%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP114^2 : 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 : n028.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 08:55:40 AM UTC 2026
% Result : Theorem 2.92s 1.57s
% Output : Refutation 2.92s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP114^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.14/0.33 % Computer : n028.cluster.edu
% 0.14/0.33 % Model : x86_64 x86_64
% 0.14/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.33 % Memory : 8042.1875MB
% 0.14/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.33 % CPULimit : 300
% 0.14/0.33 % WCLimit : 300
% 0.14/0.33 % DateTime : Tue Feb 24 08:26:42 EST 2026
% 0.14/0.33 % CPUTime :
% 0.14/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.45 ---- Original DTF file ---
% 0.18/0.45 thf(spec,logic,$$dhol).
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 % File : ITP114^2 : TPTP v9.2.1. Released v7.5.0.
% 0.18/0.45 % Domain : Interactive Theorem Proving
% 0.18/0.45 % Problem : Sledgehammer Lower_Semicontinuous problem prob_784__6255430_1
% 0.18/0.45 % Version : Especial.
% 0.18/0.45 % English :
% 0.18/0.45
% 0.18/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.18/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.18/0.45 % Source : [Des21]
% 0.18/0.45 % Names : Lower_Semicontinuous/prob_784__6255430_1 [Des21]
% 0.18/0.45
% 0.18/0.45 % Status : Theorem
% 0.18/0.45 % Rating : 0.67 v9.1.0, 0.33 v8.1.0, 0.50 v7.5.0
% 0.18/0.45 % Syntax : Number of formulae : 384 ( 153 unt; 59 typ; 0 def)
% 0.18/0.45 % Number of atoms : 888 ( 386 equ; 0 cnn)
% 0.18/0.45 % Maximal formula atoms : 81 ( 2 avg)
% 0.18/0.45 % Number of connectives : 2816 ( 98 ~; 11 |; 30 &;2243 @)
% 0.18/0.45 % ( 0 <=>; 434 =>; 0 <=; 0 <~>)
% 0.18/0.45 % Maximal formula depth : 34 ( 5 avg)
% 0.18/0.45 % Number of types : 7 ( 6 usr)
% 0.18/0.45 % Number of type conns : 118 ( 118 >; 0 *; 0 +; 0 <<)
% 0.18/0.45 % Number of symbols : 56 ( 53 usr; 5 con; 0-4 aty)
% 0.18/0.45 % Number of variables : 705 ( 29 ^; 602 !; 29 ?; 705 :)
% 0.18/0.45 % ( 45 !>; 0 ?*; 0 @-; 0 @+)
% 0.18/0.45 % SPC : TH1_THM_EQU_NAR
% 0.18/0.45
% 0.18/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 16:27:42.435
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 % Could-be-implicit typings (7)
% 0.18/0.45 thf(ty_t_Extended__Real_Oereal,type,
% 0.18/0.45 extended_ereal: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_t_Extended__Nat_Oenat,type,
% 0.18/0.45 extended_enat: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_t_Real_Oreal,type,
% 0.18/0.45 real: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_t_Set_Oset,type,
% 0.18/0.45 set: $tType > $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_t_Num_Onum,type,
% 0.18/0.45 num: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_t_Nat_Onat,type,
% 0.18/0.45 nat: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_tf_a,type,
% 0.18/0.45 a: $tType ).
% 0.18/0.45
% 0.18/0.45 % Explicit typings (52)
% 0.18/0.45 thf(sy_cl_HOL_Otype,type,
% 0.18/0.45 type:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Groups_Ozero,type,
% 0.18/0.45 zero:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Num_Onumeral,type,
% 0.18/0.45 numeral:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Groups_Ouminus,type,
% 0.18/0.45 uminus:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Oord,type,
% 0.18/0.45 ord:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Nat_Oring__char__0,type,
% 0.18/0.45 ring_char_0:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Num_Oneg__numeral,type,
% 0.18/0.45 neg_numeral:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Oorder,type,
% 0.18/0.45 order:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Groups_Ogroup__add,type,
% 0.18/0.45 group_add:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Olinorder,type,
% 0.18/0.45 linorder:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Opreorder,type,
% 0.18/0.45 preorder:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.18/0.45 semiring_char_0:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.18/0.45 linordered_idom:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Fields_Olinordered__field,type,
% 0.18/0.45 linordered_field:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 0.18/0.45 boolean_algebra:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.18/0.45 ordered_ab_group_add:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.18/0.45 linord219039673up_add:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
% 0.18/0.45 ordere142940540dd_abs:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
% 0.18/0.45 real_V2090557954_space:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.18/0.45 linord1659791738miring:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Topological__Spaces_Oorder__topology,type,
% 0.18/0.45 topolo259154727pology:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Topological__Spaces_Otopological__space,type,
% 0.18/0.45 topolo503727757_space:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.18/0.45 canoni770627133id_add:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Extended__Nat_Oenat,type,
% 0.18/0.45 extended_enat2: nat > extended_enat ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
% 0.30/0.45 extended_case_enat:
% 0.30/0.45 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
% 0.30/0.45 extended_rec_enat:
% 0.30/0.45 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
% 0.30/0.45 extend1396239628finity:
% 0.30/0.45 !>[A: $tType] : A ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Extended__Nat_Othe__enat,type,
% 0.30/0.45 extended_the_enat: extended_enat > nat ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Extended__Real_Oereal_OMInfty,type,
% 0.30/0.45 extended_MInfty: extended_ereal ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Extended__Real_Oereal_OPInfty,type,
% 0.30/0.45 extended_PInfty: extended_ereal ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Extended__Real_Oereal_Oereal,type,
% 0.30/0.45 extended_ereal2: real > extended_ereal ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Extended__Real_Oereal_Osize__ereal,type,
% 0.30/0.45 extended_size_ereal: extended_ereal > nat ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Extended__Real_Oereal__of__enat,type,
% 0.30/0.45 extend1771934483f_enat: extended_enat > extended_ereal ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Extended__Real_Oreal__of__ereal,type,
% 0.30/0.45 extend1716541707_ereal: extended_ereal > real ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Groups_Oabs__class_Oabs,type,
% 0.30/0.45 abs_abs:
% 0.30/0.45 !>[A: $tType] : ( A > A ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.30/0.45 uminus_uminus:
% 0.30/0.45 !>[A: $tType] : ( A > A ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.30/0.45 zero_zero:
% 0.30/0.45 !>[A: $tType] : A ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc,type,
% 0.30/0.45 lower_107104146le_lsc:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc__at,type,
% 0.30/0.45 lower_582600101lsc_at:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( A > ( A > B ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc__hull,type,
% 0.30/0.45 lower_1879122655c_hull:
% 0.30/0.45 !>[A: $tType] : ( ( A > extended_ereal ) > A > extended_ereal ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
% 0.30/0.45 neg_numeral_dbl:
% 0.30/0.45 !>[A: $tType] : ( A > A ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Num_Oneg__numeral__class_Osub,type,
% 0.30/0.45 neg_numeral_sub:
% 0.30/0.45 !>[A: $tType] : ( num > num > A ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Num_Onumeral__class_Onumeral,type,
% 0.30/0.45 numeral_numeral:
% 0.30/0.45 !>[A: $tType] : ( num > A ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/0.45 ord_less_eq:
% 0.30/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 0.30/0.45 order_Greatest:
% 0.30/0.45 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 0.30/0.45 order_antimono:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 0.30/0.45 order_strict_mono:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 0.30/0.45 divide_divide:
% 0.30/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_member,type,
% 0.30/0.45 member:
% 0.30/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_v_f,type,
% 0.30/0.45 f: a > extended_ereal ).
% 0.30/0.45
% 0.30/0.45 thf(sy_v_g,type,
% 0.30/0.45 g: a > extended_ereal ).
% 0.30/0.45
% 0.30/0.45 thf(sy_v_x____,type,
% 0.30/0.45 x: a ).
% 0.30/0.45
% 0.30/0.45 % Relevant facts (252)
% 0.30/0.45 thf(fact_0_assms,axiom,
% 0.30/0.45 ! [X: a] : ( ord_less_eq @ extended_ereal @ ( g @ X ) @ ( f @ X ) ) ).
% 0.30/0.45
% 0.30/0.45 % assms
% 0.30/0.45 thf(fact_1_calculation,axiom,
% 0.30/0.45 ord_less_eq @ extended_ereal @ ( lower_1879122655c_hull @ a @ g @ x ) @ ( g @ x ) ).
% 0.30/0.45
% 0.30/0.45 % calculation
% 0.30/0.45 thf(fact_2_order__refl,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( preorder @ A )
% 0.30/0.45 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_refl
% 0.30/0.45 thf(fact_3_ereal__complete__Inf,axiom,
% 0.30/0.45 ! [S: set @ extended_ereal] :
% 0.30/0.45 ? [X3: extended_ereal] :
% 0.30/0.45 ( ! [Xa: extended_ereal] :
% 0.30/0.45 ( ( member @ extended_ereal @ Xa @ S )
% 0.30/0.45 => ( ord_less_eq @ extended_ereal @ X3 @ Xa ) )
% 0.30/0.45 & ! [Z: extended_ereal] :
% 0.30/0.45 ( ! [Xa2: extended_ereal] :
% 0.30/0.45 ( ( member @ extended_ereal @ Xa2 @ S )
% 0.30/0.45 => ( ord_less_eq @ extended_ereal @ Z @ Xa2 ) )
% 0.30/0.45 => ( ord_less_eq @ extended_ereal @ Z @ X3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_complete_Inf
% 0.30/0.45 thf(fact_4_ereal__complete__Sup,axiom,
% 0.30/0.45 ! [S: set @ extended_ereal] :
% 0.30/0.45 ? [X3: extended_ereal] :
% 0.30/0.45 ( ! [Xa: extended_ereal] :
% 0.30/0.45 ( ( member @ extended_ereal @ Xa @ S )
% 0.30/0.45 => ( ord_less_eq @ extended_ereal @ Xa @ X3 ) )
% 0.30/0.45 & ! [Z: extended_ereal] :
% 0.30/0.45 ( ! [Xa2: extended_ereal] :
% 0.30/0.45 ( ( member @ extended_ereal @ Xa2 @ S )
% 0.30/0.45 => ( ord_less_eq @ extended_ereal @ Xa2 @ Z ) )
% 0.30/0.45 => ( ord_less_eq @ extended_ereal @ X3 @ Z ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_complete_Sup
% 0.30/0.45 thf(fact_5_le__funD,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ord @ B )
% 0.30/0.45 => ! [F: A > B,G: A > B,X2: A] :
% 0.30/0.45 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_funD
% 0.30/0.45 thf(fact_6_le__funE,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ord @ B )
% 0.30/0.45 => ! [F: A > B,G: A > B,X2: A] :
% 0.30/0.45 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_funE
% 0.30/0.45 thf(fact_7_le__funI,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ord @ B )
% 0.30/0.45 => ! [F: A > B,G: A > B] :
% 0.30/0.45 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.30/0.45 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_funI
% 0.30/0.45 thf(fact_8_le__fun__def,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ord @ B )
% 0.30/0.45 => ( ( ord_less_eq @ ( A > B ) )
% 0.30/0.45 = ( ^ [F2: A > B,G2: A > B] :
% 0.30/0.45 ! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_fun_def
% 0.30/0.45 thf(fact_9_order__subst1,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType] :
% 0.30/0.45 ( ( ( order @ B )
% 0.30/0.45 & ( order @ A ) )
% 0.30/0.45 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.30/0.45 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.30/0.45 => ( ! [X3: B,Y: B] :
% 0.30/0.45 ( ( ord_less_eq @ B @ X3 @ Y )
% 0.30/0.45 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.30/0.45 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_subst1
% 0.30/0.45 thf(fact_10_order__subst2,axiom,
% 0.30/0.45 ! [A: $tType,C2: $tType] :
% 0.30/0.45 ( ( ( order @ C2 )
% 0.30/0.45 & ( order @ A ) )
% 0.30/0.45 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.45 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 0.30/0.45 => ( ! [X3: A,Y: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X3 @ Y )
% 0.30/0.45 => ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.30/0.45 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_subst2
% 0.30/0.45 thf(fact_11_verit__la__disequality,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A )
% 0.30/0.45 => ! [A2: A,B2: A] :
% 0.30/0.45 ( ( A2 = B2 )
% 0.30/0.45 | ~ ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.45 | ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % verit_la_disequality
% 0.30/0.45 thf(fact_12_ord__eq__le__subst,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType] :
% 0.30/0.45 ( ( ( ord @ B )
% 0.30/0.45 & ( ord @ A ) )
% 0.30/0.45 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.30/0.45 ( ( A2
% 0.30/0.45 = ( F @ B2 ) )
% 0.30/0.45 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.30/0.45 => ( ! [X3: B,Y: B] :
% 0.30/0.45 ( ( ord_less_eq @ B @ X3 @ Y )
% 0.30/0.45 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.30/0.45 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ord_eq_le_subst
% 0.30/0.45 thf(fact_13_ord__le__eq__subst,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType] :
% 0.30/0.45 ( ( ( ord @ B )
% 0.30/0.45 & ( ord @ A ) )
% 0.30/0.45 => ! [A2: A,B2: A,F: A > B,C: B] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.45 => ( ( ( F @ B2 )
% 0.30/0.45 = C )
% 0.30/0.45 => ( ! [X3: A,Y: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X3 @ Y )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ord_le_eq_subst
% 0.30/0.45 thf(fact_14_lsc__hull__le,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( real_V2090557954_space @ A )
% 0.30/0.45 => ! [F: A > extended_ereal,X2: A] : ( ord_less_eq @ extended_ereal @ ( lower_1879122655c_hull @ A @ F @ X2 ) @ ( F @ X2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % lsc_hull_le
% 0.30/0.45 thf(fact_15_dual__order_Oantisym,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [B2: A,A2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.45 => ( A2 = B2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % dual_order.antisym
% 0.30/0.45 thf(fact_16_dual__order_Oeq__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 0.30/0.45 = ( ^ [A3: A,B3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.45 & ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % dual_order.eq_iff
% 0.30/0.45 thf(fact_17_dual__order_Otrans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [B2: A,A2: A,C: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.30/0.45 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % dual_order.trans
% 0.30/0.45 thf(fact_18_linorder__wlog,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A )
% 0.30/0.45 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.30/0.45 ( ! [A4: A,B4: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.45 => ( P @ A4 @ B4 ) )
% 0.30/0.45 => ( ! [A4: A,B4: A] :
% 0.30/0.45 ( ( P @ B4 @ A4 )
% 0.30/0.45 => ( P @ A4 @ B4 ) )
% 0.30/0.45 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % linorder_wlog
% 0.30/0.45 thf(fact_19_dual__order_Orefl,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % dual_order.refl
% 0.30/0.45 thf(fact_20_order__trans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( preorder @ A )
% 0.30/0.45 => ! [X2: A,Y3: A,Z3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ Y3 @ Z3 )
% 0.30/0.45 => ( ord_less_eq @ A @ X2 @ Z3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_trans
% 0.30/0.45 thf(fact_21_order__class_Oorder_Oantisym,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [A2: A,B2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.45 => ( A2 = B2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_class.order.antisym
% 0.30/0.45 thf(fact_22_ord__le__eq__trans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ord @ A )
% 0.30/0.45 => ! [A2: A,B2: A,C: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.45 => ( ( B2 = C )
% 0.30/0.45 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ord_le_eq_trans
% 0.30/0.45 thf(fact_23_ord__eq__le__trans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ord @ A )
% 0.30/0.45 => ! [A2: A,B2: A,C: A] :
% 0.30/0.45 ( ( A2 = B2 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.30/0.45 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ord_eq_le_trans
% 0.30/0.45 thf(fact_24_order__class_Oorder_Oeq__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 0.30/0.45 = ( ^ [A3: A,B3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.45 & ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_class.order.eq_iff
% 0.30/0.45 thf(fact_25_antisym__conv,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [Y3: A,X2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ Y3 @ X2 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.30/0.45 = ( X2 = Y3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % antisym_conv
% 0.30/0.45 thf(fact_26_le__cases3,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A )
% 0.30/0.45 => ! [X2: A,Y3: A,Z3: A] :
% 0.30/0.45 ( ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ Y3 @ Z3 ) )
% 0.30/0.45 => ( ( ( ord_less_eq @ A @ Y3 @ X2 )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ X2 @ Z3 ) )
% 0.30/0.45 => ( ( ( ord_less_eq @ A @ X2 @ Z3 )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ Z3 @ Y3 ) )
% 0.30/0.45 => ( ( ( ord_less_eq @ A @ Z3 @ Y3 )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ Y3 @ X2 ) )
% 0.30/0.45 => ( ( ( ord_less_eq @ A @ Y3 @ Z3 )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ Z3 @ X2 ) )
% 0.30/0.45 => ~ ( ( ord_less_eq @ A @ Z3 @ X2 )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_cases3
% 0.30/0.45 thf(fact_27_order_Otrans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [A2: A,B2: A,C: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.30/0.45 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order.trans
% 0.30/0.45 thf(fact_28_le__cases,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A )
% 0.30/0.45 => ! [X2: A,Y3: A] :
% 0.30/0.45 ( ~ ( ord_less_eq @ A @ X2 @ Y3 )
% 0.30/0.45 => ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_cases
% 0.30/0.45 thf(fact_29_eq__refl,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( preorder @ A )
% 0.30/0.45 => ! [X2: A,Y3: A] :
% 0.30/0.45 ( ( X2 = Y3 )
% 0.30/0.45 => ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % eq_refl
% 0.30/0.45 thf(fact_30_linear,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A )
% 0.30/0.45 => ! [X2: A,Y3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.30/0.45 | ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % linear
% 0.30/0.45 thf(fact_31_antisym,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [X2: A,Y3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ Y3 @ X2 )
% 0.30/0.45 => ( X2 = Y3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % antisym
% 0.30/0.45 thf(fact_32_eq__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 0.30/0.45 = ( ^ [X4: A,Y4: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.30/0.45 & ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % eq_iff
% 0.30/0.45 thf(fact_33_lsc__hull__iff__greatest,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( real_V2090557954_space @ A )
% 0.30/0.45 => ! [G: A > extended_ereal,F: A > extended_ereal] :
% 0.30/0.45 ( ( G
% 0.30/0.45 = ( lower_1879122655c_hull @ A @ F ) )
% 0.30/0.45 = ( ( lower_107104146le_lsc @ A @ extended_ereal @ G )
% 0.30/0.45 & ! [X4: A] : ( ord_less_eq @ extended_ereal @ ( G @ X4 ) @ ( F @ X4 ) )
% 0.30/0.45 & ! [H: A > extended_ereal] :
% 0.30/0.45 ( ( ( lower_107104146le_lsc @ A @ extended_ereal @ H )
% 0.30/0.45 & ! [X4: A] : ( ord_less_eq @ extended_ereal @ ( H @ X4 ) @ ( F @ X4 ) ) )
% 0.30/0.45 => ! [X4: A] : ( ord_less_eq @ extended_ereal @ ( H @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % lsc_hull_iff_greatest
% 0.30/0.45 thf(fact_34_lsc__hull__greatest,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( real_V2090557954_space @ A )
% 0.30/0.45 => ! [G: A > extended_ereal,F: A > extended_ereal] :
% 0.30/0.45 ( ( lower_107104146le_lsc @ A @ extended_ereal @ G )
% 0.30/0.45 => ( ! [X3: A] : ( ord_less_eq @ extended_ereal @ ( G @ X3 ) @ ( F @ X3 ) )
% 0.30/0.45 => ! [X: A] : ( ord_less_eq @ extended_ereal @ ( G @ X ) @ ( lower_1879122655c_hull @ A @ F @ X ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % lsc_hull_greatest
% 0.30/0.45 thf(fact_35_Greatest__equality,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [P: A > $o,X2: A] :
% 0.30/0.45 ( ( P @ X2 )
% 0.30/0.45 => ( ! [Y: A] :
% 0.30/0.45 ( ( P @ Y )
% 0.30/0.45 => ( ord_less_eq @ A @ Y @ X2 ) )
% 0.30/0.45 => ( ( order_Greatest @ A @ P )
% 0.30/0.45 = X2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Greatest_equality
% 0.30/0.45 thf(fact_36_GreatestI2__order,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [P: A > $o,X2: A,Q: A > $o] :
% 0.30/0.45 ( ( P @ X2 )
% 0.30/0.45 => ( ! [Y: A] :
% 0.30/0.45 ( ( P @ Y )
% 0.30/0.45 => ( ord_less_eq @ A @ Y @ X2 ) )
% 0.30/0.45 => ( ! [X3: A] :
% 0.30/0.45 ( ( P @ X3 )
% 0.30/0.45 => ( ! [Y5: A] :
% 0.30/0.45 ( ( P @ Y5 )
% 0.30/0.45 => ( ord_less_eq @ A @ Y5 @ X3 ) )
% 0.30/0.45 => ( Q @ X3 ) ) )
% 0.30/0.45 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % GreatestI2_order
% 0.30/0.45 thf(fact_37_le__rel__bool__arg__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ord @ A )
% 0.30/0.45 => ( ( ord_less_eq @ ( $o > A ) )
% 0.30/0.45 = ( ^ [X5: $o > A,Y6: $o > A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ ( X5 @ $false ) @ ( Y6 @ $false ) )
% 0.30/0.45 & ( ord_less_eq @ A @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_rel_bool_arg_iff
% 0.30/0.45 thf(fact_38_lsc__lsc__hull,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( real_V2090557954_space @ A )
% 0.30/0.45 => ! [F: A > extended_ereal] : ( lower_107104146le_lsc @ A @ extended_ereal @ ( lower_1879122655c_hull @ A @ F ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % lsc_lsc_hull
% 0.30/0.45 thf(fact_39_ereal__of__enat__le__iff,axiom,
% 0.30/0.45 ! [M: extended_enat,N: extended_enat] :
% 0.30/0.45 ( ( ord_less_eq @ extended_ereal @ ( extend1771934483f_enat @ M ) @ ( extend1771934483f_enat @ N ) )
% 0.30/0.45 = ( ord_less_eq @ extended_enat @ M @ N ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_of_enat_le_iff
% 0.30/0.45 thf(fact_40_antimono__def,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( order @ A )
% 0.30/0.45 & ( order @ B ) )
% 0.30/0.45 => ( ( order_antimono @ A @ B )
% 0.30/0.45 = ( ^ [F2: A > B] :
% 0.30/0.45 ! [X4: A,Y4: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % antimono_def
% 0.30/0.45 thf(fact_41_antimonoI,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( order @ A )
% 0.30/0.45 & ( order @ B ) )
% 0.30/0.45 => ! [F: A > B] :
% 0.30/0.45 ( ! [X3: A,Y: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X3 @ Y )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X3 ) ) )
% 0.30/0.45 => ( order_antimono @ A @ B @ F ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % antimonoI
% 0.30/0.45 thf(fact_42_antimonoE,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( order @ A )
% 0.30/0.45 & ( order @ B ) )
% 0.30/0.45 => ! [F: A > B,X2: A,Y3: A] :
% 0.30/0.45 ( ( order_antimono @ A @ B @ F )
% 0.30/0.45 => ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % antimonoE
% 0.30/0.45 thf(fact_43_antimonoD,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( order @ A )
% 0.30/0.45 & ( order @ B ) )
% 0.30/0.45 => ! [F: A > B,X2: A,Y3: A] :
% 0.30/0.45 ( ( order_antimono @ A @ B @ F )
% 0.30/0.45 => ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % antimonoD
% 0.30/0.45 thf(fact_44_lsc__def,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( topolo503727757_space @ A )
% 0.30/0.45 & ( topolo259154727pology @ B ) )
% 0.30/0.45 => ( ( lower_107104146le_lsc @ A @ B )
% 0.30/0.45 = ( ^ [F2: A > B] :
% 0.30/0.45 ! [X4: A] : ( lower_582600101lsc_at @ A @ B @ X4 @ F2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % lsc_def
% 0.30/0.45 thf(fact_45_ext,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/0.45 ( ! [X3: A] :
% 0.30/0.45 ( ( F @ X3 )
% 0.30/0.45 = ( G @ X3 ) )
% 0.30/0.45 => ( F = G ) ) ).
% 0.30/0.45
% 0.30/0.45 % ext
% 0.30/0.45 thf(fact_46_decseqD,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [F: nat > A,I: nat,J: nat] :
% 0.30/0.45 ( ( order_antimono @ nat @ A @ F )
% 0.30/0.45 => ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.45 => ( ord_less_eq @ A @ ( F @ J ) @ ( F @ I ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % decseqD
% 0.30/0.45 thf(fact_47_decseq__def,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ( ( order_antimono @ nat @ A )
% 0.30/0.45 = ( ^ [X5: nat > A] :
% 0.30/0.45 ! [M2: nat,N2: nat] :
% 0.30/0.45 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.30/0.45 => ( ord_less_eq @ A @ ( X5 @ N2 ) @ ( X5 @ M2 ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % decseq_def
% 0.30/0.45 thf(fact_48_ereal__of__enat__nonneg,axiom,
% 0.30/0.45 ! [N: extended_enat] : ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( extend1771934483f_enat @ N ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_of_enat_nonneg
% 0.30/0.45 thf(fact_49_ereal__of__enat__ge__zero__cancel__iff,axiom,
% 0.30/0.45 ! [N: extended_enat] :
% 0.30/0.45 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( extend1771934483f_enat @ N ) )
% 0.30/0.45 = ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_of_enat_ge_zero_cancel_iff
% 0.30/0.45 thf(fact_50_numeral__le__ereal__of__enat__iff,axiom,
% 0.30/0.45 ! [M: num,N: extended_enat] :
% 0.30/0.45 ( ( ord_less_eq @ extended_ereal @ ( numeral_numeral @ extended_ereal @ M ) @ ( extend1771934483f_enat @ N ) )
% 0.30/0.45 = ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ N ) ) ).
% 0.30/0.45
% 0.30/0.45 % numeral_le_ereal_of_enat_iff
% 0.30/0.45 thf(fact_51_ereal__infty__less__eq_I1_J,axiom,
% 0.30/0.45 ! [X2: extended_ereal] :
% 0.30/0.45 ( ( ord_less_eq @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ X2 )
% 0.30/0.45 = ( X2
% 0.30/0.45 = ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_infty_less_eq(1)
% 0.30/0.45 thf(fact_52_strict__mono__less__eq,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( linorder @ A )
% 0.30/0.45 & ( order @ B ) )
% 0.30/0.45 => ! [F: A > B,X2: A,Y3: A] :
% 0.30/0.45 ( ( order_strict_mono @ A @ B @ F )
% 0.30/0.45 => ( ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y3 ) )
% 0.30/0.45 = ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % strict_mono_less_eq
% 0.30/0.45 thf(fact_53_ereal__of__enat__zero,axiom,
% 0.30/0.45 ( ( extend1771934483f_enat @ ( zero_zero @ extended_enat ) )
% 0.30/0.45 = ( zero_zero @ extended_ereal ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_of_enat_zero
% 0.30/0.45 thf(fact_54_strict__mono__eq,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( linorder @ A )
% 0.30/0.45 & ( order @ B ) )
% 0.30/0.45 => ! [F: A > B,X2: A,Y3: A] :
% 0.30/0.45 ( ( order_strict_mono @ A @ B @ F )
% 0.30/0.45 => ( ( ( F @ X2 )
% 0.30/0.45 = ( F @ Y3 ) )
% 0.30/0.45 = ( X2 = Y3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % strict_mono_eq
% 0.30/0.45 thf(fact_55_Infty__neq__0_I1_J,axiom,
% 0.30/0.45 ( ( extend1396239628finity @ extended_ereal )
% 0.30/0.45 != ( zero_zero @ extended_ereal ) ) ).
% 0.30/0.45
% 0.30/0.45 % Infty_neq_0(1)
% 0.30/0.45 thf(fact_56_strict__mono__leD,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( order @ A )
% 0.30/0.45 & ( order @ B ) )
% 0.30/0.45 => ! [R: A > B,M: A,N: A] :
% 0.30/0.45 ( ( order_strict_mono @ A @ B @ R )
% 0.30/0.45 => ( ( ord_less_eq @ A @ M @ N )
% 0.30/0.45 => ( ord_less_eq @ B @ ( R @ M ) @ ( R @ N ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % strict_mono_leD
% 0.30/0.45 thf(fact_57_neq__PInf__trans,axiom,
% 0.30/0.45 ! [Y3: extended_ereal,X2: extended_ereal] :
% 0.30/0.45 ( ( Y3
% 0.30/0.45 != ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.45 => ( ( ord_less_eq @ extended_ereal @ X2 @ Y3 )
% 0.30/0.45 => ( X2
% 0.30/0.45 != ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % neq_PInf_trans
% 0.30/0.45 thf(fact_58_ereal__infty__less__eq2_I1_J,axiom,
% 0.30/0.45 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.30/0.45 ( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
% 0.30/0.45 => ( ( A2
% 0.30/0.45 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.45 => ( B2
% 0.30/0.45 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_infty_less_eq2(1)
% 0.30/0.45 thf(fact_59_ereal__less__eq_I1_J,axiom,
% 0.30/0.45 ! [X2: extended_ereal] : ( ord_less_eq @ extended_ereal @ X2 @ ( extend1396239628finity @ extended_ereal ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_less_eq(1)
% 0.30/0.45 thf(fact_60_numeral__le__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linord1659791738miring @ A )
% 0.30/0.45 => ! [M: num,N: num] :
% 0.30/0.45 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.45 = ( ord_less_eq @ num @ M @ N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % numeral_le_iff
% 0.30/0.45 thf(fact_61_le__zero__eq,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( canoni770627133id_add @ A )
% 0.30/0.45 => ! [N: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.30/0.45 = ( N
% 0.30/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_zero_eq
% 0.30/0.45 thf(fact_62_zero__le__numeral,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linord1659791738miring @ A )
% 0.30/0.45 => ! [N: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % zero_le_numeral
% 0.30/0.45 thf(fact_63_not__numeral__le__zero,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linord1659791738miring @ A )
% 0.30/0.45 => ! [N: num] :
% 0.30/0.45 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % not_numeral_le_zero
% 0.30/0.45 thf(fact_64_numeral__eq__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semiring_char_0 @ A )
% 0.30/0.45 => ! [M: num,N: num] :
% 0.30/0.45 ( ( ( numeral_numeral @ A @ M )
% 0.30/0.45 = ( numeral_numeral @ A @ N ) )
% 0.30/0.45 = ( M = N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % numeral_eq_iff
% 0.30/0.45 thf(fact_65_ile0__eq,axiom,
% 0.30/0.45 ! [N: extended_enat] :
% 0.30/0.45 ( ( ord_less_eq @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
% 0.30/0.45 = ( N
% 0.30/0.45 = ( zero_zero @ extended_enat ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ile0_eq
% 0.30/0.45 thf(fact_66_i0__lb,axiom,
% 0.30/0.45 ! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% 0.30/0.45
% 0.30/0.45 % i0_lb
% 0.30/0.45 thf(fact_67_enat__ord__number_I1_J,axiom,
% 0.30/0.45 ! [M: num,N: num] :
% 0.30/0.45 ( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N ) )
% 0.30/0.45 = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % enat_ord_number(1)
% 0.30/0.45 thf(fact_68_zero__reorient,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( zero @ A )
% 0.30/0.45 => ! [X2: A] :
% 0.30/0.45 ( ( ( zero_zero @ A )
% 0.30/0.45 = X2 )
% 0.30/0.45 = ( X2
% 0.30/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % zero_reorient
% 0.30/0.45 thf(fact_69_zero__le,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( canoni770627133id_add @ A )
% 0.30/0.45 => ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % zero_le
% 0.30/0.45 thf(fact_70_le__numeral__extra_I3_J,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linord1659791738miring @ A )
% 0.30/0.45 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_numeral_extra(3)
% 0.30/0.45 thf(fact_71_zero__neq__numeral,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semiring_char_0 @ A )
% 0.30/0.45 => ! [N: num] :
% 0.30/0.45 ( ( zero_zero @ A )
% 0.30/0.45 != ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % zero_neq_numeral
% 0.30/0.45 thf(fact_72_bot__nat__0_Oextremum,axiom,
% 0.30/0.45 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 0.30/0.45
% 0.30/0.45 % bot_nat_0.extremum
% 0.30/0.45 thf(fact_73_le0,axiom,
% 0.30/0.45 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.30/0.45
% 0.30/0.45 % le0
% 0.30/0.45 thf(fact_74_neg__numeral__le__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linordered_idom @ A )
% 0.30/0.45 => ! [M: num,N: num] :
% 0.30/0.45 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.45 = ( ord_less_eq @ num @ N @ M ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % neg_numeral_le_iff
% 0.30/0.45 thf(fact_75_numeral__le__enat__iff,axiom,
% 0.30/0.45 ! [M: num,N: nat] :
% 0.30/0.45 ( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( extended_enat2 @ N ) )
% 0.30/0.45 = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ N ) ) ).
