TPTP Problem File: SLH0758^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Cotangent_PFD_Formula/0007_Cotangent_PFD_Formula/prob_00509_020321__14108834_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1377 ( 431 unt; 100 typ; 0 def)
% Number of atoms : 4409 (1086 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 14367 ( 315 ~; 55 |; 218 &;11824 @)
% ( 0 <=>;1955 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 1060 (1060 >; 0 *; 0 +; 0 <<)
% Number of symbols : 94 ( 91 usr; 18 con; 0-3 aty)
% Number of variables : 4472 ( 676 ^;3721 !; 75 ?;4472 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:59:54.488
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
set_real_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
set_nat_real: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
formal3361831859752904756s_real: $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__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (91)
thf(sy_c_Complex__Singularities_Ois__pole_001t__Nat__Onat_001t__Real__Oreal,type,
comple320020743214073521t_real: ( nat > real ) > nat > $o ).
thf(sy_c_Complex__Singularities_Ois__pole_001t__Real__Oreal_001t__Real__Oreal,type,
comple7683793008646357389l_real: ( real > real ) > real > $o ).
thf(sy_c_Cotangent__PFD__Formula_Ocot__pfd_001t__Real__Oreal,type,
cotang1502006655779026648d_real: real > real ).
thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
invers68952373231134600s_real: formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > 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_Oeventually_001t__Nat__Onat,type,
eventually_nat: ( nat > $o ) > filter_nat > $o ).
thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
eventually_real: ( real > $o ) > filter_real > $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__Real__Oreal_001t__Nat__Onat,type,
filterlim_real_nat: ( real > nat ) > filter_nat > filter_real > $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_Filter_Omap__filter__on_001t__Nat__Onat_001t__Nat__Onat,type,
map_fi8816901555893508305at_nat: set_nat > ( nat > nat ) > filter_nat > filter_nat ).
thf(sy_c_Filter_Omap__filter__on_001t__Nat__Onat_001t__Real__Oreal,type,
map_fi9184259510650374957t_real: set_nat > ( nat > real ) > filter_nat > filter_real ).
thf(sy_c_Filter_Omap__filter__on_001t__Real__Oreal_001t__Nat__Onat,type,
map_fi4528196046924497837al_nat: set_real > ( real > nat ) > filter_real > filter_nat ).
thf(sy_c_Filter_Omap__filter__on_001t__Real__Oreal_001t__Real__Oreal,type,
map_fi4827617581384206345l_real: set_real > ( real > real ) > filter_real > filter_real ).
thf(sy_c_Formal__Power__Series_Ofps__tan_001t__Real__Oreal,type,
formal3683295897622742886n_real: real > formal3361831859752904756s_real ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
comp_nat_nat_real: ( nat > nat ) > ( real > nat ) > real > nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Real__Oreal,type,
comp_nat_real_real: ( nat > real ) > ( real > nat ) > real > real ).
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_001t__Real__Oreal_001t__Real__Oreal_001t__Nat__Onat,type,
comp_real_real_nat: ( real > real ) > ( nat > real ) > nat > real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
comp_real_real_real: ( real > real ) > ( real > real ) > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
one_on8598947968683843321s_real: formal3361831859752904756s_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
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_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Real__Oreal_J,type,
plus_plus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
zero_z7760665558314615101s_real: formal3361831859752904756s_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_HOL_ONO__MATCH_001t__Real__Oreal_001t__Real__Oreal,type,
nO_MATCH_real_real: real > real > $o ).
thf(sy_c_HOL_ONO__MATCH_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Nat__Onat_J,type,
nO_MAT504328087405689813et_nat: set_real > set_nat > $o ).
thf(sy_c_HOL_ONO__MATCH_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J,type,
nO_MAT2855227906214470577t_real: set_real > set_real > $o ).
thf(sy_c_HOL_OUniq_001t__Nat__Onat,type,
uniq_nat: ( nat > $o ) > $o ).
thf(sy_c_HOL_OUniq_001t__Real__Oreal,type,
uniq_real: ( real > $o ) > $o ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
ring_1_Ints_real: set_real ).
thf(sy_c_Landau__Symbols_Osmallo_001t__Nat__Onat_001t__Real__Oreal,type,
landau997807338407142774t_real: filter_nat > ( nat > real ) > set_nat_real ).
thf(sy_c_Landau__Symbols_Osmallo_001t__Real__Oreal_001t__Real__Oreal,type,
landau3007391416991288786l_real: filter_real > ( real > real ) > set_real_real ).
thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
bfun_nat_real: ( nat > real ) > filter_nat > $o ).
thf(sy_c_Limits_OBfun_001t__Real__Oreal_001t__Real__Oreal,type,
bfun_real_real: ( real > real ) > filter_real > $o ).
thf(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
at_infinity_real: filter_real ).
thf(sy_c_Multiseries__Expansion_Ornatmod,type,
multiseries_rnatmod: real > real > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Filter__Ofilter_It__Nat__Onat_J,type,
ord_less_filter_nat: filter_nat > filter_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Filter__Ofilter_It__Real__Oreal_J,type,
ord_less_filter_real: filter_real > filter_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Real__Oreal_J,type,
ord_le4104064031414453916r_real: filter_real > filter_real > $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_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Real__Oreal_M_Eo_J,type,
top_top_real_o: real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Filter__Ofilter_It__Nat__Onat_J,type,
top_top_filter_nat: filter_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Filter__Ofilter_It__Real__Oreal_J,type,
top_top_filter_real: filter_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
real_V7735802525324610683m_real: real > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
divide1155267253282662278s_real: formal3361831859752904756s_real > formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
set_or5849166863359141190n_real: real > set_real ).
thf(sy_c_Topological__Spaces_Ocontinuous_001t__Nat__Onat_001t__Nat__Onat,type,
topolo1306369304726495905at_nat: filter_nat > ( nat > nat ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous_001t__Nat__Onat_001t__Real__Oreal,type,
topolo3806541068715748605t_real: filter_nat > ( nat > real ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Nat__Onat,type,
topolo8373849641844647293al_nat: filter_real > ( real > nat ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Nat__Onat,type,
topolo4328251076210115529en_nat: set_nat > $o ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Real__Oreal,type,
topolo4860482606490270245n_real: set_real > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Nat__Onat,type,
topolo4659099751122792720in_nat: nat > set_nat > filter_nat ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
topolo2177554685111907308n_real: real > set_real > filter_real ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Nat__Onat,type,
topolo8926549440605965083ds_nat: nat > filter_nat ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
topolo2815343760600316023s_real: real > filter_real ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
member_nat_real: ( nat > real ) > set_nat_real > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
member_real_real: ( real > real ) > set_real_real > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_r____,type,
r: real > real ).
thf(sy_v_x,type,
x: real ).
thf(sy_v_xa____,type,
xa: real ).
% Relevant facts (1271)
thf(fact_0_assms,axiom,
~ ( member_real @ x @ ring_1_Ints_real ) ).
% assms
thf(fact_1_that,axiom,
filterlim_real_real @ r @ ( topolo2177554685111907308n_real @ ( r @ xa ) @ top_top_set_real ) @ ( topolo2177554685111907308n_real @ xa @ top_top_set_real ) ).
% that
thf(fact_2_UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% UNIV_I
thf(fact_3_iso__tuple__UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% iso_tuple_UNIV_I
thf(fact_4_eventually__frequently__const__simps_I6_J,axiom,
! [C: $o,P: real > $o,F: filter_real] :
( ( eventually_real
@ ^ [X2: real] :
( C
=> ( P @ X2 ) )
@ F )
= ( C
=> ( eventually_real @ P @ F ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_5_eventually__frequently__const__simps_I6_J,axiom,
! [C: $o,P: nat > $o,F: filter_nat] :
( ( eventually_nat
@ ^ [X2: nat] :
( C
=> ( P @ X2 ) )
@ F )
= ( C
=> ( eventually_nat @ P @ F ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_6_eventually__frequently__const__simps_I4_J,axiom,
! [C: $o,P: real > $o,F: filter_real] :
( ( eventually_real
@ ^ [X2: real] :
( C
| ( P @ X2 ) )
@ F )
= ( C
| ( eventually_real @ P @ F ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_7_eventually__frequently__const__simps_I4_J,axiom,
! [C: $o,P: nat > $o,F: filter_nat] :
( ( eventually_nat
@ ^ [X2: nat] :
( C
| ( P @ X2 ) )
@ F )
= ( C
| ( eventually_nat @ P @ F ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_8_eventually__frequently__const__simps_I3_J,axiom,
! [P: real > $o,C: $o,F: filter_real] :
( ( eventually_real
@ ^ [X2: real] :
( ( P @ X2 )
| C )
@ F )
= ( ( eventually_real @ P @ F )
| C ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_9_eventually__frequently__const__simps_I3_J,axiom,
! [P: nat > $o,C: $o,F: filter_nat] :
( ( eventually_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| C )
@ F )
= ( ( eventually_nat @ P @ F )
| C ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_10_eventually__mp,axiom,
! [P: real > $o,Q: real > $o,F: filter_real] :
( ( eventually_real
@ ^ [X2: real] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F )
=> ( ( eventually_real @ P @ F )
=> ( eventually_real @ Q @ F ) ) ) ).
% eventually_mp
thf(fact_11_eventually__mp,axiom,
! [P: nat > $o,Q: nat > $o,F: filter_nat] :
( ( eventually_nat
@ ^ [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F )
=> ( ( eventually_nat @ P @ F )
=> ( eventually_nat @ Q @ F ) ) ) ).
% eventually_mp
thf(fact_12_eventually__True,axiom,
! [F: filter_real] :
( eventually_real
@ ^ [X2: real] : $true
@ F ) ).
% eventually_True
thf(fact_13_eventually__True,axiom,
! [F: filter_nat] :
( eventually_nat
@ ^ [X2: nat] : $true
@ F ) ).
% eventually_True
thf(fact_14_eventually__conj,axiom,
! [P: real > $o,F: filter_real,Q: real > $o] :
( ( eventually_real @ P @ F )
=> ( ( eventually_real @ Q @ F )
=> ( eventually_real
@ ^ [X2: real] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ F ) ) ) ).
% eventually_conj
thf(fact_15_eventually__conj,axiom,
! [P: nat > $o,F: filter_nat,Q: nat > $o] :
( ( eventually_nat @ P @ F )
=> ( ( eventually_nat @ Q @ F )
=> ( eventually_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ F ) ) ) ).
% eventually_conj
thf(fact_16_eventually__elim2,axiom,
! [P: real > $o,F: filter_real,Q: real > $o,R: real > $o] :
( ( eventually_real @ P @ F )
=> ( ( eventually_real @ Q @ F )
=> ( ! [I: real] :
( ( P @ I )
=> ( ( Q @ I )
=> ( R @ I ) ) )
=> ( eventually_real @ R @ F ) ) ) ) ).
% eventually_elim2
thf(fact_17_eventually__elim2,axiom,
! [P: nat > $o,F: filter_nat,Q: nat > $o,R: nat > $o] :
( ( eventually_nat @ P @ F )
=> ( ( eventually_nat @ Q @ F )
=> ( ! [I: nat] :
( ( P @ I )
=> ( ( Q @ I )
=> ( R @ I ) ) )
=> ( eventually_nat @ R @ F ) ) ) ) ).
% eventually_elim2
thf(fact_18_eventually__subst,axiom,
! [P: real > $o,Q: real > $o,F: filter_real] :
( ( eventually_real
@ ^ [N: real] :
( ( P @ N )
= ( Q @ N ) )
@ F )
=> ( ( eventually_real @ P @ F )
= ( eventually_real @ Q @ F ) ) ) ).
% eventually_subst
thf(fact_19_eventually__subst,axiom,
! [P: nat > $o,Q: nat > $o,F: filter_nat] :
( ( eventually_nat
@ ^ [N: nat] :
( ( P @ N )
= ( Q @ N ) )
@ F )
=> ( ( eventually_nat @ P @ F )
= ( eventually_nat @ Q @ F ) ) ) ).
% eventually_subst
thf(fact_20_eventually__rev__mp,axiom,
! [P: real > $o,F: filter_real,Q: real > $o] :
( ( eventually_real @ P @ F )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F )
=> ( eventually_real @ Q @ F ) ) ) ).
% eventually_rev_mp
thf(fact_21_eventually__rev__mp,axiom,
! [P: nat > $o,F: filter_nat,Q: nat > $o] :
( ( eventually_nat @ P @ F )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F )
=> ( eventually_nat @ Q @ F ) ) ) ).
% eventually_rev_mp
thf(fact_22_eventually__top,axiom,
! [P: real > $o] :
( ( eventually_real @ P @ top_top_filter_real )
= ( ! [X3: real] : ( P @ X3 ) ) ) ).
% eventually_top
thf(fact_23_eventually__top,axiom,
! [P: nat > $o] :
( ( eventually_nat @ P @ top_top_filter_nat )
= ( ! [X3: nat] : ( P @ X3 ) ) ) ).
% eventually_top
thf(fact_24_filterlim__top,axiom,
! [F2: real > real,F: filter_real] : ( filterlim_real_real @ F2 @ top_top_filter_real @ F ) ).
% filterlim_top
thf(fact_25_filterlim__top,axiom,
! [F2: nat > nat,F: filter_nat] : ( filterlim_nat_nat @ F2 @ top_top_filter_nat @ F ) ).
% filterlim_top
thf(fact_26_filterlim__top,axiom,
! [F2: nat > real,F: filter_nat] : ( filterlim_nat_real @ F2 @ top_top_filter_real @ F ) ).
% filterlim_top
thf(fact_27_filterlim__ident,axiom,
! [F: filter_real] :
( filterlim_real_real
@ ^ [X2: real] : X2
@ F
@ F ) ).
