TPTP Problem File: ITP112^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP112^1 : 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 : 0.70 v8.2.0, 0.69 v8.1.0, 0.73 v7.5.0
% Syntax : Number of formulae : 451 ( 130 unt; 95 typ; 0 def)
% Number of atoms : 1312 ( 382 equ; 0 cnn)
% Maximal formula atoms : 81 ( 3 avg)
% Number of connectives : 4092 ( 116 ~; 34 |; 57 &;3135 @)
% ( 0 <=>; 750 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 8 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 500 ( 500 >; 0 *; 0 +; 0 <<)
% Number of symbols : 84 ( 83 usr; 6 con; 0-3 aty)
% Number of variables : 1297 ( 136 ^;1117 !; 44 ?;1297 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:41:16.220
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
thf(ty_n_t__Filter__Ofilter_It__Extended____Real__Oereal_J,type,
filter2049122004_ereal: $tType ).
thf(ty_n_t__Set__Oset_It__Extended____Real__Oereal_J,type,
set_Extended_ereal: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
filter_real: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
filter_nat: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Int__Oint_J,type,
filter_int: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Filter__Ofilter_Itf__a_J,type,
filter_a: $tType ).
thf(ty_n_t__Extended____Real__Oereal,type,
extended_ereal: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (83)
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Real__Oereal,type,
extend1289208545_ereal: extended_ereal ).
thf(sy_c_Extended__Real_Oereal_Oereal,type,
extended_ereal2: real > extended_ereal ).
thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
at_bot_real: filter_real ).
thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
at_top_nat: filter_nat ).
thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
at_top_real: filter_real ).
thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
filter1531173832_ereal: ( nat > extended_ereal ) > filter2049122004_ereal > filter_nat > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Int__Oint,type,
filterlim_nat_int: ( nat > int ) > filter_int > filter_nat > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001tf__a,type,
filterlim_nat_a: ( nat > a ) > filter_a > filter_nat > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
comp_E1308517939al_nat: ( extended_ereal > extended_ereal ) > ( nat > extended_ereal ) > nat > extended_ereal ).
thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001tf__a,type,
comp_E489644891real_a: ( extended_ereal > extended_ereal ) > ( a > extended_ereal ) > a > extended_ereal ).
thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001t__Int__Oint_001t__Nat__Onat,type,
comp_E1436437929nt_nat: ( extended_ereal > int ) > ( nat > extended_ereal ) > nat > int ).
thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001t__Nat__Onat_001t__Nat__Onat,type,
comp_E1523169101at_nat: ( extended_ereal > nat ) > ( nat > extended_ereal ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001t__Real__Oreal_001t__Nat__Onat,type,
comp_E1477338153al_nat: ( extended_ereal > real ) > ( nat > extended_ereal ) > nat > real ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
comp_n1096781355al_nat: ( nat > extended_ereal ) > ( nat > nat ) > nat > extended_ereal ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001tf__a_001t__Nat__Onat,type,
comp_nat_a_nat: ( nat > a ) > ( nat > nat ) > nat > a ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
comp_r1410008527al_nat: ( real > extended_ereal ) > ( nat > real ) > nat > extended_ereal ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Int__Oint_001t__Nat__Onat,type,
comp_real_int_nat: ( real > int ) > ( nat > real ) > nat > int ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Nat__Onat_001t__Nat__Onat,type,
comp_real_nat_nat: ( real > nat ) > ( nat > real ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
comp_a1112243075al_nat: ( a > extended_ereal ) > ( nat > a ) > nat > extended_ereal ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Extended____Real__Oereal_001tf__a,type,
comp_a780206603real_a: ( a > extended_ereal ) > ( a > a ) > a > extended_ereal ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Int__Oint_001t__Nat__Onat,type,
comp_a_int_nat: ( a > int ) > ( nat > a ) > nat > int ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Nat__Onat_001t__Nat__Onat,type,
comp_a_nat_nat: ( a > nat ) > ( nat > a ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Real__Oreal_001t__Nat__Onat,type,
comp_a_real_nat: ( a > real ) > ( nat > a ) > nat > real ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Nat__Onat,type,
comp_a_a_nat: ( a > a ) > ( nat > a ) > nat > a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Real__Oereal,type,
plus_p2118002693_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Extended____Real__Oereal,type,
uminus1208298309_ereal: extended_ereal > extended_ereal ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_HOL_OUniq_001t__Extended____Real__Oereal,type,
uniq_Extended_ereal: ( extended_ereal > $o ) > $o ).
thf(sy_c_HOL_OUniq_001t__Real__Oreal,type,
uniq_real: ( real > $o ) > $o ).
thf(sy_c_HOL_OUniq_001tf__a,type,
uniq_a: ( a > $o ) > $o ).
thf(sy_c_Liminf__Limsup_OLiminf_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
liminf1045857232_ereal: filter_nat > ( nat > extended_ereal ) > extended_ereal ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
lower_1087098792_ereal: extended_ereal > ( extended_ereal > extended_ereal ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Extended____Real__Oereal_001t__Int__Oint,type,
lower_48196818al_int: extended_ereal > ( extended_ereal > int ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
lower_1558406774al_nat: extended_ereal > ( extended_ereal > nat ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Extended____Real__Oereal_001t__Real__Oreal,type,
lower_1165973074l_real: extended_ereal > ( extended_ereal > real ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Real__Oreal_001t__Extended____Real__Oereal,type,
lower_551915512_ereal: real > ( real > extended_ereal ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Real__Oreal_001t__Int__Oint,type,
lower_153911426al_int: real > ( real > int ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Real__Oreal_001t__Nat__Onat,type,
lower_1664121382al_nat: real > ( real > nat ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001tf__a_001t__Extended____Real__Oereal,type,
lower_191460856_ereal: a > ( a > extended_ereal ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001tf__a_001t__Int__Oint,type,
lower_956963458_a_int: a > ( a > int ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001tf__a_001t__Nat__Onat,type,
lower_319689766_a_nat: a > ( a > nat ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001tf__a_001t__Real__Oreal,type,
lower_231615490a_real: a > ( a > real ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
lower_1071158961_ereal: extended_ereal > ( extended_ereal > extended_ereal ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Extended____Real__Oereal_001t__Int__Oint,type,
lower_637387785al_int: extended_ereal > ( extended_ereal > int ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
lower_114093al_nat: extended_ereal > ( extended_ereal > nat ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Extended____Real__Oereal_001t__Real__Oreal,type,
lower_737640969l_real: extended_ereal > ( extended_ereal > real ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Real__Oreal_001t__Int__Oint,type,
lower_1075504779al_int: real > ( real > int ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Real__Oreal_001t__Nat__Onat,type,
lower_438231087al_nat: real > ( real > nat ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001tf__a_001t__Extended____Real__Oereal,type,
lower_534855297_ereal: a > ( a > extended_ereal ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001tf__a_001t__Int__Oint,type,
lower_1672990777_a_int: a > ( a > int ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001tf__a_001t__Nat__Onat,type,
lower_1035717085_a_nat: a > ( a > nat ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001tf__a_001t__Real__Oreal,type,
lower_755922489a_real: a > ( a > real ) > $o ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri2019852685at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri2110766477t_real: nat > real ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Real__Oereal,type,
ord_le824540014_ereal: extended_ereal > extended_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
ord_le1745708096er_nat: filter_nat > filter_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Real__Oreal_J,type,
ord_le132810396r_real: filter_real > filter_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Extended____Real__Oereal,type,
topolo1069469409_ereal: ( nat > extended_ereal ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
topolo411883481eq_int: ( nat > int ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
topolo1922093437eq_nat: ( nat > nat ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
topolo144289241q_real: ( nat > real ) > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Extended____Real__Oereal,type,
topolo2140997059_ereal: extended_ereal > filter2049122004_ereal ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Int__Oint,type,
topolo54776183ds_int: int > filter_int ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Nat__Onat,type,
topolo1564986139ds_nat: nat > filter_nat ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
topolo1664202871s_real: real > filter_real ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001tf__a,type,
topolo705128563nhds_a: a > filter_a ).
thf(sy_c_member_001t__Extended____Real__Oereal,type,
member1900190071_ereal: extended_ereal > set_Extended_ereal > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $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 (355)
thf(fact_0_x__def_I1_J,axiom,
filterlim_nat_a @ x @ ( topolo705128563nhds_a @ x0 ) @ at_top_nat ).
% x_def(1)
thf(fact_1_x__def_I2_J,axiom,
filter1531173832_ereal @ ( comp_a1112243075al_nat @ f @ x ) @ ( topolo2140997059_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] :
( ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ f @ x ) @ ( topolo2140997059_ereal @ a2 ) @ F )
=> ( filter1531173832_ereal
@ ^ [X: nat] : ( uminus1208298309_ereal @ ( comp_a1112243075al_nat @ f @ x @ X ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_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_191460856_ereal @ x0
@ ^ [X: a] : ( uminus1208298309_ereal @ ( f @ X ) ) ) ).
% lsc
thf(fact_4_tendsto__uminus__ereal,axiom,
! [F2: nat > extended_ereal,X2: extended_ereal,F: filter_nat] :
( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ X2 ) @ F )
=> ( filter1531173832_ereal
@ ^ [X: nat] : ( uminus1208298309_ereal @ ( F2 @ X ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ X2 ) )
@ F ) ) ).
% tendsto_uminus_ereal
thf(fact_5_tendsto__const,axiom,
! [K: real,F: filter_real] :
( filterlim_real_real
@ ^ [X: real] : K
@ ( topolo1664202871s_real @ K )
@ F ) ).
% tendsto_const
thf(fact_6_tendsto__const,axiom,
! [K: real,F: filter_nat] :
( filterlim_nat_real
@ ^ [X: nat] : K
@ ( topolo1664202871s_real @ K )
@ F ) ).
% tendsto_const
thf(fact_7_tendsto__const,axiom,
! [K: extended_ereal,F: filter_nat] :
( filter1531173832_ereal
@ ^ [X: nat] : K
@ ( topolo2140997059_ereal @ K )
@ F ) ).
% tendsto_const
thf(fact_8_tendsto__const,axiom,
! [K: a,F: filter_nat] :
( filterlim_nat_a
@ ^ [X: nat] : K
@ ( topolo705128563nhds_a @ K )
@ F ) ).
% tendsto_const
thf(fact_9_ereal__Lim__uminus,axiom,
! [F2: nat > extended_ereal,F0: extended_ereal,Net: filter_nat] :
( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ F0 ) @ Net )
= ( filter1531173832_ereal
@ ^ [X: nat] : ( uminus1208298309_ereal @ ( F2 @ X ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ F0 ) )
@ Net ) ) ).
% ereal_Lim_uminus
thf(fact_10_LIMSEQ__const__iff,axiom,
! [K: real,L: real] :
( ( filterlim_nat_real
@ ^ [N: nat] : K
@ ( topolo1664202871s_real @ L )
@ at_top_nat )
= ( K = L ) ) ).
% LIMSEQ_const_iff
thf(fact_11_LIMSEQ__const__iff,axiom,
! [K: extended_ereal,L: extended_ereal] :
( ( filter1531173832_ereal
@ ^ [N: nat] : K
@ ( topolo2140997059_ereal @ L )
@ at_top_nat )
= ( K = L ) ) ).
% LIMSEQ_const_iff
thf(fact_12_LIMSEQ__const__iff,axiom,
! [K: a,L: a] :
( ( filterlim_nat_a
@ ^ [N: nat] : K
@ ( topolo705128563nhds_a @ L )
@ at_top_nat )
= ( K = L ) ) ).