% 0.30/0.45
% 0.30/0.45 % numeral_le_enat_iff
% 0.30/0.45 thf(fact_76_sub__non__positive,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linordered_idom @ A )
% 0.30/0.45 => ! [N: num,M: num] :
% 0.30/0.45 ( ( ord_less_eq @ A @ ( neg_numeral_sub @ A @ N @ M ) @ ( zero_zero @ A ) )
% 0.30/0.45 = ( ord_less_eq @ num @ N @ M ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sub_non_positive
% 0.30/0.45 thf(fact_77_sub__non__negative,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linordered_idom @ A )
% 0.30/0.45 => ! [N: num,M: num] :
% 0.30/0.45 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M ) )
% 0.30/0.45 = ( ord_less_eq @ num @ M @ N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sub_non_negative
% 0.30/0.45 thf(fact_78_real__of__ereal__le__0,axiom,
% 0.30/0.45 ! [X2: extended_ereal] :
% 0.30/0.45 ( ( ord_less_eq @ real @ ( extend1716541707_ereal @ X2 ) @ ( zero_zero @ real ) )
% 0.30/0.45 = ( ( ord_less_eq @ extended_ereal @ X2 @ ( zero_zero @ extended_ereal ) )
% 0.30/0.45 | ( X2
% 0.30/0.45 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % real_of_ereal_le_0
% 0.30/0.45 thf(fact_79_neg__equal__iff__equal,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( group_add @ A )
% 0.30/0.45 => ! [A2: A,B2: A] :
% 0.30/0.45 ( ( ( uminus_uminus @ A @ A2 )
% 0.30/0.45 = ( uminus_uminus @ A @ B2 ) )
% 0.30/0.45 = ( A2 = B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % neg_equal_iff_equal
% 0.30/0.45 thf(fact_80_add_Oinverse__inverse,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( group_add @ A )
% 0.30/0.45 => ! [A2: A] :
% 0.30/0.45 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 0.30/0.45 = A2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % add.inverse_inverse
% 0.30/0.45 thf(fact_81_verit__minus__simplify_I4_J,axiom,
% 0.30/0.45 ! [B: $tType] :
% 0.30/0.45 ( ( group_add @ B )
% 0.30/0.45 => ! [B2: B] :
% 0.30/0.45 ( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
% 0.30/0.45 = B2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % verit_minus_simplify(4)
% 0.30/0.45 thf(fact_82_ereal__uminus__zero__iff,axiom,
% 0.30/0.45 ! [A2: extended_ereal] :
% 0.30/0.45 ( ( ( uminus_uminus @ extended_ereal @ A2 )
% 0.30/0.45 = ( zero_zero @ extended_ereal ) )
% 0.30/0.45 = ( A2
% 0.30/0.45 = ( zero_zero @ extended_ereal ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_uminus_zero_iff
% 0.30/0.45 thf(fact_83_ereal__uminus__zero,axiom,
% 0.30/0.45 ( ( uminus_uminus @ extended_ereal @ ( zero_zero @ extended_ereal ) )
% 0.30/0.45 = ( zero_zero @ extended_ereal ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_uminus_zero
% 0.30/0.45 thf(fact_84_ereal__minus__le__minus,axiom,
% 0.30/0.45 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.30/0.45 ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) @ ( uminus_uminus @ extended_ereal @ B2 ) )
% 0.30/0.45 = ( ord_less_eq @ extended_ereal @ B2 @ A2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_minus_le_minus
% 0.30/0.45 thf(fact_85_enat_Oinject,axiom,
% 0.30/0.45 ! [Nat: nat,Nat2: nat] :
% 0.30/0.45 ( ( ( extended_enat2 @ Nat )
% 0.30/0.45 = ( extended_enat2 @ Nat2 ) )
% 0.30/0.45 = ( Nat = Nat2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % enat.inject
% 0.30/0.45 thf(fact_86_real__of__ereal,axiom,
% 0.30/0.45 ! [X2: extended_ereal] :
% 0.30/0.45 ( ( extend1716541707_ereal @ ( uminus_uminus @ extended_ereal @ X2 ) )
% 0.30/0.45 = ( uminus_uminus @ real @ ( extend1716541707_ereal @ X2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % real_of_ereal
% 0.30/0.45 thf(fact_87_neg__le__iff__le,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ordered_ab_group_add @ A )
% 0.30/0.45 => ! [B2: A,A2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 0.30/0.45 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % neg_le_iff_le
% 0.30/0.45 thf(fact_88_neg__equal__zero,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linord219039673up_add @ A )
% 0.30/0.45 => ! [A2: A] :
% 0.30/0.45 ( ( ( uminus_uminus @ A @ A2 )
% 0.30/0.45 = A2 )
% 0.30/0.45 = ( A2
% 0.30/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % neg_equal_zero
% 0.30/0.45 thf(fact_89_equal__neg__zero,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linord219039673up_add @ A )
% 0.30/0.45 => ! [A2: A] :
% 0.30/0.45 ( ( A2
% 0.30/0.45 = ( uminus_uminus @ A @ A2 ) )
% 0.30/0.45 = ( A2
% 0.30/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % equal_neg_zero
% 0.30/0.45 thf(fact_90_neg__equal__0__iff__equal,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( group_add @ A )
% 0.30/0.45 => ! [A2: A] :
% 0.30/0.45 ( ( ( uminus_uminus @ A @ A2 )
% 0.30/0.45 = ( zero_zero @ A ) )
% 0.30/0.45 = ( A2
% 0.30/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % neg_equal_0_iff_equal
% 0.30/0.45 thf(fact_91_neg__0__equal__iff__equal,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( group_add @ A )
% 0.30/0.45 => ! [A2: A] :
% 0.30/0.45 ( ( ( zero_zero @ A )
% 0.30/0.45 = ( uminus_uminus @ A @ A2 ) )
% 0.30/0.45 = ( ( zero_zero @ A )
% 0.30/0.45 = A2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % neg_0_equal_iff_equal
% 0.30/0.45 thf(fact_92_add_Oinverse__neutral,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( group_add @ A )
% 0.30/0.45 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 0.30/0.45 = ( zero_zero @ A ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % add.inverse_neutral
% 0.30/0.45 thf(fact_93_neg__numeral__eq__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ring_char_0 @ A )
% 0.30/0.45 => ! [M: num,N: num] :
% 0.30/0.45 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
% 0.30/0.45 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.45 = ( M = N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % neg_numeral_eq_iff
% 0.30/0.45 thf(fact_94_ereal__infty__less__eq_I2_J,axiom,
% 0.30/0.45 ! [X2: extended_ereal] :
% 0.30/0.45 ( ( ord_less_eq @ extended_ereal @ X2 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.45 = ( X2
% 0.30/0.45 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_infty_less_eq(2)
% 0.30/0.45 thf(fact_95_ereal__uminus__le__0__iff,axiom,
% 0.30/0.45 ! [A2: extended_ereal] :
% 0.30/0.45 ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) @ ( zero_zero @ extended_ereal ) )
% 0.30/0.45 = ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_uminus_le_0_iff
% 0.30/0.45 thf(fact_96_ereal__0__le__uminus__iff,axiom,
% 0.30/0.45 ! [A2: extended_ereal] :
% 0.30/0.45 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( uminus_uminus @ extended_ereal @ A2 ) )
% 0.30/0.45 = ( ord_less_eq @ extended_ereal @ A2 @ ( zero_zero @ extended_ereal ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_0_le_uminus_iff
% 0.30/0.45 thf(fact_97_neg__0__le__iff__le,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ordered_ab_group_add @ A )
% 0.30/0.45 => ! [A2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 0.30/0.45 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % neg_0_le_iff_le
% 0.30/0.45 thf(fact_98_neg__le__0__iff__le,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ordered_ab_group_add @ A )
% 0.30/0.45 => ! [A2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.30/0.45 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % neg_le_0_iff_le
% 0.30/0.45 thf(fact_99_less__eq__neg__nonpos,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linord219039673up_add @ A )
% 0.30/0.45 => ! [A2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 0.30/0.45 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % less_eq_neg_nonpos
% 0.30/0.45 thf(fact_100_neg__less__eq__nonneg,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linord219039673up_add @ A )
% 0.30/0.45 => ! [A2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 0.30/0.45 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % neg_less_eq_nonneg
% 0.30/0.45 thf(fact_101_real__of__ereal__0,axiom,
% 0.30/0.45 ( ( extend1716541707_ereal @ ( zero_zero @ extended_ereal ) )
% 0.30/0.45 = ( zero_zero @ real ) ) ).
% 0.30/0.45
% 0.30/0.45 % real_of_ereal_0
% 0.30/0.45 thf(fact_102_enat__ord__simps_I1_J,axiom,
% 0.30/0.45 ! [M: nat,N: nat] :
% 0.30/0.45 ( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.30/0.45 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.30/0.45
% 0.30/0.45 % enat_ord_simps(1)
% 0.30/0.45 thf(fact_103_real__of__ereal_Osimps_I3_J,axiom,
% 0.30/0.45 ( ( extend1716541707_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.45 = ( zero_zero @ real ) ) ).
% 0.30/0.45
% 0.30/0.45 % real_of_ereal.simps(3)
% 0.30/0.45 thf(fact_104_minus__equation__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( group_add @ A )
% 0.30/0.45 => ! [A2: A,B2: A] :
% 0.30/0.45 ( ( ( uminus_uminus @ A @ A2 )
% 0.30/0.45 = B2 )
% 0.30/0.45 = ( ( uminus_uminus @ A @ B2 )
% 0.30/0.45 = A2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % minus_equation_iff
% 0.30/0.45 thf(fact_105_equation__minus__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( group_add @ A )
% 0.30/0.45 => ! [A2: A,B2: A] :
% 0.30/0.45 ( ( A2
% 0.30/0.45 = ( uminus_uminus @ A @ B2 ) )
% 0.30/0.45 = ( B2
% 0.30/0.45 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % equation_minus_iff
% 0.30/0.45 thf(fact_106_verit__negate__coefficient_I3_J,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ordered_ab_group_add @ A )
% 0.30/0.45 => ! [A2: A,B2: A] :
% 0.30/0.45 ( ( A2 = B2 )
% 0.30/0.45 => ( ( uminus_uminus @ A @ A2 )
% 0.30/0.45 = ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % verit_negate_coefficient(3)
% 0.30/0.45 thf(fact_107_real__of__ereal__eq__0,axiom,
% 0.30/0.45 ! [X2: extended_ereal] :
% 0.30/0.45 ( ( ( extend1716541707_ereal @ X2 )
% 0.30/0.45 = ( zero_zero @ real ) )
% 0.30/0.45 = ( ( X2
% 0.30/0.45 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.45 | ( X2
% 0.30/0.45 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.45 | ( X2
% 0.30/0.45 = ( zero_zero @ extended_ereal ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % real_of_ereal_eq_0
% 0.30/0.45 thf(fact_108_real__of__ereal_Osimps_I2_J,axiom,
% 0.30/0.45 ( ( extend1716541707_ereal @ ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.45 = ( zero_zero @ real ) ) ).
% 0.30/0.45
% 0.30/0.45 % real_of_ereal.simps(2)
% 0.30/0.45 thf(fact_109_le__imp__neg__le,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ordered_ab_group_add @ A )
% 0.30/0.45 => ! [A2: A,B2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.45 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_imp_neg_le
% 0.30/0.45 thf(fact_110_minus__le__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ordered_ab_group_add @ A )
% 0.30/0.45 => ! [A2: A,B2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 0.30/0.45 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % minus_le_iff
% 0.30/0.45 thf(fact_111_le__minus__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ordered_ab_group_add @ A )
% 0.30/0.45 => ! [A2: A,B2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.45 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_minus_iff
% 0.30/0.45 thf(fact_112_numeral__neq__neg__numeral,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ring_char_0 @ A )
% 0.30/0.45 => ! [M: num,N: num] :
% 0.30/0.45 ( ( numeral_numeral @ A @ M )
% 0.30/0.45 != ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % numeral_neq_neg_numeral
% 0.30/0.45 thf(fact_113_neg__numeral__neq__numeral,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ring_char_0 @ A )
% 0.30/0.45 => ! [M: num,N: num] :
% 0.30/0.45 ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
% 0.30/0.45 != ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % neg_numeral_neq_numeral
% 0.30/0.45 thf(fact_114_zero__enat__def,axiom,
% 0.30/0.45 ( ( zero_zero @ extended_enat )
% 0.30/0.45 = ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % zero_enat_def
% 0.30/0.45 thf(fact_115_enat__0__iff_I1_J,axiom,
% 0.30/0.45 ! [X2: nat] :
% 0.30/0.45 ( ( ( extended_enat2 @ X2 )
% 0.30/0.45 = ( zero_zero @ extended_enat ) )
% 0.30/0.45 = ( X2
% 0.30/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % enat_0_iff(1)
% 0.30/0.45 thf(fact_116_enat__0__iff_I2_J,axiom,
% 0.30/0.45 ! [X2: nat] :
% 0.30/0.45 ( ( ( zero_zero @ extended_enat )
% 0.30/0.45 = ( extended_enat2 @ X2 ) )
% 0.30/0.45 = ( X2
% 0.30/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % enat_0_iff(2)
% 0.30/0.45 thf(fact_117_lsc__at__MInfty,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( topolo503727757_space @ A )
% 0.30/0.45 => ! [F: A > extended_ereal,X0: A] :
% 0.30/0.45 ( ( ( F @ X0 )
% 0.30/0.45 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.45 => ( lower_582600101lsc_at @ A @ extended_ereal @ X0 @ F ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % lsc_at_MInfty
% 0.30/0.45 thf(fact_118_MInfty__neq__PInfty_I1_J,axiom,
% 0.30/0.45 ( ( extend1396239628finity @ extended_ereal )
% 0.30/0.45 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % MInfty_neq_PInfty(1)
% 0.30/0.45 thf(fact_119_ereal__uminus__le__reorder,axiom,
% 0.30/0.45 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.30/0.45 ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) @ B2 )
% 0.30/0.45 = ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ B2 ) @ A2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_uminus_le_reorder
% 0.30/0.45 thf(fact_120_enat__ile,axiom,
% 0.30/0.45 ! [N: extended_enat,M: nat] :
% 0.30/0.45 ( ( ord_less_eq @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 0.30/0.45 => ? [K: nat] :
% 0.30/0.45 ( N
% 0.30/0.45 = ( extended_enat2 @ K ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % enat_ile
% 0.30/0.45 thf(fact_121_numeral__eq__enat,axiom,
% 0.30/0.45 ( ( numeral_numeral @ extended_enat )
% 0.30/0.45 = ( ^ [K2: num] : ( extended_enat2 @ ( numeral_numeral @ nat @ K2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % numeral_eq_enat
% 0.30/0.45 thf(fact_122_real__of__ereal__pos,axiom,
% 0.30/0.45 ! [X2: extended_ereal] :
% 0.30/0.45 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ X2 )
% 0.30/0.45 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( extend1716541707_ereal @ X2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % real_of_ereal_pos
% 0.30/0.45 thf(fact_123_neg__numeral__le__numeral,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linordered_idom @ A )
% 0.30/0.45 => ! [M: num,N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % neg_numeral_le_numeral
% 0.30/0.45 thf(fact_124_not__numeral__le__neg__numeral,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linordered_idom @ A )
% 0.30/0.45 => ! [M: num,N: num] :
% 0.30/0.45 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % not_numeral_le_neg_numeral
% 0.30/0.45 thf(fact_125_zero__neq__neg__numeral,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ring_char_0 @ A )
% 0.30/0.45 => ! [N: num] :
% 0.30/0.45 ( ( zero_zero @ A )
% 0.30/0.45 != ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % zero_neq_neg_numeral
% 0.30/0.45 thf(fact_126_Infty__neq__0_I3_J,axiom,
% 0.30/0.45 ( ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.45 != ( zero_zero @ extended_ereal ) ) ).
% 0.30/0.45
% 0.30/0.45 % Infty_neq_0(3)
% 0.30/0.45 thf(fact_127_ereal__infty__less__eq2_I2_J,axiom,
% 0.30/0.45 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.30/0.45 ( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
% 0.30/0.45 => ( ( B2
% 0.30/0.45 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.45 => ( A2
% 0.30/0.45 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_infty_less_eq2(2)
% 0.30/0.45 thf(fact_128_ereal__less__eq_I2_J,axiom,
% 0.30/0.45 ! [X2: extended_ereal] : ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ X2 ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_less_eq(2)
% 0.30/0.45 thf(fact_129_real__of__ereal__positive__mono,axiom,
% 0.30/0.45 ! [X2: extended_ereal,Y3: extended_ereal] :
% 0.30/0.45 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ X2 )
% 0.30/0.45 => ( ( ord_less_eq @ extended_ereal @ X2 @ Y3 )
% 0.30/0.45 => ( ( Y3
% 0.30/0.45 != ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.45 => ( ord_less_eq @ real @ ( extend1716541707_ereal @ X2 ) @ ( extend1716541707_ereal @ Y3 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % real_of_ereal_positive_mono
% 0.30/0.45 thf(fact_130_neg__numeral__le__zero,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linordered_idom @ A )
% 0.30/0.45 => ! [N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % neg_numeral_le_zero
% 0.30/0.45 thf(fact_131_not__zero__le__neg__numeral,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linordered_idom @ A )
% 0.30/0.45 => ! [N: num] :
% 0.30/0.45 ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % not_zero_le_neg_numeral
% 0.30/0.45 thf(fact_132_not__MInfty__nonneg,axiom,
% 0.30/0.45 ! [X2: extended_ereal] :
% 0.30/0.45 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ X2 )
% 0.30/0.45 => ( X2
% 0.30/0.45 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % not_MInfty_nonneg
% 0.30/0.45 thf(fact_133_le__refl,axiom,
% 0.30/0.45 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.30/0.45
% 0.30/0.45 % le_refl
% 0.30/0.45 thf(fact_134_le__trans,axiom,
% 0.30/0.45 ! [I: nat,J: nat,K3: nat] :
% 0.30/0.45 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.45 => ( ( ord_less_eq @ nat @ J @ K3 )
% 0.30/0.45 => ( ord_less_eq @ nat @ I @ K3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_trans
% 0.30/0.45 thf(fact_135_eq__imp__le,axiom,
% 0.30/0.45 ! [M: nat,N: nat] :
% 0.30/0.45 ( ( M = N )
% 0.30/0.45 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.30/0.45
% 0.30/0.45 % eq_imp_le
% 0.30/0.45 thf(fact_136_le__antisym,axiom,
% 0.30/0.45 ! [M: nat,N: nat] :
% 0.30/0.45 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.45 => ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.45 => ( M = N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_antisym
% 0.30/0.45 thf(fact_137_GreatestI__nat,axiom,
% 0.30/0.45 ! [P: nat > $o,K3: nat,B2: nat] :
% 0.30/0.45 ( ( P @ K3 )
% 0.30/0.45 => ( ! [Y: nat] :
% 0.30/0.45 ( ( P @ Y )
% 0.30/0.45 => ( ord_less_eq @ nat @ Y @ B2 ) )
% 0.30/0.45 => ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % GreatestI_nat
% 0.30/0.45 thf(fact_138_nat__le__linear,axiom,
% 0.30/0.45 ! [M: nat,N: nat] :
% 0.30/0.45 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.45 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.30/0.45
% 0.30/0.45 % nat_le_linear
% 0.30/0.45 thf(fact_139_Greatest__le__nat,axiom,
% 0.30/0.45 ! [P: nat > $o,K3: nat,B2: nat] :
% 0.30/0.45 ( ( P @ K3 )
% 0.30/0.45 => ( ! [Y: nat] :
% 0.30/0.45 ( ( P @ Y )
% 0.30/0.45 => ( ord_less_eq @ nat @ Y @ B2 ) )
% 0.30/0.45 => ( ord_less_eq @ nat @ K3 @ ( order_Greatest @ nat @ P ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Greatest_le_nat
% 0.30/0.45 thf(fact_140_GreatestI__ex__nat,axiom,
% 0.30/0.45 ! [P: nat > $o,B2: nat] :
% 0.30/0.45 ( ? [X_1: nat] : ( P @ X_1 )
% 0.30/0.45 => ( ! [Y: nat] :
% 0.30/0.45 ( ( P @ Y )
% 0.30/0.45 => ( ord_less_eq @ nat @ Y @ B2 ) )
% 0.30/0.45 => ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % GreatestI_ex_nat
% 0.30/0.45 thf(fact_141_Nat_Oex__has__greatest__nat,axiom,
% 0.30/0.45 ! [P: nat > $o,K3: nat,B2: nat] :
% 0.30/0.45 ( ( P @ K3 )
% 0.30/0.45 => ( ! [Y: nat] :
% 0.30/0.45 ( ( P @ Y )
% 0.30/0.45 => ( ord_less_eq @ nat @ Y @ B2 ) )
% 0.30/0.45 => ? [X3: nat] :
% 0.30/0.45 ( ( P @ X3 )
% 0.30/0.45 & ! [Y5: nat] :
% 0.30/0.45 ( ( P @ Y5 )
% 0.30/0.45 => ( ord_less_eq @ nat @ Y5 @ X3 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Nat.ex_has_greatest_nat
% 0.30/0.45 thf(fact_142_strict__mono__imp__increasing,axiom,
% 0.30/0.45 ! [F: nat > nat,N: nat] :
% 0.30/0.45 ( ( order_strict_mono @ nat @ nat @ F )
% 0.30/0.45 => ( ord_less_eq @ nat @ N @ ( F @ N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % strict_mono_imp_increasing
% 0.30/0.45 thf(fact_143_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.30/0.45 ! [A2: nat] :
% 0.30/0.45 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.30/0.45 => ( A2
% 0.30/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % bot_nat_0.extremum_uniqueI
% 0.30/0.45 thf(fact_144_bot__nat__0_Oextremum__unique,axiom,
% 0.30/0.45 ! [A2: nat] :
% 0.30/0.45 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.30/0.45 = ( A2
% 0.30/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % bot_nat_0.extremum_unique
% 0.30/0.45 thf(fact_145_le__0__eq,axiom,
% 0.30/0.45 ! [N: nat] :
% 0.30/0.45 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.30/0.45 = ( N
% 0.30/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_0_eq
% 0.30/0.45 thf(fact_146_less__eq__nat_Osimps_I1_J,axiom,
% 0.30/0.45 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.30/0.45
% 0.30/0.45 % less_eq_nat.simps(1)
% 0.30/0.45 thf(fact_147_compl__le__compl__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( boolean_algebra @ A )
% 0.30/0.45 => ! [X2: A,Y3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y3 ) )
% 0.30/0.45 = ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % compl_le_compl_iff
% 0.30/0.45 thf(fact_148_the__enat_Osimps,axiom,
% 0.30/0.45 ! [N: nat] :
% 0.30/0.45 ( ( extended_the_enat @ ( extended_enat2 @ N ) )
% 0.30/0.45 = N ) ).
% 0.30/0.45
% 0.30/0.45 % the_enat.simps
% 0.30/0.45 thf(fact_149_uminus__apply,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( uminus @ B )
% 0.30/0.45 => ( ( uminus_uminus @ ( A > B ) )
% 0.30/0.45 = ( ^ [A5: A > B,X4: A] : ( uminus_uminus @ B @ ( A5 @ X4 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % uminus_apply
% 0.30/0.45 thf(fact_150_double__compl,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( boolean_algebra @ A )
% 0.30/0.45 => ! [X2: A] :
% 0.30/0.45 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X2 ) )
% 0.30/0.45 = X2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % double_compl
% 0.30/0.45 thf(fact_151_compl__eq__compl__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( boolean_algebra @ A )
% 0.30/0.45 => ! [X2: A,Y3: A] :
% 0.30/0.45 ( ( ( uminus_uminus @ A @ X2 )
% 0.30/0.45 = ( uminus_uminus @ A @ Y3 ) )
% 0.30/0.45 = ( X2 = Y3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % compl_eq_compl_iff
% 0.30/0.45 thf(fact_152_enat_Osimps_I4_J,axiom,
% 0.30/0.45 ! [T: $tType,F1: nat > T,F22: T,Nat: nat] :
% 0.30/0.45 ( ( extended_case_enat @ T @ F1 @ F22 @ ( extended_enat2 @ Nat ) )
% 0.30/0.45 = ( F1 @ Nat ) ) ).
% 0.30/0.45
% 0.30/0.45 % enat.simps(4)
% 0.30/0.45 thf(fact_153_ereal__uminus__uminus,axiom,
% 0.30/0.45 ! [A2: extended_ereal] :
% 0.30/0.45 ( ( uminus_uminus @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) )
% 0.30/0.45 = A2 ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_uminus_uminus
% 0.30/0.45 thf(fact_154_ereal__uminus__eq__iff,axiom,
% 0.30/0.45 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.30/0.45 ( ( ( uminus_uminus @ extended_ereal @ A2 )
% 0.30/0.45 = ( uminus_uminus @ extended_ereal @ B2 ) )
% 0.30/0.45 = ( A2 = B2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_uminus_eq_iff
% 0.30/0.45 thf(fact_155_ereal__uminus__eq__reorder,axiom,
% 0.30/0.45 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.30/0.45 ( ( ( uminus_uminus @ extended_ereal @ A2 )
% 0.30/0.45 = B2 )
% 0.30/0.45 = ( A2
% 0.30/0.45 = ( uminus_uminus @ extended_ereal @ B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_uminus_eq_reorder
% 0.30/0.45 thf(fact_156_fun__Compl__def,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( uminus @ B )
% 0.30/0.45 => ( ( uminus_uminus @ ( A > B ) )
% 0.30/0.45 = ( ^ [A5: A > B,X4: A] : ( uminus_uminus @ B @ ( A5 @ X4 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % fun_Compl_def
% 0.30/0.45 thf(fact_157_compl__le__swap2,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( boolean_algebra @ A )
% 0.30/0.45 => ! [Y3: A,X2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X2 )
% 0.30/0.45 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ Y3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % compl_le_swap2
% 0.30/0.45 thf(fact_158_compl__le__swap1,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( boolean_algebra @ A )
% 0.30/0.45 => ! [Y3: A,X2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X2 ) )
% 0.30/0.45 => ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % compl_le_swap1
% 0.30/0.45 thf(fact_159_compl__mono,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( boolean_algebra @ A )
% 0.30/0.45 => ! [X2: A,Y3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.30/0.45 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % compl_mono
% 0.30/0.45 thf(fact_160_real__eq__0__iff__le__ge__0,axiom,
% 0.30/0.45 ! [X2: real] :
% 0.30/0.45 ( ( X2
% 0.30/0.45 = ( zero_zero @ real ) )
% 0.30/0.45 = ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
% 0.30/0.45 & ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( uminus_uminus @ real @ X2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % real_eq_0_iff_le_ge_0
% 0.30/0.45 thf(fact_161_enat_Osimps_I6_J,axiom,
% 0.30/0.45 ! [T: $tType,F1: nat > T,F22: T,Nat: nat] :
% 0.30/0.45 ( ( extended_rec_enat @ T @ F1 @ F22 @ ( extended_enat2 @ Nat ) )
% 0.30/0.45 = ( F1 @ Nat ) ) ).
% 0.30/0.45
% 0.30/0.45 % enat.simps(6)
% 0.30/0.45 thf(fact_162_dbl__simps_I1_J,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( neg_numeral @ A )
% 0.30/0.45 => ! [K3: num] :
% 0.30/0.45 ( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K3 ) ) )
% 0.30/0.45 = ( uminus_uminus @ A @ ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K3 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % dbl_simps(1)
% 0.30/0.45 thf(fact_163_dbl__simps_I2_J,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( neg_numeral @ A )
% 0.30/0.45 => ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
% 0.30/0.45 = ( zero_zero @ A ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % dbl_simps(2)
% 0.30/0.45 thf(fact_164_ereal__less__eq_I5_J,axiom,
% 0.30/0.45 ! [R: real] :
% 0.30/0.45 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( extended_ereal2 @ R ) )
% 0.30/0.45 = ( ord_less_eq @ real @ ( zero_zero @ real ) @ R ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_less_eq(5)
% 0.30/0.45 thf(fact_165_ereal__less__eq_I4_J,axiom,
% 0.30/0.45 ! [R: real] :
% 0.30/0.45 ( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ R ) @ ( zero_zero @ extended_ereal ) )
% 0.30/0.45 = ( ord_less_eq @ real @ R @ ( zero_zero @ real ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_less_eq(4)
% 0.30/0.45 thf(fact_166_MInfty__eq__minfinity,axiom,
% 0.30/0.45 ( extended_MInfty
% 0.30/0.45 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % MInfty_eq_minfinity
% 0.30/0.45 thf(fact_167_ereal__cong,axiom,
% 0.30/0.45 ! [X2: real,Y3: real] :
% 0.30/0.45 ( ( X2 = Y3 )
% 0.30/0.45 => ( ( extended_ereal2 @ X2 )
% 0.30/0.45 = ( extended_ereal2 @ Y3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_cong
% 0.30/0.45 thf(fact_168_ereal_Oinject,axiom,
% 0.30/0.45 ! [X1: real,Y1: real] :
% 0.30/0.45 ( ( ( extended_ereal2 @ X1 )
% 0.30/0.45 = ( extended_ereal2 @ Y1 ) )
% 0.30/0.45 = ( X1 = Y1 ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal.inject
% 0.30/0.45 thf(fact_169_numeral__eq__ereal,axiom,
% 0.30/0.45 ( ( numeral_numeral @ extended_ereal )
% 0.30/0.45 = ( ^ [W: num] : ( extended_ereal2 @ ( numeral_numeral @ real @ W ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % numeral_eq_ereal
% 0.30/0.45 thf(fact_170_ereal__eq__0_I1_J,axiom,
% 0.30/0.45 ! [R: real] :
% 0.30/0.45 ( ( ( extended_ereal2 @ R )
% 0.30/0.45 = ( zero_zero @ extended_ereal ) )
% 0.30/0.45 = ( R
% 0.30/0.45 = ( zero_zero @ real ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_eq_0(1)
% 0.30/0.45 thf(fact_171_ereal__eq__0_I2_J,axiom,
% 0.30/0.45 ! [R: real] :
% 0.30/0.45 ( ( ( zero_zero @ extended_ereal )
% 0.30/0.45 = ( extended_ereal2 @ R ) )
% 0.30/0.45 = ( R
% 0.30/0.45 = ( zero_zero @ real ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ereal_eq_0(2)
% 0.30/0.45 thf(fact_172_ereal__less__eq_I3_J,axiom,
% 0.30/0.45 ! [R: real,P2: real] :
% 0.30/0.45 ( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P2 ) )
% 0.30/0.45 = ( ord_less_eq @ real @ R @ P2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_less_eq(3)
% 0.30/0.46 thf(fact_173_ereal_Odistinct_I3_J,axiom,
% 0.30/0.46 ! [X1: real] :
% 0.30/0.46 ( ( extended_ereal2 @ X1 )
% 0.30/0.46 != extended_MInfty ) ).
% 0.30/0.46
% 0.30/0.46 % ereal.distinct(3)
% 0.30/0.46 thf(fact_174_real__of__ereal_Osimps_I1_J,axiom,
% 0.30/0.46 ! [R: real] :
% 0.30/0.46 ( ( extend1716541707_ereal @ ( extended_ereal2 @ R ) )
% 0.30/0.46 = R ) ).
% 0.30/0.46
% 0.30/0.46 % real_of_ereal.simps(1)
% 0.30/0.46 thf(fact_175_ereal__le__real,axiom,
% 0.30/0.46 ! [X2: extended_ereal,Y3: extended_ereal] :
% 0.30/0.46 ( ! [Z4: real] :
% 0.30/0.46 ( ( ord_less_eq @ extended_ereal @ X2 @ ( extended_ereal2 @ Z4 ) )
% 0.30/0.46 => ( ord_less_eq @ extended_ereal @ Y3 @ ( extended_ereal2 @ Z4 ) ) )
% 0.30/0.46 => ( ord_less_eq @ extended_ereal @ Y3 @ X2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_le_real
% 0.30/0.46 thf(fact_176_PInfty__neq__ereal_I1_J,axiom,
% 0.30/0.46 ! [R: real] :
% 0.30/0.46 ( ( extended_ereal2 @ R )
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) ) ).