% filterlim_ident
thf(fact_28_filterlim__ident,axiom,
! [F: filter_nat] :
( filterlim_nat_nat
@ ^ [X2: nat] : X2
@ F
@ F ) ).
% filterlim_ident
thf(fact_29_filterlim__compose,axiom,
! [G: real > nat,F3: filter_nat,F22: filter_real,F2: nat > real,F1: filter_nat] :
( ( filterlim_real_nat @ G @ F3 @ F22 )
=> ( ( filterlim_nat_real @ F2 @ F22 @ F1 )
=> ( filterlim_nat_nat
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ F3
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_30_filterlim__compose,axiom,
! [G: real > real,F3: filter_real,F22: filter_real,F2: real > real,F1: filter_real] :
( ( filterlim_real_real @ G @ F3 @ F22 )
=> ( ( filterlim_real_real @ F2 @ F22 @ F1 )
=> ( filterlim_real_real
@ ^ [X2: real] : ( G @ ( F2 @ X2 ) )
@ F3
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_31_filterlim__compose,axiom,
! [G: real > real,F3: filter_real,F22: filter_real,F2: nat > real,F1: filter_nat] :
( ( filterlim_real_real @ G @ F3 @ F22 )
=> ( ( filterlim_nat_real @ F2 @ F22 @ F1 )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ F3
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_32_filterlim__compose,axiom,
! [G: nat > nat,F3: filter_nat,F22: filter_nat,F2: nat > nat,F1: filter_nat] :
( ( filterlim_nat_nat @ G @ F3 @ F22 )
=> ( ( filterlim_nat_nat @ F2 @ F22 @ F1 )
=> ( filterlim_nat_nat
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ F3
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_33_filterlim__compose,axiom,
! [G: nat > real,F3: filter_real,F22: filter_nat,F2: real > nat,F1: filter_real] :
( ( filterlim_nat_real @ G @ F3 @ F22 )
=> ( ( filterlim_real_nat @ F2 @ F22 @ F1 )
=> ( filterlim_real_real
@ ^ [X2: real] : ( G @ ( F2 @ X2 ) )
@ F3
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_34_filterlim__compose,axiom,
! [G: nat > real,F3: filter_real,F22: filter_nat,F2: nat > nat,F1: filter_nat] :
( ( filterlim_nat_real @ G @ F3 @ F22 )
=> ( ( filterlim_nat_nat @ F2 @ F22 @ F1 )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ F3
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_35_top__set__def,axiom,
( top_top_set_real
= ( collect_real @ top_top_real_o ) ) ).
% top_set_def
thf(fact_36_eventually__compose__filterlim,axiom,
! [P: real > $o,F: filter_real,F2: real > real,G2: filter_real] :
( ( eventually_real @ P @ F )
=> ( ( filterlim_real_real @ F2 @ F @ G2 )
=> ( eventually_real
@ ^ [X2: real] : ( P @ ( F2 @ X2 ) )
@ G2 ) ) ) ).
% eventually_compose_filterlim
thf(fact_37_eventually__compose__filterlim,axiom,
! [P: real > $o,F: filter_real,F2: nat > real,G2: filter_nat] :
( ( eventually_real @ P @ F )
=> ( ( filterlim_nat_real @ F2 @ F @ G2 )
=> ( eventually_nat
@ ^ [X2: nat] : ( P @ ( F2 @ X2 ) )
@ G2 ) ) ) ).
% eventually_compose_filterlim
thf(fact_38_eventually__compose__filterlim,axiom,
! [P: nat > $o,F: filter_nat,F2: real > nat,G2: filter_real] :
( ( eventually_nat @ P @ F )
=> ( ( filterlim_real_nat @ F2 @ F @ G2 )
=> ( eventually_real
@ ^ [X2: real] : ( P @ ( F2 @ X2 ) )
@ G2 ) ) ) ).
% eventually_compose_filterlim
thf(fact_39_eventually__compose__filterlim,axiom,
! [P: nat > $o,F: filter_nat,F2: nat > nat,G2: filter_nat] :
( ( eventually_nat @ P @ F )
=> ( ( filterlim_nat_nat @ F2 @ F @ G2 )
=> ( eventually_nat
@ ^ [X2: nat] : ( P @ ( F2 @ X2 ) )
@ G2 ) ) ) ).
% eventually_compose_filterlim
thf(fact_40_filterlim__cong,axiom,
! [F1: filter_real,F12: filter_real,F22: filter_real,F23: filter_real,F2: real > real,G: real > real] :
( ( F1 = F12 )
=> ( ( F22 = F23 )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ F22 )
=> ( ( filterlim_real_real @ F2 @ F1 @ F22 )
= ( filterlim_real_real @ G @ F12 @ F23 ) ) ) ) ) ).
% filterlim_cong
thf(fact_41_filterlim__cong,axiom,
! [F1: filter_nat,F12: filter_nat,F22: filter_nat,F23: filter_nat,F2: nat > nat,G: nat > nat] :
( ( F1 = F12 )
=> ( ( F22 = F23 )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ F22 )
=> ( ( filterlim_nat_nat @ F2 @ F1 @ F22 )
= ( filterlim_nat_nat @ G @ F12 @ F23 ) ) ) ) ) ).
% filterlim_cong
thf(fact_42_filterlim__cong,axiom,
! [F1: filter_real,F12: filter_real,F22: filter_nat,F23: filter_nat,F2: nat > real,G: nat > real] :
( ( F1 = F12 )
=> ( ( F22 = F23 )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ F22 )
=> ( ( filterlim_nat_real @ F2 @ F1 @ F22 )
= ( filterlim_nat_real @ G @ F12 @ F23 ) ) ) ) ) ).
% filterlim_cong
thf(fact_43_filterlim__iff,axiom,
( filterlim_real_real
= ( ^ [F4: real > real,F24: filter_real,F13: filter_real] :
! [P2: real > $o] :
( ( eventually_real @ P2 @ F24 )
=> ( eventually_real
@ ^ [X2: real] : ( P2 @ ( F4 @ X2 ) )
@ F13 ) ) ) ) ).
% filterlim_iff
thf(fact_44_filterlim__iff,axiom,
( filterlim_nat_real
= ( ^ [F4: nat > real,F24: filter_real,F13: filter_nat] :
! [P2: real > $o] :
( ( eventually_real @ P2 @ F24 )
=> ( eventually_nat
@ ^ [X2: nat] : ( P2 @ ( F4 @ X2 ) )
@ F13 ) ) ) ) ).
% filterlim_iff
thf(fact_45_filterlim__iff,axiom,
( filterlim_real_nat
= ( ^ [F4: real > nat,F24: filter_nat,F13: filter_real] :
! [P2: nat > $o] :
( ( eventually_nat @ P2 @ F24 )
=> ( eventually_real
@ ^ [X2: real] : ( P2 @ ( F4 @ X2 ) )
@ F13 ) ) ) ) ).
% filterlim_iff
thf(fact_46_filterlim__iff,axiom,
( filterlim_nat_nat
= ( ^ [F4: nat > nat,F24: filter_nat,F13: filter_nat] :
! [P2: nat > $o] :
( ( eventually_nat @ P2 @ F24 )
=> ( eventually_nat
@ ^ [X2: nat] : ( P2 @ ( F4 @ X2 ) )
@ F13 ) ) ) ) ).
% filterlim_iff
thf(fact_47_UNIV__witness,axiom,
? [X4: real] : ( member_real @ X4 @ top_top_set_real ) ).
% UNIV_witness
thf(fact_48_UNIV__eq__I,axiom,
! [A: set_real] :
( ! [X4: real] : ( member_real @ X4 @ A )
=> ( top_top_set_real = A ) ) ).
% UNIV_eq_I
thf(fact_49_always__eventually,axiom,
! [P: real > $o,F: filter_real] :
( ! [X_1: real] : ( P @ X_1 )
=> ( eventually_real @ P @ F ) ) ).
% always_eventually
thf(fact_50_always__eventually,axiom,
! [P: nat > $o,F: filter_nat] :
( ! [X_1: nat] : ( P @ X_1 )
=> ( eventually_nat @ P @ F ) ) ).
% always_eventually
thf(fact_51_not__eventuallyD,axiom,
! [P: real > $o,F: filter_real] :
( ~ ( eventually_real @ P @ F )
=> ? [X4: real] :
~ ( P @ X4 ) ) ).
% not_eventuallyD
thf(fact_52_not__eventuallyD,axiom,
! [P: nat > $o,F: filter_nat] :
( ~ ( eventually_nat @ P @ F )
=> ? [X4: nat] :
~ ( P @ X4 ) ) ).
% not_eventuallyD
thf(fact_53_eventually__mono,axiom,
! [P: real > $o,F: filter_real,Q: real > $o] :
( ( eventually_real @ P @ F )
=> ( ! [X4: real] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( eventually_real @ Q @ F ) ) ) ).
% eventually_mono
thf(fact_54_eventually__mono,axiom,
! [P: nat > $o,F: filter_nat,Q: nat > $o] :
( ( eventually_nat @ P @ F )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( eventually_nat @ Q @ F ) ) ) ).
% eventually_mono
thf(fact_55_filter__eq__iff,axiom,
( ( ^ [Y: filter_real,Z: filter_real] : ( Y = Z ) )
= ( ^ [F5: filter_real,F6: filter_real] :
! [P2: real > $o] :
( ( eventually_real @ P2 @ F5 )
= ( eventually_real @ P2 @ F6 ) ) ) ) ).
% filter_eq_iff
thf(fact_56_filter__eq__iff,axiom,
( ( ^ [Y: filter_nat,Z: filter_nat] : ( Y = Z ) )
= ( ^ [F5: filter_nat,F6: filter_nat] :
! [P2: nat > $o] :
( ( eventually_nat @ P2 @ F5 )
= ( eventually_nat @ P2 @ F6 ) ) ) ) ).
% filter_eq_iff
thf(fact_57_eventuallyI,axiom,
! [P: real > $o,F: filter_real] :
( ! [X_1: real] : ( P @ X_1 )
=> ( eventually_real @ P @ F ) ) ).
% eventuallyI
thf(fact_58_eventuallyI,axiom,
! [P: nat > $o,F: filter_nat] :
( ! [X_1: nat] : ( P @ X_1 )
=> ( eventually_nat @ P @ F ) ) ).
% eventuallyI
thf(fact_59_UNIV__def,axiom,
( top_top_set_real
= ( collect_real
@ ^ [X2: real] : $true ) ) ).
% UNIV_def
thf(fact_60_not__eventually__impI,axiom,
! [P: real > $o,F: filter_real,Q: real > $o] :
( ( eventually_real @ P @ F )
=> ( ~ ( eventually_real @ Q @ F )
=> ~ ( eventually_real
@ ^ [X2: real] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F ) ) ) ).
% not_eventually_impI
thf(fact_61_not__eventually__impI,axiom,
! [P: nat > $o,F: filter_nat,Q: nat > $o] :
( ( eventually_nat @ P @ F )
=> ( ~ ( eventually_nat @ Q @ F )
=> ~ ( eventually_nat
@ ^ [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F ) ) ) ).
% not_eventually_impI
thf(fact_62_eventually__conj__iff,axiom,
! [P: real > $o,Q: real > $o,F: filter_real] :
( ( eventually_real
@ ^ [X2: real] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ F )
= ( ( eventually_real @ P @ F )
& ( eventually_real @ Q @ F ) ) ) ).
% eventually_conj_iff
thf(fact_63_eventually__conj__iff,axiom,
! [P: nat > $o,Q: nat > $o,F: filter_nat] :
( ( eventually_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ F )
= ( ( eventually_nat @ P @ F )
& ( eventually_nat @ Q @ F ) ) ) ).
% eventually_conj_iff
thf(fact_64_filterlim__at__within__not__equal,axiom,
! [F2: nat > nat,A2: nat,S: set_nat,F: filter_nat,B: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo4659099751122792720in_nat @ A2 @ S ) @ F )
=> ( eventually_nat
@ ^ [W: nat] :
( ( member_nat @ ( F2 @ W ) @ S )
& ( ( F2 @ W )
!= B ) )
@ F ) ) ).
% filterlim_at_within_not_equal
thf(fact_65_filterlim__at__within__not__equal,axiom,
! [F2: real > real,A2: real,S: set_real,F: filter_real,B: real] :
( ( filterlim_real_real @ F2 @ ( topolo2177554685111907308n_real @ A2 @ S ) @ F )
=> ( eventually_real
@ ^ [W: real] :
( ( member_real @ ( F2 @ W ) @ S )
& ( ( F2 @ W )
!= B ) )
@ F ) ) ).
% filterlim_at_within_not_equal
thf(fact_66_filterlim__at__within__not__equal,axiom,
! [F2: nat > real,A2: real,S: set_real,F: filter_nat,B: real] :
( ( filterlim_nat_real @ F2 @ ( topolo2177554685111907308n_real @ A2 @ S ) @ F )
=> ( eventually_nat
@ ^ [W: nat] :
( ( member_real @ ( F2 @ W ) @ S )
& ( ( F2 @ W )
!= B ) )
@ F ) ) ).
% filterlim_at_within_not_equal
thf(fact_67_filterlim__at__If,axiom,
! [F2: nat > nat,G2: filter_nat,X: nat,P: nat > $o,G: nat > nat] :
( ( filterlim_nat_nat @ F2 @ G2 @ ( topolo4659099751122792720in_nat @ X @ ( collect_nat @ P ) ) )
=> ( ( filterlim_nat_nat @ G @ G2
@ ( topolo4659099751122792720in_nat @ X
@ ( collect_nat
@ ^ [X2: nat] :
~ ( P @ X2 ) ) ) )
=> ( filterlim_nat_nat
@ ^ [X2: nat] : ( if_nat @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ G2
@ ( topolo4659099751122792720in_nat @ X @ top_top_set_nat ) ) ) ) ).