% LIMSEQ_const_iff
thf(fact_13_tendsto__minus__cancel,axiom,
! [F2: real > real,A: real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X: real] : ( uminus_uminus_real @ ( F2 @ X ) )
@ ( topolo1664202871s_real @ ( uminus_uminus_real @ A ) )
@ F )
=> ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ A ) @ F ) ) ).
% tendsto_minus_cancel
thf(fact_14_tendsto__minus__cancel,axiom,
! [F2: nat > real,A: real,F: filter_nat] :
( ( filterlim_nat_real
@ ^ [X: nat] : ( uminus_uminus_real @ ( F2 @ X ) )
@ ( topolo1664202871s_real @ ( uminus_uminus_real @ A ) )
@ F )
=> ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ A ) @ F ) ) ).
% tendsto_minus_cancel
thf(fact_15_tendsto__minus__cancel__left,axiom,
! [F2: real > real,Y: real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ ( uminus_uminus_real @ Y ) ) @ F )
= ( filterlim_real_real
@ ^ [X: real] : ( uminus_uminus_real @ ( F2 @ X ) )
@ ( topolo1664202871s_real @ Y )
@ F ) ) ).
% tendsto_minus_cancel_left
thf(fact_16_tendsto__minus__cancel__left,axiom,
! [F2: nat > real,Y: real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ ( uminus_uminus_real @ Y ) ) @ F )
= ( filterlim_nat_real
@ ^ [X: nat] : ( uminus_uminus_real @ ( F2 @ X ) )
@ ( topolo1664202871s_real @ Y )
@ F ) ) ).
% tendsto_minus_cancel_left
thf(fact_17_tendsto__minus,axiom,
! [F2: real > real,A: real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ A ) @ F )
=> ( filterlim_real_real
@ ^ [X: real] : ( uminus_uminus_real @ ( F2 @ X ) )
@ ( topolo1664202871s_real @ ( uminus_uminus_real @ A ) )
@ F ) ) ).
% tendsto_minus
thf(fact_18_tendsto__minus,axiom,
! [F2: nat > real,A: real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ A ) @ F )
=> ( filterlim_nat_real
@ ^ [X: nat] : ( uminus_uminus_real @ ( F2 @ X ) )
@ ( topolo1664202871s_real @ ( uminus_uminus_real @ A ) )
@ F ) ) ).
% tendsto_minus
thf(fact_19_LIMSEQ__unique,axiom,
! [X3: nat > extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ A ) @ at_top_nat )
=> ( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ B ) @ at_top_nat )
=> ( A = B ) ) ) ).
% LIMSEQ_unique
thf(fact_20_LIMSEQ__unique,axiom,
! [X3: nat > a,A: a,B: a] :
( ( filterlim_nat_a @ X3 @ ( topolo705128563nhds_a @ A ) @ at_top_nat )
=> ( ( filterlim_nat_a @ X3 @ ( topolo705128563nhds_a @ B ) @ at_top_nat )
=> ( A = B ) ) ) ).
% LIMSEQ_unique
thf(fact_21_LIMSEQ__unique,axiom,
! [X3: nat > real,A: real,B: real] :
( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ A ) @ at_top_nat )
=> ( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ B ) @ at_top_nat )
=> ( A = B ) ) ) ).
% LIMSEQ_unique
thf(fact_22_tendsto__uminus__nhds,axiom,
! [A: real] : ( filterlim_real_real @ uminus_uminus_real @ ( topolo1664202871s_real @ ( uminus_uminus_real @ A ) ) @ ( topolo1664202871s_real @ A ) ) ).
% tendsto_uminus_nhds
thf(fact_23_ereal__uminus__eq__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( uminus1208298309_ereal @ A )
= ( uminus1208298309_ereal @ B ) )
= ( A = B ) ) ).
% ereal_uminus_eq_iff
thf(fact_24_ereal__uminus__uminus,axiom,
! [A: extended_ereal] :
( ( uminus1208298309_ereal @ ( uminus1208298309_ereal @ A ) )
= A ) ).
% ereal_uminus_uminus
thf(fact_25_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_26_verit__negate__coefficient_I3_J,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_27_ereal__uminus__eq__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( uminus1208298309_ereal @ A )
= B )
= ( A
= ( uminus1208298309_ereal @ B ) ) ) ).
% ereal_uminus_eq_reorder
thf(fact_28_tendsto__eq__rhs,axiom,
! [F2: nat > extended_ereal,X2: extended_ereal,F: filter_nat,Y: extended_ereal] :
( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ X2 ) @ F )
=> ( ( X2 = Y )
=> ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ Y ) @ F ) ) ) ).
% tendsto_eq_rhs
thf(fact_29_tendsto__eq__rhs,axiom,
! [F2: nat > a,X2: a,F: filter_nat,Y: a] :
( ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ X2 ) @ F )
=> ( ( X2 = Y )
=> ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ Y ) @ F ) ) ) ).
% tendsto_eq_rhs
thf(fact_30_tendsto__eq__rhs,axiom,
! [F2: real > real,X2: real,F: filter_real,Y: real] :
( ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ X2 ) @ F )
=> ( ( X2 = Y )
=> ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ Y ) @ F ) ) ) ).
% tendsto_eq_rhs
thf(fact_31_tendsto__eq__rhs,axiom,
! [F2: nat > real,X2: real,F: filter_nat,Y: real] :
( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ X2 ) @ F )
=> ( ( X2 = Y )
=> ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ Y ) @ F ) ) ) ).
% tendsto_eq_rhs
thf(fact_32_tendsto__cong__limit,axiom,
! [F2: nat > extended_ereal,L: extended_ereal,F: filter_nat,K: extended_ereal] :
( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ F )
=> ( ( K = L )
=> ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ K ) @ F ) ) ) ).
% tendsto_cong_limit
thf(fact_33_tendsto__cong__limit,axiom,
! [F2: nat > a,L: a,F: filter_nat,K: a] :
( ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ L ) @ F )
=> ( ( K = L )
=> ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ K ) @ F ) ) ) ).
% tendsto_cong_limit
thf(fact_34_tendsto__cong__limit,axiom,
! [F2: real > real,L: real,F: filter_real,K: real] :
( ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ L ) @ F )
=> ( ( K = L )
=> ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ K ) @ F ) ) ) ).
% tendsto_cong_limit
thf(fact_35_tendsto__cong__limit,axiom,
! [F2: nat > real,L: real,F: filter_nat,K: real] :
( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ L ) @ F )
=> ( ( K = L )
=> ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ K ) @ F ) ) ) ).
% tendsto_cong_limit
thf(fact_36_comp__apply,axiom,
( comp_a1112243075al_nat
= ( ^ [F3: a > extended_ereal,G: nat > a,X: nat] : ( F3 @ ( G @ X ) ) ) ) ).
% comp_apply
thf(fact_37_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_38_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_39_lsc__sequentially__mem,axiom,
! [X0: real,F2: real > extended_ereal,X2: nat > real,C: nat > extended_ereal,C0: extended_ereal] :
( ( lower_551915512_ereal @ X0 @ F2 )
=> ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
=> ( ( filter1531173832_ereal @ C @ ( topolo2140997059_ereal @ C0 ) @ at_top_nat )
=> ( ! [N2: nat] : ( ord_le824540014_ereal @ ( F2 @ ( X2 @ N2 ) ) @ ( C @ N2 ) )
=> ( ord_le824540014_ereal @ ( F2 @ X0 ) @ C0 ) ) ) ) ) ).
% lsc_sequentially_mem
thf(fact_40_lsc__sequentially__mem,axiom,
! [X0: a,F2: a > extended_ereal,X2: nat > a,C: nat > extended_ereal,C0: extended_ereal] :
( ( lower_191460856_ereal @ X0 @ F2 )
=> ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
=> ( ( filter1531173832_ereal @ C @ ( topolo2140997059_ereal @ C0 ) @ at_top_nat )
=> ( ! [N2: nat] : ( ord_le824540014_ereal @ ( F2 @ ( X2 @ N2 ) ) @ ( C @ N2 ) )
=> ( ord_le824540014_ereal @ ( F2 @ X0 ) @ C0 ) ) ) ) ) ).
% lsc_sequentially_mem
thf(fact_41_lsc__sequentially__gen,axiom,
( lower_551915512_ereal
= ( ^ [X02: real,F3: real > extended_ereal] :
! [X: nat > real,C2: nat > extended_ereal,C02: extended_ereal] :
( ( ( filterlim_nat_real @ X @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
& ( filter1531173832_ereal @ C2 @ ( topolo2140997059_ereal @ C02 ) @ at_top_nat )
& ! [N: nat] : ( ord_le824540014_ereal @ ( F3 @ ( X @ N ) ) @ ( C2 @ N ) ) )
=> ( ord_le824540014_ereal @ ( F3 @ X02 ) @ C02 ) ) ) ) ).
% lsc_sequentially_gen
thf(fact_42_lsc__sequentially__gen,axiom,
( lower_191460856_ereal
= ( ^ [X02: a,F3: a > extended_ereal] :
! [X: nat > a,C2: nat > extended_ereal,C02: extended_ereal] :
( ( ( filterlim_nat_a @ X @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
& ( filter1531173832_ereal @ C2 @ ( topolo2140997059_ereal @ C02 ) @ at_top_nat )
& ! [N: nat] : ( ord_le824540014_ereal @ ( F3 @ ( X @ N ) ) @ ( C2 @ N ) ) )
=> ( ord_le824540014_ereal @ ( F3 @ X02 ) @ C02 ) ) ) ) ).
