TPTP Problem File: ITP112^2.p
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%------------------------------------------------------------------------------
% File : ITP112^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Lower_Semicontinuous problem prob_385__6250846_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_385__6250846_1 [Des21]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.50 v7.5.0
% Syntax : Number of formulae : 446 ( 181 unt; 64 typ; 0 def)
% Number of atoms : 1062 ( 375 equ; 0 cnn)
% Maximal formula atoms : 81 ( 2 avg)
% Number of connectives : 4309 ( 97 ~; 20 |; 46 &;3617 @)
% ( 0 <=>; 529 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 6 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 219 ( 219 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 59 usr; 2 con; 0-6 aty)
% Number of variables : 1126 ( 100 ^; 929 !; 39 ?;1126 :)
% ( 58 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:26:15.949
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_t_Extended__Real_Oereal,type,
extended_ereal: $tType ).
thf(ty_t_Filter_Ofilter,type,
filter: $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (57)
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Countable_Ocountable,type,
countable:
!>[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_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[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_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot2__space,type,
topological_t2_space:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__group__add,type,
topolo799126099up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__monoid__add,type,
topolo1314133330id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V2090557954_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ouniform__space,type,
topolo47006728_space:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Oorder__topology,type,
topolo259154727pology:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple1035589618norder:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Olinorder__topology,type,
topolo2117631714pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo503727757_space:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V55928688vector:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ofirst__countable__topology,type,
topolo2135403230pology:
!>[A: $tType] : $o ).
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
extend1396239628finity:
!>[A: $tType] : A ).
thf(sy_c_Extended__Real_Oereal_Ocase__ereal,type,
extended_case_ereal:
!>[A: $tType] : ( ( real > A ) > A > A > extended_ereal > A ) ).
thf(sy_c_Extended__Real_Oereal_Oereal,type,
extended_ereal2: real > extended_ereal ).
thf(sy_c_Filter_Oat__bot,type,
at_bot:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oat__top,type,
at_top:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_HOL_OEx1,type,
ex1:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_HOL_OUniq,type,
uniq:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_Liminf__Limsup_OLiminf,type,
liminf_Liminf:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( A > B ) > B ) ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc__at,type,
lower_582600101lsc_at:
!>[A: $tType,B: $tType] : ( A > ( A > B ) > $o ) ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Ousc__at,type,
lower_708585518usc_at:
!>[A: $tType,B: $tType] : ( A > ( A > B ) > $o ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Topological__Spaces_Omonoseq,type,
topological_monoseq:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
topolo1920029354e_nhds:
!>[A: $tType] : ( A > ( filter @ A ) ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocauchy__filter,type,
topolo1334162427filter:
!>[A: $tType] : ( ( filter @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_A____,type,
a2: extended_ereal ).
thf(sy_v_f,type,
f: a > extended_ereal ).
thf(sy_v_x0,type,
x0: a ).
thf(sy_v_x____,type,
x: nat > a ).
% Relevant facts (256)
thf(fact_0_x__def_I1_J,axiom,
filterlim @ nat @ a @ x @ ( topolo1920029354e_nhds @ a @ x0 ) @ ( at_top @ nat ) ).
% x_def(1)
thf(fact_1_x__def_I2_J,axiom,
filterlim @ nat @ extended_ereal @ ( comp @ a @ extended_ereal @ nat @ f @ x ) @ ( topolo1920029354e_nhds @ extended_ereal @ a2 ) @ ( at_top @ nat ) ).
% x_def(2)
thf(fact_2__092_060open_062_092_060And_062F_O_A_I_If_A_092_060circ_062_Ax_J_A_092_060longlongrightarrow_062_AA_J_AF_A_092_060Longrightarrow_062_A_I_I_092_060lambda_062xa_O_A_N_A_If_A_092_060circ_062_Ax_J_Axa_J_A_092_060longlongrightarrow_062_A_N_AA_J_AF_092_060close_062,axiom,
! [F: filter @ nat] :
( ( filterlim @ nat @ extended_ereal @ ( comp @ a @ extended_ereal @ nat @ f @ x ) @ ( topolo1920029354e_nhds @ extended_ereal @ a2 ) @ F )
=> ( filterlim @ nat @ extended_ereal
@ ^ [X: nat] : ( uminus_uminus @ extended_ereal @ ( comp @ a @ extended_ereal @ nat @ f @ x @ X ) )
@ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ a2 ) )
@ F ) ) ).
% \<open>\<And>F. ((f \<circ> x) \<longlongrightarrow> A) F \<Longrightarrow> ((\<lambda>xa. - (f \<circ> x) xa) \<longlongrightarrow> - A) F\<close>
thf(fact_3_lsc,axiom,
( lower_582600101lsc_at @ a @ extended_ereal @ x0
@ ^ [X: a] : ( uminus_uminus @ extended_ereal @ ( f @ X ) ) ) ).
% lsc
thf(fact_4_tendsto__uminus__ereal,axiom,
! [A: $tType,F2: A > extended_ereal,X2: extended_ereal,F: filter @ A] :
( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ X2 ) @ F )
=> ( filterlim @ A @ extended_ereal
@ ^ [X: A] : ( uminus_uminus @ extended_ereal @ ( F2 @ X ) )
@ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ X2 ) )
@ F ) ) ).
% tendsto_uminus_ereal
thf(fact_5_tendsto__const,axiom,
! [A: $tType,B: $tType] :
( ( topolo503727757_space @ A )
=> ! [K: A,F: filter @ B] :
( filterlim @ B @ A
@ ^ [X: B] : K
@ ( topolo1920029354e_nhds @ A @ K )
@ F ) ) ).
% tendsto_const
thf(fact_6_ereal__Lim__uminus,axiom,
! [A: $tType,F2: A > extended_ereal,F0: extended_ereal,Net: filter @ A] :
( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ F0 ) @ Net )
= ( filterlim @ A @ extended_ereal
@ ^ [X: A] : ( uminus_uminus @ extended_ereal @ ( F2 @ X ) )
@ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ F0 ) )
@ Net ) ) ).
% ereal_Lim_uminus
thf(fact_7_LIMSEQ__const__iff,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [K: A,L: A] :
( ( filterlim @ nat @ A
@ ^ [N: nat] : K
@ ( topolo1920029354e_nhds @ A @ L )
@ ( at_top @ nat ) )
= ( K = L ) ) ) ).
% LIMSEQ_const_iff
thf(fact_8_tendsto__minus__cancel,axiom,
! [A: $tType,B: $tType] :
( ( topolo799126099up_add @ A )
=> ! [F2: B > A,A2: A,F: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X: B] : ( uminus_uminus @ A @ ( F2 @ X ) )
@ ( topolo1920029354e_nhds @ A @ ( uminus_uminus @ A @ A2 ) )
@ F )
=> ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ A2 ) @ F ) ) ) ).
% tendsto_minus_cancel
thf(fact_9_tendsto__minus__cancel__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo799126099up_add @ B )
=> ! [F2: A > B,Y: B,F: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo1920029354e_nhds @ B @ ( uminus_uminus @ B @ Y ) ) @ F )
= ( filterlim @ A @ B
@ ^ [X: A] : ( uminus_uminus @ B @ ( F2 @ X ) )
@ ( topolo1920029354e_nhds @ B @ Y )
@ F ) ) ) ).
% tendsto_minus_cancel_left
thf(fact_10_tendsto__minus,axiom,
! [A: $tType,B: $tType] :
( ( topolo799126099up_add @ A )
=> ! [F2: B > A,A2: A,F: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ A2 ) @ F )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( uminus_uminus @ A @ ( F2 @ X ) )
@ ( topolo1920029354e_nhds @ A @ ( uminus_uminus @ A @ A2 ) )
@ F ) ) ) ).
% tendsto_minus
thf(fact_11_LIMSEQ__unique,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X3: nat > A,A2: A,B2: A] :
( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ B2 ) @ ( at_top @ nat ) )
=> ( A2 = B2 ) ) ) ) ).
% LIMSEQ_unique
thf(fact_12_tendsto__uminus__nhds,axiom,
! [A: $tType] :
( ( topolo799126099up_add @ A )
=> ! [A2: A] : ( filterlim @ A @ A @ ( uminus_uminus @ A ) @ ( topolo1920029354e_nhds @ A @ ( uminus_uminus @ A @ A2 ) ) @ ( topolo1920029354e_nhds @ A @ A2 ) ) ) ).
% tendsto_uminus_nhds
thf(fact_13_ereal__uminus__eq__iff,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ( uminus_uminus @ extended_ereal @ A2 )
= ( uminus_uminus @ extended_ereal @ B2 ) )
= ( A2 = B2 ) ) ).
% ereal_uminus_eq_iff
thf(fact_14_ereal__uminus__uminus,axiom,
! [A2: extended_ereal] :
( ( uminus_uminus @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) )
= A2 ) ).
% ereal_uminus_uminus
thf(fact_15_verit__minus__simplify_I4_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B2: B] :
( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
= B2 ) ) ).
% verit_minus_simplify(4)
thf(fact_16_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
=> ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_17_ereal__uminus__eq__reorder,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ( uminus_uminus @ extended_ereal @ A2 )
= B2 )
= ( A2
= ( uminus_uminus @ extended_ereal @ B2 ) ) ) ).