% filterlim_at_If
thf(fact_68_filterlim__at__If,axiom,
! [F2: nat > real,G2: filter_real,X: nat,P: nat > $o,G: nat > real] :
( ( filterlim_nat_real @ F2 @ G2 @ ( topolo4659099751122792720in_nat @ X @ ( collect_nat @ P ) ) )
=> ( ( filterlim_nat_real @ G @ G2
@ ( topolo4659099751122792720in_nat @ X
@ ( collect_nat
@ ^ [X2: nat] :
~ ( P @ X2 ) ) ) )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( if_real @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ G2
@ ( topolo4659099751122792720in_nat @ X @ top_top_set_nat ) ) ) ) ).
% filterlim_at_If
thf(fact_69_filterlim__at__If,axiom,
! [F2: real > real,G2: filter_real,X: real,P: real > $o,G: real > real] :
( ( filterlim_real_real @ F2 @ G2 @ ( topolo2177554685111907308n_real @ X @ ( collect_real @ P ) ) )
=> ( ( filterlim_real_real @ G @ G2
@ ( topolo2177554685111907308n_real @ X
@ ( collect_real
@ ^ [X2: real] :
~ ( P @ X2 ) ) ) )
=> ( filterlim_real_real
@ ^ [X2: real] : ( if_real @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ G2
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% filterlim_at_If
thf(fact_70_eventually__map__filter__on,axiom,
! [X5: set_real,F: filter_real,P: real > $o,F2: real > real] :
( ( eventually_real
@ ^ [X2: real] : ( member_real @ X2 @ X5 )
@ F )
=> ( ( eventually_real @ P @ ( map_fi4827617581384206345l_real @ X5 @ F2 @ F ) )
= ( eventually_real
@ ^ [X2: real] :
( ( P @ ( F2 @ X2 ) )
& ( member_real @ X2 @ X5 ) )
@ F ) ) ) ).
% eventually_map_filter_on
thf(fact_71_eventually__map__filter__on,axiom,
! [X5: set_real,F: filter_real,P: nat > $o,F2: real > nat] :
( ( eventually_real
@ ^ [X2: real] : ( member_real @ X2 @ X5 )
@ F )
=> ( ( eventually_nat @ P @ ( map_fi4528196046924497837al_nat @ X5 @ F2 @ F ) )
= ( eventually_real
@ ^ [X2: real] :
( ( P @ ( F2 @ X2 ) )
& ( member_real @ X2 @ X5 ) )
@ F ) ) ) ).
% eventually_map_filter_on
thf(fact_72_eventually__map__filter__on,axiom,
! [X5: set_nat,F: filter_nat,P: real > $o,F2: nat > real] :
( ( eventually_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ X5 )
@ F )
=> ( ( eventually_real @ P @ ( map_fi9184259510650374957t_real @ X5 @ F2 @ F ) )
= ( eventually_nat
@ ^ [X2: nat] :
( ( P @ ( F2 @ X2 ) )
& ( member_nat @ X2 @ X5 ) )
@ F ) ) ) ).
% eventually_map_filter_on
thf(fact_73_eventually__map__filter__on,axiom,
! [X5: set_nat,F: filter_nat,P: nat > $o,F2: nat > nat] :
( ( eventually_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ X5 )
@ F )
=> ( ( eventually_nat @ P @ ( map_fi8816901555893508305at_nat @ X5 @ F2 @ F ) )
= ( eventually_nat
@ ^ [X2: nat] :
( ( P @ ( F2 @ X2 ) )
& ( member_nat @ X2 @ X5 ) )
@ F ) ) ) ).
% eventually_map_filter_on
thf(fact_74_top__empty__eq,axiom,
( top_top_real_o
= ( ^ [X2: real] : ( member_real @ X2 @ top_top_set_real ) ) ) ).
% top_empty_eq
thf(fact_75_Lim__transform__within__set__eq,axiom,
! [S2: set_real,T: set_real,A2: real,F2: real > real,L: real] :
( ( eventually_real
@ ^ [X2: real] :
( ( member_real @ X2 @ S2 )
= ( member_real @ X2 @ T ) )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ S2 ) )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ T ) ) ) ) ).
% Lim_transform_within_set_eq
thf(fact_76_Lim__transform__within__set,axiom,
! [F2: real > real,L: real,A2: real,S2: set_real,T: set_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ S2 ) )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( member_real @ X2 @ S2 )
= ( member_real @ X2 @ T ) )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ T ) ) ) ) ).
% Lim_transform_within_set
thf(fact_77_tendsto__compose__eventually,axiom,
! [G: nat > nat,M: nat,L: nat,F2: real > nat,F: filter_real] :
( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ M ) @ ( topolo4659099751122792720in_nat @ L @ top_top_set_nat ) )
=> ( ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
!= L )
@ F )
=> ( filterlim_real_nat
@ ^ [X2: real] : ( G @ ( F2 @ X2 ) )
@ ( topolo8926549440605965083ds_nat @ M )
@ F ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_78_tendsto__compose__eventually,axiom,
! [G: nat > nat,M: nat,L: nat,F2: nat > nat,F: filter_nat] :
( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ M ) @ ( topolo4659099751122792720in_nat @ L @ top_top_set_nat ) )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
!= L )
@ F )
=> ( filterlim_nat_nat
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ ( topolo8926549440605965083ds_nat @ M )
@ F ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_79_tendsto__compose__eventually,axiom,
! [G: nat > real,M: real,L: nat,F2: real > nat,F: filter_real] :
( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ M ) @ ( topolo4659099751122792720in_nat @ L @ top_top_set_nat ) )
=> ( ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
!= L )
@ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( G @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ M )
@ F ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_80_tendsto__compose__eventually,axiom,
! [G: nat > real,M: real,L: nat,F2: nat > nat,F: filter_nat] :
( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ M ) @ ( topolo4659099751122792720in_nat @ L @ top_top_set_nat ) )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
!= L )
@ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ M )
@ F ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_81_tendsto__compose__eventually,axiom,
! [G: real > nat,M: nat,L: real,F2: nat > real,F: filter_nat] :
( ( filterlim_real_nat @ G @ ( topolo8926549440605965083ds_nat @ M ) @ ( topolo2177554685111907308n_real @ L @ top_top_set_real ) )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
!= L )
@ F )
=> ( filterlim_nat_nat
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ ( topolo8926549440605965083ds_nat @ M )
@ F ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_82_tendsto__compose__eventually,axiom,
! [G: real > real,M: real,L: real,F2: real > real,F: filter_real] :
( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ M ) @ ( topolo2177554685111907308n_real @ L @ top_top_set_real ) )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
!= L )
@ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( G @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ M )
@ F ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_83_tendsto__compose__eventually,axiom,
! [G: real > real,M: real,L: real,F2: nat > real,F: filter_nat] :
( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ M ) @ ( topolo2177554685111907308n_real @ L @ top_top_set_real ) )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
!= L )
@ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ M )
@ F ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_84_LIM__compose__eventually,axiom,
! [F2: nat > nat,B: nat,A2: nat,G: nat > nat,C2: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ B ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ C2 ) @ ( topolo4659099751122792720in_nat @ B @ top_top_set_nat ) )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
!= B )
@ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
=> ( filterlim_nat_nat
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ ( topolo8926549440605965083ds_nat @ C2 )
@ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_85_LIM__compose__eventually,axiom,
! [F2: nat > nat,B: nat,A2: nat,G: nat > real,C2: real] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ B ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ C2 ) @ ( topolo4659099751122792720in_nat @ B @ top_top_set_nat ) )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
!= B )
@ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ C2 )
@ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_86_LIM__compose__eventually,axiom,
! [F2: nat > real,B: real,A2: nat,G: real > nat,C2: nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ B ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
=> ( ( filterlim_real_nat @ G @ ( topolo8926549440605965083ds_nat @ C2 ) @ ( topolo2177554685111907308n_real @ B @ top_top_set_real ) )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
!= B )
@ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
=> ( filterlim_nat_nat
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ ( topolo8926549440605965083ds_nat @ C2 )
@ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_87_LIM__compose__eventually,axiom,
! [F2: nat > real,B: real,A2: nat,G: real > real,C2: real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ B ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ C2 ) @ ( topolo2177554685111907308n_real @ B @ top_top_set_real ) )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
!= B )
@ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ C2 )
@ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_88_LIM__compose__eventually,axiom,
! [F2: real > nat,B: nat,A2: real,G: nat > nat,C2: nat] :
( ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ B ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ C2 ) @ ( topolo4659099751122792720in_nat @ B @ top_top_set_nat ) )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
!= B )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( filterlim_real_nat
@ ^ [X2: real] : ( G @ ( F2 @ X2 ) )
@ ( topolo8926549440605965083ds_nat @ C2 )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_89_LIM__compose__eventually,axiom,
! [F2: real > nat,B: nat,A2: real,G: nat > real,C2: real] :
( ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ B ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ C2 ) @ ( topolo4659099751122792720in_nat @ B @ top_top_set_nat ) )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
!= B )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X2: real] : ( G @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ C2 )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_90_LIM__compose__eventually,axiom,
! [F2: real > real,B: real,A2: real,G: real > real,C2: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ B ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ C2 ) @ ( topolo2177554685111907308n_real @ B @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
!= B )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X2: real] : ( G @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ C2 )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_91_filterlim__atI,axiom,
! [F2: nat > nat,C2: nat,F: filter_nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ C2 ) @ F )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
!= C2 )
@ F )
=> ( filterlim_nat_nat @ F2 @ ( topolo4659099751122792720in_nat @ C2 @ top_top_set_nat ) @ F ) ) ) ).
% filterlim_atI
thf(fact_92_filterlim__atI,axiom,
! [F2: real > real,C2: real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
!= C2 )
@ F )
=> ( filterlim_real_real @ F2 @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) @ F ) ) ) ).
% filterlim_atI
thf(fact_93_filterlim__atI,axiom,
! [F2: nat > real,C2: real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
!= C2 )
@ F )
=> ( filterlim_nat_real @ F2 @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) @ F ) ) ) ).
% filterlim_atI
thf(fact_94_mem__Collect__eq,axiom,
! [A2: real,P: real > $o] :
( ( member_real @ A2 @ ( collect_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_95_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_96_tendsto__const,axiom,
! [K: nat,F: filter_nat] :
( filterlim_nat_nat
@ ^ [X2: nat] : K
@ ( topolo8926549440605965083ds_nat @ K )
@ F ) ).
% tendsto_const
thf(fact_97_tendsto__const,axiom,
! [K: real,F: filter_real] :
( filterlim_real_real
@ ^ [X2: real] : K
@ ( topolo2815343760600316023s_real @ K )
@ F ) ).
% tendsto_const
thf(fact_98_tendsto__const,axiom,
! [K: real,F: filter_nat] :
( filterlim_nat_real
@ ^ [X2: nat] : K
@ ( topolo2815343760600316023s_real @ K )
@ F ) ).
% tendsto_const
thf(fact_99_tendsto__ident__at,axiom,
! [A2: nat,S: set_nat] :
( filterlim_nat_nat
@ ^ [X2: nat] : X2
@ ( topolo8926549440605965083ds_nat @ A2 )
@ ( topolo4659099751122792720in_nat @ A2 @ S ) ) ).
% tendsto_ident_at
thf(fact_100_tendsto__ident__at,axiom,
! [A2: real,S: set_real] :
( filterlim_real_real
@ ^ [X2: real] : X2
@ ( topolo2815343760600316023s_real @ A2 )
@ ( topolo2177554685111907308n_real @ A2 @ S ) ) ).
% tendsto_ident_at
thf(fact_101_tendsto__cong__limit,axiom,
! [F2: nat > nat,L: nat,F: filter_nat,K: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
=> ( ( K = L )
=> ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ K ) @ F ) ) ) ).
% tendsto_cong_limit
thf(fact_102_tendsto__cong__limit,axiom,
! [F2: real > real,L: real,F: filter_real,K: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( ( K = L )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ K ) @ F ) ) ) ).
% tendsto_cong_limit
thf(fact_103_tendsto__cong__limit,axiom,
! [F2: nat > real,L: real,F: filter_nat,K: real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( ( K = L )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ K ) @ F ) ) ) ).
% tendsto_cong_limit
thf(fact_104_tendsto__eq__rhs,axiom,
! [F2: nat > nat,X: nat,F: filter_nat,Y2: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ X ) @ F )
=> ( ( X = Y2 )
=> ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ Y2 ) @ F ) ) ) ).
% tendsto_eq_rhs
thf(fact_105_tendsto__eq__rhs,axiom,
! [F2: real > real,X: real,F: filter_real,Y2: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ X ) @ F )
=> ( ( X = Y2 )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ Y2 ) @ F ) ) ) ).
% tendsto_eq_rhs
thf(fact_106_tendsto__eq__rhs,axiom,
! [F2: nat > real,X: real,F: filter_nat,Y2: real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ X ) @ F )
=> ( ( X = Y2 )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ Y2 ) @ F ) ) ) ).
% tendsto_eq_rhs
thf(fact_107_eventually__nhds__x__imp__x,axiom,
! [P: nat > $o,X: nat] :
( ( eventually_nat @ P @ ( topolo8926549440605965083ds_nat @ X ) )
=> ( P @ X ) ) ).
% eventually_nhds_x_imp_x
thf(fact_108_eventually__nhds__x__imp__x,axiom,
! [P: real > $o,X: real] :
( ( eventually_real @ P @ ( topolo2815343760600316023s_real @ X ) )
=> ( P @ X ) ) ).
