TPTP Problem File: ITP110^2.p
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%------------------------------------------------------------------------------
% File : ITP110^2 : 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 : 0.67 v8.1.0, 0.50 v7.5.0
% Syntax : Number of formulae : 404 ( 131 unt; 62 typ; 0 def)
% Number of atoms : 969 ( 384 equ; 0 cnn)
% Maximal formula atoms : 81 ( 2 avg)
% Number of connectives : 3379 ( 92 ~; 16 |; 48 &;2734 @)
% ( 0 <=>; 489 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 6 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 125 ( 125 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 59 usr; 2 con; 0-5 aty)
% Number of variables : 881 ( 37 ^; 757 !; 35 ?; 881 :)
% ( 52 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:29:37.519
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_t_Extended__Real_Oereal,type,
extended_ereal: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (57)
thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
ordere890947078_space:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__lattice,type,
finite_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__bits,type,
semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
real_V1076094709vector:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Space_Oeuclidean__space,type,
euclid925273238_space:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V2090557954_space:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri1193490041visors:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri1923998003cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
euclid24285859cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit378418413attice:
!>[A: $tType] : $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Convex_Oconvex,type,
convex:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
extend1396239628finity:
!>[A: $tType] : A ).
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_Oereal_Orec__ereal,type,
extended_rec_ereal:
!>[A: $tType] : ( ( real > A ) > A > A > extended_ereal > A ) ).
thf(sy_c_Extended__Real_Oreal__of__ereal,type,
extend1716541707_ereal: extended_ereal > real ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_OEpigraph,type,
lower_261390618igraph:
!>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > ( set @ ( product_prod @ A @ real ) ) ) ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Oaffine__on,type,
lower_500881736ine_on:
!>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > $o ) ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Oconcave__on,type,
lower_708069572ave_on:
!>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > $o ) ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Oconvex__on,type,
lower_673667120vex_on:
!>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > $o ) ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Odomain,type,
lower_272802190domain:
!>[A: $tType] : ( ( A > extended_ereal ) > ( set @ A ) ) ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Ofinite__on,type,
lower_728871928ite_on:
!>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_f,type,
f: a > extended_ereal ).
% Relevant facts (255)
thf(fact_0__092_060open_062convex_A_Idomain_Af_J_A_092_060Longrightarrow_062_ALower__Semicontinuous__Mirabelle__quczrylfpw_Oconvex__on_AUNIV_Af_092_060close_062,axiom,
( ( convex @ a @ ( lower_272802190domain @ a @ f ) )
=> ( lower_673667120vex_on @ a @ ( top_top @ ( set @ a ) ) @ f ) ) ).
% \<open>convex (domain f) \<Longrightarrow> Lower_Semicontinuous_Mirabelle_quczrylfpw.convex_on UNIV f\<close>
thf(fact_1_convex__domain,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ! [F: A > extended_ereal] :
( ( lower_673667120vex_on @ A @ ( top_top @ ( set @ A ) ) @ F )
=> ( convex @ A @ ( lower_272802190domain @ A @ F ) ) ) ) ).
% convex_domain
thf(fact_2_convex__on__domain,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [F: A > extended_ereal] :
( ( lower_673667120vex_on @ A @ ( lower_272802190domain @ A @ F ) @ F )
= ( lower_673667120vex_on @ A @ ( top_top @ ( set @ A ) ) @ F ) ) ) ).
% convex_on_domain
thf(fact_3_convex__on__domain2,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [F: A > extended_ereal] :
( ( lower_673667120vex_on @ A @ ( lower_272802190domain @ A @ F ) @ F )
= ( ! [S: set @ A] : ( lower_673667120vex_on @ A @ S @ F ) ) ) ) ).
% convex_on_domain2
thf(fact_4_convex__on__ereal__univ,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [F: A > extended_ereal] :
( ( lower_673667120vex_on @ A @ ( top_top @ ( set @ A ) ) @ F )
= ( ! [S: set @ A] : ( lower_673667120vex_on @ A @ S @ F ) ) ) ) ).
% convex_on_ereal_univ
thf(fact_5_convex__UNIV,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( convex @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% convex_UNIV
thf(fact_6_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_7_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_8_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_9_assms,axiom,
! [X3: a] :
( ( ( f @ X3 )
= ( extend1396239628finity @ extended_ereal ) )
| ( ( f @ X3 )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% assms
thf(fact_10_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X4: A] : ( member @ A @ X4 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_11_UNIV__witness,axiom,
! [A: $tType] :
? [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_12_convex__Epigraph,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [S2: set @ A,F: A > extended_ereal] :
( ( convex @ A @ S2 )
=> ( ( convex @ ( product_prod @ A @ real ) @ ( lower_261390618igraph @ A @ S2 @ F ) )
= ( lower_673667120vex_on @ A @ S2 @ F ) ) ) ) ).
% convex_Epigraph
thf(fact_13_convex__EpigraphI,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [S3: set @ A,F: A > extended_ereal] :
( ( lower_673667120vex_on @ A @ S3 @ F )
=> ( ( convex @ A @ S3 )
=> ( convex @ ( product_prod @ A @ real ) @ ( lower_261390618igraph @ A @ S3 @ F ) ) ) ) ) ).
% convex_EpigraphI
thf(fact_14_affine__on__def,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( ( lower_500881736ine_on @ A )
= ( ^ [S: set @ A,F2: A > extended_ereal] :
( ( lower_673667120vex_on @ A @ S @ F2 )
& ( lower_708069572ave_on @ A @ S @ F2 )
& ( lower_728871928ite_on @ A @ S @ F2 ) ) ) ) ) ).
% affine_on_def
thf(fact_15_Sup__UNIV,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Sup_UNIV
thf(fact_16_convex__on__ereal__subset,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [T: set @ A,F: A > extended_ereal,S3: set @ A] :
( ( lower_673667120vex_on @ A @ T @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ T )
=> ( lower_673667120vex_on @ A @ S3 @ F ) ) ) ) ).
% convex_on_ereal_subset
thf(fact_17_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_18_subsetI,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( member @ A @ X4 @ B ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B ) ) ).
% subsetI
thf(fact_19_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_20_in__mono,axiom,
! [A: $tType,A2: set @ A,B: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B ) ) ) ).
% in_mono
thf(fact_21_subsetD,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B ) ) ) ).
% subsetD
thf(fact_22_equalityE,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( A2 = B )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ~ ( ord_less_eq @ ( set @ A ) @ B @ A2 ) ) ) ).
% equalityE
thf(fact_23_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B2: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_24_equalityD1,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( A2 = B )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B ) ) ).
% equalityD1
thf(fact_25_equalityD2,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( A2 = B )
=> ( ord_less_eq @ ( set @ A ) @ B @ A2 ) ) ).
% equalityD2
thf(fact_26_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B2: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A3 )
=> ( member @ A @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_27_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_28_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_29_subset__trans,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_30_le__funD,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 )
=> ! [F: A > B3,G: A > B3,X: A] :
( ( ord_less_eq @ ( A > B3 ) @ F @ G )
=> ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_31_le__funE,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 )
=> ! [F: A > B3,G: A > B3,X: A] :
( ( ord_less_eq @ ( A > B3 ) @ F @ G )
=> ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_32_le__funI,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 )
=> ! [F: A > B3,G: A > B3] :
( ! [X4: A] : ( ord_less_eq @ B3 @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ ( A > B3 ) @ F @ G ) ) ) ).