% ereal_uminus_eq_reorder
thf(fact_18_tendsto__eq__rhs,axiom,
! [A: $tType,B: $tType] :
( ( topolo503727757_space @ A )
=> ! [F2: B > A,X2: A,F: filter @ B,Y: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ X2 ) @ F )
=> ( ( X2 = Y )
=> ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ Y ) @ F ) ) ) ) ).
% tendsto_eq_rhs
thf(fact_19_tendsto__cong__limit,axiom,
! [B: $tType,A: $tType] :
( ( topolo503727757_space @ B )
=> ! [F2: A > B,L: B,F: filter @ A,K: B] :
( ( filterlim @ A @ B @ F2 @ ( topolo1920029354e_nhds @ B @ L ) @ F )
=> ( ( K = L )
=> ( filterlim @ A @ B @ F2 @ ( topolo1920029354e_nhds @ B @ K ) @ F ) ) ) ) ).
% tendsto_cong_limit
thf(fact_20_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F3: B > A,G: C > B,X: C] : ( F3 @ ( G @ X ) ) ) ) ).
% comp_apply
thf(fact_21_compl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X2: A,Y: A] :
( ( ( uminus_uminus @ A @ X2 )
= ( uminus_uminus @ A @ Y ) )
= ( X2 = Y ) ) ) ).
% compl_eq_compl_iff
thf(fact_22_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X2 ) )
= X2 ) ) ).
% double_compl
thf(fact_23_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_24_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_25_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A3: A > B,X: A] : ( uminus_uminus @ B @ ( A3 @ X ) ) ) ) ) ).
% uminus_apply
thf(fact_26_lsc__sequentially__mem,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X0: A,F2: A > extended_ereal,X2: nat > A,C2: nat > extended_ereal,C0: extended_ereal] :
( ( lower_582600101lsc_at @ A @ extended_ereal @ X0 @ F2 )
=> ( ( filterlim @ nat @ A @ X2 @ ( topolo1920029354e_nhds @ A @ X0 ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ extended_ereal @ C2 @ ( topolo1920029354e_nhds @ extended_ereal @ C0 ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ extended_ereal @ ( F2 @ ( X2 @ N2 ) ) @ ( C2 @ N2 ) )
=> ( ord_less_eq @ extended_ereal @ ( F2 @ X0 ) @ C0 ) ) ) ) ) ) ).
% lsc_sequentially_mem
thf(fact_27_lsc__sequentially__gen,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ( ( lower_582600101lsc_at @ A @ extended_ereal )
= ( ^ [X02: A,F3: A > extended_ereal] :
! [X: nat > A,C3: nat > extended_ereal,C02: extended_ereal] :
( ( ( filterlim @ nat @ A @ X @ ( topolo1920029354e_nhds @ A @ X02 ) @ ( at_top @ nat ) )
& ( filterlim @ nat @ extended_ereal @ C3 @ ( topolo1920029354e_nhds @ extended_ereal @ C02 ) @ ( at_top @ nat ) )
& ! [N: nat] : ( ord_less_eq @ extended_ereal @ ( F3 @ ( X @ N ) ) @ ( C3 @ N ) ) )
=> ( ord_less_eq @ extended_ereal @ ( F3 @ X02 ) @ C02 ) ) ) ) ) ).
% lsc_sequentially_gen
thf(fact_28_lsc__at__mem,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo259154727pology @ B )
& ( topolo503727757_space @ A ) )
=> ! [X0: A,F2: A > B,X2: nat > A,A4: B] :
( ( lower_582600101lsc_at @ A @ B @ X0 @ F2 )
=> ( ( filterlim @ nat @ A @ X2 @ ( topolo1920029354e_nhds @ A @ X0 ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ B @ ( comp @ A @ B @ nat @ F2 @ X2 ) @ ( topolo1920029354e_nhds @ B @ A4 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ B @ ( F2 @ X0 ) @ A4 ) ) ) ) ) ).
% lsc_at_mem
thf(fact_29_lsc__at__def,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo259154727pology @ B ) )
=> ( ( lower_582600101lsc_at @ A @ B )
= ( ^ [X02: A,F3: A > B] :
! [X4: nat > A,L2: B] :
( ( ( filterlim @ nat @ A @ X4 @ ( topolo1920029354e_nhds @ A @ X02 ) @ ( at_top @ nat ) )
& ( filterlim @ nat @ B @ ( comp @ A @ B @ nat @ F3 @ X4 ) @ ( topolo1920029354e_nhds @ B @ L2 ) @ ( at_top @ nat ) ) )
=> ( ord_less_eq @ B @ ( F3 @ X02 ) @ L2 ) ) ) ) ) ).
% lsc_at_def
thf(fact_30_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_31_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% compl_le_compl_iff
thf(fact_32_ereal__minus__le__minus,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) @ ( uminus_uminus @ extended_ereal @ B2 ) )
= ( ord_less_eq @ extended_ereal @ B2 @ A2 ) ) ).
% ereal_minus_le_minus
thf(fact_33_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
| ~ ( ord_less_eq @ A @ A2 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% verit_la_disequality
thf(fact_34_ereal__complete__Inf,axiom,
! [S: set @ extended_ereal] :
? [X5: extended_ereal] :
( ! [Xa: extended_ereal] :
( ( member @ extended_ereal @ Xa @ S )
=> ( ord_less_eq @ extended_ereal @ X5 @ Xa ) )
& ! [Z: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member @ extended_ereal @ Xa2 @ S )
=> ( ord_less_eq @ extended_ereal @ Z @ Xa2 ) )
=> ( ord_less_eq @ extended_ereal @ Z @ X5 ) ) ) ).
% ereal_complete_Inf
thf(fact_35_ereal__complete__Sup,axiom,
! [S: set @ extended_ereal] :
? [X5: extended_ereal] :
( ! [Xa: extended_ereal] :
( ( member @ extended_ereal @ Xa @ S )
=> ( ord_less_eq @ extended_ereal @ Xa @ X5 ) )
& ! [Z: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member @ extended_ereal @ Xa2 @ S )
=> ( ord_less_eq @ extended_ereal @ Xa2 @ Z ) )
=> ( ord_less_eq @ extended_ereal @ X5 @ Z ) ) ) ).
% ereal_complete_Sup
thf(fact_36_compl__mono,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% compl_mono
thf(fact_37_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X2 ) )
=> ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_38_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_39_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_minus_iff
thf(fact_40_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_le_iff
thf(fact_41_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_imp_neg_le
thf(fact_42_ereal__uminus__le__reorder,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) @ B2 )
= ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ B2 ) @ A2 ) ) ).
% ereal_uminus_le_reorder
thf(fact_43_lim__mono,axiom,
! [A: $tType] :
( ( topolo2117631714pology @ A )
=> ! [N3: nat,X3: nat > A,Y2: nat > A,X2: A,Y: A] :
( ! [N2: nat] :
( ( ord_less_eq @ nat @ N3 @ N2 )
=> ( ord_less_eq @ A @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
=> ( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y2 @ ( topolo1920029354e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ).
% lim_mono
thf(fact_44_LIMSEQ__le,axiom,
! [A: $tType] :
( ( topolo2117631714pology @ A )
=> ! [X3: nat > A,X2: A,Y2: nat > A,Y: A] :
( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y2 @ ( topolo1920029354e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
=> ( ord_less_eq @ A @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ).
% LIMSEQ_le
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
( ! [X5: A] :
( ( F2 @ X5 )
= ( G2 @ X5 ) )
=> ( F2 = G2 ) ) ).
% ext
thf(fact_49_Lim__bounded,axiom,
! [A: $tType] :
( ( topolo2117631714pology @ A )
=> ! [F2: nat > A,L: A,M: nat,C4: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ A @ ( F2 @ N2 ) @ C4 ) )
=> ( ord_less_eq @ A @ L @ C4 ) ) ) ) ).
% Lim_bounded
thf(fact_50_Lim__bounded2,axiom,
! [A: $tType] :
( ( topolo2117631714pology @ A )
=> ! [F2: nat > A,L: A,N3: nat,C4: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N3 @ N2 )
=> ( ord_less_eq @ A @ C4 @ ( F2 @ N2 ) ) )
=> ( ord_less_eq @ A @ C4 @ L ) ) ) ) ).
% Lim_bounded2
thf(fact_51_LIMSEQ__le__const,axiom,
! [A: $tType] :
( ( topolo2117631714pology @ A )
=> ! [X3: nat > A,X2: A,A2: A] :
( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
=> ( ord_less_eq @ A @ A2 @ ( X3 @ N2 ) ) )
=> ( ord_less_eq @ A @ A2 @ X2 ) ) ) ) ).
% LIMSEQ_le_const
thf(fact_52_LIMSEQ__le__const2,axiom,
! [A: $tType] :
( ( topolo2117631714pology @ A )
=> ! [X3: nat > A,X2: A,A2: A] :
( ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
=> ( ord_less_eq @ A @ ( X3 @ N2 ) @ A2 ) )
=> ( ord_less_eq @ A @ X2 @ A2 ) ) ) ) ).
% LIMSEQ_le_const2
thf(fact_53_tendsto__mono,axiom,
! [A: $tType,B: $tType] :
( ( topolo503727757_space @ A )
=> ! [F: filter @ B,F4: filter @ B,F2: B > A,L: A] :
( ( ord_less_eq @ ( filter @ B ) @ F @ F4 )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L ) @ F4 )
=> ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L ) @ F ) ) ) ) ).
