal)]: (((((lower_673667120vex_on @ A) @ S3) @ F) => (((convex @ A) @ S3) => ((convex @ ((product_prod @ A) @ real)) @ (((lower_261390618igraph @ A) @ S3) @ F)))))))))).
% 2.31/1.57 thf(fact_14_affine__on__def, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => ((lower_500881736ine_on @ A) = (^ [S:(set @ A),F2:(A > extended_ereal)]: (((((lower_673667120vex_on @ A) @ S) @ F2) & ((((lower_708069572ave_on @ A) @ S) @ F2) & (((lower_728871928ite_on @ A) @ S) @ F2)))))))))).
% 2.31/1.57 thf(fact_15_Sup__UNIV, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (((complete_Sup_Sup @ A) @ (top_top @ (set @ A))) = (top_top @ A)))))).
% 2.31/1.57 thf(fact_16_convex__on__ereal__subset, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [T:(set @ A),F:(A > extended_ereal),S3:(set @ A)]: (((((lower_673667120vex_on @ A) @ T) @ F) => ((((ord_less_eq @ (set @ A)) @ S3) @ T) => (((lower_673667120vex_on @ A) @ S3) @ F))))))))).
% 2.31/1.57 thf(fact_17_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.31/1.57 thf(fact_18_subsetI, axiom, (! [A:$tType,A2:(set @ A),B:(set @ A)]: (((! [X4:A]: (((((member @ A) @ X4) @ A2) => (((member @ A) @ X4) @ B)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B))))).
% 2.31/1.57 thf(fact_19_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B) => ((((ord_less_eq @ (set @ A)) @ B) @ A2) => (A2 = B)))))).
% 2.31/1.57 thf(fact_20_in__mono, axiom, (! [A:$tType,A2:(set @ A),B:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B) => ((((member @ A) @ X) @ A2) => (((member @ A) @ X) @ B)))))).
% 2.31/1.57 thf(fact_21_subsetD, axiom, (! [A:$tType,A2:(set @ A),B:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B) => ((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ B)))))).
% 2.31/1.57 thf(fact_22_equalityE, axiom, (! [A:$tType,A2:(set @ A),B:(set @ A)]: (((A2 = B) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B) => (~ ((((ord_less_eq @ (set @ A)) @ B) @ A2)))))))))).
% 2.31/1.57 thf(fact_23_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B2:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ A3) => (((member @ A) @ X2) @ B2)))))))))).
% 2.31/1.57 thf(fact_24_equalityD1, axiom, (! [A:$tType,A2:(set @ A),B:(set @ A)]: (((A2 = B) => (((ord_less_eq @ (set @ A)) @ A2) @ B))))).
% 2.31/1.57 thf(fact_25_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B:(set @ A)]: (((A2 = B) => (((ord_less_eq @ (set @ A)) @ B) @ A2))))).
% 2.31/1.57 thf(fact_26_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B2:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A3) => (((member @ A) @ T2) @ B2)))))))))).
% 2.31/1.57 thf(fact_27_subset__refl, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 2.31/1.57 thf(fact_28_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) => (Q @ X4)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.31/1.57 thf(fact_29_subset__trans, axiom, (! [A:$tType,A2:(set @ A),B:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B) => ((((ord_less_eq @ (set @ A)) @ B) @ C3) => (((ord_less_eq @ (set @ A)) @ A2) @ C3)))))).
% 2.31/1.57 thf(fact_30_le__funD, axiom, (! [B3:$tType,A:$tType]: (((ord @ B3) => (! [F:(A > B3),G:(A > B3),X:A]: (((((ord_less_eq @ (A > B3)) @ F) @ G) => (((ord_less_eq @ B3) @ (F @ X)) @ (G @ X))))))))).
% 2.31/1.57 thf(fact_31_le__funE, axiom, (! [B3:$tType,A:$tType]: (((ord @ B3) => (! [F:(A > B3),G:(A > B3),X:A]: (((((ord_less_eq @ (A > B3)) @ F) @ G) => (((ord_less_eq @ B3) @ (F @ X)) @ (G @ X))))))))).
% 2.31/1.57 thf(fact_32_le__funI, axiom, (! [B3:$tType,A:$tType]: (((ord @ B3) => (! [F:(A > B3),G:(A > B3)]: (((! [X4:A]: ((((ord_less_eq @ B3) @ (F @ X4)) @ (G @ X4)))) => (((ord_less_eq @ (A > B3)) @ F) @ G)))))))).
% 2.31/1.57 thf(fact_33_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y:(set @ A),Z:(set @ A)]: ((Y = Z))) = (^ [A3:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) & (((ord_less_eq @ (set @ A)) @ B2) @ A3)))))))).
% 2.31/1.57 thf(fact_34_le__fun__def, axiom, (! [B3:$tType,A:$tType]: (((ord @ B3) => ((ord_less_eq @ (A > B3)) = (^ [F2:(A > B3),G2:(A > B3)]: ((! [X2:A]: ((((ord_less_eq @ B3) @ (F2 @ X2)) @ (G2 @ X2))))))))))).
% 2.31/1.57 thf(fact_35_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X2:A]: (((P @ X2) => (Q @ X2)))))))).
% 2.31/1.57 thf(fact_36_order__subst1, axiom, (! [A:$tType,B3:$tType]: ((((order @ B3) & (order @ A)) => (! [A4:A,F:(B3 > A),B4:B3,C2:B3]: (((((ord_less_eq @ A) @ A4) @ (F @ B4)) => ((((ord_less_eq @ B3) @ B4) @ C2) => ((! [X4:B3,Y2:B3]: (((((ord_less_eq @ B3) @ X4) @ Y2) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A4) @ (F @ C2))))))))))).
% 2.31/1.57 thf(fact_37_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A4:A,B4:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ C) @ (F @ B4)) @ C2) => ((! [X4:A,Y2:A]: (((((ord_less_eq @ A) @ X4) @ Y2) => (((ord_less_eq @ C) @ (F @ X4)) @ (F @ Y2))))) => (((ord_less_eq @ C) @ (F @ A4)) @ C2)))))))))).
% 2.31/1.57 thf(fact_38_ord__eq__le__subst, axiom, (! [A:$tType,B3:$tType]: ((((ord @ B3) & (ord @ A)) => (! [A4:A,F:(B3 > A),B4:B3,C2:B3]: (((A4 = (F @ B4)) => ((((ord_less_eq @ B3) @ B4) @ C2) => ((! [X4:B3,Y2:B3]: (((((ord_less_eq @ B3) @ X4) @ Y2) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A4) @ (F @ C2))))))))))).
% 2.31/1.57 thf(fact_39_ord__le__eq__subst, axiom, (! [A:$tType,B3:$tType]: ((((ord @ B3) & (ord @ A)) => (! [A4:A,B4:A,F:(A > B3),C2:B3]: (((((ord_less_eq @ A) @ A4) @ B4) => (((F @ B4) = C2) => ((! [X4:A,Y2:A]: (((((ord_less_eq @ A) @ X4) @ Y2) => (((ord_less_eq @ B3) @ (F @ X4)) @ (F @ Y2))))) => (((ord_less_eq @ B3) @ (F @ A4)) @ C2)))))))))).
% 2.31/1.57 thf(fact_40_Union__mono, axiom, (! [A:$tType,A2:(set @ (set @ A)),B:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ A2) @ B) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ A2)) @ ((complete_Sup_Sup @ (set @ A)) @ B)))))).
% 2.31/1.57 thf(fact_41_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X2))))))))).
% 2.31/1.57 thf(fact_42_Union__least, axiom, (! [A:$tType,A2:(set @ (set @ A)),C3:(set @ A)]: (((! [X5:(set @ A)]: (((((member @ (set @ A)) @ X5) @ A2) => (((ord_less_eq @ (set @ A)) @ X5) @ C3)))) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ A2)) @ C3))))).
% 2.31/1.57 thf(fact_43_Union__upper, axiom, (! [A:$tType,B:(set @ A),A2:(set @ (set @ A))]: (((((member @ (set @ A)) @ B) @ A2) => (((ord_less_eq @ (set @ A)) @ B) @ ((complete_Sup_Sup @ (set @ A)) @ A2)))))).
% 2.31/1.57 thf(fact_44_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ X) => (X = Y4))))))))).
% 2.31/1.57 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A4:A,P:(A > $o)]: (((((member @ A) @ A4) @ ((collect @ A) @ P)) = (P @ A4))))).
% 2.31/1.57 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A2)))) = A2)))).
% 2.31/1.57 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) = (Q @ X4)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.31/1.57 thf(fact_48_ext, axiom, (! [B3:$tType,A:$tType,F:(A > B3),G:(A > B3)]: (((! [X4:A]: (((F @ X4) = (G @ X4)))) => (F = G))))).
% 2.31/1.57 thf(fact_49_Union__subsetI, axiom, (! [A:$tType,A2:(set @ (set @ A)),B:(set @ (set @ A))]: (((! [X4:(set @ A)]: (((((member @ (set @ A)) @ X4) @ A2) => (? [Y5:(set @ A)]: (((((member @ (set @ A)) @ Y5) @ B) & (((ord_less_eq @ (set @ A)) @ X4) @ Y5))))))) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ A2)) @ ((complete_Sup_Sup @ (set @ A)) @ B)))))).
% 2.31/1.57 thf(fact_50_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) | (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 2.31/1.57 thf(fact_51_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y4:A]: (((X = Y4) => (((ord_less_eq @ A) @ X) @ Y4)))))))).
% 2.31/1.57 thf(fact_52_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y4:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y4))) => (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 2.31/1.57 thf(fact_53_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B4:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ B4) @ C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 2.31/1.57 thf(fact_54_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y4:A,Z2:A]: ((((((ord_less_eq @ A) @ X) @ Y4) => (~ ((((ord_less_eq @ A) @ Y4) @ Z2)))) => (((((ord_less_eq @ A) @ Y4) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z2)))) => (((((ord_less_eq @ A) @ X) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y4)))) => (((((ord_less_eq @ A) @ Z2) @ Y4) => (~ ((((ord_less_eq @ A) @ Y4) @ X)))) => (((((ord_less_eq @ A) @ Y4) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y4))))))))))))))))).
% 2.31/1.57 thf(fact_55_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y4:A,X:A]: (((((ord_less_eq @ A) @ Y4) @ X) => ((((ord_less_eq @ A) @ X) @ Y4) = (X = Y4))))))))).
% 2.31/1.57 thf(fact_56_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) & (((ord_less_eq @ A) @ B5) @ A5))))))))).
% 2.31/1.57 thf(fact_57_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A4:A,B4:A,C2:A]: (((A4 = B4) => ((((ord_less_eq @ A) @ B4) @ C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 2.31/1.57 thf(fact_58_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A4:A,B4:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((B4 = C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 2.31/1.57 thf(fact_59_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ B4) @ A4) => (A4 = B4))))))))).
% 2.31/1.57 thf(fact_60_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y4:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ Z2) => (((ord_less_eq @ A) @ X) @ Z2))))))))).
% 2.31/1.57 thf(fact_61_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A]: ((((ord_less_eq @ A) @ A4) @ A4))))))).
% 2.31/1.57 thf(fact_62_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A4:A,B4:A]: (((! [A6:A,B6:A]: (((((ord_less_eq @ A) @ A6) @ B6) => ((P @ A6) @ B6)))) => ((! [A6:A,B6:A]: ((((P @ B6) @ A6) => ((P @ A6) @ B6)))) => ((P @ A4) @ B4))))))))).
% 2.31/1.57 thf(fact_63_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B4:A,A4:A,C2:A]: (((((ord_less_eq @ A) @ B4) @ A4) => ((((ord_less_eq @ A) @ C2) @ B4) => (((ord_less_eq @ A) @ C2) @ A4))))))))).
% 2.31/1.57 thf(fact_64_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [A5:A,B5:A]: (((((ord_less_eq @ A) @ B5) @ A5) & (((ord_less_eq @ A) @ A5) @ B5))))))))).
% 2.31/1.57 thf(fact_65_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B4:A,A4:A]: (((((ord_less_eq @ A) @ B4) @ A4) => ((((ord_less_eq @ A) @ A4) @ B4) => (A4 = B4))))))))).
% 2.31/1.57 thf(fact_66_Sup__eqI, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [A2:(set @ A),X:A]: (((! [Y2:A]: (((((member @ A) @ Y2) @ A2) => (((ord_less_eq @ A) @ Y2) @ X)))) => ((! [Y2:A]: (((! [Z3:A]: (((((member @ A) @ Z3) @ A2) => (((ord_less_eq @ A) @ Z3) @ Y2)))) => (((ord_less_eq @ A) @ X) @ Y2)))) => (((complete_Sup_Sup @ A) @ A2) = X))))))))).
% 2.31/1.57 thf(fact_67_Sup__mono, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [A2:(set @ A),B:(set @ A)]: (((! [A6:A]: (((((member @ A) @ A6) @ A2) => (? [X3:A]: (((((member @ A) @ X3) @ B) & (((ord_less_eq @ A) @ A6) @ X3))))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A2)) @ ((complete_Sup_Sup @ A) @ B))))))))).
% 2.31/1.57 thf(fact_68_Sup__least, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [A2:(set @ A),Z2:A]: (((! [X4:A]: (((((member @ A) @ X4) @ A2) => (((ord_less_eq @ A) @ X4) @ Z2)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A2)) @ Z2)))))))).
% 2.31/1.57 thf(fact_69_Sup__upper, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [X:A,A2:(set @ A)]: (((((member @ A) @ X) @ A2) => (((ord_less_eq @ A) @ X) @ ((complete_Sup_Sup @ A) @ A2))))))))).
% 2.31/1.57 thf(fact_70_Sup__le__iff, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [A2:(set @ A),B4:A]: (((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A2)) @ B4) = (! [X2:A]: (((((member @ A) @ X2) @ A2) => (((ord_less_eq @ A) @ X2) @ B4))))))))))).
% 2.31/1.57 thf(fact_71_Sup__upper2, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [U:A,A2:(set @ A),V:A]: (((((member @ A) @ U) @ A2) => ((((ord_less_eq @ A) @ V) @ U) => (((ord_less_eq @ A) @ V) @ ((complete_Sup_Sup @ A) @ A2)))))))))).
% 2.31/1.57 thf(fact_72_finite__on__def, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => ((lower_728871928ite_on @ A) = (^ [S:(set @ A),F2:(A > extended_ereal)]: ((! [X2:A]: (((((member @ A) @ X2) @ S) => (((F2 @ X2) != (extend1396239628finity @ extended_ereal)) & ((F2 @ X2) != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))))))))).
% 2.31/1.57 thf(fact_73_Sup__subset__mono, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [A2:(set @ A),B:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A2)) @ ((complete_Sup_Sup @ A) @ B))))))))).
% 2.31/1.57 thf(fact_74_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.31/1.57 thf(fact_75_Union__UNIV, axiom, (! [A:$tType]: ((((complete_Sup_Sup @ (set @ A)) @ (top_top @ (set @ (set @ A)))) = (top_top @ (set @ A)))))).
% 2.31/1.57 thf(fact_76_top_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_top @ A) => (! [A4:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A4) => (A4 = (top_top @ A))))))))).
% 2.31/1.57 thf(fact_77_top_Oextremum__unique, axiom, (! [A:$tType]: (((order_top @ A) => (! [A4:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A4) = (A4 = (top_top @ A))))))))).
% 2.31/1.57 thf(fact_78_top__greatest, axiom, (! [A:$tType]: (((order_top @ A) => (! [A4:A]: ((((ord_less_eq @ A) @ A4) @ (top_top @ A)))))))).
% 2.31/1.57 thf(fact_79_subset__UNIV, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ (top_top @ (set @ A)))))).
% 2.31/1.57 thf(fact_80_epigraph__invertible, axiom, (! [A:$tType,F:(A > extended_ereal),G:(A > extended_ereal)]: ((((((lower_261390618igraph @ A) @ (top_top @ (set @ A))) @ F) = (((lower_261390618igraph @ A) @ (top_top @ (set @ A))) @ G)) => (F = G))))).
% 2.31/1.57 thf(fact_81_neg__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B4:A,A4:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B4)) @ ((uminus_uminus @ A) @ A4)) = (((ord_less_eq @ A) @ A4) @ B4)))))))).
% 2.31/1.57 thf(fact_82_compl__le__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X)) @ ((uminus_uminus @ A) @ Y4)) = (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 2.31/1.57 thf(fact_83_top__finite__def, axiom, (! [A:$tType]: (((finite_lattice @ A) => ((top_top @ A) = ((complete_Sup_Sup @ A) @ (top_top @ (set @ A)))))))).
% 2.31/1.57 thf(fact_84_ereal__uminus__eq__iff, axiom, (! [A4:extended_ereal,B4:extended_ereal]: (((((uminus_uminus @ extended_ereal) @ A4) = ((uminus_uminus @ extended_ereal) @ B4)) = (A4 = B4))))).
% 2.31/1.57 thf(fact_85_ereal__uminus__uminus, axiom, (! [A4:extended_ereal]: ((((uminus_uminus @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ A4)) = A4)))).
% 2.31/1.57 thf(fact_86_verit__minus__simplify_I4_J, axiom, (! [B3:$tType]: (((group_add @ B3) => (! [B4:B3]: ((((uminus_uminus @ B3) @ ((uminus_uminus @ B3) @ B4)) = B4))))))).
% 2.31/1.57 thf(fact_87_uminus__apply, axiom, (! [B3:$tType,A:$tType]: (((uminus @ B3) => ((uminus_uminus @ (A > B3)) = (^ [A3:(A > B3),X2:A]: (((uminus_uminus @ B3) @ (A3 @ X2))))))))).
% 2.31/1.57 thf(fact_88_add_Oinverse__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ A4)) = A4))))))).
% 2.31/1.57 thf(fact_89_neg__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A,B4:A]: (((((uminus_uminus @ A) @ A4) = ((uminus_uminus @ A) @ B4)) = (A4 = B4)))))))).
% 2.31/1.57 thf(fact_90_double__compl, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ X)) = X))))))).
% 2.31/1.57 thf(fact_91_compl__eq__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X:A,Y4:A]: (((((uminus_uminus @ A) @ X) = ((uminus_uminus @ A) @ Y4)) = (X = Y4)))))))).
% 2.31/1.57 thf(fact_92_Compl__subset__Compl__iff, axiom, (! [A:$tType,A2:(set @ A),B:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A2)) @ ((uminus_uminus @ (set @ A)) @ B)) = (((ord_less_eq @ (set @ A)) @ B) @ A2))))).
% 2.31/1.57 thf(fact_93_Compl__anti__mono, axiom, (! [A:$tType,A2:(set @ A),B:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ B)) @ ((uminus_uminus @ (set @ A)) @ A2)))))).
% 2.31/1.57 thf(fact_94_ereal__infty__less__eq_I1_J, axiom, (! [X:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ (extend1396239628finity @ extended_ereal)) @ X) = (X = (extend1396239628finity @ extended_ereal)))))).
% 2.31/1.57 thf(fact_95_UN__ball__bex__simps_I3_J, axiom, (! [D:$tType,A2:(set @ (set @ D)),P:(D > $o)]: (((? [X2:D]: (((((member @ D) @ X2) @ ((complete_Sup_Sup @ (set @ D)) @ A2)) & (P @ X2)))) = (? [X2:(set @ D)]: (((((member @ (set @ D)) @ X2) @ A2) & (? [Y3:D]: (((((member @ D) @ Y3) @ X2) & (P @ Y3))))))))))).
% 2.31/1.57 thf(fact_96_UN__ball__bex__simps_I1_J, axiom, (! [A:$tType,A2:(set @ (set @ A)),P:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ ((complete_Sup_Sup @ (set @ A)) @ A2)) => (P @ X2)))) = (! [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ A2) => (! [Y3:A]: (((((member @ A) @ Y3) @ X2) => (P @ Y3))))))))))).
% 2.31/1.57 thf(fact_97_UnionI, axiom, (! [A:$tType,X6:(set @ A),C3:(set @ (set @ A)),A2:A]: (((((member @ (set @ A)) @ X6) @ C3) => ((((member @ A) @ A2) @ X6) => (((member @ A) @ A2) @ ((complete_Sup_Sup @ (set @ A)) @ C3))))))).
% 2.31/1.57 thf(fact_98_Union__iff, axiom, (! [A:$tType,A2:A,C3:(set @ (set @ A))]: (((((member @ A) @ A2) @ ((complete_Sup_Sup @ (set @ A)) @ C3)) = (? [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ C3) & (((member @ A) @ A2) @ X2)))))))).
% 2.31/1.57 thf(fact_99_ereal__minus__le__minus, axiom, (! [A4:extended_ereal,B4:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ A4)) @ ((uminus_uminus @ extended_ereal) @ B4)) = (((ord_less_eq @ extended_ereal) @ B4) @ A4))))).
% 2.31/1.57 thf(fact_100_top1I, axiom, (! [A:$tType,X:A]: (((top_top @ (A > $o)) @ X)))).
% 2.31/1.57 thf(fact_101_ereal__infty__less__eq_I2_J, axiom, (! [X:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ X) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) = (X = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))).
% 2.31/1.57 thf(fact_102_UnionE, axiom, (! [A:$tType,A2:A,C3:(set @ (set @ A))]: (((((member @ A) @ A2) @ ((complete_Sup_Sup @ (set @ A)) @ C3)) => (~ ((! [X5:(set @ A)]: (((((member @ A) @ A2) @ X5) => (~ ((((member @ (set @ A)) @ X5) @ C3)))))))))))).
% 2.31/1.57 thf(fact_103_neq__PInf__trans, axiom, (! [Y4:extended_ereal,X:extended_ereal]: (((Y4 != (extend1396239628finity @ extended_ereal)) => ((((ord_less_eq @ extended_ereal) @ X) @ Y4) => (X != (extend1396239628finity @ extended_ereal))))))).
% 2.31/1.57 thf(fact_104_ereal__infty__less__eq2_I1_J, axiom, (! [A4:extended_ereal,B4:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ A4) @ B4) => ((A4 = (extend1396239628finity @ extended_ereal)) => (B4 = (extend1396239628finity @ extended_ereal))))))).
% 2.31/1.57 thf(fact_105_ereal__less__eq_I1_J, axiom, (! [X:extended_ereal]: ((((ord_less_eq @ extended_ereal) @ X) @ (extend1396239628finity @ extended_ereal))))).
% 2.31/1.57 thf(fact_106_ereal__uminus__le__reorder, axiom, (! [A4:extended_ereal,B4:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ A4)) @ B4) = (((ord_less_eq @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ B4)) @ A4))))).
% 2.31/1.57 thf(fact_107_ereal__infty__less__eq2_I2_J, axiom, (! [A4:extended_ereal,B4:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ A4) @ B4) => ((B4 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (A4 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))).
% 2.31/1.57 thf(fact_108_ereal__less__eq_I2_J, axiom, (! [X:extended_ereal]: ((((ord_less_eq @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ X)))).
% 2.31/1.57 thf(fact_109_epigraph__subset__iff, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [F:(A > extended_ereal),G:(A > extended_ereal)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ real))) @ (((lower_261390618igraph @ A) @ (top_top @ (set @ A))) @ F)) @ (((lower_261390618igraph @ A) @ (top_top @ (set @ A))) @ G)) = (! [X2:A]: ((((ord_less_eq @ extended_ereal) @ (G @ X2)) @ (F @ X2))))))))))).
% 2.31/1.57 thf(fact_110_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A4:A,B4:A]: (((A4 = B4) | ((~ ((((ord_less_eq @ A) @ A4) @ B4))) | (~ ((((ord_less_eq @ A) @ B4) @ A4))))))))))).
% 2.31/1.57 thf(fact_111_minus__equation__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A,B4:A]: (((((uminus_uminus @ A) @ A4) = B4) = (((uminus_uminus @ A) @ B4) = A4)))))))).
% 2.31/1.57 thf(fact_112_equation__minus__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A,B4:A]: (((A4 = ((uminus_uminus @ A) @ B4)) = (B4 = ((uminus_uminus @ A) @ A4))))))))).
% 2.31/1.57 thf(fact_113_fun__Compl__def, axiom, (! [B3:$tType,A:$tType]: (((uminus @ B3) => ((uminus_uminus @ (A > B3)) = (^ [A3:(A > B3),X2:A]: (((uminus_uminus @ B3) @ (A3 @ X2))))))))).
% 2.31/1.57 thf(fact_114_verit__negate__coefficient_I3_J, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B4:A]: (((A4 = B4) => (((uminus_uminus @ A) @ A4) = ((uminus_uminus @ A) @ B4))))))))).
% 2.31/1.57 thf(fact_115_top__ereal__def, axiom, ((top_top @ extended_ereal) = (extend1396239628finity @ extended_ereal))).
% 2.31/1.57 thf(fact_116_Sup__eq__PInfty, axiom, (! [S2:(set @ extended_ereal)]: (((((member @ extended_ereal) @ (extend1396239628finity @ extended_ereal)) @ S2) => (((complete_Sup_Sup @ extended_ereal) @ S2) = (extend1396239628finity @ extended_ereal)))))).
% 2.31/1.57 thf(fact_117_ereal__uminus__eq__reorder, axiom, (! [A4:extended_ereal,B4:extended_ereal]: (((((uminus_uminus @ extended_ereal) @ A4) = B4) = (A4 = ((uminus_uminus @ extended_ereal) @ B4)))))).
% 2.31/1.57 thf(fact_118_le__imp__neg__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B4)) @ ((uminus_uminus @ A) @ A4))))))))).
% 2.31/1.57 thf(fact_119_minus__le__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A4)) @ B4) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B4)) @ A4)))))))).