% eventually_nhds_x_imp_x
thf(fact_109_t1__space__nhds,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( eventually_nat
@ ^ [X2: nat] : ( X2 != Y2 )
@ ( topolo8926549440605965083ds_nat @ X ) ) ) ).
% t1_space_nhds
thf(fact_110_t1__space__nhds,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ( eventually_real
@ ^ [X2: real] : ( X2 != Y2 )
@ ( topolo2815343760600316023s_real @ X ) ) ) ).
% t1_space_nhds
thf(fact_111_eventually__eventually,axiom,
! [P: nat > $o,X: nat] :
( ( eventually_nat
@ ^ [Y3: nat] : ( eventually_nat @ P @ ( topolo8926549440605965083ds_nat @ Y3 ) )
@ ( topolo8926549440605965083ds_nat @ X ) )
= ( eventually_nat @ P @ ( topolo8926549440605965083ds_nat @ X ) ) ) ).
% eventually_eventually
thf(fact_112_eventually__eventually,axiom,
! [P: real > $o,X: real] :
( ( eventually_real
@ ^ [Y3: real] : ( eventually_real @ P @ ( topolo2815343760600316023s_real @ Y3 ) )
@ ( topolo2815343760600316023s_real @ X ) )
= ( eventually_real @ P @ ( topolo2815343760600316023s_real @ X ) ) ) ).
% eventually_eventually
thf(fact_113_eventually__at__filter,axiom,
! [P: nat > $o,A2: nat,S: set_nat] :
( ( eventually_nat @ P @ ( topolo4659099751122792720in_nat @ A2 @ S ) )
= ( eventually_nat
@ ^ [X2: nat] :
( ( X2 != A2 )
=> ( ( member_nat @ X2 @ S )
=> ( P @ X2 ) ) )
@ ( topolo8926549440605965083ds_nat @ A2 ) ) ) ).
% eventually_at_filter
thf(fact_114_eventually__at__filter,axiom,
! [P: real > $o,A2: real,S: set_real] :
( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A2 @ S ) )
= ( eventually_real
@ ^ [X2: real] :
( ( X2 != A2 )
=> ( ( member_real @ X2 @ S )
=> ( P @ X2 ) ) )
@ ( topolo2815343760600316023s_real @ A2 ) ) ) ).
% eventually_at_filter
thf(fact_115_tendsto__cong,axiom,
! [F2: nat > nat,G: nat > nat,F: filter_nat,C2: nat] :
( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ F )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ C2 ) @ F )
= ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ C2 ) @ F ) ) ) ).
% tendsto_cong
thf(fact_116_tendsto__cong,axiom,
! [F2: real > real,G: real > real,F: filter_real,C2: real] :
( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ F )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
= ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ C2 ) @ F ) ) ) ).
% tendsto_cong
thf(fact_117_tendsto__cong,axiom,
! [F2: nat > real,G: nat > real,F: filter_nat,C2: real] :
( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ F )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
= ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ C2 ) @ F ) ) ) ).
% tendsto_cong
thf(fact_118_tendsto__discrete,axiom,
! [F2: nat > nat,Y2: nat,F: filter_nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ Y2 ) @ F )
= ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
= Y2 )
@ F ) ) ).
% tendsto_discrete
thf(fact_119_tendsto__imp__eventually__ne,axiom,
! [F2: nat > nat,C2: nat,F: filter_nat,C3: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ C2 ) @ F )
=> ( ( C2 != C3 )
=> ( eventually_nat
@ ^ [Z2: nat] :
( ( F2 @ Z2 )
!= C3 )
@ F ) ) ) ).
% tendsto_imp_eventually_ne
thf(fact_120_tendsto__imp__eventually__ne,axiom,
! [F2: real > real,C2: real,F: filter_real,C3: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ( ( C2 != C3 )
=> ( eventually_real
@ ^ [Z2: real] :
( ( F2 @ Z2 )
!= C3 )
@ F ) ) ) ).
% tendsto_imp_eventually_ne
thf(fact_121_tendsto__imp__eventually__ne,axiom,
! [F2: nat > real,C2: real,F: filter_nat,C3: real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ( ( C2 != C3 )
=> ( eventually_nat
@ ^ [Z2: nat] :
( ( F2 @ Z2 )
!= C3 )
@ F ) ) ) ).
% tendsto_imp_eventually_ne
thf(fact_122_tendsto__eventually,axiom,
! [F2: nat > nat,L: nat,Net: filter_nat] :
( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
= L )
@ Net )
=> ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ Net ) ) ).
% tendsto_eventually
thf(fact_123_tendsto__eventually,axiom,
! [F2: real > real,L: real,Net: filter_real] :
( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
= L )
@ Net )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ Net ) ) ).
% tendsto_eventually
thf(fact_124_tendsto__eventually,axiom,
! [F2: nat > real,L: real,Net: filter_nat] :
( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
= L )
@ Net )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ Net ) ) ).
% tendsto_eventually
thf(fact_125_LIM__cong,axiom,
! [A2: nat,B: nat,F2: nat > nat,G: nat > nat,L: nat,M: nat] :
( ( A2 = B )
=> ( ! [X4: nat] :
( ( X4 != B )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) )
=> ( ( L = M )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
= ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ M ) @ ( topolo4659099751122792720in_nat @ B @ top_top_set_nat ) ) ) ) ) ) ).
% LIM_cong
thf(fact_126_LIM__cong,axiom,
! [A2: nat,B: nat,F2: nat > real,G: nat > real,L: real,M: real] :
( ( A2 = B )
=> ( ! [X4: nat] :
( ( X4 != B )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) )
=> ( ( L = M )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
= ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ M ) @ ( topolo4659099751122792720in_nat @ B @ top_top_set_nat ) ) ) ) ) ) ).
% LIM_cong
thf(fact_127_LIM__cong,axiom,
! [A2: real,B: real,F2: real > real,G: real > real,L: real,M: real] :
( ( A2 = B )
=> ( ! [X4: real] :
( ( X4 != B )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) )
=> ( ( L = M )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
= ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ M ) @ ( topolo2177554685111907308n_real @ B @ top_top_set_real ) ) ) ) ) ) ).
% LIM_cong
thf(fact_128_LIM__equal,axiom,
! [A2: nat,F2: nat > nat,G: nat > nat,L: nat] :
( ! [X4: nat] :
( ( X4 != A2 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
= ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ).
% LIM_equal
thf(fact_129_LIM__equal,axiom,
! [A2: nat,F2: nat > real,G: nat > real,L: real] :
( ! [X4: nat] :
( ( X4 != A2 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
= ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ).
% LIM_equal
thf(fact_130_LIM__equal,axiom,
! [A2: real,F2: real > real,G: real > real,L: real] :
( ! [X4: real] :
( ( X4 != A2 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
= ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ).
% LIM_equal
thf(fact_131_LIM__unique,axiom,
! [F2: real > real,L2: real,A2: real,M2: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ M2 ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( L2 = M2 ) ) ) ).
% LIM_unique
thf(fact_132_tendsto__at__iff__tendsto__nhds,axiom,
! [G: nat > nat,L: nat] :
( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ ( G @ L ) ) @ ( topolo4659099751122792720in_nat @ L @ top_top_set_nat ) )
= ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ ( G @ L ) ) @ ( topolo8926549440605965083ds_nat @ L ) ) ) ).
% tendsto_at_iff_tendsto_nhds
thf(fact_133_tendsto__at__iff__tendsto__nhds,axiom,
! [G: nat > real,L: nat] :
( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ ( G @ L ) ) @ ( topolo4659099751122792720in_nat @ L @ top_top_set_nat ) )
= ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ ( G @ L ) ) @ ( topolo8926549440605965083ds_nat @ L ) ) ) ).
% tendsto_at_iff_tendsto_nhds
thf(fact_134_tendsto__at__iff__tendsto__nhds,axiom,
! [G: real > real,L: real] :
( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ ( G @ L ) ) @ ( topolo2177554685111907308n_real @ L @ top_top_set_real ) )
= ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ ( G @ L ) ) @ ( topolo2815343760600316023s_real @ L ) ) ) ).
% tendsto_at_iff_tendsto_nhds
thf(fact_135_LIM__const__eq,axiom,
! [K: real,L2: real,A2: real] :
( ( filterlim_real_real
@ ^ [X2: real] : K
@ ( topolo2815343760600316023s_real @ L2 )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( K = L2 ) ) ).
% LIM_const_eq
thf(fact_136_tendsto__compose,axiom,
! [G: nat > nat,L: nat,F2: nat > nat,F: filter_nat] :
( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ ( G @ L ) ) @ ( topolo4659099751122792720in_nat @ L @ top_top_set_nat ) )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
=> ( filterlim_nat_nat
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ ( topolo8926549440605965083ds_nat @ ( G @ L ) )
@ F ) ) ) ).
% tendsto_compose
thf(fact_137_tendsto__compose,axiom,
! [G: nat > real,L: nat,F2: real > nat,F: filter_real] :
( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ ( G @ L ) ) @ ( topolo4659099751122792720in_nat @ L @ top_top_set_nat ) )
=> ( ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( G @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( G @ L ) )
@ F ) ) ) ).
% tendsto_compose
thf(fact_138_tendsto__compose,axiom,
! [G: nat > real,L: nat,F2: nat > nat,F: filter_nat] :
( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ ( G @ L ) ) @ ( topolo4659099751122792720in_nat @ L @ top_top_set_nat ) )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( G @ L ) )
@ F ) ) ) ).
% tendsto_compose
thf(fact_139_tendsto__compose,axiom,
! [G: real > nat,L: real,F2: nat > real,F: filter_nat] :
( ( filterlim_real_nat @ G @ ( topolo8926549440605965083ds_nat @ ( G @ L ) ) @ ( topolo2177554685111907308n_real @ L @ top_top_set_real ) )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( filterlim_nat_nat
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ ( topolo8926549440605965083ds_nat @ ( G @ L ) )
@ F ) ) ) ).
% tendsto_compose
thf(fact_140_tendsto__compose,axiom,
! [G: real > real,L: real,F2: real > real,F: filter_real] :
( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ ( G @ L ) ) @ ( topolo2177554685111907308n_real @ L @ top_top_set_real ) )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( G @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( G @ L ) )
@ F ) ) ) ).
% tendsto_compose
thf(fact_141_tendsto__compose,axiom,
! [G: real > real,L: real,F2: nat > real,F: filter_nat] :
( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ ( G @ L ) ) @ ( topolo2177554685111907308n_real @ L @ top_top_set_real ) )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( G @ L ) )
@ F ) ) ) ).
% tendsto_compose
thf(fact_142_LIM__const__not__eq,axiom,
! [K: real,L2: real,A2: real] :
( ( K != L2 )
=> ~ ( filterlim_real_real
@ ^ [X2: real] : K
@ ( topolo2815343760600316023s_real @ L2 )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ).
% LIM_const_not_eq
thf(fact_143_filterlim__at,axiom,
! [F2: nat > nat,B: nat,S: set_nat,F: filter_nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo4659099751122792720in_nat @ B @ S ) @ F )
= ( ( eventually_nat
@ ^ [X2: nat] :
( ( member_nat @ ( F2 @ X2 ) @ S )
& ( ( F2 @ X2 )
!= B ) )
@ F )
& ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ B ) @ F ) ) ) ).
% filterlim_at
thf(fact_144_filterlim__at,axiom,
! [F2: real > real,B: real,S: set_real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2177554685111907308n_real @ B @ S ) @ F )
= ( ( eventually_real
@ ^ [X2: real] :
( ( member_real @ ( F2 @ X2 ) @ S )
& ( ( F2 @ X2 )
!= B ) )
@ F )
& ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ B ) @ F ) ) ) ).
% filterlim_at
thf(fact_145_filterlim__at,axiom,
! [F2: nat > real,B: real,S: set_real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2177554685111907308n_real @ B @ S ) @ F )
= ( ( eventually_nat
@ ^ [X2: nat] :
( ( member_real @ ( F2 @ X2 ) @ S )
& ( ( F2 @ X2 )
!= B ) )
@ F )
& ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ B ) @ F ) ) ) ).
% filterlim_at
thf(fact_146_Lim__transform__away__at,axiom,
! [A2: nat,B: nat,F2: nat > nat,G: nat > nat,L: nat] :
( ( A2 != B )
=> ( ! [X4: nat] :
( ( ( X4 != A2 )
& ( X4 != B ) )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
=> ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ).
% Lim_transform_away_at
thf(fact_147_Lim__transform__away__at,axiom,
! [A2: nat,B: nat,F2: nat > real,G: nat > real,L: real] :
( ( A2 != B )
=> ( ! [X4: nat] :
( ( ( X4 != A2 )
& ( X4 != B ) )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
=> ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ).
% Lim_transform_away_at
thf(fact_148_Lim__transform__away__at,axiom,
! [A2: real,B: real,F2: real > real,G: real > real,L: real] :
( ( A2 != B )
=> ( ! [X4: real] :
( ( ( X4 != A2 )
& ( X4 != B ) )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ).
% Lim_transform_away_at
thf(fact_149_Lim__at__imp__Lim__at__within,axiom,
! [F2: nat > nat,L: nat,X: nat,S2: set_nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ X @ top_top_set_nat ) )
=> ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ X @ S2 ) ) ) ).
% Lim_at_imp_Lim_at_within
thf(fact_150_Lim__at__imp__Lim__at__within,axiom,
! [F2: nat > real,L: real,X: nat,S2: set_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ X @ top_top_set_nat ) )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ X @ S2 ) ) ) ).
% Lim_at_imp_Lim_at_within
thf(fact_151_Lim__at__imp__Lim__at__within,axiom,
! [F2: real > real,L: real,X: real,S2: set_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ X @ S2 ) ) ) ).