% tendsto_mono
thf(fact_54_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A3: A > B,X: A] : ( uminus_uminus @ B @ ( A3 @ X ) ) ) ) ) ).
% fun_Compl_def
thf(fact_55_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% equation_minus_iff
thf(fact_56_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A2 ) ) ) ).
% minus_equation_iff
thf(fact_57_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= C2 )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_58_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ! [V2: A] :
( ( A2 @ ( B2 @ V2 ) )
= ( C2 @ ( D2 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_59_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_60_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F2: D > B,G2: C > D,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F2 @ G2 ) @ H )
= ( comp @ D @ B @ A @ F2 @ ( comp @ C @ D @ A @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_61_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F3: B > C,G: A > B,X: A] : ( F3 @ ( G @ X ) ) ) ) ).
% comp_def
thf(fact_62_lsc__sequentially,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ( ( lower_582600101lsc_at @ A @ extended_ereal )
= ( ^ [X02: A,F3: A > extended_ereal] :
! [X: nat > A,C3: extended_ereal] :
( ( ( filterlim @ nat @ A @ X @ ( topolo1920029354e_nhds @ A @ X02 ) @ ( at_top @ nat ) )
& ! [N: nat] : ( ord_less_eq @ extended_ereal @ ( F3 @ ( X @ N ) ) @ C3 ) )
=> ( ord_less_eq @ extended_ereal @ ( F3 @ X02 ) @ C3 ) ) ) ) ) ).
% lsc_sequentially
thf(fact_63_usc__at__def,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo259154727pology @ B ) )
=> ( ( lower_708585518usc_at @ A @ B )
= ( ^ [X02: A,F3: A > B] :
! [X4: nat > A,L2: B] :
( ( ( filterlim @ nat @ A @ X4 @ ( topolo1920029354e_nhds @ A @ X02 ) @ ( at_top @ nat ) )
& ( filterlim @ nat @ B @ ( comp @ A @ B @ nat @ F3 @ X4 ) @ ( topolo1920029354e_nhds @ B @ L2 ) @ ( at_top @ nat ) ) )
=> ( ord_less_eq @ B @ L2 @ ( F3 @ X02 ) ) ) ) ) ) ).
% usc_at_def
thf(fact_64_usc__at__mem,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo259154727pology @ B ) )
=> ! [X0: A,F2: A > B,X2: nat > A,A4: B] :
( ( lower_708585518usc_at @ A @ B @ X0 @ F2 )
=> ( ( filterlim @ nat @ A @ X2 @ ( topolo1920029354e_nhds @ A @ X0 ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ B @ ( comp @ A @ B @ nat @ F2 @ X2 ) @ ( topolo1920029354e_nhds @ B @ A4 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ B @ A4 @ ( F2 @ X0 ) ) ) ) ) ) ).
% usc_at_mem
thf(fact_65_lim__decreasing__cl,axiom,
! [A: $tType] :
( ( ( comple1035589618norder @ A )
& ( topolo2117631714pology @ A ) )
=> ! [F2: nat > A] :
( ! [N2: nat,M2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( F2 @ M2 ) ) )
=> ~ ! [L3: A] :
~ ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L3 ) @ ( at_top @ nat ) ) ) ) ).
% lim_decreasing_cl
thf(fact_66_lim__increasing__cl,axiom,
! [A: $tType] :
( ( ( comple1035589618norder @ A )
& ( topolo2117631714pology @ A ) )
=> ! [F2: nat > A] :
( ! [N2: nat,M2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ A @ ( F2 @ M2 ) @ ( F2 @ N2 ) ) )
=> ~ ! [L3: A] :
~ ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ L3 ) @ ( at_top @ nat ) ) ) ) ).
% lim_increasing_cl
thf(fact_67_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_68_monoseq__le,axiom,
! [A: $tType] :
( ( topolo2117631714pology @ A )
=> ! [A2: nat > A,X2: A] :
( ( topological_monoseq @ A @ A2 )
=> ( ( filterlim @ nat @ A @ A2 @ ( topolo1920029354e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
=> ( ( ! [N5: nat] : ( ord_less_eq @ A @ ( A2 @ N5 ) @ X2 )
& ! [M3: nat,N5: nat] :
( ( ord_less_eq @ nat @ M3 @ N5 )
=> ( ord_less_eq @ A @ ( A2 @ M3 ) @ ( A2 @ N5 ) ) ) )
| ( ! [N5: nat] : ( ord_less_eq @ A @ X2 @ ( A2 @ N5 ) )
& ! [M3: nat,N5: nat] :
( ( ord_less_eq @ nat @ M3 @ N5 )
=> ( ord_less_eq @ A @ ( A2 @ N5 ) @ ( A2 @ M3 ) ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_69_lsc__liminf,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ( ( lower_582600101lsc_at @ A @ extended_ereal )
= ( ^ [X02: A,F3: A > extended_ereal] :
! [X: nat > A] :
( ( filterlim @ nat @ A @ X @ ( topolo1920029354e_nhds @ A @ X02 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ extended_ereal @ ( F3 @ X02 ) @ ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ ( comp @ A @ extended_ereal @ nat @ F3 @ X ) ) ) ) ) ) ) ).
% lsc_liminf
thf(fact_70_lsc__imp__liminf,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X0: A,F2: A > extended_ereal,X2: nat > A] :
( ( lower_582600101lsc_at @ A @ extended_ereal @ X0 @ F2 )
=> ( ( filterlim @ nat @ A @ X2 @ ( topolo1920029354e_nhds @ A @ X0 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ extended_ereal @ ( F2 @ X0 ) @ ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ ( comp @ A @ extended_ereal @ nat @ F2 @ X2 ) ) ) ) ) ) ).
% lsc_imp_liminf
thf(fact_71_LIMSEQ__Uniq,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X3: nat > A] :
( uniq @ A
@ ^ [L2: A] : ( filterlim @ nat @ A @ X3 @ ( topolo1920029354e_nhds @ A @ L2 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_Uniq
thf(fact_72_monoseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X4: nat > A] :
( ! [M4: nat,N: nat] :
( ( ord_less_eq @ nat @ M4 @ N )
=> ( ord_less_eq @ A @ ( X4 @ M4 ) @ ( X4 @ N ) ) )
| ! [M4: nat,N: nat] :
( ( ord_less_eq @ nat @ M4 @ N )
=> ( ord_less_eq @ A @ ( X4 @ N ) @ ( X4 @ M4 ) ) ) ) ) ) ) ).
% monoseq_def
thf(fact_73_monoI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: nat > A] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ A @ ( X3 @ N2 ) @ ( X3 @ M2 ) ) )
=> ( topological_monoseq @ A @ X3 ) ) ) ).
% monoI2
thf(fact_74_monoI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: nat > A] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ A @ ( X3 @ M2 ) @ ( X3 @ N2 ) ) )
=> ( topological_monoseq @ A @ X3 ) ) ) ).
% monoI1
thf(fact_75_monoseq__minus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: nat > A] :
( ( topological_monoseq @ A @ A2 )
=> ( topological_monoseq @ A
@ ^ [N: nat] : ( uminus_uminus @ A @ ( A2 @ N ) ) ) ) ) ).
% monoseq_minus
thf(fact_76_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_funD
thf(fact_77_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_funE
thf(fact_78_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B] :
( ! [X5: A] : ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( G2 @ X5 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G2 ) ) ) ).
% le_funI
thf(fact_79_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_80_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X5: B,Y3: B] :
( ( ord_less_eq @ B @ X5 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_81_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ C @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_82_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X5: B,Y3: B] :
( ( ord_less_eq @ B @ X5 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_83_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F2: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_84_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [X: A,Y5: A] :
( ( ord_less_eq @ A @ X @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% eq_iff
thf(fact_85_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( X2 = Y ) ) ) ) ).
% antisym
thf(fact_86_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% linear
thf(fact_87_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 = Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% eq_refl
thf(fact_88_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% le_cases
thf(fact_89_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_90_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A,Z3: A] :
( ( ( ord_less_eq @ A @ X2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_91_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% antisym_conv
thf(fact_92_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
& ( ord_less_eq @ A @ B3 @ A5 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_93_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_94_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_95_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_96_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less_eq @ A @ X2 @ Z3 ) ) ) ) ).
% order_trans
thf(fact_97_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_98_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A6: A,B4: A] :
( ( ord_less_eq @ A @ A6 @ B4 )
=> ( P @ A6 @ B4 ) )
=> ( ! [A6: A,B4: A] :
( ( P @ B4 @ A6 )
=> ( P @ A6 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_99_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_100_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A5: A,B3: A] :
( ( ord_less_eq @ A @ B3 @ A5 )
& ( ord_less_eq @ A @ A5 @ B3 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_101_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_102_filterlim__mono,axiom,
! [B: $tType,A: $tType,F2: A > B,F22: filter @ B,F1: filter @ A,F23: filter @ B,F12: filter @ A] :
( ( filterlim @ A @ B @ F2 @ F22 @ F1 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F22 @ F23 )
=> ( ( ord_less_eq @ ( filter @ A ) @ F12 @ F1 )
=> ( filterlim @ A @ B @ F2 @ F23 @ F12 ) ) ) ) ).