% 0.30/0.46
% 0.30/0.46 % PInfty_neq_ereal(1)
% 0.30/0.46 thf(fact_177_zero__ereal__def,axiom,
% 0.30/0.46 ( ( zero_zero @ extended_ereal )
% 0.30/0.46 = ( extended_ereal2 @ ( zero_zero @ real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_ereal_def
% 0.30/0.46 thf(fact_178_le__ereal__le,axiom,
% 0.30/0.46 ! [A2: extended_ereal,X2: real,Y3: real] :
% 0.30/0.46 ( ( ord_less_eq @ extended_ereal @ A2 @ ( extended_ereal2 @ X2 ) )
% 0.30/0.46 => ( ( ord_less_eq @ real @ X2 @ Y3 )
% 0.30/0.46 => ( ord_less_eq @ extended_ereal @ A2 @ ( extended_ereal2 @ Y3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_ereal_le
% 0.30/0.46 thf(fact_179_ereal__le__le,axiom,
% 0.30/0.46 ! [Y3: real,A2: extended_ereal,X2: real] :
% 0.30/0.46 ( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ Y3 ) @ A2 )
% 0.30/0.46 => ( ( ord_less_eq @ real @ X2 @ Y3 )
% 0.30/0.46 => ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ X2 ) @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_le_le
% 0.30/0.46 thf(fact_180_real__of__ereal_Oinduct,axiom,
% 0.30/0.46 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.30/0.46 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ( P @ ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( P @ A0 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % real_of_ereal.induct
% 0.30/0.46 thf(fact_181_real__of__ereal_Ocases,axiom,
% 0.30/0.46 ! [X2: extended_ereal] :
% 0.30/0.46 ( ! [R2: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 != ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ( X2
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( X2
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % real_of_ereal.cases
% 0.30/0.46 thf(fact_182_times__ereal_Oinduct,axiom,
% 0.30/0.46 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.30/0.46 ( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
% 0.30/0.46 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ! [R2: real] : ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( ! [R2: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % times_ereal.induct
% 0.30/0.46 thf(fact_183_plus__ereal_Oinduct,axiom,
% 0.30/0.46 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.30/0.46 ( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
% 0.30/0.46 => ( ! [X_12: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_12 )
% 0.30/0.46 => ( ! [A4: extended_ereal] : ( P @ A4 @ ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( ! [P3: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ P3 ) )
% 0.30/0.46 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % plus_ereal.induct
% 0.30/0.46 thf(fact_184_less__ereal_Oinduct,axiom,
% 0.30/0.46 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.30/0.46 ( ! [X3: real,Y: real] : ( P @ ( extended_ereal2 @ X3 ) @ ( extended_ereal2 @ Y ) )
% 0.30/0.46 => ( ! [X_12: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_12 )
% 0.30/0.46 => ( ! [A4: extended_ereal] : ( P @ A4 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( ! [X3: real] : ( P @ ( extended_ereal2 @ X3 ) @ ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ! [R2: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_ereal.induct
% 0.30/0.46 thf(fact_185_abs__ereal_Oinduct,axiom,
% 0.30/0.46 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.30/0.46 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( ( P @ ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( P @ A0 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_ereal.induct
% 0.30/0.46 thf(fact_186_ereal__all__split,axiom,
% 0.30/0.46 ( ( ^ [P4: extended_ereal > $o] :
% 0.30/0.46 ! [X6: extended_ereal] : ( P4 @ X6 ) )
% 0.30/0.46 = ( ^ [P5: extended_ereal > $o] :
% 0.30/0.46 ( ( P5 @ ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 & ! [X4: real] : ( P5 @ ( extended_ereal2 @ X4 ) )
% 0.30/0.46 & ( P5 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_all_split
% 0.30/0.46 thf(fact_187_abs__ereal_Ocases,axiom,
% 0.30/0.46 ! [X2: extended_ereal] :
% 0.30/0.46 ( ! [R2: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 != ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ( X2
% 0.30/0.46 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( X2
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_ereal.cases
% 0.30/0.46 thf(fact_188_ereal__ex__split,axiom,
% 0.30/0.46 ( ( ^ [P4: extended_ereal > $o] :
% 0.30/0.46 ? [X6: extended_ereal] : ( P4 @ X6 ) )
% 0.30/0.46 = ( ^ [P5: extended_ereal > $o] :
% 0.30/0.46 ( ( P5 @ ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 | ? [X4: real] : ( P5 @ ( extended_ereal2 @ X4 ) )
% 0.30/0.46 | ( P5 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_ex_split
% 0.30/0.46 thf(fact_189_ereal3__cases,axiom,
% 0.30/0.46 ! [X2: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
% 0.30/0.46 ( ( ? [R2: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ? [Ra: real] :
% 0.30/0.46 ( Xa3
% 0.30/0.46 = ( extended_ereal2 @ Ra ) )
% 0.30/0.46 => ! [Rb: real] :
% 0.30/0.46 ( Xb
% 0.30/0.46 != ( extended_ereal2 @ Rb ) ) ) )
% 0.30/0.46 => ( ( ? [R2: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ? [Ra: real] :
% 0.30/0.46 ( Xa3
% 0.30/0.46 = ( extended_ereal2 @ Ra ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.30/0.46 => ( ( ? [R2: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ? [Ra: real] :
% 0.30/0.46 ( Xa3
% 0.30/0.46 = ( extended_ereal2 @ Ra ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.30/0.46 => ( ( ? [R2: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ! [Ra: real] :
% 0.30/0.46 ( Xb
% 0.30/0.46 != ( extended_ereal2 @ Ra ) ) ) )
% 0.30/0.46 => ( ( ? [R2: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.30/0.46 => ( ( ? [R2: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.30/0.46 => ( ( ? [R2: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ! [Ra: real] :
% 0.30/0.46 ( Xb
% 0.30/0.46 != ( extended_ereal2 @ Ra ) ) ) )
% 0.30/0.46 => ( ( ? [R2: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.30/0.46 => ( ( ? [R2: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ? [R2: real] :
% 0.30/0.46 ( Xa3
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ! [Ra: real] :
% 0.30/0.46 ( Xb
% 0.30/0.46 != ( extended_ereal2 @ Ra ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ? [R2: real] :
% 0.30/0.46 ( Xa3
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ? [R2: real] :
% 0.30/0.46 ( Xa3
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ! [R2: real] :
% 0.30/0.46 ( Xb
% 0.30/0.46 != ( extended_ereal2 @ R2 ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ! [R2: real] :
% 0.30/0.46 ( Xb
% 0.30/0.46 != ( extended_ereal2 @ R2 ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( ? [R2: real] :
% 0.30/0.46 ( Xa3
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ! [Ra: real] :
% 0.30/0.46 ( Xb
% 0.30/0.46 != ( extended_ereal2 @ Ra ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( ? [R2: real] :
% 0.30/0.46 ( Xa3
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( ? [R2: real] :
% 0.30/0.46 ( Xa3
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ! [R2: real] :
% 0.30/0.46 ( Xb
% 0.30/0.46 != ( extended_ereal2 @ R2 ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ! [R2: real] :
% 0.30/0.46 ( Xb
% 0.30/0.46 != ( extended_ereal2 @ R2 ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.30/0.46 => ~ ( ( X2
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( ( Xa3
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( Xb
% 0.30/0.46 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal3_cases
% 0.30/0.46 thf(fact_190_ereal2__cases,axiom,
% 0.30/0.46 ! [X2: extended_ereal,Xa3: extended_ereal] :
% 0.30/0.46 ( ( ? [R2: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ! [Ra: real] :
% 0.30/0.46 ( Xa3
% 0.30/0.46 != ( extended_ereal2 @ Ra ) ) )
% 0.30/0.46 => ( ( ? [R2: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( Xa3
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( ( ? [R2: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( Xa3
% 0.30/0.46 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ! [R2: real] :
% 0.30/0.46 ( Xa3
% 0.30/0.46 != ( extended_ereal2 @ R2 ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( Xa3
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( Xa3
% 0.30/0.46 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ! [R2: real] :
% 0.30/0.46 ( Xa3
% 0.30/0.46 != ( extended_ereal2 @ R2 ) ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( Xa3
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ~ ( ( X2
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( Xa3
% 0.30/0.46 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal2_cases
% 0.30/0.46 thf(fact_191_ereal__cases,axiom,
% 0.30/0.46 ! [X2: extended_ereal] :
% 0.30/0.46 ( ! [R2: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 != ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ( X2
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( X2
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_cases
% 0.30/0.46 thf(fact_192_MInfty__neq__ereal_I1_J,axiom,
% 0.30/0.46 ! [R: real] :
% 0.30/0.46 ( ( extended_ereal2 @ R )
% 0.30/0.46 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % MInfty_neq_ereal(1)
% 0.30/0.46 thf(fact_193_ereal__top,axiom,
% 0.30/0.46 ! [X2: extended_ereal] :
% 0.30/0.46 ( ! [B5: real] : ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ B5 ) @ X2 )
% 0.30/0.46 => ( X2
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_top
% 0.30/0.46 thf(fact_194_uminus__ereal_Osimps_I1_J,axiom,
% 0.30/0.46 ! [R: real] :
% 0.30/0.46 ( ( uminus_uminus @ extended_ereal @ ( extended_ereal2 @ R ) )
% 0.30/0.46 = ( extended_ereal2 @ ( uminus_uminus @ real @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % uminus_ereal.simps(1)
% 0.30/0.46 thf(fact_195_ereal__bot,axiom,
% 0.30/0.46 ! [X2: extended_ereal] :
% 0.30/0.46 ( ! [B5: real] : ( ord_less_eq @ extended_ereal @ X2 @ ( extended_ereal2 @ B5 ) )
% 0.30/0.46 => ( X2
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_bot
% 0.30/0.46 thf(fact_196_real__of__ereal_Oelims,axiom,
% 0.30/0.46 ! [X2: extended_ereal,Y3: real] :
% 0.30/0.46 ( ( ( extend1716541707_ereal @ X2 )
% 0.30/0.46 = Y3 )
% 0.30/0.46 => ( ! [R2: real] :
% 0.30/0.46 ( ( X2
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( Y3 != R2 ) )
% 0.30/0.46 => ( ( ( X2
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( Y3
% 0.30/0.46 != ( zero_zero @ real ) ) )
% 0.30/0.46 => ~ ( ( X2
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( Y3
% 0.30/0.46 != ( zero_zero @ real ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % real_of_ereal.elims
% 0.30/0.46 thf(fact_197_ereal_Osize__gen_I3_J,axiom,
% 0.30/0.46 ( ( extended_size_ereal @ extended_MInfty )
% 0.30/0.46 = ( zero_zero @ nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal.size_gen(3)
% 0.30/0.46 thf(fact_198_ereal_Osize__gen_I1_J,axiom,
% 0.30/0.46 ! [X1: real] :
% 0.30/0.46 ( ( extended_size_ereal @ ( extended_ereal2 @ X1 ) )
% 0.30/0.46 = ( zero_zero @ nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal.size_gen(1)
% 0.30/0.46 thf(fact_199_uminus__ereal_Oelims,axiom,
% 0.30/0.46 ! [X2: extended_ereal,Y3: extended_ereal] :
% 0.30/0.46 ( ( ( uminus_uminus @ extended_ereal @ X2 )
% 0.30/0.46 = Y3 )
% 0.30/0.46 => ( ! [R2: real] :
% 0.30/0.46 ( ( X2
% 0.30/0.46 = ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( Y3
% 0.30/0.46 != ( extended_ereal2 @ ( uminus_uminus @ real @ R2 ) ) ) )
% 0.30/0.46 => ( ( ( X2 = extended_PInfty )
% 0.30/0.46 => ( Y3 != extended_MInfty ) )
% 0.30/0.46 => ~ ( ( X2 = extended_MInfty )
% 0.30/0.46 => ( Y3 != extended_PInfty ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % uminus_ereal.elims
% 0.30/0.46 thf(fact_200_ereal_Odistinct_I1_J,axiom,
% 0.30/0.46 ! [X1: real] :
% 0.30/0.46 ( ( extended_ereal2 @ X1 )
% 0.30/0.46 != extended_PInfty ) ).
% 0.30/0.46
% 0.30/0.46 % ereal.distinct(1)
% 0.30/0.46 thf(fact_201_ereal_Osize__gen_I2_J,axiom,
% 0.30/0.46 ( ( extended_size_ereal @ extended_PInfty )
% 0.30/0.46 = ( zero_zero @ nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal.size_gen(2)
% 0.30/0.46 thf(fact_202_infinity__ereal__def,axiom,
% 0.30/0.46 ( ( extend1396239628finity @ extended_ereal )
% 0.30/0.46 = extended_PInfty ) ).
% 0.30/0.46
% 0.30/0.46 % infinity_ereal_def
% 0.30/0.46 thf(fact_203_ereal_Odistinct_I5_J,axiom,
% 0.30/0.46 extended_PInfty != extended_MInfty ).
% 0.30/0.46
% 0.30/0.46 % ereal.distinct(5)
% 0.30/0.46 thf(fact_204_uminus__ereal_Oinduct,axiom,
% 0.30/0.46 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.30/0.46 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ( P @ extended_PInfty )
% 0.30/0.46 => ( ( P @ extended_MInfty )
% 0.30/0.46 => ( P @ A0 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % uminus_ereal.induct
% 0.30/0.46 thf(fact_205_uminus__ereal_Ocases,axiom,
% 0.30/0.46 ! [X2: extended_ereal] :
% 0.30/0.46 ( ! [R2: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 != ( extended_ereal2 @ R2 ) )
% 0.30/0.46 => ( ( X2 != extended_PInfty )
% 0.30/0.46 => ( X2 = extended_MInfty ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % uminus_ereal.cases
% 0.30/0.46 thf(fact_206_ereal_Oexhaust,axiom,
% 0.30/0.46 ! [Y3: extended_ereal] :
% 0.30/0.46 ( ! [X12: real] :
% 0.30/0.46 ( Y3
% 0.30/0.46 != ( extended_ereal2 @ X12 ) )
% 0.30/0.46 => ( ( Y3 != extended_PInfty )
% 0.30/0.46 => ( Y3 = extended_MInfty ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal.exhaust
% 0.30/0.46 thf(fact_207_ereal_Oinduct,axiom,
% 0.30/0.46 ! [P: extended_ereal > $o,Ereal: extended_ereal] :
% 0.30/0.46 ( ! [X3: real] : ( P @ ( extended_ereal2 @ X3 ) )
% 0.30/0.46 => ( ( P @ extended_PInfty )
% 0.30/0.46 => ( ( P @ extended_MInfty )
% 0.30/0.46 => ( P @ Ereal ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal.induct
% 0.30/0.46 thf(fact_208_uminus__ereal_Osimps_I2_J,axiom,
% 0.30/0.46 ( ( uminus_uminus @ extended_ereal @ extended_PInfty )
% 0.30/0.46 = extended_MInfty ) ).
% 0.30/0.46
% 0.30/0.46 % uminus_ereal.simps(2)
% 0.30/0.46 thf(fact_209_uminus__ereal_Osimps_I3_J,axiom,
% 0.30/0.46 ( ( uminus_uminus @ extended_ereal @ extended_MInfty )
% 0.30/0.46 = extended_PInfty ) ).
% 0.30/0.46
% 0.30/0.46 % uminus_ereal.simps(3)
% 0.30/0.46 thf(fact_210_ereal__divide__ereal,axiom,
% 0.30/0.46 ! [R: real] :
% 0.30/0.46 ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R )
% 0.30/0.46 => ( ( divide_divide @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R ) )
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 & ( ~ ( ord_less_eq @ real @ ( zero_zero @ real ) @ R )
% 0.30/0.46 => ( ( divide_divide @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R ) )
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_divide_ereal
% 0.30/0.46 thf(fact_211_ereal__le__real__iff,axiom,
% 0.30/0.46 ! [X2: real,Y3: extended_ereal] :
% 0.30/0.46 ( ( ord_less_eq @ real @ X2 @ ( extend1716541707_ereal @ Y3 ) )
% 0.30/0.46 = ( ( ( ( abs_abs @ extended_ereal @ Y3 )
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ X2 ) @ Y3 ) )
% 0.30/0.46 & ( ( ( abs_abs @ extended_ereal @ Y3 )
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_le_real_iff
% 0.30/0.46 thf(fact_212_abs__idempotent,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere142940540dd_abs @ A )
% 0.30/0.46 => ! [A2: A] :
% 0.30/0.46 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
% 0.30/0.46 = ( abs_abs @ A @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_idempotent
% 0.30/0.46 thf(fact_213_abs__zero,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere142940540dd_abs @ A )
% 0.30/0.46 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 0.30/0.46 = ( zero_zero @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_zero
% 0.30/0.46 thf(fact_214_abs__eq__0,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere142940540dd_abs @ A )
% 0.30/0.46 => ! [A2: A] :
% 0.30/0.46 ( ( ( abs_abs @ A @ A2 )
% 0.30/0.46 = ( zero_zero @ A ) )
% 0.30/0.46 = ( A2
% 0.30/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_eq_0
% 0.30/0.46 thf(fact_215_abs__0__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere142940540dd_abs @ A )
% 0.30/0.46 => ! [A2: A] :
% 0.30/0.46 ( ( ( zero_zero @ A )
% 0.30/0.46 = ( abs_abs @ A @ A2 ) )
% 0.30/0.46 = ( A2
% 0.30/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_0_eq
% 0.30/0.46 thf(fact_216_abs__numeral,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linordered_idom @ A )
% 0.30/0.46 => ! [N: num] :
% 0.30/0.46 ( ( abs_abs @ A @ ( numeral_numeral @ A @ N ) )
% 0.30/0.46 = ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_numeral
% 0.30/0.46 thf(fact_217_abs__minus__cancel,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere142940540dd_abs @ A )
% 0.30/0.46 => ! [A2: A] :
% 0.30/0.46 ( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
% 0.30/0.46 = ( abs_abs @ A @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_minus_cancel
% 0.30/0.46 thf(fact_218_abs__ereal__uminus,axiom,
% 0.30/0.46 ! [X2: extended_ereal] :
% 0.30/0.46 ( ( abs_abs @ extended_ereal @ ( uminus_uminus @ extended_ereal @ X2 ) )
% 0.30/0.46 = ( abs_abs @ extended_ereal @ X2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_ereal_uminus
% 0.30/0.46 thf(fact_219_abs__ereal__zero,axiom,
% 0.30/0.46 ( ( abs_abs @ extended_ereal @ ( zero_zero @ extended_ereal ) )
% 0.30/0.46 = ( zero_zero @ extended_ereal ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_ereal_zero
% 0.30/0.46 thf(fact_220_ereal__uminus__divide,axiom,
% 0.30/0.46 ! [X2: extended_ereal,Y3: extended_ereal] :
% 0.30/0.46 ( ( divide_divide @ extended_ereal @ ( uminus_uminus @ extended_ereal @ X2 ) @ Y3 )
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( divide_divide @ extended_ereal @ X2 @ Y3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_uminus_divide
% 0.30/0.46 thf(fact_221_ereal__divide__zero__left,axiom,
% 0.30/0.46 ! [A2: extended_ereal] :
% 0.30/0.46 ( ( divide_divide @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A2 )
% 0.30/0.46 = ( zero_zero @ extended_ereal ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_divide_zero_left
% 0.30/0.46 thf(fact_222_abs__of__nonneg,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere142940540dd_abs @ A )
% 0.30/0.46 => ! [A2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.30/0.46 => ( ( abs_abs @ A @ A2 )
% 0.30/0.46 = A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_of_nonneg
% 0.30/0.46 thf(fact_223_abs__le__self__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere142940540dd_abs @ A )
% 0.30/0.46 => ! [A2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 )
% 0.30/0.46 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_le_self_iff
% 0.30/0.46 thf(fact_224_abs__le__zero__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere142940540dd_abs @ A )
% 0.30/0.46 => ! [A2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.30/0.46 = ( A2
% 0.30/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_le_zero_iff
% 0.30/0.46 thf(fact_225_abs__neg__numeral,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linordered_idom @ A )
% 0.30/0.46 => ! [N: num] :
% 0.30/0.46 ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.46 = ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_neg_numeral
% 0.30/0.46 thf(fact_226_abs__ereal__ge0,axiom,
% 0.30/0.46 ! [X2: extended_ereal] :
% 0.30/0.46 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ X2 )
% 0.30/0.46 => ( ( abs_abs @ extended_ereal @ X2 )
% 0.30/0.46 = X2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_ereal_ge0
% 0.30/0.46 thf(fact_227_ereal__divide__Infty_I1_J,axiom,
% 0.30/0.46 ! [X2: extended_ereal] :
% 0.30/0.46 ( ( divide_divide @ extended_ereal @ X2 @ ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 = ( zero_zero @ extended_ereal ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_divide_Infty(1)
% 0.30/0.46 thf(fact_228_abs__of__nonpos,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere142940540dd_abs @ A )
% 0.30/0.46 => ! [A2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.30/0.46 => ( ( abs_abs @ A @ A2 )
% 0.30/0.46 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_of_nonpos
% 0.30/0.46 thf(fact_229_ereal__divide__Infty_I2_J,axiom,
% 0.30/0.46 ! [X2: extended_ereal] :
% 0.30/0.46 ( ( divide_divide @ extended_ereal @ X2 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 = ( zero_zero @ extended_ereal ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_divide_Infty(2)
% 0.30/0.46 thf(fact_230_not__infty__ereal,axiom,
% 0.30/0.46 ! [X2: extended_ereal] :
% 0.30/0.46 ( ( ( abs_abs @ extended_ereal @ X2 )
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 = ( ? [X7: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 = ( extended_ereal2 @ X7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % not_infty_ereal
% 0.30/0.46 thf(fact_231_abs__neq__infinity__cases,axiom,
% 0.30/0.46 ! [X2: extended_ereal] :
% 0.30/0.46 ( ( ( abs_abs @ extended_ereal @ X2 )
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ~ ! [R2: real] :
% 0.30/0.46 ( X2
% 0.30/0.46 != ( extended_ereal2 @ R2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_neq_infinity_cases
% 0.30/0.46 thf(fact_232_abs__ereal_Osimps_I2_J,axiom,
% 0.30/0.46 ( ( abs_abs @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_ereal.simps(2)
% 0.30/0.46 thf(fact_233_ereal__infinity__cases,axiom,
% 0.30/0.46 ! [A2: extended_ereal] :
% 0.30/0.46 ( ( A2
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ( A2
% 0.30/0.46 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 0.30/0.46 => ( ( abs_abs @ extended_ereal @ A2 )
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_infinity_cases
% 0.30/0.46 thf(fact_234_abs__eq__infinity__cases,axiom,
% 0.30/0.46 ! [X2: extended_ereal] :
% 0.30/0.46 ( ( ( abs_abs @ extended_ereal @ X2 )
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ( X2
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( X2
% 0.30/0.46 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_eq_infinity_cases
% 0.30/0.46 thf(fact_235_ereal__abs__leI,axiom,
% 0.30/0.46 ! [X2: extended_ereal,Y3: extended_ereal] :
% 0.30/0.46 ( ( ord_less_eq @ extended_ereal @ X2 @ Y3 )
% 0.30/0.46 => ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ X2 ) @ Y3 )
% 0.30/0.46 => ( ord_less_eq @ extended_ereal @ ( abs_abs @ extended_ereal @ X2 ) @ Y3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_abs_leI
% 0.30/0.46 thf(fact_236_abs__ereal__pos,axiom,
% 0.30/0.46 ! [X2: extended_ereal] : ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( abs_abs @ extended_ereal @ X2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_ereal_pos
% 0.30/0.46 thf(fact_237_zero__le__divide__ereal,axiom,
% 0.30/0.46 ! [A2: extended_ereal,B2: extended_ereal] :
% 0.30/0.46 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A2 )
% 0.30/0.46 => ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ B2 )
% 0.30/0.46 => ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( divide_divide @ extended_ereal @ A2 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_le_divide_ereal
% 0.30/0.46 thf(fact_238_abs__ge__zero,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere142940540dd_abs @ A )
% 0.30/0.46 => ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_ge_zero
% 0.30/0.46 thf(fact_239_abs__leI,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere142940540dd_abs @ A )
% 0.30/0.46 => ! [A2: A,B2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_leI
% 0.30/0.46 thf(fact_240_abs__le__D2,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere142940540dd_abs @ A )
% 0.30/0.46 => ! [A2: A,B2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_le_D2
% 0.30/0.46 thf(fact_241_abs__le__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere142940540dd_abs @ A )
% 0.30/0.46 => ! [A2: A,B2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
% 0.30/0.46 = ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.46 & ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_le_iff
% 0.30/0.46 thf(fact_242_abs__ge__minus__self,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere142940540dd_abs @ A )
% 0.30/0.46 => ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( abs_abs @ A @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_ge_minus_self
% 0.30/0.46 thf(fact_243_ereal__divide__eq__0__iff,axiom,
% 0.30/0.46 ! [X2: extended_ereal,Y3: extended_ereal] :
% 0.30/0.46 ( ( ( divide_divide @ extended_ereal @ X2 @ Y3 )
% 0.30/0.46 = ( zero_zero @ extended_ereal ) )
% 0.30/0.46 = ( ( X2
% 0.30/0.46 = ( zero_zero @ extended_ereal ) )
% 0.30/0.46 | ( ( abs_abs @ extended_ereal @ Y3 )
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_divide_eq_0_iff
% 0.30/0.46 thf(fact_244_abs__ereal_Osimps_I3_J,axiom,
% 0.30/0.46 ( ( abs_abs @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_ereal.simps(3)
% 0.30/0.46 thf(fact_245_abs__ge__self,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere142940540dd_abs @ A )
% 0.30/0.46 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( abs_abs @ A @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_ge_self
% 0.30/0.46 thf(fact_246_abs__le__D1,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere142940540dd_abs @ A )
% 0.30/0.46 => ! [A2: A,B2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
% 0.30/0.46 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_le_D1
% 0.30/0.46 thf(fact_247_abs__minus__le__zero,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere142940540dd_abs @ A )
% 0.30/0.46 => ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A2 ) ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_minus_le_zero
% 0.30/0.46 thf(fact_248_ereal__real_H,axiom,
% 0.30/0.46 ! [X2: extended_ereal] :
% 0.30/0.46 ( ( ( abs_abs @ extended_ereal @ X2 )
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
% 0.30/0.46 = X2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_real'
% 0.30/0.46 thf(fact_249_ereal__real,axiom,
% 0.30/0.46 ! [X2: extended_ereal] :
% 0.30/0.46 ( ( ( ( abs_abs @ extended_ereal @ X2 )
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
% 0.30/0.46 = ( zero_zero @ extended_ereal ) ) )
% 0.30/0.46 & ( ( ( abs_abs @ extended_ereal @ X2 )
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
% 0.30/0.46 = X2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ereal_real
% 0.30/0.46 thf(fact_250_real__le__ereal__iff,axiom,
% 0.30/0.46 ! [Y3: extended_ereal,X2: real] :
% 0.30/0.46 ( ( ord_less_eq @ real @ ( extend1716541707_ereal @ Y3 ) @ X2 )
% 0.30/0.46 = ( ( ( ( abs_abs @ extended_ereal @ Y3 )
% 0.30/0.46 != ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ord_less_eq @ extended_ereal @ Y3 @ ( extended_ereal2 @ X2 ) ) )
% 0.30/0.46 & ( ( ( abs_abs @ extended_ereal @ Y3 )
% 0.30/0.46 = ( extend1396239628finity @ extended_ereal ) )
% 0.30/0.46 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % real_le_ereal_iff
% 0.30/0.46 thf(fact_251_divide__le__0__abs__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linordered_field @ A )
% 0.30/0.46 => ! [A2: A,B2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) @ ( zero_zero @ A ) )
% 0.30/0.46 = ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.30/0.46 | ( B2
% 0.30/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % divide_le_0_abs_iff
% 0.30/0.46
% 0.30/0.46 % Subclasses (2)
% 0.30/0.46 thf(subcl_Real__Vector__Spaces_Ometric__space___HOL_Otype,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( real_V2090557954_space @ A )
% 0.30/0.46 => ( type @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Otopological__space,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( real_V2090557954_space @ A )
% 0.30/0.46 => ( topolo503727757_space @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % Type constructors (69)
% 0.30/0.46 thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
% 0.30/0.46 ! [A6: $tType,A7: $tType] :
% 0.30/0.46 ( ( topolo503727757_space @ A7 )
% 0.30/0.46 => ( topolo503727757_space @ ( A6 > A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 0.30/0.46 ! [A6: $tType,A7: $tType] :
% 0.30/0.46 ( ( boolean_algebra @ A7 )
% 0.30/0.46 => ( boolean_algebra @ ( A6 > A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/0.46 ! [A6: $tType,A7: $tType] :
% 0.30/0.46 ( ( preorder @ A7 )
% 0.30/0.46 => ( preorder @ ( A6 > A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/0.46 ! [A6: $tType,A7: $tType] :
% 0.30/0.46 ( ( order @ A7 )
% 0.30/0.46 => ( order @ ( A6 > A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/0.46 ! [A6: $tType,A7: $tType] :
% 0.30/0.46 ( ( ord @ A7 )
% 0.30/0.46 => ( ord @ ( A6 > A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Groups_Ouminus,axiom,
% 0.30/0.46 ! [A6: $tType,A7: $tType] :
% 0.30/0.46 ( ( uminus @ A7 )
% 0.30/0.46 => ( uminus @ ( A6 > A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.30/0.46 canoni770627133id_add @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_1,axiom,
% 0.30/0.46 topolo503727757_space @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology,axiom,
% 0.30/0.46 topolo259154727pology @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 0.30/0.46 linord1659791738miring @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Nat_Osemiring__char__0,axiom,
% 0.30/0.46 semiring_char_0 @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
% 0.30/0.46 preorder @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.30/0.46 linorder @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
% 0.30/0.46 order @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
% 0.30/0.46 ord @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Num_Onumeral,axiom,
% 0.30/0.46 numeral @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.30/0.46 zero @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Num_Onum___Orderings_Opreorder_5,axiom,
% 0.30/0.46 preorder @ num ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Num_Onum___Orderings_Olinorder_6,axiom,
% 0.30/0.46 linorder @ num ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Num_Onum___Orderings_Oorder_7,axiom,
% 0.30/0.46 order @ num ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Num_Onum___Orderings_Oord_8,axiom,
% 0.30/0.46 ord @ num ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_9,axiom,
% 0.30/0.46 topolo503727757_space @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_10,axiom,
% 0.30/0.46 topolo259154727pology @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_11,axiom,
% 0.30/0.46 boolean_algebra @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
% 0.30/0.46 preorder @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder_13,axiom,
% 0.30/0.46 linorder @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
% 0.30/0.46 order @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
% 0.30/0.46 ord @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Groups_Ouminus_16,axiom,
% 0.30/0.46 uminus @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_17,axiom,
% 0.30/0.46 topolo503727757_space @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_18,axiom,
% 0.30/0.46 topolo259154727pology @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_19,axiom,
% 0.30/0.46 linord1659791738miring @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
% 0.30/0.46 real_V2090557954_space @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs,axiom,
% 0.30/0.46 ordere142940540dd_abs @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
% 0.30/0.46 linord219039673up_add @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
% 0.30/0.46 ordered_ab_group_add @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 0.30/0.46 linordered_field @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Rings_Olinordered__idom,axiom,
% 0.30/0.46 linordered_idom @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_20,axiom,
% 0.30/0.46 semiring_char_0 @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Orderings_Opreorder_21,axiom,
% 0.30/0.46 preorder @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Orderings_Olinorder_22,axiom,
% 0.30/0.46 linorder @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 0.30/0.46 group_add @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Orderings_Oorder_23,axiom,
% 0.30/0.46 order @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Num_Oneg__numeral,axiom,
% 0.30/0.46 neg_numeral @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Nat_Oring__char__0,axiom,
% 0.30/0.46 ring_char_0 @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Orderings_Oord_24,axiom,
% 0.30/0.46 ord @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Groups_Ouminus_25,axiom,
% 0.30/0.46 uminus @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Num_Onumeral_26,axiom,
% 0.30/0.46 numeral @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Groups_Ozero_27,axiom,
% 0.30/0.46 zero @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_28,axiom,
% 0.30/0.46 canoni770627133id_add @ extended_enat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Nat_Oenat___Topological__Spaces_Otopological__space_29,axiom,
% 0.30/0.46 topolo503727757_space @ extended_enat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Nat_Oenat___Topological__Spaces_Oorder__topology_30,axiom,
% 0.30/0.46 topolo259154727pology @ extended_enat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_31,axiom,
% 0.30/0.46 linord1659791738miring @ extended_enat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_32,axiom,
% 0.30/0.46 semiring_char_0 @ extended_enat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_33,axiom,
% 0.30/0.46 preorder @ extended_enat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_34,axiom,
% 0.30/0.46 linorder @ extended_enat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_35,axiom,
% 0.30/0.46 order @ extended_enat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_36,axiom,
% 0.30/0.46 ord @ extended_enat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Nat_Oenat___Num_Onumeral_37,axiom,
% 0.30/0.46 numeral @ extended_enat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_38,axiom,
% 0.30/0.46 zero @ extended_enat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Otopological__space_39,axiom,
% 0.30/0.46 topolo503727757_space @ extended_ereal ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Oorder__topology_40,axiom,
% 0.30/0.46 topolo259154727pology @ extended_ereal ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Real_Oereal___Orderings_Opreorder_41,axiom,
% 0.30/0.46 preorder @ extended_ereal ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Real_Oereal___Orderings_Olinorder_42,axiom,
% 0.30/0.46 linorder @ extended_ereal ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Real_Oereal___Orderings_Oorder_43,axiom,
% 0.30/0.46 order @ extended_ereal ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Real_Oereal___Orderings_Oord_44,axiom,
% 0.30/0.46 ord @ extended_ereal ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Real_Oereal___Groups_Ouminus_45,axiom,
% 0.30/0.46 uminus @ extended_ereal ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Real_Oereal___Num_Onumeral_46,axiom,
% 0.30/0.46 numeral @ extended_ereal ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Extended__Real_Oereal___Groups_Ozero_47,axiom,
% 0.30/0.46 zero @ extended_ereal ).
% 0.30/0.46
% 0.30/0.46 % Free types (1)
% 0.30/0.46 thf(tfree_0,hypothesis,
% 0.30/0.46 real_V2090557954_space @ a ).
% 0.30/0.46
% 0.30/0.46 % Conjectures (1)
% 0.30/0.46 thf(conj_0,conjecture,
% 0.30/0.46 ord_less_eq @ extended_ereal @ ( g @ x ) @ ( f @ x ) ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 2.62/1.49 ---- Embedded in THF ---
% 2.62/1.49 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.62/1.49 %%% Generated on Tue Feb 24 08:26:43 EST 2026
% 2.62/1.49 %%% using '$$dhol' embedding, version 1.3.0.
% 2.62/1.49 %%% Logic specification used:
% 2.62/1.49 %%% thf(spec, logic, $$dhol).
% 2.62/1.49
% 2.62/1.49 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.pLGO0LFU7P/DTF2DTF22527.p
% 2.62/1.49 thf(ty_t_Extended__Real_Oereal, type, extended_ereal: $tType).