% 2.31/1.57 thf(fact_120_le__minus__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ ((uminus_uminus @ A) @ B4)) = (((ord_less_eq @ A) @ B4) @ ((uminus_uminus @ A) @ A4))))))))).
% 2.31/1.57 thf(fact_121_compl__le__swap2, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y4:A,X:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y4)) @ X) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X)) @ Y4)))))))).
% 2.31/1.57 thf(fact_122_compl__le__swap1, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y4:A,X:A]: (((((ord_less_eq @ A) @ Y4) @ ((uminus_uminus @ A) @ X)) => (((ord_less_eq @ A) @ X) @ ((uminus_uminus @ A) @ Y4))))))))).
% 2.31/1.57 thf(fact_123_compl__mono, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y4)) @ ((uminus_uminus @ A) @ X))))))))).
% 2.31/1.57 thf(fact_124_MInfty__neq__PInfty_I1_J, axiom, ((extend1396239628finity @ extended_ereal) != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))).
% 2.31/1.57 thf(fact_125_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A))))))))).
% 2.31/1.57 thf(fact_126_cSup__eq__maximum, axiom, (! [A:$tType]: (((condit378418413attice @ A) => (! [Z2:A,X6:(set @ A)]: (((((member @ A) @ Z2) @ X6) => ((! [X4:A]: (((((member @ A) @ X4) @ X6) => (((ord_less_eq @ A) @ X4) @ Z2)))) => (((complete_Sup_Sup @ A) @ X6) = Z2))))))))).
% 2.31/1.57 thf(fact_127_cSup__eq, axiom, (! [A:$tType]: ((((condit378418413attice @ A) & (no_bot @ A)) => (! [X6:(set @ A),A4:A]: (((! [X4:A]: (((((member @ A) @ X4) @ X6) => (((ord_less_eq @ A) @ X4) @ A4)))) => ((! [Y2:A]: (((! [X3:A]: (((((member @ A) @ X3) @ X6) => (((ord_less_eq @ A) @ X3) @ Y2)))) => (((ord_less_eq @ A) @ A4) @ Y2)))) => (((complete_Sup_Sup @ A) @ X6) = A4))))))))).
% 2.31/1.57 thf(fact_128_top__conj_I2_J, axiom, (! [A:$tType,P:$o,X:A]: (((P & ((top_top @ (A > $o)) @ X)) = P)))).
% 2.31/1.57 thf(fact_129_top__conj_I1_J, axiom, (! [A:$tType,X:A,P:$o]: (((((top_top @ (A > $o)) @ X) & P) = P)))).
% 2.31/1.57 thf(fact_130_MInfty__eq__minfinity, axiom, (extended_MInfty = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))).
% 2.31/1.57 thf(fact_131_ComplI, axiom, (! [A:$tType,C2:A,A2:(set @ A)]: (((~ ((((member @ A) @ C2) @ A2))) => (((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A2)))))).
% 2.31/1.57 thf(fact_132_Compl__iff, axiom, (! [A:$tType,C2:A,A2:(set @ A)]: (((((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A2)) = (~ ((((member @ A) @ C2) @ A2))))))).
% 2.31/1.57 thf(fact_133_Compl__eq__Compl__iff, axiom, (! [A:$tType,A2:(set @ A),B:(set @ A)]: (((((uminus_uminus @ (set @ A)) @ A2) = ((uminus_uminus @ (set @ A)) @ B)) = (A2 = B))))).
% 2.31/1.57 thf(fact_134_ComplD, axiom, (! [A:$tType,C2:A,A2:(set @ A)]: (((((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A2)) => (~ ((((member @ A) @ C2) @ A2))))))).
% 2.31/1.57 thf(fact_135_double__complement, axiom, (! [A:$tType,A2:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A2)) = A2)))).
% 2.31/1.57 thf(fact_136_ereal__complete__Inf, axiom, (! [S2:(set @ extended_ereal)]: ((? [X4:extended_ereal]: (((! [Xa:extended_ereal]: (((((member @ extended_ereal) @ Xa) @ S2) => (((ord_less_eq @ extended_ereal) @ X4) @ Xa)))) & (! [Z3:extended_ereal]: (((! [Xa2:extended_ereal]: (((((member @ extended_ereal) @ Xa2) @ S2) => (((ord_less_eq @ extended_ereal) @ Z3) @ Xa2)))) => (((ord_less_eq @ extended_ereal) @ Z3) @ X4)))))))))).
% 2.31/1.57 thf(fact_137_ereal__complete__Sup, axiom, (! [S2:(set @ extended_ereal)]: ((? [X4:extended_ereal]: (((! [Xa:extended_ereal]: (((((member @ extended_ereal) @ Xa) @ S2) => (((ord_less_eq @ extended_ereal) @ Xa) @ X4)))) & (! [Z3:extended_ereal]: (((! [Xa2:extended_ereal]: (((((member @ extended_ereal) @ Xa2) @ S2) => (((ord_less_eq @ extended_ereal) @ Xa2) @ Z3)))) => (((ord_less_eq @ extended_ereal) @ X4) @ Z3)))))))))).
% 2.31/1.57 thf(fact_138_uminus__ereal_Osimps_I3_J, axiom, (((uminus_uminus @ extended_ereal) @ extended_MInfty) = extended_PInfty)).
% 2.31/1.57 thf(fact_139_uminus__ereal_Osimps_I2_J, axiom, (((uminus_uminus @ extended_ereal) @ extended_PInfty) = extended_MInfty)).
% 2.31/1.57 thf(fact_140_exists__diff, axiom, (! [A:$tType,P:((set @ A) > $o)]: (((? [S:(set @ A)]: ((P @ ((uminus_uminus @ (set @ A)) @ S)))) = (? [X7:(set @ A)]: ((P @ X7))))))).
% 2.31/1.57 thf(fact_141_GreatestI2__order, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X:A,Q:(A > $o)]: (((P @ X) => ((! [Y2:A]: (((P @ Y2) => (((ord_less_eq @ A) @ Y2) @ X)))) => ((! [X4:A]: (((P @ X4) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ A) @ Y5) @ X4)))) => (Q @ X4))))) => (Q @ ((order_Greatest @ A) @ P))))))))))).
% 2.31/1.57 thf(fact_142_infinity__ereal__def, axiom, ((extend1396239628finity @ extended_ereal) = extended_PInfty)).
% 2.31/1.57 thf(fact_143_ereal_Odistinct_I5_J, axiom, (extended_PInfty != extended_MInfty)).
% 2.31/1.57 thf(fact_144_Greatest__equality, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X:A]: (((P @ X) => ((! [Y2:A]: (((P @ Y2) => (((ord_less_eq @ A) @ Y2) @ X)))) => (((order_Greatest @ A) @ P) = X))))))))).
% 2.31/1.57 thf(fact_145_uminus__ereal_Oelims, axiom, (! [X:extended_ereal,Y4:extended_ereal]: (((((uminus_uminus @ extended_ereal) @ X) = Y4) => ((! [R:real]: (((X = (extended_ereal2 @ R)) => (Y4 != (extended_ereal2 @ ((uminus_uminus @ real) @ R)))))) => (((X = extended_PInfty) => (Y4 != extended_MInfty)) => (~ (((X = extended_MInfty) => (Y4 != extended_PInfty)))))))))).
% 2.31/1.57 thf(fact_146_not__MInfty__nonneg, axiom, (! [X:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ X) => (X != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))).
% 2.31/1.57 thf(fact_147_ereal_Osimps_I12_J, axiom, (! [A:$tType,F1:(real > A),F22:A,F3:A]: (((((((extended_rec_ereal @ A) @ F1) @ F22) @ F3) @ extended_PInfty) = F22)))).
% 2.31/1.57 thf(fact_148_ereal_Osimps_I13_J, axiom, (! [A:$tType,F1:(real > A),F22:A,F3:A]: (((((((extended_rec_ereal @ A) @ F1) @ F22) @ F3) @ extended_MInfty) = F3)))).
% 2.31/1.57 thf(fact_149_ereal_Oinject, axiom, (! [X1:real,Y1:real]: ((((extended_ereal2 @ X1) = (extended_ereal2 @ Y1)) = (X1 = Y1))))).
% 2.31/1.57 thf(fact_150_ereal__cong, axiom, (! [X:real,Y4:real]: (((X = Y4) => ((extended_ereal2 @ X) = (extended_ereal2 @ Y4)))))).
% 2.31/1.57 thf(fact_151_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.31/1.57 thf(fact_152_neg__equal__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A4:A]: (((((uminus_uminus @ A) @ A4) = A4) = (A4 = (zero_zero @ A))))))))).
% 2.31/1.57 thf(fact_153_equal__neg__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A4:A]: (((A4 = ((uminus_uminus @ A) @ A4)) = (A4 = (zero_zero @ A))))))))).
% 2.31/1.57 thf(fact_154_neg__equal__0__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A]: (((((uminus_uminus @ A) @ A4) = (zero_zero @ A)) = (A4 = (zero_zero @ A))))))))).
% 2.31/1.57 thf(fact_155_neg__0__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A]: ((((zero_zero @ A) = ((uminus_uminus @ A) @ A4)) = ((zero_zero @ A) = A4)))))))).
% 2.31/1.57 thf(fact_156_add_Oinverse__neutral, axiom, (! [A:$tType]: (((group_add @ A) => (((uminus_uminus @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.31/1.57 thf(fact_157_ereal__less__eq_I3_J, axiom, (! [R2:real,P2:real]: (((((ord_less_eq @ extended_ereal) @ (extended_ereal2 @ R2)) @ (extended_ereal2 @ P2)) = (((ord_less_eq @ real) @ R2) @ P2))))).
% 2.31/1.57 thf(fact_158_ereal__uminus__zero, axiom, (((uminus_uminus @ extended_ereal) @ (zero_zero @ extended_ereal)) = (zero_zero @ extended_ereal))).
% 2.31/1.57 thf(fact_159_ereal__uminus__zero__iff, axiom, (! [A4:extended_ereal]: (((((uminus_uminus @ extended_ereal) @ A4) = (zero_zero @ extended_ereal)) = (A4 = (zero_zero @ extended_ereal)))))).
% 2.31/1.57 thf(fact_160_neg__0__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ A4)) = (((ord_less_eq @ A) @ A4) @ (zero_zero @ A))))))))).
% 2.31/1.57 thf(fact_161_neg__le__0__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A4)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A4)))))))).
% 2.31/1.57 thf(fact_162_less__eq__neg__nonpos, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A4:A]: (((((ord_less_eq @ A) @ A4) @ ((uminus_uminus @ A) @ A4)) = (((ord_less_eq @ A) @ A4) @ (zero_zero @ A))))))))).
% 2.31/1.57 thf(fact_163_neg__less__eq__nonneg, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A4:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A4)) @ A4) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A4)))))))).
% 2.31/1.57 thf(fact_164_ereal__uminus__le__0__iff, axiom, (! [A4:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ A4)) @ (zero_zero @ extended_ereal)) = (((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ A4))))).
% 2.31/1.57 thf(fact_165_ereal__0__le__uminus__iff, axiom, (! [A4:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ ((uminus_uminus @ extended_ereal) @ A4)) = (((ord_less_eq @ extended_ereal) @ A4) @ (zero_zero @ extended_ereal)))))).
% 2.31/1.57 thf(fact_166_ereal_Odistinct_I1_J, axiom, (! [X1:real]: (((extended_ereal2 @ X1) != extended_PInfty)))).
% 2.31/1.57 thf(fact_167_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 2.31/1.57 thf(fact_168_ereal_Osimps_I11_J, axiom, (! [A:$tType,F1:(real > A),F22:A,F3:A,X1:real]: (((((((extended_rec_ereal @ A) @ F1) @ F22) @ F3) @ (extended_ereal2 @ X1)) = (F1 @ X1))))).
% 2.31/1.57 thf(fact_169_Infty__neq__0_I1_J, axiom, ((extend1396239628finity @ extended_ereal) != (zero_zero @ extended_ereal))).
% 2.31/1.57 thf(fact_170_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 2.31/1.57 thf(fact_171_uminus__ereal_Osimps_I1_J, axiom, (! [R2:real]: ((((uminus_uminus @ extended_ereal) @ (extended_ereal2 @ R2)) = (extended_ereal2 @ ((uminus_uminus @ real) @ R2)))))).
% 2.31/1.57 thf(fact_172_PInfty__neq__ereal_I1_J, axiom, (! [R2:real]: (((extended_ereal2 @ R2) != (extend1396239628finity @ extended_ereal))))).
% 2.31/1.57 thf(fact_173_ereal_Odistinct_I3_J, axiom, (! [X1:real]: (((extended_ereal2 @ X1) != extended_MInfty)))).
% 2.31/1.57 thf(fact_174_ereal__le__real, axiom, (! [X:extended_ereal,Y4:extended_ereal]: (((! [Z4:real]: (((((ord_less_eq @ extended_ereal) @ X) @ (extended_ereal2 @ Z4)) => (((ord_less_eq @ extended_ereal) @ Y4) @ (extended_ereal2 @ Z4))))) => (((ord_less_eq @ extended_ereal) @ Y4) @ X))))).
% 2.31/1.57 thf(fact_175_le__ereal__le, axiom, (! [A4:extended_ereal,X:real,Y4:real]: (((((ord_less_eq @ extended_ereal) @ A4) @ (extended_ereal2 @ X)) => ((((ord_less_eq @ real) @ X) @ Y4) => (((ord_less_eq @ extended_ereal) @ A4) @ (extended_ereal2 @ Y4))))))).
% 2.31/1.57 thf(fact_176_ereal__le__le, axiom, (! [Y4:real,A4:extended_ereal,X:real]: (((((ord_less_eq @ extended_ereal) @ (extended_ereal2 @ Y4)) @ A4) => ((((ord_less_eq @ real) @ X) @ Y4) => (((ord_less_eq @ extended_ereal) @ (extended_ereal2 @ X)) @ A4)))))).
% 2.31/1.57 thf(fact_177_MInfty__neq__ereal_I1_J, axiom, (! [R2:real]: (((extended_ereal2 @ R2) != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))).
% 2.31/1.57 thf(fact_178_ereal__cases, axiom, (! [X:extended_ereal]: (((! [R:real]: ((X != (extended_ereal2 @ R)))) => ((X != (extend1396239628finity @ extended_ereal)) => (X = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))).
% 2.31/1.57 thf(fact_179_ereal2__cases, axiom, (! [X:extended_ereal,Xa3:extended_ereal]: ((((? [R:real]: ((X = (extended_ereal2 @ R)))) => (! [Ra:real]: ((Xa3 != (extended_ereal2 @ Ra))))) => (((? [R:real]: ((X = (extended_ereal2 @ R)))) => (Xa3 != (extend1396239628finity @ extended_ereal))) => (((? [R:real]: ((X = (extended_ereal2 @ R)))) => (Xa3 != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))) => (((X = (extend1396239628finity @ extended_ereal)) => (! [R:real]: ((Xa3 != (extended_ereal2 @ R))))) => (((X = (extend1396239628finity @ extended_ereal)) => (Xa3 != (extend1396239628finity @ extended_ereal))) => (((X = (extend1396239628finity @ extended_ereal)) => (Xa3 != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))) => (((X = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (! [R:real]: ((Xa3 != (extended_ereal2 @ R))))) => (((X = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xa3 != (extend1396239628finity @ extended_ereal))) => (~ (((X = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xa3 != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))))))))))))).
% 2.31/1.57 thf(fact_180_ereal3__cases, axiom, (! [X:extended_ereal,Xa3:extended_ereal,Xb:extended_ereal]: ((((? [R:real]: ((X = (extended_ereal2 @ R)))) => ((? [Ra:real]: ((Xa3 = (extended_ereal2 @ Ra)))) => (! [Rb:real]: ((Xb != (extended_ereal2 @ Rb)))))) => (((? [R:real]: ((X = (extended_ereal2 @ R)))) => ((? [Ra:real]: ((Xa3 = (extended_ereal2 @ Ra)))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((? [R:real]: ((X = (extended_ereal2 @ R)))) => ((? [Ra:real]: ((Xa3 = (extended_ereal2 @ Ra)))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((? [R:real]: ((X = (extended_ereal2 @ R)))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (! [Ra:real]: ((Xb != (extended_ereal2 @ Ra)))))) => (((? [R:real]: ((X = (extended_ereal2 @ R)))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((? [R:real]: ((X = (extended_ereal2 @ R)))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((? [R:real]: ((X = (extended_ereal2 @ R)))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (! [Ra:real]: ((Xb != (extended_ereal2 @ Ra)))))) => (((? [R:real]: ((X = (extended_ereal2 @ R)))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((? [R:real]: ((X = (extended_ereal2 @ R)))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X = (extend1396239628finity @ extended_ereal)) => ((? [R:real]: ((Xa3 = (extended_ereal2 @ R)))) => (! [Ra:real]: ((Xb != (extended_ereal2 @ Ra)))))) => (((X = (extend1396239628finity @ extended_ereal)) => ((? [R:real]: ((Xa3 = (extended_ereal2 @ R)))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((X = (extend1396239628finity @ extended_ereal)) => ((? [R:real]: ((Xa3 = (extended_ereal2 @ R)))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X = (extend1396239628finity @ extended_ereal)) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (! [R:real]: ((Xb != (extended_ereal2 @ R)))))) => (((X = (extend1396239628finity @ extended_ereal)) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((X = (extend1396239628finity @ extended_ereal)) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X = (extend1396239628finity @ extended_ereal)) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (! [R:real]: ((Xb != (extended_ereal2 @ R)))))) => (((X = (extend1396239628finity @ extended_ereal)) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((X = (extend1396239628finity @ extended_ereal)) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((? [R:real]: ((Xa3 = (extended_ereal2 @ R)))) => (! [Ra:real]: ((Xb != (extended_ereal2 @ Ra)))))) => (((X = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((? [R:real]: ((Xa3 = (extended_ereal2 @ R)))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((X = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((? [R:real]: ((Xa3 = (extended_ereal2 @ R)))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (! [R:real]: ((Xb != (extended_ereal2 @ R)))))) => (((X = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((X = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (! [R:real]: ((Xb != (extended_ereal2 @ R)))))) => (((X = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (~ (((X = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))))))))))))))))))))))))))))))).
% 2.31/1.57 thf(fact_181_ereal__ex__split, axiom, ((^ [P3:(extended_ereal > $o)]: ((? [X7:extended_ereal]: ((P3 @ X7))))) = (^ [P4:(extended_ereal > $o)]: (((P4 @ (extend1396239628finity @ extended_ereal)) | ((? [X2:real]: ((P4 @ (extended_ereal2 @ X2)))) | (P4 @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))))).
% 2.31/1.57 thf(fact_182_abs__ereal_Ocases, axiom, (! [X:extended_ereal]: (((! [R:real]: ((X != (extended_ereal2 @ R)))) => ((X != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (X = (extend1396239628finity @ extended_ereal))))))).
% 2.31/1.57 thf(fact_183_ereal__all__split, axiom, ((^ [P3:(extended_ereal > $o)]: ((! [X7:extended_ereal]: ((P3 @ X7))))) = (^ [P4:(extended_ereal > $o)]: (((P4 @ (extend1396239628finity @ extended_ereal)) & ((! [X2:real]: ((P4 @ (extended_ereal2 @ X2)))) & (P4 @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))))).
% 2.31/1.57 thf(fact_184_abs__ereal_Oinduct, axiom, (! [P:(extended_ereal > $o),A0:extended_ereal]: (((! [R:real]: ((P @ (extended_ereal2 @ R)))) => ((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((P @ (extend1396239628finity @ extended_ereal)) => (P @ A0))))))).
% 2.31/1.57 thf(fact_185_less__ereal_Oinduct, axiom, (! [P:(extended_ereal > (extended_ereal > $o)),A0:extended_ereal,A1:extended_ereal]: (((! [X4:real,Y2:real]: (((P @ (extended_ereal2 @ X4)) @ (extended_ereal2 @ Y2)))) => ((! [X_1:extended_ereal]: (((P @ (extend1396239628finity @ extended_ereal)) @ X_1))) => ((! [A6:extended_ereal]: (((P @ A6) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => ((! [X4:real]: (((P @ (extended_ereal2 @ X4)) @ (extend1396239628finity @ extended_ereal)))) => ((! [R:real]: (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (extended_ereal2 @ R)))) => (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (extend1396239628finity @ extended_ereal)) => ((P @ A0) @ A1)))))))))).
% 2.31/1.57 thf(fact_186_plus__ereal_Oinduct, axiom, (! [P:(extended_ereal > (extended_ereal > $o)),A0:extended_ereal,A1:extended_ereal]: (((! [R:real,P5:real]: (((P @ (extended_ereal2 @ R)) @ (extended_ereal2 @ P5)))) => ((! [X_1:extended_ereal]: (((P @ (extend1396239628finity @ extended_ereal)) @ X_1))) => ((! [A6:extended_ereal]: (((P @ A6) @ (extend1396239628finity @ extended_ereal)))) => ((! [R:real]: (((P @ (extended_ereal2 @ R)) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => ((! [P5:real]: (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (extended_ereal2 @ P5)))) => (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((P @ A0) @ A1)))))))))).
% 2.31/1.57 thf(fact_187_times__ereal_Oinduct, axiom, (! [P:(extended_ereal > (extended_ereal > $o)),A0:extended_ereal,A1:extended_ereal]: (((! [R:real,P5:real]: (((P @ (extended_ereal2 @ R)) @ (extended_ereal2 @ P5)))) => ((! [R:real]: (((P @ (extended_ereal2 @ R)) @ (extend1396239628finity @ extended_ereal)))) => ((! [R:real]: (((P @ (extend1396239628finity @ extended_ereal)) @ (extended_ereal2 @ R)))) => ((! [R:real]: (((P @ (extended_ereal2 @ R)) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => ((! [R:real]: (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (extended_ereal2 @ R)))) => (((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))))))))))))).
% 2.31/1.57 thf(fact_188_real__of__ereal_Ocases, axiom, (! [X:extended_ereal]: (((! [R:real]: ((X != (extended_ereal2 @ R)))) => ((X != (extend1396239628finity @ extended_ereal)) => (X = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))).
% 2.31/1.57 thf(fact_189_real__of__ereal_Oinduct, axiom, (! [P:(extended_ereal > $o),A0:extended_ereal]: (((! [R:real]: ((P @ (extended_ereal2 @ R)))) => ((P @ (extend1396239628finity @ extended_ereal)) => ((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (P @ A0))))))).
% 2.31/1.57 thf(fact_190_domain__Epigraph__aux, axiom, (! [X:extended_ereal]: (((X != (extend1396239628finity @ extended_ereal)) => (? [R:real]: ((((ord_less_eq @ extended_ereal) @ X) @ (extended_ereal2 @ R)))))))).
% 2.31/1.57 thf(fact_191_ereal__top, axiom, (! [X:extended_ereal]: (((! [B7:real]: ((((ord_less_eq @ extended_ereal) @ (extended_ereal2 @ B7)) @ X))) => (X = (extend1396239628finity @ extended_ereal)))))).
% 2.31/1.57 thf(fact_192_Infty__neq__0_I3_J, axiom, (((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)) != (zero_zero @ extended_ereal))).
% 2.31/1.57 thf(fact_193_ereal_Oinduct, axiom, (! [P:(extended_ereal > $o),Ereal:extended_ereal]: (((! [X4:real]: ((P @ (extended_ereal2 @ X4)))) => ((P @ extended_PInfty) => ((P @ extended_MInfty) => (P @ Ereal))))))).
% 2.31/1.57 thf(fact_194_ereal_Oexhaust, axiom, (! [Y4:extended_ereal]: (((! [X12:real]: ((Y4 != (extended_ereal2 @ X12)))) => ((Y4 != extended_PInfty) => (Y4 = extended_MInfty)))))).
% 2.31/1.57 thf(fact_195_uminus__ereal_Ocases, axiom, (! [X:extended_ereal]: (((! [R:real]: ((X != (extended_ereal2 @ R)))) => ((X != extended_PInfty) => (X = extended_MInfty)))))).
% 2.31/1.57 thf(fact_196_uminus__ereal_Oinduct, axiom, (! [P:(extended_ereal > $o),A0:extended_ereal]: (((! [R:real]: ((P @ (extended_ereal2 @ R)))) => ((P @ extended_PInfty) => ((P @ extended_MInfty) => (P @ A0))))))).
% 2.31/1.57 thf(fact_197_ereal__bot, axiom, (! [X:extended_ereal]: (((! [B7:real]: ((((ord_less_eq @ extended_ereal) @ X) @ (extended_ereal2 @ B7)))) => (X = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))).
% 2.31/1.57 thf(fact_198_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.31/1.57 thf(fact_199_ereal__less__eq_I4_J, axiom, (! [R2:real]: (((((ord_less_eq @ extended_ereal) @ (extended_ereal2 @ R2)) @ (zero_zero @ extended_ereal)) = (((ord_less_eq @ real) @ R2) @ (zero_zero @ real)))))).
% 2.31/1.57 thf(fact_200_ereal__less__eq_I5_J, axiom, (! [R2:real]: (((((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ (extended_ereal2 @ R2)) = (((ord_less_eq @ real) @ (zero_zero @ real)) @ R2))))).
% 2.31/1.57 thf(fact_201_ereal__eq__0_I2_J, axiom, (! [R2:real]: ((((zero_zero @ extended_ereal) = (extended_ereal2 @ R2)) = (R2 = (zero_zero @ real)))))).
% 2.31/1.57 thf(fact_202_ereal__eq__0_I1_J, axiom, (! [R2:real]: ((((extended_ereal2 @ R2) = (zero_zero @ extended_ereal)) = (R2 = (zero_zero @ real)))))).
% 2.31/1.57 thf(fact_203_zero__ereal__def, axiom, ((zero_zero @ extended_ereal) = (extended_ereal2 @ (zero_zero @ real)))).
