TPTP Problem File: ITP110^1.p

View Solutions - Solve Problem

%------------------------------------------------------------------------------
% File     : ITP110^1 : TPTP v9.0.0. Released v7.5.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer Lower_Semicontinuous problem prob_1321__6263368_1
% Version  : Especial.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source   : [Des21]
% Names    : Lower_Semicontinuous/prob_1321__6263368_1 [Des21]

% Status   : Theorem
% Rating   : 0.38 v9.0.0, 0.40 v8.2.0, 0.46 v8.1.0, 0.55 v7.5.0
% Syntax   : Number of formulae    :  441 ( 181 unt;  86 typ;   0 def)
%            Number of atoms       : 1033 ( 426 equ;   0 cnn)
%            Maximal formula atoms :   81 (   2 avg)
%            Number of connectives : 2543 ( 109   ~;  19   |;  26   &;1862   @)
%                                         (   0 <=>; 527  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   34 (   6 avg)
%            Number of types       :   17 (  16 usr)
%            Number of type conns  :  177 ( 177   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   71 (  70 usr;  17 con; 0-2 aty)
%            Number of variables   :  862 (  72   ^; 756   !;  34   ?; 862   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Sledgehammer 2021-02-23 15:44:43.435
%------------------------------------------------------------------------------
% Could-be-implicit typings (16)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Real__Oreal_J_J,type,
    set_Pr1158285382l_real: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Extended____Real__Oereal_Mt__Real__Oreal_J_J,type,
    set_Pr597221087l_real: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J_J,type,
    set_se1849684334_ereal: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
    set_Pr1928503567a_real: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J,type,
    set_se767749006_ereal: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Extended____Real__Oereal_M_Eo_J_J,type,
    set_Extended_ereal_o: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
    set_set_set_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
    product_prod_a_real: $tType ).

thf(ty_n_t__Set__Oset_It__Extended____Real__Oereal_J,type,
    set_Extended_ereal: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    set_set_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
    set_a_o: $tType ).

thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
    set_real: $tType ).

thf(ty_n_t__Extended____Real__Oereal,type,
    extended_ereal: $tType ).

thf(ty_n_t__Set__Oset_Itf__a_J,type,
    set_a: $tType ).

thf(ty_n_t__Real__Oreal,type,
    real: $tType ).

thf(ty_n_tf__a,type,
    a: $tType ).

% Explicit typings (70)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Extended____Real__Oereal_M_Eo_J,type,
    comple179807490real_o: set_Extended_ereal_o > extended_ereal > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_Itf__a_M_Eo_J,type,
    complete_Sup_Sup_a_o: set_a_o > a > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Extended____Real__Oereal,type,
    comple1161760187_ereal: set_Extended_ereal > extended_ereal ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
    comple2129349247p_real: set_real > real ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
    comple767846299_ereal: set_se767749006_ereal > set_Extended_ereal ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J,type,
    comple1517911419_ereal: set_se1849684334_ereal > set_se767749006_ereal ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    comple968993579_set_a: set_set_set_a > set_set_a ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
    comple1766734283_set_a: set_set_a > set_a ).

thf(sy_c_Convex_Oconvex_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
    convex291263275l_real: set_Pr1158285382l_real > $o ).

thf(sy_c_Convex_Oconvex_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
    convex78185044a_real: set_Pr1928503567a_real > $o ).

thf(sy_c_Convex_Oconvex_001tf__a,type,
    convex_a: set_a > $o ).

thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Real__Oereal,type,
    extend1289208545_ereal: extended_ereal ).

thf(sy_c_Extended__Real_Oereal_OMInfty,type,
    extended_MInfty: extended_ereal ).

thf(sy_c_Extended__Real_Oereal_OPInfty,type,
    extended_PInfty: extended_ereal ).

thf(sy_c_Extended__Real_Oereal_Oereal,type,
    extended_ereal2: real > extended_ereal ).

thf(sy_c_Extended__Real_Oreal__of__ereal,type,
    extend1716541707_ereal: extended_ereal > real ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Real__Oereal,type,
    times_1966848393_ereal: extended_ereal > extended_ereal > extended_ereal ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
    times_times_real: real > real > real ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Extended____Real__Oereal,type,
    uminus1208298309_ereal: extended_ereal > extended_ereal ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
    uminus_uminus_real: real > real ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
    uminus360668453_ereal: set_Extended_ereal > set_Extended_ereal ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
    uminus_uminus_set_a: set_a > set_a ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Real__Oereal,type,
    zero_z163181189_ereal: extended_ereal ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
    zero_zero_real: real ).

thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_OEpigraph_001t__Extended____Real__Oereal,type,
    lower_331963542_ereal: set_Extended_ereal > ( extended_ereal > extended_ereal ) > set_Pr597221087l_real ).

thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_OEpigraph_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
    lower_870348689a_real: set_Pr1928503567a_real > ( product_prod_a_real > extended_ereal ) > set_Pr1158285382l_real ).

thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_OEpigraph_001tf__a,type,
    lower_930854854raph_a: set_a > ( a > extended_ereal ) > set_Pr1928503567a_real ).

thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Oaffine__on_001tf__a,type,
    lower_128423320e_on_a: set_a > ( a > extended_ereal ) > $o ).

thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Oconcave__on_001tf__a,type,
    lower_1023700310e_on_a: set_a > ( a > extended_ereal ) > $o ).

thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Oconvex__on_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
    lower_591730471a_real: set_Pr1928503567a_real > ( product_prod_a_real > extended_ereal ) > $o ).

thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Oconvex__on_001tf__a,type,
    lower_311861424x_on_a: set_a > ( a > extended_ereal ) > $o ).

thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Odomain_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
    lower_1359521925a_real: ( product_prod_a_real > extended_ereal ) > set_Pr1928503567a_real ).

thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Odomain_001tf__a,type,
    lower_1391529426main_a: ( a > extended_ereal ) > set_a ).

thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ofinite__on_001tf__a,type,
    lower_1486931688e_on_a: set_a > ( a > extended_ereal ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Real__Oereal,type,
    ord_le824540014_ereal: extended_ereal > extended_ereal > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: real > real > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
    ord_le88246606_ereal: set_Extended_ereal > set_Extended_ereal > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
    ord_le1586073967a_real: set_Pr1928503567a_real > set_Pr1928503567a_real > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_eq_set_real: set_real > set_real > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J,type,
    ord_le1153389358_ereal: set_se767749006_ereal > set_se767749006_ereal > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    ord_le318720350_set_a: set_set_a > set_set_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
    ord_less_eq_set_a: set_a > set_a > $o ).

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Extended____Real__Oereal,type,
    order_1158471719_ereal: ( extended_ereal > $o ) > extended_ereal ).

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal,type,
    order_Greatest_real: ( real > $o ) > real ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Extended____Real__Oereal_M_Eo_J,type,
    top_to398855007real_o: extended_ereal > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
    top_top_a_o: a > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
    top_top_o: $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Extended____Real__Oereal,type,
    top_to802031902_ereal: extended_ereal ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Extended____Real__Oereal_M_Eo_J_J,type,
    top_to772344469real_o: set_Extended_ereal_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
    top_top_set_a_o: set_a_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
    top_to1767659262_ereal: set_Extended_ereal ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
    top_to2138011583a_real: set_Pr1928503567a_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J,type,
    top_to692740318_ereal: set_se767749006_ereal ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J_J,type,
    top_to1791633086_ereal: set_se1849684334_ereal ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
    top_to1486684270_set_a: set_set_set_a ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    top_top_set_set_a: set_set_a ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
    top_top_set_a: set_a ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Extended____Real__Oereal,type,
    divide595620860_ereal: extended_ereal > extended_ereal > extended_ereal ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
    divide_divide_real: real > real > real ).

thf(sy_c_Set_OCollect_001t__Extended____Real__Oereal,type,
    collec247695033_ereal: ( extended_ereal > $o ) > set_Extended_ereal ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
    collec1714955950a_real: ( product_prod_a_real > $o ) > set_Pr1928503567a_real ).

thf(sy_c_Set_OCollect_001tf__a,type,
    collect_a: ( a > $o ) > set_a ).

thf(sy_c_member_001t__Extended____Real__Oereal,type,
    member1900190071_ereal: extended_ereal > set_Extended_ereal > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
    member1103263856a_real: product_prod_a_real > set_Pr1928503567a_real > $o ).

thf(sy_c_member_001t__Real__Oreal,type,
    member_real: real > set_real > $o ).

thf(sy_c_member_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
    member450560855_ereal: set_Extended_ereal > set_se767749006_ereal > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
    member_set_a: set_a > set_set_a > $o ).

thf(sy_c_member_001tf__a,type,
    member_a: a > set_a > $o ).

thf(sy_v_f,type,
    f: a > extended_ereal ).

% Relevant facts (354)
thf(fact_0__092_060open_062convex_A_Idomain_Af_J_A_092_060Longrightarrow_062_ALower__Semicontinuous__Mirabelle__mxyexokbxt_Oconvex__on_AUNIV_Af_092_060close_062,axiom,
    ( ( convex_a @ ( lower_1391529426main_a @ f ) )
   => ( lower_311861424x_on_a @ top_top_set_a @ f ) ) ).

% \<open>convex (domain f) \<Longrightarrow> Lower_Semicontinuous_Mirabelle_mxyexokbxt.convex_on UNIV f\<close>
thf(fact_1_convex__domain,axiom,
    ! [F: product_prod_a_real > extended_ereal] :
      ( ( lower_591730471a_real @ top_to2138011583a_real @ F )
     => ( convex78185044a_real @ ( lower_1359521925a_real @ F ) ) ) ).

% convex_domain
thf(fact_2_convex__domain,axiom,
    ! [F: a > extended_ereal] :
      ( ( lower_311861424x_on_a @ top_top_set_a @ F )
     => ( convex_a @ ( lower_1391529426main_a @ F ) ) ) ).

% convex_domain
thf(fact_3_convex__on__domain,axiom,
    ! [F: a > extended_ereal] :
      ( ( lower_311861424x_on_a @ ( lower_1391529426main_a @ F ) @ F )
      = ( lower_311861424x_on_a @ top_top_set_a @ F ) ) ).

% convex_on_domain
thf(fact_4_convex__on__domain2,axiom,
    ! [F: a > extended_ereal] :
      ( ( lower_311861424x_on_a @ ( lower_1391529426main_a @ F ) @ F )
      = ( ! [S: set_a] : ( lower_311861424x_on_a @ S @ F ) ) ) ).

% convex_on_domain2
thf(fact_5_convex__on__ereal__univ,axiom,
    ! [F: a > extended_ereal] :
      ( ( lower_311861424x_on_a @ top_top_set_a @ F )
      = ( ! [S: set_a] : ( lower_311861424x_on_a @ S @ F ) ) ) ).

% convex_on_ereal_univ
thf(fact_6_convex__UNIV,axiom,
    convex78185044a_real @ top_to2138011583a_real ).

% convex_UNIV
thf(fact_7_convex__UNIV,axiom,
    convex_a @ top_top_set_a ).

% convex_UNIV
thf(fact_8_UNIV__I,axiom,
    ! [X: real] : ( member_real @ X @ top_top_set_real ) ).

% UNIV_I
thf(fact_9_UNIV__I,axiom,
    ! [X: set_Extended_ereal] : ( member450560855_ereal @ X @ top_to692740318_ereal ) ).

% UNIV_I
thf(fact_10_UNIV__I,axiom,
    ! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).

% UNIV_I
thf(fact_11_UNIV__I,axiom,
    ! [X: extended_ereal] : ( member1900190071_ereal @ X @ top_to1767659262_ereal ) ).

% UNIV_I
thf(fact_12_UNIV__I,axiom,
    ! [X: a] : ( member_a @ X @ top_top_set_a ) ).

% UNIV_I
thf(fact_13_iso__tuple__UNIV__I,axiom,
    ! [X: real] : ( member_real @ X @ top_top_set_real ) ).

% iso_tuple_UNIV_I
thf(fact_14_iso__tuple__UNIV__I,axiom,
    ! [X: set_Extended_ereal] : ( member450560855_ereal @ X @ top_to692740318_ereal ) ).

% iso_tuple_UNIV_I
thf(fact_15_iso__tuple__UNIV__I,axiom,
    ! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).

% iso_tuple_UNIV_I
thf(fact_16_iso__tuple__UNIV__I,axiom,
    ! [X: extended_ereal] : ( member1900190071_ereal @ X @ top_to1767659262_ereal ) ).

% iso_tuple_UNIV_I
thf(fact_17_iso__tuple__UNIV__I,axiom,
    ! [X: a] : ( member_a @ X @ top_top_set_a ) ).

% iso_tuple_UNIV_I
thf(fact_18_top__apply,axiom,
    ( top_to398855007real_o
    = ( ^ [X2: extended_ereal] : top_top_o ) ) ).

% top_apply
thf(fact_19_top__apply,axiom,
    ( top_top_a_o
    = ( ^ [X2: a] : top_top_o ) ) ).

% top_apply
thf(fact_20_assms,axiom,
    ! [X3: a] :
      ( ( ( f @ X3 )
        = extend1289208545_ereal )
      | ( ( f @ X3 )
        = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).

% assms
thf(fact_21_UNIV__eq__I,axiom,
    ! [A: set_real] :
      ( ! [X4: real] : ( member_real @ X4 @ A )
     => ( top_top_set_real = A ) ) ).