% 2.62/1.49 thf(ty_t_Extended__Nat_Oenat, type, extended_enat: $tType).
% 2.62/1.49 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.62/1.49 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.62/1.49 thf(ty_t_Num_Onum, type, num: $tType).
% 2.62/1.49 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.62/1.49 thf(ty_tf_a, type, a: $tType).
% 2.62/1.49 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Num_Onumeral, type, numeral: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Groups_Ouminus, type, uminus: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Nat_Oring__char__0, type, ring_char_0: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Num_Oneg__numeral, type, neg_numeral: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Lattices_Oboolean__algebra, type, boolean_algebra: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Groups_Oordered__ab__group__add__abs, type, ordere142940540dd_abs: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Real__Vector__Spaces_Ometric__space, type, real_V2090557954_space: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Topological__Spaces_Oorder__topology, type, topolo259154727pology: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Topological__Spaces_Otopological__space, type, topolo503727757_space: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.62/1.49 thf(sy_c_Extended__Nat_Oenat, type, extended_enat2: (nat > extended_enat)).
% 2.62/1.49 thf(sy_c_Extended__Nat_Oenat_Ocase__enat, type, extended_case_enat: (!> [T:$tType]: (((nat > T) > (T > (extended_enat > T)))))).
% 2.62/1.49 thf(sy_c_Extended__Nat_Oenat_Orec__enat, type, extended_rec_enat: (!> [T:$tType]: (((nat > T) > (T > (extended_enat > T)))))).
% 2.62/1.49 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity, type, extend1396239628finity: (!> [A:$tType]: (A))).
% 2.62/1.49 thf(sy_c_Extended__Nat_Othe__enat, type, extended_the_enat: (extended_enat > nat)).
% 2.62/1.49 thf(sy_c_Extended__Real_Oereal_OMInfty, type, extended_MInfty: extended_ereal).
% 2.62/1.49 thf(sy_c_Extended__Real_Oereal_OPInfty, type, extended_PInfty: extended_ereal).
% 2.62/1.49 thf(sy_c_Extended__Real_Oereal_Oereal, type, extended_ereal2: (real > extended_ereal)).
% 2.62/1.49 thf(sy_c_Extended__Real_Oereal_Osize__ereal, type, extended_size_ereal: (extended_ereal > nat)).
% 2.62/1.49 thf(sy_c_Extended__Real_Oereal__of__enat, type, extend1771934483f_enat: (extended_enat > extended_ereal)).
% 2.62/1.49 thf(sy_c_Extended__Real_Oreal__of__ereal, type, extend1716541707_ereal: (extended_ereal > real)).
% 2.62/1.49 thf(sy_c_Groups_Oabs__class_Oabs, type, abs_abs: (!> [A:$tType]: ((A > A)))).
% 2.62/1.49 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 2.62/1.49 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.62/1.49 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc, type, lower_107104146le_lsc: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.62/1.49 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc__at, type, lower_582600101lsc_at: (!> [A:$tType,B:$tType]: ((A > ((A > B) > $o))))).
% 2.62/1.49 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc__hull, type, lower_1879122655c_hull: (!> [A:$tType]: (((A > extended_ereal) > (A > extended_ereal))))).
% 2.62/1.49 thf(sy_c_Num_Oneg__numeral__class_Odbl, type, neg_numeral_dbl: (!> [A:$tType]: ((A > A)))).
% 2.62/1.49 thf(sy_c_Num_Oneg__numeral__class_Osub, type, neg_numeral_sub: (!> [A:$tType]: ((num > (num > A))))).
% 2.62/1.49 thf(sy_c_Num_Onumeral__class_Onumeral, type, numeral_numeral: (!> [A:$tType]: ((num > A)))).
% 2.62/1.49 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.62/1.49 thf(sy_c_Orderings_Oorder__class_OGreatest, type, order_Greatest: (!> [A:$tType]: (((A > $o) > A)))).
% 2.62/1.49 thf(sy_c_Orderings_Oorder__class_Oantimono, type, order_antimono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.62/1.49 thf(sy_c_Orderings_Oorder__class_Ostrict__mono, type, order_strict_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.62/1.49 thf(sy_c_Rings_Odivide__class_Odivide, type, divide_divide: (!> [A:$tType]: ((A > (A > A))))).
% 2.62/1.49 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.62/1.49 thf(sy_v_f, type, f: (a > extended_ereal)).
% 2.62/1.49 thf(sy_v_g, type, g: (a > extended_ereal)).
% 2.62/1.49 thf(sy_v_x____, type, x: a).
% 2.62/1.49 thf(fact_0_assms, axiom, (! [X:a]: ((((ord_less_eq @ extended_ereal) @ (g @ X)) @ (f @ X))))).
% 2.62/1.49 thf(fact_1_calculation, axiom, (((ord_less_eq @ extended_ereal) @ (((lower_1879122655c_hull @ a) @ g) @ x)) @ (g @ x))).
% 2.62/1.49 thf(fact_2_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ X2) @ X2))))))).
% 2.62/1.49 thf(fact_3_ereal__complete__Inf, axiom, (! [S:(set @ extended_ereal)]: ((? [X3:extended_ereal]: (((! [Xa:extended_ereal]: (((((member @ extended_ereal) @ Xa) @ S) => (((ord_less_eq @ extended_ereal) @ X3) @ Xa)))) & (! [Z:extended_ereal]: (((! [Xa2:extended_ereal]: (((((member @ extended_ereal) @ Xa2) @ S) => (((ord_less_eq @ extended_ereal) @ Z) @ Xa2)))) => (((ord_less_eq @ extended_ereal) @ Z) @ X3)))))))))).
% 2.62/1.49 thf(fact_4_ereal__complete__Sup, axiom, (! [S:(set @ extended_ereal)]: ((? [X3:extended_ereal]: (((! [Xa:extended_ereal]: (((((member @ extended_ereal) @ Xa) @ S) => (((ord_less_eq @ extended_ereal) @ Xa) @ X3)))) & (! [Z:extended_ereal]: (((! [Xa2:extended_ereal]: (((((member @ extended_ereal) @ Xa2) @ S) => (((ord_less_eq @ extended_ereal) @ Xa2) @ Z)))) => (((ord_less_eq @ extended_ereal) @ X3) @ Z)))))))))).
% 2.62/1.49 thf(fact_5_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.62/1.49 thf(fact_6_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.62/1.49 thf(fact_7_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.62/1.49 thf(fact_8_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X4:A]: ((((ord_less_eq @ B) @ (F2 @ X4)) @ (G2 @ X4))))))))))).
% 2.62/1.49 thf(fact_9_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X3:B,Y:B]: (((((ord_less_eq @ B) @ X3) @ Y) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.62/1.49 thf(fact_10_order__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C2) @ (F @ B2)) @ C) => ((! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => (((ord_less_eq @ C2) @ (F @ X3)) @ (F @ Y))))) => (((ord_less_eq @ C2) @ (F @ A2)) @ C)))))))))).
% 2.62/1.49 thf(fact_11_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((A2 = B2) | ((~ ((((ord_less_eq @ A) @ A2) @ B2))) | (~ ((((ord_less_eq @ A) @ B2) @ A2))))))))))).
% 2.62/1.49 thf(fact_12_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F:(B > A),B2:B,C:B]: (((A2 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X3:B,Y:B]: (((((ord_less_eq @ B) @ X3) @ Y) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.62/1.49 thf(fact_13_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F:(A > B),C:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F @ B2) = C) => ((! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C)))))))))).
% 2.62/1.49 thf(fact_14_lsc__hull__le, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [F:(A > extended_ereal),X2:A]: ((((ord_less_eq @ extended_ereal) @ (((lower_1879122655c_hull @ A) @ F) @ X2)) @ (F @ X2)))))))).
% 2.62/1.49 thf(fact_15_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.62/1.49 thf(fact_16_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z2:A]: ((Y2 = Z2))) = (^ [A3:A,B3:A]: (((((ord_less_eq @ A) @ B3) @ A3) & (((ord_less_eq @ A) @ A3) @ B3))))))))).
% 2.62/1.49 thf(fact_17_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less_eq @ A) @ C) @ A2))))))))).
% 2.62/1.49 thf(fact_18_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((P @ A4) @ B4)))) => ((! [A4:A,B4:A]: ((((P @ B4) @ A4) => ((P @ A4) @ B4)))) => ((P @ A2) @ B2))))))))).
% 2.62/1.49 thf(fact_19_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.62/1.49 thf(fact_20_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y3:A,Z3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ Z3) => (((ord_less_eq @ A) @ X2) @ Z3))))))))).
% 2.62/1.49 thf(fact_21_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.62/1.49 thf(fact_22_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.62/1.49 thf(fact_23_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.62/1.49 thf(fact_24_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z2:A]: ((Y2 = Z2))) = (^ [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) & (((ord_less_eq @ A) @ B3) @ A3))))))))).
% 2.62/1.49 thf(fact_25_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y3:A,X2:A]: (((((ord_less_eq @ A) @ Y3) @ X2) => ((((ord_less_eq @ A) @ X2) @ Y3) = (X2 = Y3))))))))).
% 2.62/1.49 thf(fact_26_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y3:A,Z3:A]: ((((((ord_less_eq @ A) @ X2) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ Z3)))) => (((((ord_less_eq @ A) @ Y3) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Z3)))) => (((((ord_less_eq @ A) @ X2) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ Y3)))) => (((((ord_less_eq @ A) @ Z3) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ X2)))) => (((((ord_less_eq @ A) @ Y3) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ X2)))) => (~ (((((ord_less_eq @ A) @ Z3) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Y3))))))))))))))))).
% 2.62/1.49 thf(fact_27_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.62/1.49 thf(fact_28_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y3:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y3))) => (((ord_less_eq @ A) @ Y3) @ X2)))))))).
% 2.62/1.49 thf(fact_29_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y3:A]: (((X2 = Y3) => (((ord_less_eq @ A) @ X2) @ Y3)))))))).
% 2.62/1.49 thf(fact_30_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) | (((ord_less_eq @ A) @ Y3) @ X2)))))))).
% 2.62/1.49 thf(fact_31_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ X2) => (X2 = Y3))))))))).
% 2.62/1.49 thf(fact_32_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z2:A]: ((Y2 = Z2))) = (^ [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) & (((ord_less_eq @ A) @ Y4) @ X4))))))))).
% 2.62/1.49 thf(fact_33_lsc__hull__iff__greatest, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [G:(A > extended_ereal),F:(A > extended_ereal)]: (((G = ((lower_1879122655c_hull @ A) @ F)) = ((((lower_107104146le_lsc @ A) @ extended_ereal) @ G) & ((! [X4:A]: ((((ord_less_eq @ extended_ereal) @ (G @ X4)) @ (F @ X4)))) & (! [H:(A > extended_ereal)]: ((((((lower_107104146le_lsc @ A) @ extended_ereal) @ H) & (! [X4:A]: ((((ord_less_eq @ extended_ereal) @ (H @ X4)) @ (F @ X4))))) => (! [X4:A]: ((((ord_less_eq @ extended_ereal) @ (H @ X4)) @ (G @ X4)))))))))))))))).
% 2.62/1.49 thf(fact_34_lsc__hull__greatest, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [G:(A > extended_ereal),F:(A > extended_ereal)]: (((((lower_107104146le_lsc @ A) @ extended_ereal) @ G) => ((! [X3:A]: ((((ord_less_eq @ extended_ereal) @ (G @ X3)) @ (F @ X3)))) => (! [X:A]: ((((ord_less_eq @ extended_ereal) @ (G @ X)) @ (((lower_1879122655c_hull @ A) @ F) @ X)))))))))))).
% 2.62/1.49 thf(fact_35_Greatest__equality, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X2:A]: (((P @ X2) => ((! [Y:A]: (((P @ Y) => (((ord_less_eq @ A) @ Y) @ X2)))) => (((order_Greatest @ A) @ P) = X2))))))))).
% 2.62/1.49 thf(fact_36_GreatestI2__order, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X2:A,Q:(A > $o)]: (((P @ X2) => ((! [Y:A]: (((P @ Y) => (((ord_less_eq @ A) @ Y) @ X2)))) => ((! [X3:A]: (((P @ X3) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ A) @ Y5) @ X3)))) => (Q @ X3))))) => (Q @ ((order_Greatest @ A) @ P))))))))))).
% 2.62/1.49 thf(fact_37_le__rel__bool__arg__iff, axiom, (! [A:$tType]: (((ord @ A) => ((ord_less_eq @ ($o > A)) = (^ [X5:($o > A),Y6:($o > A)]: (((((ord_less_eq @ A) @ (X5 @ $false)) @ (Y6 @ $false)) & (((ord_less_eq @ A) @ (X5 @ $true)) @ (Y6 @ $true)))))))))).
% 2.62/1.49 thf(fact_38_lsc__lsc__hull, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [F:(A > extended_ereal)]: ((((lower_107104146le_lsc @ A) @ extended_ereal) @ ((lower_1879122655c_hull @ A) @ F)))))))).
% 2.62/1.49 thf(fact_39_ereal__of__enat__le__iff, axiom, (! [M:extended_enat,N:extended_enat]: (((((ord_less_eq @ extended_ereal) @ (extend1771934483f_enat @ M)) @ (extend1771934483f_enat @ N)) = (((ord_less_eq @ extended_enat) @ M) @ N))))).
% 2.62/1.49 thf(fact_40_antimono__def, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (((order_antimono @ A) @ B) = (^ [F2:(A > B)]: ((! [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) => (((ord_less_eq @ B) @ (F2 @ Y4)) @ (F2 @ X4)))))))))))).
% 2.62/1.49 thf(fact_41_antimonoI, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F:(A > B)]: (((! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => (((ord_less_eq @ B) @ (F @ Y)) @ (F @ X3))))) => (((order_antimono @ A) @ B) @ F)))))))).
% 2.62/1.49 thf(fact_42_antimonoE, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F:(A > B),X2:A,Y3:A]: (((((order_antimono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X2) @ Y3) => (((ord_less_eq @ B) @ (F @ Y3)) @ (F @ X2)))))))))).
% 2.62/1.49 thf(fact_43_antimonoD, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F:(A > B),X2:A,Y3:A]: (((((order_antimono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X2) @ Y3) => (((ord_less_eq @ B) @ (F @ Y3)) @ (F @ X2)))))))))).
% 2.62/1.49 thf(fact_44_lsc__def, axiom, (! [B:$tType,A:$tType]: ((((topolo503727757_space @ A) & (topolo259154727pology @ B)) => (((lower_107104146le_lsc @ A) @ B) = (^ [F2:(A > B)]: ((! [X4:A]: (((((lower_582600101lsc_at @ A) @ B) @ X4) @ F2)))))))))).
% 2.62/1.49 thf(fact_45_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.62/1.49 thf(fact_46_decseqD, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),I:nat,J:nat]: (((((order_antimono @ nat) @ A) @ F) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ A) @ (F @ J)) @ (F @ I)))))))))).
% 2.62/1.49 thf(fact_47_decseq__def, axiom, (! [A:$tType]: (((order @ A) => (((order_antimono @ nat) @ A) = (^ [X5:(nat > A)]: ((! [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) => (((ord_less_eq @ A) @ (X5 @ N2)) @ (X5 @ M2)))))))))))).
% 2.62/1.49 thf(fact_48_ereal__of__enat__nonneg, axiom, (! [N:extended_enat]: ((((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ (extend1771934483f_enat @ N))))).
% 2.62/1.49 thf(fact_49_ereal__of__enat__ge__zero__cancel__iff, axiom, (! [N:extended_enat]: (((((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ (extend1771934483f_enat @ N)) = (((ord_less_eq @ extended_enat) @ (zero_zero @ extended_enat)) @ N))))).
% 2.62/1.49 thf(fact_50_numeral__le__ereal__of__enat__iff, axiom, (! [M:num,N:extended_enat]: (((((ord_less_eq @ extended_ereal) @ ((numeral_numeral @ extended_ereal) @ M)) @ (extend1771934483f_enat @ N)) = (((ord_less_eq @ extended_enat) @ ((numeral_numeral @ extended_enat) @ M)) @ N))))).
% 2.62/1.49 thf(fact_51_ereal__infty__less__eq_I1_J, axiom, (! [X2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ (extend1396239628finity @ extended_ereal)) @ X2) = (X2 = (extend1396239628finity @ extended_ereal)))))).
% 2.62/1.49 thf(fact_52_strict__mono__less__eq, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F:(A > B),X2:A,Y3:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less_eq @ B) @ (F @ X2)) @ (F @ Y3)) = (((ord_less_eq @ A) @ X2) @ Y3))))))))).
% 2.62/1.49 thf(fact_53_ereal__of__enat__zero, axiom, ((extend1771934483f_enat @ (zero_zero @ extended_enat)) = (zero_zero @ extended_ereal))).
% 2.62/1.49 thf(fact_54_strict__mono__eq, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F:(A > B),X2:A,Y3:A]: (((((order_strict_mono @ A) @ B) @ F) => (((F @ X2) = (F @ Y3)) = (X2 = Y3))))))))).
% 2.62/1.49 thf(fact_55_Infty__neq__0_I1_J, axiom, ((extend1396239628finity @ extended_ereal) != (zero_zero @ extended_ereal))).
% 2.62/1.49 thf(fact_56_strict__mono__leD, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [R:(A > B),M:A,N:A]: (((((order_strict_mono @ A) @ B) @ R) => ((((ord_less_eq @ A) @ M) @ N) => (((ord_less_eq @ B) @ (R @ M)) @ (R @ N)))))))))).
% 2.62/1.49 thf(fact_57_neq__PInf__trans, axiom, (! [Y3:extended_ereal,X2:extended_ereal]: (((Y3 != (extend1396239628finity @ extended_ereal)) => ((((ord_less_eq @ extended_ereal) @ X2) @ Y3) => (X2 != (extend1396239628finity @ extended_ereal))))))).
% 2.62/1.49 thf(fact_58_ereal__infty__less__eq2_I1_J, axiom, (! [A2:extended_ereal,B2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ A2) @ B2) => ((A2 = (extend1396239628finity @ extended_ereal)) => (B2 = (extend1396239628finity @ extended_ereal))))))).
% 2.62/1.49 thf(fact_59_ereal__less__eq_I1_J, axiom, (! [X2:extended_ereal]: ((((ord_less_eq @ extended_ereal) @ X2) @ (extend1396239628finity @ extended_ereal))))).
% 2.62/1.49 thf(fact_60_numeral__le__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:num,N:num]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N)) = (((ord_less_eq @ num) @ M) @ N)))))))).
% 2.62/1.49 thf(fact_61_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.62/1.49 thf(fact_62_zero__le__numeral, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [N:num]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ N)))))))).
% 2.62/1.49 thf(fact_63_not__numeral__le__zero, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [N:num]: ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ N)) @ (zero_zero @ A)))))))))).
% 2.62/1.49 thf(fact_64_numeral__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:num,N:num]: (((((numeral_numeral @ A) @ M) = ((numeral_numeral @ A) @ N)) = (M = N)))))))).
% 2.62/1.49 thf(fact_65_ile0__eq, axiom, (! [N:extended_enat]: (((((ord_less_eq @ extended_enat) @ N) @ (zero_zero @ extended_enat)) = (N = (zero_zero @ extended_enat)))))).
% 2.62/1.49 thf(fact_66_i0__lb, axiom, (! [N:extended_enat]: ((((ord_less_eq @ extended_enat) @ (zero_zero @ extended_enat)) @ N)))).
% 2.62/1.49 thf(fact_67_enat__ord__number_I1_J, axiom, (! [M:num,N:num]: (((((ord_less_eq @ extended_enat) @ ((numeral_numeral @ extended_enat) @ M)) @ ((numeral_numeral @ extended_enat) @ N)) = (((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ M)) @ ((numeral_numeral @ nat) @ N)))))).
% 2.62/1.49 thf(fact_68_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X2:A]: ((((zero_zero @ A) = X2) = (X2 = (zero_zero @ A))))))))).
% 2.62/1.49 thf(fact_69_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X2))))))).
% 2.62/1.49 thf(fact_70_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.62/1.49 thf(fact_71_zero__neq__numeral, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:num]: (((zero_zero @ A) != ((numeral_numeral @ A) @ N)))))))).
% 2.62/1.49 thf(fact_72_bot__nat__0_Oextremum, axiom, (! [A2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A2)))).
% 2.62/1.49 thf(fact_73_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.62/1.49 thf(fact_74_neg__numeral__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num,N:num]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = (((ord_less_eq @ num) @ N) @ M)))))))).
% 2.62/1.49 thf(fact_75_numeral__le__enat__iff, axiom, (! [M:num,N:nat]: (((((ord_less_eq @ extended_enat) @ ((numeral_numeral @ extended_enat) @ M)) @ (extended_enat2 @ N)) = (((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ M)) @ N))))).
% 2.62/1.49 thf(fact_76_sub__non__positive, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num,M:num]: (((((ord_less_eq @ A) @ (((neg_numeral_sub @ A) @ N) @ M)) @ (zero_zero @ A)) = (((ord_less_eq @ num) @ N) @ M)))))))).
% 2.62/1.49 thf(fact_77_sub__non__negative, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num,M:num]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((neg_numeral_sub @ A) @ N) @ M)) = (((ord_less_eq @ num) @ M) @ N)))))))).
% 2.62/1.49 thf(fact_78_real__of__ereal__le__0, axiom, (! [X2:extended_ereal]: (((((ord_less_eq @ real) @ (extend1716541707_ereal @ X2)) @ (zero_zero @ real)) = ((((ord_less_eq @ extended_ereal) @ X2) @ (zero_zero @ extended_ereal)) | (X2 = (extend1396239628finity @ extended_ereal))))))).
% 2.62/1.49 thf(fact_79_neg__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((uminus_uminus @ A) @ A2) = ((uminus_uminus @ A) @ B2)) = (A2 = B2)))))))).
% 2.62/1.49 thf(fact_80_add_Oinverse__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ A2)) = A2))))))).
% 2.62/1.49 thf(fact_81_verit__minus__simplify_I4_J, axiom, (! [B:$tType]: (((group_add @ B) => (! [B2:B]: ((((uminus_uminus @ B) @ ((uminus_uminus @ B) @ B2)) = B2))))))).
% 2.62/1.49 thf(fact_82_ereal__uminus__zero__iff, axiom, (! [A2:extended_ereal]: (((((uminus_uminus @ extended_ereal) @ A2) = (zero_zero @ extended_ereal)) = (A2 = (zero_zero @ extended_ereal)))))).
% 2.62/1.49 thf(fact_83_ereal__uminus__zero, axiom, (((uminus_uminus @ extended_ereal) @ (zero_zero @ extended_ereal)) = (zero_zero @ extended_ereal))).
% 2.62/1.49 thf(fact_84_ereal__minus__le__minus, axiom, (! [A2:extended_ereal,B2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ A2)) @ ((uminus_uminus @ extended_ereal) @ B2)) = (((ord_less_eq @ extended_ereal) @ B2) @ A2))))).
% 2.62/1.49 thf(fact_85_enat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((extended_enat2 @ Nat) = (extended_enat2 @ Nat2)) = (Nat = Nat2))))).
% 2.62/1.49 thf(fact_86_real__of__ereal, axiom, (! [X2:extended_ereal]: (((extend1716541707_ereal @ ((uminus_uminus @ extended_ereal) @ X2)) = ((uminus_uminus @ real) @ (extend1716541707_ereal @ X2)))))).
% 2.62/1.49 thf(fact_87_neg__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.62/1.49 thf(fact_88_neg__equal__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((uminus_uminus @ A) @ A2) = A2) = (A2 = (zero_zero @ A))))))))).
% 2.62/1.49 thf(fact_89_equal__neg__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((A2 = ((uminus_uminus @ A) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 2.62/1.49 thf(fact_90_neg__equal__0__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((uminus_uminus @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.62/1.49 thf(fact_91_neg__0__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: ((((zero_zero @ A) = ((uminus_uminus @ A) @ A2)) = ((zero_zero @ A) = A2)))))))).
% 2.62/1.49 thf(fact_92_add_Oinverse__neutral, axiom, (! [A:$tType]: (((group_add @ A) => (((uminus_uminus @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.62/1.49 thf(fact_93_neg__numeral__eq__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [M:num,N:num]: (((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = (M = N)))))))).
% 2.62/1.49 thf(fact_94_ereal__infty__less__eq_I2_J, axiom, (! [X2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ X2) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) = (X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))).
% 2.62/1.49 thf(fact_95_ereal__uminus__le__0__iff, axiom, (! [A2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ A2)) @ (zero_zero @ extended_ereal)) = (((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ A2))))).
% 2.62/1.49 thf(fact_96_ereal__0__le__uminus__iff, axiom, (! [A2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ ((uminus_uminus @ extended_ereal) @ A2)) = (((ord_less_eq @ extended_ereal) @ A2) @ (zero_zero @ extended_ereal)))))).
% 2.62/1.49 thf(fact_97_neg__0__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.62/1.49 thf(fact_98_neg__le__0__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.62/1.49 thf(fact_99_less__eq__neg__nonpos, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.62/1.49 thf(fact_100_neg__less__eq__nonneg, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.62/1.49 thf(fact_101_real__of__ereal__0, axiom, ((extend1716541707_ereal @ (zero_zero @ extended_ereal)) = (zero_zero @ real))).
% 2.62/1.49 thf(fact_102_enat__ord__simps_I1_J, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ extended_enat) @ (extended_enat2 @ M)) @ (extended_enat2 @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.62/1.49 thf(fact_103_real__of__ereal_Osimps_I3_J, axiom, ((extend1716541707_ereal @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) = (zero_zero @ real))).
% 2.62/1.49 thf(fact_104_minus__equation__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((uminus_uminus @ A) @ A2) = B2) = (((uminus_uminus @ A) @ B2) = A2)))))))).
% 2.62/1.49 thf(fact_105_equation__minus__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((A2 = ((uminus_uminus @ A) @ B2)) = (B2 = ((uminus_uminus @ A) @ A2))))))))).
% 2.62/1.49 thf(fact_106_verit__negate__coefficient_I3_J, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((A2 = B2) => (((uminus_uminus @ A) @ A2) = ((uminus_uminus @ A) @ B2))))))))).
% 2.62/1.49 thf(fact_107_real__of__ereal__eq__0, axiom, (! [X2:extended_ereal]: ((((extend1716541707_ereal @ X2) = (zero_zero @ real)) = ((X2 = (extend1396239628finity @ extended_ereal)) | ((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) | (X2 = (zero_zero @ extended_ereal)))))))).
% 2.62/1.49 thf(fact_108_real__of__ereal_Osimps_I2_J, axiom, ((extend1716541707_ereal @ (extend1396239628finity @ extended_ereal)) = (zero_zero @ real))).
% 2.62/1.49 thf(fact_109_le__imp__neg__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2))))))))).
% 2.62/1.49 thf(fact_110_minus__le__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ A2)))))))).
% 2.62/1.49 thf(fact_111_le__minus__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ B2)) = (((ord_less_eq @ A) @ B2) @ ((uminus_uminus @ A) @ A2))))))))).
% 2.62/1.49 thf(fact_112_numeral__neq__neg__numeral, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [M:num,N:num]: ((((numeral_numeral @ A) @ M) != ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))))))))).
% 2.62/1.49 thf(fact_113_neg__numeral__neq__numeral, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [M:num,N:num]: ((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M)) != ((numeral_numeral @ A) @ N)))))))).
% 2.62/1.49 thf(fact_114_zero__enat__def, axiom, ((zero_zero @ extended_enat) = (extended_enat2 @ (zero_zero @ nat)))).
% 2.62/1.49 thf(fact_115_enat__0__iff_I1_J, axiom, (! [X2:nat]: ((((extended_enat2 @ X2) = (zero_zero @ extended_enat)) = (X2 = (zero_zero @ nat)))))).
% 2.62/1.49 thf(fact_116_enat__0__iff_I2_J, axiom, (! [X2:nat]: ((((zero_zero @ extended_enat) = (extended_enat2 @ X2)) = (X2 = (zero_zero @ nat)))))).
% 2.62/1.49 thf(fact_117_lsc__at__MInfty, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [F:(A > extended_ereal),X0:A]: ((((F @ X0) = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((((lower_582600101lsc_at @ A) @ extended_ereal) @ X0) @ F)))))))).
% 2.62/1.49 thf(fact_118_MInfty__neq__PInfty_I1_J, axiom, ((extend1396239628finity @ extended_ereal) != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))).
% 2.62/1.49 thf(fact_119_ereal__uminus__le__reorder, axiom, (! [A2:extended_ereal,B2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ A2)) @ B2) = (((ord_less_eq @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ B2)) @ A2))))).
% 2.62/1.49 thf(fact_120_enat__ile, axiom, (! [N:extended_enat,M:nat]: (((((ord_less_eq @ extended_enat) @ N) @ (extended_enat2 @ M)) => (? [K:nat]: ((N = (extended_enat2 @ K)))))))).
% 2.62/1.49 thf(fact_121_numeral__eq__enat, axiom, ((numeral_numeral @ extended_enat) = (^ [K2:num]: ((extended_enat2 @ ((numeral_numeral @ nat) @ K2)))))).
% 2.62/1.49 thf(fact_122_real__of__ereal__pos, axiom, (! [X2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ X2) => (((ord_less_eq @ real) @ (zero_zero @ real)) @ (extend1716541707_ereal @ X2)))))).
% 2.62/1.49 thf(fact_123_neg__numeral__le__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num,N:num]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((numeral_numeral @ A) @ N)))))))).
% 2.62/1.49 thf(fact_124_not__numeral__le__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num,N:num]: ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))))))))))).
% 2.62/1.49 thf(fact_125_zero__neq__neg__numeral, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [N:num]: (((zero_zero @ A) != ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))))))))).
% 2.62/1.49 thf(fact_126_Infty__neq__0_I3_J, axiom, (((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)) != (zero_zero @ extended_ereal))).
% 2.62/1.49 thf(fact_127_ereal__infty__less__eq2_I2_J, axiom, (! [A2:extended_ereal,B2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ A2) @ B2) => ((B2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (A2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))).
% 2.62/1.49 thf(fact_128_ereal__less__eq_I2_J, axiom, (! [X2:extended_ereal]: ((((ord_less_eq @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ X2)))).
% 2.62/1.49 thf(fact_129_real__of__ereal__positive__mono, axiom, (! [X2:extended_ereal,Y3:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ X2) => ((((ord_less_eq @ extended_ereal) @ X2) @ Y3) => ((Y3 != (extend1396239628finity @ extended_ereal)) => (((ord_less_eq @ real) @ (extend1716541707_ereal @ X2)) @ (extend1716541707_ereal @ Y3)))))))).
% 2.62/1.49 thf(fact_130_neg__numeral__le__zero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) @ (zero_zero @ A)))))))).
% 2.62/1.49 thf(fact_131_not__zero__le__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num]: ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))))))))))).
% 2.62/1.49 thf(fact_132_not__MInfty__nonneg, axiom, (! [X2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ X2) => (X2 != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))).
% 2.62/1.49 thf(fact_133_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.62/1.49 thf(fact_134_le__trans, axiom, (! [I:nat,J:nat,K3:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K3) => (((ord_less_eq @ nat) @ I) @ K3)))))).
% 2.62/1.49 thf(fact_135_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.62/1.49 thf(fact_136_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.62/1.49 thf(fact_137_GreatestI__nat, axiom, (! [P:(nat > $o),K3:nat,B2:nat]: (((P @ K3) => ((! [Y:nat]: (((P @ Y) => (((ord_less_eq @ nat) @ Y) @ B2)))) => (P @ ((order_Greatest @ nat) @ P))))))).
% 2.62/1.49 thf(fact_138_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.62/1.49 thf(fact_139_Greatest__le__nat, axiom, (! [P:(nat > $o),K3:nat,B2:nat]: (((P @ K3) => ((! [Y:nat]: (((P @ Y) => (((ord_less_eq @ nat) @ Y) @ B2)))) => (((ord_less_eq @ nat) @ K3) @ ((order_Greatest @ nat) @ P))))))).
% 2.62/1.49 thf(fact_140_GreatestI__ex__nat, axiom, (! [P:(nat > $o),B2:nat]: (((? [X_1:nat]: ((P @ X_1))) => ((! [Y:nat]: (((P @ Y) => (((ord_less_eq @ nat) @ Y) @ B2)))) => (P @ ((order_Greatest @ nat) @ P))))))).
% 2.62/1.49 thf(fact_141_Nat_Oex__has__greatest__nat, axiom, (! [P:(nat > $o),K3:nat,B2:nat]: (((P @ K3) => ((! [Y:nat]: (((P @ Y) => (((ord_less_eq @ nat) @ Y) @ B2)))) => (? [X3:nat]: (((P @ X3) & (! [Y5:nat]: (((P @ Y5) => (((ord_less_eq @ nat) @ Y5) @ X3)))))))))))).
% 2.62/1.49 thf(fact_142_strict__mono__imp__increasing, axiom, (! [F:(nat > nat),N:nat]: (((((order_strict_mono @ nat) @ nat) @ F) => (((ord_less_eq @ nat) @ N) @ (F @ N)))))).
% 2.62/1.49 thf(fact_143_bot__nat__0_Oextremum__uniqueI, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) => (A2 = (zero_zero @ nat)))))).
% 2.62/1.49 thf(fact_144_bot__nat__0_Oextremum__unique, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) = (A2 = (zero_zero @ nat)))))).