% 2.31/1.57 thf(fact_204_real__eq__0__iff__le__ge__0, axiom, (! [X:real]: (((X = (zero_zero @ real)) = ((((ord_less_eq @ real) @ (zero_zero @ real)) @ X) & (((ord_less_eq @ real) @ (zero_zero @ real)) @ ((uminus_uminus @ real) @ X))))))).
% 2.31/1.57 thf(fact_205_ereal__divide__ereal, axiom, (! [R2:real]: ((((((ord_less_eq @ real) @ (zero_zero @ real)) @ R2) => ((((divide_divide @ extended_ereal) @ (extend1396239628finity @ extended_ereal)) @ (extended_ereal2 @ R2)) = (extend1396239628finity @ extended_ereal))) & ((~ ((((ord_less_eq @ real) @ (zero_zero @ real)) @ R2))) => ((((divide_divide @ extended_ereal) @ (extend1396239628finity @ extended_ereal)) @ (extended_ereal2 @ R2)) = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))).
% 2.31/1.57 thf(fact_206_ereal__uminus__divide, axiom, (! [X:extended_ereal,Y4:extended_ereal]: (((((divide_divide @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ X)) @ Y4) = ((uminus_uminus @ extended_ereal) @ (((divide_divide @ extended_ereal) @ X) @ Y4)))))).
% 2.31/1.57 thf(fact_207_ereal__divide__zero__left, axiom, (! [A4:extended_ereal]: (((((divide_divide @ extended_ereal) @ (zero_zero @ extended_ereal)) @ A4) = (zero_zero @ extended_ereal))))).
% 2.31/1.57 thf(fact_208_ereal__divide__Infty_I1_J, axiom, (! [X:extended_ereal]: (((((divide_divide @ extended_ereal) @ X) @ (extend1396239628finity @ extended_ereal)) = (zero_zero @ extended_ereal))))).
% 2.31/1.57 thf(fact_209_ereal__divide__Infty_I2_J, axiom, (! [X:extended_ereal]: (((((divide_divide @ extended_ereal) @ X) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) = (zero_zero @ extended_ereal))))).
% 2.31/1.57 thf(fact_210_zero__le__divide__ereal, axiom, (! [A4:extended_ereal,B4:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ A4) => ((((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ B4) => (((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ (((divide_divide @ extended_ereal) @ A4) @ B4))))))).
% 2.31/1.57 thf(fact_211_div__minus__minus, axiom, (! [A:$tType]: (((euclid24285859cancel @ A) => (! [A4:A,B4:A]: (((((divide_divide @ A) @ ((uminus_uminus @ A) @ A4)) @ ((uminus_uminus @ A) @ B4)) = (((divide_divide @ A) @ A4) @ B4)))))))).
% 2.31/1.57 thf(fact_212_div__0, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A4:A]: (((((divide_divide @ A) @ (zero_zero @ A)) @ A4) = (zero_zero @ A)))))))).
% 2.31/1.57 thf(fact_213_div__by__0, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A4:A]: (((((divide_divide @ A) @ A4) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.31/1.57 thf(fact_214_div__minus__right, axiom, (! [A:$tType]: (((euclid24285859cancel @ A) => (! [A4:A,B4:A]: (((((divide_divide @ A) @ A4) @ ((uminus_uminus @ A) @ B4)) = (((divide_divide @ A) @ ((uminus_uminus @ A) @ A4)) @ B4)))))))).
% 2.31/1.57 thf(fact_215_division__ring__divide__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A4:A]: (((((divide_divide @ A) @ A4) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.31/1.57 thf(fact_216_bits__div__by__0, axiom, (! [A:$tType]: (((semiring_bits @ A) => (! [A4:A]: (((((divide_divide @ A) @ A4) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.31/1.57 thf(fact_217_divide__eq__0__iff, axiom, (! [A:$tType]: (((field @ A) => (! [A4:A,B4:A]: ((((((divide_divide @ A) @ A4) @ B4) = (zero_zero @ A)) = ((A4 = (zero_zero @ A)) | (B4 = (zero_zero @ A)))))))))).
% 2.31/1.57 thf(fact_218_divide__cancel__left, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A4:A,B4:A]: ((((((divide_divide @ A) @ C2) @ A4) = (((divide_divide @ A) @ C2) @ B4)) = ((C2 = (zero_zero @ A)) | (A4 = B4))))))))).
% 2.31/1.57 thf(fact_219_bits__div__0, axiom, (! [A:$tType]: (((semiring_bits @ A) => (! [A4:A]: (((((divide_divide @ A) @ (zero_zero @ A)) @ A4) = (zero_zero @ A)))))))).
% 2.31/1.57 thf(fact_220_divide__cancel__right, axiom, (! [A:$tType]: (((field @ A) => (! [A4:A,C2:A,B4:A]: ((((((divide_divide @ A) @ A4) @ C2) = (((divide_divide @ A) @ B4) @ C2)) = ((C2 = (zero_zero @ A)) | (A4 = B4))))))))).
% 2.31/1.57 thf(fact_221_minus__divide__left, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A4:A,B4:A]: ((((uminus_uminus @ A) @ (((divide_divide @ A) @ A4) @ B4)) = (((divide_divide @ A) @ ((uminus_uminus @ A) @ A4)) @ B4)))))))).
% 2.31/1.57 thf(fact_222_minus__divide__divide, axiom, (! [A:$tType]: (((field @ A) => (! [A4:A,B4:A]: (((((divide_divide @ A) @ ((uminus_uminus @ A) @ A4)) @ ((uminus_uminus @ A) @ B4)) = (((divide_divide @ A) @ A4) @ B4)))))))).
% 2.31/1.57 thf(fact_223_minus__divide__right, axiom, (! [A:$tType]: (((field @ A) => (! [A4:A,B4:A]: ((((uminus_uminus @ A) @ (((divide_divide @ A) @ A4) @ B4)) = (((divide_divide @ A) @ A4) @ ((uminus_uminus @ A) @ B4))))))))).
% 2.31/1.57 thf(fact_224_divide__right__mono__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A4:A,B4:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ B4) @ C2)) @ (((divide_divide @ A) @ A4) @ C2)))))))))).
% 2.31/1.57 thf(fact_225_divide__nonpos__nonpos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ Y4) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y4)))))))))).
% 2.31/1.57 thf(fact_226_divide__nonpos__nonneg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y4) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y4)) @ (zero_zero @ A)))))))))).
% 2.31/1.57 thf(fact_227_divide__nonneg__nonpos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ Y4) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y4)) @ (zero_zero @ A)))))))))).
% 2.31/1.57 thf(fact_228_divide__nonneg__nonneg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y4) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y4)))))))))).
% 2.31/1.57 thf(fact_229_zero__le__divide__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A4) @ B4)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A4) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B4)) | ((((ord_less_eq @ A) @ A4) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B4) @ (zero_zero @ A))))))))))).
% 2.31/1.57 thf(fact_230_divide__right__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A4:A,B4:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ A4) @ C2)) @ (((divide_divide @ A) @ B4) @ C2)))))))))).
% 2.31/1.57 thf(fact_231_divide__le__0__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ A4) @ B4)) @ (zero_zero @ A)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A4) & (((ord_less_eq @ A) @ B4) @ (zero_zero @ A))) | ((((ord_less_eq @ A) @ A4) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B4)))))))))).
% 2.31/1.57 thf(fact_232_nonzero__minus__divide__divide, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B4:A,A4:A]: (((B4 != (zero_zero @ A)) => ((((divide_divide @ A) @ ((uminus_uminus @ A) @ A4)) @ ((uminus_uminus @ A) @ B4)) = (((divide_divide @ A) @ A4) @ B4))))))))).
% 2.31/1.57 thf(fact_233_nonzero__minus__divide__right, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B4:A,A4:A]: (((B4 != (zero_zero @ A)) => (((uminus_uminus @ A) @ (((divide_divide @ A) @ A4) @ B4)) = (((divide_divide @ A) @ A4) @ ((uminus_uminus @ A) @ B4)))))))))).
% 2.31/1.57 thf(fact_234_real__of__ereal__le__0, axiom, (! [X:extended_ereal]: (((((ord_less_eq @ real) @ (extend1716541707_ereal @ X)) @ (zero_zero @ real)) = ((((ord_less_eq @ extended_ereal) @ X) @ (zero_zero @ extended_ereal)) | (X = (extend1396239628finity @ extended_ereal))))))).
% 2.31/1.57 thf(fact_235_ereal__divide, axiom, (! [P2:real,R2:real]: ((((P2 = (zero_zero @ real)) => ((((divide_divide @ extended_ereal) @ (extended_ereal2 @ R2)) @ (extended_ereal2 @ P2)) = (((times_times @ extended_ereal) @ (extended_ereal2 @ R2)) @ (extend1396239628finity @ extended_ereal)))) & ((P2 != (zero_zero @ real)) => ((((divide_divide @ extended_ereal) @ (extended_ereal2 @ R2)) @ (extended_ereal2 @ P2)) = (extended_ereal2 @ (((divide_divide @ real) @ R2) @ P2)))))))).
% 2.31/1.57 thf(fact_236_mult__zero__left, axiom, (! [A:$tType]: (((mult_zero @ A) => (! [A4:A]: (((((times_times @ A) @ (zero_zero @ A)) @ A4) = (zero_zero @ A)))))))).
% 2.31/1.57 thf(fact_237_mult__zero__right, axiom, (! [A:$tType]: (((mult_zero @ A) => (! [A4:A]: (((((times_times @ A) @ A4) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.31/1.57 thf(fact_238_mult__eq__0__iff, axiom, (! [A:$tType]: (((semiri1193490041visors @ A) => (! [A4:A,B4:A]: ((((((times_times @ A) @ A4) @ B4) = (zero_zero @ A)) = ((A4 = (zero_zero @ A)) | (B4 = (zero_zero @ A)))))))))).
% 2.31/1.57 thf(fact_239_mult__cancel__left, axiom, (! [A:$tType]: (((semiri1923998003cancel @ A) => (! [C2:A,A4:A,B4:A]: ((((((times_times @ A) @ C2) @ A4) = (((times_times @ A) @ C2) @ B4)) = ((C2 = (zero_zero @ A)) | (A4 = B4))))))))).
% 2.31/1.57 thf(fact_240_mult__cancel__right, axiom, (! [A:$tType]: (((semiri1923998003cancel @ A) => (! [A4:A,C2:A,B4:A]: ((((((times_times @ A) @ A4) @ C2) = (((times_times @ A) @ B4) @ C2)) = ((C2 = (zero_zero @ A)) | (A4 = B4))))))))).
% 2.31/1.57 thf(fact_241_mult__minus__right, axiom, (! [A:$tType]: (((ring @ A) => (! [A4:A,B4:A]: (((((times_times @ A) @ A4) @ ((uminus_uminus @ A) @ B4)) = ((uminus_uminus @ A) @ (((times_times @ A) @ A4) @ B4))))))))).
% 2.31/1.57 thf(fact_242_minus__mult__minus, axiom, (! [A:$tType]: (((ring @ A) => (! [A4:A,B4:A]: (((((times_times @ A) @ ((uminus_uminus @ A) @ A4)) @ ((uminus_uminus @ A) @ B4)) = (((times_times @ A) @ A4) @ B4)))))))).
% 2.31/1.57 thf(fact_243_mult__minus__left, axiom, (! [A:$tType]: (((ring @ A) => (! [A4:A,B4:A]: (((((times_times @ A) @ ((uminus_uminus @ A) @ A4)) @ B4) = ((uminus_uminus @ A) @ (((times_times @ A) @ A4) @ B4))))))))).
% 2.31/1.57 thf(fact_244_ereal__mult__minus__right, axiom, (! [A4:extended_ereal,B4:extended_ereal]: (((((times_times @ extended_ereal) @ A4) @ ((uminus_uminus @ extended_ereal) @ B4)) = ((uminus_uminus @ extended_ereal) @ (((times_times @ extended_ereal) @ A4) @ B4)))))).
% 2.31/1.57 thf(fact_245_ereal__mult__minus__left, axiom, (! [A4:extended_ereal,B4:extended_ereal]: (((((times_times @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ A4)) @ B4) = ((uminus_uminus @ extended_ereal) @ (((times_times @ extended_ereal) @ A4) @ B4)))))).
% 2.31/1.57 thf(fact_246_ereal__mult__zero, axiom, (! [A4:extended_ereal]: (((((times_times @ extended_ereal) @ A4) @ (zero_zero @ extended_ereal)) = (zero_zero @ extended_ereal))))).
% 2.31/1.57 thf(fact_247_ereal__zero__mult, axiom, (! [A4:extended_ereal]: (((((times_times @ extended_ereal) @ (zero_zero @ extended_ereal)) @ A4) = (zero_zero @ extended_ereal))))).
% 2.31/1.57 thf(fact_248_ereal__zero__times, axiom, (! [A4:extended_ereal,B4:extended_ereal]: ((((((times_times @ extended_ereal) @ A4) @ B4) = (zero_zero @ extended_ereal)) = ((A4 = (zero_zero @ extended_ereal)) | (B4 = (zero_zero @ extended_ereal))))))).
% 2.31/1.57 thf(fact_249_ereal__times__divide__eq__left, axiom, (! [B4:extended_ereal,C2:extended_ereal,A4:extended_ereal]: (((((times_times @ extended_ereal) @ (((divide_divide @ extended_ereal) @ B4) @ C2)) @ A4) = (((divide_divide @ extended_ereal) @ (((times_times @ extended_ereal) @ B4) @ A4)) @ C2))))).
% 2.31/1.57 thf(fact_250_mult__divide__mult__cancel__left__if, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A4:A,B4:A]: ((((C2 = (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A4)) @ (((times_times @ A) @ C2) @ B4)) = (zero_zero @ A))) & ((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A4)) @ (((times_times @ A) @ C2) @ B4)) = (((divide_divide @ A) @ A4) @ B4)))))))))).
% 2.31/1.57 thf(fact_251_nonzero__mult__divide__mult__cancel__left, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A4:A,B4:A]: (((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A4)) @ (((times_times @ A) @ C2) @ B4)) = (((divide_divide @ A) @ A4) @ B4))))))))).
% 2.31/1.57 thf(fact_252_nonzero__mult__divide__mult__cancel__left2, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A4:A,B4:A]: (((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A4)) @ (((times_times @ A) @ B4) @ C2)) = (((divide_divide @ A) @ A4) @ B4))))))))).
% 2.31/1.58 thf(fact_253_nonzero__mult__divide__mult__cancel__right, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A4:A,B4:A]: (((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ A4) @ C2)) @ (((times_times @ A) @ B4) @ C2)) = (((divide_divide @ A) @ A4) @ B4))))))))).
% 2.31/1.58 thf(fact_254_nonzero__mult__divide__mult__cancel__right2, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A4:A,B4:A]: (((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ A4) @ C2)) @ (((times_times @ A) @ C2) @ B4)) = (((divide_divide @ A) @ A4) @ B4))))))))).
% 2.31/1.58 thf(subcl_Euclidean__Space_Oeuclidean__space___HOL_Otype, axiom, (! [A:$tType]: (((euclid925273238_space @ A) => (type @ A))))).
% 2.31/1.58 thf(subcl_Euclidean__Space_Oeuclidean__space___Groups_Ozero, axiom, (! [A:$tType]: (((euclid925273238_space @ A) => (zero @ A))))).
% 2.31/1.58 thf(subcl_Euclidean__Space_Oeuclidean__space___Groups_Ouminus, axiom, (! [A:$tType]: (((euclid925273238_space @ A) => (uminus @ A))))).
% 2.31/1.58 thf(subcl_Euclidean__Space_Oeuclidean__space___Groups_Ogroup__add, axiom, (! [A:$tType]: (((euclid925273238_space @ A) => (group_add @ A))))).
% 2.31/1.58 thf(subcl_Euclidean__Space_Oeuclidean__space___Real__Vector__Spaces_Oreal__vector, axiom, (! [A:$tType]: (((euclid925273238_space @ A) => (real_V1076094709vector @ A))))).
% 2.31/1.58 thf(subcl_Euclidean__Space_Oeuclidean__space___Real__Vector__Spaces_Ometric__space, axiom, (! [A:$tType]: (((euclid925273238_space @ A) => (real_V2090557954_space @ A))))).
% 2.31/1.58 thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space, axiom, (! [A7:$tType,A8:$tType]: ((((ordere890947078_space @ A7) & (ordere890947078_space @ A8)) => (ordere890947078_space @ ((product_prod @ A7) @ A8)))))).
% 2.31/1.58 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_1, axiom, (ordere890947078_space @ real)).
% 2.31/1.58 thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice, axiom, (! [A7:$tType,A8:$tType]: (((comple187826305attice @ A8) => (condit378418413attice @ (A7 > A8)))))).
% 2.31/1.58 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A7:$tType,A8:$tType]: (((comple187826305attice @ A8) => (comple187826305attice @ (A7 > A8)))))).
% 2.31/1.58 thf(tcon_fun___Lattices_Oboolean__algebra, axiom, (! [A7:$tType,A8:$tType]: (((boolean_algebra @ A8) => (boolean_algebra @ (A7 > A8)))))).
% 2.31/1.58 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A7:$tType,A8:$tType]: (((order_top @ A8) => (order_top @ (A7 > A8)))))).
% 2.31/1.58 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: (((preorder @ A8) => (preorder @ (A7 > A8)))))).
% 2.31/1.58 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: (((order @ A8) => (order @ (A7 > A8)))))).
% 2.31/1.58 thf(tcon_fun___Orderings_Otop, axiom, (! [A7:$tType,A8:$tType]: (((top @ A8) => (top @ (A7 > A8)))))).
% 2.31/1.58 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: (((ord @ A8) => (ord @ (A7 > A8)))))).
% 2.31/1.58 thf(tcon_fun___Groups_Ouminus, axiom, (! [A7:$tType,A8:$tType]: (((uminus @ A8) => (uminus @ (A7 > A8)))))).
% 2.31/1.58 thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_2, axiom, (! [A7:$tType]: ((condit378418413attice @ (set @ A7))))).
% 2.31/1.58 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_3, axiom, (! [A7:$tType]: ((comple187826305attice @ (set @ A7))))).
% 2.31/1.58 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_4, axiom, (! [A7:$tType]: ((boolean_algebra @ (set @ A7))))).
% 2.31/1.58 thf(tcon_Set_Oset___Orderings_Oorder__top_5, axiom, (! [A7:$tType]: ((order_top @ (set @ A7))))).
% 2.31/1.58 thf(tcon_Set_Oset___Orderings_Opreorder_6, axiom, (! [A7:$tType]: ((preorder @ (set @ A7))))).
% 2.31/1.58 thf(tcon_Set_Oset___Orderings_Oorder_7, axiom, (! [A7:$tType]: ((order @ (set @ A7))))).
% 2.31/1.58 thf(tcon_Set_Oset___Orderings_Otop_8, axiom, (! [A7:$tType]: ((top @ (set @ A7))))).
% 2.31/1.58 thf(tcon_Set_Oset___Orderings_Oord_9, axiom, (! [A7:$tType]: ((ord @ (set @ A7))))).
% 2.31/1.58 thf(tcon_Set_Oset___Groups_Ouminus_10, axiom, (! [A7:$tType]: ((uminus @ (set @ A7))))).
% 2.31/1.58 thf(tcon_Set_Oset___Groups_Ozero, axiom, (! [A7:$tType]: (((zero @ A7) => (zero @ (set @ A7)))))).
% 2.31/1.58 thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11, axiom, (condit378418413attice @ $o)).
% 2.31/1.58 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_12, axiom, (comple187826305attice @ $o)).
% 2.31/1.58 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_13, axiom, (boolean_algebra @ $o)).
% 2.31/1.58 thf(tcon_HOL_Obool___Orderings_Oorder__top_14, axiom, (order_top @ $o)).
% 2.31/1.58 thf(tcon_HOL_Obool___Orderings_Opreorder_15, axiom, (preorder @ $o)).
% 2.31/1.58 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 2.31/1.58 thf(tcon_HOL_Obool___Orderings_Oorder_16, axiom, (order @ $o)).
% 2.31/1.58 thf(tcon_HOL_Obool___Orderings_Otop_17, axiom, (top @ $o)).
% 2.31/1.58 thf(tcon_HOL_Obool___Orderings_Oord_18, axiom, (ord @ $o)).
% 2.31/1.58 thf(tcon_HOL_Obool___Groups_Ouminus_19, axiom, (uminus @ $o)).
% 2.31/1.58 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_20, axiom, (condit378418413attice @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel, axiom, (semiri1923998003cancel @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors, axiom, (semiri1193490041visors @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space, axiom, (real_V2090557954_space @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space, axiom, (euclid925273238_space @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector, axiom, (real_V1076094709vector @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add, axiom, (linord219039673up_add @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add, axiom, (ordered_ab_group_add @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Fields_Olinordered__field, axiom, (linordered_field @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Rings_Osemidom__divide, axiom, (semidom_divide @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Fields_Odivision__ring, axiom, (division_ring @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Orderings_Opreorder_21, axiom, (preorder @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Orderings_Olinorder_22, axiom, (linorder @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Orderings_Ono__bot, axiom, (no_bot @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Groups_Ogroup__add, axiom, (group_add @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Rings_Omult__zero, axiom, (mult_zero @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Orderings_Oorder_23, axiom, (order @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Orderings_Oord_24, axiom, (ord @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Groups_Ouminus_25, axiom, (uminus @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Fields_Ofield, axiom, (field @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Groups_Ozero_26, axiom, (zero @ real)).
% 2.31/1.58 thf(tcon_Real_Oreal___Rings_Oring, axiom, (ring @ real)).
% 2.31/1.58 thf(tcon_Product__Type_Oprod___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_27, axiom, (! [A7:$tType,A8:$tType]: ((((condit378418413attice @ A7) & (condit378418413attice @ A8)) => (condit378418413attice @ ((product_prod @ A7) @ A8)))))).
% 2.31/1.58 thf(tcon_Product__Type_Oprod___Complete__Lattices_Ocomplete__lattice_28, axiom, (! [A7:$tType,A8:$tType]: ((((comple187826305attice @ A7) & (comple187826305attice @ A8)) => (comple187826305attice @ ((product_prod @ A7) @ A8)))))).
% 2.31/1.58 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Ometric__space_29, axiom, (! [A7:$tType,A8:$tType]: ((((real_V2090557954_space @ A7) & (real_V2090557954_space @ A8)) => (real_V2090557954_space @ ((product_prod @ A7) @ A8)))))).
% 2.31/1.58 thf(tcon_Product__Type_Oprod___Euclidean__Space_Oeuclidean__space_30, axiom, (! [A7:$tType,A8:$tType]: ((((euclid925273238_space @ A7) & (euclid925273238_space @ A8)) => (euclid925273238_space @ ((product_prod @ A7) @ A8)))))).
% 2.31/1.58 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__vector_31, axiom, (! [A7:$tType,A8:$tType]: ((((real_V1076094709vector @ A7) & (real_V1076094709vector @ A8)) => (real_V1076094709vector @ ((product_prod @ A7) @ A8)))))).
% 2.31/1.58 thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__group__add_32, axiom, (! [A7:$tType,A8:$tType]: ((((ordere890947078_space @ A7) & (ordere890947078_space @ A8)) => (ordered_ab_group_add @ ((product_prod @ A7) @ A8)))))).
% 2.31/1.58 thf(tcon_Product__Type_Oprod___Lattices_Oboolean__algebra_33, axiom, (! [A7:$tType,A8:$tType]: ((((boolean_algebra @ A7) & (boolean_algebra @ A8)) => (boolean_algebra @ ((product_prod @ A7) @ A8)))))).
% 2.31/1.58 thf(tcon_Product__Type_Oprod___Orderings_Oorder__top_34, axiom, (! [A7:$tType,A8:$tType]: ((((order_top @ A7) & (order_top @ A8)) => (order_top @ ((product_prod @ A7) @ A8)))))).
% 2.31/1.58 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_35, axiom, (! [A7:$tType,A8:$tType]: ((((preorder @ A7) & (preorder @ A8)) => (preorder @ ((product_prod @ A7) @ A8)))))).
% 2.31/1.58 thf(tcon_Product__Type_Oprod___Groups_Ogroup__add_36, axiom, (! [A7:$tType,A8:$tType]: ((((group_add @ A7) & (group_add @ A8)) => (group_add @ ((product_prod @ A7) @ A8)))))).
% 2.31/1.58 thf(tcon_Product__Type_Oprod___Orderings_Oorder_37, axiom, (! [A7:$tType,A8:$tType]: ((((order @ A7) & (order @ A8)) => (order @ ((product_prod @ A7) @ A8)))))).
% 2.31/1.58 thf(tcon_Product__Type_Oprod___Orderings_Otop_38, axiom, (! [A7:$tType,A8:$tType]: ((((top @ A7) & (top @ A8)) => (top @ ((product_prod @ A7) @ A8)))))).
% 2.31/1.58 thf(tcon_Product__Type_Oprod___Orderings_Oord_39, axiom, (! [A7:$tType,A8:$tType]: ((((ord @ A7) & (ord @ A8)) => (ord @ ((product_prod @ A7) @ A8)))))).
% 2.31/1.58 thf(tcon_Product__Type_Oprod___Groups_Ouminus_40, axiom, (! [A7:$tType,A8:$tType]: ((((uminus @ A7) & (uminus @ A8)) => (uminus @ ((product_prod @ A7) @ A8)))))).
% 2.31/1.58 thf(tcon_Product__Type_Oprod___Groups_Ozero_41, axiom, (! [A7:$tType,A8:$tType]: ((((zero @ A7) & (zero @ A8)) => (zero @ ((product_prod @ A7) @ A8)))))).