% UNIV_eq_I
thf(fact_22_UNIV__eq__I,axiom,
    ! [A: set_se767749006_ereal] :
      ( ! [X4: set_Extended_ereal] : ( member450560855_ereal @ X4 @ A )
     => ( top_to692740318_ereal = A ) ) ).

% UNIV_eq_I
thf(fact_23_UNIV__eq__I,axiom,
    ! [A: set_set_a] :
      ( ! [X4: set_a] : ( member_set_a @ X4 @ A )
     => ( top_top_set_set_a = A ) ) ).

% UNIV_eq_I
thf(fact_24_UNIV__eq__I,axiom,
    ! [A: set_a] :
      ( ! [X4: a] : ( member_a @ X4 @ A )
     => ( top_top_set_a = A ) ) ).

% UNIV_eq_I
thf(fact_25_UNIV__eq__I,axiom,
    ! [A: set_Extended_ereal] :
      ( ! [X4: extended_ereal] : ( member1900190071_ereal @ X4 @ A )
     => ( top_to1767659262_ereal = A ) ) ).

% UNIV_eq_I
thf(fact_26_UNIV__witness,axiom,
    ? [X4: real] : ( member_real @ X4 @ top_top_set_real ) ).

% UNIV_witness
thf(fact_27_UNIV__witness,axiom,
    ? [X4: set_Extended_ereal] : ( member450560855_ereal @ X4 @ top_to692740318_ereal ) ).

% UNIV_witness
thf(fact_28_UNIV__witness,axiom,
    ? [X4: set_a] : ( member_set_a @ X4 @ top_top_set_set_a ) ).

% UNIV_witness
thf(fact_29_UNIV__witness,axiom,
    ? [X4: a] : ( member_a @ X4 @ top_top_set_a ) ).

% UNIV_witness
thf(fact_30_UNIV__witness,axiom,
    ? [X4: extended_ereal] : ( member1900190071_ereal @ X4 @ top_to1767659262_ereal ) ).

% UNIV_witness
thf(fact_31_convex__Epigraph,axiom,
    ! [S2: set_Pr1928503567a_real,F: product_prod_a_real > extended_ereal] :
      ( ( convex78185044a_real @ S2 )
     => ( ( convex291263275l_real @ ( lower_870348689a_real @ S2 @ F ) )
        = ( lower_591730471a_real @ S2 @ F ) ) ) ).

% convex_Epigraph
thf(fact_32_convex__Epigraph,axiom,
    ! [S2: set_a,F: a > extended_ereal] :
      ( ( convex_a @ S2 )
     => ( ( convex78185044a_real @ ( lower_930854854raph_a @ S2 @ F ) )
        = ( lower_311861424x_on_a @ S2 @ F ) ) ) ).

% convex_Epigraph
thf(fact_33_convex__EpigraphI,axiom,
    ! [S3: set_Pr1928503567a_real,F: product_prod_a_real > extended_ereal] :
      ( ( lower_591730471a_real @ S3 @ F )
     => ( ( convex78185044a_real @ S3 )
       => ( convex291263275l_real @ ( lower_870348689a_real @ S3 @ F ) ) ) ) ).

% convex_EpigraphI
thf(fact_34_convex__EpigraphI,axiom,
    ! [S3: set_a,F: a > extended_ereal] :
      ( ( lower_311861424x_on_a @ S3 @ F )
     => ( ( convex_a @ S3 )
       => ( convex78185044a_real @ ( lower_930854854raph_a @ S3 @ F ) ) ) ) ).

% convex_EpigraphI
thf(fact_35_affine__on__def,axiom,
    ( lower_128423320e_on_a
    = ( ^ [S: set_a,F2: a > extended_ereal] :
          ( ( lower_311861424x_on_a @ S @ F2 )
          & ( lower_1023700310e_on_a @ S @ F2 )
          & ( lower_1486931688e_on_a @ S @ F2 ) ) ) ) ).

% affine_on_def
thf(fact_36_Sup__UNIV,axiom,
    ( ( comple1517911419_ereal @ top_to1791633086_ereal )
    = top_to692740318_ereal ) ).

% Sup_UNIV
thf(fact_37_Sup__UNIV,axiom,
    ( ( comple968993579_set_a @ top_to1486684270_set_a )
    = top_top_set_set_a ) ).

% Sup_UNIV
thf(fact_38_Sup__UNIV,axiom,
    ( ( comple179807490real_o @ top_to772344469real_o )
    = top_to398855007real_o ) ).

% Sup_UNIV
thf(fact_39_Sup__UNIV,axiom,
    ( ( complete_Sup_Sup_a_o @ top_top_set_a_o )
    = top_top_a_o ) ).

% Sup_UNIV
thf(fact_40_Sup__UNIV,axiom,
    ( ( comple1766734283_set_a @ top_top_set_set_a )
    = top_top_set_a ) ).

% Sup_UNIV
thf(fact_41_Sup__UNIV,axiom,
    ( ( comple767846299_ereal @ top_to692740318_ereal )
    = top_to1767659262_ereal ) ).

% Sup_UNIV
thf(fact_42_Sup__UNIV,axiom,
    ( ( comple1161760187_ereal @ top_to1767659262_ereal )
    = top_to802031902_ereal ) ).

% Sup_UNIV
thf(fact_43_convex__on__ereal__subset,axiom,
    ! [T: set_Pr1928503567a_real,F: product_prod_a_real > extended_ereal,S3: set_Pr1928503567a_real] :
      ( ( lower_591730471a_real @ T @ F )
     => ( ( ord_le1586073967a_real @ S3 @ T )
       => ( lower_591730471a_real @ S3 @ F ) ) ) ).

% convex_on_ereal_subset
thf(fact_44_convex__on__ereal__subset,axiom,
    ! [T: set_a,F: a > extended_ereal,S3: set_a] :
      ( ( lower_311861424x_on_a @ T @ F )
     => ( ( ord_less_eq_set_a @ S3 @ T )
       => ( lower_311861424x_on_a @ S3 @ F ) ) ) ).

% convex_on_ereal_subset
thf(fact_45_order__refl,axiom,
    ! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).

% order_refl
thf(fact_46_order__refl,axiom,
    ! [X: set_Extended_ereal] : ( ord_le88246606_ereal @ X @ X ) ).

% order_refl
thf(fact_47_order__refl,axiom,
    ! [X: set_Pr1928503567a_real] : ( ord_le1586073967a_real @ X @ X ) ).

% order_refl
thf(fact_48_order__refl,axiom,
    ! [X: extended_ereal] : ( ord_le824540014_ereal @ X @ X ) ).

% order_refl
thf(fact_49_order__refl,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ X ) ).

% order_refl
thf(fact_50_subsetI,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ! [X4: set_a] :
          ( ( member_set_a @ X4 @ A )
         => ( member_set_a @ X4 @ B ) )
     => ( ord_le318720350_set_a @ A @ B ) ) ).

% subsetI
thf(fact_51_subsetI,axiom,
    ! [A: set_se767749006_ereal,B: set_se767749006_ereal] :
      ( ! [X4: set_Extended_ereal] :
          ( ( member450560855_ereal @ X4 @ A )
         => ( member450560855_ereal @ X4 @ B ) )
     => ( ord_le1153389358_ereal @ A @ B ) ) ).

% subsetI
thf(fact_52_subsetI,axiom,
    ! [A: set_real,B: set_real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A )
         => ( member_real @ X4 @ B ) )
     => ( ord_less_eq_set_real @ A @ B ) ) ).

% subsetI
thf(fact_53_subsetI,axiom,
    ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real] :
      ( ! [X4: product_prod_a_real] :
          ( ( member1103263856a_real @ X4 @ A )
         => ( member1103263856a_real @ X4 @ B ) )
     => ( ord_le1586073967a_real @ A @ B ) ) ).

% subsetI
thf(fact_54_subsetI,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ! [X4: extended_ereal] :
          ( ( member1900190071_ereal @ X4 @ A )
         => ( member1900190071_ereal @ X4 @ B ) )
     => ( ord_le88246606_ereal @ A @ B ) ) ).

% subsetI
thf(fact_55_subsetI,axiom,
    ! [A: set_a,B: set_a] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ A )
         => ( member_a @ X4 @ B ) )
     => ( ord_less_eq_set_a @ A @ B ) ) ).

% subsetI
thf(fact_56_subset__antisym,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_57_subset__antisym,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( ord_le88246606_ereal @ A @ B )
     => ( ( ord_le88246606_ereal @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_58_subset__antisym,axiom,
    ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real] :
      ( ( ord_le1586073967a_real @ A @ B )
     => ( ( ord_le1586073967a_real @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_59_in__mono,axiom,
    ! [A: set_set_a,B: set_set_a,X: set_a] :
      ( ( ord_le318720350_set_a @ A @ B )
     => ( ( member_set_a @ X @ A )
       => ( member_set_a @ X @ B ) ) ) ).

% in_mono
thf(fact_60_in__mono,axiom,
    ! [A: set_se767749006_ereal,B: set_se767749006_ereal,X: set_Extended_ereal] :
      ( ( ord_le1153389358_ereal @ A @ B )
     => ( ( member450560855_ereal @ X @ A )
       => ( member450560855_ereal @ X @ B ) ) ) ).

% in_mono
thf(fact_61_in__mono,axiom,
    ! [A: set_real,B: set_real,X: real] :
      ( ( ord_less_eq_set_real @ A @ B )
     => ( ( member_real @ X @ A )
       => ( member_real @ X @ B ) ) ) ).

% in_mono
thf(fact_62_in__mono,axiom,
    ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real,X: product_prod_a_real] :
      ( ( ord_le1586073967a_real @ A @ B )
     => ( ( member1103263856a_real @ X @ A )
       => ( member1103263856a_real @ X @ B ) ) ) ).

% in_mono
thf(fact_63_in__mono,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal,X: extended_ereal] :
      ( ( ord_le88246606_ereal @ A @ B )
     => ( ( member1900190071_ereal @ X @ A )
       => ( member1900190071_ereal @ X @ B ) ) ) ).

% in_mono
thf(fact_64_in__mono,axiom,
    ! [A: set_a,B: set_a,X: a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( member_a @ X @ A )
       => ( member_a @ X @ B ) ) ) ).

% in_mono
thf(fact_65_subsetD,axiom,
    ! [A: set_set_a,B: set_set_a,C: set_a] :
      ( ( ord_le318720350_set_a @ A @ B )
     => ( ( member_set_a @ C @ A )
       => ( member_set_a @ C @ B ) ) ) ).

% subsetD
thf(fact_66_subsetD,axiom,
    ! [A: set_se767749006_ereal,B: set_se767749006_ereal,C: set_Extended_ereal] :
      ( ( ord_le1153389358_ereal @ A @ B )
     => ( ( member450560855_ereal @ C @ A )
       => ( member450560855_ereal @ C @ B ) ) ) ).

% subsetD
thf(fact_67_subsetD,axiom,
    ! [A: set_real,B: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A @ B )
     => ( ( member_real @ C @ A )
       => ( member_real @ C @ B ) ) ) ).

% subsetD
thf(fact_68_subsetD,axiom,
    ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real,C: product_prod_a_real] :
      ( ( ord_le1586073967a_real @ A @ B )
     => ( ( member1103263856a_real @ C @ A )
       => ( member1103263856a_real @ C @ B ) ) ) ).

% subsetD
thf(fact_69_subsetD,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal,C: extended_ereal] :
      ( ( ord_le88246606_ereal @ A @ B )
     => ( ( member1900190071_ereal @ C @ A )
       => ( member1900190071_ereal @ C @ B ) ) ) ).

% subsetD
thf(fact_70_subsetD,axiom,
    ! [A: set_a,B: set_a,C: a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( member_a @ C @ A )
       => ( member_a @ C @ B ) ) ) ).

% subsetD
thf(fact_71_equalityE,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ~ ( ( ord_less_eq_set_a @ A @ B )
         => ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).

% equalityE
thf(fact_72_equalityE,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( A = B )
     => ~ ( ( ord_le88246606_ereal @ A @ B )
         => ~ ( ord_le88246606_ereal @ B @ A ) ) ) ).

% equalityE
thf(fact_73_equalityE,axiom,
    ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real] :
      ( ( A = B )
     => ~ ( ( ord_le1586073967a_real @ A @ B )
         => ~ ( ord_le1586073967a_real @ B @ A ) ) ) ).

% equalityE
thf(fact_74_subset__eq,axiom,
    ( ord_le318720350_set_a
    = ( ^ [A2: set_set_a,B2: set_set_a] :
        ! [X2: set_a] :
          ( ( member_set_a @ X2 @ A2 )
         => ( member_set_a @ X2 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_75_subset__eq,axiom,
    ( ord_le1153389358_ereal
    = ( ^ [A2: set_se767749006_ereal,B2: set_se767749006_ereal] :
        ! [X2: set_Extended_ereal] :
          ( ( member450560855_ereal @ X2 @ A2 )
         => ( member450560855_ereal @ X2 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_76_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A2: set_real,B2: set_real] :
        ! [X2: real] :
          ( ( member_real @ X2 @ A2 )
         => ( member_real @ X2 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_77_subset__eq,axiom,
    ( ord_le1586073967a_real
    = ( ^ [A2: set_Pr1928503567a_real,B2: set_Pr1928503567a_real] :
        ! [X2: product_prod_a_real] :
          ( ( member1103263856a_real @ X2 @ A2 )
         => ( member1103263856a_real @ X2 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_78_subset__eq,axiom,
    ( ord_le88246606_ereal
    = ( ^ [A2: set_Extended_ereal,B2: set_Extended_ereal] :
        ! [X2: extended_ereal] :
          ( ( member1900190071_ereal @ X2 @ A2 )
         => ( member1900190071_ereal @ X2 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_79_subset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A2: set_a,B2: set_a] :
        ! [X2: a] :
          ( ( member_a @ X2 @ A2 )
         => ( member_a @ X2 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_80_equalityD1,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ( ord_less_eq_set_a @ A @ B ) ) ).