% lsc_sequentially_gen
thf(fact_43_lsc__at__mem,axiom,
! [X0: extended_ereal,F2: extended_ereal > nat,X2: nat > extended_ereal,A2: nat] :
( ( lower_1558406774al_nat @ X0 @ F2 )
=> ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
=> ( ( filterlim_nat_nat @ ( comp_E1523169101at_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
=> ( ord_less_eq_nat @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% lsc_at_mem
thf(fact_44_lsc__at__mem,axiom,
! [X0: extended_ereal,F2: extended_ereal > int,X2: nat > extended_ereal,A2: int] :
( ( lower_48196818al_int @ X0 @ F2 )
=> ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
=> ( ( filterlim_nat_int @ ( comp_E1436437929nt_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
=> ( ord_less_eq_int @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% lsc_at_mem
thf(fact_45_lsc__at__mem,axiom,
! [X0: extended_ereal,F2: extended_ereal > extended_ereal,X2: nat > extended_ereal,A2: extended_ereal] :
( ( lower_1087098792_ereal @ X0 @ F2 )
=> ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
=> ( ( filter1531173832_ereal @ ( comp_E1308517939al_nat @ F2 @ X2 ) @ ( topolo2140997059_ereal @ A2 ) @ at_top_nat )
=> ( ord_le824540014_ereal @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% lsc_at_mem
thf(fact_46_lsc__at__mem,axiom,
! [X0: extended_ereal,F2: extended_ereal > real,X2: nat > extended_ereal,A2: real] :
( ( lower_1165973074l_real @ X0 @ F2 )
=> ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
=> ( ( filterlim_nat_real @ ( comp_E1477338153al_nat @ F2 @ X2 ) @ ( topolo1664202871s_real @ A2 ) @ at_top_nat )
=> ( ord_less_eq_real @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% lsc_at_mem
thf(fact_47_lsc__at__mem,axiom,
! [X0: a,F2: a > nat,X2: nat > a,A2: nat] :
( ( lower_319689766_a_nat @ X0 @ F2 )
=> ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
=> ( ( filterlim_nat_nat @ ( comp_a_nat_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
=> ( ord_less_eq_nat @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% lsc_at_mem
thf(fact_48_lsc__at__mem,axiom,
! [X0: a,F2: a > int,X2: nat > a,A2: int] :
( ( lower_956963458_a_int @ X0 @ F2 )
=> ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
=> ( ( filterlim_nat_int @ ( comp_a_int_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
=> ( ord_less_eq_int @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% lsc_at_mem
thf(fact_49_lsc__at__mem,axiom,
! [X0: a,F2: a > real,X2: nat > a,A2: real] :
( ( lower_231615490a_real @ X0 @ F2 )
=> ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
=> ( ( filterlim_nat_real @ ( comp_a_real_nat @ F2 @ X2 ) @ ( topolo1664202871s_real @ A2 ) @ at_top_nat )
=> ( ord_less_eq_real @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% lsc_at_mem
thf(fact_50_lsc__at__mem,axiom,
! [X0: real,F2: real > nat,X2: nat > real,A2: nat] :
( ( lower_1664121382al_nat @ X0 @ F2 )
=> ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
=> ( ( filterlim_nat_nat @ ( comp_real_nat_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
=> ( ord_less_eq_nat @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% lsc_at_mem
thf(fact_51_lsc__at__mem,axiom,
! [X0: real,F2: real > int,X2: nat > real,A2: int] :
( ( lower_153911426al_int @ X0 @ F2 )
=> ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
=> ( ( filterlim_nat_int @ ( comp_real_int_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
=> ( ord_less_eq_int @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% lsc_at_mem
thf(fact_52_lsc__at__mem,axiom,
! [X0: real,F2: real > extended_ereal,X2: nat > real,A2: extended_ereal] :
( ( lower_551915512_ereal @ X0 @ F2 )
=> ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
=> ( ( filter1531173832_ereal @ ( comp_r1410008527al_nat @ F2 @ X2 ) @ ( topolo2140997059_ereal @ A2 ) @ at_top_nat )
=> ( ord_le824540014_ereal @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% lsc_at_mem
thf(fact_53_lsc__at__def,axiom,
( lower_1558406774al_nat
= ( ^ [X02: extended_ereal,F3: extended_ereal > nat] :
! [X4: nat > extended_ereal,L2: nat] :
( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
& ( filterlim_nat_nat @ ( comp_E1523169101at_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
=> ( ord_less_eq_nat @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% lsc_at_def
thf(fact_54_lsc__at__def,axiom,
( lower_48196818al_int
= ( ^ [X02: extended_ereal,F3: extended_ereal > int] :
! [X4: nat > extended_ereal,L2: int] :
( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
& ( filterlim_nat_int @ ( comp_E1436437929nt_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
=> ( ord_less_eq_int @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% lsc_at_def
thf(fact_55_lsc__at__def,axiom,
( lower_1087098792_ereal
= ( ^ [X02: extended_ereal,F3: extended_ereal > extended_ereal] :
! [X4: nat > extended_ereal,L2: extended_ereal] :
( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
& ( filter1531173832_ereal @ ( comp_E1308517939al_nat @ F3 @ X4 ) @ ( topolo2140997059_ereal @ L2 ) @ at_top_nat ) )
=> ( ord_le824540014_ereal @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% lsc_at_def
thf(fact_56_lsc__at__def,axiom,
( lower_1165973074l_real
= ( ^ [X02: extended_ereal,F3: extended_ereal > real] :
! [X4: nat > extended_ereal,L2: real] :
( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
& ( filterlim_nat_real @ ( comp_E1477338153al_nat @ F3 @ X4 ) @ ( topolo1664202871s_real @ L2 ) @ at_top_nat ) )
=> ( ord_less_eq_real @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% lsc_at_def
thf(fact_57_lsc__at__def,axiom,
( lower_319689766_a_nat
= ( ^ [X02: a,F3: a > nat] :
! [X4: nat > a,L2: nat] :
( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
& ( filterlim_nat_nat @ ( comp_a_nat_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
=> ( ord_less_eq_nat @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% lsc_at_def
thf(fact_58_lsc__at__def,axiom,
( lower_956963458_a_int
= ( ^ [X02: a,F3: a > int] :
! [X4: nat > a,L2: int] :
( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
& ( filterlim_nat_int @ ( comp_a_int_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
=> ( ord_less_eq_int @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% lsc_at_def
thf(fact_59_lsc__at__def,axiom,
( lower_231615490a_real
= ( ^ [X02: a,F3: a > real] :
! [X4: nat > a,L2: real] :
( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
& ( filterlim_nat_real @ ( comp_a_real_nat @ F3 @ X4 ) @ ( topolo1664202871s_real @ L2 ) @ at_top_nat ) )
=> ( ord_less_eq_real @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% lsc_at_def
thf(fact_60_lsc__at__def,axiom,
( lower_1664121382al_nat
= ( ^ [X02: real,F3: real > nat] :
! [X4: nat > real,L2: nat] :
( ( ( filterlim_nat_real @ X4 @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
& ( filterlim_nat_nat @ ( comp_real_nat_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
=> ( ord_less_eq_nat @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% lsc_at_def
thf(fact_61_lsc__at__def,axiom,
( lower_153911426al_int
= ( ^ [X02: real,F3: real > int] :
! [X4: nat > real,L2: int] :
( ( ( filterlim_nat_real @ X4 @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
& ( filterlim_nat_int @ ( comp_real_int_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
=> ( ord_less_eq_int @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% lsc_at_def
thf(fact_62_lsc__at__def,axiom,
( lower_551915512_ereal
= ( ^ [X02: real,F3: real > extended_ereal] :
! [X4: nat > real,L2: extended_ereal] :
( ( ( filterlim_nat_real @ X4 @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
& ( filter1531173832_ereal @ ( comp_r1410008527al_nat @ F3 @ X4 ) @ ( topolo2140997059_ereal @ L2 ) @ at_top_nat ) )
=> ( ord_le824540014_ereal @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% lsc_at_def
thf(fact_63_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_64_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_65_ereal__minus__le__minus,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A ) @ ( uminus1208298309_ereal @ B ) )
= ( ord_le824540014_ereal @ B @ A ) ) ).
% ereal_minus_le_minus
thf(fact_66_verit__la__disequality,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( A = B )
| ~ ( ord_le824540014_ereal @ A @ B )
| ~ ( ord_le824540014_ereal @ B @ A ) ) ).
% verit_la_disequality
thf(fact_67_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_68_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_69_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_70_ereal__complete__Inf,axiom,
! [S: set_Extended_ereal] :
? [X5: extended_ereal] :
( ! [Xa: extended_ereal] :
( ( member1900190071_ereal @ Xa @ S )
=> ( ord_le824540014_ereal @ X5 @ Xa ) )
& ! [Z: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member1900190071_ereal @ Xa2 @ S )
=> ( ord_le824540014_ereal @ Z @ Xa2 ) )
=> ( ord_le824540014_ereal @ Z @ X5 ) ) ) ).
% ereal_complete_Inf
thf(fact_71_ereal__complete__Sup,axiom,
! [S: set_Extended_ereal] :
? [X5: extended_ereal] :
( ! [Xa: extended_ereal] :
( ( member1900190071_ereal @ Xa @ S )
=> ( ord_le824540014_ereal @ Xa @ X5 ) )
& ! [Z: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member1900190071_ereal @ Xa2 @ S )
=> ( ord_le824540014_ereal @ Xa2 @ Z ) )
=> ( ord_le824540014_ereal @ X5 @ Z ) ) ) ).
% ereal_complete_Sup
thf(fact_72_le__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_73_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_74_minus__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_75_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_76_le__imp__neg__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_77_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_78_ereal__uminus__le__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A ) @ B )
= ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ B ) @ A ) ) ).
% ereal_uminus_le_reorder
thf(fact_79_lim__mono,axiom,
! [N3: nat,X3: nat > nat,Y2: nat > nat,X2: nat,Y: nat] :
( ! [N2: nat] :
( ( ord_less_eq_nat @ N3 @ N2 )
=> ( ord_less_eq_nat @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
=> ( ( filterlim_nat_nat @ X3 @ ( topolo1564986139ds_nat @ X2 ) @ at_top_nat )
=> ( ( filterlim_nat_nat @ Y2 @ ( topolo1564986139ds_nat @ Y ) @ at_top_nat )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ).
% lim_mono
thf(fact_80_lim__mono,axiom,
! [N3: nat,X3: nat > int,Y2: nat > int,X2: int,Y: int] :
( ! [N2: nat] :
( ( ord_less_eq_nat @ N3 @ N2 )
=> ( ord_less_eq_int @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
=> ( ( filterlim_nat_int @ X3 @ ( topolo54776183ds_int @ X2 ) @ at_top_nat )
=> ( ( filterlim_nat_int @ Y2 @ ( topolo54776183ds_int @ Y ) @ at_top_nat )
=> ( ord_less_eq_int @ X2 @ Y ) ) ) ) ).
% lim_mono
thf(fact_81_lim__mono,axiom,
! [N3: nat,X3: nat > extended_ereal,Y2: nat > extended_ereal,X2: extended_ereal,Y: extended_ereal] :
( ! [N2: nat] :
( ( ord_less_eq_nat @ N3 @ N2 )
=> ( ord_le824540014_ereal @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
=> ( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ X2 ) @ at_top_nat )
=> ( ( filter1531173832_ereal @ Y2 @ ( topolo2140997059_ereal @ Y ) @ at_top_nat )
=> ( ord_le824540014_ereal @ X2 @ Y ) ) ) ) ).
% lim_mono
thf(fact_82_lim__mono,axiom,
! [N3: nat,X3: nat > real,Y2: nat > real,X2: real,Y: real] :
( ! [N2: nat] :
( ( ord_less_eq_nat @ N3 @ N2 )
=> ( ord_less_eq_real @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
=> ( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ X2 ) @ at_top_nat )
=> ( ( filterlim_nat_real @ Y2 @ ( topolo1664202871s_real @ Y ) @ at_top_nat )
=> ( ord_less_eq_real @ X2 @ Y ) ) ) ) ).
% lim_mono
thf(fact_83_LIMSEQ__le,axiom,
! [X3: nat > nat,X2: nat,Y2: nat > nat,Y: nat] :
( ( filterlim_nat_nat @ X3 @ ( topolo1564986139ds_nat @ X2 ) @ at_top_nat )
=> ( ( filterlim_nat_nat @ Y2 @ ( topolo1564986139ds_nat @ Y ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_less_eq_nat @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ).
% LIMSEQ_le
thf(fact_84_LIMSEQ__le,axiom,
! [X3: nat > int,X2: int,Y2: nat > int,Y: int] :
( ( filterlim_nat_int @ X3 @ ( topolo54776183ds_int @ X2 ) @ at_top_nat )
=> ( ( filterlim_nat_int @ Y2 @ ( topolo54776183ds_int @ Y ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_less_eq_int @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
=> ( ord_less_eq_int @ X2 @ Y ) ) ) ) ).
% LIMSEQ_le
thf(fact_85_LIMSEQ__le,axiom,
! [X3: nat > extended_ereal,X2: extended_ereal,Y2: nat > extended_ereal,Y: extended_ereal] :
( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ X2 ) @ at_top_nat )
=> ( ( filter1531173832_ereal @ Y2 @ ( topolo2140997059_ereal @ Y ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_le824540014_ereal @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
=> ( ord_le824540014_ereal @ X2 @ Y ) ) ) ) ).
% LIMSEQ_le
thf(fact_86_LIMSEQ__le,axiom,
! [X3: nat > real,X2: real,Y2: nat > real,Y: real] :
( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ X2 ) @ at_top_nat )
=> ( ( filterlim_nat_real @ Y2 @ ( topolo1664202871s_real @ Y ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_less_eq_real @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
=> ( ord_less_eq_real @ X2 @ Y ) ) ) ) ).