% 2.31/1.58 thf(tcon_Extended__Real_Oereal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_42, axiom, (condit378418413attice @ extended_ereal)).
% 2.31/1.58 thf(tcon_Extended__Real_Oereal___Complete__Lattices_Ocomplete__lattice_43, axiom, (comple187826305attice @ extended_ereal)).
% 2.31/1.58 thf(tcon_Extended__Real_Oereal___Orderings_Oorder__top_44, axiom, (order_top @ extended_ereal)).
% 2.31/1.58 thf(tcon_Extended__Real_Oereal___Orderings_Opreorder_45, axiom, (preorder @ extended_ereal)).
% 2.31/1.58 thf(tcon_Extended__Real_Oereal___Orderings_Olinorder_46, axiom, (linorder @ extended_ereal)).
% 2.31/1.58 thf(tcon_Extended__Real_Oereal___Orderings_Oorder_47, axiom, (order @ extended_ereal)).
% 2.31/1.58 thf(tcon_Extended__Real_Oereal___Orderings_Otop_48, axiom, (top @ extended_ereal)).
% 2.31/1.58 thf(tcon_Extended__Real_Oereal___Orderings_Oord_49, axiom, (ord @ extended_ereal)).
% 2.31/1.58 thf(tcon_Extended__Real_Oereal___Groups_Ouminus_50, axiom, (uminus @ extended_ereal)).
% 2.31/1.58 thf(tcon_Extended__Real_Oereal___Groups_Ozero_51, axiom, (zero @ extended_ereal)).
% 2.31/1.58 thf(tfree_0, hypothesis, (euclid925273238_space @ a)).
% 2.31/1.58 thf(conj_0, conjecture, ((((lower_673667120vex_on @ a) @ (top_top @ (set @ a))) @ f) = ((convex @ a) @ ((lower_272802190domain @ a) @ f)))).
% 2.31/1.58 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.0x1kQQmmmY/DTF2DTF23220.p
% 2.31/1.58 ------------------------
% 2.48/1.62 ---- Cleaned THF ---
% 2.48/1.62 thf(ty_t_Extended__Real_Oereal,type,
% 2.48/1.62 extended_ereal: $tType ).
% 2.48/1.62
% 2.48/1.62 thf(ty_t_Product__Type_Oprod,type,
% 2.48/1.62 product_prod: $tType > $tType > $tType ).
% 2.48/1.62
% 2.48/1.62 thf(ty_t_Real_Oreal,type,
% 2.48/1.62 real: $tType ).
% 2.48/1.62
% 2.48/1.62 thf(ty_t_Set_Oset,type,
% 2.48/1.62 set: $tType > $tType ).
% 2.48/1.62
% 2.48/1.62 thf(ty_tf_a,type,
% 2.48/1.62 a: $tType ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
% 2.48/1.62 ordere890947078_space:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_HOL_Otype,type,
% 2.48/1.62 type:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Rings_Oring,type,
% 2.48/1.62 ring:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Groups_Ozero,type,
% 2.48/1.62 zero:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Fields_Ofield,type,
% 2.48/1.62 field:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Groups_Ouminus,type,
% 2.48/1.62 uminus:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Orderings_Oord,type,
% 2.48/1.62 ord:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Orderings_Otop,type,
% 2.48/1.62 top:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Orderings_Oorder,type,
% 2.48/1.62 order:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Rings_Omult__zero,type,
% 2.48/1.62 mult_zero:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Groups_Ogroup__add,type,
% 2.48/1.62 group_add:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Orderings_Ono__bot,type,
% 2.48/1.62 no_bot:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Orderings_Olinorder,type,
% 2.48/1.62 linorder:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Orderings_Opreorder,type,
% 2.48/1.62 preorder:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Enum_Ofinite__lattice,type,
% 2.48/1.62 finite_lattice:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Orderings_Oorder__top,type,
% 2.48/1.62 order_top:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Fields_Odivision__ring,type,
% 2.48/1.62 division_ring:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Parity_Osemiring__bits,type,
% 2.48/1.62 semiring_bits:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Rings_Osemidom__divide,type,
% 2.48/1.62 semidom_divide:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Fields_Olinordered__field,type,
% 2.48/1.62 linordered_field:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 2.48/1.62 boolean_algebra:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.48/1.62 ordered_ab_group_add:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 2.48/1.62 linord219039673up_add:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
% 2.48/1.62 real_V1076094709vector:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Euclidean__Space_Oeuclidean__space,type,
% 2.48/1.62 euclid925273238_space:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
% 2.48/1.62 real_V2090557954_space:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 2.48/1.62 semiri1193490041visors:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.48/1.62 linord1659791738miring:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 2.48/1.62 comple187826305attice:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.48/1.62 canoni770627133id_add:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
% 2.48/1.62 semiri1923998003cancel:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
% 2.48/1.62 euclid24285859cancel:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
% 2.48/1.62 condit378418413attice:
% 2.48/1.62 !>[A: $tType] : $o ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
% 2.48/1.62 complete_Sup_Sup:
% 2.48/1.62 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Convex_Oconvex,type,
% 2.48/1.62 convex:
% 2.48/1.62 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
% 2.48/1.62 extend1396239628finity:
% 2.48/1.62 !>[A: $tType] : A ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Extended__Real_Oereal_OMInfty,type,
% 2.48/1.62 extended_MInfty: extended_ereal ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Extended__Real_Oereal_OPInfty,type,
% 2.48/1.62 extended_PInfty: extended_ereal ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Extended__Real_Oereal_Oereal,type,
% 2.48/1.62 extended_ereal2: real > extended_ereal ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Extended__Real_Oereal_Orec__ereal,type,
% 2.48/1.62 extended_rec_ereal:
% 2.48/1.62 !>[A: $tType] : ( ( real > A ) > A > A > extended_ereal > A ) ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Extended__Real_Oreal__of__ereal,type,
% 2.48/1.62 extend1716541707_ereal: extended_ereal > real ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 2.48/1.62 times_times:
% 2.48/1.62 !>[A: $tType] : ( A > A > A ) ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 2.48/1.62 uminus_uminus:
% 2.48/1.62 !>[A: $tType] : ( A > A ) ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.48/1.62 zero_zero:
% 2.48/1.62 !>[A: $tType] : A ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_OEpigraph,type,
% 2.48/1.62 lower_261390618igraph:
% 2.48/1.62 !>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > ( set @ ( product_prod @ A @ real ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Oaffine__on,type,
% 2.48/1.62 lower_500881736ine_on:
% 2.48/1.62 !>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > $o ) ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Oconcave__on,type,
% 2.48/1.62 lower_708069572ave_on:
% 2.48/1.62 !>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > $o ) ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Oconvex__on,type,
% 2.48/1.62 lower_673667120vex_on:
% 2.48/1.62 !>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > $o ) ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Odomain,type,
% 2.48/1.62 lower_272802190domain:
% 2.48/1.62 !>[A: $tType] : ( ( A > extended_ereal ) > ( set @ A ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Ofinite__on,type,
% 2.48/1.62 lower_728871928ite_on:
% 2.48/1.62 !>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > $o ) ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.48/1.62 ord_less_eq:
% 2.48/1.62 !>[A: $tType] : ( A > A > $o ) ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 2.48/1.62 order_Greatest:
% 2.48/1.62 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.48/1.62 top_top:
% 2.48/1.62 !>[A: $tType] : A ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 2.48/1.62 divide_divide:
% 2.48/1.62 !>[A: $tType] : ( A > A > A ) ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_Set_OCollect,type,
% 2.48/1.62 collect:
% 2.48/1.62 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(sy_c_member,type,
% 2.48/1.62 member:
% 2.48/1.62 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.48/1.62
% 2.48/1.62 thf(sy_v_f,type,
% 2.48/1.62 f: a > extended_ereal ).
% 2.48/1.62
% 2.48/1.62 thf(fact_0__092_060open_062convex_A_Idomain_Af_J_A_092_060Longrightarrow_062_ALower__Semicontinuous__Mirabelle__quczrylfpw_Oconvex__on_AUNIV_Af_092_060close_062,axiom,
% 2.48/1.62 ( ( convex @ a @ ( lower_272802190domain @ a @ f ) )
% 2.48/1.62 => ( lower_673667120vex_on @ a @ ( top_top @ ( set @ a ) ) @ f ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_1_convex__domain,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( euclid925273238_space @ A )
% 2.48/1.62 => ! [F: A > extended_ereal] :
% 2.48/1.62 ( ( lower_673667120vex_on @ A @ ( top_top @ ( set @ A ) ) @ F )
% 2.48/1.62 => ( convex @ A @ ( lower_272802190domain @ A @ F ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_2_convex__on__domain,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( real_V1076094709vector @ A )
% 2.48/1.62 => ! [F: A > extended_ereal] :
% 2.48/1.62 ( ( lower_673667120vex_on @ A @ ( lower_272802190domain @ A @ F ) @ F )
% 2.48/1.62 = ( lower_673667120vex_on @ A @ ( top_top @ ( set @ A ) ) @ F ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_3_convex__on__domain2,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( real_V1076094709vector @ A )
% 2.48/1.62 => ! [F: A > extended_ereal] :
% 2.48/1.62 ( ( lower_673667120vex_on @ A @ ( lower_272802190domain @ A @ F ) @ F )
% 2.48/1.62 = ( ! [S: set @ A] : ( lower_673667120vex_on @ A @ S @ F ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_4_convex__on__ereal__univ,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( real_V1076094709vector @ A )
% 2.48/1.62 => ! [F: A > extended_ereal] :
% 2.48/1.62 ( ( lower_673667120vex_on @ A @ ( top_top @ ( set @ A ) ) @ F )
% 2.48/1.62 = ( ! [S: set @ A] : ( lower_673667120vex_on @ A @ S @ F ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_5_convex__UNIV,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( real_V1076094709vector @ A )
% 2.48/1.62 => ( convex @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_6_UNIV__I,axiom,
% 2.48/1.62 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_7_iso__tuple__UNIV__I,axiom,
% 2.48/1.62 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_8_top__apply,axiom,
% 2.48/1.62 ! [C: $tType,D: $tType] :
% 2.48/1.62 ( ( top @ C )
% 2.48/1.62 => ( ( top_top @ ( D > C ) )
% 2.48/1.62 = ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_9_assms,axiom,
% 2.48/1.62 ! [X3: a] :
% 2.48/1.62 ( ( ( f @ X3 )
% 2.48/1.62 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.62 | ( ( f @ X3 )
% 2.48/1.62 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_10_UNIV__eq__I,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ A] :
% 2.48/1.62 ( ! [X4: A] : ( member @ A @ X4 @ A2 )
% 2.48/1.62 => ( ( top_top @ ( set @ A ) )
% 2.48/1.62 = A2 ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_11_UNIV__witness,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ? [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_12_convex__Epigraph,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( real_V1076094709vector @ A )
% 2.48/1.62 => ! [S2: set @ A,F: A > extended_ereal] :
% 2.48/1.62 ( ( convex @ A @ S2 )
% 2.48/1.62 => ( ( convex @ ( product_prod @ A @ real ) @ ( lower_261390618igraph @ A @ S2 @ F ) )
% 2.48/1.62 = ( lower_673667120vex_on @ A @ S2 @ F ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_13_convex__EpigraphI,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( real_V1076094709vector @ A )
% 2.48/1.62 => ! [S3: set @ A,F: A > extended_ereal] :
% 2.48/1.62 ( ( lower_673667120vex_on @ A @ S3 @ F )
% 2.48/1.62 => ( ( convex @ A @ S3 )
% 2.48/1.62 => ( convex @ ( product_prod @ A @ real ) @ ( lower_261390618igraph @ A @ S3 @ F ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_14_affine__on__def,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( real_V1076094709vector @ A )
% 2.48/1.62 => ( ( lower_500881736ine_on @ A )
% 2.48/1.62 = ( ^ [S: set @ A,F2: A > extended_ereal] :
% 2.48/1.62 ( ( lower_673667120vex_on @ A @ S @ F2 )
% 2.48/1.62 & ( lower_708069572ave_on @ A @ S @ F2 )
% 2.48/1.62 & ( lower_728871928ite_on @ A @ S @ F2 ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_15_Sup__UNIV,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( comple187826305attice @ A )
% 2.48/1.62 => ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
% 2.48/1.62 = ( top_top @ A ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_16_convex__on__ereal__subset,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( real_V1076094709vector @ A )
% 2.48/1.62 => ! [T: set @ A,F: A > extended_ereal,S3: set @ A] :
% 2.48/1.62 ( ( lower_673667120vex_on @ A @ T @ F )
% 2.48/1.62 => ( ( ord_less_eq @ ( set @ A ) @ S3 @ T )
% 2.48/1.62 => ( lower_673667120vex_on @ A @ S3 @ F ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_17_order__refl,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( preorder @ A )
% 2.48/1.62 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_18_subsetI,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ A,B: set @ A] :
% 2.48/1.62 ( ! [X4: A] :
% 2.48/1.62 ( ( member @ A @ X4 @ A2 )
% 2.48/1.62 => ( member @ A @ X4 @ B ) )
% 2.48/1.62 => ( ord_less_eq @ ( set @ A ) @ A2 @ B ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_19_subset__antisym,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ A,B: set @ A] :
% 2.48/1.62 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
% 2.48/1.62 => ( ( ord_less_eq @ ( set @ A ) @ B @ A2 )
% 2.48/1.62 => ( A2 = B ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_20_in__mono,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ A,B: set @ A,X: A] :
% 2.48/1.62 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
% 2.48/1.62 => ( ( member @ A @ X @ A2 )
% 2.48/1.62 => ( member @ A @ X @ B ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_21_subsetD,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ A,B: set @ A,C2: A] :
% 2.48/1.62 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
% 2.48/1.62 => ( ( member @ A @ C2 @ A2 )
% 2.48/1.62 => ( member @ A @ C2 @ B ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_22_equalityE,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ A,B: set @ A] :
% 2.48/1.62 ( ( A2 = B )
% 2.48/1.62 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
% 2.48/1.62 => ~ ( ord_less_eq @ ( set @ A ) @ B @ A2 ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_23_subset__eq,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( ord_less_eq @ ( set @ A ) )
% 2.48/1.62 = ( ^ [A3: set @ A,B2: set @ A] :
% 2.48/1.62 ! [X2: A] :
% 2.48/1.62 ( ( member @ A @ X2 @ A3 )
% 2.48/1.62 => ( member @ A @ X2 @ B2 ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_24_equalityD1,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ A,B: set @ A] :
% 2.48/1.62 ( ( A2 = B )
% 2.48/1.62 => ( ord_less_eq @ ( set @ A ) @ A2 @ B ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_25_equalityD2,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ A,B: set @ A] :
% 2.48/1.62 ( ( A2 = B )
% 2.48/1.62 => ( ord_less_eq @ ( set @ A ) @ B @ A2 ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_26_subset__iff,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( ord_less_eq @ ( set @ A ) )
% 2.48/1.62 = ( ^ [A3: set @ A,B2: set @ A] :
% 2.48/1.62 ! [T2: A] :
% 2.48/1.62 ( ( member @ A @ T2 @ A3 )
% 2.48/1.62 => ( member @ A @ T2 @ B2 ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_27_subset__refl,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_28_Collect__mono,axiom,
% 2.48/1.62 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.48/1.62 ( ! [X4: A] :
% 2.48/1.62 ( ( P @ X4 )
% 2.48/1.62 => ( Q @ X4 ) )
% 2.48/1.62 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_29_subset__trans,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ A,B: set @ A,C3: set @ A] :
% 2.48/1.62 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
% 2.48/1.62 => ( ( ord_less_eq @ ( set @ A ) @ B @ C3 )
% 2.48/1.62 => ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_30_le__funD,axiom,
% 2.48/1.62 ! [B3: $tType,A: $tType] :
% 2.48/1.62 ( ( ord @ B3 )
% 2.48/1.62 => ! [F: A > B3,G: A > B3,X: A] :
% 2.48/1.62 ( ( ord_less_eq @ ( A > B3 ) @ F @ G )
% 2.48/1.62 => ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_31_le__funE,axiom,
% 2.48/1.62 ! [B3: $tType,A: $tType] :
% 2.48/1.62 ( ( ord @ B3 )
% 2.48/1.62 => ! [F: A > B3,G: A > B3,X: A] :
% 2.48/1.62 ( ( ord_less_eq @ ( A > B3 ) @ F @ G )
% 2.48/1.62 => ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_32_le__funI,axiom,
% 2.48/1.62 ! [B3: $tType,A: $tType] :
% 2.48/1.62 ( ( ord @ B3 )
% 2.48/1.62 => ! [F: A > B3,G: A > B3] :
% 2.48/1.62 ( ! [X4: A] : ( ord_less_eq @ B3 @ ( F @ X4 ) @ ( G @ X4 ) )
% 2.48/1.62 => ( ord_less_eq @ ( A > B3 ) @ F @ G ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_33_set__eq__subset,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( ^ [Y: set @ A,Z: set @ A] : ( Y = Z ) )
% 2.48/1.62 = ( ^ [A3: set @ A,B2: set @ A] :
% 2.48/1.62 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.48/1.62 & ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_34_le__fun__def,axiom,
% 2.48/1.62 ! [B3: $tType,A: $tType] :
% 2.48/1.62 ( ( ord @ B3 )
% 2.48/1.62 => ( ( ord_less_eq @ ( A > B3 ) )
% 2.48/1.62 = ( ^ [F2: A > B3,G2: A > B3] :
% 2.48/1.62 ! [X2: A] : ( ord_less_eq @ B3 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_35_Collect__mono__iff,axiom,
% 2.48/1.62 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.48/1.62 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.48/1.62 = ( ! [X2: A] :
% 2.48/1.62 ( ( P @ X2 )
% 2.48/1.62 => ( Q @ X2 ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_36_order__subst1,axiom,
% 2.48/1.62 ! [A: $tType,B3: $tType] :
% 2.48/1.62 ( ( ( order @ B3 )
% 2.48/1.62 & ( order @ A ) )
% 2.48/1.62 => ! [A4: A,F: B3 > A,B4: B3,C2: B3] :
% 2.48/1.62 ( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
% 2.48/1.62 => ( ( ord_less_eq @ B3 @ B4 @ C2 )
% 2.48/1.62 => ( ! [X4: B3,Y2: B3] :
% 2.48/1.62 ( ( ord_less_eq @ B3 @ X4 @ Y2 )
% 2.48/1.62 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 2.48/1.62 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_37_order__subst2,axiom,
% 2.48/1.62 ! [A: $tType,C: $tType] :
% 2.48/1.62 ( ( ( order @ C )
% 2.48/1.62 & ( order @ A ) )
% 2.48/1.62 => ! [A4: A,B4: A,F: A > C,C2: C] :
% 2.48/1.62 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.48/1.62 => ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
% 2.48/1.62 => ( ! [X4: A,Y2: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 2.48/1.62 => ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 2.48/1.62 => ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_38_ord__eq__le__subst,axiom,
% 2.48/1.62 ! [A: $tType,B3: $tType] :
% 2.48/1.62 ( ( ( ord @ B3 )
% 2.48/1.62 & ( ord @ A ) )
% 2.48/1.62 => ! [A4: A,F: B3 > A,B4: B3,C2: B3] :
% 2.48/1.62 ( ( A4
% 2.48/1.62 = ( F @ B4 ) )
% 2.48/1.62 => ( ( ord_less_eq @ B3 @ B4 @ C2 )
% 2.48/1.62 => ( ! [X4: B3,Y2: B3] :
% 2.48/1.62 ( ( ord_less_eq @ B3 @ X4 @ Y2 )
% 2.48/1.62 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 2.48/1.62 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_39_ord__le__eq__subst,axiom,
% 2.48/1.62 ! [A: $tType,B3: $tType] :
% 2.48/1.62 ( ( ( ord @ B3 )
% 2.48/1.62 & ( ord @ A ) )
% 2.48/1.62 => ! [A4: A,B4: A,F: A > B3,C2: B3] :
% 2.48/1.62 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.48/1.62 => ( ( ( F @ B4 )
% 2.48/1.62 = C2 )
% 2.48/1.62 => ( ! [X4: A,Y2: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 2.48/1.62 => ( ord_less_eq @ B3 @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 2.48/1.62 => ( ord_less_eq @ B3 @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_40_Union__mono,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ ( set @ A ),B: set @ ( set @ A )] :
% 2.48/1.62 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ B )
% 2.48/1.62 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_41_eq__iff,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( order @ A )
% 2.48/1.62 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 2.48/1.62 = ( ^ [X2: A,Y3: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.48/1.62 & ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_42_Union__least,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ ( set @ A ),C3: set @ A] :
% 2.48/1.62 ( ! [X5: set @ A] :
% 2.48/1.62 ( ( member @ ( set @ A ) @ X5 @ A2 )
% 2.48/1.62 => ( ord_less_eq @ ( set @ A ) @ X5 @ C3 ) )
% 2.48/1.62 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ C3 ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_43_Union__upper,axiom,
% 2.48/1.62 ! [A: $tType,B: set @ A,A2: set @ ( set @ A )] :
% 2.48/1.62 ( ( member @ ( set @ A ) @ B @ A2 )
% 2.48/1.62 => ( ord_less_eq @ ( set @ A ) @ B @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_44_antisym,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( order @ A )
% 2.48/1.62 => ! [X: A,Y4: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.48/1.62 => ( ( ord_less_eq @ A @ Y4 @ X )
% 2.48/1.62 => ( X = Y4 ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_45_mem__Collect__eq,axiom,
% 2.48/1.62 ! [A: $tType,A4: A,P: A > $o] :
% 2.48/1.62 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 2.48/1.62 = ( P @ A4 ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_46_Collect__mem__eq,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ A] :
% 2.48/1.62 ( ( collect @ A
% 2.48/1.62 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 2.48/1.62 = A2 ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_47_Collect__cong,axiom,
% 2.48/1.62 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.48/1.62 ( ! [X4: A] :
% 2.48/1.62 ( ( P @ X4 )
% 2.48/1.62 = ( Q @ X4 ) )
% 2.48/1.62 => ( ( collect @ A @ P )
% 2.48/1.62 = ( collect @ A @ Q ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_48_ext,axiom,
% 2.48/1.62 ! [B3: $tType,A: $tType,F: A > B3,G: A > B3] :
% 2.48/1.62 ( ! [X4: A] :
% 2.48/1.62 ( ( F @ X4 )
% 2.48/1.62 = ( G @ X4 ) )
% 2.48/1.62 => ( F = G ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_49_Union__subsetI,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ ( set @ A ),B: set @ ( set @ A )] :
% 2.48/1.62 ( ! [X4: set @ A] :
% 2.48/1.62 ( ( member @ ( set @ A ) @ X4 @ A2 )
% 2.48/1.62 => ? [Y5: set @ A] :
% 2.48/1.62 ( ( member @ ( set @ A ) @ Y5 @ B )
% 2.48/1.62 & ( ord_less_eq @ ( set @ A ) @ X4 @ Y5 ) ) )
% 2.48/1.62 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_50_linear,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( linorder @ A )
% 2.48/1.62 => ! [X: A,Y4: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.48/1.62 | ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_51_eq__refl,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( preorder @ A )
% 2.48/1.62 => ! [X: A,Y4: A] :
% 2.48/1.62 ( ( X = Y4 )
% 2.48/1.62 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_52_le__cases,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( linorder @ A )
% 2.48/1.62 => ! [X: A,Y4: A] :
% 2.48/1.62 ( ~ ( ord_less_eq @ A @ X @ Y4 )
% 2.48/1.62 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_53_order_Otrans,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( order @ A )
% 2.48/1.62 => ! [A4: A,B4: A,C2: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.48/1.62 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 2.48/1.62 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_54_le__cases3,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( linorder @ A )
% 2.48/1.62 => ! [X: A,Y4: A,Z2: A] :
% 2.48/1.62 ( ( ( ord_less_eq @ A @ X @ Y4 )
% 2.48/1.62 => ~ ( ord_less_eq @ A @ Y4 @ Z2 ) )
% 2.48/1.62 => ( ( ( ord_less_eq @ A @ Y4 @ X )
% 2.48/1.62 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 2.48/1.62 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 2.48/1.62 => ~ ( ord_less_eq @ A @ Z2 @ Y4 ) )
% 2.48/1.62 => ( ( ( ord_less_eq @ A @ Z2 @ Y4 )
% 2.48/1.62 => ~ ( ord_less_eq @ A @ Y4 @ X ) )
% 2.48/1.62 => ( ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 2.48/1.62 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 2.48/1.62 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 2.48/1.62 => ~ ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_55_antisym__conv,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( order @ A )
% 2.48/1.62 => ! [Y4: A,X: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ Y4 @ X )
% 2.48/1.62 => ( ( ord_less_eq @ A @ X @ Y4 )