% equalityD1
thf(fact_81_equalityD1,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( A = B )
     => ( ord_le88246606_ereal @ A @ B ) ) ).

% equalityD1
thf(fact_82_equalityD1,axiom,
    ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real] :
      ( ( A = B )
     => ( ord_le1586073967a_real @ A @ B ) ) ).

% equalityD1
thf(fact_83_equalityD2,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ( ord_less_eq_set_a @ B @ A ) ) ).

% equalityD2
thf(fact_84_equalityD2,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( A = B )
     => ( ord_le88246606_ereal @ B @ A ) ) ).

% equalityD2
thf(fact_85_equalityD2,axiom,
    ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real] :
      ( ( A = B )
     => ( ord_le1586073967a_real @ B @ A ) ) ).

% equalityD2
thf(fact_86_subset__iff,axiom,
    ( ord_le318720350_set_a
    = ( ^ [A2: set_set_a,B2: set_set_a] :
        ! [T2: set_a] :
          ( ( member_set_a @ T2 @ A2 )
         => ( member_set_a @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_87_subset__iff,axiom,
    ( ord_le1153389358_ereal
    = ( ^ [A2: set_se767749006_ereal,B2: set_se767749006_ereal] :
        ! [T2: set_Extended_ereal] :
          ( ( member450560855_ereal @ T2 @ A2 )
         => ( member450560855_ereal @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_88_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A2: set_real,B2: set_real] :
        ! [T2: real] :
          ( ( member_real @ T2 @ A2 )
         => ( member_real @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_89_subset__iff,axiom,
    ( ord_le1586073967a_real
    = ( ^ [A2: set_Pr1928503567a_real,B2: set_Pr1928503567a_real] :
        ! [T2: product_prod_a_real] :
          ( ( member1103263856a_real @ T2 @ A2 )
         => ( member1103263856a_real @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_90_subset__iff,axiom,
    ( ord_le88246606_ereal
    = ( ^ [A2: set_Extended_ereal,B2: set_Extended_ereal] :
        ! [T2: extended_ereal] :
          ( ( member1900190071_ereal @ T2 @ A2 )
         => ( member1900190071_ereal @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_91_subset__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A2: set_a,B2: set_a] :
        ! [T2: a] :
          ( ( member_a @ T2 @ A2 )
         => ( member_a @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_92_subset__refl,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).

% subset_refl
thf(fact_93_subset__refl,axiom,
    ! [A: set_Extended_ereal] : ( ord_le88246606_ereal @ A @ A ) ).

% subset_refl
thf(fact_94_subset__refl,axiom,
    ! [A: set_Pr1928503567a_real] : ( ord_le1586073967a_real @ A @ A ) ).

% subset_refl
thf(fact_95_Collect__mono,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ! [X4: a] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).

% Collect_mono
thf(fact_96_Collect__mono,axiom,
    ! [P: extended_ereal > $o,Q: extended_ereal > $o] :
      ( ! [X4: extended_ereal] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le88246606_ereal @ ( collec247695033_ereal @ P ) @ ( collec247695033_ereal @ Q ) ) ) ).

% Collect_mono
thf(fact_97_Collect__mono,axiom,
    ! [P: product_prod_a_real > $o,Q: product_prod_a_real > $o] :
      ( ! [X4: product_prod_a_real] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le1586073967a_real @ ( collec1714955950a_real @ P ) @ ( collec1714955950a_real @ Q ) ) ) ).

% Collect_mono
thf(fact_98_subset__trans,axiom,
    ! [A: set_a,B: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ B @ C2 )
       => ( ord_less_eq_set_a @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_99_subset__trans,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal,C2: set_Extended_ereal] :
      ( ( ord_le88246606_ereal @ A @ B )
     => ( ( ord_le88246606_ereal @ B @ C2 )
       => ( ord_le88246606_ereal @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_100_subset__trans,axiom,
    ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real,C2: set_Pr1928503567a_real] :
      ( ( ord_le1586073967a_real @ A @ B )
     => ( ( ord_le1586073967a_real @ B @ C2 )
       => ( ord_le1586073967a_real @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_101_set__eq__subset,axiom,
    ( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
    = ( ^ [A2: set_a,B2: set_a] :
          ( ( ord_less_eq_set_a @ A2 @ B2 )
          & ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).

% set_eq_subset
thf(fact_102_set__eq__subset,axiom,
    ( ( ^ [Y: set_Extended_ereal,Z: set_Extended_ereal] : ( Y = Z ) )
    = ( ^ [A2: set_Extended_ereal,B2: set_Extended_ereal] :
          ( ( ord_le88246606_ereal @ A2 @ B2 )
          & ( ord_le88246606_ereal @ B2 @ A2 ) ) ) ) ).

% set_eq_subset
thf(fact_103_set__eq__subset,axiom,
    ( ( ^ [Y: set_Pr1928503567a_real,Z: set_Pr1928503567a_real] : ( Y = Z ) )
    = ( ^ [A2: set_Pr1928503567a_real,B2: set_Pr1928503567a_real] :
          ( ( ord_le1586073967a_real @ A2 @ B2 )
          & ( ord_le1586073967a_real @ B2 @ A2 ) ) ) ) ).

% set_eq_subset
thf(fact_104_Collect__mono__iff,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
      = ( ! [X2: a] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_105_Collect__mono__iff,axiom,
    ! [P: extended_ereal > $o,Q: extended_ereal > $o] :
      ( ( ord_le88246606_ereal @ ( collec247695033_ereal @ P ) @ ( collec247695033_ereal @ Q ) )
      = ( ! [X2: extended_ereal] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_106_Collect__mono__iff,axiom,
    ! [P: product_prod_a_real > $o,Q: product_prod_a_real > $o] :
      ( ( ord_le1586073967a_real @ ( collec1714955950a_real @ P ) @ ( collec1714955950a_real @ Q ) )
      = ( ! [X2: product_prod_a_real] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_107_order__subst1,axiom,
    ! [A3: extended_ereal,F: extended_ereal > extended_ereal,B3: extended_ereal,C: extended_ereal] :
      ( ( ord_le824540014_ereal @ A3 @ ( F @ B3 ) )
     => ( ( ord_le824540014_ereal @ B3 @ C )
       => ( ! [X4: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le824540014_ereal @ X4 @ Y2 )
             => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_108_order__subst1,axiom,
    ! [A3: extended_ereal,F: real > extended_ereal,B3: real,C: real] :
      ( ( ord_le824540014_ereal @ A3 @ ( F @ B3 ) )
     => ( ( ord_less_eq_real @ B3 @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_109_order__subst1,axiom,
    ! [A3: real,F: extended_ereal > real,B3: extended_ereal,C: extended_ereal] :
      ( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
     => ( ( ord_le824540014_ereal @ B3 @ C )
       => ( ! [X4: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le824540014_ereal @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_110_order__subst1,axiom,
    ! [A3: real,F: real > real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
     => ( ( ord_less_eq_real @ B3 @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_111_order__subst1,axiom,
    ! [A3: extended_ereal,F: set_a > extended_ereal,B3: set_a,C: set_a] :
      ( ( ord_le824540014_ereal @ A3 @ ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_112_order__subst1,axiom,
    ! [A3: extended_ereal,F: set_Extended_ereal > extended_ereal,B3: set_Extended_ereal,C: set_Extended_ereal] :
      ( ( ord_le824540014_ereal @ A3 @ ( F @ B3 ) )
     => ( ( ord_le88246606_ereal @ B3 @ C )
       => ( ! [X4: set_Extended_ereal,Y2: set_Extended_ereal] :
              ( ( ord_le88246606_ereal @ X4 @ Y2 )
             => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_113_order__subst1,axiom,
    ! [A3: real,F: set_a > real,B3: set_a,C: set_a] :
      ( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_114_order__subst1,axiom,
    ! [A3: real,F: set_Extended_ereal > real,B3: set_Extended_ereal,C: set_Extended_ereal] :
      ( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
     => ( ( ord_le88246606_ereal @ B3 @ C )
       => ( ! [X4: set_Extended_ereal,Y2: set_Extended_ereal] :
              ( ( ord_le88246606_ereal @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_115_order__subst1,axiom,
    ! [A3: set_a,F: extended_ereal > set_a,B3: extended_ereal,C: extended_ereal] :
      ( ( ord_less_eq_set_a @ A3 @ ( F @ B3 ) )
     => ( ( ord_le824540014_ereal @ B3 @ C )
       => ( ! [X4: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le824540014_ereal @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_116_order__subst1,axiom,
    ! [A3: set_a,F: real > set_a,B3: real,C: real] :
      ( ( ord_less_eq_set_a @ A3 @ ( F @ B3 ) )
     => ( ( ord_less_eq_real @ B3 @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_117_order__subst2,axiom,
    ! [A3: set_Pr1928503567a_real,B3: set_Pr1928503567a_real,F: set_Pr1928503567a_real > set_a,C: set_a] :
      ( ( ord_le1586073967a_real @ A3 @ B3 )
     => ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
       => ( ! [X4: set_Pr1928503567a_real,Y2: set_Pr1928503567a_real] :
              ( ( ord_le1586073967a_real @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_118_order__subst2,axiom,
    ! [A3: set_Pr1928503567a_real,B3: set_Pr1928503567a_real,F: set_Pr1928503567a_real > set_Extended_ereal,C: set_Extended_ereal] :
      ( ( ord_le1586073967a_real @ A3 @ B3 )
     => ( ( ord_le88246606_ereal @ ( F @ B3 ) @ C )
       => ( ! [X4: set_Pr1928503567a_real,Y2: set_Pr1928503567a_real] :
              ( ( ord_le1586073967a_real @ X4 @ Y2 )
             => ( ord_le88246606_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le88246606_ereal @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_119_order__subst2,axiom,
    ! [A3: set_Pr1928503567a_real,B3: set_Pr1928503567a_real,F: set_Pr1928503567a_real > set_Pr1928503567a_real,C: set_Pr1928503567a_real] :
      ( ( ord_le1586073967a_real @ A3 @ B3 )
     => ( ( ord_le1586073967a_real @ ( F @ B3 ) @ C )
       => ( ! [X4: set_Pr1928503567a_real,Y2: set_Pr1928503567a_real] :
              ( ( ord_le1586073967a_real @ X4 @ Y2 )
             => ( ord_le1586073967a_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le1586073967a_real @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_120_order__subst2,axiom,
    ! [A3: extended_ereal,B3: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
      ( ( ord_le824540014_ereal @ A3 @ B3 )
     => ( ( ord_le824540014_ereal @ ( F @ B3 ) @ C )
       => ( ! [X4: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le824540014_ereal @ X4 @ Y2 )
             => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le824540014_ereal @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_121_order__subst2,axiom,
    ! [A3: extended_ereal,B3: extended_ereal,F: extended_ereal > real,C: real] :
      ( ( ord_le824540014_ereal @ A3 @ B3 )
     => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
       => ( ! [X4: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le824540014_ereal @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_122_order__subst2,axiom,
    ! [A3: real,B3: real,F: real > extended_ereal,C: extended_ereal] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_le824540014_ereal @ ( F @ B3 ) @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le824540014_ereal @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_123_order__subst2,axiom,
    ! [A3: real,B3: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_124_ord__eq__le__subst,axiom,
    ! [A3: extended_ereal,F: extended_ereal > extended_ereal,B3: extended_ereal,C: extended_ereal] :
      ( ( A3
        = ( F @ B3 ) )
     => ( ( ord_le824540014_ereal @ B3 @ C )
       => ( ! [X4: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le824540014_ereal @ X4 @ Y2 )
             => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_125_ord__eq__le__subst,axiom,
    ! [A3: real,F: extended_ereal > real,B3: extended_ereal,C: extended_ereal] :
      ( ( A3
        = ( F @ B3 ) )
     => ( ( ord_le824540014_ereal @ B3 @ C )
       => ( ! [X4: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le824540014_ereal @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_126_ord__eq__le__subst,axiom,
    ! [A3: extended_ereal,F: real > extended_ereal,B3: real,C: real] :
      ( ( A3
        = ( F @ B3 ) )
     => ( ( ord_less_eq_real @ B3 @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_127_ord__eq__le__subst,axiom,
    ! [A3: real,F: real > real,B3: real,C: real] :
      ( ( A3
        = ( F @ B3 ) )
     => ( ( ord_less_eq_real @ B3 @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_128_ord__le__eq__subst,axiom,
    ! [A3: extended_ereal,B3: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
      ( ( ord_le824540014_ereal @ A3 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X4: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le824540014_ereal @ X4 @ Y2 )
             => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le824540014_ereal @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_129_ord__le__eq__subst,axiom,
    ! [A3: extended_ereal,B3: extended_ereal,F: extended_ereal > real,C: real] :
      ( ( ord_le824540014_ereal @ A3 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X4: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le824540014_ereal @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_130_ord__le__eq__subst,axiom,
    ! [A3: real,B3: real,F: real > extended_ereal,C: extended_ereal] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le824540014_ereal @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_131_ord__le__eq__subst,axiom,
    ! [A3: real,B3: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_132_eq__iff,axiom,
    ( ( ^ [Y: extended_ereal,Z: extended_ereal] : ( Y = Z ) )
    = ( ^ [X2: extended_ereal,Y3: extended_ereal] :
          ( ( ord_le824540014_ereal @ X2 @ Y3 )
          & ( ord_le824540014_ereal @ Y3 @ X2 ) ) ) ) ).