% le_funI
thf(fact_33_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y: set @ A,Z: set @ A] : Y = Z )
= ( ^ [A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
& ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_34_le__fun__def,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 )
=> ( ( ord_less_eq @ ( A > B3 ) )
= ( ^ [F2: A > B3,G2: A > B3] :
! [X2: A] : ( ord_less_eq @ B3 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_35_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_36_order__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 )
& ( order @ A ) )
=> ! [A4: A,F: B3 > A,B4: B3,C2: B3] :
( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq @ B3 @ B4 @ C2 )
=> ( ! [X4: B3,Y2: B3] :
( ( ord_less_eq @ B3 @ X4 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_37_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B4: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
=> ( ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_38_ord__eq__le__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 )
& ( ord @ A ) )
=> ! [A4: A,F: B3 > A,B4: B3,C2: B3] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq @ B3 @ B4 @ C2 )
=> ( ! [X4: B3,Y2: B3] :
( ( ord_less_eq @ B3 @ X4 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_39_ord__le__eq__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 )
& ( ord @ A ) )
=> ! [A4: A,B4: A,F: A > B3,C2: B3] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ord_less_eq @ B3 @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B3 @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_40_Union__mono,axiom,
! [A: $tType,A2: set @ ( set @ A ),B: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ B )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B ) ) ) ).
% Union_mono
thf(fact_41_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_42_Union__least,axiom,
! [A: $tType,A2: set @ ( set @ A ),C3: set @ A] :
( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ X5 @ C3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ C3 ) ) ).
% Union_least
thf(fact_43_Union__upper,axiom,
! [A: $tType,B: set @ A,A2: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ B @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% Union_upper
thf(fact_44_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ X )
=> ( X = Y4 ) ) ) ) ).
% antisym
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B3: $tType,A: $tType,F: A > B3,G: A > B3] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_Union__subsetI,axiom,
! [A: $tType,A2: set @ ( set @ A ),B: set @ ( set @ A )] :
( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A2 )
=> ? [Y5: set @ A] :
( ( member @ ( set @ A ) @ Y5 @ B )
& ( ord_less_eq @ ( set @ A ) @ X4 @ Y5 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B ) ) ) ).
% Union_subsetI
thf(fact_50_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
| ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% linear
thf(fact_51_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y4: A] :
( ( X = Y4 )
=> ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% eq_refl
thf(fact_52_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y4: A] :
( ~ ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% le_cases
thf(fact_53_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% order.trans
thf(fact_54_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y4: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y4 )
=> ~ ( ord_less_eq @ A @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y4 @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y4 )
=> ~ ( ord_less_eq @ A @ Y4 @ X ) )
=> ( ( ( ord_less_eq @ A @ Y4 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_55_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y4: A,X: A] :
( ( ord_less_eq @ A @ Y4 @ X )
=> ( ( ord_less_eq @ A @ X @ Y4 )
= ( X = Y4 ) ) ) ) ).
% antisym_conv
thf(fact_56_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_57_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( A4 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_58_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_59_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ) ).
% order_class.order.antisym
thf(fact_60_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y4: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_61_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_62_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A4: A,B4: A] :
( ! [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: A,B6: A] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A4 @ B4 ) ) ) ) ).
% linorder_wlog
thf(fact_63_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ B4 )
=> ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_64_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_65_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ) ).
% dual_order.antisym
thf(fact_66_Sup__eqI,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [A2: set @ A,X: A] :
( ! [Y2: A] :
( ( member @ A @ Y2 @ A2 )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ! [Y2: A] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ A2 )
=> ( ord_less_eq @ A @ Z3 @ Y2 ) )
=> ( ord_less_eq @ A @ X @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ A2 )
= X ) ) ) ) ).
% Sup_eqI
thf(fact_67_Sup__mono,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [A2: set @ A,B: set @ A] :
( ! [A6: A] :
( ( member @ A @ A6 @ A2 )
=> ? [X3: A] :
( ( member @ A @ X3 @ B )
& ( ord_less_eq @ A @ A6 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B ) ) ) ) ).
% Sup_mono
thf(fact_68_Sup__least,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [A2: set @ A,Z2: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( ord_less_eq @ A @ X4 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ Z2 ) ) ) ).
% Sup_least
thf(fact_69_Sup__upper,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [X: A,A2: set @ A] :
( ( member @ A @ X @ A2 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ).
% Sup_upper
thf(fact_70_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [A2: set @ A,B4: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ B4 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( ord_less_eq @ A @ X2 @ B4 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_71_Sup__upper2,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [U: A,A2: set @ A,V: A] :
( ( member @ A @ U @ A2 )
=> ( ( ord_less_eq @ A @ V @ U )
=> ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% Sup_upper2
thf(fact_72_finite__on__def,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( ( lower_728871928ite_on @ A )
= ( ^ [S: set @ A,F2: A > extended_ereal] :
! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( ( ( F2 @ X2 )
!= ( extend1396239628finity @ extended_ereal ) )
& ( ( F2 @ X2 )
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ).
% finite_on_def
thf(fact_73_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [A2: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B ) ) ) ) ).
% Sup_subset_mono
thf(fact_74_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_75_Union__UNIV,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Union_UNIV
thf(fact_76_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
=> ( A4
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_77_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
= ( A4
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_78_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_79_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_80_epigraph__invertible,axiom,
! [A: $tType,F: A > extended_ereal,G: A > extended_ereal] :
( ( ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ F )
= ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ G ) )
=> ( F = G ) ) ).
% epigraph_invertible
thf(fact_81_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B4: A,A4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% neg_le_iff_le
thf(fact_82_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y4 ) )
= ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_83_top__finite__def,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ( ( top_top @ A )
= ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_finite_def
thf(fact_84_ereal__uminus__eq__iff,axiom,
! [A4: extended_ereal,B4: extended_ereal] :
( ( ( uminus_uminus @ extended_ereal @ A4 )
= ( uminus_uminus @ extended_ereal @ B4 ) )
= ( A4 = B4 ) ) ).
% ereal_uminus_eq_iff
thf(fact_85_ereal__uminus__uminus,axiom,
! [A4: extended_ereal] :
( ( uminus_uminus @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A4 ) )
= A4 ) ).
% ereal_uminus_uminus
thf(fact_86_verit__minus__simplify_I4_J,axiom,
! [B3: $tType] :
( ( group_add @ B3 )
=> ! [B4: B3] :
( ( uminus_uminus @ B3 @ ( uminus_uminus @ B3 @ B4 ) )
= B4 ) ) ).
% verit_minus_simplify(4)
thf(fact_87_uminus__apply,axiom,
! [B3: $tType,A: $tType] :
( ( uminus @ B3 )
=> ( ( uminus_uminus @ ( A > B3 ) )
= ( ^ [A3: A > B3,X2: A] : ( uminus_uminus @ B3 @ ( A3 @ X2 ) ) ) ) ) ).
% uminus_apply
thf(fact_88_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A4 ) )
= A4 ) ) ).
% add.inverse_inverse
thf(fact_89_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B4: A] :
( ( ( uminus_uminus @ A @ A4 )
= ( uminus_uminus @ A @ B4 ) )
= ( A4 = B4 ) ) ) ).
% neg_equal_iff_equal
thf(fact_90_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
= X ) ) ).
% double_compl
thf(fact_91_compl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X: A,Y4: A] :
( ( ( uminus_uminus @ A @ X )
= ( uminus_uminus @ A @ Y4 ) )
= ( X = Y4 ) ) ) ).