% LIMSEQ_le
thf(fact_87_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X: real] : ( member_real @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_89_Collect__cong,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X5: real] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_real @ P )
= ( collect_real @ Q ) ) ) ).
% Collect_cong
thf(fact_90_Lim__bounded,axiom,
! [F2: nat > nat,L: nat,M: nat,C3: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo1564986139ds_nat @ L ) @ at_top_nat )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( F2 @ N2 ) @ C3 ) )
=> ( ord_less_eq_nat @ L @ C3 ) ) ) ).
% Lim_bounded
thf(fact_91_Lim__bounded,axiom,
! [F2: nat > int,L: int,M: nat,C3: int] :
( ( filterlim_nat_int @ F2 @ ( topolo54776183ds_int @ L ) @ at_top_nat )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_int @ ( F2 @ N2 ) @ C3 ) )
=> ( ord_less_eq_int @ L @ C3 ) ) ) ).
% Lim_bounded
thf(fact_92_Lim__bounded,axiom,
! [F2: nat > extended_ereal,L: extended_ereal,M: nat,C3: extended_ereal] :
( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ at_top_nat )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_le824540014_ereal @ ( F2 @ N2 ) @ C3 ) )
=> ( ord_le824540014_ereal @ L @ C3 ) ) ) ).
% Lim_bounded
thf(fact_93_Lim__bounded,axiom,
! [F2: nat > real,L: real,M: nat,C3: real] :
( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ L ) @ at_top_nat )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_real @ ( F2 @ N2 ) @ C3 ) )
=> ( ord_less_eq_real @ L @ C3 ) ) ) ).
% Lim_bounded
thf(fact_94_Lim__bounded2,axiom,
! [F2: nat > nat,L: nat,N3: nat,C3: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo1564986139ds_nat @ L ) @ at_top_nat )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ N3 @ N2 )
=> ( ord_less_eq_nat @ C3 @ ( F2 @ N2 ) ) )
=> ( ord_less_eq_nat @ C3 @ L ) ) ) ).
% Lim_bounded2
thf(fact_95_Lim__bounded2,axiom,
! [F2: nat > int,L: int,N3: nat,C3: int] :
( ( filterlim_nat_int @ F2 @ ( topolo54776183ds_int @ L ) @ at_top_nat )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ N3 @ N2 )
=> ( ord_less_eq_int @ C3 @ ( F2 @ N2 ) ) )
=> ( ord_less_eq_int @ C3 @ L ) ) ) ).
% Lim_bounded2
thf(fact_96_Lim__bounded2,axiom,
! [F2: nat > extended_ereal,L: extended_ereal,N3: nat,C3: extended_ereal] :
( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ at_top_nat )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ N3 @ N2 )
=> ( ord_le824540014_ereal @ C3 @ ( F2 @ N2 ) ) )
=> ( ord_le824540014_ereal @ C3 @ L ) ) ) ).
% Lim_bounded2
thf(fact_97_Lim__bounded2,axiom,
! [F2: nat > real,L: real,N3: nat,C3: real] :
( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ L ) @ at_top_nat )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ N3 @ N2 )
=> ( ord_less_eq_real @ C3 @ ( F2 @ N2 ) ) )
=> ( ord_less_eq_real @ C3 @ L ) ) ) ).
% Lim_bounded2
thf(fact_98_LIMSEQ__le__const,axiom,
! [X3: nat > nat,X2: nat,A: nat] :
( ( filterlim_nat_nat @ X3 @ ( topolo1564986139ds_nat @ X2 ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_less_eq_nat @ A @ ( X3 @ N2 ) ) )
=> ( ord_less_eq_nat @ A @ X2 ) ) ) ).
% LIMSEQ_le_const
thf(fact_99_LIMSEQ__le__const,axiom,
! [X3: nat > int,X2: int,A: int] :
( ( filterlim_nat_int @ X3 @ ( topolo54776183ds_int @ X2 ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_less_eq_int @ A @ ( X3 @ N2 ) ) )
=> ( ord_less_eq_int @ A @ X2 ) ) ) ).
% LIMSEQ_le_const
thf(fact_100_LIMSEQ__le__const,axiom,
! [X3: nat > extended_ereal,X2: extended_ereal,A: extended_ereal] :
( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ X2 ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_le824540014_ereal @ A @ ( X3 @ N2 ) ) )
=> ( ord_le824540014_ereal @ A @ X2 ) ) ) ).
% LIMSEQ_le_const
thf(fact_101_LIMSEQ__le__const,axiom,
! [X3: nat > real,X2: real,A: real] :
( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ X2 ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_less_eq_real @ A @ ( X3 @ N2 ) ) )
=> ( ord_less_eq_real @ A @ X2 ) ) ) ).
% LIMSEQ_le_const
thf(fact_102_LIMSEQ__le__const2,axiom,
! [X3: nat > nat,X2: nat,A: nat] :
( ( filterlim_nat_nat @ X3 @ ( topolo1564986139ds_nat @ X2 ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_less_eq_nat @ ( X3 @ N2 ) @ A ) )
=> ( ord_less_eq_nat @ X2 @ A ) ) ) ).
% LIMSEQ_le_const2
thf(fact_103_LIMSEQ__le__const2,axiom,
! [X3: nat > int,X2: int,A: int] :
( ( filterlim_nat_int @ X3 @ ( topolo54776183ds_int @ X2 ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_less_eq_int @ ( X3 @ N2 ) @ A ) )
=> ( ord_less_eq_int @ X2 @ A ) ) ) ).
% LIMSEQ_le_const2
thf(fact_104_LIMSEQ__le__const2,axiom,
! [X3: nat > extended_ereal,X2: extended_ereal,A: extended_ereal] :
( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ X2 ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_le824540014_ereal @ ( X3 @ N2 ) @ A ) )
=> ( ord_le824540014_ereal @ X2 @ A ) ) ) ).
% LIMSEQ_le_const2
thf(fact_105_LIMSEQ__le__const2,axiom,
! [X3: nat > real,X2: real,A: real] :
( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ X2 ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_less_eq_real @ ( X3 @ N2 ) @ A ) )
=> ( ord_less_eq_real @ X2 @ A ) ) ) ).
% LIMSEQ_le_const2
thf(fact_106_tendsto__mono,axiom,
! [F: filter_nat,F4: filter_nat,F2: nat > extended_ereal,L: extended_ereal] :
( ( ord_le1745708096er_nat @ F @ F4 )
=> ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ F4 )
=> ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ F ) ) ) ).
% tendsto_mono
thf(fact_107_tendsto__mono,axiom,
! [F: filter_nat,F4: filter_nat,F2: nat > a,L: a] :
( ( ord_le1745708096er_nat @ F @ F4 )
=> ( ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ L ) @ F4 )
=> ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ L ) @ F ) ) ) ).
% tendsto_mono
thf(fact_108_tendsto__mono,axiom,
! [F: filter_real,F4: filter_real,F2: real > real,L: real] :
( ( ord_le132810396r_real @ F @ F4 )
=> ( ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ L ) @ F4 )
=> ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ L ) @ F ) ) ) ).
% tendsto_mono
thf(fact_109_tendsto__mono,axiom,
! [F: filter_nat,F4: filter_nat,F2: nat > real,L: real] :
( ( ord_le1745708096er_nat @ F @ F4 )
=> ( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ L ) @ F4 )
=> ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ L ) @ F ) ) ) ).
% tendsto_mono
thf(fact_110_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_111_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_112_comp__eq__dest__lhs,axiom,
! [A: a > extended_ereal,B: nat > a,C: nat > extended_ereal,V: nat] :
( ( ( comp_a1112243075al_nat @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_113_comp__eq__elim,axiom,
! [A: a > extended_ereal,B: nat > a,C: a > extended_ereal,D: nat > a] :
( ( ( comp_a1112243075al_nat @ A @ B )
= ( comp_a1112243075al_nat @ C @ D ) )
=> ! [V2: nat] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_114_comp__eq__dest,axiom,
! [A: a > extended_ereal,B: nat > a,C: a > extended_ereal,D: nat > a,V: nat] :
( ( ( comp_a1112243075al_nat @ A @ B )
= ( comp_a1112243075al_nat @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_115_comp__assoc,axiom,
! [F2: a > extended_ereal,G2: nat > a,H: nat > nat] :
( ( comp_n1096781355al_nat @ ( comp_a1112243075al_nat @ F2 @ G2 ) @ H )
= ( comp_a1112243075al_nat @ F2 @ ( comp_nat_a_nat @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_116_comp__assoc,axiom,
! [F2: extended_ereal > extended_ereal,G2: a > extended_ereal,H: nat > a] :
( ( comp_a1112243075al_nat @ ( comp_E489644891real_a @ F2 @ G2 ) @ H )
= ( comp_E1308517939al_nat @ F2 @ ( comp_a1112243075al_nat @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_117_comp__assoc,axiom,
! [F2: a > extended_ereal,G2: a > a,H: nat > a] :
( ( comp_a1112243075al_nat @ ( comp_a780206603real_a @ F2 @ G2 ) @ H )
= ( comp_a1112243075al_nat @ F2 @ ( comp_a_a_nat @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_118_comp__def,axiom,
( comp_a1112243075al_nat
= ( ^ [F3: a > extended_ereal,G: nat > a,X: nat] : ( F3 @ ( G @ X ) ) ) ) ).
% comp_def
thf(fact_119_lsc__sequentially,axiom,
( lower_551915512_ereal
= ( ^ [X02: real,F3: real > extended_ereal] :
! [X: nat > real,C2: extended_ereal] :
( ( ( filterlim_nat_real @ X @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
& ! [N: nat] : ( ord_le824540014_ereal @ ( F3 @ ( X @ N ) ) @ C2 ) )
=> ( ord_le824540014_ereal @ ( F3 @ X02 ) @ C2 ) ) ) ) ).
% lsc_sequentially
thf(fact_120_lsc__sequentially,axiom,
( lower_191460856_ereal
= ( ^ [X02: a,F3: a > extended_ereal] :
! [X: nat > a,C2: extended_ereal] :
( ( ( filterlim_nat_a @ X @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
& ! [N: nat] : ( ord_le824540014_ereal @ ( F3 @ ( X @ N ) ) @ C2 ) )
=> ( ord_le824540014_ereal @ ( F3 @ X02 ) @ C2 ) ) ) ) ).