% eq_iff
thf(fact_133_eq__iff,axiom,
    ( ( ^ [Y: real,Z: real] : ( Y = Z ) )
    = ( ^ [X2: real,Y3: real] :
          ( ( ord_less_eq_real @ X2 @ Y3 )
          & ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).

% eq_iff
thf(fact_134_antisym,axiom,
    ! [X: extended_ereal,Y4: extended_ereal] :
      ( ( ord_le824540014_ereal @ X @ Y4 )
     => ( ( ord_le824540014_ereal @ Y4 @ X )
       => ( X = Y4 ) ) ) ).

% antisym
thf(fact_135_antisym,axiom,
    ! [X: real,Y4: real] :
      ( ( ord_less_eq_real @ X @ Y4 )
     => ( ( ord_less_eq_real @ Y4 @ X )
       => ( X = Y4 ) ) ) ).

% antisym
thf(fact_136_mem__Collect__eq,axiom,
    ! [A3: extended_ereal,P: extended_ereal > $o] :
      ( ( member1900190071_ereal @ A3 @ ( collec247695033_ereal @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_137_mem__Collect__eq,axiom,
    ! [A3: a,P: a > $o] :
      ( ( member_a @ A3 @ ( collect_a @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_138_Collect__mem__eq,axiom,
    ! [A: set_Extended_ereal] :
      ( ( collec247695033_ereal
        @ ^ [X2: extended_ereal] : ( member1900190071_ereal @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_139_Collect__mem__eq,axiom,
    ! [A: set_a] :
      ( ( collect_a
        @ ^ [X2: a] : ( member_a @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_140_linear,axiom,
    ! [X: extended_ereal,Y4: extended_ereal] :
      ( ( ord_le824540014_ereal @ X @ Y4 )
      | ( ord_le824540014_ereal @ Y4 @ X ) ) ).

% linear
thf(fact_141_linear,axiom,
    ! [X: real,Y4: real] :
      ( ( ord_less_eq_real @ X @ Y4 )
      | ( ord_less_eq_real @ Y4 @ X ) ) ).

% linear
thf(fact_142_eq__refl,axiom,
    ! [X: extended_ereal,Y4: extended_ereal] :
      ( ( X = Y4 )
     => ( ord_le824540014_ereal @ X @ Y4 ) ) ).

% eq_refl
thf(fact_143_eq__refl,axiom,
    ! [X: real,Y4: real] :
      ( ( X = Y4 )
     => ( ord_less_eq_real @ X @ Y4 ) ) ).

% eq_refl
thf(fact_144_le__cases,axiom,
    ! [X: extended_ereal,Y4: extended_ereal] :
      ( ~ ( ord_le824540014_ereal @ X @ Y4 )
     => ( ord_le824540014_ereal @ Y4 @ X ) ) ).

% le_cases
thf(fact_145_le__cases,axiom,
    ! [X: real,Y4: real] :
      ( ~ ( ord_less_eq_real @ X @ Y4 )
     => ( ord_less_eq_real @ Y4 @ X ) ) ).

% le_cases
thf(fact_146_order_Otrans,axiom,
    ! [A3: extended_ereal,B3: extended_ereal,C: extended_ereal] :
      ( ( ord_le824540014_ereal @ A3 @ B3 )
     => ( ( ord_le824540014_ereal @ B3 @ C )
       => ( ord_le824540014_ereal @ A3 @ C ) ) ) ).

% order.trans
thf(fact_147_order_Otrans,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ B3 @ C )
       => ( ord_less_eq_real @ A3 @ C ) ) ) ).

% order.trans
thf(fact_148_le__cases3,axiom,
    ! [X: extended_ereal,Y4: extended_ereal,Z2: extended_ereal] :
      ( ( ( ord_le824540014_ereal @ X @ Y4 )
       => ~ ( ord_le824540014_ereal @ Y4 @ Z2 ) )
     => ( ( ( ord_le824540014_ereal @ Y4 @ X )
         => ~ ( ord_le824540014_ereal @ X @ Z2 ) )
       => ( ( ( ord_le824540014_ereal @ X @ Z2 )
           => ~ ( ord_le824540014_ereal @ Z2 @ Y4 ) )
         => ( ( ( ord_le824540014_ereal @ Z2 @ Y4 )
             => ~ ( ord_le824540014_ereal @ Y4 @ X ) )
           => ( ( ( ord_le824540014_ereal @ Y4 @ Z2 )
               => ~ ( ord_le824540014_ereal @ Z2 @ X ) )
             => ~ ( ( ord_le824540014_ereal @ Z2 @ X )
                 => ~ ( ord_le824540014_ereal @ X @ Y4 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_149_le__cases3,axiom,
    ! [X: real,Y4: real,Z2: real] :
      ( ( ( ord_less_eq_real @ X @ Y4 )
       => ~ ( ord_less_eq_real @ Y4 @ Z2 ) )
     => ( ( ( ord_less_eq_real @ Y4 @ X )
         => ~ ( ord_less_eq_real @ X @ Z2 ) )
       => ( ( ( ord_less_eq_real @ X @ Z2 )
           => ~ ( ord_less_eq_real @ Z2 @ Y4 ) )
         => ( ( ( ord_less_eq_real @ Z2 @ Y4 )
             => ~ ( ord_less_eq_real @ Y4 @ X ) )
           => ( ( ( ord_less_eq_real @ Y4 @ Z2 )
               => ~ ( ord_less_eq_real @ Z2 @ X ) )
             => ~ ( ( ord_less_eq_real @ Z2 @ X )
                 => ~ ( ord_less_eq_real @ X @ Y4 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_150_antisym__conv,axiom,
    ! [Y4: extended_ereal,X: extended_ereal] :
      ( ( ord_le824540014_ereal @ Y4 @ X )
     => ( ( ord_le824540014_ereal @ X @ Y4 )
        = ( X = Y4 ) ) ) ).

% antisym_conv
thf(fact_151_antisym__conv,axiom,
    ! [Y4: real,X: real] :
      ( ( ord_less_eq_real @ Y4 @ X )
     => ( ( ord_less_eq_real @ X @ Y4 )
        = ( X = Y4 ) ) ) ).

% antisym_conv
thf(fact_152_order__class_Oorder_Oeq__iff,axiom,
    ( ( ^ [Y: extended_ereal,Z: extended_ereal] : ( Y = Z ) )
    = ( ^ [A4: extended_ereal,B4: extended_ereal] :
          ( ( ord_le824540014_ereal @ A4 @ B4 )
          & ( ord_le824540014_ereal @ B4 @ A4 ) ) ) ) ).

% order_class.order.eq_iff
thf(fact_153_order__class_Oorder_Oeq__iff,axiom,
    ( ( ^ [Y: real,Z: real] : ( Y = Z ) )
    = ( ^ [A4: real,B4: real] :
          ( ( ord_less_eq_real @ A4 @ B4 )
          & ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).

% order_class.order.eq_iff
thf(fact_154_ord__eq__le__trans,axiom,
    ! [A3: extended_ereal,B3: extended_ereal,C: extended_ereal] :
      ( ( A3 = B3 )
     => ( ( ord_le824540014_ereal @ B3 @ C )
       => ( ord_le824540014_ereal @ A3 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_155_ord__eq__le__trans,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( A3 = B3 )
     => ( ( ord_less_eq_real @ B3 @ C )
       => ( ord_less_eq_real @ A3 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_156_ord__le__eq__trans,axiom,
    ! [A3: extended_ereal,B3: extended_ereal,C: extended_ereal] :
      ( ( ord_le824540014_ereal @ A3 @ B3 )
     => ( ( B3 = C )
       => ( ord_le824540014_ereal @ A3 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_157_ord__le__eq__trans,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( B3 = C )
       => ( ord_less_eq_real @ A3 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_158_order__class_Oorder_Oantisym,axiom,
    ! [A3: extended_ereal,B3: extended_ereal] :
      ( ( ord_le824540014_ereal @ A3 @ B3 )
     => ( ( ord_le824540014_ereal @ B3 @ A3 )
       => ( A3 = B3 ) ) ) ).

% order_class.order.antisym
thf(fact_159_order__class_Oorder_Oantisym,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ B3 @ A3 )
       => ( A3 = B3 ) ) ) ).

% order_class.order.antisym
thf(fact_160_order__trans,axiom,
    ! [X: extended_ereal,Y4: extended_ereal,Z2: extended_ereal] :
      ( ( ord_le824540014_ereal @ X @ Y4 )
     => ( ( ord_le824540014_ereal @ Y4 @ Z2 )
       => ( ord_le824540014_ereal @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_161_order__trans,axiom,
    ! [X: real,Y4: real,Z2: real] :
      ( ( ord_less_eq_real @ X @ Y4 )
     => ( ( ord_less_eq_real @ Y4 @ Z2 )
       => ( ord_less_eq_real @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_162_dual__order_Orefl,axiom,
    ! [A3: extended_ereal] : ( ord_le824540014_ereal @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_163_dual__order_Orefl,axiom,
    ! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_164_linorder__wlog,axiom,
    ! [P: extended_ereal > extended_ereal > $o,A3: extended_ereal,B3: extended_ereal] :
      ( ! [A5: extended_ereal,B5: extended_ereal] :
          ( ( ord_le824540014_ereal @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: extended_ereal,B5: extended_ereal] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A3 @ B3 ) ) ) ).

% linorder_wlog
thf(fact_165_linorder__wlog,axiom,
    ! [P: real > real > $o,A3: real,B3: real] :
      ( ! [A5: real,B5: real] :
          ( ( ord_less_eq_real @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: real,B5: real] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A3 @ B3 ) ) ) ).

% linorder_wlog
thf(fact_166_dual__order_Otrans,axiom,
    ! [B3: extended_ereal,A3: extended_ereal,C: extended_ereal] :
      ( ( ord_le824540014_ereal @ B3 @ A3 )
     => ( ( ord_le824540014_ereal @ C @ B3 )
       => ( ord_le824540014_ereal @ C @ A3 ) ) ) ).

% dual_order.trans
thf(fact_167_dual__order_Otrans,axiom,
    ! [B3: real,A3: real,C: real] :
      ( ( ord_less_eq_real @ B3 @ A3 )
     => ( ( ord_less_eq_real @ C @ B3 )
       => ( ord_less_eq_real @ C @ A3 ) ) ) ).

% dual_order.trans
thf(fact_168_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y: extended_ereal,Z: extended_ereal] : ( Y = Z ) )
    = ( ^ [A4: extended_ereal,B4: extended_ereal] :
          ( ( ord_le824540014_ereal @ B4 @ A4 )
          & ( ord_le824540014_ereal @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_169_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y: real,Z: real] : ( Y = Z ) )
    = ( ^ [A4: real,B4: real] :
          ( ( ord_less_eq_real @ B4 @ A4 )
          & ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_170_dual__order_Oantisym,axiom,
    ! [B3: extended_ereal,A3: extended_ereal] :
      ( ( ord_le824540014_ereal @ B3 @ A3 )
     => ( ( ord_le824540014_ereal @ A3 @ B3 )
       => ( A3 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_171_dual__order_Oantisym,axiom,
    ! [B3: real,A3: real] :
      ( ( ord_less_eq_real @ B3 @ A3 )
     => ( ( ord_less_eq_real @ A3 @ B3 )
       => ( A3 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_172_Sup__eqI,axiom,
    ! [A: set_Extended_ereal,X: extended_ereal] :
      ( ! [Y2: extended_ereal] :
          ( ( member1900190071_ereal @ Y2 @ A )
         => ( ord_le824540014_ereal @ Y2 @ X ) )
     => ( ! [Y2: extended_ereal] :
            ( ! [Z3: extended_ereal] :
                ( ( member1900190071_ereal @ Z3 @ A )
               => ( ord_le824540014_ereal @ Z3 @ Y2 ) )
           => ( ord_le824540014_ereal @ X @ Y2 ) )
       => ( ( comple1161760187_ereal @ A )
          = X ) ) ) ).

% Sup_eqI
thf(fact_173_Sup__mono,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ! [A5: extended_ereal] :
          ( ( member1900190071_ereal @ A5 @ A )
         => ? [X3: extended_ereal] :
              ( ( member1900190071_ereal @ X3 @ B )
              & ( ord_le824540014_ereal @ A5 @ X3 ) ) )
     => ( ord_le824540014_ereal @ ( comple1161760187_ereal @ A ) @ ( comple1161760187_ereal @ B ) ) ) ).

% Sup_mono
thf(fact_174_Sup__least,axiom,
    ! [A: set_Extended_ereal,Z2: extended_ereal] :
      ( ! [X4: extended_ereal] :
          ( ( member1900190071_ereal @ X4 @ A )
         => ( ord_le824540014_ereal @ X4 @ Z2 ) )
     => ( ord_le824540014_ereal @ ( comple1161760187_ereal @ A ) @ Z2 ) ) ).

% Sup_least
thf(fact_175_Sup__upper,axiom,
    ! [X: extended_ereal,A: set_Extended_ereal] :
      ( ( member1900190071_ereal @ X @ A )
     => ( ord_le824540014_ereal @ X @ ( comple1161760187_ereal @ A ) ) ) ).