% Lim_at_imp_Lim_at_within
thf(fact_152_tendsto__discrete__iff,axiom,
! [F2: nat > nat,C2: nat,F: filter_nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ C2 ) @ F )
= ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
= C2 )
@ F ) ) ).
% tendsto_discrete_iff
thf(fact_153_Lim__transform__eventually,axiom,
! [F2: nat > nat,L: nat,F: filter_nat,G: nat > nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ F )
=> ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ L ) @ F ) ) ) ).
% Lim_transform_eventually
thf(fact_154_Lim__transform__eventually,axiom,
! [F2: real > real,L: real,F: filter_real,G: real > real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ F )
=> ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ) ).
% Lim_transform_eventually
thf(fact_155_Lim__transform__eventually,axiom,
! [F2: nat > real,L: real,F: filter_nat,G: nat > real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ F )
=> ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ) ).
% Lim_transform_eventually
thf(fact_156_Lim__transform__away__within,axiom,
! [A2: nat,B: nat,S2: set_nat,F2: nat > nat,G: nat > nat,L: nat] :
( ( A2 != B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( ( ( X4 != A2 )
& ( X4 != B ) )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) )
=> ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) ) ) ) ) ).
% Lim_transform_away_within
thf(fact_157_Lim__transform__away__within,axiom,
! [A2: nat,B: nat,S2: set_nat,F2: nat > real,G: nat > real,L: real] :
( ( A2 != B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( ( ( X4 != A2 )
& ( X4 != B ) )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) )
=> ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) ) ) ) ) ).
% Lim_transform_away_within
thf(fact_158_Lim__transform__away__within,axiom,
! [A2: real,B: real,S2: set_real,F2: real > real,G: real > real,L: real] :
( ( A2 != B )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( ( X4 != A2 )
& ( X4 != B ) )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ S2 ) )
=> ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ S2 ) ) ) ) ) ).
% Lim_transform_away_within
thf(fact_159_Lim__cong__within,axiom,
! [A2: nat,B: nat,X: nat,Y2: nat,S2: set_nat,T: set_nat,F2: nat > nat,G: nat > nat] :
( ( A2 = B )
=> ( ( X = Y2 )
=> ( ( S2 = T )
=> ( ! [X4: nat] :
( ( X4 != B )
=> ( ( member_nat @ X4 @ T )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ X ) @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) )
= ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ Y2 ) @ ( topolo4659099751122792720in_nat @ B @ T ) ) ) ) ) ) ) ).
% Lim_cong_within
thf(fact_160_Lim__cong__within,axiom,
! [A2: nat,B: nat,X: real,Y2: real,S2: set_nat,T: set_nat,F2: nat > real,G: nat > real] :
( ( A2 = B )
=> ( ( X = Y2 )
=> ( ( S2 = T )
=> ( ! [X4: nat] :
( ( X4 != B )
=> ( ( member_nat @ X4 @ T )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ X ) @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) )
= ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ Y2 ) @ ( topolo4659099751122792720in_nat @ B @ T ) ) ) ) ) ) ) ).
% Lim_cong_within
thf(fact_161_Lim__cong__within,axiom,
! [A2: real,B: real,X: real,Y2: real,S2: set_real,T: set_real,F2: real > real,G: real > real] :
( ( A2 = B )
=> ( ( X = Y2 )
=> ( ( S2 = T )
=> ( ! [X4: real] :
( ( X4 != B )
=> ( ( member_real @ X4 @ T )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ X ) @ ( topolo2177554685111907308n_real @ A2 @ S2 ) )
= ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ Y2 ) @ ( topolo2177554685111907308n_real @ B @ T ) ) ) ) ) ) ) ).
% Lim_cong_within
thf(fact_162_LIM__Uniq,axiom,
! [F2: real > real,A2: real] :
( uniq_real
@ ^ [L3: real] : ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L3 ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ).
% LIM_Uniq
thf(fact_163_is__pole__def,axiom,
( comple320020743214073521t_real
= ( ^ [F4: nat > real,A3: nat] : ( filterlim_nat_real @ F4 @ at_infinity_real @ ( topolo4659099751122792720in_nat @ A3 @ top_top_set_nat ) ) ) ) ).
% is_pole_def
thf(fact_164_is__pole__def,axiom,
( comple7683793008646357389l_real
= ( ^ [F4: real > real,A3: real] : ( filterlim_real_real @ F4 @ at_infinity_real @ ( topolo2177554685111907308n_real @ A3 @ top_top_set_real ) ) ) ) ).
% is_pole_def
thf(fact_165_non__zero__neighbour__pole,axiom,
! [F2: nat > real,Z3: nat] :
( ( comple320020743214073521t_real @ F2 @ Z3 )
=> ( eventually_nat
@ ^ [W: nat] :
( ( F2 @ W )
!= zero_zero_real )
@ ( topolo4659099751122792720in_nat @ Z3 @ top_top_set_nat ) ) ) ).
% non_zero_neighbour_pole
thf(fact_166_non__zero__neighbour__pole,axiom,
! [F2: real > real,Z3: real] :
( ( comple7683793008646357389l_real @ F2 @ Z3 )
=> ( eventually_real
@ ^ [W: real] :
( ( F2 @ W )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) ).
% non_zero_neighbour_pole
thf(fact_167_tendsto__compose__at,axiom,
! [F2: real > nat,Y2: nat,F: filter_real,G: nat > nat,Z3: nat] :
( ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ Y2 ) @ F )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ Z3 ) @ ( topolo4659099751122792720in_nat @ Y2 @ top_top_set_nat ) )
=> ( ( eventually_real
@ ^ [W: real] :
( ( ( F2 @ W )
= Y2 )
=> ( ( G @ Y2 )
= Z3 ) )
@ F )
=> ( filterlim_real_nat @ ( comp_nat_nat_real @ G @ F2 ) @ ( topolo8926549440605965083ds_nat @ Z3 ) @ F ) ) ) ) ).
% tendsto_compose_at
thf(fact_168_tendsto__compose__at,axiom,
! [F2: nat > nat,Y2: nat,F: filter_nat,G: nat > nat,Z3: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ Y2 ) @ F )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ Z3 ) @ ( topolo4659099751122792720in_nat @ Y2 @ top_top_set_nat ) )
=> ( ( eventually_nat
@ ^ [W: nat] :
( ( ( F2 @ W )
= Y2 )
=> ( ( G @ Y2 )
= Z3 ) )
@ F )
=> ( filterlim_nat_nat @ ( comp_nat_nat_nat @ G @ F2 ) @ ( topolo8926549440605965083ds_nat @ Z3 ) @ F ) ) ) ) ).
% tendsto_compose_at
thf(fact_169_tendsto__compose__at,axiom,
! [F2: real > nat,Y2: nat,F: filter_real,G: nat > real,Z3: real] :
( ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ Y2 ) @ F )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ Z3 ) @ ( topolo4659099751122792720in_nat @ Y2 @ top_top_set_nat ) )
=> ( ( eventually_real
@ ^ [W: real] :
( ( ( F2 @ W )
= Y2 )
=> ( ( G @ Y2 )
= Z3 ) )
@ F )
=> ( filterlim_real_real @ ( comp_nat_real_real @ G @ F2 ) @ ( topolo2815343760600316023s_real @ Z3 ) @ F ) ) ) ) ).
% tendsto_compose_at
thf(fact_170_tendsto__compose__at,axiom,
! [F2: nat > nat,Y2: nat,F: filter_nat,G: nat > real,Z3: real] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ Y2 ) @ F )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ Z3 ) @ ( topolo4659099751122792720in_nat @ Y2 @ top_top_set_nat ) )
=> ( ( eventually_nat
@ ^ [W: nat] :
( ( ( F2 @ W )
= Y2 )
=> ( ( G @ Y2 )
= Z3 ) )
@ F )
=> ( filterlim_nat_real @ ( comp_nat_real_nat @ G @ F2 ) @ ( topolo2815343760600316023s_real @ Z3 ) @ F ) ) ) ) ).
% tendsto_compose_at
thf(fact_171_tendsto__compose__at,axiom,
! [F2: nat > real,Y2: real,F: filter_nat,G: real > nat,Z3: nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ Y2 ) @ F )
=> ( ( filterlim_real_nat @ G @ ( topolo8926549440605965083ds_nat @ Z3 ) @ ( topolo2177554685111907308n_real @ Y2 @ top_top_set_real ) )
=> ( ( eventually_nat
@ ^ [W: nat] :
( ( ( F2 @ W )
= Y2 )
=> ( ( G @ Y2 )
= Z3 ) )
@ F )
=> ( filterlim_nat_nat @ ( comp_real_nat_nat @ G @ F2 ) @ ( topolo8926549440605965083ds_nat @ Z3 ) @ F ) ) ) ) ).
% tendsto_compose_at
thf(fact_172_tendsto__compose__at,axiom,
! [F2: real > real,Y2: real,F: filter_real,G: real > real,Z3: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ Y2 ) @ F )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ Z3 ) @ ( topolo2177554685111907308n_real @ Y2 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [W: real] :
( ( ( F2 @ W )
= Y2 )
=> ( ( G @ Y2 )
= Z3 ) )
@ F )
=> ( filterlim_real_real @ ( comp_real_real_real @ G @ F2 ) @ ( topolo2815343760600316023s_real @ Z3 ) @ F ) ) ) ) ).
% tendsto_compose_at
thf(fact_173_tendsto__compose__at,axiom,
! [F2: nat > real,Y2: real,F: filter_nat,G: real > real,Z3: real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ Y2 ) @ F )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ Z3 ) @ ( topolo2177554685111907308n_real @ Y2 @ top_top_set_real ) )
=> ( ( eventually_nat
@ ^ [W: nat] :
( ( ( F2 @ W )
= Y2 )
=> ( ( G @ Y2 )
= Z3 ) )
@ F )
=> ( filterlim_nat_real @ ( comp_real_real_nat @ G @ F2 ) @ ( topolo2815343760600316023s_real @ Z3 ) @ F ) ) ) ) ).
% tendsto_compose_at
thf(fact_174_eventually__not__equal__at__infinity,axiom,
! [A2: real] :
( eventually_real
@ ^ [X2: real] : ( X2 != A2 )
@ at_infinity_real ) ).
% eventually_not_equal_at_infinity
thf(fact_175_filterlim__at__infinity__imp__eventually__ne,axiom,
! [F2: real > real,F: filter_real,C2: real] :
( ( filterlim_real_real @ F2 @ at_infinity_real @ F )
=> ( eventually_real
@ ^ [Z2: real] :
( ( F2 @ Z2 )
!= C2 )
@ F ) ) ).
% filterlim_at_infinity_imp_eventually_ne
thf(fact_176_filterlim__at__infinity__imp__eventually__ne,axiom,
! [F2: nat > real,F: filter_nat,C2: real] :
( ( filterlim_nat_real @ F2 @ at_infinity_real @ F )
=> ( eventually_nat
@ ^ [Z2: nat] :
( ( F2 @ Z2 )
!= C2 )
@ F ) ) ).
% filterlim_at_infinity_imp_eventually_ne
thf(fact_177_LIM__not__zero,axiom,
! [K: real,A2: real] :
( ( K != zero_zero_real )
=> ~ ( filterlim_real_real
@ ^ [X2: real] : K
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ).
% LIM_not_zero
thf(fact_178_Ints__0,axiom,
member_real @ zero_zero_real @ ring_1_Ints_real ).
% Ints_0
thf(fact_179_is__pole__tendsto,axiom,
! [F2: nat > real,X: nat] :
( ( comple320020743214073521t_real @ F2 @ X )
=> ( filterlim_nat_real @ ( comp_real_real_nat @ inverse_inverse_real @ F2 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo4659099751122792720in_nat @ X @ top_top_set_nat ) ) ) ).
% is_pole_tendsto
thf(fact_180_is__pole__tendsto,axiom,
! [F2: real > real,X: real] :
( ( comple7683793008646357389l_real @ F2 @ X )
=> ( filterlim_real_real @ ( comp_real_real_real @ inverse_inverse_real @ F2 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% is_pole_tendsto
thf(fact_181_lim__at__infinity__0,axiom,
! [F2: real > real,L: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ at_infinity_real )
= ( filterlim_real_real @ ( comp_real_real_real @ F2 @ inverse_inverse_real ) @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ).
% lim_at_infinity_0
thf(fact_182_filterlim__divide__at__infinity,axiom,
! [F2: real > real,C2: real,F: filter_real,G: real > real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ( ( filterlim_real_real @ G @ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) @ F )
=> ( ( C2 != zero_zero_real )
=> ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ at_infinity_real
@ F ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_183_LIM__isCont__iff,axiom,
! [F2: real > real,A2: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
= ( filterlim_real_real
@ ^ [H: real] : ( F2 @ ( plus_plus_real @ A2 @ H ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ).
% LIM_isCont_iff
thf(fact_184_add__right__cancel,axiom,
! [B: real,A2: real,C2: real] :
( ( ( plus_plus_real @ B @ A2 )
= ( plus_plus_real @ C2 @ A2 ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_185_add__right__cancel,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C2 @ A2 ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_186_add__left__cancel,axiom,
! [A2: real,B: real,C2: real] :
( ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ A2 @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_187_add__left__cancel,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_188_double__eq__0__iff,axiom,
! [A2: real] :
( ( ( plus_plus_real @ A2 @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_189_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_190_add__0,axiom,
! [A2: real] :
( ( plus_plus_real @ zero_zero_real @ A2 )
= A2 ) ).