% filterlim_mono
thf(fact_103_nhds__imp__cauchy__filter,axiom,
! [A: $tType] :
( ( topolo47006728_space @ A )
=> ! [F: filter @ A,X2: A] :
( ( ord_less_eq @ ( filter @ A ) @ F @ ( topolo1920029354e_nhds @ A @ X2 ) )
=> ( topolo1334162427filter @ A @ F ) ) ) ).
% nhds_imp_cauchy_filter
thf(fact_104_alt__ex1E_H,axiom,
! [A: $tType,P: A > $o] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X6 ) ) )
=> ~ ( ? [X_1: A] : ( P @ X_1 )
=> ~ ( uniq @ A @ P ) ) ) ).
% alt_ex1E'
thf(fact_105_ex1__iff__ex__Uniq,axiom,
! [A: $tType] :
( ( ex1 @ A )
= ( ^ [P2: A > $o] :
( ? [X7: A] : ( P2 @ X7 )
& ( uniq @ A @ P2 ) ) ) ) ).
% ex1_iff_ex_Uniq
thf(fact_106_liminf__PInfty,axiom,
! [X3: nat > extended_ereal] :
( ( filterlim @ nat @ extended_ereal @ X3 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( at_top @ nat ) )
= ( ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ X3 )
= ( extend1396239628finity @ extended_ereal ) ) ) ).
% liminf_PInfty
thf(fact_107_fun_Omap__ident,axiom,
! [A: $tType,D: $tType,T: D > A] :
( ( comp @ A @ A @ D
@ ^ [X: A] : X
@ T )
= T ) ).
% fun.map_ident
thf(fact_108_K__record__comp,axiom,
! [C: $tType,B: $tType,A: $tType,C2: B,F2: A > C] :
( ( comp @ C @ B @ A
@ ^ [X: C] : C2
@ F2 )
= ( ^ [X: A] : C2 ) ) ).
% K_record_comp
thf(fact_109_ereal__infty__less__eq_I1_J,axiom,
! [X2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ X2 )
= ( X2
= ( extend1396239628finity @ extended_ereal ) ) ) ).
% ereal_infty_less_eq(1)
thf(fact_110_ereal__infty__less__eq_I2_J,axiom,
! [X2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ X2 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
= ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% ereal_infty_less_eq(2)
thf(fact_111_MInfty__neq__PInfty_I1_J,axiom,
( ( extend1396239628finity @ extended_ereal )
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% MInfty_neq_PInfty(1)
thf(fact_112_ereal__less__eq_I1_J,axiom,
! [X2: extended_ereal] : ( ord_less_eq @ extended_ereal @ X2 @ ( extend1396239628finity @ extended_ereal ) ) ).
% ereal_less_eq(1)
thf(fact_113_ereal__infty__less__eq2_I1_J,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
=> ( ( A2
= ( extend1396239628finity @ extended_ereal ) )
=> ( B2
= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% ereal_infty_less_eq2(1)
thf(fact_114_neq__PInf__trans,axiom,
! [Y: extended_ereal,X2: extended_ereal] :
( ( Y
!= ( extend1396239628finity @ extended_ereal ) )
=> ( ( ord_less_eq @ extended_ereal @ X2 @ Y )
=> ( X2
!= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% neq_PInf_trans
thf(fact_115_ereal__infty__less__eq2_I2_J,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
=> ( ( B2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( A2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% ereal_infty_less_eq2(2)
thf(fact_116_ereal__less__eq_I2_J,axiom,
! [X2: extended_ereal] : ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ X2 ) ).
% ereal_less_eq(2)
thf(fact_117_lsc__at__MInfty,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [F2: A > extended_ereal,X0: A] :
( ( ( F2 @ X0 )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( lower_582600101lsc_at @ A @ extended_ereal @ X0 @ F2 ) ) ) ).
% lsc_at_MInfty
thf(fact_118_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G2: B > C,F2: A > B,V: D > A] :
( ( comp @ B @ C @ D @ G2 @ ( comp @ A @ B @ D @ F2 @ V ) )
= ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_119_Uniq__def,axiom,
! [A: $tType] :
( ( uniq @ A )
= ( ^ [P2: A > $o] :
! [X: A,Y5: A] :
( ( P2 @ X )
=> ( ( P2 @ Y5 )
=> ( Y5 = X ) ) ) ) ) ).
% Uniq_def
thf(fact_120_Uniq__I,axiom,
! [A: $tType,P: A > $o] :
( ! [X5: A,Y3: A] :
( ( P @ X5 )
=> ( ( P @ Y3 )
=> ( Y3 = X5 ) ) )
=> ( uniq @ A @ P ) ) ).
% Uniq_I
thf(fact_121_Uniq__D,axiom,
! [A: $tType,P: A > $o,A2: A,B2: A] :
( ( uniq @ A @ P )
=> ( ( P @ A2 )
=> ( ( P @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% Uniq_D
thf(fact_122_filterlim__ident,axiom,
! [A: $tType,F: filter @ A] :
( filterlim @ A @ A
@ ^ [X: A] : X
@ F
@ F ) ).
% filterlim_ident
thf(fact_123_filterlim__compose,axiom,
! [B: $tType,A: $tType,C: $tType,G2: A > B,F32: filter @ B,F22: filter @ A,F2: C > A,F1: filter @ C] :
( ( filterlim @ A @ B @ G2 @ F32 @ F22 )
=> ( ( filterlim @ C @ A @ F2 @ F22 @ F1 )
=> ( filterlim @ C @ B
@ ^ [X: C] : ( G2 @ ( F2 @ X ) )
@ F32
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_124_Lim__MInfty,axiom,
! [F2: nat > extended_ereal] :
( ( filterlim @ nat @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) @ ( at_top @ nat ) )
= ( ! [B5: real] :
? [N6: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N6 @ N )
=> ( ord_less_eq @ extended_ereal @ ( F2 @ N ) @ ( extended_ereal2 @ B5 ) ) ) ) ) ).
% Lim_MInfty
thf(fact_125_MInfty__cases,axiom,
! [A: $tType,F2: real > A,Y: A,Z3: A] :
( ( extended_case_ereal @ A @ F2 @ Y @ Z3 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
= Z3 ) ).
% MInfty_cases
thf(fact_126_Lim__bounded__MInfty,axiom,
! [F2: nat > extended_ereal,L: extended_ereal,B6: real] :
( ( filterlim @ nat @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ L ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ B6 ) @ ( F2 @ N2 ) )
=> ( L
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% Lim_bounded_MInfty
thf(fact_127_Lim__PInfty,axiom,
! [F2: nat > extended_ereal] :
( ( filterlim @ nat @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( at_top @ nat ) )
= ( ! [B5: real] :
? [N6: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N6 @ N )
=> ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ B5 ) @ ( F2 @ N ) ) ) ) ) ).
% Lim_PInfty
thf(fact_128_Lim__bounded__PInfty2,axiom,
! [F2: nat > extended_ereal,L: extended_ereal,N3: nat,B6: real] :
( ( filterlim @ nat @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ L ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N3 @ N2 )
=> ( ord_less_eq @ extended_ereal @ ( F2 @ N2 ) @ ( extended_ereal2 @ B6 ) ) )
=> ( L
!= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% Lim_bounded_PInfty2
thf(fact_129_ereal_Oinject,axiom,
! [X1: real,Y1: real] :
( ( ( extended_ereal2 @ X1 )
= ( extended_ereal2 @ Y1 ) )
= ( X1 = Y1 ) ) ).
% ereal.inject
thf(fact_130_ereal__cong,axiom,
! [X2: real,Y: real] :
( ( X2 = Y )
=> ( ( extended_ereal2 @ X2 )
= ( extended_ereal2 @ Y ) ) ) ).
% ereal_cong
thf(fact_131_ereal__less__eq_I3_J,axiom,
! [R: real,P3: real] :
( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P3 ) )
= ( ord_less_eq @ real @ R @ P3 ) ) ).
% ereal_less_eq(3)
thf(fact_132_PInfty__cases,axiom,
! [A: $tType,F2: real > A,Y: A,Z3: A] :
( ( extended_case_ereal @ A @ F2 @ Y @ Z3 @ ( extend1396239628finity @ extended_ereal ) )
= Y ) ).
% PInfty_cases
thf(fact_133_tendsto__ereal,axiom,
! [A: $tType,F2: A > real,X2: real,F: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo1920029354e_nhds @ real @ X2 ) @ F )
=> ( filterlim @ A @ extended_ereal
@ ^ [X: A] : ( extended_ereal2 @ ( F2 @ X ) )
@ ( topolo1920029354e_nhds @ extended_ereal @ ( extended_ereal2 @ X2 ) )
@ F ) ) ).
% tendsto_ereal
thf(fact_134_lim__ereal,axiom,
! [A: $tType,F2: A > real,X2: real,Net: filter @ A] :
( ( filterlim @ A @ extended_ereal
@ ^ [N: A] : ( extended_ereal2 @ ( F2 @ N ) )
@ ( topolo1920029354e_nhds @ extended_ereal @ ( extended_ereal2 @ X2 ) )
@ Net )
= ( filterlim @ A @ real @ F2 @ ( topolo1920029354e_nhds @ real @ X2 ) @ Net ) ) ).
% lim_ereal
thf(fact_135_PInfty__neq__ereal_I1_J,axiom,
! [R: real] :
( ( extended_ereal2 @ R )
!= ( extend1396239628finity @ extended_ereal ) ) ).