% compl_eq_compl_iff
thf(fact_92_Compl__subset__Compl__iff,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B ) )
= ( ord_less_eq @ ( set @ A ) @ B @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_93_Compl__anti__mono,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_94_ereal__infty__less__eq_I1_J,axiom,
! [X: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ X )
= ( X
= ( extend1396239628finity @ extended_ereal ) ) ) ).
% ereal_infty_less_eq(1)
thf(fact_95_UN__ball__bex__simps_I3_J,axiom,
! [D: $tType,A2: set @ ( set @ D ),P: D > $o] :
( ( ? [X2: D] :
( ( member @ D @ X2 @ ( complete_Sup_Sup @ ( set @ D ) @ A2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set @ D] :
( ( member @ ( set @ D ) @ X2 @ A2 )
& ? [Y3: D] :
( ( member @ D @ Y3 @ X2 )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_96_UN__ball__bex__simps_I1_J,axiom,
! [A: $tType,A2: set @ ( set @ A ),P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A2 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ X2 )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_97_UnionI,axiom,
! [A: $tType,X6: set @ A,C3: set @ ( set @ A ),A2: A] :
( ( member @ ( set @ A ) @ X6 @ C3 )
=> ( ( member @ A @ A2 @ X6 )
=> ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) ) ) ) ).
% UnionI
thf(fact_98_Union__iff,axiom,
! [A: $tType,A2: A,C3: set @ ( set @ A )] :
( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
= ( ? [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C3 )
& ( member @ A @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_99_ereal__minus__le__minus,axiom,
! [A4: extended_ereal,B4: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A4 ) @ ( uminus_uminus @ extended_ereal @ B4 ) )
= ( ord_less_eq @ extended_ereal @ B4 @ A4 ) ) ).
% ereal_minus_le_minus
thf(fact_100_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_101_ereal__infty__less__eq_I2_J,axiom,
! [X: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ X @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
= ( X
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% ereal_infty_less_eq(2)
thf(fact_102_UnionE,axiom,
! [A: $tType,A2: A,C3: set @ ( set @ A )] :
( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
=> ~ ! [X5: set @ A] :
( ( member @ A @ A2 @ X5 )
=> ~ ( member @ ( set @ A ) @ X5 @ C3 ) ) ) ).
% UnionE
thf(fact_103_neq__PInf__trans,axiom,
! [Y4: extended_ereal,X: extended_ereal] :
( ( Y4
!= ( extend1396239628finity @ extended_ereal ) )
=> ( ( ord_less_eq @ extended_ereal @ X @ Y4 )
=> ( X
!= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% neq_PInf_trans
thf(fact_104_ereal__infty__less__eq2_I1_J,axiom,
! [A4: extended_ereal,B4: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ A4 @ B4 )
=> ( ( A4
= ( extend1396239628finity @ extended_ereal ) )
=> ( B4
= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% ereal_infty_less_eq2(1)
thf(fact_105_ereal__less__eq_I1_J,axiom,
! [X: extended_ereal] : ( ord_less_eq @ extended_ereal @ X @ ( extend1396239628finity @ extended_ereal ) ) ).
% ereal_less_eq(1)
thf(fact_106_ereal__uminus__le__reorder,axiom,
! [A4: extended_ereal,B4: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A4 ) @ B4 )
= ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ B4 ) @ A4 ) ) ).
% ereal_uminus_le_reorder
thf(fact_107_ereal__infty__less__eq2_I2_J,axiom,
! [A4: extended_ereal,B4: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ A4 @ B4 )
=> ( ( B4
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( A4
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% ereal_infty_less_eq2(2)
thf(fact_108_ereal__less__eq_I2_J,axiom,
! [X: extended_ereal] : ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ X ) ).
% ereal_less_eq(2)
thf(fact_109_epigraph__subset__iff,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [F: A > extended_ereal,G: A > extended_ereal] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ real ) ) @ ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ F ) @ ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ G ) )
= ( ! [X2: A] : ( ord_less_eq @ extended_ereal @ ( G @ X2 ) @ ( F @ X2 ) ) ) ) ) ).
% epigraph_subset_iff
thf(fact_110_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B4: A] :
( ( A4 = B4 )
| ~ ( ord_less_eq @ A @ A4 @ B4 )
| ~ ( ord_less_eq @ A @ B4 @ A4 ) ) ) ).
% verit_la_disequality
thf(fact_111_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B4: A] :
( ( ( uminus_uminus @ A @ A4 )
= B4 )
= ( ( uminus_uminus @ A @ B4 )
= A4 ) ) ) ).
% minus_equation_iff
thf(fact_112_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B4: A] :
( ( A4
= ( uminus_uminus @ A @ B4 ) )
= ( B4
= ( uminus_uminus @ A @ A4 ) ) ) ) ).
% equation_minus_iff
thf(fact_113_fun__Compl__def,axiom,
! [B3: $tType,A: $tType] :
( ( uminus @ B3 )
=> ( ( uminus_uminus @ ( A > B3 ) )
= ( ^ [A3: A > B3,X2: A] : ( uminus_uminus @ B3 @ ( A3 @ X2 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_114_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B4: A] :
( ( A4 = B4 )
=> ( ( uminus_uminus @ A @ A4 )
= ( uminus_uminus @ A @ B4 ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_115_top__ereal__def,axiom,
( ( top_top @ extended_ereal )
= ( extend1396239628finity @ extended_ereal ) ) ).
% top_ereal_def
thf(fact_116_Sup__eq__PInfty,axiom,
! [S2: set @ extended_ereal] :
( ( member @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ S2 )
=> ( ( complete_Sup_Sup @ extended_ereal @ S2 )
= ( extend1396239628finity @ extended_ereal ) ) ) ).
% Sup_eq_PInfty
thf(fact_117_ereal__uminus__eq__reorder,axiom,
! [A4: extended_ereal,B4: extended_ereal] :
( ( ( uminus_uminus @ extended_ereal @ A4 )
= B4 )
= ( A4
= ( uminus_uminus @ extended_ereal @ B4 ) ) ) ).
% ereal_uminus_eq_reorder
thf(fact_118_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% le_imp_neg_le
thf(fact_119_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ B4 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ A4 ) ) ) ).
% minus_le_iff
thf(fact_120_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ B4 ) )
= ( ord_less_eq @ A @ B4 @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% le_minus_iff
thf(fact_121_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y4: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y4 ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y4 ) ) ) ).
% compl_le_swap2
thf(fact_122_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y4: A,X: A] :
( ( ord_less_eq @ A @ Y4 @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y4 ) ) ) ) ).
% compl_le_swap1
thf(fact_123_compl__mono,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y4 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_124_MInfty__neq__PInfty_I1_J,axiom,
( ( extend1396239628finity @ extended_ereal )
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% MInfty_neq_PInfty(1)
thf(fact_125_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_126_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit378418413attice @ A )
=> ! [Z2: A,X6: set @ A] :
( ( member @ A @ Z2 @ X6 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X6 )
=> ( ord_less_eq @ A @ X4 @ Z2 ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= Z2 ) ) ) ) ).
% cSup_eq_maximum
thf(fact_127_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit378418413attice @ A )
& ( no_bot @ A ) )
=> ! [X6: set @ A,A4: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X6 )
=> ( ord_less_eq @ A @ X4 @ A4 ) )
=> ( ! [Y2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ X3 @ Y2 ) )
=> ( ord_less_eq @ A @ A4 @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= A4 ) ) ) ) ).