% Sup_upper
thf(fact_176_Sup__le__iff,axiom,
    ! [A: set_Extended_ereal,B3: extended_ereal] :
      ( ( ord_le824540014_ereal @ ( comple1161760187_ereal @ A ) @ B3 )
      = ( ! [X2: extended_ereal] :
            ( ( member1900190071_ereal @ X2 @ A )
           => ( ord_le824540014_ereal @ X2 @ B3 ) ) ) ) ).

% Sup_le_iff
thf(fact_177_Sup__upper2,axiom,
    ! [U: extended_ereal,A: set_Extended_ereal,V: extended_ereal] :
      ( ( member1900190071_ereal @ U @ A )
     => ( ( ord_le824540014_ereal @ V @ U )
       => ( ord_le824540014_ereal @ V @ ( comple1161760187_ereal @ A ) ) ) ) ).

% Sup_upper2
thf(fact_178_Sup__subset__mono,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( ord_le88246606_ereal @ A @ B )
     => ( ord_le824540014_ereal @ ( comple1161760187_ereal @ A ) @ ( comple1161760187_ereal @ B ) ) ) ).

% Sup_subset_mono
thf(fact_179_top__set__def,axiom,
    ( top_top_set_a
    = ( collect_a @ top_top_a_o ) ) ).

% top_set_def
thf(fact_180_top__set__def,axiom,
    ( top_to1767659262_ereal
    = ( collec247695033_ereal @ top_to398855007real_o ) ) ).

% top_set_def
thf(fact_181_Union__UNIV,axiom,
    ( ( comple1766734283_set_a @ top_top_set_set_a )
    = top_top_set_a ) ).

% Union_UNIV
thf(fact_182_Union__UNIV,axiom,
    ( ( comple767846299_ereal @ top_to692740318_ereal )
    = top_to1767659262_ereal ) ).

% Union_UNIV
thf(fact_183_top_Oextremum__uniqueI,axiom,
    ! [A3: set_a] :
      ( ( ord_less_eq_set_a @ top_top_set_a @ A3 )
     => ( A3 = top_top_set_a ) ) ).

% top.extremum_uniqueI
thf(fact_184_top_Oextremum__uniqueI,axiom,
    ! [A3: set_Extended_ereal] :
      ( ( ord_le88246606_ereal @ top_to1767659262_ereal @ A3 )
     => ( A3 = top_to1767659262_ereal ) ) ).

% top.extremum_uniqueI
thf(fact_185_top_Oextremum__uniqueI,axiom,
    ! [A3: extended_ereal] :
      ( ( ord_le824540014_ereal @ top_to802031902_ereal @ A3 )
     => ( A3 = top_to802031902_ereal ) ) ).

% top.extremum_uniqueI
thf(fact_186_top_Oextremum__unique,axiom,
    ! [A3: set_a] :
      ( ( ord_less_eq_set_a @ top_top_set_a @ A3 )
      = ( A3 = top_top_set_a ) ) ).

% top.extremum_unique
thf(fact_187_top_Oextremum__unique,axiom,
    ! [A3: set_Extended_ereal] :
      ( ( ord_le88246606_ereal @ top_to1767659262_ereal @ A3 )
      = ( A3 = top_to1767659262_ereal ) ) ).

% top.extremum_unique
thf(fact_188_top_Oextremum__unique,axiom,
    ! [A3: extended_ereal] :
      ( ( ord_le824540014_ereal @ top_to802031902_ereal @ A3 )
      = ( A3 = top_to802031902_ereal ) ) ).

% top.extremum_unique
thf(fact_189_top__greatest,axiom,
    ! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ top_top_set_a ) ).

% top_greatest
thf(fact_190_top__greatest,axiom,
    ! [A3: set_Extended_ereal] : ( ord_le88246606_ereal @ A3 @ top_to1767659262_ereal ) ).

% top_greatest
thf(fact_191_top__greatest,axiom,
    ! [A3: extended_ereal] : ( ord_le824540014_ereal @ A3 @ top_to802031902_ereal ) ).

% top_greatest
thf(fact_192_subset__UNIV,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).

% subset_UNIV
thf(fact_193_subset__UNIV,axiom,
    ! [A: set_Extended_ereal] : ( ord_le88246606_ereal @ A @ top_to1767659262_ereal ) ).

% subset_UNIV
thf(fact_194_epigraph__invertible,axiom,
    ! [F: a > extended_ereal,G: a > extended_ereal] :
      ( ( ( lower_930854854raph_a @ top_top_set_a @ F )
        = ( lower_930854854raph_a @ top_top_set_a @ G ) )
     => ( F = G ) ) ).

% epigraph_invertible
thf(fact_195_epigraph__invertible,axiom,
    ! [F: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
      ( ( ( lower_331963542_ereal @ top_to1767659262_ereal @ F )
        = ( lower_331963542_ereal @ top_to1767659262_ereal @ G ) )
     => ( F = G ) ) ).

% epigraph_invertible
thf(fact_196_neg__le__iff__le,axiom,
    ! [B3: real,A3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) )
      = ( ord_less_eq_real @ A3 @ B3 ) ) ).

% neg_le_iff_le
thf(fact_197_ereal__uminus__eq__iff,axiom,
    ! [A3: extended_ereal,B3: extended_ereal] :
      ( ( ( uminus1208298309_ereal @ A3 )
        = ( uminus1208298309_ereal @ B3 ) )
      = ( A3 = B3 ) ) ).

% ereal_uminus_eq_iff
thf(fact_198_ereal__uminus__uminus,axiom,
    ! [A3: extended_ereal] :
      ( ( uminus1208298309_ereal @ ( uminus1208298309_ereal @ A3 ) )
      = A3 ) ).

% ereal_uminus_uminus
thf(fact_199_verit__minus__simplify_I4_J,axiom,
    ! [B3: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ B3 ) )
      = B3 ) ).

% verit_minus_simplify(4)
thf(fact_200_add_Oinverse__inverse,axiom,
    ! [A3: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A3 ) )
      = A3 ) ).

% add.inverse_inverse
thf(fact_201_neg__equal__iff__equal,axiom,
    ! [A3: real,B3: real] :
      ( ( ( uminus_uminus_real @ A3 )
        = ( uminus_uminus_real @ B3 ) )
      = ( A3 = B3 ) ) ).

% neg_equal_iff_equal
thf(fact_202_ereal__infty__less__eq_I1_J,axiom,
    ! [X: extended_ereal] :
      ( ( ord_le824540014_ereal @ extend1289208545_ereal @ X )
      = ( X = extend1289208545_ereal ) ) ).

% ereal_infty_less_eq(1)
thf(fact_203_UnionI,axiom,
    ! [X5: set_Extended_ereal,C2: set_se767749006_ereal,A: extended_ereal] :
      ( ( member450560855_ereal @ X5 @ C2 )
     => ( ( member1900190071_ereal @ A @ X5 )
       => ( member1900190071_ereal @ A @ ( comple767846299_ereal @ C2 ) ) ) ) ).

% UnionI
thf(fact_204_UnionI,axiom,
    ! [X5: set_a,C2: set_set_a,A: a] :
      ( ( member_set_a @ X5 @ C2 )
     => ( ( member_a @ A @ X5 )
       => ( member_a @ A @ ( comple1766734283_set_a @ C2 ) ) ) ) ).

% UnionI
thf(fact_205_Union__iff,axiom,
    ! [A: extended_ereal,C2: set_se767749006_ereal] :
      ( ( member1900190071_ereal @ A @ ( comple767846299_ereal @ C2 ) )
      = ( ? [X2: set_Extended_ereal] :
            ( ( member450560855_ereal @ X2 @ C2 )
            & ( member1900190071_ereal @ A @ X2 ) ) ) ) ).

% Union_iff
thf(fact_206_Union__iff,axiom,
    ! [A: a,C2: set_set_a] :
      ( ( member_a @ A @ ( comple1766734283_set_a @ C2 ) )
      = ( ? [X2: set_a] :
            ( ( member_set_a @ X2 @ C2 )
            & ( member_a @ A @ X2 ) ) ) ) ).

% Union_iff
thf(fact_207_ereal__minus__le__minus,axiom,
    ! [A3: extended_ereal,B3: extended_ereal] :
      ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A3 ) @ ( uminus1208298309_ereal @ B3 ) )
      = ( ord_le824540014_ereal @ B3 @ A3 ) ) ).

% ereal_minus_le_minus
thf(fact_208_ereal__infty__less__eq_I2_J,axiom,
    ! [X: extended_ereal] :
      ( ( ord_le824540014_ereal @ X @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
      = ( X
        = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).

% ereal_infty_less_eq(2)
thf(fact_209_UnionE,axiom,
    ! [A: extended_ereal,C2: set_se767749006_ereal] :
      ( ( member1900190071_ereal @ A @ ( comple767846299_ereal @ C2 ) )
     => ~ ! [X6: set_Extended_ereal] :
            ( ( member1900190071_ereal @ A @ X6 )
           => ~ ( member450560855_ereal @ X6 @ C2 ) ) ) ).

% UnionE
thf(fact_210_UnionE,axiom,
    ! [A: a,C2: set_set_a] :
      ( ( member_a @ A @ ( comple1766734283_set_a @ C2 ) )
     => ~ ! [X6: set_a] :
            ( ( member_a @ A @ X6 )
           => ~ ( member_set_a @ X6 @ C2 ) ) ) ).

% UnionE
thf(fact_211_neq__PInf__trans,axiom,
    ! [Y4: extended_ereal,X: extended_ereal] :
      ( ( Y4 != extend1289208545_ereal )
     => ( ( ord_le824540014_ereal @ X @ Y4 )
       => ( X != extend1289208545_ereal ) ) ) ).

% neq_PInf_trans
thf(fact_212_ereal__infty__less__eq2_I1_J,axiom,
    ! [A3: extended_ereal,B3: extended_ereal] :
      ( ( ord_le824540014_ereal @ A3 @ B3 )
     => ( ( A3 = extend1289208545_ereal )
       => ( B3 = extend1289208545_ereal ) ) ) ).

% ereal_infty_less_eq2(1)
thf(fact_213_ereal__less__eq_I1_J,axiom,
    ! [X: extended_ereal] : ( ord_le824540014_ereal @ X @ extend1289208545_ereal ) ).

% ereal_less_eq(1)
thf(fact_214_ereal__uminus__le__reorder,axiom,
    ! [A3: extended_ereal,B3: extended_ereal] :
      ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A3 ) @ B3 )
      = ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ B3 ) @ A3 ) ) ).

% ereal_uminus_le_reorder
thf(fact_215_ereal__infty__less__eq2_I2_J,axiom,
    ! [A3: extended_ereal,B3: extended_ereal] :
      ( ( ord_le824540014_ereal @ A3 @ B3 )
     => ( ( B3
          = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
       => ( A3
          = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).

% ereal_infty_less_eq2(2)
thf(fact_216_ereal__less__eq_I2_J,axiom,
    ! [X: extended_ereal] : ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ X ) ).

% ereal_less_eq(2)
thf(fact_217_epigraph__subset__iff,axiom,
    ! [F: a > extended_ereal,G: a > extended_ereal] :
      ( ( ord_le1586073967a_real @ ( lower_930854854raph_a @ top_top_set_a @ F ) @ ( lower_930854854raph_a @ top_top_set_a @ G ) )
      = ( ! [X2: a] : ( ord_le824540014_ereal @ ( G @ X2 ) @ ( F @ X2 ) ) ) ) ).

% epigraph_subset_iff
thf(fact_218_verit__la__disequality,axiom,
    ! [A3: extended_ereal,B3: extended_ereal] :
      ( ( A3 = B3 )
      | ~ ( ord_le824540014_ereal @ A3 @ B3 )
      | ~ ( ord_le824540014_ereal @ B3 @ A3 ) ) ).

% verit_la_disequality
thf(fact_219_verit__la__disequality,axiom,
    ! [A3: real,B3: real] :
      ( ( A3 = B3 )
      | ~ ( ord_less_eq_real @ A3 @ B3 )
      | ~ ( ord_less_eq_real @ B3 @ A3 ) ) ).

% verit_la_disequality
thf(fact_220_minus__equation__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ( uminus_uminus_real @ A3 )
        = B3 )
      = ( ( uminus_uminus_real @ B3 )
        = A3 ) ) ).

% minus_equation_iff
thf(fact_221_equation__minus__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( A3
        = ( uminus_uminus_real @ B3 ) )
      = ( B3
        = ( uminus_uminus_real @ A3 ) ) ) ).

% equation_minus_iff
thf(fact_222_verit__negate__coefficient_I3_J,axiom,
    ! [A3: real,B3: real] :
      ( ( A3 = B3 )
     => ( ( uminus_uminus_real @ A3 )
        = ( uminus_uminus_real @ B3 ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_223_top__ereal__def,axiom,
    top_to802031902_ereal = extend1289208545_ereal ).

% top_ereal_def
thf(fact_224_Sup__eq__PInfty,axiom,
    ! [S2: set_Extended_ereal] :
      ( ( member1900190071_ereal @ extend1289208545_ereal @ S2 )
     => ( ( comple1161760187_ereal @ S2 )
        = extend1289208545_ereal ) ) ).

% Sup_eq_PInfty
thf(fact_225_ereal__uminus__eq__reorder,axiom,
    ! [A3: extended_ereal,B3: extended_ereal] :
      ( ( ( uminus1208298309_ereal @ A3 )
        = B3 )
      = ( A3
        = ( uminus1208298309_ereal @ B3 ) ) ) ).