% 2.48/1.62 = ( X = Y4 ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_56_order__class_Oorder_Oeq__iff,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( order @ A )
% 2.48/1.62 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 2.48/1.62 = ( ^ [A5: A,B5: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ A5 @ B5 )
% 2.48/1.62 & ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_57_ord__eq__le__trans,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( ord @ A )
% 2.48/1.62 => ! [A4: A,B4: A,C2: A] :
% 2.48/1.62 ( ( A4 = B4 )
% 2.48/1.62 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 2.48/1.62 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_58_ord__le__eq__trans,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( ord @ A )
% 2.48/1.62 => ! [A4: A,B4: A,C2: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.48/1.62 => ( ( B4 = C2 )
% 2.48/1.62 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_59_order__class_Oorder_Oantisym,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( order @ A )
% 2.48/1.62 => ! [A4: A,B4: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.48/1.62 => ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.48/1.62 => ( A4 = B4 ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_60_order__trans,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( preorder @ A )
% 2.48/1.62 => ! [X: A,Y4: A,Z2: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.48/1.62 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 2.48/1.62 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_61_dual__order_Orefl,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( order @ A )
% 2.48/1.62 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_62_linorder__wlog,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( linorder @ A )
% 2.48/1.62 => ! [P: A > A > $o,A4: A,B4: A] :
% 2.48/1.62 ( ! [A6: A,B6: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ A6 @ B6 )
% 2.48/1.62 => ( P @ A6 @ B6 ) )
% 2.48/1.62 => ( ! [A6: A,B6: A] :
% 2.48/1.62 ( ( P @ B6 @ A6 )
% 2.48/1.62 => ( P @ A6 @ B6 ) )
% 2.48/1.62 => ( P @ A4 @ B4 ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_63_dual__order_Otrans,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( order @ A )
% 2.48/1.62 => ! [B4: A,A4: A,C2: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.48/1.62 => ( ( ord_less_eq @ A @ C2 @ B4 )
% 2.48/1.62 => ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_64_dual__order_Oeq__iff,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( order @ A )
% 2.48/1.62 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 2.48/1.62 = ( ^ [A5: A,B5: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ B5 @ A5 )
% 2.48/1.62 & ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_65_dual__order_Oantisym,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( order @ A )
% 2.48/1.62 => ! [B4: A,A4: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.48/1.62 => ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.48/1.62 => ( A4 = B4 ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_66_Sup__eqI,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( comple187826305attice @ A )
% 2.48/1.62 => ! [A2: set @ A,X: A] :
% 2.48/1.62 ( ! [Y2: A] :
% 2.48/1.62 ( ( member @ A @ Y2 @ A2 )
% 2.48/1.62 => ( ord_less_eq @ A @ Y2 @ X ) )
% 2.48/1.62 => ( ! [Y2: A] :
% 2.48/1.62 ( ! [Z3: A] :
% 2.48/1.62 ( ( member @ A @ Z3 @ A2 )
% 2.48/1.62 => ( ord_less_eq @ A @ Z3 @ Y2 ) )
% 2.48/1.62 => ( ord_less_eq @ A @ X @ Y2 ) )
% 2.48/1.62 => ( ( complete_Sup_Sup @ A @ A2 )
% 2.48/1.62 = X ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_67_Sup__mono,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( comple187826305attice @ A )
% 2.48/1.62 => ! [A2: set @ A,B: set @ A] :
% 2.48/1.62 ( ! [A6: A] :
% 2.48/1.62 ( ( member @ A @ A6 @ A2 )
% 2.48/1.62 => ? [X3: A] :
% 2.48/1.62 ( ( member @ A @ X3 @ B )
% 2.48/1.62 & ( ord_less_eq @ A @ A6 @ X3 ) ) )
% 2.48/1.62 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_68_Sup__least,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( comple187826305attice @ A )
% 2.48/1.62 => ! [A2: set @ A,Z2: A] :
% 2.48/1.62 ( ! [X4: A] :
% 2.48/1.62 ( ( member @ A @ X4 @ A2 )
% 2.48/1.62 => ( ord_less_eq @ A @ X4 @ Z2 ) )
% 2.48/1.62 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ Z2 ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_69_Sup__upper,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( comple187826305attice @ A )
% 2.48/1.62 => ! [X: A,A2: set @ A] :
% 2.48/1.62 ( ( member @ A @ X @ A2 )
% 2.48/1.62 => ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_70_Sup__le__iff,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( comple187826305attice @ A )
% 2.48/1.62 => ! [A2: set @ A,B4: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ B4 )
% 2.48/1.62 = ( ! [X2: A] :
% 2.48/1.62 ( ( member @ A @ X2 @ A2 )
% 2.48/1.62 => ( ord_less_eq @ A @ X2 @ B4 ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_71_Sup__upper2,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( comple187826305attice @ A )
% 2.48/1.62 => ! [U: A,A2: set @ A,V: A] :
% 2.48/1.62 ( ( member @ A @ U @ A2 )
% 2.48/1.62 => ( ( ord_less_eq @ A @ V @ U )
% 2.48/1.62 => ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_72_finite__on__def,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( real_V1076094709vector @ A )
% 2.48/1.62 => ( ( lower_728871928ite_on @ A )
% 2.48/1.62 = ( ^ [S: set @ A,F2: A > extended_ereal] :
% 2.48/1.62 ! [X2: A] :
% 2.48/1.62 ( ( member @ A @ X2 @ S )
% 2.48/1.62 => ( ( ( F2 @ X2 )
% 2.48/1.62 != ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.62 & ( ( F2 @ X2 )
% 2.48/1.62 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_73_Sup__subset__mono,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( comple187826305attice @ A )
% 2.48/1.62 => ! [A2: set @ A,B: set @ A] :
% 2.48/1.62 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
% 2.48/1.62 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_74_top__set__def,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( top_top @ ( set @ A ) )
% 2.48/1.62 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_75_Union__UNIV,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( complete_Sup_Sup @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
% 2.48/1.62 = ( top_top @ ( set @ A ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_76_top_Oextremum__uniqueI,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( order_top @ A )
% 2.48/1.62 => ! [A4: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
% 2.48/1.62 => ( A4
% 2.48/1.62 = ( top_top @ A ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_77_top_Oextremum__unique,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( order_top @ A )
% 2.48/1.62 => ! [A4: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
% 2.48/1.62 = ( A4
% 2.48/1.62 = ( top_top @ A ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_78_top__greatest,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( order_top @ A )
% 2.48/1.62 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( top_top @ A ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_79_subset__UNIV,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_80_epigraph__invertible,axiom,
% 2.48/1.62 ! [A: $tType,F: A > extended_ereal,G: A > extended_ereal] :
% 2.48/1.62 ( ( ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ F )
% 2.48/1.62 = ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ G ) )
% 2.48/1.62 => ( F = G ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_81_neg__le__iff__le,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( ordered_ab_group_add @ A )
% 2.48/1.62 => ! [B4: A,A4: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A4 ) )
% 2.48/1.62 = ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_82_compl__le__compl__iff,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( boolean_algebra @ A )
% 2.48/1.62 => ! [X: A,Y4: A] :
% 2.48/1.62 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y4 ) )
% 2.48/1.62 = ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_83_top__finite__def,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( finite_lattice @ A )
% 2.48/1.62 => ( ( top_top @ A )
% 2.48/1.62 = ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_84_ereal__uminus__eq__iff,axiom,
% 2.48/1.62 ! [A4: extended_ereal,B4: extended_ereal] :
% 2.48/1.62 ( ( ( uminus_uminus @ extended_ereal @ A4 )
% 2.48/1.62 = ( uminus_uminus @ extended_ereal @ B4 ) )
% 2.48/1.62 = ( A4 = B4 ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_85_ereal__uminus__uminus,axiom,
% 2.48/1.62 ! [A4: extended_ereal] :
% 2.48/1.62 ( ( uminus_uminus @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A4 ) )
% 2.48/1.62 = A4 ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_86_verit__minus__simplify_I4_J,axiom,
% 2.48/1.62 ! [B3: $tType] :
% 2.48/1.62 ( ( group_add @ B3 )
% 2.48/1.62 => ! [B4: B3] :
% 2.48/1.62 ( ( uminus_uminus @ B3 @ ( uminus_uminus @ B3 @ B4 ) )
% 2.48/1.62 = B4 ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_87_uminus__apply,axiom,
% 2.48/1.62 ! [B3: $tType,A: $tType] :
% 2.48/1.62 ( ( uminus @ B3 )
% 2.48/1.62 => ( ( uminus_uminus @ ( A > B3 ) )
% 2.48/1.62 = ( ^ [A3: A > B3,X2: A] : ( uminus_uminus @ B3 @ ( A3 @ X2 ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_88_add_Oinverse__inverse,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( group_add @ A )
% 2.48/1.62 => ! [A4: A] :
% 2.48/1.62 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A4 ) )
% 2.48/1.62 = A4 ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_89_neg__equal__iff__equal,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( group_add @ A )
% 2.48/1.62 => ! [A4: A,B4: A] :
% 2.48/1.62 ( ( ( uminus_uminus @ A @ A4 )
% 2.48/1.62 = ( uminus_uminus @ A @ B4 ) )
% 2.48/1.62 = ( A4 = B4 ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_90_double__compl,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( boolean_algebra @ A )
% 2.48/1.62 => ! [X: A] :
% 2.48/1.62 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
% 2.48/1.62 = X ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_91_compl__eq__compl__iff,axiom,
% 2.48/1.62 ! [A: $tType] :
% 2.48/1.62 ( ( boolean_algebra @ A )
% 2.48/1.62 => ! [X: A,Y4: A] :
% 2.48/1.62 ( ( ( uminus_uminus @ A @ X )
% 2.48/1.62 = ( uminus_uminus @ A @ Y4 ) )
% 2.48/1.62 = ( X = Y4 ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_92_Compl__subset__Compl__iff,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ A,B: set @ A] :
% 2.48/1.62 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B ) )
% 2.48/1.62 = ( ord_less_eq @ ( set @ A ) @ B @ A2 ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_93_Compl__anti__mono,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ A,B: set @ A] :
% 2.48/1.62 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
% 2.48/1.62 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_94_ereal__infty__less__eq_I1_J,axiom,
% 2.48/1.62 ! [X: extended_ereal] :
% 2.48/1.62 ( ( ord_less_eq @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ X )
% 2.48/1.62 = ( X
% 2.48/1.62 = ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_95_UN__ball__bex__simps_I3_J,axiom,
% 2.48/1.62 ! [D: $tType,A2: set @ ( set @ D ),P: D > $o] :
% 2.48/1.62 ( ( ? [X2: D] :
% 2.48/1.62 ( ( member @ D @ X2 @ ( complete_Sup_Sup @ ( set @ D ) @ A2 ) )
% 2.48/1.62 & ( P @ X2 ) ) )
% 2.48/1.62 = ( ? [X2: set @ D] :
% 2.48/1.62 ( ( member @ ( set @ D ) @ X2 @ A2 )
% 2.48/1.62 & ? [Y3: D] :
% 2.48/1.62 ( ( member @ D @ Y3 @ X2 )
% 2.48/1.62 & ( P @ Y3 ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_96_UN__ball__bex__simps_I1_J,axiom,
% 2.48/1.62 ! [A: $tType,A2: set @ ( set @ A ),P: A > $o] :
% 2.48/1.62 ( ( ! [X2: A] :
% 2.48/1.62 ( ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
% 2.48/1.62 => ( P @ X2 ) ) )
% 2.48/1.62 = ( ! [X2: set @ A] :
% 2.48/1.62 ( ( member @ ( set @ A ) @ X2 @ A2 )
% 2.48/1.62 => ! [Y3: A] :
% 2.48/1.62 ( ( member @ A @ Y3 @ X2 )
% 2.48/1.62 => ( P @ Y3 ) ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_97_UnionI,axiom,
% 2.48/1.62 ! [A: $tType,X6: set @ A,C3: set @ ( set @ A ),A2: A] :
% 2.48/1.62 ( ( member @ ( set @ A ) @ X6 @ C3 )
% 2.48/1.62 => ( ( member @ A @ A2 @ X6 )
% 2.48/1.62 => ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_98_Union__iff,axiom,
% 2.48/1.62 ! [A: $tType,A2: A,C3: set @ ( set @ A )] :
% 2.48/1.62 ( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
% 2.48/1.62 = ( ? [X2: set @ A] :
% 2.48/1.62 ( ( member @ ( set @ A ) @ X2 @ C3 )
% 2.48/1.62 & ( member @ A @ A2 @ X2 ) ) ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_99_ereal__minus__le__minus,axiom,
% 2.48/1.62 ! [A4: extended_ereal,B4: extended_ereal] :
% 2.48/1.62 ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A4 ) @ ( uminus_uminus @ extended_ereal @ B4 ) )
% 2.48/1.62 = ( ord_less_eq @ extended_ereal @ B4 @ A4 ) ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_100_top1I,axiom,
% 2.48/1.62 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 2.48/1.62
% 2.48/1.62 thf(fact_101_ereal__infty__less__eq_I2_J,axiom,
% 2.48/1.62 ! [X: extended_ereal] :
% 2.48/1.62 ( ( ord_less_eq @ extended_ereal @ X @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 = ( X
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_102_UnionE,axiom,
% 2.48/1.63 ! [A: $tType,A2: A,C3: set @ ( set @ A )] :
% 2.48/1.63 ( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
% 2.48/1.63 => ~ ! [X5: set @ A] :
% 2.48/1.63 ( ( member @ A @ A2 @ X5 )
% 2.48/1.63 => ~ ( member @ ( set @ A ) @ X5 @ C3 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_103_neq__PInf__trans,axiom,
% 2.48/1.63 ! [Y4: extended_ereal,X: extended_ereal] :
% 2.48/1.63 ( ( Y4
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( ( ord_less_eq @ extended_ereal @ X @ Y4 )
% 2.48/1.63 => ( X
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_104_ereal__infty__less__eq2_I1_J,axiom,
% 2.48/1.63 ! [A4: extended_ereal,B4: extended_ereal] :
% 2.48/1.63 ( ( ord_less_eq @ extended_ereal @ A4 @ B4 )
% 2.48/1.63 => ( ( A4
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( B4
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_105_ereal__less__eq_I1_J,axiom,
% 2.48/1.63 ! [X: extended_ereal] : ( ord_less_eq @ extended_ereal @ X @ ( extend1396239628finity @ extended_ereal ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_106_ereal__uminus__le__reorder,axiom,
% 2.48/1.63 ! [A4: extended_ereal,B4: extended_ereal] :
% 2.48/1.63 ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A4 ) @ B4 )
% 2.48/1.63 = ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ B4 ) @ A4 ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_107_ereal__infty__less__eq2_I2_J,axiom,
% 2.48/1.63 ! [A4: extended_ereal,B4: extended_ereal] :
% 2.48/1.63 ( ( ord_less_eq @ extended_ereal @ A4 @ B4 )
% 2.48/1.63 => ( ( B4
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( A4
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_108_ereal__less__eq_I2_J,axiom,
% 2.48/1.63 ! [X: extended_ereal] : ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ X ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_109_epigraph__subset__iff,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( real_V2090557954_space @ A )
% 2.48/1.63 => ! [F: A > extended_ereal,G: A > extended_ereal] :
% 2.48/1.63 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ real ) ) @ ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ F ) @ ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ G ) )
% 2.48/1.63 = ( ! [X2: A] : ( ord_less_eq @ extended_ereal @ ( G @ X2 ) @ ( F @ X2 ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_110_verit__la__disequality,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( linorder @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( A4 = B4 )
% 2.48/1.63 | ~ ( ord_less_eq @ A @ A4 @ B4 )
% 2.48/1.63 | ~ ( ord_less_eq @ A @ B4 @ A4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_111_minus__equation__iff,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( group_add @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( ( uminus_uminus @ A @ A4 )
% 2.48/1.63 = B4 )
% 2.48/1.63 = ( ( uminus_uminus @ A @ B4 )
% 2.48/1.63 = A4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_112_equation__minus__iff,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( group_add @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( A4
% 2.48/1.63 = ( uminus_uminus @ A @ B4 ) )
% 2.48/1.63 = ( B4
% 2.48/1.63 = ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_113_fun__Compl__def,axiom,
% 2.48/1.63 ! [B3: $tType,A: $tType] :
% 2.48/1.63 ( ( uminus @ B3 )
% 2.48/1.63 => ( ( uminus_uminus @ ( A > B3 ) )
% 2.48/1.63 = ( ^ [A3: A > B3,X2: A] : ( uminus_uminus @ B3 @ ( A3 @ X2 ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_114_verit__negate__coefficient_I3_J,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( ordered_ab_group_add @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( A4 = B4 )
% 2.48/1.63 => ( ( uminus_uminus @ A @ A4 )
% 2.48/1.63 = ( uminus_uminus @ A @ B4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_115_top__ereal__def,axiom,
% 2.48/1.63 ( ( top_top @ extended_ereal )
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_116_Sup__eq__PInfty,axiom,
% 2.48/1.63 ! [S2: set @ extended_ereal] :
% 2.48/1.63 ( ( member @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ S2 )
% 2.48/1.63 => ( ( complete_Sup_Sup @ extended_ereal @ S2 )
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_117_ereal__uminus__eq__reorder,axiom,
% 2.48/1.63 ! [A4: extended_ereal,B4: extended_ereal] :
% 2.48/1.63 ( ( ( uminus_uminus @ extended_ereal @ A4 )
% 2.48/1.63 = B4 )
% 2.48/1.63 = ( A4
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ B4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_118_le__imp__neg__le,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( ordered_ab_group_add @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.48/1.63 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_119_minus__le__iff,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( ordered_ab_group_add @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ B4 )
% 2.48/1.63 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ A4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_120_le__minus__iff,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( ordered_ab_group_add @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ B4 ) )
% 2.48/1.63 = ( ord_less_eq @ A @ B4 @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_121_compl__le__swap2,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( boolean_algebra @ A )
% 2.48/1.63 => ! [Y4: A,X: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y4 ) @ X )
% 2.48/1.63 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_122_compl__le__swap1,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( boolean_algebra @ A )
% 2.48/1.63 => ! [Y4: A,X: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ Y4 @ ( uminus_uminus @ A @ X ) )
% 2.48/1.63 => ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_123_compl__mono,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( boolean_algebra @ A )
% 2.48/1.63 => ! [X: A,Y4: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.48/1.63 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y4 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_124_MInfty__neq__PInfty_I1_J,axiom,
% 2.48/1.63 ( ( extend1396239628finity @ extended_ereal )
% 2.48/1.63 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_125_top__empty__eq,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( top_top @ ( A > $o ) )
% 2.48/1.63 = ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_126_cSup__eq__maximum,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( condit378418413attice @ A )
% 2.48/1.63 => ! [Z2: A,X6: set @ A] :
% 2.48/1.63 ( ( member @ A @ Z2 @ X6 )
% 2.48/1.63 => ( ! [X4: A] :
% 2.48/1.63 ( ( member @ A @ X4 @ X6 )
% 2.48/1.63 => ( ord_less_eq @ A @ X4 @ Z2 ) )
% 2.48/1.63 => ( ( complete_Sup_Sup @ A @ X6 )
% 2.48/1.63 = Z2 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_127_cSup__eq,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( ( condit378418413attice @ A )
% 2.48/1.63 & ( no_bot @ A ) )
% 2.48/1.63 => ! [X6: set @ A,A4: A] :
% 2.48/1.63 ( ! [X4: A] :
% 2.48/1.63 ( ( member @ A @ X4 @ X6 )
% 2.48/1.63 => ( ord_less_eq @ A @ X4 @ A4 ) )
% 2.48/1.63 => ( ! [Y2: A] :
% 2.48/1.63 ( ! [X3: A] :
% 2.48/1.63 ( ( member @ A @ X3 @ X6 )
% 2.48/1.63 => ( ord_less_eq @ A @ X3 @ Y2 ) )
% 2.48/1.63 => ( ord_less_eq @ A @ A4 @ Y2 ) )
% 2.48/1.63 => ( ( complete_Sup_Sup @ A @ X6 )
% 2.48/1.63 = A4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_128_top__conj_I2_J,axiom,
% 2.48/1.63 ! [A: $tType,P: $o,X: A] :
% 2.48/1.63 ( ( P
% 2.48/1.63 & ( top_top @ ( A > $o ) @ X ) )
% 2.48/1.63 = P ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_129_top__conj_I1_J,axiom,
% 2.48/1.63 ! [A: $tType,X: A,P: $o] :
% 2.48/1.63 ( ( ( top_top @ ( A > $o ) @ X )
% 2.48/1.63 & P )
% 2.48/1.63 = P ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_130_MInfty__eq__minfinity,axiom,
% 2.48/1.63 ( extended_MInfty
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_131_ComplI,axiom,
% 2.48/1.63 ! [A: $tType,C2: A,A2: set @ A] :
% 2.48/1.63 ( ~ ( member @ A @ C2 @ A2 )
% 2.48/1.63 => ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_132_Compl__iff,axiom,
% 2.48/1.63 ! [A: $tType,C2: A,A2: set @ A] :
% 2.48/1.63 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.48/1.63 = ( ~ ( member @ A @ C2 @ A2 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_133_Compl__eq__Compl__iff,axiom,
% 2.48/1.63 ! [A: $tType,A2: set @ A,B: set @ A] :
% 2.48/1.63 ( ( ( uminus_uminus @ ( set @ A ) @ A2 )
% 2.48/1.63 = ( uminus_uminus @ ( set @ A ) @ B ) )
% 2.48/1.63 = ( A2 = B ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_134_ComplD,axiom,
% 2.48/1.63 ! [A: $tType,C2: A,A2: set @ A] :
% 2.48/1.63 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.48/1.63 => ~ ( member @ A @ C2 @ A2 ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_135_double__complement,axiom,
% 2.48/1.63 ! [A: $tType,A2: set @ A] :
% 2.48/1.63 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.48/1.63 = A2 ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_136_ereal__complete__Inf,axiom,
% 2.48/1.63 ! [S2: set @ extended_ereal] :
% 2.48/1.63 ? [X4: extended_ereal] :
% 2.48/1.63 ( ! [Xa: extended_ereal] :
% 2.48/1.63 ( ( member @ extended_ereal @ Xa @ S2 )
% 2.48/1.63 => ( ord_less_eq @ extended_ereal @ X4 @ Xa ) )
% 2.48/1.63 & ! [Z3: extended_ereal] :
% 2.48/1.63 ( ! [Xa2: extended_ereal] :
% 2.48/1.63 ( ( member @ extended_ereal @ Xa2 @ S2 )
% 2.48/1.63 => ( ord_less_eq @ extended_ereal @ Z3 @ Xa2 ) )
% 2.48/1.63 => ( ord_less_eq @ extended_ereal @ Z3 @ X4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_137_ereal__complete__Sup,axiom,
% 2.48/1.63 ! [S2: set @ extended_ereal] :
% 2.48/1.63 ? [X4: extended_ereal] :
% 2.48/1.63 ( ! [Xa: extended_ereal] :
% 2.48/1.63 ( ( member @ extended_ereal @ Xa @ S2 )
% 2.48/1.63 => ( ord_less_eq @ extended_ereal @ Xa @ X4 ) )
% 2.48/1.63 & ! [Z3: extended_ereal] :
% 2.48/1.63 ( ! [Xa2: extended_ereal] :
% 2.48/1.63 ( ( member @ extended_ereal @ Xa2 @ S2 )
% 2.48/1.63 => ( ord_less_eq @ extended_ereal @ Xa2 @ Z3 ) )
% 2.48/1.63 => ( ord_less_eq @ extended_ereal @ X4 @ Z3 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_138_uminus__ereal_Osimps_I3_J,axiom,
% 2.48/1.63 ( ( uminus_uminus @ extended_ereal @ extended_MInfty )
% 2.48/1.63 = extended_PInfty ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_139_uminus__ereal_Osimps_I2_J,axiom,
% 2.48/1.63 ( ( uminus_uminus @ extended_ereal @ extended_PInfty )
% 2.48/1.63 = extended_MInfty ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_140_exists__diff,axiom,
% 2.48/1.63 ! [A: $tType,P: ( set @ A ) > $o] :
% 2.48/1.63 ( ( ? [S: set @ A] : ( P @ ( uminus_uminus @ ( set @ A ) @ S ) ) )
% 2.48/1.63 = ( ? [X7: set @ A] : ( P @ X7 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_141_GreatestI2__order,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( order @ A )
% 2.48/1.63 => ! [P: A > $o,X: A,Q: A > $o] :
% 2.48/1.63 ( ( P @ X )
% 2.48/1.63 => ( ! [Y2: A] :
% 2.48/1.63 ( ( P @ Y2 )
% 2.48/1.63 => ( ord_less_eq @ A @ Y2 @ X ) )
% 2.48/1.63 => ( ! [X4: A] :
% 2.48/1.63 ( ( P @ X4 )
% 2.48/1.63 => ( ! [Y5: A] :
% 2.48/1.63 ( ( P @ Y5 )
% 2.48/1.63 => ( ord_less_eq @ A @ Y5 @ X4 ) )
% 2.48/1.63 => ( Q @ X4 ) ) )
% 2.48/1.63 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_142_infinity__ereal__def,axiom,
% 2.48/1.63 ( ( extend1396239628finity @ extended_ereal )
% 2.48/1.63 = extended_PInfty ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_143_ereal_Odistinct_I5_J,axiom,
% 2.48/1.63 extended_PInfty != extended_MInfty ).