% cSup_eq
thf(fact_128_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X: A] :
( ( P
& ( top_top @ ( A > $o ) @ X ) )
= P ) ).
% top_conj(2)
thf(fact_129_top__conj_I1_J,axiom,
! [A: $tType,X: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X )
& P )
= P ) ).
% top_conj(1)
thf(fact_130_MInfty__eq__minfinity,axiom,
( extended_MInfty
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% MInfty_eq_minfinity
thf(fact_131_ComplI,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ~ ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% ComplI
thf(fact_132_Compl__iff,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= ( ~ ( member @ A @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_133_Compl__eq__Compl__iff,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A2 )
= ( uminus_uminus @ ( set @ A ) @ B ) )
= ( A2 = B ) ) ).
% Compl_eq_Compl_iff
thf(fact_134_ComplD,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
=> ~ ( member @ A @ C2 @ A2 ) ) ).
% ComplD
thf(fact_135_double__complement,axiom,
! [A: $tType,A2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= A2 ) ).
% double_complement
thf(fact_136_ereal__complete__Inf,axiom,
! [S2: set @ extended_ereal] :
? [X4: extended_ereal] :
( ! [Xa: extended_ereal] :
( ( member @ extended_ereal @ Xa @ S2 )
=> ( ord_less_eq @ extended_ereal @ X4 @ Xa ) )
& ! [Z3: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member @ extended_ereal @ Xa2 @ S2 )
=> ( ord_less_eq @ extended_ereal @ Z3 @ Xa2 ) )
=> ( ord_less_eq @ extended_ereal @ Z3 @ X4 ) ) ) ).
% ereal_complete_Inf
thf(fact_137_ereal__complete__Sup,axiom,
! [S2: set @ extended_ereal] :
? [X4: extended_ereal] :
( ! [Xa: extended_ereal] :
( ( member @ extended_ereal @ Xa @ S2 )
=> ( ord_less_eq @ extended_ereal @ Xa @ X4 ) )
& ! [Z3: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member @ extended_ereal @ Xa2 @ S2 )
=> ( ord_less_eq @ extended_ereal @ Xa2 @ Z3 ) )
=> ( ord_less_eq @ extended_ereal @ X4 @ Z3 ) ) ) ).
% ereal_complete_Sup
thf(fact_138_uminus__ereal_Osimps_I3_J,axiom,
( ( uminus_uminus @ extended_ereal @ extended_MInfty )
= extended_PInfty ) ).
% uminus_ereal.simps(3)
thf(fact_139_uminus__ereal_Osimps_I2_J,axiom,
( ( uminus_uminus @ extended_ereal @ extended_PInfty )
= extended_MInfty ) ).
% uminus_ereal.simps(2)
thf(fact_140_exists__diff,axiom,
! [A: $tType,P: ( set @ A ) > $o] :
( ( ? [S: set @ A] : ( P @ ( uminus_uminus @ ( set @ A ) @ S ) ) )
= ( ? [X7: set @ A] : ( P @ X7 ) ) ) ).
% exists_diff
thf(fact_141_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_142_infinity__ereal__def,axiom,
( ( extend1396239628finity @ extended_ereal )
= extended_PInfty ) ).
% infinity_ereal_def
thf(fact_143_ereal_Odistinct_I5_J,axiom,
extended_PInfty != extended_MInfty ).
% ereal.distinct(5)
thf(fact_144_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ( order_Greatest @ A @ P )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_145_uminus__ereal_Oelims,axiom,
! [X: extended_ereal,Y4: extended_ereal] :
( ( ( uminus_uminus @ extended_ereal @ X )
= Y4 )
=> ( ! [R: real] :
( ( X
= ( extended_ereal2 @ R ) )
=> ( Y4
!= ( extended_ereal2 @ ( uminus_uminus @ real @ R ) ) ) )
=> ( ( ( X = extended_PInfty )
=> ( Y4 != extended_MInfty ) )
=> ~ ( ( X = extended_MInfty )
=> ( Y4 != extended_PInfty ) ) ) ) ) ).
% uminus_ereal.elims
thf(fact_146_not__MInfty__nonneg,axiom,
! [X: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ X )
=> ( X
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% not_MInfty_nonneg
thf(fact_147_ereal_Osimps_I12_J,axiom,
! [A: $tType,F1: real > A,F22: A,F3: A] :
( ( extended_rec_ereal @ A @ F1 @ F22 @ F3 @ extended_PInfty )
= F22 ) ).
% ereal.simps(12)
thf(fact_148_ereal_Osimps_I13_J,axiom,
! [A: $tType,F1: real > A,F22: A,F3: A] :
( ( extended_rec_ereal @ A @ F1 @ F22 @ F3 @ extended_MInfty )
= F3 ) ).
% ereal.simps(13)
thf(fact_149_ereal_Oinject,axiom,
! [X1: real,Y1: real] :
( ( ( extended_ereal2 @ X1 )
= ( extended_ereal2 @ Y1 ) )
= ( X1 = Y1 ) ) ).
% ereal.inject
thf(fact_150_ereal__cong,axiom,
! [X: real,Y4: real] :
( ( X = Y4 )
=> ( ( extended_ereal2 @ X )
= ( extended_ereal2 @ Y4 ) ) ) ).
% ereal_cong
thf(fact_151_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_152_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A4: A] :
( ( ( uminus_uminus @ A @ A4 )
= A4 )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_153_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A4: A] :
( ( A4
= ( uminus_uminus @ A @ A4 ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_154_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( ( uminus_uminus @ A @ A4 )
= ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_155_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A4 ) )
= ( ( zero_zero @ A )
= A4 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_156_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_157_ereal__less__eq_I3_J,axiom,
! [R2: real,P2: real] :
( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
= ( ord_less_eq @ real @ R2 @ P2 ) ) ).
% ereal_less_eq(3)
thf(fact_158_ereal__uminus__zero,axiom,
( ( uminus_uminus @ extended_ereal @ ( zero_zero @ extended_ereal ) )
= ( zero_zero @ extended_ereal ) ) ).
% ereal_uminus_zero
thf(fact_159_ereal__uminus__zero__iff,axiom,
! [A4: extended_ereal] :
( ( ( uminus_uminus @ extended_ereal @ A4 )
= ( zero_zero @ extended_ereal ) )
= ( A4
= ( zero_zero @ extended_ereal ) ) ) ).
% ereal_uminus_zero_iff
thf(fact_160_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_161_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% neg_le_0_iff_le
thf(fact_162_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_163_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ A4 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_164_ereal__uminus__le__0__iff,axiom,
! [A4: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A4 ) @ ( zero_zero @ extended_ereal ) )
= ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A4 ) ) ).
% ereal_uminus_le_0_iff
thf(fact_165_ereal__0__le__uminus__iff,axiom,
! [A4: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( uminus_uminus @ extended_ereal @ A4 ) )
= ( ord_less_eq @ extended_ereal @ A4 @ ( zero_zero @ extended_ereal ) ) ) ).
% ereal_0_le_uminus_iff
thf(fact_166_ereal_Odistinct_I1_J,axiom,
! [X1: real] :
( ( extended_ereal2 @ X1 )
!= extended_PInfty ) ).
% ereal.distinct(1)
thf(fact_167_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_168_ereal_Osimps_I11_J,axiom,
! [A: $tType,F1: real > A,F22: A,F3: A,X1: real] :
( ( extended_rec_ereal @ A @ F1 @ F22 @ F3 @ ( extended_ereal2 @ X1 ) )
= ( F1 @ X1 ) ) ).