% ereal_uminus_eq_reorder
thf(fact_226_le__imp__neg__le,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) ) ) ).

% le_imp_neg_le
thf(fact_227_minus__le__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ B3 )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ A3 ) ) ).

% minus_le_iff
thf(fact_228_le__minus__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ B3 ) )
      = ( ord_less_eq_real @ B3 @ ( uminus_uminus_real @ A3 ) ) ) ).

% le_minus_iff
thf(fact_229_MInfty__neq__PInfty_I1_J,axiom,
    ( extend1289208545_ereal
   != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).

% MInfty_neq_PInfty(1)
thf(fact_230_top__empty__eq,axiom,
    ( top_top_a_o
    = ( ^ [X2: a] : ( member_a @ X2 @ top_top_set_a ) ) ) ).

% top_empty_eq
thf(fact_231_top__empty__eq,axiom,
    ( top_to398855007real_o
    = ( ^ [X2: extended_ereal] : ( member1900190071_ereal @ X2 @ top_to1767659262_ereal ) ) ) ).

% top_empty_eq
thf(fact_232_cSup__eq__maximum,axiom,
    ! [Z2: real,X5: set_real] :
      ( ( member_real @ Z2 @ X5 )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ X5 )
           => ( ord_less_eq_real @ X4 @ Z2 ) )
       => ( ( comple2129349247p_real @ X5 )
          = Z2 ) ) ) ).

% cSup_eq_maximum
thf(fact_233_cSup__eq__maximum,axiom,
    ! [Z2: extended_ereal,X5: set_Extended_ereal] :
      ( ( member1900190071_ereal @ Z2 @ X5 )
     => ( ! [X4: extended_ereal] :
            ( ( member1900190071_ereal @ X4 @ X5 )
           => ( ord_le824540014_ereal @ X4 @ Z2 ) )
       => ( ( comple1161760187_ereal @ X5 )
          = Z2 ) ) ) ).

% cSup_eq_maximum
thf(fact_234_cSup__eq,axiom,
    ! [X5: set_real,A3: real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ X5 )
         => ( ord_less_eq_real @ X4 @ A3 ) )
     => ( ! [Y2: real] :
            ( ! [X3: real] :
                ( ( member_real @ X3 @ X5 )
               => ( ord_less_eq_real @ X3 @ Y2 ) )
           => ( ord_less_eq_real @ A3 @ Y2 ) )
       => ( ( comple2129349247p_real @ X5 )
          = A3 ) ) ) ).

% cSup_eq
thf(fact_235_MInfty__eq__minfinity,axiom,
    ( extended_MInfty
    = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).

% MInfty_eq_minfinity
thf(fact_236_ComplI,axiom,
    ! [C: extended_ereal,A: set_Extended_ereal] :
      ( ~ ( member1900190071_ereal @ C @ A )
     => ( member1900190071_ereal @ C @ ( uminus360668453_ereal @ A ) ) ) ).

% ComplI
thf(fact_237_ComplI,axiom,
    ! [C: a,A: set_a] :
      ( ~ ( member_a @ C @ A )
     => ( member_a @ C @ ( uminus_uminus_set_a @ A ) ) ) ).

% ComplI
thf(fact_238_Compl__iff,axiom,
    ! [C: extended_ereal,A: set_Extended_ereal] :
      ( ( member1900190071_ereal @ C @ ( uminus360668453_ereal @ A ) )
      = ( ~ ( member1900190071_ereal @ C @ A ) ) ) ).

% Compl_iff
thf(fact_239_Compl__iff,axiom,
    ! [C: a,A: set_a] :
      ( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
      = ( ~ ( member_a @ C @ A ) ) ) ).

% Compl_iff
thf(fact_240_ComplD,axiom,
    ! [C: extended_ereal,A: set_Extended_ereal] :
      ( ( member1900190071_ereal @ C @ ( uminus360668453_ereal @ A ) )
     => ~ ( member1900190071_ereal @ C @ A ) ) ).

% ComplD
thf(fact_241_ComplD,axiom,
    ! [C: a,A: set_a] :
      ( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
     => ~ ( member_a @ C @ A ) ) ).

% ComplD
thf(fact_242_ereal__complete__Inf,axiom,
    ! [S2: set_Extended_ereal] :
    ? [X4: extended_ereal] :
      ( ! [Xa: extended_ereal] :
          ( ( member1900190071_ereal @ Xa @ S2 )
         => ( ord_le824540014_ereal @ X4 @ Xa ) )
      & ! [Z3: extended_ereal] :
          ( ! [Xa2: extended_ereal] :
              ( ( member1900190071_ereal @ Xa2 @ S2 )
             => ( ord_le824540014_ereal @ Z3 @ Xa2 ) )
         => ( ord_le824540014_ereal @ Z3 @ X4 ) ) ) ).

% ereal_complete_Inf
thf(fact_243_ereal__complete__Sup,axiom,
    ! [S2: set_Extended_ereal] :
    ? [X4: extended_ereal] :
      ( ! [Xa: extended_ereal] :
          ( ( member1900190071_ereal @ Xa @ S2 )
         => ( ord_le824540014_ereal @ Xa @ X4 ) )
      & ! [Z3: extended_ereal] :
          ( ! [Xa2: extended_ereal] :
              ( ( member1900190071_ereal @ Xa2 @ S2 )
             => ( ord_le824540014_ereal @ Xa2 @ Z3 ) )
         => ( ord_le824540014_ereal @ X4 @ Z3 ) ) ) ).

% ereal_complete_Sup
thf(fact_244_uminus__ereal_Osimps_I3_J,axiom,
    ( ( uminus1208298309_ereal @ extended_MInfty )
    = extended_PInfty ) ).

% uminus_ereal.simps(3)
thf(fact_245_uminus__ereal_Osimps_I2_J,axiom,
    ( ( uminus1208298309_ereal @ extended_PInfty )
    = extended_MInfty ) ).

% uminus_ereal.simps(2)
thf(fact_246_GreatestI2__order,axiom,
    ! [P: extended_ereal > $o,X: extended_ereal,Q: extended_ereal > $o] :
      ( ( P @ X )
     => ( ! [Y2: extended_ereal] :
            ( ( P @ Y2 )
           => ( ord_le824540014_ereal @ Y2 @ X ) )
       => ( ! [X4: extended_ereal] :
              ( ( P @ X4 )
             => ( ! [Y5: extended_ereal] :
                    ( ( P @ Y5 )
                   => ( ord_le824540014_ereal @ Y5 @ X4 ) )
               => ( Q @ X4 ) ) )
         => ( Q @ ( order_1158471719_ereal @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_247_GreatestI2__order,axiom,
    ! [P: real > $o,X: real,Q: real > $o] :
      ( ( P @ X )
     => ( ! [Y2: real] :
            ( ( P @ Y2 )
           => ( ord_less_eq_real @ Y2 @ X ) )
       => ( ! [X4: real] :
              ( ( P @ X4 )
             => ( ! [Y5: real] :
                    ( ( P @ Y5 )
                   => ( ord_less_eq_real @ Y5 @ X4 ) )
               => ( Q @ X4 ) ) )
         => ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_248_infinity__ereal__def,axiom,
    extend1289208545_ereal = extended_PInfty ).

% infinity_ereal_def
thf(fact_249_ereal_Odistinct_I5_J,axiom,
    extended_PInfty != extended_MInfty ).

% ereal.distinct(5)
thf(fact_250_Greatest__equality,axiom,
    ! [P: extended_ereal > $o,X: extended_ereal] :
      ( ( P @ X )
     => ( ! [Y2: extended_ereal] :
            ( ( P @ Y2 )
           => ( ord_le824540014_ereal @ Y2 @ X ) )
       => ( ( order_1158471719_ereal @ P )
          = X ) ) ) ).

% Greatest_equality
thf(fact_251_Greatest__equality,axiom,
    ! [P: real > $o,X: real] :
      ( ( P @ X )
     => ( ! [Y2: real] :
            ( ( P @ Y2 )
           => ( ord_less_eq_real @ Y2 @ X ) )
       => ( ( order_Greatest_real @ P )
          = X ) ) ) ).

% Greatest_equality
thf(fact_252_uminus__ereal_Oelims,axiom,
    ! [X: extended_ereal,Y4: extended_ereal] :
      ( ( ( uminus1208298309_ereal @ X )
        = Y4 )
     => ( ! [R: real] :
            ( ( X
              = ( extended_ereal2 @ R ) )
           => ( Y4
             != ( extended_ereal2 @ ( uminus_uminus_real @ R ) ) ) )
       => ( ( ( X = extended_PInfty )
           => ( Y4 != extended_MInfty ) )
         => ~ ( ( X = extended_MInfty )
             => ( Y4 != extended_PInfty ) ) ) ) ) ).

% uminus_ereal.elims
thf(fact_253_not__MInfty__nonneg,axiom,
    ! [X: extended_ereal] :
      ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ X )
     => ( X
       != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).

% not_MInfty_nonneg
thf(fact_254_ereal_Oinject,axiom,
    ! [X1: real,Y1: real] :
      ( ( ( extended_ereal2 @ X1 )
        = ( extended_ereal2 @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% ereal.inject
thf(fact_255_ereal__cong,axiom,
    ! [X: real,Y4: real] :
      ( ( X = Y4 )
     => ( ( extended_ereal2 @ X )
        = ( extended_ereal2 @ Y4 ) ) ) ).

% ereal_cong
thf(fact_256_neg__equal__zero,axiom,
    ! [A3: real] :
      ( ( ( uminus_uminus_real @ A3 )
        = A3 )
      = ( A3 = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_257_equal__neg__zero,axiom,
    ! [A3: real] :
      ( ( A3
        = ( uminus_uminus_real @ A3 ) )
      = ( A3 = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_258_neg__equal__0__iff__equal,axiom,
    ! [A3: real] :
      ( ( ( uminus_uminus_real @ A3 )
        = zero_zero_real )
      = ( A3 = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
thf(fact_259_neg__0__equal__iff__equal,axiom,
    ! [A3: real] :
      ( ( zero_zero_real
        = ( uminus_uminus_real @ A3 ) )
      = ( zero_zero_real = A3 ) ) ).

% neg_0_equal_iff_equal
thf(fact_260_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_261_ereal__less__eq_I3_J,axiom,
    ! [R2: real,P2: real] :
      ( ( ord_le824540014_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
      = ( ord_less_eq_real @ R2 @ P2 ) ) ).

% ereal_less_eq(3)
thf(fact_262_ereal__uminus__zero,axiom,
    ( ( uminus1208298309_ereal @ zero_z163181189_ereal )
    = zero_z163181189_ereal ) ).

% ereal_uminus_zero
thf(fact_263_ereal__uminus__zero__iff,axiom,
    ! [A3: extended_ereal] :
      ( ( ( uminus1208298309_ereal @ A3 )
        = zero_z163181189_ereal )
      = ( A3 = zero_z163181189_ereal ) ) ).

% ereal_uminus_zero_iff
thf(fact_264_neg__0__le__iff__le,axiom,
    ! [A3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A3 ) )
      = ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).

% neg_0_le_iff_le
thf(fact_265_neg__le__0__iff__le,axiom,
    ! [A3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ zero_zero_real )
      = ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).

% neg_le_0_iff_le
thf(fact_266_less__eq__neg__nonpos,axiom,
    ! [A3: real] :
      ( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ A3 ) )
      = ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).

% less_eq_neg_nonpos
thf(fact_267_neg__less__eq__nonneg,axiom,
    ! [A3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ A3 )
      = ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).

% neg_less_eq_nonneg
thf(fact_268_ereal__uminus__le__0__iff,axiom,
    ! [A3: extended_ereal] :
      ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A3 ) @ zero_z163181189_ereal )
      = ( ord_le824540014_ereal @ zero_z163181189_ereal @ A3 ) ) ).

% ereal_uminus_le_0_iff
thf(fact_269_ereal__0__le__uminus__iff,axiom,
    ! [A3: extended_ereal] :
      ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( uminus1208298309_ereal @ A3 ) )
      = ( ord_le824540014_ereal @ A3 @ zero_z163181189_ereal ) ) ).

% ereal_0_le_uminus_iff
thf(fact_270_ereal_Odistinct_I1_J,axiom,
    ! [X1: real] :
      ( ( extended_ereal2 @ X1 )
     != extended_PInfty ) ).

% ereal.distinct(1)
thf(fact_271_zero__reorient,axiom,
    ! [X: extended_ereal] :
      ( ( zero_z163181189_ereal = X )
      = ( X = zero_z163181189_ereal ) ) ).

% zero_reorient
thf(fact_272_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
      = ( X = zero_zero_real ) ) ).

% zero_reorient
thf(fact_273_Infty__neq__0_I1_J,axiom,
    extend1289208545_ereal != zero_z163181189_ereal ).

% Infty_neq_0(1)
thf(fact_274_uminus__ereal_Osimps_I1_J,axiom,
    ! [R2: real] :
      ( ( uminus1208298309_ereal @ ( extended_ereal2 @ R2 ) )
      = ( extended_ereal2 @ ( uminus_uminus_real @ R2 ) ) ) ).

% uminus_ereal.simps(1)
thf(fact_275_PInfty__neq__ereal_I1_J,axiom,
    ! [R2: real] :
      ( ( extended_ereal2 @ R2 )
     != extend1289208545_ereal ) ).

% PInfty_neq_ereal(1)
thf(fact_276_ereal_Odistinct_I3_J,axiom,
    ! [X1: real] :
      ( ( extended_ereal2 @ X1 )
     != extended_MInfty ) ).