% add_0
thf(fact_191_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y2 ) )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_192_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_193_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_194_add__cancel__right__right,axiom,
! [A2: real,B: real] :
( ( A2
= ( plus_plus_real @ A2 @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_195_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_196_add__cancel__right__left,axiom,
! [A2: real,B: real] :
( ( A2
= ( plus_plus_real @ B @ A2 ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_197_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_198_add__cancel__left__right,axiom,
! [A2: real,B: real] :
( ( ( plus_plus_real @ A2 @ B )
= A2 )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_199_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_200_add__cancel__left__left,axiom,
! [B: real,A2: real] :
( ( ( plus_plus_real @ B @ A2 )
= A2 )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_201_double__zero__sym,axiom,
! [A2: real] :
( ( zero_zero_real
= ( plus_plus_real @ A2 @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_202_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_203_add_Oright__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% add.right_neutral
thf(fact_204_Ints__add__iff1,axiom,
! [X: real,Y2: real] :
( ( member_real @ X @ ring_1_Ints_real )
=> ( ( member_real @ ( plus_plus_real @ X @ Y2 ) @ ring_1_Ints_real )
= ( member_real @ Y2 @ ring_1_Ints_real ) ) ) ).
% Ints_add_iff1
thf(fact_205_Ints__add__iff2,axiom,
! [Y2: real,X: real] :
( ( member_real @ Y2 @ ring_1_Ints_real )
=> ( ( member_real @ ( plus_plus_real @ X @ Y2 ) @ ring_1_Ints_real )
= ( member_real @ X @ ring_1_Ints_real ) ) ) ).
% Ints_add_iff2
thf(fact_206_add__right__imp__eq,axiom,
! [B: real,A2: real,C2: real] :
( ( ( plus_plus_real @ B @ A2 )
= ( plus_plus_real @ C2 @ A2 ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_207_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C2 @ A2 ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_208_add__left__imp__eq,axiom,
! [A2: real,B: real,C2: real] :
( ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ A2 @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_209_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_210_add_Oleft__commute,axiom,
! [B: real,A2: real,C2: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A2 @ C2 ) )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_211_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_212_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B2: real] : ( plus_plus_real @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_213_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_214_add_Oright__cancel,axiom,
! [B: real,A2: real,C2: real] :
( ( ( plus_plus_real @ B @ A2 )
= ( plus_plus_real @ C2 @ A2 ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_215_add_Oleft__cancel,axiom,
! [A2: real,B: real,C2: real] :
( ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ A2 @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_216_add_Oassoc,axiom,
! [A2: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A2 @ B ) @ C2 )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_217_add_Oassoc,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_218_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A2: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A2 @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_219_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A2: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_220_group__cancel_Oadd1,axiom,
! [A: real,K: real,A2: real,B: real] :
( ( A
= ( plus_plus_real @ K @ A2 ) )
=> ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_221_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_222_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: real,J: real,K: real,L: real] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I2 @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_223_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_224_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A2 @ B ) @ C2 )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_225_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_226_add_Ogroup__left__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ zero_zero_real @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_227_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_228_add_Ocomm__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% add.comm_neutral
thf(fact_229_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_230_comm__monoid__add__class_Oadd__0,axiom,
! [A2: real] :
( ( plus_plus_real @ zero_zero_real @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_231_Ints__add,axiom,
! [A2: real,B: real] :
( ( member_real @ A2 @ ring_1_Ints_real )
=> ( ( member_real @ B @ ring_1_Ints_real )
=> ( member_real @ ( plus_plus_real @ A2 @ B ) @ ring_1_Ints_real ) ) ) ).
% Ints_add
thf(fact_232_Ints__double__eq__0__iff,axiom,
! [A2: real] :
( ( member_real @ A2 @ ring_1_Ints_real )
=> ( ( ( plus_plus_real @ A2 @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_233_tendsto__add__const__iff,axiom,
! [C2: real,F2: real > real,D: real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X2: real] : ( plus_plus_real @ C2 @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( plus_plus_real @ C2 @ D ) )
@ F )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ D ) @ F ) ) ).
% tendsto_add_const_iff
thf(fact_234_tendsto__add__const__iff,axiom,
! [C2: real,F2: nat > real,D: real,F: filter_nat] :
( ( filterlim_nat_real
@ ^ [X2: nat] : ( plus_plus_real @ C2 @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( plus_plus_real @ C2 @ D ) )
@ F )
= ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ D ) @ F ) ) ).
% tendsto_add_const_iff
thf(fact_235_tendsto__add,axiom,
! [F2: nat > nat,A2: nat,F: filter_nat,G: nat > nat,B: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ A2 ) @ F )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ B ) @ F )
=> ( filterlim_nat_nat
@ ^ [X2: nat] : ( plus_plus_nat @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo8926549440605965083ds_nat @ ( plus_plus_nat @ A2 @ B ) )
@ F ) ) ) ).
% tendsto_add
thf(fact_236_tendsto__add,axiom,
! [F2: real > real,A2: real,F: filter_real,G: real > real,B: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ B ) @ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( plus_plus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( plus_plus_real @ A2 @ B ) )
@ F ) ) ) ).
% tendsto_add
thf(fact_237_tendsto__add,axiom,
! [F2: nat > real,A2: real,F: filter_nat,G: nat > real,B: real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ B ) @ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( plus_plus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( plus_plus_real @ A2 @ B ) )
@ F ) ) ) ).
% tendsto_add
thf(fact_238_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_239_tendsto__inverse,axiom,
! [F2: real > real,A2: real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( A2 != zero_zero_real )
=> ( filterlim_real_real
@ ^ [X2: real] : ( inverse_inverse_real @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( inverse_inverse_real @ A2 ) )
@ F ) ) ) ).
% tendsto_inverse
thf(fact_240_tendsto__inverse,axiom,
! [F2: nat > real,A2: real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( A2 != zero_zero_real )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( inverse_inverse_real @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( inverse_inverse_real @ A2 ) )
@ F ) ) ) ).
% tendsto_inverse
thf(fact_241_tendsto__divide,axiom,
! [F2: real > real,A2: real,F: filter_real,G: real > real,B: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ B ) @ F )
=> ( ( B != zero_zero_real )
=> ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( divide_divide_real @ A2 @ B ) )
@ F ) ) ) ) ).
% tendsto_divide
thf(fact_242_tendsto__divide,axiom,
! [F2: nat > real,A2: real,F: filter_nat,G: nat > real,B: real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ B ) @ F )
=> ( ( B != zero_zero_real )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( divide_divide_real @ A2 @ B ) )
@ F ) ) ) ) ).
% tendsto_divide
thf(fact_243_tendsto__divide__zero,axiom,
! [F2: real > real,F: filter_real,C2: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F2 @ X2 ) @ C2 )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ).
% tendsto_divide_zero
thf(fact_244_tendsto__divide__zero,axiom,
! [F2: nat > real,F: filter_nat,C2: real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( divide_divide_real @ ( F2 @ X2 ) @ C2 )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ).
% tendsto_divide_zero
thf(fact_245_tendsto__add__zero,axiom,
! [F2: nat > nat,F: filter_nat,G: nat > nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ zero_zero_nat ) @ F )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ zero_zero_nat ) @ F )
=> ( filterlim_nat_nat
@ ^ [X2: nat] : ( plus_plus_nat @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo8926549440605965083ds_nat @ zero_zero_nat )
@ F ) ) ) ).
% tendsto_add_zero
thf(fact_246_tendsto__add__zero,axiom,
! [F2: real > real,F: filter_real,G: real > real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( plus_plus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ) ).
% tendsto_add_zero
thf(fact_247_tendsto__add__zero,axiom,
! [F2: nat > real,F: filter_nat,G: nat > real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( plus_plus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ) ).
% tendsto_add_zero
thf(fact_248_tendsto__add__filterlim__at__infinity,axiom,
! [F2: real > real,C2: real,F: filter_real,G: real > real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ( ( filterlim_real_real @ G @ at_infinity_real @ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( plus_plus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ at_infinity_real
@ F ) ) ) ).
% tendsto_add_filterlim_at_infinity
thf(fact_249_tendsto__add__filterlim__at__infinity,axiom,
! [F2: nat > real,C2: real,F: filter_nat,G: nat > real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ( ( filterlim_nat_real @ G @ at_infinity_real @ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( plus_plus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ at_infinity_real
@ F ) ) ) ).
% tendsto_add_filterlim_at_infinity
thf(fact_250_tendsto__add__filterlim__at__infinity_H,axiom,
! [F2: real > real,F: filter_real,G: real > real,C2: real] :
( ( filterlim_real_real @ F2 @ at_infinity_real @ F )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( plus_plus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ at_infinity_real
@ F ) ) ) ).
% tendsto_add_filterlim_at_infinity'
thf(fact_251_tendsto__add__filterlim__at__infinity_H,axiom,
! [F2: nat > real,F: filter_nat,G: nat > real,C2: real] :
( ( filterlim_nat_real @ F2 @ at_infinity_real @ F )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( plus_plus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ at_infinity_real
@ F ) ) ) ).
% tendsto_add_filterlim_at_infinity'
thf(fact_252_tendsto__inverse__0,axiom,
filterlim_real_real @ inverse_inverse_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_infinity_real ).
% tendsto_inverse_0
thf(fact_253_filterlim__at__to__0,axiom,
! [F2: real > real,F: filter_real,A2: real] :
( ( filterlim_real_real @ F2 @ F @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
= ( filterlim_real_real
@ ^ [X2: real] : ( F2 @ ( plus_plus_real @ X2 @ A2 ) )
@ F
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ).
% filterlim_at_to_0
thf(fact_254_eventually__at__to__0,axiom,
! [P: real > $o,A2: real] :
( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
= ( eventually_real
@ ^ [X2: real] : ( P @ ( plus_plus_real @ X2 @ A2 ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ).
% eventually_at_to_0
thf(fact_255_tendsto__divide__0,axiom,
! [F2: real > real,C2: real,F: filter_real,G: real > real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ( ( filterlim_real_real @ G @ at_infinity_real @ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ) ).
% tendsto_divide_0
thf(fact_256_tendsto__divide__0,axiom,
! [F2: nat > real,C2: real,F: filter_nat,G: nat > real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ( ( filterlim_nat_real @ G @ at_infinity_real @ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ) ).
% tendsto_divide_0
thf(fact_257_filterlim__inverse__at__infinity,axiom,
filterlim_real_real @ inverse_inverse_real @ at_infinity_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ).
% filterlim_inverse_at_infinity
thf(fact_258_filterlim__inverse__at__iff,axiom,
! [G: real > real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X2: real] : ( inverse_inverse_real @ ( G @ X2 ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real )
@ F )
= ( filterlim_real_real @ G @ at_infinity_real @ F ) ) ).
% filterlim_inverse_at_iff
thf(fact_259_filterlim__inverse__at__iff,axiom,
! [G: nat > real,F: filter_nat] :
( ( filterlim_nat_real
@ ^ [X2: nat] : ( inverse_inverse_real @ ( G @ X2 ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real )
@ F )
= ( filterlim_nat_real @ G @ at_infinity_real @ F ) ) ).
% filterlim_inverse_at_iff
thf(fact_260_Lim__at__zero,axiom,
! [F2: real > real,L: real,A2: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
= ( filterlim_real_real
@ ^ [X2: real] : ( F2 @ ( plus_plus_real @ A2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ L )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ).
% Lim_at_zero
thf(fact_261_LIM__offset__zero__cancel,axiom,
! [F2: real > real,A2: real,L2: real] :
( ( filterlim_real_real
@ ^ [H: real] : ( F2 @ ( plus_plus_real @ A2 @ H ) )
@ ( topolo2815343760600316023s_real @ L2 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ).
% LIM_offset_zero_cancel
thf(fact_262_LIM__offset__zero,axiom,
! [F2: real > real,L2: real,A2: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [H: real] : ( F2 @ ( plus_plus_real @ A2 @ H ) )
@ ( topolo2815343760600316023s_real @ L2 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ).
% LIM_offset_zero
thf(fact_263_inverse__divide,axiom,
! [A2: real,B: real] :
( ( inverse_inverse_real @ ( divide_divide_real @ A2 @ B ) )
= ( divide_divide_real @ B @ A2 ) ) ).
% inverse_divide
thf(fact_264_inverse__nonzero__iff__nonzero,axiom,
! [A2: real] :
( ( ( inverse_inverse_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_265_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_266_division__ring__divide__zero,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_267_divide__cancel__right,axiom,
! [A2: real,C2: real,B: real] :
( ( ( divide_divide_real @ A2 @ C2 )
= ( divide_divide_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A2 = B ) ) ) ).
% divide_cancel_right
thf(fact_268_divide__cancel__left,axiom,
! [C2: real,A2: real,B: real] :
( ( ( divide_divide_real @ C2 @ A2 )
= ( divide_divide_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( A2 = B ) ) ) ).
% divide_cancel_left
thf(fact_269_div__by__0,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_270_divide__eq__0__iff,axiom,
! [A2: real,B: real] :
( ( ( divide_divide_real @ A2 @ B )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_271_inverse__eq__iff__eq,axiom,
! [A2: real,B: real] :
( ( ( inverse_inverse_real @ A2 )
= ( inverse_inverse_real @ B ) )
= ( A2 = B ) ) ).
% inverse_eq_iff_eq
thf(fact_272_inverse__inverse__eq,axiom,
! [A2: real] :
( ( inverse_inverse_real @ ( inverse_inverse_real @ A2 ) )
= A2 ) ).
% inverse_inverse_eq
thf(fact_273_div__0,axiom,
! [A2: real] :
( ( divide_divide_real @ zero_zero_real @ A2 )
= zero_zero_real ) ).
% div_0
thf(fact_274_inverse__eq__imp__eq,axiom,
! [A2: real,B: real] :
( ( ( inverse_inverse_real @ A2 )
= ( inverse_inverse_real @ B ) )
=> ( A2 = B ) ) ).