% ereal.simps(11)
thf(fact_169_Infty__neq__0_I1_J,axiom,
( ( extend1396239628finity @ extended_ereal )
!= ( zero_zero @ extended_ereal ) ) ).
% Infty_neq_0(1)
thf(fact_170_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_171_uminus__ereal_Osimps_I1_J,axiom,
! [R2: real] :
( ( uminus_uminus @ extended_ereal @ ( extended_ereal2 @ R2 ) )
= ( extended_ereal2 @ ( uminus_uminus @ real @ R2 ) ) ) ).
% uminus_ereal.simps(1)
thf(fact_172_PInfty__neq__ereal_I1_J,axiom,
! [R2: real] :
( ( extended_ereal2 @ R2 )
!= ( extend1396239628finity @ extended_ereal ) ) ).
% PInfty_neq_ereal(1)
thf(fact_173_ereal_Odistinct_I3_J,axiom,
! [X1: real] :
( ( extended_ereal2 @ X1 )
!= extended_MInfty ) ).
% ereal.distinct(3)
thf(fact_174_ereal__le__real,axiom,
! [X: extended_ereal,Y4: extended_ereal] :
( ! [Z4: real] :
( ( ord_less_eq @ extended_ereal @ X @ ( extended_ereal2 @ Z4 ) )
=> ( ord_less_eq @ extended_ereal @ Y4 @ ( extended_ereal2 @ Z4 ) ) )
=> ( ord_less_eq @ extended_ereal @ Y4 @ X ) ) ).
% ereal_le_real
thf(fact_175_le__ereal__le,axiom,
! [A4: extended_ereal,X: real,Y4: real] :
( ( ord_less_eq @ extended_ereal @ A4 @ ( extended_ereal2 @ X ) )
=> ( ( ord_less_eq @ real @ X @ Y4 )
=> ( ord_less_eq @ extended_ereal @ A4 @ ( extended_ereal2 @ Y4 ) ) ) ) ).
% le_ereal_le
thf(fact_176_ereal__le__le,axiom,
! [Y4: real,A4: extended_ereal,X: real] :
( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ Y4 ) @ A4 )
=> ( ( ord_less_eq @ real @ X @ Y4 )
=> ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ X ) @ A4 ) ) ) ).
% ereal_le_le
thf(fact_177_MInfty__neq__ereal_I1_J,axiom,
! [R2: real] :
( ( extended_ereal2 @ R2 )
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% MInfty_neq_ereal(1)
thf(fact_178_ereal__cases,axiom,
! [X: extended_ereal] :
( ! [R: real] :
( X
!= ( extended_ereal2 @ R ) )
=> ( ( X
!= ( extend1396239628finity @ extended_ereal ) )
=> ( X
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% ereal_cases
thf(fact_179_ereal2__cases,axiom,
! [X: extended_ereal,Xa3: extended_ereal] :
( ( ? [R: real] :
( X
= ( extended_ereal2 @ R ) )
=> ! [Ra: real] :
( Xa3
!= ( extended_ereal2 @ Ra ) ) )
=> ( ( ? [R: real] :
( X
= ( extended_ereal2 @ R ) )
=> ( Xa3
!= ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( ? [R: real] :
( X
= ( extended_ereal2 @ R ) )
=> ( Xa3
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X
= ( extend1396239628finity @ extended_ereal ) )
=> ! [R: real] :
( Xa3
!= ( extended_ereal2 @ R ) ) )
=> ( ( ( X
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xa3
!= ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( ( X
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xa3
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ! [R: real] :
( Xa3
!= ( extended_ereal2 @ R ) ) )
=> ( ( ( X
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xa3
!= ( extend1396239628finity @ extended_ereal ) ) )
=> ~ ( ( X
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xa3
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ).
% ereal2_cases
thf(fact_180_ereal3__cases,axiom,
! [X: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
( ( ? [R: real] :
( X
= ( extended_ereal2 @ R ) )
=> ( ? [Ra: real] :
( Xa3
= ( extended_ereal2 @ Ra ) )
=> ! [Rb: real] :
( Xb
!= ( extended_ereal2 @ Rb ) ) ) )
=> ( ( ? [R: real] :
( X
= ( extended_ereal2 @ R ) )
=> ( ? [Ra: real] :
( Xa3
= ( extended_ereal2 @ Ra ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ? [R: real] :
( X
= ( extended_ereal2 @ R ) )
=> ( ? [Ra: real] :
( Xa3
= ( extended_ereal2 @ Ra ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ? [R: real] :
( X
= ( extended_ereal2 @ R ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ? [R: real] :
( X
= ( extended_ereal2 @ R ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ? [R: real] :
( X
= ( extended_ereal2 @ R ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ? [R: real] :
( X
= ( extended_ereal2 @ R ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ? [R: real] :
( X
= ( extended_ereal2 @ R ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ? [R: real] :
( X
= ( extended_ereal2 @ R ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X
= ( extend1396239628finity @ extended_ereal ) )
=> ( ? [R: real] :
( Xa3
= ( extended_ereal2 @ R ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ( X
= ( extend1396239628finity @ extended_ereal ) )
=> ( ? [R: real] :
( Xa3
= ( extended_ereal2 @ R ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X
= ( extend1396239628finity @ extended_ereal ) )
=> ( ? [R: real] :
( Xa3
= ( extended_ereal2 @ R ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ! [R: real] :
( Xb
!= ( extended_ereal2 @ R ) ) ) )
=> ( ( ( X
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ! [R: real] :
( Xb
!= ( extended_ereal2 @ R ) ) ) )
=> ( ( ( X
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ? [R: real] :
( Xa3
= ( extended_ereal2 @ R ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ( X
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ? [R: real] :
( Xa3
= ( extended_ereal2 @ R ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ? [R: real] :
( Xa3
= ( extended_ereal2 @ R ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ! [R: real] :
( Xb
!= ( extended_ereal2 @ R ) ) ) )
=> ( ( ( X
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ! [R: real] :
( Xb
!= ( extended_ereal2 @ R ) ) ) )
=> ( ( ( X
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ~ ( ( X
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ereal3_cases
thf(fact_181_ereal__ex__split,axiom,
( ( ^ [P3: extended_ereal > $o] :
? [X7: extended_ereal] : ( P3 @ X7 ) )
= ( ^ [P4: extended_ereal > $o] :
( ( P4 @ ( extend1396239628finity @ extended_ereal ) )
| ? [X2: real] : ( P4 @ ( extended_ereal2 @ X2 ) )
| ( P4 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% ereal_ex_split
thf(fact_182_abs__ereal_Ocases,axiom,
! [X: extended_ereal] :
( ! [R: real] :
( X
!= ( extended_ereal2 @ R ) )
=> ( ( X
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( X
= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% abs_ereal.cases
thf(fact_183_ereal__all__split,axiom,
( ( ^ [P3: extended_ereal > $o] :
! [X7: extended_ereal] : ( P3 @ X7 ) )
= ( ^ [P4: extended_ereal > $o] :
( ( P4 @ ( extend1396239628finity @ extended_ereal ) )
& ! [X2: real] : ( P4 @ ( extended_ereal2 @ X2 ) )
& ( P4 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% ereal_all_split
thf(fact_184_abs__ereal_Oinduct,axiom,
! [P: extended_ereal > $o,A0: extended_ereal] :
( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( P @ ( extend1396239628finity @ extended_ereal ) )
=> ( P @ A0 ) ) ) ) ).