% ereal.distinct(3)
thf(fact_277_ereal__le__real,axiom,
    ! [X: extended_ereal,Y4: extended_ereal] :
      ( ! [Z4: real] :
          ( ( ord_le824540014_ereal @ X @ ( extended_ereal2 @ Z4 ) )
         => ( ord_le824540014_ereal @ Y4 @ ( extended_ereal2 @ Z4 ) ) )
     => ( ord_le824540014_ereal @ Y4 @ X ) ) ).

% ereal_le_real
thf(fact_278_le__ereal__le,axiom,
    ! [A3: extended_ereal,X: real,Y4: real] :
      ( ( ord_le824540014_ereal @ A3 @ ( extended_ereal2 @ X ) )
     => ( ( ord_less_eq_real @ X @ Y4 )
       => ( ord_le824540014_ereal @ A3 @ ( extended_ereal2 @ Y4 ) ) ) ) ).

% le_ereal_le
thf(fact_279_ereal__le__le,axiom,
    ! [Y4: real,A3: extended_ereal,X: real] :
      ( ( ord_le824540014_ereal @ ( extended_ereal2 @ Y4 ) @ A3 )
     => ( ( ord_less_eq_real @ X @ Y4 )
       => ( ord_le824540014_ereal @ ( extended_ereal2 @ X ) @ A3 ) ) ) ).

% ereal_le_le
thf(fact_280_MInfty__neq__ereal_I1_J,axiom,
    ! [R2: real] :
      ( ( extended_ereal2 @ R2 )
     != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).

% MInfty_neq_ereal(1)
thf(fact_281_ereal__cases,axiom,
    ! [X: extended_ereal] :
      ( ! [R: real] :
          ( X
         != ( extended_ereal2 @ R ) )
     => ( ( X != extend1289208545_ereal )
       => ( X
          = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).

% ereal_cases
thf(fact_282_ereal2__cases,axiom,
    ! [X: extended_ereal,Xa3: extended_ereal] :
      ( ( ? [R: real] :
            ( X
            = ( extended_ereal2 @ R ) )
       => ! [Ra: real] :
            ( Xa3
           != ( extended_ereal2 @ Ra ) ) )
     => ( ( ? [R: real] :
              ( X
              = ( extended_ereal2 @ R ) )
         => ( Xa3 != extend1289208545_ereal ) )
       => ( ( ? [R: real] :
                ( X
                = ( extended_ereal2 @ R ) )
           => ( Xa3
             != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
         => ( ( ( X = extend1289208545_ereal )
             => ! [R: real] :
                  ( Xa3
                 != ( extended_ereal2 @ R ) ) )
           => ( ( ( X = extend1289208545_ereal )
               => ( Xa3 != extend1289208545_ereal ) )
             => ( ( ( X = extend1289208545_ereal )
                 => ( Xa3
                   != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
               => ( ( ( X
                      = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                   => ! [R: real] :
                        ( Xa3
                       != ( extended_ereal2 @ R ) ) )
                 => ( ( ( X
                        = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                     => ( Xa3 != extend1289208545_ereal ) )
                   => ~ ( ( X
                          = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                       => ( Xa3
                         != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ).

% ereal2_cases
thf(fact_283_ereal3__cases,axiom,
    ! [X: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
      ( ( ? [R: real] :
            ( X
            = ( extended_ereal2 @ R ) )
       => ( ? [Ra: real] :
              ( Xa3
              = ( extended_ereal2 @ Ra ) )
         => ! [Rb: real] :
              ( Xb
             != ( extended_ereal2 @ Rb ) ) ) )
     => ( ( ? [R: real] :
              ( X
              = ( extended_ereal2 @ R ) )
         => ( ? [Ra: real] :
                ( Xa3
                = ( extended_ereal2 @ Ra ) )
           => ( Xb != extend1289208545_ereal ) ) )
       => ( ( ? [R: real] :
                ( X
                = ( extended_ereal2 @ R ) )
           => ( ? [Ra: real] :
                  ( Xa3
                  = ( extended_ereal2 @ Ra ) )
             => ( Xb
               != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
         => ( ( ? [R: real] :
                  ( X
                  = ( extended_ereal2 @ R ) )
             => ( ( Xa3 = extend1289208545_ereal )
               => ! [Ra: real] :
                    ( Xb
                   != ( extended_ereal2 @ Ra ) ) ) )
           => ( ( ? [R: real] :
                    ( X
                    = ( extended_ereal2 @ R ) )
               => ( ( Xa3 = extend1289208545_ereal )
                 => ( Xb != extend1289208545_ereal ) ) )
             => ( ( ? [R: real] :
                      ( X
                      = ( extended_ereal2 @ R ) )
                 => ( ( Xa3 = extend1289208545_ereal )
                   => ( Xb
                     != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
               => ( ( ? [R: real] :
                        ( X
                        = ( extended_ereal2 @ R ) )
                   => ( ( Xa3
                        = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                     => ! [Ra: real] :
                          ( Xb
                         != ( extended_ereal2 @ Ra ) ) ) )
                 => ( ( ? [R: real] :
                          ( X
                          = ( extended_ereal2 @ R ) )
                     => ( ( Xa3
                          = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                       => ( Xb != extend1289208545_ereal ) ) )
                   => ( ( ? [R: real] :
                            ( X
                            = ( extended_ereal2 @ R ) )
                       => ( ( Xa3
                            = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                         => ( Xb
                           != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
                     => ( ( ( X = extend1289208545_ereal )
                         => ( ? [R: real] :
                                ( Xa3
                                = ( extended_ereal2 @ R ) )
                           => ! [Ra: real] :
                                ( Xb
                               != ( extended_ereal2 @ Ra ) ) ) )
                       => ( ( ( X = extend1289208545_ereal )
                           => ( ? [R: real] :
                                  ( Xa3
                                  = ( extended_ereal2 @ R ) )
                             => ( Xb != extend1289208545_ereal ) ) )
                         => ( ( ( X = extend1289208545_ereal )
                             => ( ? [R: real] :
                                    ( Xa3
                                    = ( extended_ereal2 @ R ) )
                               => ( Xb
                                 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
                           => ( ( ( X = extend1289208545_ereal )
                               => ( ( Xa3 = extend1289208545_ereal )
                                 => ! [R: real] :
                                      ( Xb
                                     != ( extended_ereal2 @ R ) ) ) )
                             => ( ( ( X = extend1289208545_ereal )
                                 => ( ( Xa3 = extend1289208545_ereal )
                                   => ( Xb != extend1289208545_ereal ) ) )
                               => ( ( ( X = extend1289208545_ereal )
                                   => ( ( Xa3 = extend1289208545_ereal )
                                     => ( Xb
                                       != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
                                 => ( ( ( X = extend1289208545_ereal )
                                     => ( ( Xa3
                                          = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                                       => ! [R: real] :
                                            ( Xb
                                           != ( extended_ereal2 @ R ) ) ) )
                                   => ( ( ( X = extend1289208545_ereal )
                                       => ( ( Xa3
                                            = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                                         => ( Xb != extend1289208545_ereal ) ) )
                                     => ( ( ( X = extend1289208545_ereal )
                                         => ( ( Xa3
                                              = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                                           => ( Xb
                                             != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
                                       => ( ( ( X
                                              = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                                           => ( ? [R: real] :
                                                  ( Xa3
                                                  = ( extended_ereal2 @ R ) )
                                             => ! [Ra: real] :
                                                  ( Xb
                                                 != ( extended_ereal2 @ Ra ) ) ) )
                                         => ( ( ( X
                                                = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                                             => ( ? [R: real] :
                                                    ( Xa3
                                                    = ( extended_ereal2 @ R ) )
                                               => ( Xb != extend1289208545_ereal ) ) )
                                           => ( ( ( X
                                                  = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                                               => ( ? [R: real] :
                                                      ( Xa3
                                                      = ( extended_ereal2 @ R ) )
                                                 => ( Xb
                                                   != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
                                             => ( ( ( X
                                                    = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                                                 => ( ( Xa3 = extend1289208545_ereal )
                                                   => ! [R: real] :
                                                        ( Xb
                                                       != ( extended_ereal2 @ R ) ) ) )
                                               => ( ( ( X
                                                      = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                                                   => ( ( Xa3 = extend1289208545_ereal )
                                                     => ( Xb != extend1289208545_ereal ) ) )
                                                 => ( ( ( X
                                                        = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                                                     => ( ( Xa3 = extend1289208545_ereal )
                                                       => ( Xb
                                                         != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
                                                   => ( ( ( X
                                                          = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                                                       => ( ( Xa3
                                                            = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                                                         => ! [R: real] :
                                                              ( Xb
                                                             != ( extended_ereal2 @ R ) ) ) )
                                                     => ( ( ( X
                                                            = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                                                         => ( ( Xa3
                                                              = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                                                           => ( Xb != extend1289208545_ereal ) ) )
                                                       => ~ ( ( X
                                                              = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                                                           => ( ( Xa3
                                                                = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                                                             => ( Xb
                                                               != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% ereal3_cases
thf(fact_284_ereal__ex__split,axiom,
    ( ( ^ [P3: extended_ereal > $o] :
        ? [X7: extended_ereal] : ( P3 @ X7 ) )
    = ( ^ [P4: extended_ereal > $o] :
          ( ( P4 @ extend1289208545_ereal )
          | ? [X2: real] : ( P4 @ ( extended_ereal2 @ X2 ) )
          | ( P4 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).

% ereal_ex_split
thf(fact_285_abs__ereal_Ocases,axiom,
    ! [X: extended_ereal] :
      ( ! [R: real] :
          ( X
         != ( extended_ereal2 @ R ) )
     => ( ( X
         != ( uminus1208298309_ereal @ extend1289208545_ereal ) )
       => ( X = extend1289208545_ereal ) ) ) ).

% abs_ereal.cases
thf(fact_286_ereal__all__split,axiom,
    ( ( ^ [P3: extended_ereal > $o] :
        ! [X7: extended_ereal] : ( P3 @ X7 ) )
    = ( ^ [P4: extended_ereal > $o] :
          ( ( P4 @ extend1289208545_ereal )
          & ! [X2: real] : ( P4 @ ( extended_ereal2 @ X2 ) )
          & ( P4 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).

% ereal_all_split
thf(fact_287_abs__ereal_Oinduct,axiom,
    ! [P: extended_ereal > $o,A0: extended_ereal] :
      ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
     => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
       => ( ( P @ extend1289208545_ereal )
         => ( P @ A0 ) ) ) ) ).

% abs_ereal.induct
thf(fact_288_less__ereal_Oinduct,axiom,
    ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
      ( ! [X4: real,Y2: real] : ( P @ ( extended_ereal2 @ X4 ) @ ( extended_ereal2 @ Y2 ) )
     => ( ! [X_1: extended_ereal] : ( P @ extend1289208545_ereal @ X_1 )
       => ( ! [A5: extended_ereal] : ( P @ A5 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
         => ( ! [X4: real] : ( P @ ( extended_ereal2 @ X4 ) @ extend1289208545_ereal )
           => ( ! [R: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R ) )
             => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
               => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).

% less_ereal.induct
thf(fact_289_plus__ereal_Oinduct,axiom,
    ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
      ( ! [R: real,P5: real] : ( P @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P5 ) )
     => ( ! [X_1: extended_ereal] : ( P @ extend1289208545_ereal @ X_1 )
       => ( ! [A5: extended_ereal] : ( P @ A5 @ extend1289208545_ereal )
         => ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
           => ( ! [P5: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ P5 ) )
             => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
               => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).

% plus_ereal.induct
thf(fact_290_times__ereal_Oinduct,axiom,
    ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
      ( ! [R: real,P5: real] : ( P @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P5 ) )
     => ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ extend1289208545_ereal )
       => ( ! [R: real] : ( P @ extend1289208545_ereal @ ( extended_ereal2 @ R ) )
         => ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
           => ( ! [R: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R ) )
             => ( ( P @ extend1289208545_ereal @ extend1289208545_ereal )
               => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
                 => ( ( P @ extend1289208545_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                   => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
                     => ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).

% times_ereal.induct
thf(fact_291_real__of__ereal_Ocases,axiom,
    ! [X: extended_ereal] :
      ( ! [R: real] :
          ( X
         != ( extended_ereal2 @ R ) )
     => ( ( X != extend1289208545_ereal )
       => ( X
          = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).

% real_of_ereal.cases
thf(fact_292_real__of__ereal_Oinduct,axiom,
    ! [P: extended_ereal > $o,A0: extended_ereal] :
      ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
     => ( ( P @ extend1289208545_ereal )
       => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
         => ( P @ A0 ) ) ) ) ).

% real_of_ereal.induct
thf(fact_293_domain__Epigraph__aux,axiom,
    ! [X: extended_ereal] :
      ( ( X != extend1289208545_ereal )
     => ? [R: real] : ( ord_le824540014_ereal @ X @ ( extended_ereal2 @ R ) ) ) ).

% domain_Epigraph_aux
thf(fact_294_ereal__top,axiom,
    ! [X: extended_ereal] :
      ( ! [B6: real] : ( ord_le824540014_ereal @ ( extended_ereal2 @ B6 ) @ X )
     => ( X = extend1289208545_ereal ) ) ).

% ereal_top
thf(fact_295_Infty__neq__0_I3_J,axiom,
    ( ( uminus1208298309_ereal @ extend1289208545_ereal )
   != zero_z163181189_ereal ) ).