% lsc_sequentially
thf(fact_121_usc__at__def,axiom,
( lower_114093al_nat
= ( ^ [X02: extended_ereal,F3: extended_ereal > nat] :
! [X4: nat > extended_ereal,L2: nat] :
( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
& ( filterlim_nat_nat @ ( comp_E1523169101at_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
=> ( ord_less_eq_nat @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% usc_at_def
thf(fact_122_usc__at__def,axiom,
( lower_637387785al_int
= ( ^ [X02: extended_ereal,F3: extended_ereal > int] :
! [X4: nat > extended_ereal,L2: int] :
( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
& ( filterlim_nat_int @ ( comp_E1436437929nt_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
=> ( ord_less_eq_int @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% usc_at_def
thf(fact_123_usc__at__def,axiom,
( lower_1071158961_ereal
= ( ^ [X02: extended_ereal,F3: extended_ereal > extended_ereal] :
! [X4: nat > extended_ereal,L2: extended_ereal] :
( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
& ( filter1531173832_ereal @ ( comp_E1308517939al_nat @ F3 @ X4 ) @ ( topolo2140997059_ereal @ L2 ) @ at_top_nat ) )
=> ( ord_le824540014_ereal @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% usc_at_def
thf(fact_124_usc__at__def,axiom,
( lower_737640969l_real
= ( ^ [X02: extended_ereal,F3: extended_ereal > real] :
! [X4: nat > extended_ereal,L2: real] :
( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
& ( filterlim_nat_real @ ( comp_E1477338153al_nat @ F3 @ X4 ) @ ( topolo1664202871s_real @ L2 ) @ at_top_nat ) )
=> ( ord_less_eq_real @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% usc_at_def
thf(fact_125_usc__at__def,axiom,
( lower_1035717085_a_nat
= ( ^ [X02: a,F3: a > nat] :
! [X4: nat > a,L2: nat] :
( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
& ( filterlim_nat_nat @ ( comp_a_nat_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
=> ( ord_less_eq_nat @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% usc_at_def
thf(fact_126_usc__at__def,axiom,
( lower_1672990777_a_int
= ( ^ [X02: a,F3: a > int] :
! [X4: nat > a,L2: int] :
( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
& ( filterlim_nat_int @ ( comp_a_int_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
=> ( ord_less_eq_int @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% usc_at_def
thf(fact_127_usc__at__def,axiom,
( lower_534855297_ereal
= ( ^ [X02: a,F3: a > extended_ereal] :
! [X4: nat > a,L2: extended_ereal] :
( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
& ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ F3 @ X4 ) @ ( topolo2140997059_ereal @ L2 ) @ at_top_nat ) )
=> ( ord_le824540014_ereal @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% usc_at_def
thf(fact_128_usc__at__def,axiom,
( lower_755922489a_real
= ( ^ [X02: a,F3: a > real] :
! [X4: nat > a,L2: real] :
( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
& ( filterlim_nat_real @ ( comp_a_real_nat @ F3 @ X4 ) @ ( topolo1664202871s_real @ L2 ) @ at_top_nat ) )
=> ( ord_less_eq_real @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% usc_at_def
thf(fact_129_usc__at__def,axiom,
( lower_438231087al_nat
= ( ^ [X02: real,F3: real > nat] :
! [X4: nat > real,L2: nat] :
( ( ( filterlim_nat_real @ X4 @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
& ( filterlim_nat_nat @ ( comp_real_nat_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
=> ( ord_less_eq_nat @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% usc_at_def
thf(fact_130_usc__at__def,axiom,
( lower_1075504779al_int
= ( ^ [X02: real,F3: real > int] :
! [X4: nat > real,L2: int] :
( ( ( filterlim_nat_real @ X4 @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
& ( filterlim_nat_int @ ( comp_real_int_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
=> ( ord_less_eq_int @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% usc_at_def
thf(fact_131_usc__at__mem,axiom,
! [X0: extended_ereal,F2: extended_ereal > nat,X2: nat > extended_ereal,A2: nat] :
( ( lower_114093al_nat @ X0 @ F2 )
=> ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
=> ( ( filterlim_nat_nat @ ( comp_E1523169101at_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% usc_at_mem
thf(fact_132_usc__at__mem,axiom,
! [X0: extended_ereal,F2: extended_ereal > int,X2: nat > extended_ereal,A2: int] :
( ( lower_637387785al_int @ X0 @ F2 )
=> ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
=> ( ( filterlim_nat_int @ ( comp_E1436437929nt_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
=> ( ord_less_eq_int @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% usc_at_mem
thf(fact_133_usc__at__mem,axiom,
! [X0: extended_ereal,F2: extended_ereal > extended_ereal,X2: nat > extended_ereal,A2: extended_ereal] :
( ( lower_1071158961_ereal @ X0 @ F2 )
=> ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
=> ( ( filter1531173832_ereal @ ( comp_E1308517939al_nat @ F2 @ X2 ) @ ( topolo2140997059_ereal @ A2 ) @ at_top_nat )
=> ( ord_le824540014_ereal @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% usc_at_mem
thf(fact_134_usc__at__mem,axiom,
! [X0: extended_ereal,F2: extended_ereal > real,X2: nat > extended_ereal,A2: real] :
( ( lower_737640969l_real @ X0 @ F2 )
=> ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
=> ( ( filterlim_nat_real @ ( comp_E1477338153al_nat @ F2 @ X2 ) @ ( topolo1664202871s_real @ A2 ) @ at_top_nat )
=> ( ord_less_eq_real @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% usc_at_mem
thf(fact_135_usc__at__mem,axiom,
! [X0: a,F2: a > nat,X2: nat > a,A2: nat] :
( ( lower_1035717085_a_nat @ X0 @ F2 )
=> ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
=> ( ( filterlim_nat_nat @ ( comp_a_nat_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% usc_at_mem
thf(fact_136_usc__at__mem,axiom,
! [X0: a,F2: a > int,X2: nat > a,A2: int] :
( ( lower_1672990777_a_int @ X0 @ F2 )
=> ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
=> ( ( filterlim_nat_int @ ( comp_a_int_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
=> ( ord_less_eq_int @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% usc_at_mem
thf(fact_137_usc__at__mem,axiom,
! [X0: a,F2: a > extended_ereal,X2: nat > a,A2: extended_ereal] :
( ( lower_534855297_ereal @ X0 @ F2 )
=> ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
=> ( ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ F2 @ X2 ) @ ( topolo2140997059_ereal @ A2 ) @ at_top_nat )
=> ( ord_le824540014_ereal @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% usc_at_mem
thf(fact_138_usc__at__mem,axiom,
! [X0: a,F2: a > real,X2: nat > a,A2: real] :
( ( lower_755922489a_real @ X0 @ F2 )
=> ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
=> ( ( filterlim_nat_real @ ( comp_a_real_nat @ F2 @ X2 ) @ ( topolo1664202871s_real @ A2 ) @ at_top_nat )
=> ( ord_less_eq_real @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% usc_at_mem
thf(fact_139_usc__at__mem,axiom,
! [X0: real,F2: real > nat,X2: nat > real,A2: nat] :
( ( lower_438231087al_nat @ X0 @ F2 )
=> ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
=> ( ( filterlim_nat_nat @ ( comp_real_nat_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% usc_at_mem
thf(fact_140_usc__at__mem,axiom,
! [X0: real,F2: real > int,X2: nat > real,A2: int] :
( ( lower_1075504779al_int @ X0 @ F2 )
=> ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
=> ( ( filterlim_nat_int @ ( comp_real_int_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
=> ( ord_less_eq_int @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% usc_at_mem
thf(fact_141_lim__decreasing__cl,axiom,
! [F2: nat > extended_ereal] :
( ! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_le824540014_ereal @ ( F2 @ N2 ) @ ( F2 @ M2 ) ) )
=> ~ ! [L3: extended_ereal] :
~ ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L3 ) @ at_top_nat ) ) ).
% lim_decreasing_cl
thf(fact_142_lim__increasing__cl,axiom,
! [F2: nat > extended_ereal] :
( ! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_le824540014_ereal @ ( F2 @ M2 ) @ ( F2 @ N2 ) ) )
=> ~ ! [L3: extended_ereal] :
~ ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L3 ) @ at_top_nat ) ) ).
% lim_increasing_cl
thf(fact_143_order__refl,axiom,
! [X2: extended_ereal] : ( ord_le824540014_ereal @ X2 @ X2 ) ).
% order_refl
thf(fact_144_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_145_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_146_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_147_monoseq__le,axiom,
! [A: nat > nat,X2: nat] :
( ( topolo1922093437eq_nat @ A )
=> ( ( filterlim_nat_nat @ A @ ( topolo1564986139ds_nat @ X2 ) @ at_top_nat )
=> ( ( ! [N5: nat] : ( ord_less_eq_nat @ ( A @ N5 ) @ X2 )
& ! [M3: nat,N5: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
=> ( ord_less_eq_nat @ ( A @ M3 ) @ ( A @ N5 ) ) ) )
| ( ! [N5: nat] : ( ord_less_eq_nat @ X2 @ ( A @ N5 ) )
& ! [M3: nat,N5: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
=> ( ord_less_eq_nat @ ( A @ N5 ) @ ( A @ M3 ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_148_monoseq__le,axiom,
! [A: nat > int,X2: int] :
( ( topolo411883481eq_int @ A )
=> ( ( filterlim_nat_int @ A @ ( topolo54776183ds_int @ X2 ) @ at_top_nat )
=> ( ( ! [N5: nat] : ( ord_less_eq_int @ ( A @ N5 ) @ X2 )
& ! [M3: nat,N5: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
=> ( ord_less_eq_int @ ( A @ M3 ) @ ( A @ N5 ) ) ) )
| ( ! [N5: nat] : ( ord_less_eq_int @ X2 @ ( A @ N5 ) )
& ! [M3: nat,N5: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
=> ( ord_less_eq_int @ ( A @ N5 ) @ ( A @ M3 ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_149_monoseq__le,axiom,
! [A: nat > extended_ereal,X2: extended_ereal] :
( ( topolo1069469409_ereal @ A )
=> ( ( filter1531173832_ereal @ A @ ( topolo2140997059_ereal @ X2 ) @ at_top_nat )
=> ( ( ! [N5: nat] : ( ord_le824540014_ereal @ ( A @ N5 ) @ X2 )
& ! [M3: nat,N5: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
=> ( ord_le824540014_ereal @ ( A @ M3 ) @ ( A @ N5 ) ) ) )
| ( ! [N5: nat] : ( ord_le824540014_ereal @ X2 @ ( A @ N5 ) )
& ! [M3: nat,N5: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
=> ( ord_le824540014_ereal @ ( A @ N5 ) @ ( A @ M3 ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_150_monoseq__le,axiom,
! [A: nat > real,X2: real] :
( ( topolo144289241q_real @ A )
=> ( ( filterlim_nat_real @ A @ ( topolo1664202871s_real @ X2 ) @ at_top_nat )
=> ( ( ! [N5: nat] : ( ord_less_eq_real @ ( A @ N5 ) @ X2 )
& ! [M3: nat,N5: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
=> ( ord_less_eq_real @ ( A @ M3 ) @ ( A @ N5 ) ) ) )
| ( ! [N5: nat] : ( ord_less_eq_real @ X2 @ ( A @ N5 ) )
& ! [M3: nat,N5: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
=> ( ord_less_eq_real @ ( A @ N5 ) @ ( A @ M3 ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_151_lsc__liminf,axiom,
( lower_551915512_ereal
= ( ^ [X02: real,F3: real > extended_ereal] :
! [X: nat > real] :
( ( filterlim_nat_real @ X @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
=> ( ord_le824540014_ereal @ ( F3 @ X02 ) @ ( liminf1045857232_ereal @ at_top_nat @ ( comp_r1410008527al_nat @ F3 @ X ) ) ) ) ) ) ).
% lsc_liminf
thf(fact_152_lsc__liminf,axiom,
( lower_191460856_ereal
= ( ^ [X02: a,F3: a > extended_ereal] :
! [X: nat > a] :
( ( filterlim_nat_a @ X @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
=> ( ord_le824540014_ereal @ ( F3 @ X02 ) @ ( liminf1045857232_ereal @ at_top_nat @ ( comp_a1112243075al_nat @ F3 @ X ) ) ) ) ) ) ).
% lsc_liminf
thf(fact_153_lsc__imp__liminf,axiom,
! [X0: real,F2: real > extended_ereal,X2: nat > real] :
( ( lower_551915512_ereal @ X0 @ F2 )
=> ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
=> ( ord_le824540014_ereal @ ( F2 @ X0 ) @ ( liminf1045857232_ereal @ at_top_nat @ ( comp_r1410008527al_nat @ F2 @ X2 ) ) ) ) ) ).
% lsc_imp_liminf
thf(fact_154_lsc__imp__liminf,axiom,
! [X0: a,F2: a > extended_ereal,X2: nat > a] :
( ( lower_191460856_ereal @ X0 @ F2 )
=> ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
=> ( ord_le824540014_ereal @ ( F2 @ X0 ) @ ( liminf1045857232_ereal @ at_top_nat @ ( comp_a1112243075al_nat @ F2 @ X2 ) ) ) ) ) ).