% abs_ereal.induct
thf(fact_185_less__ereal_Oinduct,axiom,
! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
( ! [X4: real,Y2: real] : ( P @ ( extended_ereal2 @ X4 ) @ ( extended_ereal2 @ Y2 ) )
=> ( ! [X_1: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_1 )
=> ( ! [A6: extended_ereal] : ( P @ A6 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ! [X4: real] : ( P @ ( extended_ereal2 @ X4 ) @ ( extend1396239628finity @ extended_ereal ) )
=> ( ! [R: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% less_ereal.induct
thf(fact_186_plus__ereal_Oinduct,axiom,
! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
( ! [R: real,P5: real] : ( P @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P5 ) )
=> ( ! [X_1: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_1 )
=> ( ! [A6: extended_ereal] : ( P @ A6 @ ( extend1396239628finity @ extended_ereal ) )
=> ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ! [P5: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ P5 ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% plus_ereal.induct
thf(fact_187_times__ereal_Oinduct,axiom,
! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
( ! [R: real,P5: real] : ( P @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P5 ) )
=> ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ ( extend1396239628finity @ extended_ereal ) )
=> ( ! [R: real] : ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R ) )
=> ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ! [R: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R ) )
=> ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extend1396239628finity @ extended_ereal ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
=> ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).
% times_ereal.induct
thf(fact_188_real__of__ereal_Ocases,axiom,
! [X: extended_ereal] :
( ! [R: real] :
( X
!= ( extended_ereal2 @ R ) )
=> ( ( X
!= ( extend1396239628finity @ extended_ereal ) )
=> ( X
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% real_of_ereal.cases
thf(fact_189_real__of__ereal_Oinduct,axiom,
! [P: extended_ereal > $o,A0: extended_ereal] :
( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
=> ( ( P @ ( extend1396239628finity @ extended_ereal ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( P @ A0 ) ) ) ) ).
% real_of_ereal.induct
thf(fact_190_domain__Epigraph__aux,axiom,
! [X: extended_ereal] :
( ( X
!= ( extend1396239628finity @ extended_ereal ) )
=> ? [R: real] : ( ord_less_eq @ extended_ereal @ X @ ( extended_ereal2 @ R ) ) ) ).
% domain_Epigraph_aux
thf(fact_191_ereal__top,axiom,
! [X: extended_ereal] :
( ! [B7: real] : ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ B7 ) @ X )
=> ( X
= ( extend1396239628finity @ extended_ereal ) ) ) ).
% ereal_top
thf(fact_192_Infty__neq__0_I3_J,axiom,
( ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
!= ( zero_zero @ extended_ereal ) ) ).
% Infty_neq_0(3)
thf(fact_193_ereal_Oinduct,axiom,
! [P: extended_ereal > $o,Ereal: extended_ereal] :
( ! [X4: real] : ( P @ ( extended_ereal2 @ X4 ) )
=> ( ( P @ extended_PInfty )
=> ( ( P @ extended_MInfty )
=> ( P @ Ereal ) ) ) ) ).
% ereal.induct
thf(fact_194_ereal_Oexhaust,axiom,
! [Y4: extended_ereal] :
( ! [X12: real] :
( Y4
!= ( extended_ereal2 @ X12 ) )
=> ( ( Y4 != extended_PInfty )
=> ( Y4 = extended_MInfty ) ) ) ).
% ereal.exhaust
thf(fact_195_uminus__ereal_Ocases,axiom,
! [X: extended_ereal] :
( ! [R: real] :
( X
!= ( extended_ereal2 @ R ) )
=> ( ( X != extended_PInfty )
=> ( X = extended_MInfty ) ) ) ).
% uminus_ereal.cases
thf(fact_196_uminus__ereal_Oinduct,axiom,
! [P: extended_ereal > $o,A0: extended_ereal] :
( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
=> ( ( P @ extended_PInfty )
=> ( ( P @ extended_MInfty )
=> ( P @ A0 ) ) ) ) ).
% uminus_ereal.induct
thf(fact_197_ereal__bot,axiom,
! [X: extended_ereal] :
( ! [B7: real] : ( ord_less_eq @ extended_ereal @ X @ ( extended_ereal2 @ B7 ) )
=> ( X
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% ereal_bot
thf(fact_198_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_199_ereal__less__eq_I4_J,axiom,
! [R2: real] :
( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ R2 ) @ ( zero_zero @ extended_ereal ) )
= ( ord_less_eq @ real @ R2 @ ( zero_zero @ real ) ) ) ).
% ereal_less_eq(4)
thf(fact_200_ereal__less__eq_I5_J,axiom,
! [R2: real] :
( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( extended_ereal2 @ R2 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 ) ) ).
% ereal_less_eq(5)
thf(fact_201_ereal__eq__0_I2_J,axiom,
! [R2: real] :
( ( ( zero_zero @ extended_ereal )
= ( extended_ereal2 @ R2 ) )
= ( R2
= ( zero_zero @ real ) ) ) ).
% ereal_eq_0(2)
thf(fact_202_ereal__eq__0_I1_J,axiom,
! [R2: real] :
( ( ( extended_ereal2 @ R2 )
= ( zero_zero @ extended_ereal ) )
= ( R2
= ( zero_zero @ real ) ) ) ).
% ereal_eq_0(1)
thf(fact_203_zero__ereal__def,axiom,
( ( zero_zero @ extended_ereal )
= ( extended_ereal2 @ ( zero_zero @ real ) ) ) ).
% zero_ereal_def
thf(fact_204_real__eq__0__iff__le__ge__0,axiom,
! [X: real] :
( ( X
= ( zero_zero @ real ) )
= ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( uminus_uminus @ real @ X ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_205_ereal__divide__ereal,axiom,
! [R2: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( divide_divide @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R2 ) )
= ( extend1396239628finity @ extended_ereal ) ) )
& ( ~ ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( divide_divide @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R2 ) )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% ereal_divide_ereal
thf(fact_206_ereal__uminus__divide,axiom,
! [X: extended_ereal,Y4: extended_ereal] :
( ( divide_divide @ extended_ereal @ ( uminus_uminus @ extended_ereal @ X ) @ Y4 )
= ( uminus_uminus @ extended_ereal @ ( divide_divide @ extended_ereal @ X @ Y4 ) ) ) ).
% ereal_uminus_divide
thf(fact_207_ereal__divide__zero__left,axiom,
! [A4: extended_ereal] :
( ( divide_divide @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A4 )
= ( zero_zero @ extended_ereal ) ) ).
% ereal_divide_zero_left
thf(fact_208_ereal__divide__Infty_I1_J,axiom,
! [X: extended_ereal] :
( ( divide_divide @ extended_ereal @ X @ ( extend1396239628finity @ extended_ereal ) )
= ( zero_zero @ extended_ereal ) ) ).
% ereal_divide_Infty(1)
thf(fact_209_ereal__divide__Infty_I2_J,axiom,
! [X: extended_ereal] :
( ( divide_divide @ extended_ereal @ X @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
= ( zero_zero @ extended_ereal ) ) ).
% ereal_divide_Infty(2)
thf(fact_210_zero__le__divide__ereal,axiom,
! [A4: extended_ereal,B4: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A4 )
=> ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ B4 )
=> ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( divide_divide @ extended_ereal @ A4 @ B4 ) ) ) ) ).
% zero_le_divide_ereal
thf(fact_211_div__minus__minus,axiom,
! [A: $tType] :
( ( euclid24285859cancel @ A )
=> ! [A4: A,B4: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ A @ B4 ) )
= ( divide_divide @ A @ A4 @ B4 ) ) ) ).