% inverse_eq_imp_eq
thf(fact_275_add__divide__distrib,axiom,
! [A2: real,B: real,C2: real] :
( ( divide_divide_real @ ( plus_plus_real @ A2 @ B ) @ C2 )
= ( plus_plus_real @ ( divide_divide_real @ A2 @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ).
% add_divide_distrib
thf(fact_276_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_277_inverse__zero__imp__zero,axiom,
! [A2: real] :
( ( ( inverse_inverse_real @ A2 )
= zero_zero_real )
=> ( A2 = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_278_nonzero__inverse__eq__imp__eq,axiom,
! [A2: real,B: real] :
( ( ( inverse_inverse_real @ A2 )
= ( inverse_inverse_real @ B ) )
=> ( ( A2 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( A2 = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_279_nonzero__inverse__inverse__eq,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( inverse_inverse_real @ ( inverse_inverse_real @ A2 ) )
= A2 ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_280_nonzero__imp__inverse__nonzero,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( inverse_inverse_real @ A2 )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_281_set__plus__intro,axiom,
! [A2: real,C: set_real,B: real,D2: set_real] :
( ( member_real @ A2 @ C )
=> ( ( member_real @ B @ D2 )
=> ( member_real @ ( plus_plus_real @ A2 @ B ) @ ( plus_plus_set_real @ C @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_282_set__plus__intro,axiom,
! [A2: nat,C: set_nat,B: nat,D2: set_nat] :
( ( member_nat @ A2 @ C )
=> ( ( member_nat @ B @ D2 )
=> ( member_nat @ ( plus_plus_nat @ A2 @ B ) @ ( plus_plus_set_nat @ C @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_283_lim__zero__infinity,axiom,
! [F2: real > real,L: real] :
( ( filterlim_real_real
@ ^ [X2: real] : ( F2 @ ( divide_divide_real @ one_one_real @ X2 ) )
@ ( topolo2815343760600316023s_real @ L )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ at_infinity_real ) ) ).
% lim_zero_infinity
thf(fact_284_Bfun__inverse,axiom,
! [F2: real > real,A2: real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( A2 != zero_zero_real )
=> ( bfun_real_real
@ ^ [X2: real] : ( inverse_inverse_real @ ( F2 @ X2 ) )
@ F ) ) ) ).
% Bfun_inverse
thf(fact_285_Bfun__inverse,axiom,
! [F2: nat > real,A2: real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( A2 != zero_zero_real )
=> ( bfun_nat_real
@ ^ [X2: nat] : ( inverse_inverse_real @ ( F2 @ X2 ) )
@ F ) ) ) ).
% Bfun_inverse
thf(fact_286_LIM__offset__zero__iff,axiom,
! [A2: real,F2: real > real,L2: real] :
( ( nO_MATCH_real_real @ zero_zero_real @ A2 )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
= ( filterlim_real_real
@ ^ [H: real] : ( F2 @ ( plus_plus_real @ A2 @ H ) )
@ ( topolo2815343760600316023s_real @ L2 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ) ).
% LIM_offset_zero_iff
thf(fact_287_fps__inverse__zero_H,axiom,
( ( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real )
=> ( ( invers68952373231134600s_real @ zero_z7760665558314615101s_real )
= zero_z7760665558314615101s_real ) ) ).
% fps_inverse_zero'
thf(fact_288_fps__div__by__zero_H,axiom,
! [G: formal3361831859752904756s_real] :
( ( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real )
=> ( ( divide1155267253282662278s_real @ G @ zero_z7760665558314615101s_real )
= zero_z7760665558314615101s_real ) ) ).
% fps_div_by_zero'
thf(fact_289_pth__7_I1_J,axiom,
! [X: real] :
( ( plus_plus_real @ zero_zero_real @ X )
= X ) ).
% pth_7(1)
thf(fact_290_eq__add__iff,axiom,
! [X: real,Y2: real] :
( ( X
= ( plus_plus_real @ X @ Y2 ) )
= ( Y2 = zero_zero_real ) ) ).
% eq_add_iff
thf(fact_291_div__by__1,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ one_one_real )
= A2 ) ).
% div_by_1
thf(fact_292_inverse__1,axiom,
( ( inverse_inverse_real @ one_one_real )
= one_one_real ) ).
% inverse_1
thf(fact_293_inverse__eq__1__iff,axiom,
! [X: real] :
( ( ( inverse_inverse_real @ X )
= one_one_real )
= ( X = one_one_real ) ) ).
% inverse_eq_1_iff
thf(fact_294_divide__eq__1__iff,axiom,
! [A2: real,B: real] :
( ( ( divide_divide_real @ A2 @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A2 = B ) ) ) ).
% divide_eq_1_iff
thf(fact_295_div__self,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= one_one_real ) ) ).
% div_self
thf(fact_296_one__eq__divide__iff,axiom,
! [A2: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A2 @ B ) )
= ( ( B != zero_zero_real )
& ( A2 = B ) ) ) ).
% one_eq_divide_iff
thf(fact_297_divide__self,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= one_one_real ) ) ).
% divide_self
thf(fact_298_divide__self__if,axiom,
! [A2: real] :
( ( ( A2 = zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= zero_zero_real ) )
& ( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_299_divide__eq__eq__1,axiom,
! [B: real,A2: real] :
( ( ( divide_divide_real @ B @ A2 )
= one_one_real )
= ( ( A2 != zero_zero_real )
& ( A2 = B ) ) ) ).
% divide_eq_eq_1
thf(fact_300_eq__divide__eq__1,axiom,
! [B: real,A2: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A2 ) )
= ( ( A2 != zero_zero_real )
& ( A2 = B ) ) ) ).
% eq_divide_eq_1
thf(fact_301_one__divide__eq__0__iff,axiom,
! [A2: real] :
( ( ( divide_divide_real @ one_one_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_302_zero__eq__1__divide__iff,axiom,
! [A2: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_303_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_304_fps__divide__1_H,axiom,
! [A2: formal3361831859752904756s_real] :
( ( ( inverse_inverse_real @ one_one_real )
= one_one_real )
=> ( ( divide1155267253282662278s_real @ A2 @ one_on8598947968683843321s_real )
= A2 ) ) ).
% fps_divide_1'
thf(fact_305_fps__inverse__one_H,axiom,
( ( ( inverse_inverse_real @ one_one_real )
= one_one_real )
=> ( ( invers68952373231134600s_real @ one_on8598947968683843321s_real )
= one_on8598947968683843321s_real ) ) ).
% fps_inverse_one'
thf(fact_306_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_307_Ints__1,axiom,
member_real @ one_one_real @ ring_1_Ints_real ).
% Ints_1
thf(fact_308_right__inverse__eq,axiom,
! [B: real,A2: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A2 @ B )
= one_one_real )
= ( A2 = B ) ) ) ).
% right_inverse_eq
thf(fact_309_inverse__eq__divide,axiom,
( inverse_inverse_real
= ( divide_divide_real @ one_one_real ) ) ).
% inverse_eq_divide
thf(fact_310_set__plus__elim,axiom,
! [X: real,A: set_real,B3: set_real] :
( ( member_real @ X @ ( plus_plus_set_real @ A @ B3 ) )
=> ~ ! [A4: real,B4: real] :
( ( X
= ( plus_plus_real @ A4 @ B4 ) )
=> ( ( member_real @ A4 @ A )
=> ~ ( member_real @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_311_set__plus__elim,axiom,
! [X: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ X @ ( plus_plus_set_nat @ A @ B3 ) )
=> ~ ! [A4: nat,B4: nat] :
( ( X
= ( plus_plus_nat @ A4 @ B4 ) )
=> ( ( member_nat @ A4 @ A )
=> ~ ( member_nat @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_312_nonzero__inverse__eq__divide,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( inverse_inverse_real @ A2 )
= ( divide_divide_real @ one_one_real @ A2 ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_313_Ints__odd__nonzero,axiom,
! [A2: real] :
( ( member_real @ A2 @ ring_1_Ints_real )
=> ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A2 ) @ A2 )
!= zero_zero_real ) ) ).
% Ints_odd_nonzero
thf(fact_314_pth__d,axiom,
! [X: real] :
( ( plus_plus_real @ X @ zero_zero_real )
= X ) ).
% pth_d
thf(fact_315_div__add__self1,axiom,
! [B: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_316_div__add__self2,axiom,
! [B: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_317_fps__tan__0,axiom,
( ( formal3683295897622742886n_real @ zero_zero_real )
= zero_z7760665558314615101s_real ) ).
% fps_tan_0
thf(fact_318_tendsto__offset__zero__iff,axiom,
! [A2: real,S2: set_real,F2: real > real,L2: real] :
( ( nO_MATCH_real_real @ zero_zero_real @ A2 )
=> ( ( member_real @ A2 @ S2 )
=> ( ( topolo4860482606490270245n_real @ S2 )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ A2 @ S2 ) )
= ( filterlim_real_real
@ ^ [H: real] : ( F2 @ ( plus_plus_real @ A2 @ H ) )
@ ( topolo2815343760600316023s_real @ L2 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ) ) ) ).
% tendsto_offset_zero_iff
thf(fact_319_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_320_verit__sum__simplify,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% verit_sum_simplify
thf(fact_321_add__0__iff,axiom,
! [B: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ B @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_322_add__0__iff,axiom,
! [B: real,A2: real] :
( ( B
= ( plus_plus_real @ B @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% add_0_iff
thf(fact_323_open__UNIV,axiom,
topolo4860482606490270245n_real @ top_top_set_real ).
% open_UNIV
thf(fact_324_separation__t1,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
= ( ? [U: set_real] :
( ( topolo4860482606490270245n_real @ U )
& ( member_real @ X @ U )
& ~ ( member_real @ Y2 @ U ) ) ) ) ).
% separation_t1
thf(fact_325_separation__t0,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
= ( ? [U: set_real] :
( ( topolo4860482606490270245n_real @ U )
& ( ( member_real @ X @ U )
!= ( member_real @ Y2 @ U ) ) ) ) ) ).
% separation_t0
thf(fact_326_t1__space,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ? [U2: set_real] :
( ( topolo4860482606490270245n_real @ U2 )
& ( member_real @ X @ U2 )
& ~ ( member_real @ Y2 @ U2 ) ) ) ).
% t1_space
thf(fact_327_t0__space,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ? [U2: set_real] :
( ( topolo4860482606490270245n_real @ U2 )
& ( ( member_real @ X @ U2 )
!= ( member_real @ Y2 @ U2 ) ) ) ) ).
% t0_space
thf(fact_328_at__within__open,axiom,
! [A2: real,S2: set_real] :
( ( member_real @ A2 @ S2 )
=> ( ( topolo4860482606490270245n_real @ S2 )
=> ( ( topolo2177554685111907308n_real @ A2 @ S2 )
= ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ).
% at_within_open
thf(fact_329_at__within__open__NO__MATCH,axiom,
! [A2: real,S: set_real] :
( ( member_real @ A2 @ S )
=> ( ( topolo4860482606490270245n_real @ S )
=> ( ( nO_MAT2855227906214470577t_real @ top_top_set_real @ S )
=> ( ( topolo2177554685111907308n_real @ A2 @ S )
= ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ).
% at_within_open_NO_MATCH
thf(fact_330_eventually__at__topological,axiom,
! [P: nat > $o,A2: nat,S: set_nat] :
( ( eventually_nat @ P @ ( topolo4659099751122792720in_nat @ A2 @ S ) )
= ( ? [S3: set_nat] :
( ( topolo4328251076210115529en_nat @ S3 )
& ( member_nat @ A2 @ S3 )
& ! [X2: nat] :
( ( member_nat @ X2 @ S3 )
=> ( ( X2 != A2 )
=> ( ( member_nat @ X2 @ S )
=> ( P @ X2 ) ) ) ) ) ) ) ).
% eventually_at_topological
thf(fact_331_eventually__at__topological,axiom,
! [P: real > $o,A2: real,S: set_real] :
( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A2 @ S ) )
= ( ? [S3: set_real] :
( ( topolo4860482606490270245n_real @ S3 )
& ( member_real @ A2 @ S3 )
& ! [X2: real] :
( ( member_real @ X2 @ S3 )
=> ( ( X2 != A2 )
=> ( ( member_real @ X2 @ S )
=> ( P @ X2 ) ) ) ) ) ) ) ).
% eventually_at_topological
thf(fact_332_eventually__nhds,axiom,
! [P: nat > $o,A2: nat] :
( ( eventually_nat @ P @ ( topolo8926549440605965083ds_nat @ A2 ) )
= ( ? [S3: set_nat] :
( ( topolo4328251076210115529en_nat @ S3 )
& ( member_nat @ A2 @ S3 )
& ! [X2: nat] :
( ( member_nat @ X2 @ S3 )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_nhds
thf(fact_333_eventually__nhds,axiom,
! [P: real > $o,A2: real] :
( ( eventually_real @ P @ ( topolo2815343760600316023s_real @ A2 ) )
= ( ? [S3: set_real] :
( ( topolo4860482606490270245n_real @ S3 )
& ( member_real @ A2 @ S3 )
& ! [X2: real] :
( ( member_real @ X2 @ S3 )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_nhds
thf(fact_334_eventually__nhds__in__open,axiom,
! [S: set_nat,X: nat] :
( ( topolo4328251076210115529en_nat @ S )
=> ( ( member_nat @ X @ S )
=> ( eventually_nat
@ ^ [Y3: nat] : ( member_nat @ Y3 @ S )
@ ( topolo8926549440605965083ds_nat @ X ) ) ) ) ).
% eventually_nhds_in_open
thf(fact_335_eventually__nhds__in__open,axiom,
! [S: set_real,X: real] :
( ( topolo4860482606490270245n_real @ S )
=> ( ( member_real @ X @ S )
=> ( eventually_real
@ ^ [Y3: real] : ( member_real @ Y3 @ S )
@ ( topolo2815343760600316023s_real @ X ) ) ) ) ).