% lsc_imp_liminf
thf(fact_155_LIMSEQ__Uniq,axiom,
! [X3: nat > extended_ereal] :
( uniq_Extended_ereal
@ ^ [L2: extended_ereal] : ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ L2 ) @ at_top_nat ) ) ).
% LIMSEQ_Uniq
thf(fact_156_LIMSEQ__Uniq,axiom,
! [X3: nat > a] :
( uniq_a
@ ^ [L2: a] : ( filterlim_nat_a @ X3 @ ( topolo705128563nhds_a @ L2 ) @ at_top_nat ) ) ).
% LIMSEQ_Uniq
thf(fact_157_LIMSEQ__Uniq,axiom,
! [X3: nat > real] :
( uniq_real
@ ^ [L2: real] : ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ L2 ) @ at_top_nat ) ) ).
% LIMSEQ_Uniq
thf(fact_158_monoseq__def,axiom,
( topolo1069469409_ereal
= ( ^ [X4: nat > extended_ereal] :
( ! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_le824540014_ereal @ ( X4 @ M4 ) @ ( X4 @ N ) ) )
| ! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_le824540014_ereal @ ( X4 @ N ) @ ( X4 @ M4 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_159_monoseq__def,axiom,
( topolo1922093437eq_nat
= ( ^ [X4: nat > nat] :
( ! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_less_eq_nat @ ( X4 @ M4 ) @ ( X4 @ N ) ) )
| ! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_less_eq_nat @ ( X4 @ N ) @ ( X4 @ M4 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_160_monoseq__def,axiom,
( topolo144289241q_real
= ( ^ [X4: nat > real] :
( ! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_less_eq_real @ ( X4 @ M4 ) @ ( X4 @ N ) ) )
| ! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_less_eq_real @ ( X4 @ N ) @ ( X4 @ M4 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_161_monoseq__def,axiom,
( topolo411883481eq_int
= ( ^ [X4: nat > int] :
( ! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_less_eq_int @ ( X4 @ M4 ) @ ( X4 @ N ) ) )
| ! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_less_eq_int @ ( X4 @ N ) @ ( X4 @ M4 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_162_monoI2,axiom,
! [X3: nat > extended_ereal] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_le824540014_ereal @ ( X3 @ N2 ) @ ( X3 @ M2 ) ) )
=> ( topolo1069469409_ereal @ X3 ) ) ).
% monoI2
thf(fact_163_monoI2,axiom,
! [X3: nat > nat] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( X3 @ N2 ) @ ( X3 @ M2 ) ) )
=> ( topolo1922093437eq_nat @ X3 ) ) ).
% monoI2
thf(fact_164_monoI2,axiom,
! [X3: nat > real] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_real @ ( X3 @ N2 ) @ ( X3 @ M2 ) ) )
=> ( topolo144289241q_real @ X3 ) ) ).
% monoI2
thf(fact_165_monoI2,axiom,
! [X3: nat > int] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_int @ ( X3 @ N2 ) @ ( X3 @ M2 ) ) )
=> ( topolo411883481eq_int @ X3 ) ) ).
% monoI2
thf(fact_166_monoI1,axiom,
! [X3: nat > extended_ereal] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_le824540014_ereal @ ( X3 @ M2 ) @ ( X3 @ N2 ) ) )
=> ( topolo1069469409_ereal @ X3 ) ) ).
% monoI1
thf(fact_167_monoI1,axiom,
! [X3: nat > nat] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( X3 @ M2 ) @ ( X3 @ N2 ) ) )
=> ( topolo1922093437eq_nat @ X3 ) ) ).
% monoI1
thf(fact_168_monoI1,axiom,
! [X3: nat > real] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_real @ ( X3 @ M2 ) @ ( X3 @ N2 ) ) )
=> ( topolo144289241q_real @ X3 ) ) ).
% monoI1
thf(fact_169_monoI1,axiom,
! [X3: nat > int] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_int @ ( X3 @ M2 ) @ ( X3 @ N2 ) ) )
=> ( topolo411883481eq_int @ X3 ) ) ).
% monoI1
thf(fact_170_monoseq__minus,axiom,
! [A: nat > real] :
( ( topolo144289241q_real @ A )
=> ( topolo144289241q_real
@ ^ [N: nat] : ( uminus_uminus_real @ ( A @ N ) ) ) ) ).
% monoseq_minus
thf(fact_171_order__subst1,axiom,
! [A: extended_ereal,F2: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le824540014_ereal @ A @ ( F2 @ B ) )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ! [X5: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X5 @ Y3 )
=> ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_172_order__subst1,axiom,
! [A: extended_ereal,F2: nat > extended_ereal,B: nat,C: nat] :
( ( ord_le824540014_ereal @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_173_order__subst1,axiom,
! [A: extended_ereal,F2: real > extended_ereal,B: real,C: real] :
( ( ord_le824540014_ereal @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X5: real,Y3: real] :
( ( ord_less_eq_real @ X5 @ Y3 )
=> ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_174_order__subst1,axiom,
! [A: extended_ereal,F2: int > extended_ereal,B: int,C: int] :
( ( ord_le824540014_ereal @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_175_order__subst1,axiom,
! [A: nat,F2: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ! [X5: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_176_order__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_177_order__subst1,axiom,
! [A: nat,F2: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X5: real,Y3: real] :
( ( ord_less_eq_real @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_178_order__subst1,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_179_order__subst1,axiom,
! [A: real,F2: extended_ereal > real,B: extended_ereal,C: extended_ereal] :
( ( ord_less_eq_real @ A @ ( F2 @ B ) )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ! [X5: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X5 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_180_order__subst1,axiom,
! [A: real,F2: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_181_order__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > extended_ereal,C: extended_ereal] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ord_le824540014_ereal @ ( F2 @ B ) @ C )
=> ( ! [X5: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X5 @ Y3 )
=> ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_182_order__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > nat,C: nat] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X5: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_183_order__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > real,C: real] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ord_less_eq_real @ ( F2 @ B ) @ C )
=> ( ! [X5: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X5 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_184_order__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > int,C: int] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X5: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_185_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > extended_ereal,C: extended_ereal] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le824540014_ereal @ ( F2 @ B ) @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_186_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_187_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F2 @ B ) @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_188_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_189_order__subst2,axiom,
! [A: real,B: real,F2: real > extended_ereal,C: extended_ereal] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_le824540014_ereal @ ( F2 @ B ) @ C )
=> ( ! [X5: real,Y3: real] :
( ( ord_less_eq_real @ X5 @ Y3 )
=> ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_190_order__subst2,axiom,
! [A: real,B: real,F2: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X5: real,Y3: real] :
( ( ord_less_eq_real @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_191_ord__eq__le__subst,axiom,
! [A: extended_ereal,F2: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ! [X5: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X5 @ Y3 )
=> ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_192_ord__eq__le__subst,axiom,
! [A: nat,F2: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ! [X5: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_193_ord__eq__le__subst,axiom,
! [A: real,F2: extended_ereal > real,B: extended_ereal,C: extended_ereal] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ! [X5: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X5 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_194_ord__eq__le__subst,axiom,
! [A: int,F2: extended_ereal > int,B: extended_ereal,C: extended_ereal] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ! [X5: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_195_ord__eq__le__subst,axiom,
! [A: extended_ereal,F2: nat > extended_ereal,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_196_ord__eq__le__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_197_ord__eq__le__subst,axiom,
! [A: real,F2: nat > real,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_198_ord__eq__le__subst,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_199_ord__eq__le__subst,axiom,
! [A: extended_ereal,F2: real > extended_ereal,B: real,C: real] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X5: real,Y3: real] :
( ( ord_less_eq_real @ X5 @ Y3 )
=> ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_200_ord__eq__le__subst,axiom,
! [A: nat,F2: real > nat,B: real,C: real] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X5: real,Y3: real] :
( ( ord_less_eq_real @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_201_ord__le__eq__subst,axiom,
! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > extended_ereal,C: extended_ereal] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X5: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X5 @ Y3 )
=> ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_202_ord__le__eq__subst,axiom,
! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > nat,C: nat] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X5: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_203_ord__le__eq__subst,axiom,
! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > real,C: real] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X5: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X5 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_204_ord__le__eq__subst,axiom,
! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > int,C: int] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X5: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_205_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > extended_ereal,C: extended_ereal] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_206_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_207_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_208_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_209_ord__le__eq__subst,axiom,
! [A: real,B: real,F2: real > extended_ereal,C: extended_ereal] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X5: real,Y3: real] :
( ( ord_less_eq_real @ X5 @ Y3 )
=> ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_210_ord__le__eq__subst,axiom,
! [A: real,B: real,F2: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X5: real,Y3: real] :
( ( ord_less_eq_real @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_211_eq__iff,axiom,
( ( ^ [Y4: extended_ereal,Z2: extended_ereal] : Y4 = Z2 )
= ( ^ [X: extended_ereal,Y5: extended_ereal] :
( ( ord_le824540014_ereal @ X @ Y5 )
& ( ord_le824540014_ereal @ Y5 @ X ) ) ) ) ).
% eq_iff
thf(fact_212_eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
= ( ^ [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ).
% eq_iff
thf(fact_213_eq__iff,axiom,
( ( ^ [Y4: real,Z2: real] : Y4 = Z2 )
= ( ^ [X: real,Y5: real] :
( ( ord_less_eq_real @ X @ Y5 )
& ( ord_less_eq_real @ Y5 @ X ) ) ) ) ).
% eq_iff
thf(fact_214_eq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
= ( ^ [X: int,Y5: int] :
( ( ord_less_eq_int @ X @ Y5 )
& ( ord_less_eq_int @ Y5 @ X ) ) ) ) ).
% eq_iff
thf(fact_215_antisym,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X2 @ Y )
=> ( ( ord_le824540014_ereal @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% antisym
thf(fact_216_antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% antisym
thf(fact_217_antisym,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% antisym
thf(fact_218_antisym,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% antisym
thf(fact_219_linear,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X2 @ Y )
| ( ord_le824540014_ereal @ Y @ X2 ) ) ).
% linear
thf(fact_220_linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linear
thf(fact_221_linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
| ( ord_less_eq_real @ Y @ X2 ) ) ).
% linear
thf(fact_222_linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_eq_int @ Y @ X2 ) ) ).
% linear
thf(fact_223_eq__refl,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( X2 = Y )
=> ( ord_le824540014_ereal @ X2 @ Y ) ) ).
% eq_refl
thf(fact_224_eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% eq_refl
thf(fact_225_eq__refl,axiom,
! [X2: real,Y: real] :
( ( X2 = Y )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% eq_refl
thf(fact_226_eq__refl,axiom,
! [X2: int,Y: int] :
( ( X2 = Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% eq_refl
thf(fact_227_le__cases,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ~ ( ord_le824540014_ereal @ X2 @ Y )
=> ( ord_le824540014_ereal @ Y @ X2 ) ) ).
% le_cases
thf(fact_228_le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% le_cases
thf(fact_229_le__cases,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% le_cases
thf(fact_230_le__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% le_cases
thf(fact_231_order_Otrans,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ord_le824540014_ereal @ A @ C ) ) ) ).