% div_minus_minus
thf(fact_212_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A4: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A4 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_213_div__by__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A4: A] :
( ( divide_divide @ A @ A4 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% div_by_0
thf(fact_214_div__minus__right,axiom,
! [A: $tType] :
( ( euclid24285859cancel @ A )
=> ! [A4: A,B4: A] :
( ( divide_divide @ A @ A4 @ ( uminus_uminus @ A @ B4 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A4 ) @ B4 ) ) ) ).
% div_minus_right
thf(fact_215_division__ring__divide__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A] :
( ( divide_divide @ A @ A4 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% division_ring_divide_zero
thf(fact_216_bits__div__by__0,axiom,
! [A: $tType] :
( ( semiring_bits @ A )
=> ! [A4: A] :
( ( divide_divide @ A @ A4 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_div_by_0
thf(fact_217_divide__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B4: A] :
( ( ( divide_divide @ A @ A4 @ B4 )
= ( zero_zero @ A ) )
= ( ( A4
= ( zero_zero @ A ) )
| ( B4
= ( zero_zero @ A ) ) ) ) ) ).
% divide_eq_0_iff
thf(fact_218_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A4: A,B4: A] :
( ( ( divide_divide @ A @ C2 @ A4 )
= ( divide_divide @ A @ C2 @ B4 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A4 = B4 ) ) ) ) ).
% divide_cancel_left
thf(fact_219_bits__div__0,axiom,
! [A: $tType] :
( ( semiring_bits @ A )
=> ! [A4: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A4 )
= ( zero_zero @ A ) ) ) ).
% bits_div_0
thf(fact_220_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,C2: A,B4: A] :
( ( ( divide_divide @ A @ A4 @ C2 )
= ( divide_divide @ A @ B4 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A4 = B4 ) ) ) ) ).
% divide_cancel_right
thf(fact_221_minus__divide__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B4: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ B4 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A4 ) @ B4 ) ) ) ).
% minus_divide_left
thf(fact_222_minus__divide__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B4: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ A @ B4 ) )
= ( divide_divide @ A @ A4 @ B4 ) ) ) ).
% minus_divide_divide
thf(fact_223_minus__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B4: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ B4 ) )
= ( divide_divide @ A @ A4 @ ( uminus_uminus @ A @ B4 ) ) ) ) ).
% minus_divide_right
thf(fact_224_divide__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ B4 @ C2 ) @ ( divide_divide @ A @ A4 @ C2 ) ) ) ) ) ).
% divide_right_mono_neg
thf(fact_225_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y4 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y4 ) ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_226_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y4 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_nonneg
thf(fact_227_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ Y4 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_nonpos
thf(fact_228_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y4 ) ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_229_zero__le__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A4 @ B4 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) )
| ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_divide_iff
thf(fact_230_divide__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ A4 @ C2 ) @ ( divide_divide @ A @ B4 @ C2 ) ) ) ) ) ).
% divide_right_mono
thf(fact_231_divide__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A4 @ B4 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) ) ) ) ) ).
% divide_le_0_iff
thf(fact_232_nonzero__minus__divide__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B4: A,A4: A] :
( ( B4
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ A @ B4 ) )
= ( divide_divide @ A @ A4 @ B4 ) ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_233_nonzero__minus__divide__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B4: A,A4: A] :
( ( B4
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ B4 ) )
= ( divide_divide @ A @ A4 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_234_real__of__ereal__le__0,axiom,
! [X: extended_ereal] :
( ( ord_less_eq @ real @ ( extend1716541707_ereal @ X ) @ ( zero_zero @ real ) )
= ( ( ord_less_eq @ extended_ereal @ X @ ( zero_zero @ extended_ereal ) )
| ( X
= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% real_of_ereal_le_0
thf(fact_235_ereal__divide,axiom,
! [P2: real,R2: real] :
( ( ( P2
= ( zero_zero @ real ) )
=> ( ( divide_divide @ extended_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
= ( times_times @ extended_ereal @ ( extended_ereal2 @ R2 ) @ ( extend1396239628finity @ extended_ereal ) ) ) )
& ( ( P2
!= ( zero_zero @ real ) )
=> ( ( divide_divide @ extended_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
= ( extended_ereal2 @ ( divide_divide @ real @ R2 @ P2 ) ) ) ) ) ).
% ereal_divide
thf(fact_236_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A4 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_237_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A4: A] :
( ( times_times @ A @ A4 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_238_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A4: A,B4: A] :
( ( ( times_times @ A @ A4 @ B4 )
= ( zero_zero @ A ) )
= ( ( A4
= ( zero_zero @ A ) )
| ( B4
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_239_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C2: A,A4: A,B4: A] :
( ( ( times_times @ A @ C2 @ A4 )
= ( times_times @ A @ C2 @ B4 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A4 = B4 ) ) ) ) ).
% mult_cancel_left
thf(fact_240_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [A4: A,C2: A,B4: A] :
( ( ( times_times @ A @ A4 @ C2 )
= ( times_times @ A @ B4 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A4 = B4 ) ) ) ) ).
% mult_cancel_right
thf(fact_241_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B4: A] :
( ( times_times @ A @ A4 @ ( uminus_uminus @ A @ B4 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A4 @ B4 ) ) ) ) ).
% mult_minus_right
thf(fact_242_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B4: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ A @ B4 ) )
= ( times_times @ A @ A4 @ B4 ) ) ) ).
% minus_mult_minus
thf(fact_243_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B4: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A4 ) @ B4 )
= ( uminus_uminus @ A @ ( times_times @ A @ A4 @ B4 ) ) ) ) ).
% mult_minus_left
thf(fact_244_ereal__mult__minus__right,axiom,
! [A4: extended_ereal,B4: extended_ereal] :
( ( times_times @ extended_ereal @ A4 @ ( uminus_uminus @ extended_ereal @ B4 ) )
= ( uminus_uminus @ extended_ereal @ ( times_times @ extended_ereal @ A4 @ B4 ) ) ) ).
% ereal_mult_minus_right
thf(fact_245_ereal__mult__minus__left,axiom,
! [A4: extended_ereal,B4: extended_ereal] :
( ( times_times @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A4 ) @ B4 )
= ( uminus_uminus @ extended_ereal @ ( times_times @ extended_ereal @ A4 @ B4 ) ) ) ).
% ereal_mult_minus_left
thf(fact_246_ereal__mult__zero,axiom,
! [A4: extended_ereal] :
( ( times_times @ extended_ereal @ A4 @ ( zero_zero @ extended_ereal ) )
= ( zero_zero @ extended_ereal ) ) ).
% ereal_mult_zero
thf(fact_247_ereal__zero__mult,axiom,
! [A4: extended_ereal] :
( ( times_times @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A4 )
= ( zero_zero @ extended_ereal ) ) ).
% ereal_zero_mult
thf(fact_248_ereal__zero__times,axiom,
! [A4: extended_ereal,B4: extended_ereal] :
( ( ( times_times @ extended_ereal @ A4 @ B4 )
= ( zero_zero @ extended_ereal ) )
= ( ( A4
= ( zero_zero @ extended_ereal ) )
| ( B4
= ( zero_zero @ extended_ereal ) ) ) ) ).