% 2.62/1.49 thf(fact_145_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 2.62/1.49 thf(fact_146_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.62/1.49 thf(fact_147_compl__le__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X2:A,Y3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X2)) @ ((uminus_uminus @ A) @ Y3)) = (((ord_less_eq @ A) @ Y3) @ X2)))))))).
% 2.62/1.49 thf(fact_148_the__enat_Osimps, axiom, (! [N:nat]: (((extended_the_enat @ (extended_enat2 @ N)) = N)))).
% 2.62/1.49 thf(fact_149_uminus__apply, axiom, (! [B:$tType,A:$tType]: (((uminus @ B) => ((uminus_uminus @ (A > B)) = (^ [A5:(A > B),X4:A]: (((uminus_uminus @ B) @ (A5 @ X4))))))))).
% 2.62/1.49 thf(fact_150_double__compl, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X2:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ X2)) = X2))))))).
% 2.62/1.49 thf(fact_151_compl__eq__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X2:A,Y3:A]: (((((uminus_uminus @ A) @ X2) = ((uminus_uminus @ A) @ Y3)) = (X2 = Y3)))))))).
% 2.62/1.49 thf(fact_152_enat_Osimps_I4_J, axiom, (! [T:$tType,F1:(nat > T),F22:T,Nat:nat]: ((((((extended_case_enat @ T) @ F1) @ F22) @ (extended_enat2 @ Nat)) = (F1 @ Nat))))).
% 2.62/1.49 thf(fact_153_ereal__uminus__uminus, axiom, (! [A2:extended_ereal]: ((((uminus_uminus @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ A2)) = A2)))).
% 2.62/1.49 thf(fact_154_ereal__uminus__eq__iff, axiom, (! [A2:extended_ereal,B2:extended_ereal]: (((((uminus_uminus @ extended_ereal) @ A2) = ((uminus_uminus @ extended_ereal) @ B2)) = (A2 = B2))))).
% 2.62/1.49 thf(fact_155_ereal__uminus__eq__reorder, axiom, (! [A2:extended_ereal,B2:extended_ereal]: (((((uminus_uminus @ extended_ereal) @ A2) = B2) = (A2 = ((uminus_uminus @ extended_ereal) @ B2)))))).
% 2.62/1.49 thf(fact_156_fun__Compl__def, axiom, (! [B:$tType,A:$tType]: (((uminus @ B) => ((uminus_uminus @ (A > B)) = (^ [A5:(A > B),X4:A]: (((uminus_uminus @ B) @ (A5 @ X4))))))))).
% 2.62/1.49 thf(fact_157_compl__le__swap2, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y3:A,X2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y3)) @ X2) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X2)) @ Y3)))))))).
% 2.62/1.49 thf(fact_158_compl__le__swap1, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y3:A,X2:A]: (((((ord_less_eq @ A) @ Y3) @ ((uminus_uminus @ A) @ X2)) => (((ord_less_eq @ A) @ X2) @ ((uminus_uminus @ A) @ Y3))))))))).
% 2.62/1.49 thf(fact_159_compl__mono, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y3)) @ ((uminus_uminus @ A) @ X2))))))))).
% 2.62/1.49 thf(fact_160_real__eq__0__iff__le__ge__0, axiom, (! [X2:real]: (((X2 = (zero_zero @ real)) = ((((ord_less_eq @ real) @ (zero_zero @ real)) @ X2) & (((ord_less_eq @ real) @ (zero_zero @ real)) @ ((uminus_uminus @ real) @ X2))))))).
% 2.62/1.49 thf(fact_161_enat_Osimps_I6_J, axiom, (! [T:$tType,F1:(nat > T),F22:T,Nat:nat]: ((((((extended_rec_enat @ T) @ F1) @ F22) @ (extended_enat2 @ Nat)) = (F1 @ Nat))))).
% 2.62/1.49 thf(fact_162_dbl__simps_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [K3:num]: ((((neg_numeral_dbl @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ K3))) = ((uminus_uminus @ A) @ ((neg_numeral_dbl @ A) @ ((numeral_numeral @ A) @ K3)))))))))).
% 2.62/1.49 thf(fact_163_dbl__simps_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.62/1.49 thf(fact_164_ereal__less__eq_I5_J, axiom, (! [R:real]: (((((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ (extended_ereal2 @ R)) = (((ord_less_eq @ real) @ (zero_zero @ real)) @ R))))).
% 2.62/1.49 thf(fact_165_ereal__less__eq_I4_J, axiom, (! [R:real]: (((((ord_less_eq @ extended_ereal) @ (extended_ereal2 @ R)) @ (zero_zero @ extended_ereal)) = (((ord_less_eq @ real) @ R) @ (zero_zero @ real)))))).
% 2.62/1.49 thf(fact_166_MInfty__eq__minfinity, axiom, (extended_MInfty = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))).
% 2.62/1.49 thf(fact_167_ereal__cong, axiom, (! [X2:real,Y3:real]: (((X2 = Y3) => ((extended_ereal2 @ X2) = (extended_ereal2 @ Y3)))))).
% 2.62/1.49 thf(fact_168_ereal_Oinject, axiom, (! [X1:real,Y1:real]: ((((extended_ereal2 @ X1) = (extended_ereal2 @ Y1)) = (X1 = Y1))))).
% 2.62/1.49 thf(fact_169_numeral__eq__ereal, axiom, ((numeral_numeral @ extended_ereal) = (^ [W:num]: ((extended_ereal2 @ ((numeral_numeral @ real) @ W)))))).
% 2.62/1.49 thf(fact_170_ereal__eq__0_I1_J, axiom, (! [R:real]: ((((extended_ereal2 @ R) = (zero_zero @ extended_ereal)) = (R = (zero_zero @ real)))))).
% 2.62/1.49 thf(fact_171_ereal__eq__0_I2_J, axiom, (! [R:real]: ((((zero_zero @ extended_ereal) = (extended_ereal2 @ R)) = (R = (zero_zero @ real)))))).
% 2.62/1.49 thf(fact_172_ereal__less__eq_I3_J, axiom, (! [R:real,P2:real]: (((((ord_less_eq @ extended_ereal) @ (extended_ereal2 @ R)) @ (extended_ereal2 @ P2)) = (((ord_less_eq @ real) @ R) @ P2))))).
% 2.62/1.49 thf(fact_173_ereal_Odistinct_I3_J, axiom, (! [X1:real]: (((extended_ereal2 @ X1) != extended_MInfty)))).
% 2.62/1.49 thf(fact_174_real__of__ereal_Osimps_I1_J, axiom, (! [R:real]: (((extend1716541707_ereal @ (extended_ereal2 @ R)) = R)))).
% 2.62/1.49 thf(fact_175_ereal__le__real, axiom, (! [X2:extended_ereal,Y3:extended_ereal]: (((! [Z4:real]: (((((ord_less_eq @ extended_ereal) @ X2) @ (extended_ereal2 @ Z4)) => (((ord_less_eq @ extended_ereal) @ Y3) @ (extended_ereal2 @ Z4))))) => (((ord_less_eq @ extended_ereal) @ Y3) @ X2))))).
% 2.62/1.49 thf(fact_176_PInfty__neq__ereal_I1_J, axiom, (! [R:real]: (((extended_ereal2 @ R) != (extend1396239628finity @ extended_ereal))))).
% 2.62/1.49 thf(fact_177_zero__ereal__def, axiom, ((zero_zero @ extended_ereal) = (extended_ereal2 @ (zero_zero @ real)))).
% 2.62/1.49 thf(fact_178_le__ereal__le, axiom, (! [A2:extended_ereal,X2:real,Y3:real]: (((((ord_less_eq @ extended_ereal) @ A2) @ (extended_ereal2 @ X2)) => ((((ord_less_eq @ real) @ X2) @ Y3) => (((ord_less_eq @ extended_ereal) @ A2) @ (extended_ereal2 @ Y3))))))).
% 2.62/1.49 thf(fact_179_ereal__le__le, axiom, (! [Y3:real,A2:extended_ereal,X2:real]: (((((ord_less_eq @ extended_ereal) @ (extended_ereal2 @ Y3)) @ A2) => ((((ord_less_eq @ real) @ X2) @ Y3) => (((ord_less_eq @ extended_ereal) @ (extended_ereal2 @ X2)) @ A2)))))).
% 2.62/1.49 thf(fact_180_real__of__ereal_Oinduct, axiom, (! [P:(extended_ereal > $o),A0:extended_ereal]: (((! [R2:real]: ((P @ (extended_ereal2 @ R2)))) => ((P @ (extend1396239628finity @ extended_ereal)) => ((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (P @ A0))))))).
% 2.62/1.49 thf(fact_181_real__of__ereal_Ocases, axiom, (! [X2:extended_ereal]: (((! [R2:real]: ((X2 != (extended_ereal2 @ R2)))) => ((X2 != (extend1396239628finity @ extended_ereal)) => (X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))).
% 2.62/1.49 thf(fact_182_times__ereal_Oinduct, axiom, (! [P:(extended_ereal > (extended_ereal > $o)),A0:extended_ereal,A1:extended_ereal]: (((! [R2:real,P3:real]: (((P @ (extended_ereal2 @ R2)) @ (extended_ereal2 @ P3)))) => ((! [R2:real]: (((P @ (extended_ereal2 @ R2)) @ (extend1396239628finity @ extended_ereal)))) => ((! [R2:real]: (((P @ (extend1396239628finity @ extended_ereal)) @ (extended_ereal2 @ R2)))) => ((! [R2:real]: (((P @ (extended_ereal2 @ R2)) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => ((! [R2:real]: (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (extended_ereal2 @ R2)))) => (((P @ (extend1396239628finity @ extended_ereal)) @ (extend1396239628finity @ extended_ereal)) => (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (extend1396239628finity @ extended_ereal)) => (((P @ (extend1396239628finity @ extended_ereal)) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((P @ A0) @ A1))))))))))))).
% 2.62/1.49 thf(fact_183_plus__ereal_Oinduct, axiom, (! [P:(extended_ereal > (extended_ereal > $o)),A0:extended_ereal,A1:extended_ereal]: (((! [R2:real,P3:real]: (((P @ (extended_ereal2 @ R2)) @ (extended_ereal2 @ P3)))) => ((! [X_12:extended_ereal]: (((P @ (extend1396239628finity @ extended_ereal)) @ X_12))) => ((! [A4:extended_ereal]: (((P @ A4) @ (extend1396239628finity @ extended_ereal)))) => ((! [R2:real]: (((P @ (extended_ereal2 @ R2)) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => ((! [P3:real]: (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (extended_ereal2 @ P3)))) => (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((P @ A0) @ A1)))))))))).
% 2.62/1.49 thf(fact_184_less__ereal_Oinduct, axiom, (! [P:(extended_ereal > (extended_ereal > $o)),A0:extended_ereal,A1:extended_ereal]: (((! [X3:real,Y:real]: (((P @ (extended_ereal2 @ X3)) @ (extended_ereal2 @ Y)))) => ((! [X_12:extended_ereal]: (((P @ (extend1396239628finity @ extended_ereal)) @ X_12))) => ((! [A4:extended_ereal]: (((P @ A4) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => ((! [X3:real]: (((P @ (extended_ereal2 @ X3)) @ (extend1396239628finity @ extended_ereal)))) => ((! [R2:real]: (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (extended_ereal2 @ R2)))) => (((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) @ (extend1396239628finity @ extended_ereal)) => ((P @ A0) @ A1)))))))))).
% 2.62/1.49 thf(fact_185_abs__ereal_Oinduct, axiom, (! [P:(extended_ereal > $o),A0:extended_ereal]: (((! [R2:real]: ((P @ (extended_ereal2 @ R2)))) => ((P @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((P @ (extend1396239628finity @ extended_ereal)) => (P @ A0))))))).
% 2.62/1.49 thf(fact_186_ereal__all__split, axiom, ((^ [P4:(extended_ereal > $o)]: ((! [X6:extended_ereal]: ((P4 @ X6))))) = (^ [P5:(extended_ereal > $o)]: (((P5 @ (extend1396239628finity @ extended_ereal)) & ((! [X4:real]: ((P5 @ (extended_ereal2 @ X4)))) & (P5 @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))))).
% 2.62/1.49 thf(fact_187_abs__ereal_Ocases, axiom, (! [X2:extended_ereal]: (((! [R2:real]: ((X2 != (extended_ereal2 @ R2)))) => ((X2 != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (X2 = (extend1396239628finity @ extended_ereal))))))).
% 2.62/1.49 thf(fact_188_ereal__ex__split, axiom, ((^ [P4:(extended_ereal > $o)]: ((? [X6:extended_ereal]: ((P4 @ X6))))) = (^ [P5:(extended_ereal > $o)]: (((P5 @ (extend1396239628finity @ extended_ereal)) | ((? [X4:real]: ((P5 @ (extended_ereal2 @ X4)))) | (P5 @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))))).
% 2.62/1.49 thf(fact_189_ereal3__cases, axiom, (! [X2:extended_ereal,Xa3:extended_ereal,Xb:extended_ereal]: ((((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((? [Ra:real]: ((Xa3 = (extended_ereal2 @ Ra)))) => (! [Rb:real]: ((Xb != (extended_ereal2 @ Rb)))))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((? [Ra:real]: ((Xa3 = (extended_ereal2 @ Ra)))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((? [Ra:real]: ((Xa3 = (extended_ereal2 @ Ra)))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (! [Ra:real]: ((Xb != (extended_ereal2 @ Ra)))))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (! [Ra:real]: ((Xb != (extended_ereal2 @ Ra)))))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((? [R2:real]: ((Xa3 = (extended_ereal2 @ R2)))) => (! [Ra:real]: ((Xb != (extended_ereal2 @ Ra)))))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((? [R2:real]: ((Xa3 = (extended_ereal2 @ R2)))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((? [R2:real]: ((Xa3 = (extended_ereal2 @ R2)))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (! [R2:real]: ((Xb != (extended_ereal2 @ R2)))))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (! [R2:real]: ((Xb != (extended_ereal2 @ R2)))))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((X2 = (extend1396239628finity @ extended_ereal)) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((? [R2:real]: ((Xa3 = (extended_ereal2 @ R2)))) => (! [Ra:real]: ((Xb != (extended_ereal2 @ Ra)))))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((? [R2:real]: ((Xa3 = (extended_ereal2 @ R2)))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((? [R2:real]: ((Xa3 = (extended_ereal2 @ R2)))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (! [R2:real]: ((Xb != (extended_ereal2 @ R2)))))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != (extend1396239628finity @ extended_ereal)))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = (extend1396239628finity @ extended_ereal)) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (! [R2:real]: ((Xb != (extended_ereal2 @ R2)))))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != (extend1396239628finity @ extended_ereal)))) => (~ (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => ((Xa3 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xb != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))))))))))))))))))))))))))))))).
% 2.62/1.49 thf(fact_190_ereal2__cases, axiom, (! [X2:extended_ereal,Xa3:extended_ereal]: ((((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => (! [Ra:real]: ((Xa3 != (extended_ereal2 @ Ra))))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => (Xa3 != (extend1396239628finity @ extended_ereal))) => (((? [R2:real]: ((X2 = (extended_ereal2 @ R2)))) => (Xa3 != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))) => (((X2 = (extend1396239628finity @ extended_ereal)) => (! [R2:real]: ((Xa3 != (extended_ereal2 @ R2))))) => (((X2 = (extend1396239628finity @ extended_ereal)) => (Xa3 != (extend1396239628finity @ extended_ereal))) => (((X2 = (extend1396239628finity @ extended_ereal)) => (Xa3 != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (! [R2:real]: ((Xa3 != (extended_ereal2 @ R2))))) => (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xa3 != (extend1396239628finity @ extended_ereal))) => (~ (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Xa3 != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))))))))))))).
% 2.62/1.49 thf(fact_191_ereal__cases, axiom, (! [X2:extended_ereal]: (((! [R2:real]: ((X2 != (extended_ereal2 @ R2)))) => ((X2 != (extend1396239628finity @ extended_ereal)) => (X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))).
% 2.62/1.49 thf(fact_192_MInfty__neq__ereal_I1_J, axiom, (! [R:real]: (((extended_ereal2 @ R) != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))).
% 2.62/1.49 thf(fact_193_ereal__top, axiom, (! [X2:extended_ereal]: (((! [B5:real]: ((((ord_less_eq @ extended_ereal) @ (extended_ereal2 @ B5)) @ X2))) => (X2 = (extend1396239628finity @ extended_ereal)))))).
% 2.62/1.49 thf(fact_194_uminus__ereal_Osimps_I1_J, axiom, (! [R:real]: ((((uminus_uminus @ extended_ereal) @ (extended_ereal2 @ R)) = (extended_ereal2 @ ((uminus_uminus @ real) @ R)))))).
% 2.62/1.49 thf(fact_195_ereal__bot, axiom, (! [X2:extended_ereal]: (((! [B5:real]: ((((ord_less_eq @ extended_ereal) @ X2) @ (extended_ereal2 @ B5)))) => (X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))))))).
% 2.62/1.49 thf(fact_196_real__of__ereal_Oelims, axiom, (! [X2:extended_ereal,Y3:real]: ((((extend1716541707_ereal @ X2) = Y3) => ((! [R2:real]: (((X2 = (extended_ereal2 @ R2)) => (Y3 != R2)))) => (((X2 = (extend1396239628finity @ extended_ereal)) => (Y3 != (zero_zero @ real))) => (~ (((X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (Y3 != (zero_zero @ real))))))))))).
% 2.62/1.49 thf(fact_197_ereal_Osize__gen_I3_J, axiom, ((extended_size_ereal @ extended_MInfty) = (zero_zero @ nat))).
% 2.62/1.49 thf(fact_198_ereal_Osize__gen_I1_J, axiom, (! [X1:real]: (((extended_size_ereal @ (extended_ereal2 @ X1)) = (zero_zero @ nat))))).
% 2.62/1.49 thf(fact_199_uminus__ereal_Oelims, axiom, (! [X2:extended_ereal,Y3:extended_ereal]: (((((uminus_uminus @ extended_ereal) @ X2) = Y3) => ((! [R2:real]: (((X2 = (extended_ereal2 @ R2)) => (Y3 != (extended_ereal2 @ ((uminus_uminus @ real) @ R2)))))) => (((X2 = extended_PInfty) => (Y3 != extended_MInfty)) => (~ (((X2 = extended_MInfty) => (Y3 != extended_PInfty)))))))))).
% 2.62/1.49 thf(fact_200_ereal_Odistinct_I1_J, axiom, (! [X1:real]: (((extended_ereal2 @ X1) != extended_PInfty)))).
% 2.62/1.49 thf(fact_201_ereal_Osize__gen_I2_J, axiom, ((extended_size_ereal @ extended_PInfty) = (zero_zero @ nat))).
% 2.62/1.49 thf(fact_202_infinity__ereal__def, axiom, ((extend1396239628finity @ extended_ereal) = extended_PInfty)).
% 2.62/1.49 thf(fact_203_ereal_Odistinct_I5_J, axiom, (extended_PInfty != extended_MInfty)).
% 2.62/1.49 thf(fact_204_uminus__ereal_Oinduct, axiom, (! [P:(extended_ereal > $o),A0:extended_ereal]: (((! [R2:real]: ((P @ (extended_ereal2 @ R2)))) => ((P @ extended_PInfty) => ((P @ extended_MInfty) => (P @ A0))))))).
% 2.62/1.49 thf(fact_205_uminus__ereal_Ocases, axiom, (! [X2:extended_ereal]: (((! [R2:real]: ((X2 != (extended_ereal2 @ R2)))) => ((X2 != extended_PInfty) => (X2 = extended_MInfty)))))).
% 2.62/1.49 thf(fact_206_ereal_Oexhaust, axiom, (! [Y3:extended_ereal]: (((! [X12:real]: ((Y3 != (extended_ereal2 @ X12)))) => ((Y3 != extended_PInfty) => (Y3 = extended_MInfty)))))).
% 2.62/1.49 thf(fact_207_ereal_Oinduct, axiom, (! [P:(extended_ereal > $o),Ereal:extended_ereal]: (((! [X3:real]: ((P @ (extended_ereal2 @ X3)))) => ((P @ extended_PInfty) => ((P @ extended_MInfty) => (P @ Ereal))))))).
% 2.62/1.49 thf(fact_208_uminus__ereal_Osimps_I2_J, axiom, (((uminus_uminus @ extended_ereal) @ extended_PInfty) = extended_MInfty)).
% 2.62/1.49 thf(fact_209_uminus__ereal_Osimps_I3_J, axiom, (((uminus_uminus @ extended_ereal) @ extended_MInfty) = extended_PInfty)).
% 2.62/1.49 thf(fact_210_ereal__divide__ereal, axiom, (! [R:real]: ((((((ord_less_eq @ real) @ (zero_zero @ real)) @ R) => ((((divide_divide @ extended_ereal) @ (extend1396239628finity @ extended_ereal)) @ (extended_ereal2 @ R)) = (extend1396239628finity @ extended_ereal))) & ((~ ((((ord_less_eq @ real) @ (zero_zero @ real)) @ R))) => ((((divide_divide @ extended_ereal) @ (extend1396239628finity @ extended_ereal)) @ (extended_ereal2 @ R)) = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))).
% 2.62/1.49 thf(fact_211_ereal__le__real__iff, axiom, (! [X2:real,Y3:extended_ereal]: (((((ord_less_eq @ real) @ X2) @ (extend1716541707_ereal @ Y3)) = (((((abs_abs @ extended_ereal) @ Y3) != (extend1396239628finity @ extended_ereal)) => (((ord_less_eq @ extended_ereal) @ (extended_ereal2 @ X2)) @ Y3)) & ((((abs_abs @ extended_ereal) @ Y3) = (extend1396239628finity @ extended_ereal)) => (((ord_less_eq @ real) @ X2) @ (zero_zero @ real)))))))).
% 2.62/1.49 thf(fact_212_abs__idempotent, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: ((((abs_abs @ A) @ ((abs_abs @ A) @ A2)) = ((abs_abs @ A) @ A2)))))))).
% 2.62/1.49 thf(fact_213_abs__zero, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (((abs_abs @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.62/1.49 thf(fact_214_abs__eq__0, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((abs_abs @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.62/1.49 thf(fact_215_abs__0__eq, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: ((((zero_zero @ A) = ((abs_abs @ A) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 2.62/1.49 thf(fact_216_abs__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num]: ((((abs_abs @ A) @ ((numeral_numeral @ A) @ N)) = ((numeral_numeral @ A) @ N)))))))).
% 2.62/1.49 thf(fact_217_abs__minus__cancel, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: ((((abs_abs @ A) @ ((uminus_uminus @ A) @ A2)) = ((abs_abs @ A) @ A2)))))))).
% 2.62/1.49 thf(fact_218_abs__ereal__uminus, axiom, (! [X2:extended_ereal]: ((((abs_abs @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ X2)) = ((abs_abs @ extended_ereal) @ X2))))).
% 2.62/1.49 thf(fact_219_abs__ereal__zero, axiom, (((abs_abs @ extended_ereal) @ (zero_zero @ extended_ereal)) = (zero_zero @ extended_ereal))).
% 2.62/1.49 thf(fact_220_ereal__uminus__divide, axiom, (! [X2:extended_ereal,Y3:extended_ereal]: (((((divide_divide @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ X2)) @ Y3) = ((uminus_uminus @ extended_ereal) @ (((divide_divide @ extended_ereal) @ X2) @ Y3)))))).
% 2.62/1.49 thf(fact_221_ereal__divide__zero__left, axiom, (! [A2:extended_ereal]: (((((divide_divide @ extended_ereal) @ (zero_zero @ extended_ereal)) @ A2) = (zero_zero @ extended_ereal))))).
% 2.62/1.49 thf(fact_222_abs__of__nonneg, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((abs_abs @ A) @ A2) = A2)))))))).
% 2.62/1.49 thf(fact_223_abs__le__self__iff, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A2)) @ A2) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.62/1.49 thf(fact_224_abs__le__zero__iff, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A2)) @ (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.62/1.49 thf(fact_225_abs__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num]: ((((abs_abs @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = ((numeral_numeral @ A) @ N)))))))).
% 2.62/1.49 thf(fact_226_abs__ereal__ge0, axiom, (! [X2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ X2) => (((abs_abs @ extended_ereal) @ X2) = X2))))).
% 2.62/1.49 thf(fact_227_ereal__divide__Infty_I1_J, axiom, (! [X2:extended_ereal]: (((((divide_divide @ extended_ereal) @ X2) @ (extend1396239628finity @ extended_ereal)) = (zero_zero @ extended_ereal))))).
% 2.62/1.49 thf(fact_228_abs__of__nonpos, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => (((abs_abs @ A) @ A2) = ((uminus_uminus @ A) @ A2))))))))).
% 2.62/1.49 thf(fact_229_ereal__divide__Infty_I2_J, axiom, (! [X2:extended_ereal]: (((((divide_divide @ extended_ereal) @ X2) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) = (zero_zero @ extended_ereal))))).
% 2.62/1.49 thf(fact_230_not__infty__ereal, axiom, (! [X2:extended_ereal]: (((((abs_abs @ extended_ereal) @ X2) != (extend1396239628finity @ extended_ereal)) = (? [X7:real]: ((X2 = (extended_ereal2 @ X7)))))))).
% 2.62/1.49 thf(fact_231_abs__neq__infinity__cases, axiom, (! [X2:extended_ereal]: (((((abs_abs @ extended_ereal) @ X2) != (extend1396239628finity @ extended_ereal)) => (~ ((! [R2:real]: ((X2 != (extended_ereal2 @ R2)))))))))).
% 2.62/1.49 thf(fact_232_abs__ereal_Osimps_I2_J, axiom, (((abs_abs @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) = (extend1396239628finity @ extended_ereal))).
% 2.62/1.49 thf(fact_233_ereal__infinity__cases, axiom, (! [A2:extended_ereal]: (((A2 != (extend1396239628finity @ extended_ereal)) => ((A2 != ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal))) => (((abs_abs @ extended_ereal) @ A2) != (extend1396239628finity @ extended_ereal))))))).
% 2.62/1.49 thf(fact_234_abs__eq__infinity__cases, axiom, (! [X2:extended_ereal]: (((((abs_abs @ extended_ereal) @ X2) = (extend1396239628finity @ extended_ereal)) => ((X2 != (extend1396239628finity @ extended_ereal)) => (X2 = ((uminus_uminus @ extended_ereal) @ (extend1396239628finity @ extended_ereal)))))))).
% 2.62/1.49 thf(fact_235_ereal__abs__leI, axiom, (! [X2:extended_ereal,Y3:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ X2) @ Y3) => ((((ord_less_eq @ extended_ereal) @ ((uminus_uminus @ extended_ereal) @ X2)) @ Y3) => (((ord_less_eq @ extended_ereal) @ ((abs_abs @ extended_ereal) @ X2)) @ Y3)))))).
% 2.62/1.49 thf(fact_236_abs__ereal__pos, axiom, (! [X2:extended_ereal]: ((((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ ((abs_abs @ extended_ereal) @ X2))))).
% 2.62/1.49 thf(fact_237_zero__le__divide__ereal, axiom, (! [A2:extended_ereal,B2:extended_ereal]: (((((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ A2) => ((((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ B2) => (((ord_less_eq @ extended_ereal) @ (zero_zero @ extended_ereal)) @ (((divide_divide @ extended_ereal) @ A2) @ B2))))))).
% 2.62/1.49 thf(fact_238_abs__ge__zero, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((abs_abs @ A) @ A2)))))))).
% 2.62/1.49 thf(fact_239_abs__leI, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ A2)) @ B2))))))))).
% 2.62/1.49 thf(fact_240_abs__le__D2, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A2)) @ B2) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ B2)))))))).
% 2.62/1.49 thf(fact_241_abs__le__iff, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A2)) @ B2) = ((((ord_less_eq @ A) @ A2) @ B2) & (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ B2))))))))).
% 2.62/1.49 thf(fact_242_abs__ge__minus__self, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ ((abs_abs @ A) @ A2)))))))).
% 2.62/1.49 thf(fact_243_ereal__divide__eq__0__iff, axiom, (! [X2:extended_ereal,Y3:extended_ereal]: ((((((divide_divide @ extended_ereal) @ X2) @ Y3) = (zero_zero @ extended_ereal)) = ((X2 = (zero_zero @ extended_ereal)) | (((abs_abs @ extended_ereal) @ Y3) = (extend1396239628finity @ extended_ereal))))))).
% 2.62/1.49 thf(fact_244_abs__ereal_Osimps_I3_J, axiom, (((abs_abs @ extended_ereal) @ (extend1396239628finity @ extended_ereal)) = (extend1396239628finity @ extended_ereal))).
% 2.62/1.49 thf(fact_245_abs__ge__self, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ ((abs_abs @ A) @ A2)))))))).
% 2.62/1.49 thf(fact_246_abs__le__D1, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A2)) @ B2) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.62/1.49 thf(fact_247_abs__minus__le__zero, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((abs_abs @ A) @ A2))) @ (zero_zero @ A)))))))).
% 2.62/1.49 thf(fact_248_ereal__real_H, axiom, (! [X2:extended_ereal]: (((((abs_abs @ extended_ereal) @ X2) != (extend1396239628finity @ extended_ereal)) => ((extended_ereal2 @ (extend1716541707_ereal @ X2)) = X2))))).
% 2.62/1.49 thf(fact_249_ereal__real, axiom, (! [X2:extended_ereal]: ((((((abs_abs @ extended_ereal) @ X2) = (extend1396239628finity @ extended_ereal)) => ((extended_ereal2 @ (extend1716541707_ereal @ X2)) = (zero_zero @ extended_ereal))) & ((((abs_abs @ extended_ereal) @ X2) != (extend1396239628finity @ extended_ereal)) => ((extended_ereal2 @ (extend1716541707_ereal @ X2)) = X2)))))).
% 2.62/1.49 thf(fact_250_real__le__ereal__iff, axiom, (! [Y3:extended_ereal,X2:real]: (((((ord_less_eq @ real) @ (extend1716541707_ereal @ Y3)) @ X2) = (((((abs_abs @ extended_ereal) @ Y3) != (extend1396239628finity @ extended_ereal)) => (((ord_less_eq @ extended_ereal) @ Y3) @ (extended_ereal2 @ X2))) & ((((abs_abs @ extended_ereal) @ Y3) = (extend1396239628finity @ extended_ereal)) => (((ord_less_eq @ real) @ (zero_zero @ real)) @ X2))))))).
% 2.62/1.49 thf(fact_251_divide__le__0__abs__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ A2) @ ((abs_abs @ A) @ B2))) @ (zero_zero @ A)) = ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) | (B2 = (zero_zero @ A)))))))))).
% 2.62/1.49 thf(subcl_Real__Vector__Spaces_Ometric__space___HOL_Otype, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (type @ A))))).
% 2.62/1.49 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Otopological__space, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (topolo503727757_space @ A))))).
% 2.62/1.49 thf(tcon_fun___Topological__Spaces_Otopological__space, axiom, (! [A6:$tType,A7:$tType]: (((topolo503727757_space @ A7) => (topolo503727757_space @ (A6 > A7)))))).
% 2.62/1.49 thf(tcon_fun___Lattices_Oboolean__algebra, axiom, (! [A6:$tType,A7:$tType]: (((boolean_algebra @ A7) => (boolean_algebra @ (A6 > A7)))))).
% 2.62/1.49 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A6:$tType,A7:$tType]: (((preorder @ A7) => (preorder @ (A6 > A7)))))).
% 2.62/1.49 thf(tcon_fun___Orderings_Oorder, axiom, (! [A6:$tType,A7:$tType]: (((order @ A7) => (order @ (A6 > A7)))))).
% 2.62/1.49 thf(tcon_fun___Orderings_Oord, axiom, (! [A6:$tType,A7:$tType]: (((ord @ A7) => (ord @ (A6 > A7)))))).
% 2.62/1.49 thf(tcon_fun___Groups_Ouminus, axiom, (! [A6:$tType,A7:$tType]: (((uminus @ A7) => (uminus @ (A6 > A7)))))).
% 2.62/1.49 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.62/1.49 thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_1, axiom, (topolo503727757_space @ nat)).
% 2.62/1.49 thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology, axiom, (topolo259154727pology @ nat)).
% 2.62/1.49 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ nat)).
% 2.62/1.49 thf(tcon_Nat_Onat___Nat_Osemiring__char__0, axiom, (semiring_char_0 @ nat)).
% 2.62/1.49 thf(tcon_Nat_Onat___Orderings_Opreorder_2, axiom, (preorder @ nat)).
% 2.62/1.49 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 2.62/1.49 thf(tcon_Nat_Onat___Orderings_Oorder_3, axiom, (order @ nat)).
% 2.62/1.49 thf(tcon_Nat_Onat___Orderings_Oord_4, axiom, (ord @ nat)).
% 2.62/1.49 thf(tcon_Nat_Onat___Num_Onumeral, axiom, (numeral @ nat)).
% 2.62/1.49 thf(tcon_Nat_Onat___Groups_Ozero, axiom, (zero @ nat)).
% 2.62/1.49 thf(tcon_Num_Onum___Orderings_Opreorder_5, axiom, (preorder @ num)).
% 2.62/1.49 thf(tcon_Num_Onum___Orderings_Olinorder_6, axiom, (linorder @ num)).