% 2.48/1.63
% 2.48/1.63 thf(fact_144_Greatest__equality,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( order @ A )
% 2.48/1.63 => ! [P: A > $o,X: A] :
% 2.48/1.63 ( ( P @ X )
% 2.48/1.63 => ( ! [Y2: A] :
% 2.48/1.63 ( ( P @ Y2 )
% 2.48/1.63 => ( ord_less_eq @ A @ Y2 @ X ) )
% 2.48/1.63 => ( ( order_Greatest @ A @ P )
% 2.48/1.63 = X ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_145_uminus__ereal_Oelims,axiom,
% 2.48/1.63 ! [X: extended_ereal,Y4: extended_ereal] :
% 2.48/1.63 ( ( ( uminus_uminus @ extended_ereal @ X )
% 2.48/1.63 = Y4 )
% 2.48/1.63 => ( ! [R: real] :
% 2.48/1.63 ( ( X
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( Y4
% 2.48/1.63 != ( extended_ereal2 @ ( uminus_uminus @ real @ R ) ) ) )
% 2.48/1.63 => ( ( ( X = extended_PInfty )
% 2.48/1.63 => ( Y4 != extended_MInfty ) )
% 2.48/1.63 => ~ ( ( X = extended_MInfty )
% 2.48/1.63 => ( Y4 != extended_PInfty ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_146_not__MInfty__nonneg,axiom,
% 2.48/1.63 ! [X: extended_ereal] :
% 2.48/1.63 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ X )
% 2.48/1.63 => ( X
% 2.48/1.63 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_147_ereal_Osimps_I12_J,axiom,
% 2.48/1.63 ! [A: $tType,F1: real > A,F22: A,F3: A] :
% 2.48/1.63 ( ( extended_rec_ereal @ A @ F1 @ F22 @ F3 @ extended_PInfty )
% 2.48/1.63 = F22 ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_148_ereal_Osimps_I13_J,axiom,
% 2.48/1.63 ! [A: $tType,F1: real > A,F22: A,F3: A] :
% 2.48/1.63 ( ( extended_rec_ereal @ A @ F1 @ F22 @ F3 @ extended_MInfty )
% 2.48/1.63 = F3 ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_149_ereal_Oinject,axiom,
% 2.48/1.63 ! [X1: real,Y1: real] :
% 2.48/1.63 ( ( ( extended_ereal2 @ X1 )
% 2.48/1.63 = ( extended_ereal2 @ Y1 ) )
% 2.48/1.63 = ( X1 = Y1 ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_150_ereal__cong,axiom,
% 2.48/1.63 ! [X: real,Y4: real] :
% 2.48/1.63 ( ( X = Y4 )
% 2.48/1.63 => ( ( extended_ereal2 @ X )
% 2.48/1.63 = ( extended_ereal2 @ Y4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_151_le__zero__eq,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( canoni770627133id_add @ A )
% 2.48/1.63 => ! [N: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.48/1.63 = ( N
% 2.48/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_152_neg__equal__zero,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( linord219039673up_add @ A )
% 2.48/1.63 => ! [A4: A] :
% 2.48/1.63 ( ( ( uminus_uminus @ A @ A4 )
% 2.48/1.63 = A4 )
% 2.48/1.63 = ( A4
% 2.48/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_153_equal__neg__zero,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( linord219039673up_add @ A )
% 2.48/1.63 => ! [A4: A] :
% 2.48/1.63 ( ( A4
% 2.48/1.63 = ( uminus_uminus @ A @ A4 ) )
% 2.48/1.63 = ( A4
% 2.48/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_154_neg__equal__0__iff__equal,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( group_add @ A )
% 2.48/1.63 => ! [A4: A] :
% 2.48/1.63 ( ( ( uminus_uminus @ A @ A4 )
% 2.48/1.63 = ( zero_zero @ A ) )
% 2.48/1.63 = ( A4
% 2.48/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_155_neg__0__equal__iff__equal,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( group_add @ A )
% 2.48/1.63 => ! [A4: A] :
% 2.48/1.63 ( ( ( zero_zero @ A )
% 2.48/1.63 = ( uminus_uminus @ A @ A4 ) )
% 2.48/1.63 = ( ( zero_zero @ A )
% 2.48/1.63 = A4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_156_add_Oinverse__neutral,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( group_add @ A )
% 2.48/1.63 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 2.48/1.63 = ( zero_zero @ A ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_157_ereal__less__eq_I3_J,axiom,
% 2.48/1.63 ! [R2: real,P2: real] :
% 2.48/1.63 ( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
% 2.48/1.63 = ( ord_less_eq @ real @ R2 @ P2 ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_158_ereal__uminus__zero,axiom,
% 2.48/1.63 ( ( uminus_uminus @ extended_ereal @ ( zero_zero @ extended_ereal ) )
% 2.48/1.63 = ( zero_zero @ extended_ereal ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_159_ereal__uminus__zero__iff,axiom,
% 2.48/1.63 ! [A4: extended_ereal] :
% 2.48/1.63 ( ( ( uminus_uminus @ extended_ereal @ A4 )
% 2.48/1.63 = ( zero_zero @ extended_ereal ) )
% 2.48/1.63 = ( A4
% 2.48/1.63 = ( zero_zero @ extended_ereal ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_160_neg__0__le__iff__le,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( ordered_ab_group_add @ A )
% 2.48/1.63 => ! [A4: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A4 ) )
% 2.48/1.63 = ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_161_neg__le__0__iff__le,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( ordered_ab_group_add @ A )
% 2.48/1.63 => ! [A4: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ ( zero_zero @ A ) )
% 2.48/1.63 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_162_less__eq__neg__nonpos,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( linord219039673up_add @ A )
% 2.48/1.63 => ! [A4: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ A4 ) )
% 2.48/1.63 = ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_163_neg__less__eq__nonneg,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( linord219039673up_add @ A )
% 2.48/1.63 => ! [A4: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ A4 )
% 2.48/1.63 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_164_ereal__uminus__le__0__iff,axiom,
% 2.48/1.63 ! [A4: extended_ereal] :
% 2.48/1.63 ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A4 ) @ ( zero_zero @ extended_ereal ) )
% 2.48/1.63 = ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A4 ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_165_ereal__0__le__uminus__iff,axiom,
% 2.48/1.63 ! [A4: extended_ereal] :
% 2.48/1.63 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( uminus_uminus @ extended_ereal @ A4 ) )
% 2.48/1.63 = ( ord_less_eq @ extended_ereal @ A4 @ ( zero_zero @ extended_ereal ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_166_ereal_Odistinct_I1_J,axiom,
% 2.48/1.63 ! [X1: real] :
% 2.48/1.63 ( ( extended_ereal2 @ X1 )
% 2.48/1.63 != extended_PInfty ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_167_zero__reorient,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( zero @ A )
% 2.48/1.63 => ! [X: A] :
% 2.48/1.63 ( ( ( zero_zero @ A )
% 2.48/1.63 = X )
% 2.48/1.63 = ( X
% 2.48/1.63 = ( zero_zero @ A ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_168_ereal_Osimps_I11_J,axiom,
% 2.48/1.63 ! [A: $tType,F1: real > A,F22: A,F3: A,X1: real] :
% 2.48/1.63 ( ( extended_rec_ereal @ A @ F1 @ F22 @ F3 @ ( extended_ereal2 @ X1 ) )
% 2.48/1.63 = ( F1 @ X1 ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_169_Infty__neq__0_I1_J,axiom,
% 2.48/1.63 ( ( extend1396239628finity @ extended_ereal )
% 2.48/1.63 != ( zero_zero @ extended_ereal ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_170_zero__le,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( canoni770627133id_add @ A )
% 2.48/1.63 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_171_uminus__ereal_Osimps_I1_J,axiom,
% 2.48/1.63 ! [R2: real] :
% 2.48/1.63 ( ( uminus_uminus @ extended_ereal @ ( extended_ereal2 @ R2 ) )
% 2.48/1.63 = ( extended_ereal2 @ ( uminus_uminus @ real @ R2 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_172_PInfty__neq__ereal_I1_J,axiom,
% 2.48/1.63 ! [R2: real] :
% 2.48/1.63 ( ( extended_ereal2 @ R2 )
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_173_ereal_Odistinct_I3_J,axiom,
% 2.48/1.63 ! [X1: real] :
% 2.48/1.63 ( ( extended_ereal2 @ X1 )
% 2.48/1.63 != extended_MInfty ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_174_ereal__le__real,axiom,
% 2.48/1.63 ! [X: extended_ereal,Y4: extended_ereal] :
% 2.48/1.63 ( ! [Z4: real] :
% 2.48/1.63 ( ( ord_less_eq @ extended_ereal @ X @ ( extended_ereal2 @ Z4 ) )
% 2.48/1.63 => ( ord_less_eq @ extended_ereal @ Y4 @ ( extended_ereal2 @ Z4 ) ) )
% 2.48/1.63 => ( ord_less_eq @ extended_ereal @ Y4 @ X ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_175_le__ereal__le,axiom,
% 2.48/1.63 ! [A4: extended_ereal,X: real,Y4: real] :
% 2.48/1.63 ( ( ord_less_eq @ extended_ereal @ A4 @ ( extended_ereal2 @ X ) )
% 2.48/1.63 => ( ( ord_less_eq @ real @ X @ Y4 )
% 2.48/1.63 => ( ord_less_eq @ extended_ereal @ A4 @ ( extended_ereal2 @ Y4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_176_ereal__le__le,axiom,
% 2.48/1.63 ! [Y4: real,A4: extended_ereal,X: real] :
% 2.48/1.63 ( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ Y4 ) @ A4 )
% 2.48/1.63 => ( ( ord_less_eq @ real @ X @ Y4 )
% 2.48/1.63 => ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ X ) @ A4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_177_MInfty__neq__ereal_I1_J,axiom,
% 2.48/1.63 ! [R2: real] :
% 2.48/1.63 ( ( extended_ereal2 @ R2 )
% 2.48/1.63 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_178_ereal__cases,axiom,
% 2.48/1.63 ! [X: extended_ereal] :
% 2.48/1.63 ( ! [R: real] :
% 2.48/1.63 ( X
% 2.48/1.63 != ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ( X
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( X
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_179_ereal2__cases,axiom,
% 2.48/1.63 ! [X: extended_ereal,Xa3: extended_ereal] :
% 2.48/1.63 ( ( ? [R: real] :
% 2.48/1.63 ( X
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ! [Ra: real] :
% 2.48/1.63 ( Xa3
% 2.48/1.63 != ( extended_ereal2 @ Ra ) ) )
% 2.48/1.63 => ( ( ? [R: real] :
% 2.48/1.63 ( X
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( Xa3
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( ( ? [R: real] :
% 2.48/1.63 ( X
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( Xa3
% 2.48/1.63 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ! [R: real] :
% 2.48/1.63 ( Xa3
% 2.48/1.63 != ( extended_ereal2 @ R ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( Xa3
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( Xa3
% 2.48/1.63 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ! [R: real] :
% 2.48/1.63 ( Xa3
% 2.48/1.63 != ( extended_ereal2 @ R ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( Xa3
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ~ ( ( X
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( Xa3
% 2.48/1.63 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_180_ereal3__cases,axiom,
% 2.48/1.63 ! [X: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
% 2.48/1.63 ( ( ? [R: real] :
% 2.48/1.63 ( X
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ? [Ra: real] :
% 2.48/1.63 ( Xa3
% 2.48/1.63 = ( extended_ereal2 @ Ra ) )
% 2.48/1.63 => ! [Rb: real] :
% 2.48/1.63 ( Xb
% 2.48/1.63 != ( extended_ereal2 @ Rb ) ) ) )
% 2.48/1.63 => ( ( ? [R: real] :
% 2.48/1.63 ( X
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ? [Ra: real] :
% 2.48/1.63 ( Xa3
% 2.48/1.63 = ( extended_ereal2 @ Ra ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.48/1.63 => ( ( ? [R: real] :
% 2.48/1.63 ( X
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ? [Ra: real] :
% 2.48/1.63 ( Xa3
% 2.48/1.63 = ( extended_ereal2 @ Ra ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.48/1.63 => ( ( ? [R: real] :
% 2.48/1.63 ( X
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ! [Ra: real] :
% 2.48/1.63 ( Xb
% 2.48/1.63 != ( extended_ereal2 @ Ra ) ) ) )
% 2.48/1.63 => ( ( ? [R: real] :
% 2.48/1.63 ( X
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.48/1.63 => ( ( ? [R: real] :
% 2.48/1.63 ( X
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.48/1.63 => ( ( ? [R: real] :
% 2.48/1.63 ( X
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ! [Ra: real] :
% 2.48/1.63 ( Xb
% 2.48/1.63 != ( extended_ereal2 @ Ra ) ) ) )
% 2.48/1.63 => ( ( ? [R: real] :
% 2.48/1.63 ( X
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.48/1.63 => ( ( ? [R: real] :
% 2.48/1.63 ( X
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( ? [R: real] :
% 2.48/1.63 ( Xa3
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ! [Ra: real] :
% 2.48/1.63 ( Xb
% 2.48/1.63 != ( extended_ereal2 @ Ra ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( ? [R: real] :
% 2.48/1.63 ( Xa3
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( ? [R: real] :
% 2.48/1.63 ( Xa3
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ! [R: real] :
% 2.48/1.63 ( Xb
% 2.48/1.63 != ( extended_ereal2 @ R ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ! [R: real] :
% 2.48/1.63 ( Xb
% 2.48/1.63 != ( extended_ereal2 @ R ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( ? [R: real] :
% 2.48/1.63 ( Xa3
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ! [Ra: real] :
% 2.48/1.63 ( Xb
% 2.48/1.63 != ( extended_ereal2 @ Ra ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( ? [R: real] :
% 2.48/1.63 ( Xa3
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( ? [R: real] :
% 2.48/1.63 ( Xa3
% 2.48/1.63 = ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ! [R: real] :
% 2.48/1.63 ( Xb
% 2.48/1.63 != ( extended_ereal2 @ R ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ! [R: real] :
% 2.48/1.63 ( Xb
% 2.48/1.63 != ( extended_ereal2 @ R ) ) ) )
% 2.48/1.63 => ( ( ( X
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.48/1.63 => ~ ( ( X
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( ( Xa3
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( Xb
% 2.48/1.63 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_181_ereal__ex__split,axiom,
% 2.48/1.63 ( ( ^ [P3: extended_ereal > $o] :
% 2.48/1.63 ? [X7: extended_ereal] : ( P3 @ X7 ) )
% 2.48/1.63 = ( ^ [P4: extended_ereal > $o] :
% 2.48/1.63 ( ( P4 @ ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 | ? [X2: real] : ( P4 @ ( extended_ereal2 @ X2 ) )
% 2.48/1.63 | ( P4 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_182_abs__ereal_Ocases,axiom,
% 2.48/1.63 ! [X: extended_ereal] :
% 2.48/1.63 ( ! [R: real] :
% 2.48/1.63 ( X
% 2.48/1.63 != ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ( X
% 2.48/1.63 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( X
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_183_ereal__all__split,axiom,
% 2.48/1.63 ( ( ^ [P3: extended_ereal > $o] :
% 2.48/1.63 ! [X7: extended_ereal] : ( P3 @ X7 ) )
% 2.48/1.63 = ( ^ [P4: extended_ereal > $o] :
% 2.48/1.63 ( ( P4 @ ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 & ! [X2: real] : ( P4 @ ( extended_ereal2 @ X2 ) )
% 2.48/1.63 & ( P4 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_184_abs__ereal_Oinduct,axiom,
% 2.48/1.63 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 2.48/1.63 ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( ( P @ ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( P @ A0 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_185_less__ereal_Oinduct,axiom,
% 2.48/1.63 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 2.48/1.63 ( ! [X4: real,Y2: real] : ( P @ ( extended_ereal2 @ X4 ) @ ( extended_ereal2 @ Y2 ) )
% 2.48/1.63 => ( ! [X_1: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_1 )
% 2.48/1.63 => ( ! [A6: extended_ereal] : ( P @ A6 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( ! [X4: real] : ( P @ ( extended_ereal2 @ X4 ) @ ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( ! [R: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_186_plus__ereal_Oinduct,axiom,
% 2.48/1.63 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 2.48/1.63 ( ! [R: real,P5: real] : ( P @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P5 ) )
% 2.48/1.63 => ( ! [X_1: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_1 )
% 2.48/1.63 => ( ! [A6: extended_ereal] : ( P @ A6 @ ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( ! [P5: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ P5 ) )
% 2.48/1.63 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_187_times__ereal_Oinduct,axiom,
% 2.48/1.63 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 2.48/1.63 ( ! [R: real,P5: real] : ( P @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P5 ) )
% 2.48/1.63 => ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( ! [R: real] : ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( ! [R: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_188_real__of__ereal_Ocases,axiom,
% 2.48/1.63 ! [X: extended_ereal] :
% 2.48/1.63 ( ! [R: real] :
% 2.48/1.63 ( X
% 2.48/1.63 != ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ( X
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( X
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_189_real__of__ereal_Oinduct,axiom,
% 2.48/1.63 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 2.48/1.63 ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ( P @ ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 => ( P @ A0 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_190_domain__Epigraph__aux,axiom,
% 2.48/1.63 ! [X: extended_ereal] :
% 2.48/1.63 ( ( X
% 2.48/1.63 != ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 => ? [R: real] : ( ord_less_eq @ extended_ereal @ X @ ( extended_ereal2 @ R ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_191_ereal__top,axiom,
% 2.48/1.63 ! [X: extended_ereal] :
% 2.48/1.63 ( ! [B7: real] : ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ B7 ) @ X )
% 2.48/1.63 => ( X
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_192_Infty__neq__0_I3_J,axiom,
% 2.48/1.63 ( ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 != ( zero_zero @ extended_ereal ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_193_ereal_Oinduct,axiom,
% 2.48/1.63 ! [P: extended_ereal > $o,Ereal: extended_ereal] :
% 2.48/1.63 ( ! [X4: real] : ( P @ ( extended_ereal2 @ X4 ) )
% 2.48/1.63 => ( ( P @ extended_PInfty )
% 2.48/1.63 => ( ( P @ extended_MInfty )
% 2.48/1.63 => ( P @ Ereal ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_194_ereal_Oexhaust,axiom,
% 2.48/1.63 ! [Y4: extended_ereal] :
% 2.48/1.63 ( ! [X12: real] :
% 2.48/1.63 ( Y4
% 2.48/1.63 != ( extended_ereal2 @ X12 ) )
% 2.48/1.63 => ( ( Y4 != extended_PInfty )
% 2.48/1.63 => ( Y4 = extended_MInfty ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_195_uminus__ereal_Ocases,axiom,
% 2.48/1.63 ! [X: extended_ereal] :
% 2.48/1.63 ( ! [R: real] :
% 2.48/1.63 ( X
% 2.48/1.63 != ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ( X != extended_PInfty )
% 2.48/1.63 => ( X = extended_MInfty ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_196_uminus__ereal_Oinduct,axiom,
% 2.48/1.63 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 2.48/1.63 ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
% 2.48/1.63 => ( ( P @ extended_PInfty )
% 2.48/1.63 => ( ( P @ extended_MInfty )
% 2.48/1.63 => ( P @ A0 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_197_ereal__bot,axiom,
% 2.48/1.63 ! [X: extended_ereal] :
% 2.48/1.63 ( ! [B7: real] : ( ord_less_eq @ extended_ereal @ X @ ( extended_ereal2 @ B7 ) )
% 2.48/1.63 => ( X
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_198_le__numeral__extra_I3_J,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( linord1659791738miring @ A )
% 2.48/1.63 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_199_ereal__less__eq_I4_J,axiom,
% 2.48/1.63 ! [R2: real] :
% 2.48/1.63 ( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ R2 ) @ ( zero_zero @ extended_ereal ) )
% 2.48/1.63 = ( ord_less_eq @ real @ R2 @ ( zero_zero @ real ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_200_ereal__less__eq_I5_J,axiom,
% 2.48/1.63 ! [R2: real] :
% 2.48/1.63 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( extended_ereal2 @ R2 ) )
% 2.48/1.63 = ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_201_ereal__eq__0_I2_J,axiom,
% 2.48/1.63 ! [R2: real] :
% 2.48/1.63 ( ( ( zero_zero @ extended_ereal )
% 2.48/1.63 = ( extended_ereal2 @ R2 ) )
% 2.48/1.63 = ( R2
% 2.48/1.63 = ( zero_zero @ real ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_202_ereal__eq__0_I1_J,axiom,
% 2.48/1.63 ! [R2: real] :
% 2.48/1.63 ( ( ( extended_ereal2 @ R2 )
% 2.48/1.63 = ( zero_zero @ extended_ereal ) )
% 2.48/1.63 = ( R2
% 2.48/1.63 = ( zero_zero @ real ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_203_zero__ereal__def,axiom,
% 2.48/1.63 ( ( zero_zero @ extended_ereal )
% 2.48/1.63 = ( extended_ereal2 @ ( zero_zero @ real ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_204_real__eq__0__iff__le__ge__0,axiom,
% 2.48/1.63 ! [X: real] :
% 2.48/1.63 ( ( X
% 2.48/1.63 = ( zero_zero @ real ) )
% 2.48/1.63 = ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
% 2.48/1.63 & ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( uminus_uminus @ real @ X ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_205_ereal__divide__ereal,axiom,
% 2.48/1.63 ! [R2: real] :
% 2.48/1.63 ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 )
% 2.48/1.63 => ( ( divide_divide @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R2 ) )
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 & ( ~ ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 )
% 2.48/1.63 => ( ( divide_divide @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R2 ) )
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_206_ereal__uminus__divide,axiom,
% 2.48/1.63 ! [X: extended_ereal,Y4: extended_ereal] :
% 2.48/1.63 ( ( divide_divide @ extended_ereal @ ( uminus_uminus @ extended_ereal @ X ) @ Y4 )
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( divide_divide @ extended_ereal @ X @ Y4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_207_ereal__divide__zero__left,axiom,
% 2.48/1.63 ! [A4: extended_ereal] :
% 2.48/1.63 ( ( divide_divide @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A4 )
% 2.48/1.63 = ( zero_zero @ extended_ereal ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_208_ereal__divide__Infty_I1_J,axiom,
% 2.48/1.63 ! [X: extended_ereal] :
% 2.48/1.63 ( ( divide_divide @ extended_ereal @ X @ ( extend1396239628finity @ extended_ereal ) )
% 2.48/1.63 = ( zero_zero @ extended_ereal ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_209_ereal__divide__Infty_I2_J,axiom,
% 2.48/1.63 ! [X: extended_ereal] :
% 2.48/1.63 ( ( divide_divide @ extended_ereal @ X @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.48/1.63 = ( zero_zero @ extended_ereal ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_210_zero__le__divide__ereal,axiom,
% 2.48/1.63 ! [A4: extended_ereal,B4: extended_ereal] :
% 2.48/1.63 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A4 )
% 2.48/1.63 => ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ B4 )
% 2.48/1.63 => ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( divide_divide @ extended_ereal @ A4 @ B4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_211_div__minus__minus,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( euclid24285859cancel @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( divide_divide @ A @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ A @ B4 ) )
% 2.48/1.63 = ( divide_divide @ A @ A4 @ B4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_212_div__0,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( semidom_divide @ A )
% 2.48/1.63 => ! [A4: A] :
% 2.48/1.63 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A4 )
% 2.48/1.63 = ( zero_zero @ A ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_213_div__by__0,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( semidom_divide @ A )
% 2.48/1.63 => ! [A4: A] :
% 2.48/1.63 ( ( divide_divide @ A @ A4 @ ( zero_zero @ A ) )
% 2.48/1.63 = ( zero_zero @ A ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_214_div__minus__right,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( euclid24285859cancel @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( divide_divide @ A @ A4 @ ( uminus_uminus @ A @ B4 ) )
% 2.48/1.63 = ( divide_divide @ A @ ( uminus_uminus @ A @ A4 ) @ B4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_215_division__ring__divide__zero,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( division_ring @ A )
% 2.48/1.63 => ! [A4: A] :
% 2.48/1.63 ( ( divide_divide @ A @ A4 @ ( zero_zero @ A ) )
% 2.48/1.63 = ( zero_zero @ A ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_216_bits__div__by__0,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( semiring_bits @ A )
% 2.48/1.63 => ! [A4: A] :
% 2.48/1.63 ( ( divide_divide @ A @ A4 @ ( zero_zero @ A ) )
% 2.48/1.63 = ( zero_zero @ A ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_217_divide__eq__0__iff,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( field @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( ( divide_divide @ A @ A4 @ B4 )
% 2.48/1.63 = ( zero_zero @ A ) )
% 2.48/1.63 = ( ( A4
% 2.48/1.63 = ( zero_zero @ A ) )
% 2.48/1.63 | ( B4
% 2.48/1.63 = ( zero_zero @ A ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_218_divide__cancel__left,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( field @ A )
% 2.48/1.63 => ! [C2: A,A4: A,B4: A] :
% 2.48/1.63 ( ( ( divide_divide @ A @ C2 @ A4 )
% 2.48/1.63 = ( divide_divide @ A @ C2 @ B4 ) )
% 2.48/1.63 = ( ( C2
% 2.48/1.63 = ( zero_zero @ A ) )
% 2.48/1.63 | ( A4 = B4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_219_bits__div__0,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( semiring_bits @ A )
% 2.48/1.63 => ! [A4: A] :
% 2.48/1.63 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A4 )
% 2.48/1.63 = ( zero_zero @ A ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_220_divide__cancel__right,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( field @ A )
% 2.48/1.63 => ! [A4: A,C2: A,B4: A] :
% 2.48/1.63 ( ( ( divide_divide @ A @ A4 @ C2 )
% 2.48/1.63 = ( divide_divide @ A @ B4 @ C2 ) )
% 2.48/1.63 = ( ( C2
% 2.48/1.63 = ( zero_zero @ A ) )
% 2.48/1.63 | ( A4 = B4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_221_minus__divide__left,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( division_ring @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ B4 ) )
% 2.48/1.63 = ( divide_divide @ A @ ( uminus_uminus @ A @ A4 ) @ B4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_222_minus__divide__divide,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( field @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( divide_divide @ A @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ A @ B4 ) )
% 2.48/1.63 = ( divide_divide @ A @ A4 @ B4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_223_minus__divide__right,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( field @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ B4 ) )
% 2.48/1.63 = ( divide_divide @ A @ A4 @ ( uminus_uminus @ A @ B4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_224_divide__right__mono__neg,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( linordered_field @ A )
% 2.48/1.63 => ! [A4: A,B4: A,C2: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.48/1.63 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 2.48/1.63 => ( ord_less_eq @ A @ ( divide_divide @ A @ B4 @ C2 ) @ ( divide_divide @ A @ A4 @ C2 ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_225_divide__nonpos__nonpos,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( linordered_field @ A )
% 2.48/1.63 => ! [X: A,Y4: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 2.48/1.63 => ( ( ord_less_eq @ A @ Y4 @ ( zero_zero @ A ) )