% ereal_zero_times
thf(fact_249_ereal__times__divide__eq__left,axiom,
! [B4: extended_ereal,C2: extended_ereal,A4: extended_ereal] :
( ( times_times @ extended_ereal @ ( divide_divide @ extended_ereal @ B4 @ C2 ) @ A4 )
= ( divide_divide @ extended_ereal @ ( times_times @ extended_ereal @ B4 @ A4 ) @ C2 ) ) ).
% ereal_times_divide_eq_left
thf(fact_250_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A4: A,B4: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) )
= ( divide_divide @ A @ A4 @ B4 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_251_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A4: A,B4: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) )
= ( divide_divide @ A @ A4 @ B4 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_252_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A4: A,B4: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ B4 @ C2 ) )
= ( divide_divide @ A @ A4 @ B4 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_253_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A4: A,B4: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ C2 ) )
= ( divide_divide @ A @ A4 @ B4 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_254_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A4: A,B4: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ C2 @ B4 ) )
= ( divide_divide @ A @ A4 @ B4 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
% Subclasses (6)
thf(subcl_Euclidean__Space_Oeuclidean__space___HOL_Otype,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ( type @ A ) ) ).
thf(subcl_Euclidean__Space_Oeuclidean__space___Groups_Ozero,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ( zero @ A ) ) ).
thf(subcl_Euclidean__Space_Oeuclidean__space___Groups_Ouminus,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ( uminus @ A ) ) ).
thf(subcl_Euclidean__Space_Oeuclidean__space___Groups_Ogroup__add,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ( group_add @ A ) ) ).
thf(subcl_Euclidean__Space_Oeuclidean__space___Real__Vector__Spaces_Oreal__vector,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ( real_V1076094709vector @ A ) ) ).
thf(subcl_Euclidean__Space_Oeuclidean__space___Real__Vector__Spaces_Ometric__space,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ( real_V2090557954_space @ A ) ) ).
% Type constructors (79)
thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
! [A7: $tType,A8: $tType] :
( ( ( ordere890947078_space @ A7 )
& ( ordere890947078_space @ A8 ) )
=> ( ordere890947078_space @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_1,axiom,
ordere890947078_space @ real ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A7: $tType,A8: $tType] :
( ( comple187826305attice @ A8 )
=> ( condit378418413attice @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A7: $tType,A8: $tType] :
( ( comple187826305attice @ A8 )
=> ( comple187826305attice @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A7: $tType,A8: $tType] :
( ( boolean_algebra @ A8 )
=> ( boolean_algebra @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A7: $tType,A8: $tType] :
( ( order_top @ A8 )
=> ( order_top @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A7: $tType,A8: $tType] :
( ( top @ A8 )
=> ( top @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A7: $tType,A8: $tType] :
( ( uminus @ A8 )
=> ( uminus @ ( A7 > A8 ) ) ) ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_2,axiom,
! [A7: $tType] : ( condit378418413attice @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_3,axiom,
! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_4,axiom,
! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_5,axiom,
! [A7: $tType] : ( order_top @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_6,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_7,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_8,axiom,
! [A7: $tType] : ( top @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_10,axiom,
! [A7: $tType] : ( uminus @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Groups_Ozero,axiom,
! [A7: $tType] :
( ( zero @ A7 )
=> ( zero @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11,axiom,
condit378418413attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_12,axiom,
comple187826305attice @ $o ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_13,axiom,
boolean_algebra @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_14,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_15,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_17,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_18,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_19,axiom,
uminus @ $o ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_20,axiom,
condit378418413attice @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri1923998003cancel @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors,axiom,
semiri1193490041visors @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
real_V2090557954_space @ real ).
thf(tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space,axiom,
euclid925273238_space @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
real_V1076094709vector @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_21,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_22,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Omult__zero,axiom,
mult_zero @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_23,axiom,
order @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_24,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_25,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Fields_Ofield,axiom,
field @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_26,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Rings_Oring,axiom,
ring @ real ).
thf(tcon_Product__Type_Oprod___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_27,axiom,
! [A7: $tType,A8: $tType] :
( ( ( condit378418413attice @ A7 )
& ( condit378418413attice @ A8 ) )
=> ( condit378418413attice @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Complete__Lattices_Ocomplete__lattice_28,axiom,
! [A7: $tType,A8: $tType] :
( ( ( comple187826305attice @ A7 )
& ( comple187826305attice @ A8 ) )
=> ( comple187826305attice @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Ometric__space_29,axiom,
! [A7: $tType,A8: $tType] :
( ( ( real_V2090557954_space @ A7 )
& ( real_V2090557954_space @ A8 ) )
=> ( real_V2090557954_space @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Euclidean__Space_Oeuclidean__space_30,axiom,
! [A7: $tType,A8: $tType] :
( ( ( euclid925273238_space @ A7 )
& ( euclid925273238_space @ A8 ) )
=> ( euclid925273238_space @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__vector_31,axiom,
! [A7: $tType,A8: $tType] :
( ( ( real_V1076094709vector @ A7 )
& ( real_V1076094709vector @ A8 ) )
=> ( real_V1076094709vector @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__group__add_32,axiom,
! [A7: $tType,A8: $tType] :
( ( ( ordere890947078_space @ A7 )
& ( ordere890947078_space @ A8 ) )
=> ( ordered_ab_group_add @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Lattices_Oboolean__algebra_33,axiom,
! [A7: $tType,A8: $tType] :
( ( ( boolean_algebra @ A7 )
& ( boolean_algebra @ A8 ) )
=> ( boolean_algebra @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder__top_34,axiom,
! [A7: $tType,A8: $tType] :
( ( ( order_top @ A7 )
& ( order_top @ A8 ) )
=> ( order_top @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Opreorder_35,axiom,
! [A7: $tType,A8: $tType] :
( ( ( preorder @ A7 )
& ( preorder @ A8 ) )
=> ( preorder @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ogroup__add_36,axiom,
! [A7: $tType,A8: $tType] :
( ( ( group_add @ A7 )
& ( group_add @ A8 ) )
=> ( group_add @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder_37,axiom,
! [A7: $tType,A8: $tType] :
( ( ( order @ A7 )
& ( order @ A8 ) )
=> ( order @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Otop_38,axiom,
! [A7: $tType,A8: $tType] :
( ( ( top @ A7 )
& ( top @ A8 ) )
=> ( top @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oord_39,axiom,
! [A7: $tType,A8: $tType] :
( ( ( ord @ A7 )
& ( ord @ A8 ) )
=> ( ord @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ouminus_40,axiom,
! [A7: $tType,A8: $tType] :
( ( ( uminus @ A7 )
& ( uminus @ A8 ) )
=> ( uminus @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ozero_41,axiom,
! [A7: $tType,A8: $tType] :
( ( ( zero @ A7 )
& ( zero @ A8 ) )
=> ( zero @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Extended__Real_Oereal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_42,axiom,
condit378418413attice @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Complete__Lattices_Ocomplete__lattice_43,axiom,
comple187826305attice @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Oorder__top_44,axiom,
order_top @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Opreorder_45,axiom,
preorder @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Olinorder_46,axiom,
linorder @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Oorder_47,axiom,
order @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Otop_48,axiom,
top @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Oord_49,axiom,
ord @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Ouminus_50,axiom,
uminus @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Ozero_51,axiom,
zero @ extended_ereal ).
% Free types (1)
thf(tfree_0,hypothesis,
euclid925273238_space @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( lower_673667120vex_on @ a @ ( top_top @ ( set @ a ) ) @ f )
= ( convex @ a @ ( lower_272802190domain @ a @ f ) ) ) ).
%------------------------------------------------------------------------------