% 2.62/1.49 thf(tcon_Num_Onum___Orderings_Oorder_7, axiom, (order @ num)).
% 2.62/1.49 thf(tcon_Num_Onum___Orderings_Oord_8, axiom, (ord @ num)).
% 2.62/1.49 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_9, axiom, (topolo503727757_space @ $o)).
% 2.62/1.49 thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_10, axiom, (topolo259154727pology @ $o)).
% 2.62/1.49 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_11, axiom, (boolean_algebra @ $o)).
% 2.62/1.49 thf(tcon_HOL_Obool___Orderings_Opreorder_12, axiom, (preorder @ $o)).
% 2.62/1.49 thf(tcon_HOL_Obool___Orderings_Olinorder_13, axiom, (linorder @ $o)).
% 2.62/1.49 thf(tcon_HOL_Obool___Orderings_Oorder_14, axiom, (order @ $o)).
% 2.62/1.49 thf(tcon_HOL_Obool___Orderings_Oord_15, axiom, (ord @ $o)).
% 2.62/1.49 thf(tcon_HOL_Obool___Groups_Ouminus_16, axiom, (uminus @ $o)).
% 2.62/1.49 thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_17, axiom, (topolo503727757_space @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_18, axiom, (topolo259154727pology @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_19, axiom, (linord1659791738miring @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space, axiom, (real_V2090557954_space @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs, axiom, (ordere142940540dd_abs @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add, axiom, (linord219039673up_add @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add, axiom, (ordered_ab_group_add @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Fields_Olinordered__field, axiom, (linordered_field @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Rings_Olinordered__idom, axiom, (linordered_idom @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_20, axiom, (semiring_char_0 @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Orderings_Opreorder_21, axiom, (preorder @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Orderings_Olinorder_22, axiom, (linorder @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Groups_Ogroup__add, axiom, (group_add @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Orderings_Oorder_23, axiom, (order @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Num_Oneg__numeral, axiom, (neg_numeral @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Nat_Oring__char__0, axiom, (ring_char_0 @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Orderings_Oord_24, axiom, (ord @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Groups_Ouminus_25, axiom, (uminus @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Num_Onumeral_26, axiom, (numeral @ real)).
% 2.62/1.49 thf(tcon_Real_Oreal___Groups_Ozero_27, axiom, (zero @ real)).
% 2.62/1.49 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_28, axiom, (canoni770627133id_add @ extended_enat)).
% 2.62/1.49 thf(tcon_Extended__Nat_Oenat___Topological__Spaces_Otopological__space_29, axiom, (topolo503727757_space @ extended_enat)).
% 2.62/1.49 thf(tcon_Extended__Nat_Oenat___Topological__Spaces_Oorder__topology_30, axiom, (topolo259154727pology @ extended_enat)).
% 2.62/1.49 thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_31, axiom, (linord1659791738miring @ extended_enat)).
% 2.62/1.49 thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_32, axiom, (semiring_char_0 @ extended_enat)).
% 2.62/1.49 thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_33, axiom, (preorder @ extended_enat)).
% 2.62/1.49 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_34, axiom, (linorder @ extended_enat)).
% 2.62/1.49 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_35, axiom, (order @ extended_enat)).
% 2.62/1.49 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_36, axiom, (ord @ extended_enat)).
% 2.62/1.49 thf(tcon_Extended__Nat_Oenat___Num_Onumeral_37, axiom, (numeral @ extended_enat)).
% 2.62/1.49 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_38, axiom, (zero @ extended_enat)).
% 2.62/1.49 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Otopological__space_39, axiom, (topolo503727757_space @ extended_ereal)).
% 2.62/1.49 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Oorder__topology_40, axiom, (topolo259154727pology @ extended_ereal)).
% 2.62/1.49 thf(tcon_Extended__Real_Oereal___Orderings_Opreorder_41, axiom, (preorder @ extended_ereal)).
% 2.62/1.49 thf(tcon_Extended__Real_Oereal___Orderings_Olinorder_42, axiom, (linorder @ extended_ereal)).
% 2.62/1.49 thf(tcon_Extended__Real_Oereal___Orderings_Oorder_43, axiom, (order @ extended_ereal)).
% 2.62/1.49 thf(tcon_Extended__Real_Oereal___Orderings_Oord_44, axiom, (ord @ extended_ereal)).
% 2.62/1.49 thf(tcon_Extended__Real_Oereal___Groups_Ouminus_45, axiom, (uminus @ extended_ereal)).
% 2.62/1.49 thf(tcon_Extended__Real_Oereal___Num_Onumeral_46, axiom, (numeral @ extended_ereal)).
% 2.62/1.49 thf(tcon_Extended__Real_Oereal___Groups_Ozero_47, axiom, (zero @ extended_ereal)).
% 2.62/1.49 thf(tfree_0, hypothesis, (real_V2090557954_space @ a)).
% 2.62/1.49 thf(conj_0, conjecture, (((ord_less_eq @ extended_ereal) @ (g @ x)) @ (f @ x))).
% 2.62/1.49 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.pLGO0LFU7P/DTF2DTF22527.p
% 2.62/1.49 ------------------------
% 2.92/1.53 ---- Cleaned THF ---
% 2.92/1.53 thf(ty_t_Extended__Real_Oereal,type,
% 2.92/1.53 extended_ereal: $tType ).
% 2.92/1.53
% 2.92/1.53 thf(ty_t_Extended__Nat_Oenat,type,
% 2.92/1.53 extended_enat: $tType ).
% 2.92/1.53
% 2.92/1.53 thf(ty_t_Real_Oreal,type,
% 2.92/1.53 real: $tType ).
% 2.92/1.53
% 2.92/1.53 thf(ty_t_Set_Oset,type,
% 2.92/1.53 set: $tType > $tType ).
% 2.92/1.53
% 2.92/1.53 thf(ty_t_Num_Onum,type,
% 2.92/1.53 num: $tType ).
% 2.92/1.53
% 2.92/1.53 thf(ty_t_Nat_Onat,type,
% 2.92/1.53 nat: $tType ).
% 2.92/1.53
% 2.92/1.53 thf(ty_tf_a,type,
% 2.92/1.53 a: $tType ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_HOL_Otype,type,
% 2.92/1.53 type:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Groups_Ozero,type,
% 2.92/1.53 zero:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Num_Onumeral,type,
% 2.92/1.53 numeral:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Groups_Ouminus,type,
% 2.92/1.53 uminus:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Orderings_Oord,type,
% 2.92/1.53 ord:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Nat_Oring__char__0,type,
% 2.92/1.53 ring_char_0:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Num_Oneg__numeral,type,
% 2.92/1.53 neg_numeral:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Orderings_Oorder,type,
% 2.92/1.53 order:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Groups_Ogroup__add,type,
% 2.92/1.53 group_add:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Orderings_Olinorder,type,
% 2.92/1.53 linorder:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Orderings_Opreorder,type,
% 2.92/1.53 preorder:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Nat_Osemiring__char__0,type,
% 2.92/1.53 semiring_char_0:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.92/1.53 linordered_idom:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Fields_Olinordered__field,type,
% 2.92/1.53 linordered_field:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 2.92/1.53 boolean_algebra:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.92/1.53 ordered_ab_group_add:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 2.92/1.53 linord219039673up_add:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
% 2.92/1.53 ordere142940540dd_abs:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
% 2.92/1.53 real_V2090557954_space:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.92/1.53 linord1659791738miring:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Topological__Spaces_Oorder__topology,type,
% 2.92/1.53 topolo259154727pology:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Topological__Spaces_Otopological__space,type,
% 2.92/1.53 topolo503727757_space:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.92/1.53 canoni770627133id_add:
% 2.92/1.53 !>[A: $tType] : $o ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Extended__Nat_Oenat,type,
% 2.92/1.53 extended_enat2: nat > extended_enat ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
% 2.92/1.53 extended_case_enat:
% 2.92/1.53 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
% 2.92/1.53 extended_rec_enat:
% 2.92/1.53 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
% 2.92/1.53 extend1396239628finity:
% 2.92/1.53 !>[A: $tType] : A ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Extended__Nat_Othe__enat,type,
% 2.92/1.53 extended_the_enat: extended_enat > nat ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Extended__Real_Oereal_OMInfty,type,
% 2.92/1.53 extended_MInfty: extended_ereal ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Extended__Real_Oereal_OPInfty,type,
% 2.92/1.53 extended_PInfty: extended_ereal ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Extended__Real_Oereal_Oereal,type,
% 2.92/1.53 extended_ereal2: real > extended_ereal ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Extended__Real_Oereal_Osize__ereal,type,
% 2.92/1.53 extended_size_ereal: extended_ereal > nat ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Extended__Real_Oereal__of__enat,type,
% 2.92/1.53 extend1771934483f_enat: extended_enat > extended_ereal ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Extended__Real_Oreal__of__ereal,type,
% 2.92/1.53 extend1716541707_ereal: extended_ereal > real ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Groups_Oabs__class_Oabs,type,
% 2.92/1.53 abs_abs:
% 2.92/1.53 !>[A: $tType] : ( A > A ) ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 2.92/1.53 uminus_uminus:
% 2.92/1.53 !>[A: $tType] : ( A > A ) ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.92/1.53 zero_zero:
% 2.92/1.53 !>[A: $tType] : A ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc,type,
% 2.92/1.53 lower_107104146le_lsc:
% 2.92/1.53 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc__at,type,
% 2.92/1.53 lower_582600101lsc_at:
% 2.92/1.53 !>[A: $tType,B: $tType] : ( A > ( A > B ) > $o ) ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc__hull,type,
% 2.92/1.53 lower_1879122655c_hull:
% 2.92/1.53 !>[A: $tType] : ( ( A > extended_ereal ) > A > extended_ereal ) ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
% 2.92/1.53 neg_numeral_dbl:
% 2.92/1.53 !>[A: $tType] : ( A > A ) ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Num_Oneg__numeral__class_Osub,type,
% 2.92/1.53 neg_numeral_sub:
% 2.92/1.53 !>[A: $tType] : ( num > num > A ) ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Num_Onumeral__class_Onumeral,type,
% 2.92/1.53 numeral_numeral:
% 2.92/1.53 !>[A: $tType] : ( num > A ) ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.92/1.53 ord_less_eq:
% 2.92/1.53 !>[A: $tType] : ( A > A > $o ) ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 2.92/1.53 order_Greatest:
% 2.92/1.53 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 2.92/1.53 order_antimono:
% 2.92/1.53 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 2.92/1.53 order_strict_mono:
% 2.92/1.53 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 2.92/1.53 divide_divide:
% 2.92/1.53 !>[A: $tType] : ( A > A > A ) ).
% 2.92/1.53
% 2.92/1.53 thf(sy_c_member,type,
% 2.92/1.53 member:
% 2.92/1.53 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.92/1.53
% 2.92/1.53 thf(sy_v_f,type,
% 2.92/1.53 f: a > extended_ereal ).
% 2.92/1.53
% 2.92/1.53 thf(sy_v_g,type,
% 2.92/1.53 g: a > extended_ereal ).
% 2.92/1.53
% 2.92/1.53 thf(sy_v_x____,type,
% 2.92/1.53 x: a ).
% 2.92/1.53
% 2.92/1.53 thf(fact_0_assms,axiom,
% 2.92/1.53 ! [X: a] : ( ord_less_eq @ extended_ereal @ ( g @ X ) @ ( f @ X ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_1_calculation,axiom,
% 2.92/1.53 ord_less_eq @ extended_ereal @ ( lower_1879122655c_hull @ a @ g @ x ) @ ( g @ x ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_2_order__refl,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( preorder @ A )
% 2.92/1.53 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_3_ereal__complete__Inf,axiom,
% 2.92/1.53 ! [S: set @ extended_ereal] :
% 2.92/1.53 ? [X3: extended_ereal] :
% 2.92/1.53 ( ! [Xa: extended_ereal] :
% 2.92/1.53 ( ( member @ extended_ereal @ Xa @ S )
% 2.92/1.53 => ( ord_less_eq @ extended_ereal @ X3 @ Xa ) )
% 2.92/1.53 & ! [Z: extended_ereal] :
% 2.92/1.53 ( ! [Xa2: extended_ereal] :
% 2.92/1.53 ( ( member @ extended_ereal @ Xa2 @ S )
% 2.92/1.53 => ( ord_less_eq @ extended_ereal @ Z @ Xa2 ) )
% 2.92/1.53 => ( ord_less_eq @ extended_ereal @ Z @ X3 ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_4_ereal__complete__Sup,axiom,
% 2.92/1.53 ! [S: set @ extended_ereal] :
% 2.92/1.53 ? [X3: extended_ereal] :
% 2.92/1.53 ( ! [Xa: extended_ereal] :
% 2.92/1.53 ( ( member @ extended_ereal @ Xa @ S )
% 2.92/1.53 => ( ord_less_eq @ extended_ereal @ Xa @ X3 ) )
% 2.92/1.53 & ! [Z: extended_ereal] :
% 2.92/1.53 ( ! [Xa2: extended_ereal] :
% 2.92/1.53 ( ( member @ extended_ereal @ Xa2 @ S )
% 2.92/1.53 => ( ord_less_eq @ extended_ereal @ Xa2 @ Z ) )
% 2.92/1.53 => ( ord_less_eq @ extended_ereal @ X3 @ Z ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_5_le__funD,axiom,
% 2.92/1.53 ! [B: $tType,A: $tType] :
% 2.92/1.53 ( ( ord @ B )
% 2.92/1.53 => ! [F: A > B,G: A > B,X2: A] :
% 2.92/1.53 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.92/1.53 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_6_le__funE,axiom,
% 2.92/1.53 ! [B: $tType,A: $tType] :
% 2.92/1.53 ( ( ord @ B )
% 2.92/1.53 => ! [F: A > B,G: A > B,X2: A] :
% 2.92/1.53 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.92/1.53 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_7_le__funI,axiom,
% 2.92/1.53 ! [B: $tType,A: $tType] :
% 2.92/1.53 ( ( ord @ B )
% 2.92/1.53 => ! [F: A > B,G: A > B] :
% 2.92/1.53 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 2.92/1.53 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_8_le__fun__def,axiom,
% 2.92/1.53 ! [B: $tType,A: $tType] :
% 2.92/1.53 ( ( ord @ B )
% 2.92/1.53 => ( ( ord_less_eq @ ( A > B ) )
% 2.92/1.53 = ( ^ [F2: A > B,G2: A > B] :
% 2.92/1.53 ! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_9_order__subst1,axiom,
% 2.92/1.53 ! [A: $tType,B: $tType] :
% 2.92/1.53 ( ( ( order @ B )
% 2.92/1.53 & ( order @ A ) )
% 2.92/1.53 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.92/1.53 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 2.92/1.53 => ( ( ord_less_eq @ B @ B2 @ C )
% 2.92/1.53 => ( ! [X3: B,Y: B] :
% 2.92/1.53 ( ( ord_less_eq @ B @ X3 @ Y )
% 2.92/1.53 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.92/1.53 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_10_order__subst2,axiom,
% 2.92/1.53 ! [A: $tType,C2: $tType] :
% 2.92/1.53 ( ( ( order @ C2 )
% 2.92/1.53 & ( order @ A ) )
% 2.92/1.53 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 2.92/1.53 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.92/1.53 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 2.92/1.53 => ( ! [X3: A,Y: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ X3 @ Y )
% 2.92/1.53 => ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.92/1.53 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_11_verit__la__disequality,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( linorder @ A )
% 2.92/1.53 => ! [A2: A,B2: A] :
% 2.92/1.53 ( ( A2 = B2 )
% 2.92/1.53 | ~ ( ord_less_eq @ A @ A2 @ B2 )
% 2.92/1.53 | ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_12_ord__eq__le__subst,axiom,
% 2.92/1.53 ! [A: $tType,B: $tType] :
% 2.92/1.53 ( ( ( ord @ B )
% 2.92/1.53 & ( ord @ A ) )
% 2.92/1.53 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.92/1.53 ( ( A2
% 2.92/1.53 = ( F @ B2 ) )
% 2.92/1.53 => ( ( ord_less_eq @ B @ B2 @ C )
% 2.92/1.53 => ( ! [X3: B,Y: B] :
% 2.92/1.53 ( ( ord_less_eq @ B @ X3 @ Y )
% 2.92/1.53 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.92/1.53 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_13_ord__le__eq__subst,axiom,
% 2.92/1.53 ! [A: $tType,B: $tType] :
% 2.92/1.53 ( ( ( ord @ B )
% 2.92/1.53 & ( ord @ A ) )
% 2.92/1.53 => ! [A2: A,B2: A,F: A > B,C: B] :
% 2.92/1.53 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.92/1.53 => ( ( ( F @ B2 )
% 2.92/1.53 = C )
% 2.92/1.53 => ( ! [X3: A,Y: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ X3 @ Y )
% 2.92/1.53 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.92/1.53 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_14_lsc__hull__le,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( real_V2090557954_space @ A )
% 2.92/1.53 => ! [F: A > extended_ereal,X2: A] : ( ord_less_eq @ extended_ereal @ ( lower_1879122655c_hull @ A @ F @ X2 ) @ ( F @ X2 ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_15_dual__order_Oantisym,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( order @ A )
% 2.92/1.53 => ! [B2: A,A2: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.92/1.53 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.92/1.53 => ( A2 = B2 ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_16_dual__order_Oeq__iff,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( order @ A )
% 2.92/1.53 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 2.92/1.53 = ( ^ [A3: A,B3: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.92/1.53 & ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_17_dual__order_Otrans,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( order @ A )
% 2.92/1.53 => ! [B2: A,A2: A,C: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.92/1.53 => ( ( ord_less_eq @ A @ C @ B2 )
% 2.92/1.53 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_18_linorder__wlog,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( linorder @ A )
% 2.92/1.53 => ! [P: A > A > $o,A2: A,B2: A] :
% 2.92/1.53 ( ! [A4: A,B4: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.92/1.53 => ( P @ A4 @ B4 ) )
% 2.92/1.53 => ( ! [A4: A,B4: A] :
% 2.92/1.53 ( ( P @ B4 @ A4 )
% 2.92/1.53 => ( P @ A4 @ B4 ) )
% 2.92/1.53 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_19_dual__order_Orefl,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( order @ A )
% 2.92/1.53 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_20_order__trans,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( preorder @ A )
% 2.92/1.53 => ! [X2: A,Y3: A,Z3: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.92/1.53 => ( ( ord_less_eq @ A @ Y3 @ Z3 )
% 2.92/1.53 => ( ord_less_eq @ A @ X2 @ Z3 ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_21_order__class_Oorder_Oantisym,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( order @ A )
% 2.92/1.53 => ! [A2: A,B2: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.92/1.53 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.92/1.53 => ( A2 = B2 ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_22_ord__le__eq__trans,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( ord @ A )
% 2.92/1.53 => ! [A2: A,B2: A,C: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.92/1.53 => ( ( B2 = C )
% 2.92/1.53 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_23_ord__eq__le__trans,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( ord @ A )
% 2.92/1.53 => ! [A2: A,B2: A,C: A] :
% 2.92/1.53 ( ( A2 = B2 )
% 2.92/1.53 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.92/1.53 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_24_order__class_Oorder_Oeq__iff,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( order @ A )
% 2.92/1.53 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 2.92/1.53 = ( ^ [A3: A,B3: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.92/1.53 & ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_25_antisym__conv,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( order @ A )
% 2.92/1.53 => ! [Y3: A,X2: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ Y3 @ X2 )
% 2.92/1.53 => ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.92/1.53 = ( X2 = Y3 ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_26_le__cases3,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( linorder @ A )
% 2.92/1.53 => ! [X2: A,Y3: A,Z3: A] :
% 2.92/1.53 ( ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.92/1.53 => ~ ( ord_less_eq @ A @ Y3 @ Z3 ) )
% 2.92/1.53 => ( ( ( ord_less_eq @ A @ Y3 @ X2 )
% 2.92/1.53 => ~ ( ord_less_eq @ A @ X2 @ Z3 ) )
% 2.92/1.53 => ( ( ( ord_less_eq @ A @ X2 @ Z3 )
% 2.92/1.53 => ~ ( ord_less_eq @ A @ Z3 @ Y3 ) )
% 2.92/1.53 => ( ( ( ord_less_eq @ A @ Z3 @ Y3 )
% 2.92/1.53 => ~ ( ord_less_eq @ A @ Y3 @ X2 ) )
% 2.92/1.53 => ( ( ( ord_less_eq @ A @ Y3 @ Z3 )
% 2.92/1.53 => ~ ( ord_less_eq @ A @ Z3 @ X2 ) )
% 2.92/1.53 => ~ ( ( ord_less_eq @ A @ Z3 @ X2 )
% 2.92/1.53 => ~ ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_27_order_Otrans,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( order @ A )
% 2.92/1.53 => ! [A2: A,B2: A,C: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.92/1.53 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.92/1.53 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_28_le__cases,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( linorder @ A )
% 2.92/1.53 => ! [X2: A,Y3: A] :
% 2.92/1.53 ( ~ ( ord_less_eq @ A @ X2 @ Y3 )
% 2.92/1.53 => ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_29_eq__refl,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( preorder @ A )
% 2.92/1.53 => ! [X2: A,Y3: A] :
% 2.92/1.53 ( ( X2 = Y3 )
% 2.92/1.53 => ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_30_linear,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( linorder @ A )
% 2.92/1.53 => ! [X2: A,Y3: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.92/1.53 | ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_31_antisym,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( order @ A )
% 2.92/1.53 => ! [X2: A,Y3: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.92/1.53 => ( ( ord_less_eq @ A @ Y3 @ X2 )
% 2.92/1.53 => ( X2 = Y3 ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_32_eq__iff,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( order @ A )
% 2.92/1.53 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 2.92/1.53 = ( ^ [X4: A,Y4: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 2.92/1.53 & ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_33_lsc__hull__iff__greatest,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( real_V2090557954_space @ A )
% 2.92/1.53 => ! [G: A > extended_ereal,F: A > extended_ereal] :
% 2.92/1.53 ( ( G
% 2.92/1.53 = ( lower_1879122655c_hull @ A @ F ) )
% 2.92/1.53 = ( ( lower_107104146le_lsc @ A @ extended_ereal @ G )
% 2.92/1.53 & ! [X4: A] : ( ord_less_eq @ extended_ereal @ ( G @ X4 ) @ ( F @ X4 ) )
% 2.92/1.53 & ! [H: A > extended_ereal] :
% 2.92/1.53 ( ( ( lower_107104146le_lsc @ A @ extended_ereal @ H )
% 2.92/1.53 & ! [X4: A] : ( ord_less_eq @ extended_ereal @ ( H @ X4 ) @ ( F @ X4 ) ) )
% 2.92/1.53 => ! [X4: A] : ( ord_less_eq @ extended_ereal @ ( H @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_34_lsc__hull__greatest,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( real_V2090557954_space @ A )
% 2.92/1.53 => ! [G: A > extended_ereal,F: A > extended_ereal] :
% 2.92/1.53 ( ( lower_107104146le_lsc @ A @ extended_ereal @ G )
% 2.92/1.53 => ( ! [X3: A] : ( ord_less_eq @ extended_ereal @ ( G @ X3 ) @ ( F @ X3 ) )
% 2.92/1.53 => ! [X: A] : ( ord_less_eq @ extended_ereal @ ( G @ X ) @ ( lower_1879122655c_hull @ A @ F @ X ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_35_Greatest__equality,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( order @ A )
% 2.92/1.53 => ! [P: A > $o,X2: A] :
% 2.92/1.53 ( ( P @ X2 )
% 2.92/1.53 => ( ! [Y: A] :
% 2.92/1.53 ( ( P @ Y )
% 2.92/1.53 => ( ord_less_eq @ A @ Y @ X2 ) )
% 2.92/1.53 => ( ( order_Greatest @ A @ P )
% 2.92/1.53 = X2 ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_36_GreatestI2__order,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( order @ A )
% 2.92/1.53 => ! [P: A > $o,X2: A,Q: A > $o] :
% 2.92/1.53 ( ( P @ X2 )
% 2.92/1.53 => ( ! [Y: A] :
% 2.92/1.53 ( ( P @ Y )
% 2.92/1.53 => ( ord_less_eq @ A @ Y @ X2 ) )
% 2.92/1.53 => ( ! [X3: A] :
% 2.92/1.53 ( ( P @ X3 )
% 2.92/1.53 => ( ! [Y5: A] :
% 2.92/1.53 ( ( P @ Y5 )
% 2.92/1.53 => ( ord_less_eq @ A @ Y5 @ X3 ) )
% 2.92/1.53 => ( Q @ X3 ) ) )
% 2.92/1.53 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_37_le__rel__bool__arg__iff,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( ord @ A )
% 2.92/1.53 => ( ( ord_less_eq @ ( $o > A ) )
% 2.92/1.53 = ( ^ [X5: $o > A,Y6: $o > A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ ( X5 @ $false ) @ ( Y6 @ $false ) )
% 2.92/1.53 & ( ord_less_eq @ A @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_38_lsc__lsc__hull,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( real_V2090557954_space @ A )
% 2.92/1.53 => ! [F: A > extended_ereal] : ( lower_107104146le_lsc @ A @ extended_ereal @ ( lower_1879122655c_hull @ A @ F ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_39_ereal__of__enat__le__iff,axiom,
% 2.92/1.53 ! [M: extended_enat,N: extended_enat] :
% 2.92/1.53 ( ( ord_less_eq @ extended_ereal @ ( extend1771934483f_enat @ M ) @ ( extend1771934483f_enat @ N ) )
% 2.92/1.53 = ( ord_less_eq @ extended_enat @ M @ N ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_40_antimono__def,axiom,
% 2.92/1.53 ! [B: $tType,A: $tType] :
% 2.92/1.53 ( ( ( order @ A )
% 2.92/1.53 & ( order @ B ) )
% 2.92/1.53 => ( ( order_antimono @ A @ B )
% 2.92/1.53 = ( ^ [F2: A > B] :
% 2.92/1.53 ! [X4: A,Y4: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 2.92/1.53 => ( ord_less_eq @ B @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_41_antimonoI,axiom,
% 2.92/1.53 ! [B: $tType,A: $tType] :
% 2.92/1.53 ( ( ( order @ A )
% 2.92/1.53 & ( order @ B ) )
% 2.92/1.53 => ! [F: A > B] :
% 2.92/1.53 ( ! [X3: A,Y: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ X3 @ Y )
% 2.92/1.53 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X3 ) ) )
% 2.92/1.53 => ( order_antimono @ A @ B @ F ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_42_antimonoE,axiom,
% 2.92/1.53 ! [B: $tType,A: $tType] :
% 2.92/1.53 ( ( ( order @ A )
% 2.92/1.53 & ( order @ B ) )
% 2.92/1.53 => ! [F: A > B,X2: A,Y3: A] :
% 2.92/1.53 ( ( order_antimono @ A @ B @ F )
% 2.92/1.53 => ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.92/1.53 => ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_43_antimonoD,axiom,
% 2.92/1.53 ! [B: $tType,A: $tType] :
% 2.92/1.53 ( ( ( order @ A )
% 2.92/1.53 & ( order @ B ) )
% 2.92/1.53 => ! [F: A > B,X2: A,Y3: A] :
% 2.92/1.53 ( ( order_antimono @ A @ B @ F )
% 2.92/1.53 => ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.92/1.53 => ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_44_lsc__def,axiom,
% 2.92/1.53 ! [B: $tType,A: $tType] :
% 2.92/1.53 ( ( ( topolo503727757_space @ A )
% 2.92/1.53 & ( topolo259154727pology @ B ) )
% 2.92/1.53 => ( ( lower_107104146le_lsc @ A @ B )
% 2.92/1.53 = ( ^ [F2: A > B] :
% 2.92/1.53 ! [X4: A] : ( lower_582600101lsc_at @ A @ B @ X4 @ F2 ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_45_ext,axiom,
% 2.92/1.53 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.92/1.53 ( ! [X3: A] :
% 2.92/1.53 ( ( F @ X3 )
% 2.92/1.53 = ( G @ X3 ) )
% 2.92/1.53 => ( F = G ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_46_decseqD,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( order @ A )
% 2.92/1.53 => ! [F: nat > A,I: nat,J: nat] :
% 2.92/1.53 ( ( order_antimono @ nat @ A @ F )
% 2.92/1.53 => ( ( ord_less_eq @ nat @ I @ J )
% 2.92/1.53 => ( ord_less_eq @ A @ ( F @ J ) @ ( F @ I ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_47_decseq__def,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( order @ A )
% 2.92/1.53 => ( ( order_antimono @ nat @ A )
% 2.92/1.53 = ( ^ [X5: nat > A] :
% 2.92/1.53 ! [M2: nat,N2: nat] :
% 2.92/1.53 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.92/1.53 => ( ord_less_eq @ A @ ( X5 @ N2 ) @ ( X5 @ M2 ) ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_48_ereal__of__enat__nonneg,axiom,
% 2.92/1.53 ! [N: extended_enat] : ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( extend1771934483f_enat @ N ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_49_ereal__of__enat__ge__zero__cancel__iff,axiom,
% 2.92/1.53 ! [N: extended_enat] :
% 2.92/1.53 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( extend1771934483f_enat @ N ) )
% 2.92/1.53 = ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_50_numeral__le__ereal__of__enat__iff,axiom,
% 2.92/1.53 ! [M: num,N: extended_enat] :
% 2.92/1.53 ( ( ord_less_eq @ extended_ereal @ ( numeral_numeral @ extended_ereal @ M ) @ ( extend1771934483f_enat @ N ) )
% 2.92/1.53 = ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ N ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_51_ereal__infty__less__eq_I1_J,axiom,
% 2.92/1.53 ! [X2: extended_ereal] :
% 2.92/1.53 ( ( ord_less_eq @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ X2 )
% 2.92/1.53 = ( X2
% 2.92/1.53 = ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_52_strict__mono__less__eq,axiom,
% 2.92/1.53 ! [B: $tType,A: $tType] :
% 2.92/1.53 ( ( ( linorder @ A )
% 2.92/1.53 & ( order @ B ) )
% 2.92/1.53 => ! [F: A > B,X2: A,Y3: A] :
% 2.92/1.53 ( ( order_strict_mono @ A @ B @ F )
% 2.92/1.53 => ( ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y3 ) )
% 2.92/1.53 = ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_53_ereal__of__enat__zero,axiom,
% 2.92/1.53 ( ( extend1771934483f_enat @ ( zero_zero @ extended_enat ) )
% 2.92/1.53 = ( zero_zero @ extended_ereal ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_54_strict__mono__eq,axiom,
% 2.92/1.53 ! [B: $tType,A: $tType] :
% 2.92/1.53 ( ( ( linorder @ A )
% 2.92/1.53 & ( order @ B ) )
% 2.92/1.53 => ! [F: A > B,X2: A,Y3: A] :
% 2.92/1.53 ( ( order_strict_mono @ A @ B @ F )
% 2.92/1.53 => ( ( ( F @ X2 )
% 2.92/1.53 = ( F @ Y3 ) )
% 2.92/1.53 = ( X2 = Y3 ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_55_Infty__neq__0_I1_J,axiom,
% 2.92/1.53 ( ( extend1396239628finity @ extended_ereal )
% 2.92/1.53 != ( zero_zero @ extended_ereal ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_56_strict__mono__leD,axiom,
% 2.92/1.53 ! [B: $tType,A: $tType] :
% 2.92/1.53 ( ( ( order @ A )
% 2.92/1.53 & ( order @ B ) )
% 2.92/1.53 => ! [R: A > B,M: A,N: A] :
% 2.92/1.53 ( ( order_strict_mono @ A @ B @ R )
% 2.92/1.53 => ( ( ord_less_eq @ A @ M @ N )
% 2.92/1.53 => ( ord_less_eq @ B @ ( R @ M ) @ ( R @ N ) ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_57_neq__PInf__trans,axiom,
% 2.92/1.53 ! [Y3: extended_ereal,X2: extended_ereal] :
% 2.92/1.53 ( ( Y3
% 2.92/1.53 != ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.53 => ( ( ord_less_eq @ extended_ereal @ X2 @ Y3 )
% 2.92/1.53 => ( X2
% 2.92/1.53 != ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_58_ereal__infty__less__eq2_I1_J,axiom,
% 2.92/1.53 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.92/1.53 ( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
% 2.92/1.53 => ( ( A2
% 2.92/1.53 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.53 => ( B2
% 2.92/1.53 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_59_ereal__less__eq_I1_J,axiom,
% 2.92/1.53 ! [X2: extended_ereal] : ( ord_less_eq @ extended_ereal @ X2 @ ( extend1396239628finity @ extended_ereal ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_60_numeral__le__iff,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( linord1659791738miring @ A )
% 2.92/1.53 => ! [M: num,N: num] :
% 2.92/1.53 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 2.92/1.53 = ( ord_less_eq @ num @ M @ N ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_61_le__zero__eq,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( canoni770627133id_add @ A )
% 2.92/1.53 => ! [N: A] :
% 2.92/1.53 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.92/1.53 = ( N
% 2.92/1.53 = ( zero_zero @ A ) ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_62_zero__le__numeral,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( linord1659791738miring @ A )
% 2.92/1.53 => ! [N: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_63_not__numeral__le__zero,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( linord1659791738miring @ A )
% 2.92/1.53 => ! [N: num] :
% 2.92/1.53 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% 2.92/1.53
% 2.92/1.53 thf(fact_64_numeral__eq__iff,axiom,
% 2.92/1.53 ! [A: $tType] :
% 2.92/1.53 ( ( semiring_char_0 @ A )
% 2.92/1.54 => ! [M: num,N: num] :
% 2.92/1.54 ( ( ( numeral_numeral @ A @ M )
% 2.92/1.54 = ( numeral_numeral @ A @ N ) )
% 2.92/1.54 = ( M = N ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_65_ile0__eq,axiom,
% 2.92/1.54 ! [N: extended_enat] :
% 2.92/1.54 ( ( ord_less_eq @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
% 2.92/1.54 = ( N
% 2.92/1.54 = ( zero_zero @ extended_enat ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_66_i0__lb,axiom,
% 2.92/1.54 ! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_67_enat__ord__number_I1_J,axiom,
% 2.92/1.54 ! [M: num,N: num] :
% 2.92/1.54 ( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N ) )
% 2.92/1.54 = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_68_zero__reorient,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( zero @ A )
% 2.92/1.54 => ! [X2: A] :
% 2.92/1.54 ( ( ( zero_zero @ A )
% 2.92/1.54 = X2 )
% 2.92/1.54 = ( X2
% 2.92/1.54 = ( zero_zero @ A ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_69_zero__le,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( canoni770627133id_add @ A )
% 2.92/1.54 => ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_70_le__numeral__extra_I3_J,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( linord1659791738miring @ A )
% 2.92/1.54 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_71_zero__neq__numeral,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( semiring_char_0 @ A )
% 2.92/1.54 => ! [N: num] :
% 2.92/1.54 ( ( zero_zero @ A )
% 2.92/1.54 != ( numeral_numeral @ A @ N ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_72_bot__nat__0_Oextremum,axiom,
% 2.92/1.54 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_73_le0,axiom,
% 2.92/1.54 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_74_neg__numeral__le__iff,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( linordered_idom @ A )
% 2.92/1.54 => ! [M: num,N: num] :
% 2.92/1.54 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 2.92/1.54 = ( ord_less_eq @ num @ N @ M ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_75_numeral__le__enat__iff,axiom,
% 2.92/1.54 ! [M: num,N: nat] :
% 2.92/1.54 ( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( extended_enat2 @ N ) )
% 2.92/1.54 = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ N ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_76_sub__non__positive,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( linordered_idom @ A )
% 2.92/1.54 => ! [N: num,M: num] :
% 2.92/1.54 ( ( ord_less_eq @ A @ ( neg_numeral_sub @ A @ N @ M ) @ ( zero_zero @ A ) )
% 2.92/1.54 = ( ord_less_eq @ num @ N @ M ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_77_sub__non__negative,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( linordered_idom @ A )
% 2.92/1.54 => ! [N: num,M: num] :
% 2.92/1.54 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M ) )
% 2.92/1.54 = ( ord_less_eq @ num @ M @ N ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_78_real__of__ereal__le__0,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ( ord_less_eq @ real @ ( extend1716541707_ereal @ X2 ) @ ( zero_zero @ real ) )
