TPTP Problem File: ITP110^1.p
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%------------------------------------------------------------------------------
% File : ITP110^1 : TPTP v8.2.0. 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 : 0.40 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 ) ) ) ).
%------------------------------------------------------------------------------