% order.trans
thf(fact_232_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_233_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_234_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_235_le__cases3,axiom,
! [X2: extended_ereal,Y: extended_ereal,Z3: extended_ereal] :
( ( ( ord_le824540014_ereal @ X2 @ Y )
=> ~ ( ord_le824540014_ereal @ Y @ Z3 ) )
=> ( ( ( ord_le824540014_ereal @ Y @ X2 )
=> ~ ( ord_le824540014_ereal @ X2 @ Z3 ) )
=> ( ( ( ord_le824540014_ereal @ X2 @ Z3 )
=> ~ ( ord_le824540014_ereal @ Z3 @ Y ) )
=> ( ( ( ord_le824540014_ereal @ Z3 @ Y )
=> ~ ( ord_le824540014_ereal @ Y @ X2 ) )
=> ( ( ( ord_le824540014_ereal @ Y @ Z3 )
=> ~ ( ord_le824540014_ereal @ Z3 @ X2 ) )
=> ~ ( ( ord_le824540014_ereal @ Z3 @ X2 )
=> ~ ( ord_le824540014_ereal @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_236_le__cases3,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_237_le__cases3,axiom,
! [X2: real,Y: real,Z3: real] :
( ( ( ord_less_eq_real @ X2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_238_le__cases3,axiom,
! [X2: int,Y: int,Z3: int] :
( ( ( ord_less_eq_int @ X2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_239_antisym__conv,axiom,
! [Y: extended_ereal,X2: extended_ereal] :
( ( ord_le824540014_ereal @ Y @ X2 )
=> ( ( ord_le824540014_ereal @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_240_antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_241_antisym__conv,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_242_antisym__conv,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_243_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: extended_ereal,Z2: extended_ereal] : Y4 = Z2 )
= ( ^ [A3: extended_ereal,B2: extended_ereal] :
( ( ord_le824540014_ereal @ A3 @ B2 )
& ( ord_le824540014_ereal @ B2 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_244_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_245_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: real,Z2: real] : Y4 = Z2 )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_246_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_247_ord__eq__le__trans,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( A = B )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ord_le824540014_ereal @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_248_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_249_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_250_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_251_ord__le__eq__trans,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( B = C )
=> ( ord_le824540014_ereal @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_252_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_253_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_254_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_255_order__class_Oorder_Oantisym,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ord_le824540014_ereal @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_256_order__class_Oorder_Oantisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_257_order__class_Oorder_Oantisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_258_order__class_Oorder_Oantisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_259_order__trans,axiom,
! [X2: extended_ereal,Y: extended_ereal,Z3: extended_ereal] :
( ( ord_le824540014_ereal @ X2 @ Y )
=> ( ( ord_le824540014_ereal @ Y @ Z3 )
=> ( ord_le824540014_ereal @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_260_order__trans,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_261_order__trans,axiom,
! [X2: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_eq_real @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_262_order__trans,axiom,
! [X2: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_eq_int @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_263_dual__order_Orefl,axiom,
! [A: extended_ereal] : ( ord_le824540014_ereal @ A @ A ) ).
% dual_order.refl
thf(fact_264_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_265_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_266_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_267_linorder__wlog,axiom,
! [P: extended_ereal > extended_ereal > $o,A: extended_ereal,B: extended_ereal] :
( ! [A4: extended_ereal,B3: extended_ereal] :
( ( ord_le824540014_ereal @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: extended_ereal,B3: extended_ereal] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_268_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_269_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_270_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_271_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_272_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_273_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_274_liminf__PInfty,axiom,
! [X3: nat > extended_ereal] :
( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ extend1289208545_ereal ) @ at_top_nat )
= ( ( liminf1045857232_ereal @ at_top_nat @ X3 )
= extend1289208545_ereal ) ) ).
% liminf_PInfty
thf(fact_275_ereal__infty__less__eq_I1_J,axiom,
! [X2: extended_ereal] :
( ( ord_le824540014_ereal @ extend1289208545_ereal @ X2 )
= ( X2 = extend1289208545_ereal ) ) ).
% ereal_infty_less_eq(1)
thf(fact_276_ereal__infty__less__eq_I2_J,axiom,
! [X2: extended_ereal] :
( ( ord_le824540014_ereal @ X2 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
= ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% ereal_infty_less_eq(2)
thf(fact_277_MInfty__neq__PInfty_I1_J,axiom,
( extend1289208545_ereal
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% MInfty_neq_PInfty(1)
thf(fact_278_ereal__less__eq_I1_J,axiom,
! [X2: extended_ereal] : ( ord_le824540014_ereal @ X2 @ extend1289208545_ereal ) ).
% ereal_less_eq(1)
thf(fact_279_ereal__infty__less__eq2_I1_J,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( A = extend1289208545_ereal )
=> ( B = extend1289208545_ereal ) ) ) ).
% ereal_infty_less_eq2(1)
thf(fact_280_neq__PInf__trans,axiom,
! [Y: extended_ereal,X2: extended_ereal] :
( ( Y != extend1289208545_ereal )
=> ( ( ord_le824540014_ereal @ X2 @ Y )
=> ( X2 != extend1289208545_ereal ) ) ) ).
% neq_PInf_trans
thf(fact_281_ereal__infty__less__eq2_I2_J,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( B
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( A
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% ereal_infty_less_eq2(2)
thf(fact_282_ereal__less__eq_I2_J,axiom,
! [X2: extended_ereal] : ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ X2 ) ).
% ereal_less_eq(2)
thf(fact_283_Lim__MInfty,axiom,
! [F2: nat > extended_ereal] :
( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) @ at_top_nat )
= ( ! [B4: real] :
? [N6: nat] :
! [N: nat] :
( ( ord_less_eq_nat @ N6 @ N )
=> ( ord_le824540014_ereal @ ( F2 @ N ) @ ( extended_ereal2 @ B4 ) ) ) ) ) ).
% Lim_MInfty
thf(fact_284_Lim__bounded__MInfty,axiom,
! [F2: nat > extended_ereal,L: extended_ereal,B5: real] :
( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ at_top_nat )
=> ( ! [N2: nat] : ( ord_le824540014_ereal @ ( extended_ereal2 @ B5 ) @ ( F2 @ N2 ) )
=> ( L
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% Lim_bounded_MInfty
thf(fact_285_Lim__PInfty,axiom,
! [F2: nat > extended_ereal] :
( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ extend1289208545_ereal ) @ at_top_nat )
= ( ! [B4: real] :
? [N6: nat] :
! [N: nat] :
( ( ord_less_eq_nat @ N6 @ N )
=> ( ord_le824540014_ereal @ ( extended_ereal2 @ B4 ) @ ( F2 @ N ) ) ) ) ) ).
% Lim_PInfty
thf(fact_286_Lim__bounded__PInfty2,axiom,
! [F2: nat > extended_ereal,L: extended_ereal,N3: nat,B5: real] :
( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ at_top_nat )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ N3 @ N2 )
=> ( ord_le824540014_ereal @ ( F2 @ N2 ) @ ( extended_ereal2 @ B5 ) ) )
=> ( L != extend1289208545_ereal ) ) ) ).
% Lim_bounded_PInfty2
thf(fact_287_ereal_Oinject,axiom,
! [X1: real,Y1: real] :
( ( ( extended_ereal2 @ X1 )
= ( extended_ereal2 @ Y1 ) )
= ( X1 = Y1 ) ) ).
% ereal.inject
thf(fact_288_ereal__cong,axiom,
! [X2: real,Y: real] :
( ( X2 = Y )
=> ( ( extended_ereal2 @ X2 )
= ( extended_ereal2 @ Y ) ) ) ).
% ereal_cong
thf(fact_289_ereal__less__eq_I3_J,axiom,
! [R: real,P2: real] :
( ( ord_le824540014_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P2 ) )
= ( ord_less_eq_real @ R @ P2 ) ) ).
% ereal_less_eq(3)
thf(fact_290_PInfty__neq__ereal_I1_J,axiom,
! [R: real] :
( ( extended_ereal2 @ R )
!= extend1289208545_ereal ) ).
% PInfty_neq_ereal(1)
thf(fact_291_uminus__ereal_Osimps_I1_J,axiom,
! [R: real] :
( ( uminus1208298309_ereal @ ( extended_ereal2 @ R ) )
= ( extended_ereal2 @ ( uminus_uminus_real @ R ) ) ) ).
% uminus_ereal.simps(1)
thf(fact_292_ereal__le__le,axiom,
! [Y: real,A: extended_ereal,X2: real] :
( ( ord_le824540014_ereal @ ( extended_ereal2 @ Y ) @ A )
=> ( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_le824540014_ereal @ ( extended_ereal2 @ X2 ) @ A ) ) ) ).
% ereal_le_le
thf(fact_293_le__ereal__le,axiom,
! [A: extended_ereal,X2: real,Y: real] :
( ( ord_le824540014_ereal @ A @ ( extended_ereal2 @ X2 ) )
=> ( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_le824540014_ereal @ A @ ( extended_ereal2 @ Y ) ) ) ) ).
% le_ereal_le
thf(fact_294_ereal__le__real,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ! [Z4: real] :
( ( ord_le824540014_ereal @ X2 @ ( extended_ereal2 @ Z4 ) )
=> ( ord_le824540014_ereal @ Y @ ( extended_ereal2 @ Z4 ) ) )
=> ( ord_le824540014_ereal @ Y @ X2 ) ) ).
% ereal_le_real
thf(fact_295_ereal__semiline__unique,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( collect_real
@ ^ [Y5: real] : ( ord_le824540014_ereal @ A @ ( extended_ereal2 @ Y5 ) ) )
= ( collect_real
@ ^ [Y5: real] : ( ord_le824540014_ereal @ B @ ( extended_ereal2 @ Y5 ) ) ) )
= ( A = B ) ) ).
% ereal_semiline_unique
thf(fact_296_real__of__ereal_Oinduct,axiom,
! [P: extended_ereal > $o,A0: extended_ereal] :
( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ extend1289208545_ereal )
=> ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( P @ A0 ) ) ) ) ).
% real_of_ereal.induct
thf(fact_297_real__of__ereal_Ocases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2 != extend1289208545_ereal )
=> ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% real_of_ereal.cases
thf(fact_298_times__ereal_Oinduct,axiom,
! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
=> ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ extend1289208545_ereal )
=> ( ! [R2: real] : ( P @ extend1289208545_ereal @ ( extended_ereal2 @ R2 ) )
=> ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ! [R2: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ extend1289208545_ereal @ extend1289208545_ereal )
=> ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
=> ( ( P @ extend1289208545_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).
% times_ereal.induct
thf(fact_299_plus__ereal_Oinduct,axiom,
! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
=> ( ! [X_1: extended_ereal] : ( P @ extend1289208545_ereal @ X_1 )
=> ( ! [A4: extended_ereal] : ( P @ A4 @ extend1289208545_ereal )
=> ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ! [P3: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ P3 ) )
=> ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% plus_ereal.induct
thf(fact_300_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 @ extend1289208545_ereal @ X_1 )
=> ( ! [A4: extended_ereal] : ( P @ A4 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ! [X5: real] : ( P @ ( extended_ereal2 @ X5 ) @ extend1289208545_ereal )
=> ( ! [R2: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% less_ereal.induct
thf(fact_301_abs__ereal_Oinduct,axiom,
! [P: extended_ereal > $o,A0: extended_ereal] :
( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( P @ extend1289208545_ereal )
=> ( P @ A0 ) ) ) ) ).
% abs_ereal.induct
thf(fact_302_ereal__all__split,axiom,
( ( ^ [P4: extended_ereal > $o] :
! [X6: extended_ereal] : ( P4 @ X6 ) )
= ( ^ [P5: extended_ereal > $o] :
( ( P5 @ extend1289208545_ereal )
& ! [X: real] : ( P5 @ ( extended_ereal2 @ X ) )
& ( P5 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% ereal_all_split
thf(fact_303_abs__ereal_Ocases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( X2 = extend1289208545_ereal ) ) ) ).