% 2.92/1.54 = ( ( ord_less_eq @ extended_ereal @ X2 @ ( zero_zero @ extended_ereal ) )
% 2.92/1.54 | ( X2
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_79_neg__equal__iff__equal,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( group_add @ A )
% 2.92/1.54 => ! [A2: A,B2: A] :
% 2.92/1.54 ( ( ( uminus_uminus @ A @ A2 )
% 2.92/1.54 = ( uminus_uminus @ A @ B2 ) )
% 2.92/1.54 = ( A2 = B2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_80_add_Oinverse__inverse,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( group_add @ A )
% 2.92/1.54 => ! [A2: A] :
% 2.92/1.54 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 2.92/1.54 = A2 ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_81_verit__minus__simplify_I4_J,axiom,
% 2.92/1.54 ! [B: $tType] :
% 2.92/1.54 ( ( group_add @ B )
% 2.92/1.54 => ! [B2: B] :
% 2.92/1.54 ( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
% 2.92/1.54 = B2 ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_82_ereal__uminus__zero__iff,axiom,
% 2.92/1.54 ! [A2: extended_ereal] :
% 2.92/1.54 ( ( ( uminus_uminus @ extended_ereal @ A2 )
% 2.92/1.54 = ( zero_zero @ extended_ereal ) )
% 2.92/1.54 = ( A2
% 2.92/1.54 = ( zero_zero @ extended_ereal ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_83_ereal__uminus__zero,axiom,
% 2.92/1.54 ( ( uminus_uminus @ extended_ereal @ ( zero_zero @ extended_ereal ) )
% 2.92/1.54 = ( zero_zero @ extended_ereal ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_84_ereal__minus__le__minus,axiom,
% 2.92/1.54 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) @ ( uminus_uminus @ extended_ereal @ B2 ) )
% 2.92/1.54 = ( ord_less_eq @ extended_ereal @ B2 @ A2 ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_85_enat_Oinject,axiom,
% 2.92/1.54 ! [Nat: nat,Nat2: nat] :
% 2.92/1.54 ( ( ( extended_enat2 @ Nat )
% 2.92/1.54 = ( extended_enat2 @ Nat2 ) )
% 2.92/1.54 = ( Nat = Nat2 ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_86_real__of__ereal,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ( extend1716541707_ereal @ ( uminus_uminus @ extended_ereal @ X2 ) )
% 2.92/1.54 = ( uminus_uminus @ real @ ( extend1716541707_ereal @ X2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_87_neg__le__iff__le,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordered_ab_group_add @ A )
% 2.92/1.54 => ! [B2: A,A2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 2.92/1.54 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_88_neg__equal__zero,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( linord219039673up_add @ A )
% 2.92/1.54 => ! [A2: A] :
% 2.92/1.54 ( ( ( uminus_uminus @ A @ A2 )
% 2.92/1.54 = A2 )
% 2.92/1.54 = ( A2
% 2.92/1.54 = ( zero_zero @ A ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_89_equal__neg__zero,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( linord219039673up_add @ A )
% 2.92/1.54 => ! [A2: A] :
% 2.92/1.54 ( ( A2
% 2.92/1.54 = ( uminus_uminus @ A @ A2 ) )
% 2.92/1.54 = ( A2
% 2.92/1.54 = ( zero_zero @ A ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_90_neg__equal__0__iff__equal,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( group_add @ A )
% 2.92/1.54 => ! [A2: A] :
% 2.92/1.54 ( ( ( uminus_uminus @ A @ A2 )
% 2.92/1.54 = ( zero_zero @ A ) )
% 2.92/1.54 = ( A2
% 2.92/1.54 = ( zero_zero @ A ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_91_neg__0__equal__iff__equal,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( group_add @ A )
% 2.92/1.54 => ! [A2: A] :
% 2.92/1.54 ( ( ( zero_zero @ A )
% 2.92/1.54 = ( uminus_uminus @ A @ A2 ) )
% 2.92/1.54 = ( ( zero_zero @ A )
% 2.92/1.54 = A2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_92_add_Oinverse__neutral,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( group_add @ A )
% 2.92/1.54 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 2.92/1.54 = ( zero_zero @ A ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_93_neg__numeral__eq__iff,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ring_char_0 @ A )
% 2.92/1.54 => ! [M: num,N: num] :
% 2.92/1.54 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
% 2.92/1.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 2.92/1.54 = ( M = N ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_94_ereal__infty__less__eq_I2_J,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ X2 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 = ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_95_ereal__uminus__le__0__iff,axiom,
% 2.92/1.54 ! [A2: extended_ereal] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) @ ( zero_zero @ extended_ereal ) )
% 2.92/1.54 = ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A2 ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_96_ereal__0__le__uminus__iff,axiom,
% 2.92/1.54 ! [A2: extended_ereal] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( uminus_uminus @ extended_ereal @ A2 ) )
% 2.92/1.54 = ( ord_less_eq @ extended_ereal @ A2 @ ( zero_zero @ extended_ereal ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_97_neg__0__le__iff__le,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordered_ab_group_add @ A )
% 2.92/1.54 => ! [A2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 2.92/1.54 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_98_neg__le__0__iff__le,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordered_ab_group_add @ A )
% 2.92/1.54 => ! [A2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.92/1.54 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_99_less__eq__neg__nonpos,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( linord219039673up_add @ A )
% 2.92/1.54 => ! [A2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 2.92/1.54 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_100_neg__less__eq__nonneg,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( linord219039673up_add @ A )
% 2.92/1.54 => ! [A2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 2.92/1.54 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_101_real__of__ereal__0,axiom,
% 2.92/1.54 ( ( extend1716541707_ereal @ ( zero_zero @ extended_ereal ) )
% 2.92/1.54 = ( zero_zero @ real ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_102_enat__ord__simps_I1_J,axiom,
% 2.92/1.54 ! [M: nat,N: nat] :
% 2.92/1.54 ( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 2.92/1.54 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_103_real__of__ereal_Osimps_I3_J,axiom,
% 2.92/1.54 ( ( extend1716541707_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 = ( zero_zero @ real ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_104_minus__equation__iff,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( group_add @ A )
% 2.92/1.54 => ! [A2: A,B2: A] :
% 2.92/1.54 ( ( ( uminus_uminus @ A @ A2 )
% 2.92/1.54 = B2 )
% 2.92/1.54 = ( ( uminus_uminus @ A @ B2 )
% 2.92/1.54 = A2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_105_equation__minus__iff,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( group_add @ A )
% 2.92/1.54 => ! [A2: A,B2: A] :
% 2.92/1.54 ( ( A2
% 2.92/1.54 = ( uminus_uminus @ A @ B2 ) )
% 2.92/1.54 = ( B2
% 2.92/1.54 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_106_verit__negate__coefficient_I3_J,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordered_ab_group_add @ A )
% 2.92/1.54 => ! [A2: A,B2: A] :
% 2.92/1.54 ( ( A2 = B2 )
% 2.92/1.54 => ( ( uminus_uminus @ A @ A2 )
% 2.92/1.54 = ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_107_real__of__ereal__eq__0,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ( ( extend1716541707_ereal @ X2 )
% 2.92/1.54 = ( zero_zero @ real ) )
% 2.92/1.54 = ( ( X2
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 | ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 | ( X2
% 2.92/1.54 = ( zero_zero @ extended_ereal ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_108_real__of__ereal_Osimps_I2_J,axiom,
% 2.92/1.54 ( ( extend1716541707_ereal @ ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 = ( zero_zero @ real ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_109_le__imp__neg__le,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordered_ab_group_add @ A )
% 2.92/1.54 => ! [A2: A,B2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.92/1.54 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_110_minus__le__iff,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordered_ab_group_add @ A )
% 2.92/1.54 => ! [A2: A,B2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 2.92/1.54 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_111_le__minus__iff,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordered_ab_group_add @ A )
% 2.92/1.54 => ! [A2: A,B2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 2.92/1.54 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_112_numeral__neq__neg__numeral,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ring_char_0 @ A )
% 2.92/1.54 => ! [M: num,N: num] :
% 2.92/1.54 ( ( numeral_numeral @ A @ M )
% 2.92/1.54 != ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_113_neg__numeral__neq__numeral,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ring_char_0 @ A )
% 2.92/1.54 => ! [M: num,N: num] :
% 2.92/1.54 ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
% 2.92/1.54 != ( numeral_numeral @ A @ N ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_114_zero__enat__def,axiom,
% 2.92/1.54 ( ( zero_zero @ extended_enat )
% 2.92/1.54 = ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_115_enat__0__iff_I1_J,axiom,
% 2.92/1.54 ! [X2: nat] :
% 2.92/1.54 ( ( ( extended_enat2 @ X2 )
% 2.92/1.54 = ( zero_zero @ extended_enat ) )
% 2.92/1.54 = ( X2
% 2.92/1.54 = ( zero_zero @ nat ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_116_enat__0__iff_I2_J,axiom,
% 2.92/1.54 ! [X2: nat] :
% 2.92/1.54 ( ( ( zero_zero @ extended_enat )
% 2.92/1.54 = ( extended_enat2 @ X2 ) )
% 2.92/1.54 = ( X2
% 2.92/1.54 = ( zero_zero @ nat ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_117_lsc__at__MInfty,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( topolo503727757_space @ A )
% 2.92/1.54 => ! [F: A > extended_ereal,X0: A] :
% 2.92/1.54 ( ( ( F @ X0 )
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( lower_582600101lsc_at @ A @ extended_ereal @ X0 @ F ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_118_MInfty__neq__PInfty_I1_J,axiom,
% 2.92/1.54 ( ( extend1396239628finity @ extended_ereal )
% 2.92/1.54 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_119_ereal__uminus__le__reorder,axiom,
% 2.92/1.54 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) @ B2 )
% 2.92/1.54 = ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ B2 ) @ A2 ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_120_enat__ile,axiom,
% 2.92/1.54 ! [N: extended_enat,M: nat] :
% 2.92/1.54 ( ( ord_less_eq @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 2.92/1.54 => ? [K: nat] :
% 2.92/1.54 ( N
% 2.92/1.54 = ( extended_enat2 @ K ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_121_numeral__eq__enat,axiom,
% 2.92/1.54 ( ( numeral_numeral @ extended_enat )
% 2.92/1.54 = ( ^ [K2: num] : ( extended_enat2 @ ( numeral_numeral @ nat @ K2 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_122_real__of__ereal__pos,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ X2 )
% 2.92/1.54 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( extend1716541707_ereal @ X2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_123_neg__numeral__le__numeral,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( linordered_idom @ A )
% 2.92/1.54 => ! [M: num,N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_124_not__numeral__le__neg__numeral,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( linordered_idom @ A )
% 2.92/1.54 => ! [M: num,N: num] :
% 2.92/1.54 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_125_zero__neq__neg__numeral,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ring_char_0 @ A )
% 2.92/1.54 => ! [N: num] :
% 2.92/1.54 ( ( zero_zero @ A )
% 2.92/1.54 != ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_126_Infty__neq__0_I3_J,axiom,
% 2.92/1.54 ( ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 != ( zero_zero @ extended_ereal ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_127_ereal__infty__less__eq2_I2_J,axiom,
% 2.92/1.54 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
% 2.92/1.54 => ( ( B2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( A2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_128_ereal__less__eq_I2_J,axiom,
% 2.92/1.54 ! [X2: extended_ereal] : ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ X2 ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_129_real__of__ereal__positive__mono,axiom,
% 2.92/1.54 ! [X2: extended_ereal,Y3: extended_ereal] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ X2 )
% 2.92/1.54 => ( ( ord_less_eq @ extended_ereal @ X2 @ Y3 )
% 2.92/1.54 => ( ( Y3
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ord_less_eq @ real @ ( extend1716541707_ereal @ X2 ) @ ( extend1716541707_ereal @ Y3 ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_130_neg__numeral__le__zero,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( linordered_idom @ A )
% 2.92/1.54 => ! [N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ ( zero_zero @ A ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_131_not__zero__le__neg__numeral,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( linordered_idom @ A )
% 2.92/1.54 => ! [N: num] :
% 2.92/1.54 ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_132_not__MInfty__nonneg,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ X2 )
% 2.92/1.54 => ( X2
% 2.92/1.54 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_133_le__refl,axiom,
% 2.92/1.54 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_134_le__trans,axiom,
% 2.92/1.54 ! [I: nat,J: nat,K3: nat] :
% 2.92/1.54 ( ( ord_less_eq @ nat @ I @ J )
% 2.92/1.54 => ( ( ord_less_eq @ nat @ J @ K3 )
% 2.92/1.54 => ( ord_less_eq @ nat @ I @ K3 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_135_eq__imp__le,axiom,
% 2.92/1.54 ! [M: nat,N: nat] :
% 2.92/1.54 ( ( M = N )
% 2.92/1.54 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_136_le__antisym,axiom,
% 2.92/1.54 ! [M: nat,N: nat] :
% 2.92/1.54 ( ( ord_less_eq @ nat @ M @ N )
% 2.92/1.54 => ( ( ord_less_eq @ nat @ N @ M )
% 2.92/1.54 => ( M = N ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_137_GreatestI__nat,axiom,
% 2.92/1.54 ! [P: nat > $o,K3: nat,B2: nat] :
% 2.92/1.54 ( ( P @ K3 )
% 2.92/1.54 => ( ! [Y: nat] :
% 2.92/1.54 ( ( P @ Y )
% 2.92/1.54 => ( ord_less_eq @ nat @ Y @ B2 ) )
% 2.92/1.54 => ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_138_nat__le__linear,axiom,
% 2.92/1.54 ! [M: nat,N: nat] :
% 2.92/1.54 ( ( ord_less_eq @ nat @ M @ N )
% 2.92/1.54 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_139_Greatest__le__nat,axiom,
% 2.92/1.54 ! [P: nat > $o,K3: nat,B2: nat] :
% 2.92/1.54 ( ( P @ K3 )
% 2.92/1.54 => ( ! [Y: nat] :
% 2.92/1.54 ( ( P @ Y )
% 2.92/1.54 => ( ord_less_eq @ nat @ Y @ B2 ) )
% 2.92/1.54 => ( ord_less_eq @ nat @ K3 @ ( order_Greatest @ nat @ P ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_140_GreatestI__ex__nat,axiom,
% 2.92/1.54 ! [P: nat > $o,B2: nat] :
% 2.92/1.54 ( ? [X_1: nat] : ( P @ X_1 )
% 2.92/1.54 => ( ! [Y: nat] :
% 2.92/1.54 ( ( P @ Y )
% 2.92/1.54 => ( ord_less_eq @ nat @ Y @ B2 ) )
% 2.92/1.54 => ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_141_Nat_Oex__has__greatest__nat,axiom,
% 2.92/1.54 ! [P: nat > $o,K3: nat,B2: nat] :
% 2.92/1.54 ( ( P @ K3 )
% 2.92/1.54 => ( ! [Y: nat] :
% 2.92/1.54 ( ( P @ Y )
% 2.92/1.54 => ( ord_less_eq @ nat @ Y @ B2 ) )
% 2.92/1.54 => ? [X3: nat] :
% 2.92/1.54 ( ( P @ X3 )
% 2.92/1.54 & ! [Y5: nat] :
% 2.92/1.54 ( ( P @ Y5 )
% 2.92/1.54 => ( ord_less_eq @ nat @ Y5 @ X3 ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_142_strict__mono__imp__increasing,axiom,
% 2.92/1.54 ! [F: nat > nat,N: nat] :
% 2.92/1.54 ( ( order_strict_mono @ nat @ nat @ F )
% 2.92/1.54 => ( ord_less_eq @ nat @ N @ ( F @ N ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_143_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.92/1.54 ! [A2: nat] :
% 2.92/1.54 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.92/1.54 => ( A2
% 2.92/1.54 = ( zero_zero @ nat ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_144_bot__nat__0_Oextremum__unique,axiom,
% 2.92/1.54 ! [A2: nat] :
% 2.92/1.54 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.92/1.54 = ( A2
% 2.92/1.54 = ( zero_zero @ nat ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_145_le__0__eq,axiom,
% 2.92/1.54 ! [N: nat] :
% 2.92/1.54 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 2.92/1.54 = ( N
% 2.92/1.54 = ( zero_zero @ nat ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_146_less__eq__nat_Osimps_I1_J,axiom,
% 2.92/1.54 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_147_compl__le__compl__iff,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( boolean_algebra @ A )
% 2.92/1.54 => ! [X2: A,Y3: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y3 ) )
% 2.92/1.54 = ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_148_the__enat_Osimps,axiom,
% 2.92/1.54 ! [N: nat] :
% 2.92/1.54 ( ( extended_the_enat @ ( extended_enat2 @ N ) )
% 2.92/1.54 = N ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_149_uminus__apply,axiom,
% 2.92/1.54 ! [B: $tType,A: $tType] :
% 2.92/1.54 ( ( uminus @ B )
% 2.92/1.54 => ( ( uminus_uminus @ ( A > B ) )
% 2.92/1.54 = ( ^ [A5: A > B,X4: A] : ( uminus_uminus @ B @ ( A5 @ X4 ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_150_double__compl,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( boolean_algebra @ A )
% 2.92/1.54 => ! [X2: A] :
% 2.92/1.54 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X2 ) )
% 2.92/1.54 = X2 ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_151_compl__eq__compl__iff,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( boolean_algebra @ A )
% 2.92/1.54 => ! [X2: A,Y3: A] :
% 2.92/1.54 ( ( ( uminus_uminus @ A @ X2 )
% 2.92/1.54 = ( uminus_uminus @ A @ Y3 ) )
% 2.92/1.54 = ( X2 = Y3 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_152_enat_Osimps_I4_J,axiom,
% 2.92/1.54 ! [T: $tType,F1: nat > T,F22: T,Nat: nat] :
% 2.92/1.54 ( ( extended_case_enat @ T @ F1 @ F22 @ ( extended_enat2 @ Nat ) )
% 2.92/1.54 = ( F1 @ Nat ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_153_ereal__uminus__uminus,axiom,
% 2.92/1.54 ! [A2: extended_ereal] :
% 2.92/1.54 ( ( uminus_uminus @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) )
% 2.92/1.54 = A2 ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_154_ereal__uminus__eq__iff,axiom,
% 2.92/1.54 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.92/1.54 ( ( ( uminus_uminus @ extended_ereal @ A2 )
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ B2 ) )
% 2.92/1.54 = ( A2 = B2 ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_155_ereal__uminus__eq__reorder,axiom,
% 2.92/1.54 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.92/1.54 ( ( ( uminus_uminus @ extended_ereal @ A2 )
% 2.92/1.54 = B2 )
% 2.92/1.54 = ( A2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ B2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_156_fun__Compl__def,axiom,
% 2.92/1.54 ! [B: $tType,A: $tType] :
% 2.92/1.54 ( ( uminus @ B )
% 2.92/1.54 => ( ( uminus_uminus @ ( A > B ) )
% 2.92/1.54 = ( ^ [A5: A > B,X4: A] : ( uminus_uminus @ B @ ( A5 @ X4 ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_157_compl__le__swap2,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( boolean_algebra @ A )
% 2.92/1.54 => ! [Y3: A,X2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X2 )
% 2.92/1.54 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ Y3 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_158_compl__le__swap1,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( boolean_algebra @ A )
% 2.92/1.54 => ! [Y3: A,X2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X2 ) )
% 2.92/1.54 => ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_159_compl__mono,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( boolean_algebra @ A )
% 2.92/1.54 => ! [X2: A,Y3: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.92/1.54 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_160_real__eq__0__iff__le__ge__0,axiom,
% 2.92/1.54 ! [X2: real] :
% 2.92/1.54 ( ( X2
% 2.92/1.54 = ( zero_zero @ real ) )
% 2.92/1.54 = ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
% 2.92/1.54 & ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( uminus_uminus @ real @ X2 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_161_enat_Osimps_I6_J,axiom,
% 2.92/1.54 ! [T: $tType,F1: nat > T,F22: T,Nat: nat] :
% 2.92/1.54 ( ( extended_rec_enat @ T @ F1 @ F22 @ ( extended_enat2 @ Nat ) )
% 2.92/1.54 = ( F1 @ Nat ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_162_dbl__simps_I1_J,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( neg_numeral @ A )
% 2.92/1.54 => ! [K3: num] :
% 2.92/1.54 ( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K3 ) ) )
% 2.92/1.54 = ( uminus_uminus @ A @ ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K3 ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_163_dbl__simps_I2_J,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( neg_numeral @ A )
% 2.92/1.54 => ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
% 2.92/1.54 = ( zero_zero @ A ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_164_ereal__less__eq_I5_J,axiom,
% 2.92/1.54 ! [R: real] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( extended_ereal2 @ R ) )
% 2.92/1.54 = ( ord_less_eq @ real @ ( zero_zero @ real ) @ R ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_165_ereal__less__eq_I4_J,axiom,
% 2.92/1.54 ! [R: real] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ R ) @ ( zero_zero @ extended_ereal ) )
% 2.92/1.54 = ( ord_less_eq @ real @ R @ ( zero_zero @ real ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_166_MInfty__eq__minfinity,axiom,
% 2.92/1.54 ( extended_MInfty
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_167_ereal__cong,axiom,
% 2.92/1.54 ! [X2: real,Y3: real] :
% 2.92/1.54 ( ( X2 = Y3 )
% 2.92/1.54 => ( ( extended_ereal2 @ X2 )
% 2.92/1.54 = ( extended_ereal2 @ Y3 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_168_ereal_Oinject,axiom,
% 2.92/1.54 ! [X1: real,Y1: real] :
% 2.92/1.54 ( ( ( extended_ereal2 @ X1 )
% 2.92/1.54 = ( extended_ereal2 @ Y1 ) )
% 2.92/1.54 = ( X1 = Y1 ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_169_numeral__eq__ereal,axiom,
% 2.92/1.54 ( ( numeral_numeral @ extended_ereal )
% 2.92/1.54 = ( ^ [W: num] : ( extended_ereal2 @ ( numeral_numeral @ real @ W ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_170_ereal__eq__0_I1_J,axiom,
% 2.92/1.54 ! [R: real] :
% 2.92/1.54 ( ( ( extended_ereal2 @ R )
% 2.92/1.54 = ( zero_zero @ extended_ereal ) )
% 2.92/1.54 = ( R
% 2.92/1.54 = ( zero_zero @ real ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_171_ereal__eq__0_I2_J,axiom,
% 2.92/1.54 ! [R: real] :
% 2.92/1.54 ( ( ( zero_zero @ extended_ereal )
% 2.92/1.54 = ( extended_ereal2 @ R ) )
% 2.92/1.54 = ( R
% 2.92/1.54 = ( zero_zero @ real ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_172_ereal__less__eq_I3_J,axiom,
% 2.92/1.54 ! [R: real,P2: real] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P2 ) )
% 2.92/1.54 = ( ord_less_eq @ real @ R @ P2 ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_173_ereal_Odistinct_I3_J,axiom,
% 2.92/1.54 ! [X1: real] :
% 2.92/1.54 ( ( extended_ereal2 @ X1 )
% 2.92/1.54 != extended_MInfty ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_174_real__of__ereal_Osimps_I1_J,axiom,
% 2.92/1.54 ! [R: real] :
% 2.92/1.54 ( ( extend1716541707_ereal @ ( extended_ereal2 @ R ) )
% 2.92/1.54 = R ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_175_ereal__le__real,axiom,
% 2.92/1.54 ! [X2: extended_ereal,Y3: extended_ereal] :
% 2.92/1.54 ( ! [Z4: real] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ X2 @ ( extended_ereal2 @ Z4 ) )
% 2.92/1.54 => ( ord_less_eq @ extended_ereal @ Y3 @ ( extended_ereal2 @ Z4 ) ) )
% 2.92/1.54 => ( ord_less_eq @ extended_ereal @ Y3 @ X2 ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_176_PInfty__neq__ereal_I1_J,axiom,
% 2.92/1.54 ! [R: real] :
% 2.92/1.54 ( ( extended_ereal2 @ R )
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_177_zero__ereal__def,axiom,
% 2.92/1.54 ( ( zero_zero @ extended_ereal )
% 2.92/1.54 = ( extended_ereal2 @ ( zero_zero @ real ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_178_le__ereal__le,axiom,
% 2.92/1.54 ! [A2: extended_ereal,X2: real,Y3: real] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ A2 @ ( extended_ereal2 @ X2 ) )
% 2.92/1.54 => ( ( ord_less_eq @ real @ X2 @ Y3 )
% 2.92/1.54 => ( ord_less_eq @ extended_ereal @ A2 @ ( extended_ereal2 @ Y3 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_179_ereal__le__le,axiom,
% 2.92/1.54 ! [Y3: real,A2: extended_ereal,X2: real] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ Y3 ) @ A2 )
% 2.92/1.54 => ( ( ord_less_eq @ real @ X2 @ Y3 )
% 2.92/1.54 => ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ X2 ) @ A2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_180_real__of__ereal_Oinduct,axiom,
% 2.92/1.54 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 2.92/1.54 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ( P @ ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( P @ A0 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_181_real__of__ereal_Ocases,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ! [R2: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 != ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ( X2
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_182_times__ereal_Oinduct,axiom,
% 2.92/1.54 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 2.92/1.54 ( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
% 2.92/1.54 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ! [R2: real] : ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( ! [R2: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_183_plus__ereal_Oinduct,axiom,
% 2.92/1.54 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 2.92/1.54 ( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
% 2.92/1.54 => ( ! [X_12: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_12 )
% 2.92/1.54 => ( ! [A4: extended_ereal] : ( P @ A4 @ ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( ! [P3: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ P3 ) )
% 2.92/1.54 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_184_less__ereal_Oinduct,axiom,
% 2.92/1.54 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 2.92/1.54 ( ! [X3: real,Y: real] : ( P @ ( extended_ereal2 @ X3 ) @ ( extended_ereal2 @ Y ) )
% 2.92/1.54 => ( ! [X_12: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_12 )
% 2.92/1.54 => ( ! [A4: extended_ereal] : ( P @ A4 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( ! [X3: real] : ( P @ ( extended_ereal2 @ X3 ) @ ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ! [R2: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_185_abs__ereal_Oinduct,axiom,
% 2.92/1.54 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 2.92/1.54 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( ( P @ ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( P @ A0 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_186_ereal__all__split,axiom,
% 2.92/1.54 ( ( ^ [P4: extended_ereal > $o] :
% 2.92/1.54 ! [X6: extended_ereal] : ( P4 @ X6 ) )
% 2.92/1.54 = ( ^ [P5: extended_ereal > $o] :
% 2.92/1.54 ( ( P5 @ ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 & ! [X4: real] : ( P5 @ ( extended_ereal2 @ X4 ) )
% 2.92/1.54 & ( P5 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_187_abs__ereal_Ocases,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ! [R2: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 != ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ( X2
% 2.92/1.54 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( X2
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_188_ereal__ex__split,axiom,
% 2.92/1.54 ( ( ^ [P4: extended_ereal > $o] :
% 2.92/1.54 ? [X6: extended_ereal] : ( P4 @ X6 ) )
% 2.92/1.54 = ( ^ [P5: extended_ereal > $o] :
% 2.92/1.54 ( ( P5 @ ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 | ? [X4: real] : ( P5 @ ( extended_ereal2 @ X4 ) )
% 2.92/1.54 | ( P5 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_189_ereal3__cases,axiom,
% 2.92/1.54 ! [X2: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
% 2.92/1.54 ( ( ? [R2: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ? [Ra: real] :
% 2.92/1.54 ( Xa3
% 2.92/1.54 = ( extended_ereal2 @ Ra ) )
% 2.92/1.54 => ! [Rb: real] :
% 2.92/1.54 ( Xb
% 2.92/1.54 != ( extended_ereal2 @ Rb ) ) ) )
% 2.92/1.54 => ( ( ? [R2: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ? [Ra: real] :
% 2.92/1.54 ( Xa3
% 2.92/1.54 = ( extended_ereal2 @ Ra ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.92/1.54 => ( ( ? [R2: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ? [Ra: real] :
% 2.92/1.54 ( Xa3
% 2.92/1.54 = ( extended_ereal2 @ Ra ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.92/1.54 => ( ( ? [R2: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ! [Ra: real] :
% 2.92/1.54 ( Xb
% 2.92/1.54 != ( extended_ereal2 @ Ra ) ) ) )
% 2.92/1.54 => ( ( ? [R2: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.92/1.54 => ( ( ? [R2: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.92/1.54 => ( ( ? [R2: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ! [Ra: real] :
% 2.92/1.54 ( Xb
% 2.92/1.54 != ( extended_ereal2 @ Ra ) ) ) )
% 2.92/1.54 => ( ( ? [R2: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.92/1.54 => ( ( ? [R2: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ? [R2: real] :
% 2.92/1.54 ( Xa3
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ! [Ra: real] :
% 2.92/1.54 ( Xb
% 2.92/1.54 != ( extended_ereal2 @ Ra ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ? [R2: real] :
% 2.92/1.54 ( Xa3
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ? [R2: real] :
% 2.92/1.54 ( Xa3
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ! [R2: real] :
% 2.92/1.54 ( Xb
% 2.92/1.54 != ( extended_ereal2 @ R2 ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ! [R2: real] :
% 2.92/1.54 ( Xb
% 2.92/1.54 != ( extended_ereal2 @ R2 ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( ? [R2: real] :
% 2.92/1.54 ( Xa3
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ! [Ra: real] :
% 2.92/1.54 ( Xb
% 2.92/1.54 != ( extended_ereal2 @ Ra ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( ? [R2: real] :
% 2.92/1.54 ( Xa3
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( ? [R2: real] :
% 2.92/1.54 ( Xa3
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ! [R2: real] :
% 2.92/1.54 ( Xb
% 2.92/1.54 != ( extended_ereal2 @ R2 ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ! [R2: real] :
% 2.92/1.54 ( Xb
% 2.92/1.54 != ( extended_ereal2 @ R2 ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.92/1.54 => ~ ( ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( ( Xa3
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( Xb
% 2.92/1.54 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_190_ereal2__cases,axiom,
% 2.92/1.54 ! [X2: extended_ereal,Xa3: extended_ereal] :
% 2.92/1.54 ( ( ? [R2: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ! [Ra: real] :
% 2.92/1.54 ( Xa3
% 2.92/1.54 != ( extended_ereal2 @ Ra ) ) )
% 2.92/1.54 => ( ( ? [R2: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( Xa3
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( ( ? [R2: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( Xa3
% 2.92/1.54 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ! [R2: real] :
% 2.92/1.54 ( Xa3
% 2.92/1.54 != ( extended_ereal2 @ R2 ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( Xa3
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( Xa3
% 2.92/1.54 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ! [R2: real] :
% 2.92/1.54 ( Xa3
% 2.92/1.54 != ( extended_ereal2 @ R2 ) ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( Xa3
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ~ ( ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( Xa3
% 2.92/1.54 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_191_ereal__cases,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ! [R2: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 != ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ( X2
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_192_MInfty__neq__ereal_I1_J,axiom,
% 2.92/1.54 ! [R: real] :
% 2.92/1.54 ( ( extended_ereal2 @ R )
% 2.92/1.54 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_193_ereal__top,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ! [B5: real] : ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ B5 ) @ X2 )
% 2.92/1.54 => ( X2
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_194_uminus__ereal_Osimps_I1_J,axiom,
% 2.92/1.54 ! [R: real] :
% 2.92/1.54 ( ( uminus_uminus @ extended_ereal @ ( extended_ereal2 @ R ) )
% 2.92/1.54 = ( extended_ereal2 @ ( uminus_uminus @ real @ R ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_195_ereal__bot,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ! [B5: real] : ( ord_less_eq @ extended_ereal @ X2 @ ( extended_ereal2 @ B5 ) )
% 2.92/1.54 => ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_196_real__of__ereal_Oelims,axiom,
% 2.92/1.54 ! [X2: extended_ereal,Y3: real] :
% 2.92/1.54 ( ( ( extend1716541707_ereal @ X2 )
% 2.92/1.54 = Y3 )
% 2.92/1.54 => ( ! [R2: real] :
% 2.92/1.54 ( ( X2
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( Y3 != R2 ) )
% 2.92/1.54 => ( ( ( X2
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( Y3
% 2.92/1.54 != ( zero_zero @ real ) ) )
% 2.92/1.54 => ~ ( ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( Y3
% 2.92/1.54 != ( zero_zero @ real ) ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_197_ereal_Osize__gen_I3_J,axiom,
% 2.92/1.54 ( ( extended_size_ereal @ extended_MInfty )
% 2.92/1.54 = ( zero_zero @ nat ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_198_ereal_Osize__gen_I1_J,axiom,
% 2.92/1.54 ! [X1: real] :
% 2.92/1.54 ( ( extended_size_ereal @ ( extended_ereal2 @ X1 ) )
% 2.92/1.54 = ( zero_zero @ nat ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_199_uminus__ereal_Oelims,axiom,
% 2.92/1.54 ! [X2: extended_ereal,Y3: extended_ereal] :
% 2.92/1.54 ( ( ( uminus_uminus @ extended_ereal @ X2 )
% 2.92/1.54 = Y3 )
% 2.92/1.54 => ( ! [R2: real] :
% 2.92/1.54 ( ( X2
% 2.92/1.54 = ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( Y3
% 2.92/1.54 != ( extended_ereal2 @ ( uminus_uminus @ real @ R2 ) ) ) )
% 2.92/1.54 => ( ( ( X2 = extended_PInfty )
% 2.92/1.54 => ( Y3 != extended_MInfty ) )
% 2.92/1.54 => ~ ( ( X2 = extended_MInfty )
% 2.92/1.54 => ( Y3 != extended_PInfty ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_200_ereal_Odistinct_I1_J,axiom,
% 2.92/1.54 ! [X1: real] :
% 2.92/1.54 ( ( extended_ereal2 @ X1 )
% 2.92/1.54 != extended_PInfty ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_201_ereal_Osize__gen_I2_J,axiom,
% 2.92/1.54 ( ( extended_size_ereal @ extended_PInfty )
% 2.92/1.54 = ( zero_zero @ nat ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_202_infinity__ereal__def,axiom,
% 2.92/1.54 ( ( extend1396239628finity @ extended_ereal )
% 2.92/1.54 = extended_PInfty ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_203_ereal_Odistinct_I5_J,axiom,
% 2.92/1.54 extended_PInfty != extended_MInfty ).