% PInfty_neq_ereal(1)
thf(fact_136_ereal_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: A > B,F13: real > A,F24: A,F33: A,Ereal: extended_ereal] :
( ( H @ ( extended_case_ereal @ A @ F13 @ F24 @ F33 @ Ereal ) )
= ( extended_case_ereal @ B
@ ^ [X: real] : ( H @ ( F13 @ X ) )
@ ( H @ F24 )
@ ( H @ F33 )
@ Ereal ) ) ).
% ereal.case_distrib
thf(fact_137_uminus__ereal_Osimps_I1_J,axiom,
! [R: real] :
( ( uminus_uminus @ extended_ereal @ ( extended_ereal2 @ R ) )
= ( extended_ereal2 @ ( uminus_uminus @ real @ R ) ) ) ).
% uminus_ereal.simps(1)
thf(fact_138_ereal__le__le,axiom,
! [Y: real,A2: extended_ereal,X2: real] :
( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ Y ) @ A2 )
=> ( ( ord_less_eq @ real @ X2 @ Y )
=> ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ X2 ) @ A2 ) ) ) ).
% ereal_le_le
thf(fact_139_le__ereal__le,axiom,
! [A2: extended_ereal,X2: real,Y: real] :
( ( ord_less_eq @ extended_ereal @ A2 @ ( extended_ereal2 @ X2 ) )
=> ( ( ord_less_eq @ real @ X2 @ Y )
=> ( ord_less_eq @ extended_ereal @ A2 @ ( extended_ereal2 @ Y ) ) ) ) ).
% le_ereal_le
thf(fact_140_ereal__le__real,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ! [Z4: real] :
( ( ord_less_eq @ extended_ereal @ X2 @ ( extended_ereal2 @ Z4 ) )
=> ( ord_less_eq @ extended_ereal @ Y @ ( extended_ereal2 @ Z4 ) ) )
=> ( ord_less_eq @ extended_ereal @ Y @ X2 ) ) ).
% ereal_le_real
thf(fact_141_ereal_Osimps_I8_J,axiom,
! [A: $tType,F13: real > A,F24: A,F33: A,X1: real] :
( ( extended_case_ereal @ A @ F13 @ F24 @ F33 @ ( extended_ereal2 @ X1 ) )
= ( F13 @ X1 ) ) ).
% ereal.simps(8)
thf(fact_142_ereal__semiline__unique,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ( collect @ real
@ ^ [Y5: real] : ( ord_less_eq @ extended_ereal @ A2 @ ( extended_ereal2 @ Y5 ) ) )
= ( collect @ real
@ ^ [Y5: real] : ( ord_less_eq @ extended_ereal @ B2 @ ( extended_ereal2 @ Y5 ) ) ) )
= ( A2 = B2 ) ) ).
% ereal_semiline_unique
thf(fact_143_real__of__ereal_Oinduct,axiom,
! [P: extended_ereal > $o,A0: extended_ereal] :
( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ ( extend1396239628finity @ extended_ereal ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( P @ A0 ) ) ) ) ).
% real_of_ereal.induct
thf(fact_144_real__of__ereal_Ocases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2
!= ( extend1396239628finity @ extended_ereal ) )
=> ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% real_of_ereal.cases
thf(fact_145_times__ereal_Oinduct,axiom,
! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
( ! [R2: real,P4: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P4 ) )
=> ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extend1396239628finity @ extended_ereal ) )
=> ( ! [R2: real] : ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R2 ) )
=> ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ! [R2: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extend1396239628finity @ extended_ereal ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
=> ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).
% times_ereal.induct
thf(fact_146_plus__ereal_Oinduct,axiom,
! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
( ! [R2: real,P4: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P4 ) )
=> ( ! [X_1: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_1 )
=> ( ! [A6: extended_ereal] : ( P @ A6 @ ( extend1396239628finity @ extended_ereal ) )
=> ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ! [P4: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ P4 ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% plus_ereal.induct
thf(fact_147_less__ereal_Oinduct,axiom,
! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
( ! [X5: real,Y3: real] : ( P @ ( extended_ereal2 @ X5 ) @ ( extended_ereal2 @ Y3 ) )
=> ( ! [X_1: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_1 )
=> ( ! [A6: extended_ereal] : ( P @ A6 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ! [X5: real] : ( P @ ( extended_ereal2 @ X5 ) @ ( extend1396239628finity @ extended_ereal ) )
=> ( ! [R2: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% less_ereal.induct
thf(fact_148_abs__ereal_Oinduct,axiom,
! [P: extended_ereal > $o,A0: extended_ereal] :
( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( P @ ( extend1396239628finity @ extended_ereal ) )
=> ( P @ A0 ) ) ) ) ).
% abs_ereal.induct
thf(fact_149_ereal__all__split,axiom,
( ( ^ [P5: extended_ereal > $o] :
! [X7: extended_ereal] : ( P5 @ X7 ) )
= ( ^ [P2: extended_ereal > $o] :
( ( P2 @ ( extend1396239628finity @ extended_ereal ) )
& ! [X: real] : ( P2 @ ( extended_ereal2 @ X ) )
& ( P2 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% ereal_all_split
thf(fact_150_abs__ereal_Ocases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( X2
= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% abs_ereal.cases
thf(fact_151_ereal__ex__split,axiom,
( ( ^ [P5: extended_ereal > $o] :
? [X7: extended_ereal] : ( P5 @ X7 ) )
= ( ^ [P2: extended_ereal > $o] :
( ( P2 @ ( extend1396239628finity @ extended_ereal ) )
| ? [X: real] : ( P2 @ ( extended_ereal2 @ X ) )
| ( P2 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% ereal_ex_split
thf(fact_152_ereal3__cases,axiom,
! [X2: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ? [Ra: real] :
( Xa3
= ( extended_ereal2 @ Ra ) )
=> ! [Rb: real] :
( Xb
!= ( extended_ereal2 @ Rb ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ? [Ra: real] :
( Xa3
= ( extended_ereal2 @ Ra ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ? [Ra: real] :
( Xa3
= ( extended_ereal2 @ Ra ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ~ ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ereal3_cases
thf(fact_153_ereal2__cases,axiom,
! [X2: extended_ereal,Xa3: extended_ereal] :
( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ! [Ra: real] :
( Xa3
!= ( extended_ereal2 @ Ra ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( Xa3
!= ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( Xa3
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ! [R2: real] :
( Xa3
!= ( extended_ereal2 @ R2 ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xa3
!= ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xa3
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ! [R2: real] :
( Xa3
!= ( extended_ereal2 @ R2 ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xa3
!= ( extend1396239628finity @ extended_ereal ) ) )
=> ~ ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xa3
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ).
% ereal2_cases
thf(fact_154_ereal__cases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2
!= ( extend1396239628finity @ extended_ereal ) )
=> ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% ereal_cases
thf(fact_155_MInfty__neq__ereal_I1_J,axiom,
! [R: real] :
( ( extended_ereal2 @ R )
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% MInfty_neq_ereal(1)
thf(fact_156_ereal__top,axiom,
! [X2: extended_ereal] :
( ! [B7: real] : ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ B7 ) @ X2 )
=> ( X2
= ( extend1396239628finity @ extended_ereal ) ) ) ).
% ereal_top
thf(fact_157_ereal__tendsto__simps2_I1_J,axiom,
! [A: $tType,F2: A > real,A2: real,F: filter @ A] :
( ( filterlim @ A @ extended_ereal @ ( comp @ real @ extended_ereal @ A @ extended_ereal2 @ F2 ) @ ( topolo1920029354e_nhds @ extended_ereal @ ( extended_ereal2 @ A2 ) ) @ F )
= ( filterlim @ A @ real @ F2 @ ( topolo1920029354e_nhds @ real @ A2 ) @ F ) ) ).
% ereal_tendsto_simps2(1)
thf(fact_158_ereal__bot,axiom,
! [X2: extended_ereal] :
( ! [B7: real] : ( ord_less_eq @ extended_ereal @ X2 @ ( extended_ereal2 @ B7 ) )
=> ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% ereal_bot
thf(fact_159_Lim__bounded__PInfty,axiom,
! [F2: nat > extended_ereal,L: extended_ereal,B6: real] :
( ( filterlim @ nat @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ L ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ extended_ereal @ ( F2 @ N2 ) @ ( extended_ereal2 @ B6 ) )
=> ( L
!= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% Lim_bounded_PInfty
thf(fact_160_ereal__tendsto__simps2_I3_J,axiom,
! [A: $tType,F2: A > real,F: filter @ A] :
( ( filterlim @ A @ extended_ereal @ ( comp @ real @ extended_ereal @ A @ extended_ereal2 @ F2 ) @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) @ F )
= ( filterlim @ A @ real @ F2 @ ( at_bot @ real ) @ F ) ) ).
% ereal_tendsto_simps2(3)
thf(fact_161_tendsto__cadd__ereal,axiom,
! [A: $tType,Y: extended_ereal,X2: extended_ereal,F2: A > extended_ereal,F: filter @ A] :
( ( Y
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( X2
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ X2 ) @ F )
=> ( filterlim @ A @ extended_ereal
@ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F2 @ X ) @ Y )
@ ( topolo1920029354e_nhds @ extended_ereal @ ( plus_plus @ extended_ereal @ X2 @ Y ) )
@ F ) ) ) ) ).