% Infty_neq_0(3)
thf(fact_296_ereal_Oinduct,axiom,
    ! [P: extended_ereal > $o,Ereal: extended_ereal] :
      ( ! [X4: real] : ( P @ ( extended_ereal2 @ X4 ) )
     => ( ( P @ extended_PInfty )
       => ( ( P @ extended_MInfty )
         => ( P @ Ereal ) ) ) ) ).

% ereal.induct
thf(fact_297_ereal_Oexhaust,axiom,
    ! [Y4: extended_ereal] :
      ( ! [X12: real] :
          ( Y4
         != ( extended_ereal2 @ X12 ) )
     => ( ( Y4 != extended_PInfty )
       => ( Y4 = extended_MInfty ) ) ) ).

% ereal.exhaust
thf(fact_298_uminus__ereal_Ocases,axiom,
    ! [X: extended_ereal] :
      ( ! [R: real] :
          ( X
         != ( extended_ereal2 @ R ) )
     => ( ( X != extended_PInfty )
       => ( X = extended_MInfty ) ) ) ).

% uminus_ereal.cases
thf(fact_299_uminus__ereal_Oinduct,axiom,
    ! [P: extended_ereal > $o,A0: extended_ereal] :
      ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
     => ( ( P @ extended_PInfty )
       => ( ( P @ extended_MInfty )
         => ( P @ A0 ) ) ) ) ).

% uminus_ereal.induct
thf(fact_300_ereal__bot,axiom,
    ! [X: extended_ereal] :
      ( ! [B6: real] : ( ord_le824540014_ereal @ X @ ( extended_ereal2 @ B6 ) )
     => ( X
        = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).

% ereal_bot
thf(fact_301_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_302_ereal__less__eq_I4_J,axiom,
    ! [R2: real] :
      ( ( ord_le824540014_ereal @ ( extended_ereal2 @ R2 ) @ zero_z163181189_ereal )
      = ( ord_less_eq_real @ R2 @ zero_zero_real ) ) ).

% ereal_less_eq(4)
thf(fact_303_ereal__less__eq_I5_J,axiom,
    ! [R2: real] :
      ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( extended_ereal2 @ R2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ R2 ) ) ).

% ereal_less_eq(5)
thf(fact_304_ereal__eq__0_I2_J,axiom,
    ! [R2: real] :
      ( ( zero_z163181189_ereal
        = ( extended_ereal2 @ R2 ) )
      = ( R2 = zero_zero_real ) ) ).

% ereal_eq_0(2)
thf(fact_305_ereal__eq__0_I1_J,axiom,
    ! [R2: real] :
      ( ( ( extended_ereal2 @ R2 )
        = zero_z163181189_ereal )
      = ( R2 = zero_zero_real ) ) ).

% ereal_eq_0(1)
thf(fact_306_zero__ereal__def,axiom,
    ( zero_z163181189_ereal
    = ( extended_ereal2 @ zero_zero_real ) ) ).

% zero_ereal_def
thf(fact_307_real__eq__0__iff__le__ge__0,axiom,
    ! [X: real] :
      ( ( X = zero_zero_real )
      = ( ( ord_less_eq_real @ zero_zero_real @ X )
        & ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X ) ) ) ) ).

% real_eq_0_iff_le_ge_0
thf(fact_308_ereal__divide__ereal,axiom,
    ! [R2: real] :
      ( ( ( ord_less_eq_real @ zero_zero_real @ R2 )
       => ( ( divide595620860_ereal @ extend1289208545_ereal @ ( extended_ereal2 @ R2 ) )
          = extend1289208545_ereal ) )
      & ( ~ ( ord_less_eq_real @ zero_zero_real @ R2 )
       => ( ( divide595620860_ereal @ extend1289208545_ereal @ ( extended_ereal2 @ R2 ) )
          = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).

% ereal_divide_ereal
thf(fact_309_ereal__uminus__divide,axiom,
    ! [X: extended_ereal,Y4: extended_ereal] :
      ( ( divide595620860_ereal @ ( uminus1208298309_ereal @ X ) @ Y4 )
      = ( uminus1208298309_ereal @ ( divide595620860_ereal @ X @ Y4 ) ) ) ).

% ereal_uminus_divide
thf(fact_310_ereal__divide__zero__left,axiom,
    ! [A3: extended_ereal] :
      ( ( divide595620860_ereal @ zero_z163181189_ereal @ A3 )
      = zero_z163181189_ereal ) ).

% ereal_divide_zero_left
thf(fact_311_ereal__divide__Infty_I1_J,axiom,
    ! [X: extended_ereal] :
      ( ( divide595620860_ereal @ X @ extend1289208545_ereal )
      = zero_z163181189_ereal ) ).

% ereal_divide_Infty(1)
thf(fact_312_ereal__divide__Infty_I2_J,axiom,
    ! [X: extended_ereal] :
      ( ( divide595620860_ereal @ X @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
      = zero_z163181189_ereal ) ).

% ereal_divide_Infty(2)
thf(fact_313_zero__le__divide__ereal,axiom,
    ! [A3: extended_ereal,B3: extended_ereal] :
      ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ A3 )
     => ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ B3 )
       => ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( divide595620860_ereal @ A3 @ B3 ) ) ) ) ).

% zero_le_divide_ereal
thf(fact_314_div__0,axiom,
    ! [A3: real] :
      ( ( divide_divide_real @ zero_zero_real @ A3 )
      = zero_zero_real ) ).

% div_0
thf(fact_315_div__by__0,axiom,
    ! [A3: real] :
      ( ( divide_divide_real @ A3 @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_316_division__ring__divide__zero,axiom,
    ! [A3: real] :
      ( ( divide_divide_real @ A3 @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_317_divide__eq__0__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ( divide_divide_real @ A3 @ B3 )
        = zero_zero_real )
      = ( ( A3 = zero_zero_real )
        | ( B3 = zero_zero_real ) ) ) ).

% divide_eq_0_iff
thf(fact_318_divide__cancel__left,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ( divide_divide_real @ C @ A3 )
        = ( divide_divide_real @ C @ B3 ) )
      = ( ( C = zero_zero_real )
        | ( A3 = B3 ) ) ) ).

% divide_cancel_left
thf(fact_319_divide__cancel__right,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ( divide_divide_real @ A3 @ C )
        = ( divide_divide_real @ B3 @ C ) )
      = ( ( C = zero_zero_real )
        | ( A3 = B3 ) ) ) ).

% divide_cancel_right
thf(fact_320_minus__divide__left,axiom,
    ! [A3: real,B3: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ B3 ) ) ).

% minus_divide_left
thf(fact_321_minus__divide__divide,axiom,
    ! [A3: real,B3: real] :
      ( ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
      = ( divide_divide_real @ A3 @ B3 ) ) ).

% minus_divide_divide
thf(fact_322_minus__divide__right,axiom,
    ! [A3: real,B3: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) )
      = ( divide_divide_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ).

% minus_divide_right
thf(fact_323_divide__right__mono__neg,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( divide_divide_real @ A3 @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_324_divide__nonpos__nonpos,axiom,
    ! [X: real,Y4: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_325_divide__nonpos__nonneg,axiom,
    ! [X: real,Y4: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).

% divide_nonpos_nonneg
thf(fact_326_divide__nonneg__nonpos,axiom,
    ! [X: real,Y4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).

% divide_nonneg_nonpos
thf(fact_327_divide__nonneg__nonneg,axiom,
    ! [X: real,Y4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_328_zero__le__divide__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A3 @ B3 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
          & ( ord_less_eq_real @ zero_zero_real @ B3 ) )
        | ( ( ord_less_eq_real @ A3 @ zero_zero_real )
          & ( ord_less_eq_real @ B3 @ zero_zero_real ) ) ) ) ).

% zero_le_divide_iff
thf(fact_329_divide__right__mono,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( divide_divide_real @ A3 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).

% divide_right_mono
thf(fact_330_divide__le__0__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A3 @ B3 ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
          & ( ord_less_eq_real @ B3 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A3 @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ) ) ).

% divide_le_0_iff
thf(fact_331_nonzero__minus__divide__divide,axiom,
    ! [B3: real,A3: real] :
      ( ( B3 != zero_zero_real )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
        = ( divide_divide_real @ A3 @ B3 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_332_nonzero__minus__divide__right,axiom,
    ! [B3: real,A3: real] :
      ( ( B3 != zero_zero_real )
     => ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) )
        = ( divide_divide_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_333_real__of__ereal__le__0,axiom,
    ! [X: extended_ereal] :
      ( ( ord_less_eq_real @ ( extend1716541707_ereal @ X ) @ zero_zero_real )
      = ( ( ord_le824540014_ereal @ X @ zero_z163181189_ereal )
        | ( X = extend1289208545_ereal ) ) ) ).

% real_of_ereal_le_0
thf(fact_334_ereal__divide,axiom,
    ! [P2: real,R2: real] :
      ( ( ( P2 = zero_zero_real )
       => ( ( divide595620860_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
          = ( times_1966848393_ereal @ ( extended_ereal2 @ R2 ) @ extend1289208545_ereal ) ) )
      & ( ( P2 != zero_zero_real )
       => ( ( divide595620860_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
          = ( extended_ereal2 @ ( divide_divide_real @ R2 @ P2 ) ) ) ) ) ).

% ereal_divide
thf(fact_335_mult__zero__left,axiom,
    ! [A3: real] :
      ( ( times_times_real @ zero_zero_real @ A3 )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_336_mult__zero__right,axiom,
    ! [A3: real] :
      ( ( times_times_real @ A3 @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_337_mult__eq__0__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ( times_times_real @ A3 @ B3 )
        = zero_zero_real )
      = ( ( A3 = zero_zero_real )
        | ( B3 = zero_zero_real ) ) ) ).

% mult_eq_0_iff
thf(fact_338_mult__cancel__left,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ( times_times_real @ C @ A3 )
        = ( times_times_real @ C @ B3 ) )
      = ( ( C = zero_zero_real )
        | ( A3 = B3 ) ) ) ).

% mult_cancel_left
thf(fact_339_mult__cancel__right,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ( times_times_real @ A3 @ C )
        = ( times_times_real @ B3 @ C ) )
      = ( ( C = zero_zero_real )
        | ( A3 = B3 ) ) ) ).

% mult_cancel_right
thf(fact_340_mult__minus__right,axiom,
    ! [A3: real,B3: real] :
      ( ( times_times_real @ A3 @ ( uminus_uminus_real @ B3 ) )
      = ( uminus_uminus_real @ ( times_times_real @ A3 @ B3 ) ) ) ).

% mult_minus_right
thf(fact_341_minus__mult__minus,axiom,
    ! [A3: real,B3: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
      = ( times_times_real @ A3 @ B3 ) ) ).

% minus_mult_minus
thf(fact_342_mult__minus__left,axiom,
    ! [A3: real,B3: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ B3 )
      = ( uminus_uminus_real @ ( times_times_real @ A3 @ B3 ) ) ) ).

% mult_minus_left
thf(fact_343_ereal__mult__minus__right,axiom,
    ! [A3: extended_ereal,B3: extended_ereal] :
      ( ( times_1966848393_ereal @ A3 @ ( uminus1208298309_ereal @ B3 ) )
      = ( uminus1208298309_ereal @ ( times_1966848393_ereal @ A3 @ B3 ) ) ) ).

% ereal_mult_minus_right
thf(fact_344_ereal__mult__minus__left,axiom,
    ! [A3: extended_ereal,B3: extended_ereal] :
      ( ( times_1966848393_ereal @ ( uminus1208298309_ereal @ A3 ) @ B3 )
      = ( uminus1208298309_ereal @ ( times_1966848393_ereal @ A3 @ B3 ) ) ) ).

% ereal_mult_minus_left
thf(fact_345_ereal__mult__zero,axiom,
    ! [A3: extended_ereal] :
      ( ( times_1966848393_ereal @ A3 @ zero_z163181189_ereal )
      = zero_z163181189_ereal ) ).

% ereal_mult_zero
thf(fact_346_ereal__zero__mult,axiom,
    ! [A3: extended_ereal] :
      ( ( times_1966848393_ereal @ zero_z163181189_ereal @ A3 )
      = zero_z163181189_ereal ) ).

% ereal_zero_mult
thf(fact_347_ereal__zero__times,axiom,
    ! [A3: extended_ereal,B3: extended_ereal] :
      ( ( ( times_1966848393_ereal @ A3 @ B3 )
        = zero_z163181189_ereal )
      = ( ( A3 = zero_z163181189_ereal )
        | ( B3 = zero_z163181189_ereal ) ) ) ).

% ereal_zero_times
thf(fact_348_ereal__times__divide__eq__left,axiom,
    ! [B3: extended_ereal,C: extended_ereal,A3: extended_ereal] :
      ( ( times_1966848393_ereal @ ( divide595620860_ereal @ B3 @ C ) @ A3 )
      = ( divide595620860_ereal @ ( times_1966848393_ereal @ B3 @ A3 ) @ C ) ) ).

% ereal_times_divide_eq_left
thf(fact_349_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ( C = zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
          = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
          = ( divide_divide_real @ A3 @ B3 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_350_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
        = ( divide_divide_real @ A3 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_351_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ B3 @ C ) )
        = ( divide_divide_real @ A3 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_352_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
        = ( divide_divide_real @ A3 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_353_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ C @ B3 ) )
        = ( divide_divide_real @ A3 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( lower_311861424x_on_a @ top_top_set_a @ f )
    = ( convex_a @ ( lower_1391529426main_a @ f ) ) ) ).

%------------------------------------------------------------------------------