% abs_ereal.cases
thf(fact_304_ereal__ex__split,axiom,
( ( ^ [P4: extended_ereal > $o] :
? [X6: extended_ereal] : ( P4 @ X6 ) )
= ( ^ [P5: extended_ereal > $o] :
( ( P5 @ extend1289208545_ereal )
| ? [X: real] : ( P5 @ ( extended_ereal2 @ X ) )
| ( P5 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% ereal_ex_split
thf(fact_305_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 != extend1289208545_ereal ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ? [Ra: real] :
( Xa3
= ( extended_ereal2 @ Ra ) )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3 = extend1289208545_ereal )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3 = extend1289208545_ereal )
=> ( Xb != extend1289208545_ereal ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3 = extend1289208545_ereal )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Xb != extend1289208545_ereal ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ( Xb != extend1289208545_ereal ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ( Xa3 = extend1289208545_ereal )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ( Xa3 = extend1289208545_ereal )
=> ( Xb != extend1289208545_ereal ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ( Xa3 = extend1289208545_ereal )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Xb != extend1289208545_ereal ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ( Xb != extend1289208545_ereal ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( Xa3 = extend1289208545_ereal )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( Xa3 = extend1289208545_ereal )
=> ( Xb != extend1289208545_ereal ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( Xa3 = extend1289208545_ereal )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Xb != extend1289208545_ereal ) ) )
=> ~ ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ereal3_cases
thf(fact_306_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 != extend1289208545_ereal ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( Xa3
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ! [R2: real] :
( Xa3
!= ( extended_ereal2 @ R2 ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( Xa3 != extend1289208545_ereal ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( Xa3
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ! [R2: real] :
( Xa3
!= ( extended_ereal2 @ R2 ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Xa3 != extend1289208545_ereal ) )
=> ~ ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Xa3
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ).
% ereal2_cases
thf(fact_307_ereal__cases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2 != extend1289208545_ereal )
=> ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% ereal_cases
thf(fact_308_MInfty__neq__ereal_I1_J,axiom,
! [R: real] :
( ( extended_ereal2 @ R )
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% MInfty_neq_ereal(1)
thf(fact_309_ereal__top,axiom,
! [X2: extended_ereal] :
( ! [B6: real] : ( ord_le824540014_ereal @ ( extended_ereal2 @ B6 ) @ X2 )
=> ( X2 = extend1289208545_ereal ) ) ).
% ereal_top
thf(fact_310_ereal__bot,axiom,
! [X2: extended_ereal] :
( ! [B6: real] : ( ord_le824540014_ereal @ X2 @ ( extended_ereal2 @ B6 ) )
=> ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% ereal_bot
thf(fact_311_Lim__bounded__PInfty,axiom,
! [F2: nat > extended_ereal,L: extended_ereal,B5: real] :
( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ at_top_nat )
=> ( ! [N2: nat] : ( ord_le824540014_ereal @ ( F2 @ N2 ) @ ( extended_ereal2 @ B5 ) )
=> ( L != extend1289208545_ereal ) ) ) ).
% Lim_bounded_PInfty
thf(fact_312_ereal__minus__real__tendsto__MInf,axiom,
( filter1531173832_ereal
@ ^ [X: nat] : ( extended_ereal2 @ ( uminus_uminus_real @ ( semiri2110766477t_real @ X ) ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
@ at_top_nat ) ).
% ereal_minus_real_tendsto_MInf
thf(fact_313_ereal__PInfty__eq__plus,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( extend1289208545_ereal
= ( plus_p2118002693_ereal @ A @ B ) )
= ( ( A = extend1289208545_ereal )
| ( B = extend1289208545_ereal ) ) ) ).
% ereal_PInfty_eq_plus
thf(fact_314_ereal__plus__eq__PInfty,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( plus_p2118002693_ereal @ A @ B )
= extend1289208545_ereal )
= ( ( A = extend1289208545_ereal )
| ( B = extend1289208545_ereal ) ) ) ).
% ereal_plus_eq_PInfty
thf(fact_315_ereal__MInfty__eq__plus,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( uminus1208298309_ereal @ extend1289208545_ereal )
= ( plus_p2118002693_ereal @ A @ B ) )
= ( ( ( A
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
& ( B != extend1289208545_ereal ) )
| ( ( B
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
& ( A != extend1289208545_ereal ) ) ) ) ).
% ereal_MInfty_eq_plus
thf(fact_316_ereal__plus__eq__MInfty,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( plus_p2118002693_ereal @ A @ B )
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
= ( ( ( A
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
| ( B
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
& ( A != extend1289208545_ereal )
& ( B != extend1289208545_ereal ) ) ) ).
% ereal_plus_eq_MInfty
thf(fact_317_plus__ereal_Osimps_I2_J,axiom,
! [A: extended_ereal] :
( ( plus_p2118002693_ereal @ extend1289208545_ereal @ A )
= extend1289208545_ereal ) ).
% plus_ereal.simps(2)
thf(fact_318_plus__ereal_Osimps_I3_J,axiom,
! [A: extended_ereal] :
( ( plus_p2118002693_ereal @ A @ extend1289208545_ereal )
= extend1289208545_ereal ) ).
% plus_ereal.simps(3)
thf(fact_319_plus__ereal_Osimps_I6_J,axiom,
( ( plus_p2118002693_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% plus_ereal.simps(6)
thf(fact_320_ereal__add__cancel__left,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( A
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( ( plus_p2118002693_ereal @ A @ B )
= ( plus_p2118002693_ereal @ A @ C ) )
= ( ( A = extend1289208545_ereal )
| ( B = C ) ) ) ) ).
% ereal_add_cancel_left
thf(fact_321_ereal__add__cancel__right,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( A
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( ( plus_p2118002693_ereal @ B @ A )
= ( plus_p2118002693_ereal @ C @ A ) )
= ( ( A = extend1289208545_ereal )
| ( B = C ) ) ) ) ).
% ereal_add_cancel_right
thf(fact_322_plus__ereal_Osimps_I5_J,axiom,
! [P2: real] :
( ( plus_p2118002693_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ P2 ) )
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% plus_ereal.simps(5)
thf(fact_323_plus__ereal_Osimps_I4_J,axiom,
! [R: real] :
( ( plus_p2118002693_ereal @ ( extended_ereal2 @ R ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% plus_ereal.simps(4)
thf(fact_324_ereal__add__le__add__iff,axiom,
! [C: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( ord_le824540014_ereal @ ( plus_p2118002693_ereal @ C @ A ) @ ( plus_p2118002693_ereal @ C @ B ) )
= ( ( ord_le824540014_ereal @ A @ B )
| ( C = extend1289208545_ereal )
| ( ( C
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
& ( A != extend1289208545_ereal )
& ( B != extend1289208545_ereal ) ) ) ) ).
% ereal_add_le_add_iff
thf(fact_325_ereal__add__le__add__iff2,axiom,
! [A: extended_ereal,C: extended_ereal,B: extended_ereal] :
( ( ord_le824540014_ereal @ ( plus_p2118002693_ereal @ A @ C ) @ ( plus_p2118002693_ereal @ B @ C ) )
= ( ( ord_le824540014_ereal @ A @ B )
| ( C = extend1289208545_ereal )
| ( ( C
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
& ( A != extend1289208545_ereal )
& ( B != extend1289208545_ereal ) ) ) ) ).
% ereal_add_le_add_iff2
thf(fact_326_id__nat__ereal__tendsto__PInf,axiom,
( filter1531173832_ereal
@ ^ [X: nat] : ( extended_ereal2 @ ( semiri2110766477t_real @ X ) )
@ ( topolo2140997059_ereal @ extend1289208545_ereal )
@ at_top_nat ) ).
% id_nat_ereal_tendsto_PInf
thf(fact_327_ereal__liminf__add__mono,axiom,
! [U: nat > extended_ereal,V: nat > extended_ereal] :
( ~ ( ( ( ( liminf1045857232_ereal @ at_top_nat @ U )
= extend1289208545_ereal )
& ( ( liminf1045857232_ereal @ at_top_nat @ V )
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
| ( ( ( liminf1045857232_ereal @ at_top_nat @ U )
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
& ( ( liminf1045857232_ereal @ at_top_nat @ V )
= extend1289208545_ereal ) ) )
=> ( ord_le824540014_ereal @ ( plus_p2118002693_ereal @ ( liminf1045857232_ereal @ at_top_nat @ U ) @ ( liminf1045857232_ereal @ at_top_nat @ V ) )
@ ( liminf1045857232_ereal @ at_top_nat
@ ^ [N: nat] : ( plus_p2118002693_ereal @ ( U @ N ) @ ( V @ N ) ) ) ) ) ).
% ereal_liminf_add_mono
thf(fact_328_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_329_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri2019852685at_int @ A3 ) @ ( semiri2019852685at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_330_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_331_le__add1,axiom,
! [N7: nat,M5: nat] : ( ord_less_eq_nat @ N7 @ ( plus_plus_nat @ N7 @ M5 ) ) ).
% le_add1
thf(fact_332_le__add2,axiom,
! [N7: nat,M5: nat] : ( ord_less_eq_nat @ N7 @ ( plus_plus_nat @ M5 @ N7 ) ) ).
% le_add2
thf(fact_333_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_334_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_335_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_336_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_337_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_338_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_339_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_340_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_341_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri2019852685at_int @ A3 ) @ ( semiri2019852685at_int @ B2 ) ) ) ) ).
% nat_leq_as_int
thf(fact_342_plus__ereal_Osimps_I1_J,axiom,
! [R: real,P2: real] :
( ( plus_p2118002693_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P2 ) )
= ( extended_ereal2 @ ( plus_plus_real @ R @ P2 ) ) ) ).
% plus_ereal.simps(1)
thf(fact_343_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_344_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_345_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_346_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_347_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_348_eq__imp__le,axiom,
! [M5: nat,N7: nat] :
( ( M5 = N7 )
=> ( ord_less_eq_nat @ M5 @ N7 ) ) ).
% eq_imp_le
thf(fact_349_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_350_le__refl,axiom,
! [N7: nat] : ( ord_less_eq_nat @ N7 @ N7 ) ).
% le_refl
thf(fact_351_plus__ereal_Oelims,axiom,
! [X2: extended_ereal,Xa3: extended_ereal,Y: extended_ereal] :
( ( ( plus_p2118002693_ereal @ X2 @ Xa3 )
= Y )
=> ( ! [R2: real] :
( ( X2
= ( extended_ereal2 @ R2 ) )
=> ! [P3: real] :
( ( Xa3
= ( extended_ereal2 @ P3 ) )
=> ( Y
!= ( extended_ereal2 @ ( plus_plus_real @ R2 @ P3 ) ) ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( Y != extend1289208545_ereal ) )
=> ( ( ( Xa3 = extend1289208545_ereal )
=> ( Y != extend1289208545_ereal ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Y
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ? [P3: real] :
( Xa3
= ( extended_ereal2 @ P3 ) )
=> ( Y
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ~ ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Y
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ).
% plus_ereal.elims
thf(fact_352_filterlim__real__sequentially,axiom,
filterlim_nat_real @ semiri2110766477t_real @ at_top_real @ at_top_nat ).
% filterlim_real_sequentially
thf(fact_353_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri2019852685at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri2019852685at_int @ A ) @ ( semiri2019852685at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_354_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
= ( ^ [A3: nat,B2: nat] :
( ( semiri2019852685at_int @ A3 )
= ( semiri2019852685at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
% Conjectures (1)
thf(conj_0,conjecture,
( filter1531173832_ereal
@ ^ [I2: nat] : ( uminus1208298309_ereal @ ( f @ ( x @ I2 ) ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
@ at_top_nat ) ).
%------------------------------------------------------------------------------