% tendsto_cadd_ereal
thf(fact_162_ereal__minus__real__tendsto__MInf,axiom,
( filterlim @ nat @ extended_ereal
@ ^ [X: nat] : ( extended_ereal2 @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ X ) ) )
@ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
@ ( at_top @ nat ) ) ).
% ereal_minus_real_tendsto_MInf
thf(fact_163_tendsto__add__ereal__nonneg,axiom,
! [A: $tType,X2: extended_ereal,Y: extended_ereal,F2: A > extended_ereal,F: filter @ A,G2: A > extended_ereal] :
( ( X2
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Y
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ X2 ) @ F )
=> ( ( filterlim @ A @ extended_ereal @ G2 @ ( topolo1920029354e_nhds @ extended_ereal @ Y ) @ F )
=> ( filterlim @ A @ extended_ereal
@ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F2 @ X ) @ ( G2 @ X ) )
@ ( topolo1920029354e_nhds @ extended_ereal @ ( plus_plus @ extended_ereal @ X2 @ Y ) )
@ F ) ) ) ) ) ).
% tendsto_add_ereal_nonneg
thf(fact_164_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add_left_cancel
thf(fact_165_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add_right_cancel
thf(fact_166_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_167_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_168_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_add_distrib
thf(fact_169_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
= B2 ) ) ).
% minus_add_cancel
thf(fact_170_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
= B2 ) ) ).
% add_minus_cancel
thf(fact_171_ereal__PInfty__eq__plus,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ( extend1396239628finity @ extended_ereal )
= ( plus_plus @ extended_ereal @ A2 @ B2 ) )
= ( ( A2
= ( extend1396239628finity @ extended_ereal ) )
| ( B2
= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% ereal_PInfty_eq_plus
thf(fact_172_ereal__plus__eq__PInfty,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ( plus_plus @ extended_ereal @ A2 @ B2 )
= ( extend1396239628finity @ extended_ereal ) )
= ( ( A2
= ( extend1396239628finity @ extended_ereal ) )
| ( B2
= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% ereal_plus_eq_PInfty
thf(fact_173_ereal__MInfty__eq__plus,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
= ( plus_plus @ extended_ereal @ A2 @ B2 ) )
= ( ( ( A2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
& ( B2
!= ( extend1396239628finity @ extended_ereal ) ) )
| ( ( B2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
& ( A2
!= ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% ereal_MInfty_eq_plus
thf(fact_174_ereal__plus__eq__MInfty,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ( plus_plus @ extended_ereal @ A2 @ B2 )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
= ( ( ( A2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
| ( B2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
& ( A2
!= ( extend1396239628finity @ extended_ereal ) )
& ( B2
!= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% ereal_plus_eq_MInfty
thf(fact_175_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_176_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_177_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A,K: A,A2: A,B2: A] :
( ( A4
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A4 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_178_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B6: A,K: A,B2: A,A2: A] :
( ( B6
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B6 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_179_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_180_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add.left_cancel
thf(fact_181_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add.right_cancel
thf(fact_182_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B3: A] : ( plus_plus @ A @ B3 @ A5 ) ) ) ) ).
% add.commute
thf(fact_183_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.left_commute
thf(fact_184_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_185_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_186_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_187_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,K: A,A2: A] :
( ( A4
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( uminus_uminus @ A @ A4 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_188_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_189_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_190_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_191_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_mono
thf(fact_192_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_193_less__eqE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ! [C5: A] :
( B2
!= ( plus_plus @ A @ A2 @ C5 ) ) ) ) ).
% less_eqE
thf(fact_194_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_195_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
? [C3: A] :
( B3
= ( plus_plus @ A @ A5 @ C3 ) ) ) ) ) ).
% le_iff_add
thf(fact_196_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_197_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_198_plus__ereal_Osimps_I2_J,axiom,
! [A2: extended_ereal] :
( ( plus_plus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ A2 )
= ( extend1396239628finity @ extended_ereal ) ) ).
% plus_ereal.simps(2)
thf(fact_199_plus__ereal_Osimps_I3_J,axiom,
! [A2: extended_ereal] :
( ( plus_plus @ extended_ereal @ A2 @ ( extend1396239628finity @ extended_ereal ) )
= ( extend1396239628finity @ extended_ereal ) ) ).
% plus_ereal.simps(3)
thf(fact_200_plus__ereal_Osimps_I6_J,axiom,
( ( plus_plus @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% plus_ereal.simps(6)
thf(fact_201_ereal__add__cancel__left,axiom,
! [A2: extended_ereal,B2: extended_ereal,C2: extended_ereal] :
( ( A2
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( ( plus_plus @ extended_ereal @ A2 @ B2 )
= ( plus_plus @ extended_ereal @ A2 @ C2 ) )
= ( ( A2
= ( extend1396239628finity @ extended_ereal ) )
| ( B2 = C2 ) ) ) ) ).
% ereal_add_cancel_left
thf(fact_202_ereal__add__cancel__right,axiom,
! [A2: extended_ereal,B2: extended_ereal,C2: extended_ereal] :
( ( A2
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( ( plus_plus @ extended_ereal @ B2 @ A2 )
= ( plus_plus @ extended_ereal @ C2 @ A2 ) )
= ( ( A2
= ( extend1396239628finity @ extended_ereal ) )
| ( B2 = C2 ) ) ) ) ).
% ereal_add_cancel_right
thf(fact_203_tendsto__add,axiom,
! [A: $tType,B: $tType] :
( ( topolo1314133330id_add @ A )
=> ! [F2: B > A,A2: A,F: filter @ B,G2: B > A,B2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ A2 ) @ F )
=> ( ( filterlim @ B @ A @ G2 @ ( topolo1920029354e_nhds @ A @ B2 ) @ F )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ ( F2 @ X ) @ ( G2 @ X ) )
@ ( topolo1920029354e_nhds @ A @ ( plus_plus @ A @ A2 @ B2 ) )
@ F ) ) ) ) ).
% tendsto_add
thf(fact_204_tendsto__add__const__iff,axiom,
! [A: $tType,B: $tType] :
( ( real_V55928688vector @ A )
=> ! [C2: A,F2: B > A,D2: A,F: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ C2 @ ( F2 @ X ) )
@ ( topolo1920029354e_nhds @ A @ ( plus_plus @ A @ C2 @ D2 ) )
@ F )
= ( filterlim @ B @ A @ F2 @ ( topolo1920029354e_nhds @ A @ D2 ) @ F ) ) ) ).
% tendsto_add_const_iff
thf(fact_205_plus__ereal_Osimps_I5_J,axiom,
! [P3: real] :
( ( plus_plus @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ P3 ) )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% plus_ereal.simps(5)
thf(fact_206_plus__ereal_Osimps_I4_J,axiom,
! [R: real] :
( ( plus_plus @ extended_ereal @ ( extended_ereal2 @ R ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% plus_ereal.simps(4)
thf(fact_207_ereal__add__le__add__iff,axiom,
! [C2: extended_ereal,A2: extended_ereal,B2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( plus_plus @ extended_ereal @ C2 @ A2 ) @ ( plus_plus @ extended_ereal @ C2 @ B2 ) )
= ( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
| ( C2
= ( extend1396239628finity @ extended_ereal ) )
| ( ( C2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
& ( A2
!= ( extend1396239628finity @ extended_ereal ) )
& ( B2
!= ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% ereal_add_le_add_iff
thf(fact_208_ereal__add__le__add__iff2,axiom,
! [A2: extended_ereal,C2: extended_ereal,B2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( plus_plus @ extended_ereal @ A2 @ C2 ) @ ( plus_plus @ extended_ereal @ B2 @ C2 ) )
= ( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
| ( C2
= ( extend1396239628finity @ extended_ereal ) )
| ( ( C2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
& ( A2
!= ( extend1396239628finity @ extended_ereal ) )
& ( B2
!= ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% ereal_add_le_add_iff2
thf(fact_209_tendsto__add__ereal__MInf,axiom,
! [A: $tType,Y: extended_ereal,F2: A > extended_ereal,F: filter @ A,G2: A > extended_ereal] :
( ( Y
!= ( extend1396239628finity @ extended_ereal ) )
=> ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) @ F )
=> ( ( filterlim @ A @ extended_ereal @ G2 @ ( topolo1920029354e_nhds @ extended_ereal @ Y ) @ F )
=> ( filterlim @ A @ extended_ereal
@ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F2 @ X ) @ ( G2 @ X ) )
@ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
@ F ) ) ) ) ).
% tendsto_add_ereal_MInf
thf(fact_210_tendsto__add__ereal__PInf,axiom,
! [A: $tType,Y: extended_ereal,F2: A > extended_ereal,F: filter @ A,G2: A > extended_ereal] :
( ( Y
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ F )
=> ( ( filterlim @ A @ extended_ereal @ G2 @ ( topolo1920029354e_nhds @ extended_ereal @ Y ) @ F )
=> ( filterlim @ A @ extended_ereal
@ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F2 @ X ) @ ( G2 @ X ) )
@ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
@ F ) ) ) ) ).