% 2.48/1.63 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y4 ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_226_divide__nonpos__nonneg,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( linordered_field @ A )
% 2.48/1.63 => ! [X: A,Y4: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 2.48/1.63 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y4 )
% 2.48/1.63 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_227_divide__nonneg__nonpos,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( linordered_field @ A )
% 2.48/1.63 => ! [X: A,Y4: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.48/1.63 => ( ( ord_less_eq @ A @ Y4 @ ( zero_zero @ A ) )
% 2.48/1.63 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_228_divide__nonneg__nonneg,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( linordered_field @ A )
% 2.48/1.63 => ! [X: A,Y4: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.48/1.63 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y4 )
% 2.48/1.63 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y4 ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_229_zero__le__divide__iff,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( linordered_field @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A4 @ B4 ) )
% 2.48/1.63 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 2.48/1.63 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) )
% 2.48/1.63 | ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
% 2.48/1.63 & ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_230_divide__right__mono,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( linordered_field @ A )
% 2.48/1.63 => ! [A4: A,B4: A,C2: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.48/1.63 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 2.48/1.63 => ( ord_less_eq @ A @ ( divide_divide @ A @ A4 @ C2 ) @ ( divide_divide @ A @ B4 @ C2 ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_231_divide__le__0__iff,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( linordered_field @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A4 @ B4 ) @ ( zero_zero @ A ) )
% 2.48/1.63 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 2.48/1.63 & ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) )
% 2.48/1.63 | ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
% 2.48/1.63 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_232_nonzero__minus__divide__divide,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( division_ring @ A )
% 2.48/1.63 => ! [B4: A,A4: A] :
% 2.48/1.63 ( ( B4
% 2.48/1.63 != ( zero_zero @ A ) )
% 2.48/1.63 => ( ( divide_divide @ A @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ A @ B4 ) )
% 2.48/1.63 = ( divide_divide @ A @ A4 @ B4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_233_nonzero__minus__divide__right,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( division_ring @ A )
% 2.48/1.63 => ! [B4: A,A4: A] :
% 2.48/1.63 ( ( B4
% 2.48/1.63 != ( zero_zero @ A ) )
% 2.48/1.63 => ( ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ B4 ) )
% 2.48/1.63 = ( divide_divide @ A @ A4 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_234_real__of__ereal__le__0,axiom,
% 2.48/1.63 ! [X: extended_ereal] :
% 2.48/1.63 ( ( ord_less_eq @ real @ ( extend1716541707_ereal @ X ) @ ( zero_zero @ real ) )
% 2.48/1.63 = ( ( ord_less_eq @ extended_ereal @ X @ ( zero_zero @ extended_ereal ) )
% 2.48/1.63 | ( X
% 2.48/1.63 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_235_ereal__divide,axiom,
% 2.48/1.63 ! [P2: real,R2: real] :
% 2.48/1.63 ( ( ( P2
% 2.48/1.63 = ( zero_zero @ real ) )
% 2.48/1.63 => ( ( divide_divide @ extended_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
% 2.48/1.63 = ( times_times @ extended_ereal @ ( extended_ereal2 @ R2 ) @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.48/1.63 & ( ( P2
% 2.48/1.63 != ( zero_zero @ real ) )
% 2.48/1.63 => ( ( divide_divide @ extended_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
% 2.48/1.63 = ( extended_ereal2 @ ( divide_divide @ real @ R2 @ P2 ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_236_mult__zero__left,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( mult_zero @ A )
% 2.48/1.63 => ! [A4: A] :
% 2.48/1.63 ( ( times_times @ A @ ( zero_zero @ A ) @ A4 )
% 2.48/1.63 = ( zero_zero @ A ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_237_mult__zero__right,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( mult_zero @ A )
% 2.48/1.63 => ! [A4: A] :
% 2.48/1.63 ( ( times_times @ A @ A4 @ ( zero_zero @ A ) )
% 2.48/1.63 = ( zero_zero @ A ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_238_mult__eq__0__iff,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( semiri1193490041visors @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( ( times_times @ A @ A4 @ B4 )
% 2.48/1.63 = ( zero_zero @ A ) )
% 2.48/1.63 = ( ( A4
% 2.48/1.63 = ( zero_zero @ A ) )
% 2.48/1.63 | ( B4
% 2.48/1.63 = ( zero_zero @ A ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_239_mult__cancel__left,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( semiri1923998003cancel @ A )
% 2.48/1.63 => ! [C2: A,A4: A,B4: A] :
% 2.48/1.63 ( ( ( times_times @ A @ C2 @ A4 )
% 2.48/1.63 = ( times_times @ A @ C2 @ B4 ) )
% 2.48/1.63 = ( ( C2
% 2.48/1.63 = ( zero_zero @ A ) )
% 2.48/1.63 | ( A4 = B4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_240_mult__cancel__right,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( semiri1923998003cancel @ A )
% 2.48/1.63 => ! [A4: A,C2: A,B4: A] :
% 2.48/1.63 ( ( ( times_times @ A @ A4 @ C2 )
% 2.48/1.63 = ( times_times @ A @ B4 @ C2 ) )
% 2.48/1.63 = ( ( C2
% 2.48/1.63 = ( zero_zero @ A ) )
% 2.48/1.63 | ( A4 = B4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_241_mult__minus__right,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( ring @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( times_times @ A @ A4 @ ( uminus_uminus @ A @ B4 ) )
% 2.48/1.63 = ( uminus_uminus @ A @ ( times_times @ A @ A4 @ B4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_242_minus__mult__minus,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( ring @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( times_times @ A @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ A @ B4 ) )
% 2.48/1.63 = ( times_times @ A @ A4 @ B4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_243_mult__minus__left,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( ring @ A )
% 2.48/1.63 => ! [A4: A,B4: A] :
% 2.48/1.63 ( ( times_times @ A @ ( uminus_uminus @ A @ A4 ) @ B4 )
% 2.48/1.63 = ( uminus_uminus @ A @ ( times_times @ A @ A4 @ B4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_244_ereal__mult__minus__right,axiom,
% 2.48/1.63 ! [A4: extended_ereal,B4: extended_ereal] :
% 2.48/1.63 ( ( times_times @ extended_ereal @ A4 @ ( uminus_uminus @ extended_ereal @ B4 ) )
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( times_times @ extended_ereal @ A4 @ B4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_245_ereal__mult__minus__left,axiom,
% 2.48/1.63 ! [A4: extended_ereal,B4: extended_ereal] :
% 2.48/1.63 ( ( times_times @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A4 ) @ B4 )
% 2.48/1.63 = ( uminus_uminus @ extended_ereal @ ( times_times @ extended_ereal @ A4 @ B4 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_246_ereal__mult__zero,axiom,
% 2.48/1.63 ! [A4: extended_ereal] :
% 2.48/1.63 ( ( times_times @ extended_ereal @ A4 @ ( zero_zero @ extended_ereal ) )
% 2.48/1.63 = ( zero_zero @ extended_ereal ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_247_ereal__zero__mult,axiom,
% 2.48/1.63 ! [A4: extended_ereal] :
% 2.48/1.63 ( ( times_times @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A4 )
% 2.48/1.63 = ( zero_zero @ extended_ereal ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_248_ereal__zero__times,axiom,
% 2.48/1.63 ! [A4: extended_ereal,B4: extended_ereal] :
% 2.48/1.63 ( ( ( times_times @ extended_ereal @ A4 @ B4 )
% 2.48/1.63 = ( zero_zero @ extended_ereal ) )
% 2.48/1.63 = ( ( A4
% 2.48/1.63 = ( zero_zero @ extended_ereal ) )
% 2.48/1.63 | ( B4
% 2.48/1.63 = ( zero_zero @ extended_ereal ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_249_ereal__times__divide__eq__left,axiom,
% 2.48/1.63 ! [B4: extended_ereal,C2: extended_ereal,A4: extended_ereal] :
% 2.48/1.63 ( ( times_times @ extended_ereal @ ( divide_divide @ extended_ereal @ B4 @ C2 ) @ A4 )
% 2.48/1.63 = ( divide_divide @ extended_ereal @ ( times_times @ extended_ereal @ B4 @ A4 ) @ C2 ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_250_mult__divide__mult__cancel__left__if,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( field @ A )
% 2.48/1.63 => ! [C2: A,A4: A,B4: A] :
% 2.48/1.63 ( ( ( C2
% 2.48/1.63 = ( zero_zero @ A ) )
% 2.48/1.63 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) )
% 2.48/1.63 = ( zero_zero @ A ) ) )
% 2.48/1.63 & ( ( C2
% 2.48/1.63 != ( zero_zero @ A ) )
% 2.48/1.63 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) )
% 2.48/1.63 = ( divide_divide @ A @ A4 @ B4 ) ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_251_nonzero__mult__divide__mult__cancel__left,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( field @ A )
% 2.48/1.63 => ! [C2: A,A4: A,B4: A] :
% 2.48/1.63 ( ( C2
% 2.48/1.63 != ( zero_zero @ A ) )
% 2.48/1.63 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) )
% 2.48/1.63 = ( divide_divide @ A @ A4 @ B4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_252_nonzero__mult__divide__mult__cancel__left2,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( field @ A )
% 2.48/1.63 => ! [C2: A,A4: A,B4: A] :
% 2.48/1.63 ( ( C2
% 2.48/1.63 != ( zero_zero @ A ) )
% 2.48/1.63 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ B4 @ C2 ) )
% 2.48/1.63 = ( divide_divide @ A @ A4 @ B4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_253_nonzero__mult__divide__mult__cancel__right,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( field @ A )
% 2.48/1.63 => ! [C2: A,A4: A,B4: A] :
% 2.48/1.63 ( ( C2
% 2.48/1.63 != ( zero_zero @ A ) )
% 2.48/1.63 => ( ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ C2 ) )
% 2.48/1.63 = ( divide_divide @ A @ A4 @ B4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(fact_254_nonzero__mult__divide__mult__cancel__right2,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( field @ A )
% 2.48/1.63 => ! [C2: A,A4: A,B4: A] :
% 2.48/1.63 ( ( C2
% 2.48/1.63 != ( zero_zero @ A ) )
% 2.48/1.63 => ( ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ C2 @ B4 ) )
% 2.48/1.63 = ( divide_divide @ A @ A4 @ B4 ) ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(subcl_Euclidean__Space_Oeuclidean__space___HOL_Otype,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( euclid925273238_space @ A )
% 2.48/1.63 => ( type @ A ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(subcl_Euclidean__Space_Oeuclidean__space___Groups_Ozero,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( euclid925273238_space @ A )
% 2.48/1.63 => ( zero @ A ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(subcl_Euclidean__Space_Oeuclidean__space___Groups_Ouminus,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( euclid925273238_space @ A )
% 2.48/1.63 => ( uminus @ A ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(subcl_Euclidean__Space_Oeuclidean__space___Groups_Ogroup__add,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( euclid925273238_space @ A )
% 2.48/1.63 => ( group_add @ A ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(subcl_Euclidean__Space_Oeuclidean__space___Real__Vector__Spaces_Oreal__vector,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( euclid925273238_space @ A )
% 2.48/1.63 => ( real_V1076094709vector @ A ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(subcl_Euclidean__Space_Oeuclidean__space___Real__Vector__Spaces_Ometric__space,axiom,
% 2.48/1.63 ! [A: $tType] :
% 2.48/1.63 ( ( euclid925273238_space @ A )
% 2.48/1.63 => ( real_V2090557954_space @ A ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( ( ordere890947078_space @ A7 )
% 2.48/1.63 & ( ordere890947078_space @ A8 ) )
% 2.48/1.63 => ( ordere890947078_space @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_1,axiom,
% 2.48/1.63 ordere890947078_space @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( comple187826305attice @ A8 )
% 2.48/1.63 => ( condit378418413attice @ ( A7 > A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( comple187826305attice @ A8 )
% 2.48/1.63 => ( comple187826305attice @ ( A7 > A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( boolean_algebra @ A8 )
% 2.48/1.63 => ( boolean_algebra @ ( A7 > A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( order_top @ A8 )
% 2.48/1.63 => ( order_top @ ( A7 > A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( preorder @ A8 )
% 2.48/1.63 => ( preorder @ ( A7 > A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( order @ A8 )
% 2.48/1.63 => ( order @ ( A7 > A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_fun___Orderings_Otop,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( top @ A8 )
% 2.48/1.63 => ( top @ ( A7 > A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_fun___Orderings_Oord,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( ord @ A8 )
% 2.48/1.63 => ( ord @ ( A7 > A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_fun___Groups_Ouminus,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( uminus @ A8 )
% 2.48/1.63 => ( uminus @ ( A7 > A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_2,axiom,
% 2.48/1.63 ! [A7: $tType] : ( condit378418413attice @ ( set @ A7 ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_3,axiom,
% 2.48/1.63 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_4,axiom,
% 2.48/1.63 ! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Set_Oset___Orderings_Oorder__top_5,axiom,
% 2.48/1.63 ! [A7: $tType] : ( order_top @ ( set @ A7 ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Set_Oset___Orderings_Opreorder_6,axiom,
% 2.48/1.63 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Set_Oset___Orderings_Oorder_7,axiom,
% 2.48/1.63 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Set_Oset___Orderings_Otop_8,axiom,
% 2.48/1.63 ! [A7: $tType] : ( top @ ( set @ A7 ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
% 2.48/1.63 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Set_Oset___Groups_Ouminus_10,axiom,
% 2.48/1.63 ! [A7: $tType] : ( uminus @ ( set @ A7 ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Set_Oset___Groups_Ozero,axiom,
% 2.48/1.63 ! [A7: $tType] :
% 2.48/1.63 ( ( zero @ A7 )
% 2.48/1.63 => ( zero @ ( set @ A7 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11,axiom,
% 2.48/1.63 condit378418413attice @ $o ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_12,axiom,
% 2.48/1.63 comple187826305attice @ $o ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_13,axiom,
% 2.48/1.63 boolean_algebra @ $o ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_HOL_Obool___Orderings_Oorder__top_14,axiom,
% 2.48/1.63 order_top @ $o ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_HOL_Obool___Orderings_Opreorder_15,axiom,
% 2.48/1.63 preorder @ $o ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.48/1.63 linorder @ $o ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
% 2.48/1.63 order @ $o ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_HOL_Obool___Orderings_Otop_17,axiom,
% 2.48/1.63 top @ $o ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_HOL_Obool___Orderings_Oord_18,axiom,
% 2.48/1.63 ord @ $o ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_HOL_Obool___Groups_Ouminus_19,axiom,
% 2.48/1.63 uminus @ $o ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_20,axiom,
% 2.48/1.63 condit378418413attice @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel,axiom,
% 2.48/1.63 semiri1923998003cancel @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring,axiom,
% 2.48/1.63 linord1659791738miring @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors,axiom,
% 2.48/1.63 semiri1193490041visors @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
% 2.48/1.63 real_V2090557954_space @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space,axiom,
% 2.48/1.63 euclid925273238_space @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
% 2.48/1.63 real_V1076094709vector @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
% 2.48/1.63 linord219039673up_add @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
% 2.48/1.63 ordered_ab_group_add @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 2.48/1.63 linordered_field @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Rings_Osemidom__divide,axiom,
% 2.48/1.63 semidom_divide @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
% 2.48/1.63 division_ring @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Orderings_Opreorder_21,axiom,
% 2.48/1.63 preorder @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Orderings_Olinorder_22,axiom,
% 2.48/1.63 linorder @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Orderings_Ono__bot,axiom,
% 2.48/1.63 no_bot @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 2.48/1.63 group_add @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Rings_Omult__zero,axiom,
% 2.48/1.63 mult_zero @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Orderings_Oorder_23,axiom,
% 2.48/1.63 order @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Orderings_Oord_24,axiom,
% 2.48/1.63 ord @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Groups_Ouminus_25,axiom,
% 2.48/1.63 uminus @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Fields_Ofield,axiom,
% 2.48/1.63 field @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Groups_Ozero_26,axiom,
% 2.48/1.63 zero @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Real_Oreal___Rings_Oring,axiom,
% 2.48/1.63 ring @ real ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Product__Type_Oprod___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_27,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( ( condit378418413attice @ A7 )
% 2.48/1.63 & ( condit378418413attice @ A8 ) )
% 2.48/1.63 => ( condit378418413attice @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Product__Type_Oprod___Complete__Lattices_Ocomplete__lattice_28,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( ( comple187826305attice @ A7 )
% 2.48/1.63 & ( comple187826305attice @ A8 ) )
% 2.48/1.63 => ( comple187826305attice @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Ometric__space_29,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( ( real_V2090557954_space @ A7 )
% 2.48/1.63 & ( real_V2090557954_space @ A8 ) )
% 2.48/1.63 => ( real_V2090557954_space @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Product__Type_Oprod___Euclidean__Space_Oeuclidean__space_30,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( ( euclid925273238_space @ A7 )
% 2.48/1.63 & ( euclid925273238_space @ A8 ) )
% 2.48/1.63 => ( euclid925273238_space @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__vector_31,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( ( real_V1076094709vector @ A7 )
% 2.48/1.63 & ( real_V1076094709vector @ A8 ) )
% 2.48/1.63 => ( real_V1076094709vector @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__group__add_32,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( ( ordere890947078_space @ A7 )
% 2.48/1.63 & ( ordere890947078_space @ A8 ) )
% 2.48/1.63 => ( ordered_ab_group_add @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Product__Type_Oprod___Lattices_Oboolean__algebra_33,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( ( boolean_algebra @ A7 )
% 2.48/1.63 & ( boolean_algebra @ A8 ) )
% 2.48/1.63 => ( boolean_algebra @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Product__Type_Oprod___Orderings_Oorder__top_34,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( ( order_top @ A7 )
% 2.48/1.63 & ( order_top @ A8 ) )
% 2.48/1.63 => ( order_top @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_35,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( ( preorder @ A7 )
% 2.48/1.63 & ( preorder @ A8 ) )
% 2.48/1.63 => ( preorder @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Product__Type_Oprod___Groups_Ogroup__add_36,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( ( group_add @ A7 )
% 2.48/1.63 & ( group_add @ A8 ) )
% 2.48/1.63 => ( group_add @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Product__Type_Oprod___Orderings_Oorder_37,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( ( order @ A7 )
% 2.48/1.63 & ( order @ A8 ) )
% 2.48/1.63 => ( order @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Product__Type_Oprod___Orderings_Otop_38,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( ( top @ A7 )
% 2.48/1.63 & ( top @ A8 ) )
% 2.48/1.63 => ( top @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Product__Type_Oprod___Orderings_Oord_39,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( ( ord @ A7 )
% 2.48/1.63 & ( ord @ A8 ) )
% 2.48/1.63 => ( ord @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Product__Type_Oprod___Groups_Ouminus_40,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( ( uminus @ A7 )
% 2.48/1.63 & ( uminus @ A8 ) )
% 2.48/1.63 => ( uminus @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Product__Type_Oprod___Groups_Ozero_41,axiom,
% 2.48/1.63 ! [A7: $tType,A8: $tType] :
% 2.48/1.63 ( ( ( zero @ A7 )
% 2.48/1.63 & ( zero @ A8 ) )
% 2.48/1.63 => ( zero @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Extended__Real_Oereal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_42,axiom,
% 2.48/1.63 condit378418413attice @ extended_ereal ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Extended__Real_Oereal___Complete__Lattices_Ocomplete__lattice_43,axiom,
% 2.48/1.63 comple187826305attice @ extended_ereal ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Extended__Real_Oereal___Orderings_Oorder__top_44,axiom,
% 2.48/1.63 order_top @ extended_ereal ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Extended__Real_Oereal___Orderings_Opreorder_45,axiom,
% 2.48/1.63 preorder @ extended_ereal ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Extended__Real_Oereal___Orderings_Olinorder_46,axiom,
% 2.48/1.63 linorder @ extended_ereal ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Extended__Real_Oereal___Orderings_Oorder_47,axiom,
% 2.48/1.63 order @ extended_ereal ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Extended__Real_Oereal___Orderings_Otop_48,axiom,
% 2.48/1.63 top @ extended_ereal ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Extended__Real_Oereal___Orderings_Oord_49,axiom,
% 2.48/1.63 ord @ extended_ereal ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Extended__Real_Oereal___Groups_Ouminus_50,axiom,
% 2.48/1.63 uminus @ extended_ereal ).
% 2.48/1.63
% 2.48/1.63 thf(tcon_Extended__Real_Oereal___Groups_Ozero_51,axiom,
% 2.48/1.63 zero @ extended_ereal ).
% 2.48/1.63
% 2.48/1.63 thf(tfree_0,hypothesis,
% 2.48/1.63 euclid925273238_space @ a ).
% 2.48/1.63
% 2.48/1.63 thf(conj_0,conjecture,
% 2.48/1.63 ( ( lower_673667120vex_on @ a @ ( top_top @ ( set @ a ) ) @ f )
% 2.48/1.63 = ( convex @ a @ ( lower_272802190domain @ a @ f ) ) ) ).
% 2.48/1.63 ------------------------
% 2.48/1.66 % (23472)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_23220 for (2999ds/2Mi)
% 2.48/1.66 % (23472)Instruction limit reached!
% 2.48/1.66 % (23472)------------------------------
% 2.48/1.66 % (23472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.48/1.66 % (23472)Termination reason: Unknown
% 2.48/1.66 % (23472)Termination phase: shuffling
% 2.48/1.66
% 2.48/1.66 % (23472)Memory used [KB]: 1407
% 2.48/1.66 % (23472)Time elapsed: 0.003 s
% 2.48/1.66 % (23472)Instructions burned: 5 (million)
% 2.48/1.66 % (23472)------------------------------
% 2.48/1.66 % (23472)------------------------------
% 2.48/1.67 % (23468)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_23220 for (2999ds/183Mi)
% 2.48/1.67 % (23469)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_23220 for (2999ds/4Mi)
% 2.48/1.67 % (23470)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_23220 for (2999ds/27Mi)
% 2.48/1.67 % (23471)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_23220 for (2999ds/2Mi)
% 2.48/1.67 % (23473)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_23220 for (2999ds/275Mi)
% 2.48/1.67 % (23474)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_23220 for (2999ds/18Mi)
% 2.48/1.67 % (23471)Instruction limit reached!
% 2.48/1.67 % (23471)------------------------------
% 2.48/1.67 % (23471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.48/1.67 % (23471)Termination reason: Unknown
% 2.48/1.67 % (23471)Termination phase: shuffling
% 2.48/1.67
% 2.48/1.67 % (23471)Memory used [KB]: 1407
% 2.48/1.67 % (23471)Time elapsed: 0.004 s
% 2.48/1.67 % (23471)Instructions burned: 2 (million)
% 2.48/1.67 % (23471)------------------------------
% 2.48/1.67 % (23471)------------------------------
% 2.48/1.67 % (23469)Instruction limit reached!
% 2.48/1.67 % (23469)------------------------------
% 2.48/1.67 % (23469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.48/1.67 % (23469)Termination reason: Unknown
% 2.48/1.67 % (23469)Termination phase: shuffling
% 2.48/1.67
% 2.48/1.67 % (23469)Memory used [KB]: 1407
% 2.48/1.67 % (23469)Time elapsed: 0.005 s
% 2.48/1.67 % (23469)Instructions burned: 5 (million)
% 2.48/1.67 % (23469)------------------------------
% 2.48/1.67 % (23469)------------------------------
% 2.48/1.67 % (23480)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_23220 for (2999ds/3Mi)
% 2.48/1.67 % (23480)Instruction limit reached!
% 2.48/1.67 % (23480)------------------------------
% 2.48/1.67 % (23480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.48/1.67 % (23480)Termination reason: Unknown
% 2.48/1.67 % (23480)Termination phase: shuffling
% 2.48/1.67
% 2.48/1.67 % (23480)Memory used [KB]: 1407
% 2.48/1.67 % (23480)Time elapsed: 0.002 s
% 2.48/1.67 % (23480)Instructions burned: 3 (million)
% 2.48/1.67 % (23480)------------------------------
% 2.48/1.67 % (23480)------------------------------
% 2.48/1.68 % (23474)Instruction limit reached!
% 2.48/1.68 % (23474)------------------------------
% 2.48/1.68 % (23474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.48/1.68 % (23474)Termination reason: Unknown
% 2.48/1.68 % (23474)Termination phase: Property scanning
% 2.48/1.68
% 2.48/1.68 % (23474)Memory used [KB]: 1663
% 2.48/1.68 % (23474)Time elapsed: 0.014 s
% 2.48/1.68 % (23474)Instructions burned: 19 (million)
% 2.48/1.68 % (23474)------------------------------
% 2.48/1.68 % (23474)------------------------------
% 2.48/1.68 % (23486)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_23220 for (2999ds/3Mi)
% 2.48/1.68 % (23486)Instruction limit reached!
% 2.48/1.68 % (23486)------------------------------
% 2.48/1.68 % (23486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.48/1.68 % (23486)Termination reason: Unknown
% 2.48/1.68 % (23486)Termination phase: shuffling
% 2.48/1.68
% 2.48/1.68 % (23486)Memory used [KB]: 1407
% 2.48/1.68 % (23486)Time elapsed: 0.002 s
% 2.48/1.68 % (23486)Instructions burned: 3 (million)
% 2.48/1.68 % (23486)------------------------------
% 2.48/1.68 % (23486)------------------------------
% 2.48/1.68 % (23470)Instruction limit reached!