% 2.92/1.54
% 2.92/1.54 thf(fact_204_uminus__ereal_Oinduct,axiom,
% 2.92/1.54 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 2.92/1.54 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ( P @ extended_PInfty )
% 2.92/1.54 => ( ( P @ extended_MInfty )
% 2.92/1.54 => ( P @ A0 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_205_uminus__ereal_Ocases,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ! [R2: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 != ( extended_ereal2 @ R2 ) )
% 2.92/1.54 => ( ( X2 != extended_PInfty )
% 2.92/1.54 => ( X2 = extended_MInfty ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_206_ereal_Oexhaust,axiom,
% 2.92/1.54 ! [Y3: extended_ereal] :
% 2.92/1.54 ( ! [X12: real] :
% 2.92/1.54 ( Y3
% 2.92/1.54 != ( extended_ereal2 @ X12 ) )
% 2.92/1.54 => ( ( Y3 != extended_PInfty )
% 2.92/1.54 => ( Y3 = extended_MInfty ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_207_ereal_Oinduct,axiom,
% 2.92/1.54 ! [P: extended_ereal > $o,Ereal: extended_ereal] :
% 2.92/1.54 ( ! [X3: real] : ( P @ ( extended_ereal2 @ X3 ) )
% 2.92/1.54 => ( ( P @ extended_PInfty )
% 2.92/1.54 => ( ( P @ extended_MInfty )
% 2.92/1.54 => ( P @ Ereal ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_208_uminus__ereal_Osimps_I2_J,axiom,
% 2.92/1.54 ( ( uminus_uminus @ extended_ereal @ extended_PInfty )
% 2.92/1.54 = extended_MInfty ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_209_uminus__ereal_Osimps_I3_J,axiom,
% 2.92/1.54 ( ( uminus_uminus @ extended_ereal @ extended_MInfty )
% 2.92/1.54 = extended_PInfty ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_210_ereal__divide__ereal,axiom,
% 2.92/1.54 ! [R: real] :
% 2.92/1.54 ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R )
% 2.92/1.54 => ( ( divide_divide @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R ) )
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 & ( ~ ( ord_less_eq @ real @ ( zero_zero @ real ) @ R )
% 2.92/1.54 => ( ( divide_divide @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R ) )
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_211_ereal__le__real__iff,axiom,
% 2.92/1.54 ! [X2: real,Y3: extended_ereal] :
% 2.92/1.54 ( ( ord_less_eq @ real @ X2 @ ( extend1716541707_ereal @ Y3 ) )
% 2.92/1.54 = ( ( ( ( abs_abs @ extended_ereal @ Y3 )
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ X2 ) @ Y3 ) )
% 2.92/1.54 & ( ( ( abs_abs @ extended_ereal @ Y3 )
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_212_abs__idempotent,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordere142940540dd_abs @ A )
% 2.92/1.54 => ! [A2: A] :
% 2.92/1.54 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
% 2.92/1.54 = ( abs_abs @ A @ A2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_213_abs__zero,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordere142940540dd_abs @ A )
% 2.92/1.54 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 2.92/1.54 = ( zero_zero @ A ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_214_abs__eq__0,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordere142940540dd_abs @ A )
% 2.92/1.54 => ! [A2: A] :
% 2.92/1.54 ( ( ( abs_abs @ A @ A2 )
% 2.92/1.54 = ( zero_zero @ A ) )
% 2.92/1.54 = ( A2
% 2.92/1.54 = ( zero_zero @ A ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_215_abs__0__eq,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordere142940540dd_abs @ A )
% 2.92/1.54 => ! [A2: A] :
% 2.92/1.54 ( ( ( zero_zero @ A )
% 2.92/1.54 = ( abs_abs @ A @ A2 ) )
% 2.92/1.54 = ( A2
% 2.92/1.54 = ( zero_zero @ A ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_216_abs__numeral,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( linordered_idom @ A )
% 2.92/1.54 => ! [N: num] :
% 2.92/1.54 ( ( abs_abs @ A @ ( numeral_numeral @ A @ N ) )
% 2.92/1.54 = ( numeral_numeral @ A @ N ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_217_abs__minus__cancel,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordere142940540dd_abs @ A )
% 2.92/1.54 => ! [A2: A] :
% 2.92/1.54 ( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
% 2.92/1.54 = ( abs_abs @ A @ A2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_218_abs__ereal__uminus,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ( abs_abs @ extended_ereal @ ( uminus_uminus @ extended_ereal @ X2 ) )
% 2.92/1.54 = ( abs_abs @ extended_ereal @ X2 ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_219_abs__ereal__zero,axiom,
% 2.92/1.54 ( ( abs_abs @ extended_ereal @ ( zero_zero @ extended_ereal ) )
% 2.92/1.54 = ( zero_zero @ extended_ereal ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_220_ereal__uminus__divide,axiom,
% 2.92/1.54 ! [X2: extended_ereal,Y3: extended_ereal] :
% 2.92/1.54 ( ( divide_divide @ extended_ereal @ ( uminus_uminus @ extended_ereal @ X2 ) @ Y3 )
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( divide_divide @ extended_ereal @ X2 @ Y3 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_221_ereal__divide__zero__left,axiom,
% 2.92/1.54 ! [A2: extended_ereal] :
% 2.92/1.54 ( ( divide_divide @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A2 )
% 2.92/1.54 = ( zero_zero @ extended_ereal ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_222_abs__of__nonneg,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordere142940540dd_abs @ A )
% 2.92/1.54 => ! [A2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.92/1.54 => ( ( abs_abs @ A @ A2 )
% 2.92/1.54 = A2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_223_abs__le__self__iff,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordere142940540dd_abs @ A )
% 2.92/1.54 => ! [A2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 )
% 2.92/1.54 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_224_abs__le__zero__iff,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordere142940540dd_abs @ A )
% 2.92/1.54 => ! [A2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.92/1.54 = ( A2
% 2.92/1.54 = ( zero_zero @ A ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_225_abs__neg__numeral,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( linordered_idom @ A )
% 2.92/1.54 => ! [N: num] :
% 2.92/1.54 ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 2.92/1.54 = ( numeral_numeral @ A @ N ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_226_abs__ereal__ge0,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ X2 )
% 2.92/1.54 => ( ( abs_abs @ extended_ereal @ X2 )
% 2.92/1.54 = X2 ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_227_ereal__divide__Infty_I1_J,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ( divide_divide @ extended_ereal @ X2 @ ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 = ( zero_zero @ extended_ereal ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_228_abs__of__nonpos,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordere142940540dd_abs @ A )
% 2.92/1.54 => ! [A2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.92/1.54 => ( ( abs_abs @ A @ A2 )
% 2.92/1.54 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_229_ereal__divide__Infty_I2_J,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ( divide_divide @ extended_ereal @ X2 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 = ( zero_zero @ extended_ereal ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_230_not__infty__ereal,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ( ( abs_abs @ extended_ereal @ X2 )
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 = ( ? [X7: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 = ( extended_ereal2 @ X7 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_231_abs__neq__infinity__cases,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ( ( abs_abs @ extended_ereal @ X2 )
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ~ ! [R2: real] :
% 2.92/1.54 ( X2
% 2.92/1.54 != ( extended_ereal2 @ R2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_232_abs__ereal_Osimps_I2_J,axiom,
% 2.92/1.54 ( ( abs_abs @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_233_ereal__infinity__cases,axiom,
% 2.92/1.54 ! [A2: extended_ereal] :
% 2.92/1.54 ( ( A2
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ( A2
% 2.92/1.54 != ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
% 2.92/1.54 => ( ( abs_abs @ extended_ereal @ A2 )
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_234_abs__eq__infinity__cases,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ( ( abs_abs @ extended_ereal @ X2 )
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ( X2
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( X2
% 2.92/1.54 = ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_235_ereal__abs__leI,axiom,
% 2.92/1.54 ! [X2: extended_ereal,Y3: extended_ereal] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ X2 @ Y3 )
% 2.92/1.54 => ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ X2 ) @ Y3 )
% 2.92/1.54 => ( ord_less_eq @ extended_ereal @ ( abs_abs @ extended_ereal @ X2 ) @ Y3 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_236_abs__ereal__pos,axiom,
% 2.92/1.54 ! [X2: extended_ereal] : ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( abs_abs @ extended_ereal @ X2 ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_237_zero__le__divide__ereal,axiom,
% 2.92/1.54 ! [A2: extended_ereal,B2: extended_ereal] :
% 2.92/1.54 ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A2 )
% 2.92/1.54 => ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ B2 )
% 2.92/1.54 => ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( divide_divide @ extended_ereal @ A2 @ B2 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_238_abs__ge__zero,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordere142940540dd_abs @ A )
% 2.92/1.54 => ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_239_abs__leI,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordere142940540dd_abs @ A )
% 2.92/1.54 => ! [A2: A,B2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.92/1.54 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 2.92/1.54 => ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_240_abs__le__D2,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordere142940540dd_abs @ A )
% 2.92/1.54 => ! [A2: A,B2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
% 2.92/1.54 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_241_abs__le__iff,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordere142940540dd_abs @ A )
% 2.92/1.54 => ! [A2: A,B2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
% 2.92/1.54 = ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.92/1.54 & ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_242_abs__ge__minus__self,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordere142940540dd_abs @ A )
% 2.92/1.54 => ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( abs_abs @ A @ A2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_243_ereal__divide__eq__0__iff,axiom,
% 2.92/1.54 ! [X2: extended_ereal,Y3: extended_ereal] :
% 2.92/1.54 ( ( ( divide_divide @ extended_ereal @ X2 @ Y3 )
% 2.92/1.54 = ( zero_zero @ extended_ereal ) )
% 2.92/1.54 = ( ( X2
% 2.92/1.54 = ( zero_zero @ extended_ereal ) )
% 2.92/1.54 | ( ( abs_abs @ extended_ereal @ Y3 )
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_244_abs__ereal_Osimps_I3_J,axiom,
% 2.92/1.54 ( ( abs_abs @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_245_abs__ge__self,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordere142940540dd_abs @ A )
% 2.92/1.54 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( abs_abs @ A @ A2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_246_abs__le__D1,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordere142940540dd_abs @ A )
% 2.92/1.54 => ! [A2: A,B2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
% 2.92/1.54 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_247_abs__minus__le__zero,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( ordere142940540dd_abs @ A )
% 2.92/1.54 => ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A2 ) ) @ ( zero_zero @ A ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_248_ereal__real_H,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ( ( abs_abs @ extended_ereal @ X2 )
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
% 2.92/1.54 = X2 ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_249_ereal__real,axiom,
% 2.92/1.54 ! [X2: extended_ereal] :
% 2.92/1.54 ( ( ( ( abs_abs @ extended_ereal @ X2 )
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
% 2.92/1.54 = ( zero_zero @ extended_ereal ) ) )
% 2.92/1.54 & ( ( ( abs_abs @ extended_ereal @ X2 )
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
% 2.92/1.54 = X2 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_250_real__le__ereal__iff,axiom,
% 2.92/1.54 ! [Y3: extended_ereal,X2: real] :
% 2.92/1.54 ( ( ord_less_eq @ real @ ( extend1716541707_ereal @ Y3 ) @ X2 )
% 2.92/1.54 = ( ( ( ( abs_abs @ extended_ereal @ Y3 )
% 2.92/1.54 != ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ord_less_eq @ extended_ereal @ Y3 @ ( extended_ereal2 @ X2 ) ) )
% 2.92/1.54 & ( ( ( abs_abs @ extended_ereal @ Y3 )
% 2.92/1.54 = ( extend1396239628finity @ extended_ereal ) )
% 2.92/1.54 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(fact_251_divide__le__0__abs__iff,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( linordered_field @ A )
% 2.92/1.54 => ! [A2: A,B2: A] :
% 2.92/1.54 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) @ ( zero_zero @ A ) )
% 2.92/1.54 = ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.92/1.54 | ( B2
% 2.92/1.54 = ( zero_zero @ A ) ) ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(subcl_Real__Vector__Spaces_Ometric__space___HOL_Otype,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( real_V2090557954_space @ A )
% 2.92/1.54 => ( type @ A ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Otopological__space,axiom,
% 2.92/1.54 ! [A: $tType] :
% 2.92/1.54 ( ( real_V2090557954_space @ A )
% 2.92/1.54 => ( topolo503727757_space @ A ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
% 2.92/1.54 ! [A6: $tType,A7: $tType] :
% 2.92/1.54 ( ( topolo503727757_space @ A7 )
% 2.92/1.54 => ( topolo503727757_space @ ( A6 > A7 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 2.92/1.54 ! [A6: $tType,A7: $tType] :
% 2.92/1.54 ( ( boolean_algebra @ A7 )
% 2.92/1.54 => ( boolean_algebra @ ( A6 > A7 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.92/1.54 ! [A6: $tType,A7: $tType] :
% 2.92/1.54 ( ( preorder @ A7 )
% 2.92/1.54 => ( preorder @ ( A6 > A7 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.92/1.54 ! [A6: $tType,A7: $tType] :
% 2.92/1.54 ( ( order @ A7 )
% 2.92/1.54 => ( order @ ( A6 > A7 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_fun___Orderings_Oord,axiom,
% 2.92/1.54 ! [A6: $tType,A7: $tType] :
% 2.92/1.54 ( ( ord @ A7 )
% 2.92/1.54 => ( ord @ ( A6 > A7 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_fun___Groups_Ouminus,axiom,
% 2.92/1.54 ! [A6: $tType,A7: $tType] :
% 2.92/1.54 ( ( uminus @ A7 )
% 2.92/1.54 => ( uminus @ ( A6 > A7 ) ) ) ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.92/1.54 canoni770627133id_add @ nat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_1,axiom,
% 2.92/1.54 topolo503727757_space @ nat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology,axiom,
% 2.92/1.54 topolo259154727pology @ nat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 2.92/1.54 linord1659791738miring @ nat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Nat_Onat___Nat_Osemiring__char__0,axiom,
% 2.92/1.54 semiring_char_0 @ nat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
% 2.92/1.54 preorder @ nat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.92/1.54 linorder @ nat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
% 2.92/1.54 order @ nat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
% 2.92/1.54 ord @ nat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Nat_Onat___Num_Onumeral,axiom,
% 2.92/1.54 numeral @ nat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 2.92/1.54 zero @ nat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Num_Onum___Orderings_Opreorder_5,axiom,
% 2.92/1.54 preorder @ num ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Num_Onum___Orderings_Olinorder_6,axiom,
% 2.92/1.54 linorder @ num ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Num_Onum___Orderings_Oorder_7,axiom,
% 2.92/1.54 order @ num ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Num_Onum___Orderings_Oord_8,axiom,
% 2.92/1.54 ord @ num ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_9,axiom,
% 2.92/1.54 topolo503727757_space @ $o ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_10,axiom,
% 2.92/1.54 topolo259154727pology @ $o ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_11,axiom,
% 2.92/1.54 boolean_algebra @ $o ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
% 2.92/1.54 preorder @ $o ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_HOL_Obool___Orderings_Olinorder_13,axiom,
% 2.92/1.54 linorder @ $o ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
% 2.92/1.54 order @ $o ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
% 2.92/1.54 ord @ $o ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_HOL_Obool___Groups_Ouminus_16,axiom,
% 2.92/1.54 uminus @ $o ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_17,axiom,
% 2.92/1.54 topolo503727757_space @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_18,axiom,
% 2.92/1.54 topolo259154727pology @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_19,axiom,
% 2.92/1.54 linord1659791738miring @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
% 2.92/1.54 real_V2090557954_space @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs,axiom,
% 2.92/1.54 ordere142940540dd_abs @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
% 2.92/1.54 linord219039673up_add @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
% 2.92/1.54 ordered_ab_group_add @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 2.92/1.54 linordered_field @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Rings_Olinordered__idom,axiom,
% 2.92/1.54 linordered_idom @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_20,axiom,
% 2.92/1.54 semiring_char_0 @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Orderings_Opreorder_21,axiom,
% 2.92/1.54 preorder @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Orderings_Olinorder_22,axiom,
% 2.92/1.54 linorder @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 2.92/1.54 group_add @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Orderings_Oorder_23,axiom,
% 2.92/1.54 order @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Num_Oneg__numeral,axiom,
% 2.92/1.54 neg_numeral @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Nat_Oring__char__0,axiom,
% 2.92/1.54 ring_char_0 @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Orderings_Oord_24,axiom,
% 2.92/1.54 ord @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Groups_Ouminus_25,axiom,
% 2.92/1.54 uminus @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Num_Onumeral_26,axiom,
% 2.92/1.54 numeral @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Real_Oreal___Groups_Ozero_27,axiom,
% 2.92/1.54 zero @ real ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_28,axiom,
% 2.92/1.54 canoni770627133id_add @ extended_enat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Nat_Oenat___Topological__Spaces_Otopological__space_29,axiom,
% 2.92/1.54 topolo503727757_space @ extended_enat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Nat_Oenat___Topological__Spaces_Oorder__topology_30,axiom,
% 2.92/1.54 topolo259154727pology @ extended_enat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_31,axiom,
% 2.92/1.54 linord1659791738miring @ extended_enat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_32,axiom,
% 2.92/1.54 semiring_char_0 @ extended_enat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_33,axiom,
% 2.92/1.54 preorder @ extended_enat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_34,axiom,
% 2.92/1.54 linorder @ extended_enat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_35,axiom,
% 2.92/1.54 order @ extended_enat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_36,axiom,
% 2.92/1.54 ord @ extended_enat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Nat_Oenat___Num_Onumeral_37,axiom,
% 2.92/1.54 numeral @ extended_enat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_38,axiom,
% 2.92/1.54 zero @ extended_enat ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Otopological__space_39,axiom,
% 2.92/1.54 topolo503727757_space @ extended_ereal ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Real_Oereal___Topological__Spaces_Oorder__topology_40,axiom,
% 2.92/1.54 topolo259154727pology @ extended_ereal ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Real_Oereal___Orderings_Opreorder_41,axiom,
% 2.92/1.54 preorder @ extended_ereal ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Real_Oereal___Orderings_Olinorder_42,axiom,
% 2.92/1.54 linorder @ extended_ereal ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Real_Oereal___Orderings_Oorder_43,axiom,
% 2.92/1.54 order @ extended_ereal ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Real_Oereal___Orderings_Oord_44,axiom,
% 2.92/1.54 ord @ extended_ereal ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Real_Oereal___Groups_Ouminus_45,axiom,
% 2.92/1.54 uminus @ extended_ereal ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Real_Oereal___Num_Onumeral_46,axiom,
% 2.92/1.54 numeral @ extended_ereal ).
% 2.92/1.54
% 2.92/1.54 thf(tcon_Extended__Real_Oereal___Groups_Ozero_47,axiom,
% 2.92/1.54 zero @ extended_ereal ).
% 2.92/1.54
% 2.92/1.54 thf(tfree_0,hypothesis,
% 2.92/1.54 real_V2090557954_space @ a ).
% 2.92/1.54
% 2.92/1.54 thf(conj_0,conjecture,
% 2.92/1.54 ord_less_eq @ extended_ereal @ ( g @ x ) @ ( f @ x ) ).
% 2.92/1.54 ------------------------
% 2.92/1.56 % (22632)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22527 for (2999ds/18Mi)
% 2.92/1.56 % (22631)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_22527 for (2999ds/275Mi)
% 2.92/1.56 % (22629)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22527 for (2999ds/2Mi)
% 2.92/1.56 % (22630)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_22527 for (2999ds/2Mi)
% 2.92/1.56 % (22629)Instruction limit reached!
% 2.92/1.56 % (22629)------------------------------
% 2.92/1.56 % (22629)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.56 % (22629)Termination reason: Unknown
% 2.92/1.56 % (22629)Termination phase: shuffling
% 2.92/1.56
% 2.92/1.56 % (22629)Memory used [KB]: 1407
% 2.92/1.56 % (22629)Time elapsed: 0.002 s
% 2.92/1.56 % (22629)Instructions burned: 3 (million)
% 2.92/1.56 % (22630)Instruction limit reached!
% 2.92/1.56 % (22630)------------------------------
% 2.92/1.56 % (22630)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.56 % (22630)Termination reason: Unknown
% 2.92/1.56 % (22630)Termination phase: shuffling
% 2.92/1.56
% 2.92/1.56 % (22630)Memory used [KB]: 1407
% 2.92/1.56 % (22630)Time elapsed: 0.002 s
% 2.92/1.56 % (22630)Instructions burned: 3 (million)
% 2.92/1.56 % (22630)------------------------------
% 2.92/1.56 % (22630)------------------------------
% 2.92/1.56 % (22629)------------------------------
% 2.92/1.56 % (22629)------------------------------
% 2.92/1.56 % (22632)Instruction limit reached!
% 2.92/1.56 % (22632)------------------------------
% 2.92/1.56 % (22632)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.56 % (22632)Termination reason: Unknown
% 2.92/1.56 % (22632)Termination phase: Property scanning
% 2.92/1.56
% 2.92/1.56 % (22632)Memory used [KB]: 1663
% 2.92/1.56 % (22632)Time elapsed: 0.007 s
% 2.92/1.56 % (22632)Instructions burned: 18 (million)
% 2.92/1.56 % (22632)------------------------------
% 2.92/1.56 % (22632)------------------------------
% 2.92/1.56 % (22626)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22527 for (2999ds/183Mi)
% 2.92/1.56 % (22628)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_22527 for (2999ds/27Mi)
% 2.92/1.56 % (22627)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_22527 for (2999ds/4Mi)
% 2.92/1.56 % (22631)First to succeed.
% 2.92/1.57 % (22631)Refutation found. Thanks to Tanya!
% 2.92/1.57 % SZS status Theorem for DTF2THF_22527
% 2.92/1.57 % SZS output start Proof for DTF2THF_22527
% 2.92/1.57 thf(type_def_5, type, nat: $tType).
% 2.92/1.57 thf(type_def_6, type, extended_enat: $tType).
% 2.92/1.57 thf(type_def_8, type, extended_ereal: $tType).
% 2.92/1.57 thf(type_def_9, type, real: $tType).
% 2.92/1.57 thf(type_def_10, type, num: $tType).
% 2.92/1.57 thf(type_def_11, type, set: ($tType) > $tType).
% 2.92/1.57 thf(type_def_12, type, a: $tType).
% 2.92/1.57 thf(func_def_0, type, extended_ereal: $tType).
% 2.92/1.57 thf(func_def_1, type, extended_enat: $tType).
% 2.92/1.57 thf(func_def_2, type, real: $tType).
% 2.92/1.57 thf(func_def_3, type, set: ($tType) > $tType).
% 2.92/1.57 thf(func_def_4, type, num: $tType).
% 2.92/1.57 thf(func_def_5, type, nat: $tType).
% 2.92/1.57 thf(func_def_6, type, a: $tType).
% 2.92/1.57 thf(func_def_7, type, type: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_8, type, zero: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_9, type, numeral: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_10, type, uminus: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_11, type, ord: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_12, type, ring_char_0: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_13, type, neg_numeral: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_14, type, order: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_15, type, group_add: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_16, type, linorder: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_17, type, preorder: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_18, type, semiring_char_0: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_19, type, linordered_idom: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_20, type, linordered_field: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_21, type, boolean_algebra: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_22, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_23, type, linord219039673up_add: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_24, type, ordere142940540dd_abs: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_25, type, real_V2090557954_space: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_26, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_27, type, topolo259154727pology: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_28, type, topolo503727757_space: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_29, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.92/1.57 thf(func_def_30, type, extended_enat2: nat > extended_enat).
% 2.92/1.57 thf(func_def_31, type, extended_case_enat: !>[X0: $tType]:((nat > X0) > X0 > extended_enat > X0)).
% 2.92/1.57 thf(func_def_32, type, extended_rec_enat: !>[X0: $tType]:((nat > X0) > X0 > extended_enat > X0)).
% 2.92/1.57 thf(func_def_33, type, extend1396239628finity: !>[X0: $tType]:(X0)).
% 2.92/1.57 thf(func_def_34, type, extended_the_enat: extended_enat > nat).
% 2.92/1.57 thf(func_def_35, type, extended_MInfty: extended_ereal).
% 2.92/1.57 thf(func_def_36, type, extended_PInfty: extended_ereal).
% 2.92/1.57 thf(func_def_37, type, extended_ereal2: real > extended_ereal).
% 2.92/1.57 thf(func_def_38, type, extended_size_ereal: extended_ereal > nat).
% 2.92/1.57 thf(func_def_39, type, extend1771934483f_enat: extended_enat > extended_ereal).
% 2.92/1.57 thf(func_def_40, type, extend1716541707_ereal: extended_ereal > real).
% 2.92/1.57 thf(func_def_41, type, abs_abs: !>[X0: $tType]:(X0 > X0)).
% 2.92/1.57 thf(func_def_42, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 2.92/1.57 thf(func_def_43, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.92/1.57 thf(func_def_44, type, lower_107104146le_lsc: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 2.92/1.57 thf(func_def_45, type, lower_582600101lsc_at: !>[X0: $tType, X1: $tType]:(X0 > (X0 > X1) > $o)).
% 2.92/1.57 thf(func_def_46, type, lower_1879122655c_hull: !>[X0: $tType]:((X0 > extended_ereal) > X0 > extended_ereal)).
% 2.92/1.57 thf(func_def_47, type, neg_numeral_dbl: !>[X0: $tType]:(X0 > X0)).
% 2.92/1.57 thf(func_def_48, type, neg_numeral_sub: !>[X0: $tType]:(num > num > X0)).
% 2.92/1.57 thf(func_def_49, type, numeral_numeral: !>[X0: $tType]:(num > X0)).
% 2.92/1.57 thf(func_def_50, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.92/1.57 thf(func_def_51, type, order_Greatest: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.92/1.57 thf(func_def_52, type, order_antimono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 2.92/1.57 thf(func_def_53, type, order_strict_mono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 2.92/1.57 thf(func_def_54, type, divide_divide: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.92/1.57 thf(func_def_55, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.92/1.57 thf(func_def_56, type, f: a > extended_ereal).
% 2.92/1.57 thf(func_def_57, type, g: a > extended_ereal).
% 2.92/1.57 thf(func_def_58, type, x: a).
% 2.92/1.57 thf(func_def_72, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.92/1.57 thf(func_def_74, type, ph2: !>[X0: $tType]:(X0)).
% 2.92/1.57 thf(f875,plain,(
% 2.92/1.57 $false),
% 2.92/1.57 inference(subsumption_resolution,[status(thm)],[f867,f859])).
% 2.92/1.57 thf(f859,plain,(
% 2.92/1.57 ( ! [X0 : a] : (($true = (ord_less_eq @ extended_ereal @ (g @ X0) @ (f @ X0)))) )),
% 2.92/1.57 inference(cnf_transformation,[status(thm)],[f701])).
% 2.92/1.57 thf(f701,plain,(
% 2.92/1.57 ! [X0 : a] : ($true = (ord_less_eq @ extended_ereal @ (g @ X0) @ (f @ X0)))),
% 2.92/1.57 inference(fool_elimination,[status(thm)],[f700])).
% 2.92/1.57 thf(f700,plain,(
% 2.92/1.57 ! [X0 : a] : (ord_less_eq @ extended_ereal @ (g @ X0) @ (f @ X0))),
% 2.92/1.57 inference(rectify,[status(thm)],[f1])).
% 2.92/1.57 thf(f1,axiom,(
% 2.92/1.57 ! [X3 : a] : (ord_less_eq @ extended_ereal @ (g @ X3) @ (f @ X3))),
% 2.92/1.57 file('/export/starexec/sandbox/tmp/tmp.pLGO0LFU7P/DTF2THF_22527.p',fact_0_assms)).
% 2.92/1.57 thf(f867,plain,(
% 2.92/1.57 ((ord_less_eq @ extended_ereal @ (g @ x) @ (f @ x)) != $true)),
% 2.92/1.57 inference(cnf_transformation,[status(thm)],[f846])).
% 2.92/1.57 thf(f846,plain,(
% 2.92/1.57 ((ord_less_eq @ extended_ereal @ (g @ x) @ (f @ x)) != $true)),
% 2.92/1.57 inference(flattening,[status(thm)],[f729])).
% 2.92/1.57 thf(f729,plain,(
% 2.92/1.57 ~((ord_less_eq @ extended_ereal @ (g @ x) @ (f @ x)) = $true)),
% 2.92/1.57 inference(fool_elimination,[status(thm)],[f728])).
% 2.92/1.57 thf(f728,plain,(
% 2.92/1.57 ~(ord_less_eq @ extended_ereal @ (g @ x) @ (f @ x))),
% 2.92/1.57 inference(rectify,[status(thm)],[f326])).
% 2.92/1.57 thf(f326,negated_conjecture,(
% 2.92/1.57 ~(ord_less_eq @ extended_ereal @ (g @ x) @ (f @ x))),
% 2.92/1.57 inference(negated_conjecture,[status(cth)],[f325])).
% 2.92/1.57 thf(f325,conjecture,(
% 2.92/1.57 (ord_less_eq @ extended_ereal @ (g @ x) @ (f @ x))),
% 2.92/1.57 file('/export/starexec/sandbox/tmp/tmp.pLGO0LFU7P/DTF2THF_22527.p',conj_0)).
% 2.92/1.57 % SZS output end Proof for DTF2THF_22527
% 2.92/1.57 % (22631)------------------------------
% 2.92/1.57 % (22631)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.57 % (22631)Termination reason: Refutation
% 2.92/1.57
% 2.92/1.57 % (22631)Memory used [KB]: 6140
% 2.92/1.57 % (22631)Time elapsed: 0.010 s
% 2.92/1.57 % (22631)Instructions burned: 25 (million)
% 2.92/1.57 % (22631)------------------------------
% 2.92/1.57 % (22631)------------------------------
% 2.92/1.57 % (22625)Success in time 0.022 s
%------------------------------------------------------------------------------