% tendsto_add_ereal_PInf
thf(fact_211_tendsto__add__ereal__general,axiom,
! [A: $tType,X2: extended_ereal,Y: extended_ereal,F2: A > extended_ereal,F: filter @ A,G2: A > extended_ereal] :
( ~ ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
& ( Y
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
| ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
& ( Y
= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( filterlim @ A @ extended_ereal @ F2 @ ( topolo1920029354e_nhds @ extended_ereal @ X2 ) @ F )
=> ( ( filterlim @ A @ extended_ereal @ G2 @ ( topolo1920029354e_nhds @ extended_ereal @ Y ) @ F )
=> ( filterlim @ A @ extended_ereal
@ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F2 @ X ) @ ( G2 @ X ) )
@ ( topolo1920029354e_nhds @ extended_ereal @ ( plus_plus @ extended_ereal @ X2 @ Y ) )
@ F ) ) ) ) ).
% tendsto_add_ereal_general
thf(fact_212_tendsto__at__botI__sequentially,axiom,
! [B: $tType] :
( ( topolo2135403230pology @ B )
=> ! [F2: real > B,Y: B] :
( ! [X8: nat > real] :
( ( filterlim @ nat @ real @ X8 @ ( at_bot @ real ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ B
@ ^ [N: nat] : ( F2 @ ( X8 @ N ) )
@ ( topolo1920029354e_nhds @ B @ Y )
@ ( at_top @ nat ) ) )
=> ( filterlim @ real @ B @ F2 @ ( topolo1920029354e_nhds @ B @ Y ) @ ( at_bot @ real ) ) ) ) ).
% tendsto_at_botI_sequentially
thf(fact_213_id__nat__ereal__tendsto__PInf,axiom,
( filterlim @ nat @ extended_ereal
@ ^ [X: nat] : ( extended_ereal2 @ ( semiring_1_of_nat @ real @ X ) )
@ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
@ ( at_top @ nat ) ) ).
% id_nat_ereal_tendsto_PInf
thf(fact_214_ereal__liminf__add__mono,axiom,
! [U: nat > extended_ereal,V: nat > extended_ereal] :
( ~ ( ( ( ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ U )
= ( extend1396239628finity @ extended_ereal ) )
& ( ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ V )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
| ( ( ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ U )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
& ( ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ V )
= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ord_less_eq @ extended_ereal @ ( plus_plus @ extended_ereal @ ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ U ) @ ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat ) @ V ) )
@ ( liminf_Liminf @ nat @ extended_ereal @ ( at_top @ nat )
@ ^ [N: nat] : ( plus_plus @ extended_ereal @ ( U @ N ) @ ( V @ N ) ) ) ) ) ).
% ereal_liminf_add_mono
thf(fact_215_of__nat__le__iff,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ! [M5: nat,N7: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M5 ) @ ( semiring_1_of_nat @ A @ N7 ) )
= ( ord_less_eq @ nat @ M5 @ N7 ) ) ) ).
% of_nat_le_iff
thf(fact_216_of__nat__mono,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ).
% of_nat_mono
thf(fact_217_ereal__tendsto__simps2_I2_J,axiom,
! [A: $tType,F2: A > real,F: filter @ A] :
( ( filterlim @ A @ extended_ereal @ ( comp @ real @ extended_ereal @ A @ extended_ereal2 @ F2 ) @ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ F )
= ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F ) ) ).
% ereal_tendsto_simps2(2)
thf(fact_218_nat__add__left__cancel__le,axiom,
! [K: nat,M5: nat,N7: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M5 ) @ ( plus_plus @ nat @ K @ N7 ) )
= ( ord_less_eq @ nat @ M5 @ N7 ) ) ).
% nat_add_left_cancel_le
thf(fact_219_filterlim__at__top__add__at__top,axiom,
! [A: $tType,F2: A > real,F: filter @ A,G2: A > real] :
( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F )
=> ( ( filterlim @ A @ real @ G2 @ ( at_top @ real ) @ F )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( plus_plus @ real @ ( F2 @ X ) @ ( G2 @ X ) )
@ ( at_top @ real )
@ F ) ) ) ).
% filterlim_at_top_add_at_top
thf(fact_220_filterlim__tendsto__add__at__top,axiom,
! [A: $tType,F2: A > real,C2: real,F: filter @ A,G2: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo1920029354e_nhds @ real @ C2 ) @ F )
=> ( ( filterlim @ A @ real @ G2 @ ( at_top @ real ) @ F )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( plus_plus @ real @ ( F2 @ X ) @ ( G2 @ X ) )
@ ( at_top @ real )
@ F ) ) ) ).
% filterlim_tendsto_add_at_top
thf(fact_221_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_222_add__leE,axiom,
! [M5: nat,K: nat,N7: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M5 @ K ) @ N7 )
=> ~ ( ( ord_less_eq @ nat @ M5 @ N7 )
=> ~ ( ord_less_eq @ nat @ K @ N7 ) ) ) ).
% add_leE
thf(fact_223_le__add1,axiom,
! [N7: nat,M5: nat] : ( ord_less_eq @ nat @ N7 @ ( plus_plus @ nat @ N7 @ M5 ) ) ).
% le_add1
thf(fact_224_le__add2,axiom,
! [N7: nat,M5: nat] : ( ord_less_eq @ nat @ N7 @ ( plus_plus @ nat @ M5 @ N7 ) ) ).
% le_add2
thf(fact_225_add__leD1,axiom,
! [M5: nat,K: nat,N7: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M5 @ K ) @ N7 )
=> ( ord_less_eq @ nat @ M5 @ N7 ) ) ).
% add_leD1
thf(fact_226_add__leD2,axiom,
! [M5: nat,K: nat,N7: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M5 @ K ) @ N7 )
=> ( ord_less_eq @ nat @ K @ N7 ) ) ).
% add_leD2
thf(fact_227_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq @ nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus @ nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_228_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_229_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_230_trans__le__add1,axiom,
! [I: nat,J: nat,M5: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M5 ) ) ) ).
% trans_le_add1
thf(fact_231_trans__le__add2,axiom,
! [I: nat,J: nat,M5: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M5 @ J ) ) ) ).
% trans_le_add2
thf(fact_232_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M4: nat,N: nat] :
? [K2: nat] :
( N
= ( plus_plus @ nat @ M4 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_233_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_leq_as_int
thf(fact_234_plus__ereal_Osimps_I1_J,axiom,
! [R: real,P3: real] :
( ( plus_plus @ extended_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P3 ) )
= ( extended_ereal2 @ ( plus_plus @ real @ R @ P3 ) ) ) ).
% plus_ereal.simps(1)
thf(fact_235_filterlim__at__bot__mirror,axiom,
! [A: $tType,F2: real > A,F: filter @ A] :
( ( filterlim @ real @ A @ F2 @ F @ ( at_bot @ real ) )
= ( filterlim @ real @ A
@ ^ [X: real] : ( F2 @ ( uminus_uminus @ real @ X ) )
@ F
@ ( at_top @ real ) ) ) ).
% filterlim_at_bot_mirror
thf(fact_236_filterlim__at__top__mirror,axiom,
! [A: $tType,F2: real > A,F: filter @ A] :
( ( filterlim @ real @ A @ F2 @ F @ ( at_top @ real ) )
= ( filterlim @ real @ A
@ ^ [X: real] : ( F2 @ ( uminus_uminus @ real @ X ) )
@ F
@ ( at_bot @ real ) ) ) ).
% filterlim_at_top_mirror
thf(fact_237_filterlim__uminus__at__bot__at__top,axiom,
filterlim @ real @ real @ ( uminus_uminus @ real ) @ ( at_bot @ real ) @ ( at_top @ real ) ).
% filterlim_uminus_at_bot_at_top
thf(fact_238_filterlim__uminus__at__top__at__bot,axiom,
filterlim @ real @ real @ ( uminus_uminus @ real ) @ ( at_top @ real ) @ ( at_bot @ real ) ).
% filterlim_uminus_at_top_at_bot
thf(fact_239_liminf__shift__k,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [U: nat > A,K: nat] :
( ( liminf_Liminf @ nat @ A @ ( at_top @ nat )
@ ^ [N: nat] : ( U @ ( plus_plus @ nat @ N @ K ) ) )
= ( liminf_Liminf @ nat @ A @ ( at_top @ nat ) @ U ) ) ) ).
% liminf_shift_k
thf(fact_240_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq @ nat @ Y6 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_241_nat__le__linear,axiom,
! [M5: nat,N7: nat] :
( ( ord_less_eq @ nat @ M5 @ N7 )
| ( ord_less_eq @ nat @ N7 @ M5 ) ) ).
% nat_le_linear
thf(fact_242_le__antisym,axiom,
! [M5: nat,N7: nat] :
( ( ord_less_eq @ nat @ M5 @ N7 )
=> ( ( ord_less_eq @ nat @ N7 @ M5 )
=> ( M5 = N7 ) ) ) ).
% le_antisym
thf(fact_243_eq__imp__le,axiom,
! [M5: nat,N7: nat] :
( ( M5 = N7 )
=> ( ord_less_eq @ nat @ M5 @ N7 ) ) ).
% eq_imp_le
thf(fact_244_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_245_le__refl,axiom,
! [N7: nat] : ( ord_less_eq @ nat @ N7 @ N7 ) ).
% le_refl
thf(fact_246_filterlim__uminus__at__bot,axiom,
! [A: $tType,F2: A > real,F: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_bot @ real ) @ F )
= ( filterlim @ A @ real
@ ^ [X: A] : ( uminus_uminus @ real @ ( F2 @ X ) )
@ ( at_top @ real )
@ F ) ) ).