% 2.48/1.68 % (23470)------------------------------
% 2.48/1.68 % (23470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.48/1.68 % (23470)Termination reason: Unknown
% 2.48/1.68 % (23470)Termination phase: Preprocessing 1
% 2.48/1.68
% 2.48/1.68 % (23470)Memory used [KB]: 1663
% 2.48/1.68 % (23470)Time elapsed: 0.018 s
% 2.48/1.68 % (23470)Instructions burned: 27 (million)
% 2.48/1.68 % (23470)------------------------------
% 2.48/1.68 % (23470)------------------------------
% 2.48/1.68 % (23483)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_23220 for (2999ds/37Mi)
% 2.48/1.68 % (23484)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_23220 for (2999ds/15Mi)
% 2.48/1.69 % (23492)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_23220 for (2999ds/7Mi)
% 2.48/1.69 % (23492)Instruction limit reached!
% 2.48/1.69 % (23492)------------------------------
% 2.48/1.69 % (23492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.48/1.69 % (23492)Termination reason: Unknown
% 2.48/1.69 % (23492)Termination phase: shuffling
% 2.48/1.69
% 2.48/1.69 % (23492)Memory used [KB]: 1535
% 2.48/1.69 % (23492)Time elapsed: 0.004 s
% 2.48/1.69 % (23492)Instructions burned: 9 (million)
% 2.48/1.69 % (23492)------------------------------
% 2.48/1.69 % (23492)------------------------------
% 2.48/1.69 % (23484)Instruction limit reached!
% 2.48/1.69 % (23484)------------------------------
% 2.48/1.69 % (23484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.48/1.69 % (23484)Termination reason: Unknown
% 2.48/1.69 % (23484)Termination phase: shuffling
% 2.48/1.69
% 2.48/1.69 % (23484)Memory used [KB]: 1663
% 2.48/1.69 % (23484)Time elapsed: 0.009 s
% 2.48/1.69 % (23484)Instructions burned: 15 (million)
% 2.48/1.69 % (23484)------------------------------
% 2.48/1.69 % (23484)------------------------------
% 2.48/1.70 % (23490)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_23220 for (2999ds/1041Mi)
% 2.52/1.70 % (23473)First to succeed.
% 2.52/1.70 % (23493)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_23220 for (2999ds/16Mi)
% 2.52/1.70 % (23473)Refutation found. Thanks to Tanya!
% 2.52/1.70 % SZS status Theorem for DTF2THF_23220
% 2.52/1.70 % SZS output start Proof for DTF2THF_23220
% 2.52/1.70 thf(type_def_5, type, set: ($tType) > $tType).
% 2.52/1.70 thf(type_def_7, type, extended_ereal: $tType).
% 2.52/1.70 thf(type_def_8, type, real: $tType).
% 2.52/1.70 thf(type_def_9, type, product_prod: ($tType * $tType) > $tType).
% 2.52/1.70 thf(type_def_10, type, a: $tType).
% 2.52/1.70 thf(func_def_0, type, extended_ereal: $tType).
% 2.52/1.70 thf(func_def_1, type, product_prod: ($tType * $tType) > $tType).
% 2.52/1.70 thf(func_def_2, type, real: $tType).
% 2.52/1.70 thf(func_def_3, type, set: ($tType) > $tType).
% 2.52/1.70 thf(func_def_4, type, a: $tType).
% 2.52/1.70 thf(func_def_5, type, ordere890947078_space: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_6, type, type: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_7, type, ring: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_8, type, zero: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_9, type, field: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_10, type, uminus: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_11, type, ord: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_12, type, top: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_13, type, order: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_14, type, mult_zero: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_15, type, group_add: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_16, type, no_bot: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_17, type, linorder: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_18, type, preorder: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_19, type, finite_lattice: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_20, type, order_top: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_21, type, division_ring: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_22, type, semiring_bits: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_23, type, semidom_divide: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_24, type, linordered_field: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_25, type, boolean_algebra: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_26, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_27, type, linord219039673up_add: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_28, type, real_V1076094709vector: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_29, type, euclid925273238_space: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_30, type, real_V2090557954_space: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_31, type, semiri1193490041visors: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_32, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_33, type, comple187826305attice: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_34, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_35, type, semiri1923998003cancel: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_36, type, euclid24285859cancel: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_37, type, condit378418413attice: !>[X0: $tType]:($o)).
% 2.52/1.70 thf(func_def_38, type, complete_Sup_Sup: !>[X0: $tType]:(set @ X0 > X0)).
% 2.52/1.70 thf(func_def_39, type, convex: !>[X0: $tType]:(set @ X0 > $o)).
% 2.52/1.70 thf(func_def_40, type, extend1396239628finity: !>[X0: $tType]:(X0)).
% 2.52/1.70 thf(func_def_41, type, extended_MInfty: extended_ereal).
% 2.52/1.70 thf(func_def_42, type, extended_PInfty: extended_ereal).
% 2.52/1.70 thf(func_def_43, type, extended_ereal2: real > extended_ereal).
% 2.52/1.70 thf(func_def_44, type, extended_rec_ereal: !>[X0: $tType]:((real > X0) > X0 > X0 > extended_ereal > X0)).
% 2.52/1.70 thf(func_def_45, type, extend1716541707_ereal: extended_ereal > real).
% 2.52/1.70 thf(func_def_46, type, times_times: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.52/1.70 thf(func_def_47, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 2.52/1.70 thf(func_def_48, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.52/1.70 thf(func_def_49, type, lower_261390618igraph: !>[X0: $tType]:(set @ X0 > (X0 > extended_ereal) > set @ product_prod @ X0 @ real)).
% 2.52/1.70 thf(func_def_50, type, lower_500881736ine_on: !>[X0: $tType]:(set @ X0 > (X0 > extended_ereal) > $o)).
% 2.52/1.70 thf(func_def_51, type, lower_708069572ave_on: !>[X0: $tType]:(set @ X0 > (X0 > extended_ereal) > $o)).
% 2.52/1.70 thf(func_def_52, type, lower_673667120vex_on: !>[X0: $tType]:(set @ X0 > (X0 > extended_ereal) > $o)).
% 2.52/1.70 thf(func_def_53, type, lower_272802190domain: !>[X0: $tType]:((X0 > extended_ereal) > set @ X0)).
% 2.52/1.70 thf(func_def_54, type, lower_728871928ite_on: !>[X0: $tType]:(set @ X0 > (X0 > extended_ereal) > $o)).
% 2.52/1.70 thf(func_def_55, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.52/1.70 thf(func_def_56, type, order_Greatest: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.52/1.70 thf(func_def_57, type, top_top: !>[X0: $tType]:(X0)).
% 2.52/1.70 thf(func_def_58, type, divide_divide: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.52/1.70 thf(func_def_59, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.52/1.70 thf(func_def_60, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.52/1.70 thf(func_def_61, type, f: a > extended_ereal).
% 2.52/1.70 thf(func_def_76, type, sP0: extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_77, type, sP1: extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_78, type, sP2: extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_79, type, sP3: extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_80, type, sP4: extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_81, type, sP5: extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_82, type, sP6: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_83, type, sP7: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_84, type, sP8: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_85, type, sP9: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_86, type, sP10: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_87, type, sP11: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_88, type, sP12: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_89, type, sP13: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_90, type, sP14: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_91, type, sP15: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_92, type, sP16: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_93, type, sP17: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_94, type, sP18: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_95, type, sP19: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_96, type, sP20: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_97, type, sP21: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_98, type, sP22: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_99, type, sP23: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_100, type, sP24: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_101, type, sP25: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_102, type, sP26: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_103, type, sP27: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_104, type, sP28: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_105, type, sP29: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_106, type, sP30: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_107, type, sP31: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.52/1.70 thf(func_def_108, type, sK32: extended_ereal > real).
% 2.52/1.70 thf(func_def_109, type, sK33: extended_ereal > real).
% 2.52/1.70 thf(func_def_110, type, sK34: extended_ereal > real).
% 2.52/1.70 thf(func_def_111, type, sK35: extended_ereal > real).
% 2.52/1.70 thf(func_def_112, type, sK36: extended_ereal > real).
% 2.52/1.70 thf(func_def_113, type, sK37: extended_ereal > real).
% 2.52/1.70 thf(func_def_114, type, sK38: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.52/1.70 thf(func_def_115, type, sK39: extended_ereal > real).
% 2.52/1.70 thf(func_def_116, type, sK40: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 2.52/1.70 thf(func_def_117, type, sK41: (extended_ereal > extended_ereal > $o) > real).
% 2.52/1.70 thf(func_def_118, type, sK42: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 2.52/1.70 thf(func_def_119, type, sK43: (extended_ereal > extended_ereal > $o) > real).
% 2.52/1.70 thf(func_def_120, type, sK44: (extended_ereal > extended_ereal > $o) > real).
% 2.52/1.70 thf(func_def_121, type, sK45: (extended_ereal > extended_ereal > $o) > real).
% 2.52/1.70 thf(func_def_122, type, sK46: set @ extended_ereal > extended_ereal).
% 2.52/1.70 thf(func_def_123, type, sK47: extended_ereal > set @ extended_ereal > extended_ereal).
% 2.52/1.70 thf(func_def_124, type, sK48: !>[X0: $tType]:((X0 > extended_ereal) > set @ X0)).
% 2.52/1.70 thf(func_def_125, type, sK49: (extended_ereal > extended_ereal > $o) > real).
% 2.52/1.70 thf(func_def_126, type, sK50: (extended_ereal > extended_ereal > $o) > real).
% 2.52/1.70 thf(func_def_127, type, sK51: (extended_ereal > extended_ereal > $o) > real).
% 2.52/1.70 thf(func_def_128, type, sK52: (extended_ereal > extended_ereal > $o) > real).
% 2.52/1.70 thf(func_def_129, type, sK53: (extended_ereal > extended_ereal > $o) > real).
% 2.52/1.70 thf(func_def_130, type, sK54: (extended_ereal > extended_ereal > $o) > real).
% 2.52/1.70 thf(func_def_131, type, sK55: extended_ereal > real).
% 2.52/1.70 thf(func_def_132, type, sK56: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 2.52/1.70 thf(func_def_133, type, sK57: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 2.52/1.70 thf(func_def_134, type, sK58: extended_ereal > real).
% 2.52/1.70 thf(func_def_135, type, sK59: !>[X0: $tType]:((X0 > extended_ereal) > set @ X0)).
% 2.52/1.70 thf(func_def_136, type, sK60: (extended_ereal > $o) > real).
% 2.52/1.70 thf(func_def_137, type, sK61: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.52/1.70 thf(func_def_138, type, sK62: extended_ereal > real).
% 2.52/1.70 thf(func_def_139, type, sK63: (extended_ereal > $o) > real).
% 2.52/1.70 thf(func_def_140, type, sK64: set @ extended_ereal > extended_ereal).
% 2.52/1.70 thf(func_def_141, type, sK65: extended_ereal > set @ extended_ereal > extended_ereal).
% 2.52/1.70 thf(func_def_142, type, sK66: !>[X0: $tType]:(X0)).
% 2.52/1.70 thf(func_def_143, type, sK67: !>[X0: $tType]:(set @ X0 > X0)).
% 2.52/1.70 thf(func_def_144, type, sK68: (extended_ereal > extended_ereal > $o) > real).
% 2.52/1.70 thf(func_def_145, type, sK69: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 2.52/1.70 thf(func_def_146, type, sK70: (extended_ereal > extended_ereal > $o) > real).
% 2.52/1.70 thf(func_def_147, type, sK71: (extended_ereal > extended_ereal > $o) > real).
% 2.52/1.70 thf(func_def_148, type, sK72: (extended_ereal > extended_ereal > $o) > real).
% 2.52/1.70 thf(func_def_149, type, sK73: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 2.52/1.70 thf(func_def_150, type, sK74: extended_ereal > real).
% 2.52/1.70 thf(func_def_151, type, sK75: extended_ereal > real).
% 2.52/1.70 thf(func_def_152, type, sK76: extended_ereal > real).
% 2.52/1.70 thf(func_def_153, type, sK77: extended_ereal > real).
% 2.52/1.70 thf(func_def_154, type, sK78: extended_ereal > real).
% 2.52/1.70 thf(func_def_155, type, sK79: extended_ereal > real).
% 2.52/1.70 thf(func_def_156, type, sK80: extended_ereal > real).
% 2.52/1.70 thf(func_def_157, type, sK81: extended_ereal > real).
% 2.52/1.70 thf(func_def_158, type, sK82: extended_ereal > real).
% 2.52/1.70 thf(func_def_159, type, sK83: extended_ereal > real).
% 2.52/1.70 thf(func_def_160, type, sK84: extended_ereal > real).
% 2.52/1.70 thf(func_def_161, type, sK85: extended_ereal > real).
% 2.52/1.70 thf(func_def_162, type, sK86: extended_ereal > real).
% 2.52/1.70 thf(func_def_163, type, sK87: extended_ereal > real).
% 2.52/1.70 thf(func_def_164, type, sK88: extended_ereal > real).
% 2.52/1.70 thf(func_def_165, type, sK89: extended_ereal > real).
% 2.52/1.70 thf(func_def_166, type, sK90: extended_ereal > real).
% 2.52/1.70 thf(func_def_167, type, sK91: extended_ereal > real).
% 2.52/1.70 thf(func_def_168, type, sK92: extended_ereal > real).
% 2.52/1.70 thf(func_def_169, type, sK93: extended_ereal > real).
% 2.52/1.70 thf(func_def_170, type, sK94: extended_ereal > real).
% 2.52/1.70 thf(func_def_171, type, sK95: extended_ereal > real).
% 2.52/1.70 thf(func_def_172, type, sK96: extended_ereal > real).
% 2.52/1.70 thf(func_def_173, type, sK97: extended_ereal > real).
% 2.52/1.70 thf(func_def_174, type, sK98: extended_ereal > real).
% 2.52/1.70 thf(func_def_175, type, sK99: extended_ereal > real).
% 2.52/1.70 thf(func_def_176, type, sK100: extended_ereal > real).
% 2.52/1.70 thf(func_def_177, type, sK101: extended_ereal > real).
% 2.52/1.70 thf(func_def_178, type, sK102: extended_ereal > real).
% 2.52/1.70 thf(func_def_180, type, ph104: !>[X0: $tType]:(X0)).
% 2.52/1.70 thf(f1535,plain,(
% 2.52/1.70 $false),
% 2.52/1.70 inference(avatar_sat_refutation,[status(thm)],[f1492,f1508,f1534])).
% 2.52/1.70 thf(f1534,plain,(
% 2.52/1.70 ~spl103_1),
% 2.52/1.70 inference(avatar_contradiction_clause,[status(thm)],[f1533])).
% 2.52/1.70 thf(f1533,plain,(
% 2.52/1.70 $false | ~spl103_1),
% 2.52/1.70 inference(subsumption_resolution,[status(thm)],[f1521,f1505])).
% 2.52/1.70 thf(f1505,plain,(
% 2.52/1.70 ((convex @ a @ (lower_272802190domain @ a @ f)) = $true)),
% 2.52/1.70 inference(subsumption_resolution,[status(thm)],[f1495,f1266])).
% 2.52/1.70 thf(f1266,plain,(
% 2.52/1.70 (euclid925273238_space @ a = $true)),
% 2.52/1.70 inference(cnf_transformation,[status(thm)],[f694])).
% 2.52/1.70 thf(f694,plain,(
% 2.52/1.70 (euclid925273238_space @ a = $true)),
% 2.52/1.70 inference(fool_elimination,[status(thm)],[f693])).
% 2.52/1.70 thf(f693,plain,(
% 2.52/1.70 euclid925273238_space @ a),
% 2.52/1.70 inference(rectify,[status(thm)],[f341])).
% 2.52/1.70 thf(f341,axiom,(
% 2.52/1.70 euclid925273238_space @ a),
% 2.52/1.70 file('/export/starexec/sandbox/tmp/tmp.0x1kQQmmmY/DTF2THF_23220.p',tfree_0)).
% 2.52/1.70 thf(f1495,plain,(
% 2.52/1.70 ((convex @ a @ (lower_272802190domain @ a @ f)) = $true) | (euclid925273238_space @ a != $true)),
% 2.52/1.70 inference(trivial_inequality_removal,[status(thm)],[f1493])).
% 2.52/1.70 thf(f1493,plain,(
% 2.52/1.70 (euclid925273238_space @ a != $true) | ($true != $true) | ((convex @ a @ (lower_272802190domain @ a @ f)) = $true)),
% 2.52/1.70 inference(superposition,[status(thm)],[f1304,f1483])).
% 2.52/1.70 thf(f1483,plain,(
% 2.52/1.70 ((lower_673667120vex_on @ a @ top_top @ set @ a @ f) = $true)),
% 2.52/1.70 inference(subsumption_resolution,[status(thm)],[f1481,f1336])).
% 2.52/1.70 thf(f1336,plain,(
% 2.52/1.70 ((lower_673667120vex_on @ a @ top_top @ set @ a @ f) = $true) | ((convex @ a @ (lower_272802190domain @ a @ f)) != $true)),
% 2.52/1.70 inference(cnf_transformation,[status(thm)],[f962])).
% 2.52/1.70 thf(f962,plain,(
% 2.52/1.70 ((convex @ a @ (lower_272802190domain @ a @ f)) != $true) | ((lower_673667120vex_on @ a @ top_top @ set @ a @ f) = $true)),
% 2.52/1.70 inference(ennf_transformation,[status(thm)],[f704])).
% 2.52/1.70 thf(f704,plain,(
% 2.52/1.70 ((convex @ a @ (lower_272802190domain @ a @ f)) = $true) => ((lower_673667120vex_on @ a @ top_top @ set @ a @ f) = $true)),
% 2.52/1.70 inference(fool_elimination,[status(thm)],[f703])).
% 2.52/1.70 thf(f703,plain,(
% 2.52/1.70 (convex @ a @ (lower_272802190domain @ a @ f)) => (lower_673667120vex_on @ a @ top_top @ set @ a @ f)),
% 2.52/1.70 inference(rectify,[status(thm)],[f1])).
% 2.52/1.70 thf(f1,axiom,(
% 2.52/1.70 (convex @ a @ (lower_272802190domain @ a @ f)) => (lower_673667120vex_on @ a @ top_top @ set @ a @ f)),
% 2.52/1.70 file('/export/starexec/sandbox/tmp/tmp.0x1kQQmmmY/DTF2THF_23220.p',fact_0__092_060open_062convex_A_Idomain_Af_J_A_092_060Longrightarrow_062_ALower__Semicontinuous__Mirabelle__quczrylfpw_Oconvex__on_AUNIV_Af_092_060close_062)).
% 2.52/1.70 thf(f1481,plain,(
% 2.52/1.70 ((convex @ a @ (lower_272802190domain @ a @ f)) = $true) | ((lower_673667120vex_on @ a @ top_top @ set @ a @ f) = $true)),
% 2.52/1.70 inference(binary_proxy_clausification,[status(thm)],[f1446])).
% 2.52/1.70 thf(f1446,plain,(
% 2.52/1.70 ((convex @ a @ (lower_272802190domain @ a @ f)) != (lower_673667120vex_on @ a @ top_top @ set @ a @ f))),
% 2.52/1.70 inference(cnf_transformation,[status(thm)],[f917])).
% 2.52/1.70 thf(f917,plain,(
% 2.52/1.70 ((convex @ a @ (lower_272802190domain @ a @ f)) != (lower_673667120vex_on @ a @ top_top @ set @ a @ f))),
% 2.52/1.70 inference(flattening,[status(thm)],[f343])).
% 2.52/1.70 thf(f343,negated_conjecture,(
% 2.52/1.70 ~((convex @ a @ (lower_272802190domain @ a @ f)) = (lower_673667120vex_on @ a @ top_top @ set @ a @ f))),
% 2.52/1.70 inference(negated_conjecture,[status(cth)],[f342])).
% 2.52/1.70 thf(f342,conjecture,(
% 2.52/1.70 ((convex @ a @ (lower_272802190domain @ a @ f)) = (lower_673667120vex_on @ a @ top_top @ set @ a @ f))),
% 2.52/1.70 file('/export/starexec/sandbox/tmp/tmp.0x1kQQmmmY/DTF2THF_23220.p',conj_0)).
% 2.52/1.70 thf(f1304,plain,(
% 2.52/1.70 ( ! [X0 : $tType,X1 : X0 > extended_ereal] : (((lower_673667120vex_on @ X0 @ top_top @ set @ X0 @ X1) != $true) | ((convex @ X0 @ (lower_272802190domain @ X0 @ X1)) = $true) | (euclid925273238_space @ X0 != $true)) )),
% 2.52/1.70 inference(cnf_transformation,[status(thm)],[f944])).
% 2.52/1.70 thf(f944,plain,(
% 2.52/1.70 ! [X0 : $tType] : ((euclid925273238_space @ X0 != $true) | ! [X1 : X0 > extended_ereal] : (((convex @ X0 @ (lower_272802190domain @ X0 @ X1)) = $true) | ((lower_673667120vex_on @ X0 @ top_top @ set @ X0 @ X1) != $true)))),
% 2.52/1.70 inference(ennf_transformation,[status(thm)],[f692])).
% 2.52/1.70 thf(f692,plain,(
% 2.52/1.70 ! [X0 : $tType] : ((euclid925273238_space @ X0 = $true) => ! [X1 : X0 > extended_ereal] : (((lower_673667120vex_on @ X0 @ top_top @ set @ X0 @ X1) = $true) => ((convex @ X0 @ (lower_272802190domain @ X0 @ X1)) = $true)))),
% 2.52/1.70 inference(fool_elimination,[status(thm)],[f691])).
% 2.52/1.70 thf(f691,plain,(
% 2.52/1.70 ! [X0 : $tType] : (euclid925273238_space @ X0 => ! [X1 : X0 > extended_ereal] : ((lower_673667120vex_on @ X0 @ top_top @ set @ X0 @ X1) => (convex @ X0 @ (lower_272802190domain @ X0 @ X1))))),
% 2.52/1.70 inference(rectify,[status(thm)],[f2])).
% 2.52/1.70 thf(f2,axiom,(
% 2.52/1.70 ! [X0 : $tType] : (euclid925273238_space @ X0 => ! [X1 : X0 > extended_ereal] : ((lower_673667120vex_on @ X0 @ top_top @ set @ X0 @ X1) => (convex @ X0 @ (lower_272802190domain @ X0 @ X1))))),
% 2.52/1.70 file('/export/starexec/sandbox/tmp/tmp.0x1kQQmmmY/DTF2THF_23220.p',fact_1_convex__domain)).
% 2.52/1.70 thf(f1521,plain,(
% 2.52/1.70 ((convex @ a @ (lower_272802190domain @ a @ f)) != $true) | ~spl103_1),
% 2.52/1.70 inference(trivial_inequality_removal,[status(thm)],[f1517])).
% 2.52/1.70 thf(f1517,plain,(
% 2.52/1.70 ((convex @ a @ (lower_272802190domain @ a @ f)) != $true) | ($true = $false) | ~spl103_1),
% 2.52/1.70 inference(superposition,[status(thm)],[f1336,f1487])).
% 2.52/1.70 thf(f1487,plain,(
% 2.52/1.70 ((lower_673667120vex_on @ a @ top_top @ set @ a @ f) = $false) | ~spl103_1),
% 2.52/1.70 inference(avatar_component_clause,[status(thm)],[f1485])).
% 2.52/1.70 thf(f1485,definition,(
% 2.52/1.70 spl103_1 <=> ((lower_673667120vex_on @ a @ top_top @ set @ a @ f) = $false)),
% 2.52/1.70 introduced(definition,[new_symbols(naming,[spl103_1])],[avatar_definition])).
% 2.52/1.70 thf(f1508,plain,(
% 2.52/1.70 ~spl103_2),
% 2.52/1.70 inference(avatar_contradiction_clause,[status(thm)],[f1507])).
% 2.52/1.70 thf(f1507,plain,(
% 2.52/1.70 $false | ~spl103_2),
% 2.52/1.70 inference(trivial_inequality_removal,[status(thm)],[f1506])).
% 2.52/1.70 thf(f1506,plain,(
% 2.52/1.70 ($true = $false) | ~spl103_2),
% 2.52/1.70 inference(forward_demodulation,[status(thm)],[f1505,f1491])).
% 2.52/1.70 thf(f1491,plain,(
% 2.52/1.70 ((convex @ a @ (lower_272802190domain @ a @ f)) = $false) | ~spl103_2),
% 2.52/1.70 inference(avatar_component_clause,[status(thm)],[f1489])).
% 2.52/1.70 thf(f1489,definition,(
% 2.52/1.70 spl103_2 <=> ((convex @ a @ (lower_272802190domain @ a @ f)) = $false)),
% 2.52/1.70 introduced(definition,[new_symbols(naming,[spl103_2])],[avatar_definition])).
% 2.52/1.70 thf(f1492,plain,(
% 2.52/1.70 spl103_1 | spl103_2),
% 2.52/1.70 inference(avatar_split_clause,[status(thm)],[f1482,f1489,f1485])).
% 2.52/1.70 thf(f1482,plain,(
% 2.52/1.70 ((lower_673667120vex_on @ a @ top_top @ set @ a @ f) = $false) | ((convex @ a @ (lower_272802190domain @ a @ f)) = $false)),
% 2.52/1.70 inference(binary_proxy_clausification,[status(thm)],[f1446])).
% 2.52/1.70 % SZS output end Proof for DTF2THF_23220
% 2.52/1.70 % (23473)------------------------------
% 2.52/1.70 % (23473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.70 % (23473)Termination reason: Refutation
% 2.52/1.70
% 2.52/1.70 % (23473)Memory used [KB]: 6780
% 2.52/1.70 % (23473)Time elapsed: 0.035 s
% 2.52/1.70 % (23473)Instructions burned: 68 (million)
% 2.52/1.70 % (23473)------------------------------
% 2.52/1.70 % (23473)------------------------------
% 2.52/1.70 % (23459)Success in time 0.065 s
%------------------------------------------------------------------------------