% filterlim_uminus_at_bot
thf(fact_247_filterlim__uminus__at__top,axiom,
! [A: $tType,F2: A > real,F: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F )
= ( filterlim @ A @ real
@ ^ [X: A] : ( uminus_uminus @ real @ ( F2 @ X ) )
@ ( at_bot @ real )
@ F ) ) ).
% filterlim_uminus_at_top
thf(fact_248_LIMSEQ__ignore__initial__segment,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [F2: nat > A,A2: A,K: nat] :
( ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) )
@ ( topolo1920029354e_nhds @ A @ A2 )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_249_LIMSEQ__offset,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [F2: nat > A,K: nat,A2: A] :
( ( filterlim @ nat @ A
@ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) )
@ ( topolo1920029354e_nhds @ A @ A2 )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo1920029354e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_offset
thf(fact_250_plus__ereal_Oelims,axiom,
! [X2: extended_ereal,Xa3: extended_ereal,Y: extended_ereal] :
( ( ( plus_plus @ extended_ereal @ X2 @ Xa3 )
= Y )
=> ( ! [R2: real] :
( ( X2
= ( extended_ereal2 @ R2 ) )
=> ! [P4: real] :
( ( Xa3
= ( extended_ereal2 @ P4 ) )
=> ( Y
!= ( extended_ereal2 @ ( plus_plus @ real @ R2 @ P4 ) ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( Y
!= ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Y
!= ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Y
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ? [P4: real] :
( Xa3
= ( extended_ereal2 @ P4 ) )
=> ( Y
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ~ ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Y
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ).
% plus_ereal.elims
thf(fact_251_tendsto__PInfty__eq__at__top,axiom,
! [A: $tType,F2: A > real,F: filter @ A] :
( ( filterlim @ A @ extended_ereal
@ ^ [Z5: A] : ( extended_ereal2 @ ( F2 @ Z5 ) )
@ ( topolo1920029354e_nhds @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
@ F )
= ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F ) ) ).
% tendsto_PInfty_eq_at_top
thf(fact_252_tendsto__at__topI__sequentially,axiom,
! [B: $tType] :
( ( topolo2135403230pology @ B )
=> ! [F2: real > B,Y: B] :
( ! [X8: nat > real] :
( ( filterlim @ nat @ real @ X8 @ ( at_top @ real ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ B
@ ^ [N: nat] : ( F2 @ ( X8 @ N ) )
@ ( topolo1920029354e_nhds @ B @ Y )
@ ( at_top @ nat ) ) )
=> ( filterlim @ real @ B @ F2 @ ( topolo1920029354e_nhds @ B @ Y ) @ ( at_top @ real ) ) ) ) ).
% tendsto_at_topI_sequentially
thf(fact_253_filterlim__real__sequentially,axiom,
filterlim @ nat @ real @ ( semiring_1_of_nat @ real ) @ ( at_top @ real ) @ ( at_top @ nat ) ).
% filterlim_real_sequentially
thf(fact_254_int__ops_I5_J,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A2 @ B2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_255_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
= ( ^ [A5: nat,B3: nat] :
( ( semiring_1_of_nat @ int @ A5 )
= ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
% Subclasses (5)
thf(subcl_Real__Vector__Spaces_Ometric__space___HOL_Otype,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ( type @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Ot2__space,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ( topological_t2_space @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Ouniform__space,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ( topolo47006728_space @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Otopological__space,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ( topolo503727757_space @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Ofirst__countable__topology,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ( topolo2135403230pology @ A ) ) ).
% Type constructors (119)
thf(tcon_HOL_Obool___Countable_Ocountable,axiom,
countable @ $o ).
thf(tcon_Set_Oset___Countable_Ocountable_1,axiom,
! [A7: $tType] :
( ( finite_finite @ A7 )
=> ( countable @ ( set @ A7 ) ) ) ).
thf(tcon_Nat_Onat___Countable_Ocountable_2,axiom,
countable @ nat ).
thf(tcon_Int_Oint___Countable_Ocountable_3,axiom,
countable @ int ).
thf(tcon_fun___Countable_Ocountable_4,axiom,
! [A7: $tType,A8: $tType] :
( ( ( finite_finite @ A7 )
& ( countable @ A8 ) )
=> ( countable @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A7: $tType,A8: $tType] :
( ( ( finite_finite @ A7 )
& ( finite_finite @ A8 ) )
=> ( finite_finite @ ( A7 > A8 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
! [A7: $tType] :
( ( finite_finite @ A7 )
=> ( finite_finite @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_6,axiom,
finite_finite @ $o ).
thf(tcon_fun___Topological__Spaces_Ofirst__countable__topology,axiom,
! [A7: $tType,A8: $tType] :
( ( ( countable @ A7 )
& ( topolo2135403230pology @ A8 ) )
=> ( topolo2135403230pology @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
! [A7: $tType,A8: $tType] :
( ( topolo503727757_space @ A8 )
=> ( topolo503727757_space @ ( 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_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_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_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Otopological__space_7,axiom,
topolo503727757_space @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
topolo2117631714pology @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
topolo259154727pology @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ int ).
thf(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom,
topolo1314133330id_add @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
topological_t2_space @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_8,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_9,axiom,
order @ int ).
thf(tcon_Int_Oint___Orderings_Oord_10,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_11,axiom,
uminus @ int ).
thf(tcon_Nat_Onat___Topological__Spaces_Ofirst__countable__topology_12,axiom,
topolo2135403230pology @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_13,axiom,
ordere236663937imp_le @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_14,axiom,
topolo503727757_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_15,axiom,
topolo2117631714pology @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_16,axiom,
topolo259154727pology @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_17,axiom,
linord1659791738miring @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_18,axiom,
ordere779506340up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_19,axiom,
topolo1314133330id_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_20,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_21,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_22,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_23,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_24,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_25,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_26,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_27,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_28,axiom,
ord @ nat ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_29,axiom,
! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_30,axiom,
! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Groups_Oab__semigroup__add_31,axiom,
! [A7: $tType] :
( ( ab_semigroup_add @ A7 )
=> ( ab_semigroup_add @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ocomm__monoid__add_32,axiom,
! [A7: $tType] :
( ( comm_monoid_add @ A7 )
=> ( comm_monoid_add @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Groups_Osemigroup__add_33,axiom,
! [A7: $tType] :
( ( semigroup_add @ A7 )
=> ( semigroup_add @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_34,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_35,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_36,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_37,axiom,
! [A7: $tType] : ( uminus @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_38,axiom,
topolo503727757_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_39,axiom,
topolo2117631714pology @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_40,axiom,
comple187826305attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_41,axiom,
topolo259154727pology @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_42,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_43,axiom,
boolean_algebra @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_44,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_45,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_46,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_47,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_48,axiom,
uminus @ $o ).
thf(tcon_Real_Oreal___Topological__Spaces_Ofirst__countable__topology_49,axiom,
topolo2135403230pology @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_50,axiom,
ordere236663937imp_le @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
real_V55928688vector @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_51,axiom,
topolo503727757_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_52,axiom,
topolo2117631714pology @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_53,axiom,
topolo259154727pology @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_54,axiom,
linord1659791738miring @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
topolo47006728_space @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
real_V2090557954_space @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_55,axiom,
ordere779506340up_add @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__monoid__add_56,axiom,
topolo1314133330id_add @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
topolo799126099up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_57,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_58,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_59,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_60,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_61,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_62,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_63,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_64,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_65,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_66,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_67,axiom,
order @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_68,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_69,axiom,
uminus @ real ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_70,axiom,
! [A7: $tType] : ( comple187826305attice @ ( filter @ A7 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_71,axiom,
! [A7: $tType] : ( preorder @ ( filter @ A7 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_72,axiom,
! [A7: $tType] : ( order @ ( filter @ A7 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_73,axiom,
! [A7: $tType] : ( ord @ ( filter @ A7 ) ) ).
thf(tcon_Extended__Real_Oereal___Topological__Spaces_Ofirst__countable__topology_74,axiom,
topolo2135403230pology @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Topological__Spaces_Otopological__space_75,axiom,
topolo503727757_space @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Topological__Spaces_Olinorder__topology_76,axiom,
topolo2117631714pology @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Complete__Lattices_Ocomplete__linorder,axiom,
comple1035589618norder @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Complete__Lattices_Ocomplete__lattice_77,axiom,
comple187826305attice @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Topological__Spaces_Oorder__topology_78,axiom,
topolo259154727pology @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Oordered__ab__semigroup__add_79,axiom,
ordere779506340up_add @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Topological__Spaces_Ot2__space_80,axiom,
topological_t2_space @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Oab__semigroup__add_81,axiom,
ab_semigroup_add @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Ocomm__monoid__add_82,axiom,
comm_monoid_add @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Osemigroup__add_83,axiom,
semigroup_add @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Opreorder_84,axiom,
preorder @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Olinorder_85,axiom,
linorder @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Oorder_86,axiom,
order @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Oord_87,axiom,
ord @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Ouminus_88,axiom,
uminus @ extended_ereal ).
% Free types (1)
thf(tfree_0,hypothesis,
real_V2090557954_space @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
( filterlim @ nat @ extended_ereal
@ ^ [I2: nat] : ( uminus_uminus @ extended_ereal @ ( f @ ( x @ I2 ) ) )
@ ( topolo1920029354e_nhds @ extended_ereal @ ( uminus_uminus @ extended_ereal @ a2 ) )
@ ( at_top @ nat ) ) ).
%------------------------------------------------------------------------------