% eventually_nhds_in_open
thf(fact_336_Lim__transform__within__open,axiom,
! [F2: nat > nat,L: nat,A2: nat,T: set_nat,S: set_nat,G: nat > nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ T ) )
=> ( ( topolo4328251076210115529en_nat @ S )
=> ( ( member_nat @ A2 @ S )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S )
=> ( ( X4 != A2 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ T ) ) ) ) ) ) ).
% Lim_transform_within_open
thf(fact_337_Lim__transform__within__open,axiom,
! [F2: nat > real,L: real,A2: nat,T: set_nat,S: set_nat,G: nat > real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ T ) )
=> ( ( topolo4328251076210115529en_nat @ S )
=> ( ( member_nat @ A2 @ S )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S )
=> ( ( X4 != A2 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ T ) ) ) ) ) ) ).
% Lim_transform_within_open
thf(fact_338_Lim__transform__within__open,axiom,
! [F2: real > real,L: real,A2: real,T: set_real,S: set_real,G: real > real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ T ) )
=> ( ( topolo4860482606490270245n_real @ S )
=> ( ( member_real @ A2 @ S )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ( X4 != A2 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ T ) ) ) ) ) ) ).
% Lim_transform_within_open
thf(fact_339_tendsto__def,axiom,
! [F2: nat > nat,L: nat,F: filter_nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
= ( ! [S3: set_nat] :
( ( topolo4328251076210115529en_nat @ S3 )
=> ( ( member_nat @ L @ S3 )
=> ( eventually_nat
@ ^ [X2: nat] : ( member_nat @ ( F2 @ X2 ) @ S3 )
@ F ) ) ) ) ) ).
% tendsto_def
thf(fact_340_tendsto__def,axiom,
! [F2: real > real,L: real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
= ( ! [S3: set_real] :
( ( topolo4860482606490270245n_real @ S3 )
=> ( ( member_real @ L @ S3 )
=> ( eventually_real
@ ^ [X2: real] : ( member_real @ ( F2 @ X2 ) @ S3 )
@ F ) ) ) ) ) ).
% tendsto_def
thf(fact_341_tendsto__def,axiom,
! [F2: nat > real,L: real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
= ( ! [S3: set_real] :
( ( topolo4860482606490270245n_real @ S3 )
=> ( ( member_real @ L @ S3 )
=> ( eventually_nat
@ ^ [X2: nat] : ( member_real @ ( F2 @ X2 ) @ S3 )
@ F ) ) ) ) ) ).
% tendsto_def
thf(fact_342_topological__tendstoD,axiom,
! [F2: nat > nat,L: nat,F: filter_nat,S2: set_nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
=> ( ( topolo4328251076210115529en_nat @ S2 )
=> ( ( member_nat @ L @ S2 )
=> ( eventually_nat
@ ^ [X2: nat] : ( member_nat @ ( F2 @ X2 ) @ S2 )
@ F ) ) ) ) ).
% topological_tendstoD
thf(fact_343_topological__tendstoD,axiom,
! [F2: real > real,L: real,F: filter_real,S2: set_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( ( topolo4860482606490270245n_real @ S2 )
=> ( ( member_real @ L @ S2 )
=> ( eventually_real
@ ^ [X2: real] : ( member_real @ ( F2 @ X2 ) @ S2 )
@ F ) ) ) ) ).
% topological_tendstoD
thf(fact_344_topological__tendstoD,axiom,
! [F2: nat > real,L: real,F: filter_nat,S2: set_real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( ( topolo4860482606490270245n_real @ S2 )
=> ( ( member_real @ L @ S2 )
=> ( eventually_nat
@ ^ [X2: nat] : ( member_real @ ( F2 @ X2 ) @ S2 )
@ F ) ) ) ) ).
% topological_tendstoD
thf(fact_345_topological__tendstoI,axiom,
! [L: nat,F2: nat > nat,F: filter_nat] :
( ! [S4: set_nat] :
( ( topolo4328251076210115529en_nat @ S4 )
=> ( ( member_nat @ L @ S4 )
=> ( eventually_nat
@ ^ [X2: nat] : ( member_nat @ ( F2 @ X2 ) @ S4 )
@ F ) ) )
=> ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F ) ) ).
% topological_tendstoI
thf(fact_346_topological__tendstoI,axiom,
! [L: real,F2: real > real,F: filter_real] :
( ! [S4: set_real] :
( ( topolo4860482606490270245n_real @ S4 )
=> ( ( member_real @ L @ S4 )
=> ( eventually_real
@ ^ [X2: real] : ( member_real @ ( F2 @ X2 ) @ S4 )
@ F ) ) )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ).
% topological_tendstoI
thf(fact_347_topological__tendstoI,axiom,
! [L: real,F2: nat > real,F: filter_nat] :
( ! [S4: set_real] :
( ( topolo4860482606490270245n_real @ S4 )
=> ( ( member_real @ L @ S4 )
=> ( eventually_nat
@ ^ [X2: nat] : ( member_real @ ( F2 @ X2 ) @ S4 )
@ F ) ) )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ).
% topological_tendstoI
thf(fact_348_tendsto__within__open,axiom,
! [A2: nat,S2: set_nat,F2: nat > nat,L: nat] :
( ( member_nat @ A2 @ S2 )
=> ( ( topolo4328251076210115529en_nat @ S2 )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) )
= ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ).
% tendsto_within_open
thf(fact_349_tendsto__within__open,axiom,
! [A2: nat,S2: set_nat,F2: nat > real,L: real] :
( ( member_nat @ A2 @ S2 )
=> ( ( topolo4328251076210115529en_nat @ S2 )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) )
= ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ).
% tendsto_within_open
thf(fact_350_tendsto__within__open,axiom,
! [A2: real,S2: set_real,F2: real > real,L: real] :
( ( member_real @ A2 @ S2 )
=> ( ( topolo4860482606490270245n_real @ S2 )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ S2 ) )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ).
% tendsto_within_open
thf(fact_351_tendsto__within__open__NO__MATCH,axiom,
! [A2: nat,S2: set_nat,F2: nat > nat,L: nat] :
( ( member_nat @ A2 @ S2 )
=> ( ( nO_MAT504328087405689813et_nat @ top_top_set_real @ S2 )
=> ( ( topolo4328251076210115529en_nat @ S2 )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) )
= ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ) ).
% tendsto_within_open_NO_MATCH
thf(fact_352_tendsto__within__open__NO__MATCH,axiom,
! [A2: nat,S2: set_nat,F2: nat > real,L: real] :
( ( member_nat @ A2 @ S2 )
=> ( ( nO_MAT504328087405689813et_nat @ top_top_set_real @ S2 )
=> ( ( topolo4328251076210115529en_nat @ S2 )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) )
= ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ) ).
% tendsto_within_open_NO_MATCH
thf(fact_353_tendsto__within__open__NO__MATCH,axiom,
! [A2: real,S2: set_real,F2: real > real,L: real] :
( ( member_real @ A2 @ S2 )
=> ( ( nO_MAT2855227906214470577t_real @ top_top_set_real @ S2 )
=> ( ( topolo4860482606490270245n_real @ S2 )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ S2 ) )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ) ).
% tendsto_within_open_NO_MATCH
thf(fact_354_is__pole__divide,axiom,
! [Z3: nat,F2: nat > real,G: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ Z3 @ top_top_set_nat ) @ F2 )
=> ( ( filterlim_nat_real @ G @ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) @ ( topolo4659099751122792720in_nat @ Z3 @ top_top_set_nat ) )
=> ( ( ( F2 @ Z3 )
!= zero_zero_real )
=> ( comple320020743214073521t_real
@ ^ [Z2: nat] : ( divide_divide_real @ ( F2 @ Z2 ) @ ( G @ Z2 ) )
@ Z3 ) ) ) ) ).
% is_pole_divide
thf(fact_355_is__pole__divide,axiom,
! [Z3: real,F2: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F2 )
=> ( ( filterlim_real_real @ G @ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) )
=> ( ( ( F2 @ Z3 )
!= zero_zero_real )
=> ( comple7683793008646357389l_real
@ ^ [Z2: real] : ( divide_divide_real @ ( F2 @ Z2 ) @ ( G @ Z2 ) )
@ Z3 ) ) ) ) ).
% is_pole_divide
thf(fact_356_isCont__iff,axiom,
! [X: real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F2 )
= ( filterlim_real_real
@ ^ [H: real] : ( F2 @ ( plus_plus_real @ X @ H ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ X ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ).
% isCont_iff
thf(fact_357_tendsto__divide__smallo,axiom,
! [F14: real > real,G1: real > real,A2: real,F: filter_real,F25: real > real,G22: real > real] :
( ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F14 @ X2 ) @ ( G1 @ X2 ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F )
=> ( ( member_real_real @ F25 @ ( landau3007391416991288786l_real @ F @ F14 ) )
=> ( ( member_real_real @ G22 @ ( landau3007391416991288786l_real @ F @ G1 ) )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( G1 @ X2 )
!= zero_zero_real )
@ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( plus_plus_real @ ( F14 @ X2 ) @ ( F25 @ X2 ) ) @ ( plus_plus_real @ ( G1 @ X2 ) @ ( G22 @ X2 ) ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F ) ) ) ) ) ).
% tendsto_divide_smallo
thf(fact_358_tendsto__divide__smallo,axiom,
! [F14: nat > real,G1: nat > real,A2: real,F: filter_nat,F25: nat > real,G22: nat > real] :
( ( filterlim_nat_real
@ ^ [X2: nat] : ( divide_divide_real @ ( F14 @ X2 ) @ ( G1 @ X2 ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F )
=> ( ( member_nat_real @ F25 @ ( landau997807338407142774t_real @ F @ F14 ) )
=> ( ( member_nat_real @ G22 @ ( landau997807338407142774t_real @ F @ G1 ) )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( G1 @ X2 )
!= zero_zero_real )
@ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( divide_divide_real @ ( plus_plus_real @ ( F14 @ X2 ) @ ( F25 @ X2 ) ) @ ( plus_plus_real @ ( G1 @ X2 ) @ ( G22 @ X2 ) ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F ) ) ) ) ) ).
% tendsto_divide_smallo
thf(fact_359_LIM__offset,axiom,
! [F2: real > real,L2: real,A2: real,K: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X2: real] : ( F2 @ ( plus_plus_real @ X2 @ K ) )
@ ( topolo2815343760600316023s_real @ L2 )
@ ( topolo2177554685111907308n_real @ ( minus_minus_real @ A2 @ K ) @ top_top_set_real ) ) ) ).
% LIM_offset
thf(fact_360_DERIV__LIM__iff,axiom,
! [F2: real > real,A2: real,D2: real] :
( ( filterlim_real_real
@ ^ [H: real] : ( divide_divide_real @ ( minus_minus_real @ ( F2 @ ( plus_plus_real @ A2 @ H ) ) @ ( F2 @ A2 ) ) @ H )
@ ( topolo2815343760600316023s_real @ D2 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) )
= ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( minus_minus_real @ ( F2 @ X2 ) @ ( F2 @ A2 ) ) @ ( minus_minus_real @ X2 @ A2 ) )
@ ( topolo2815343760600316023s_real @ D2 )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ).
% DERIV_LIM_iff
thf(fact_361_cot__pfd__real__tendsto__0,axiom,
filterlim_real_real @ cotang1502006655779026648d_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ).
% cot_pfd_real_tendsto_0
thf(fact_362_filterlim__shift,axiom,
! [F2: real > real,F: filter_real,A2: real,D: real] :
( ( filterlim_real_real @ F2 @ F @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( filterlim_real_real @ ( comp_real_real_real @ F2 @ ( plus_plus_real @ D ) ) @ F @ ( topolo2177554685111907308n_real @ ( minus_minus_real @ A2 @ D ) @ top_top_set_real ) ) ) ).
% filterlim_shift
thf(fact_363_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_364_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_365_diff__zero,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_zero
thf(fact_366_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_367_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_368_diff__0__right,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_0_right
thf(fact_369_diff__self,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% diff_self
thf(fact_370_add__diff__cancel__right_H,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_371_add__diff__cancel__right_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_372_add__diff__cancel__right,axiom,
! [A2: real,C2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( minus_minus_real @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_373_add__diff__cancel__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_374_add__diff__cancel__left_H,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_375_add__diff__cancel__left_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_376_add__diff__cancel__left,axiom,
! [C2: real,A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C2 @ A2 ) @ ( plus_plus_real @ C2 @ B ) )
= ( minus_minus_real @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_377_add__diff__cancel__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_378_diff__add__cancel,axiom,
! [A2: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ B )
= A2 ) ).
% diff_add_cancel
thf(fact_379_add__diff__cancel,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel
thf(fact_380_diff__add__zero,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_381_in__smallo__zero__iff,axiom,
! [F2: real > real,F: filter_real] :
( ( member_real_real @ F2
@ ( landau3007391416991288786l_real @ F
@ ^ [Uu: real] : zero_zero_real ) )
= ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
= zero_zero_real )
@ F ) ) ).
% in_smallo_zero_iff
thf(fact_382_in__smallo__zero__iff,axiom,
! [F2: nat > real,F: filter_nat] :
( ( member_nat_real @ F2
@ ( landau997807338407142774t_real @ F
@ ^ [Uu: nat] : zero_zero_real ) )
= ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
= zero_zero_real )
@ F ) ) ).
% in_smallo_zero_iff
thf(fact_383_continuous__diff,axiom,
! [F: filter_real,F2: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ F @ F2 )
=> ( ( topolo4422821103128117721l_real @ F @ G )
=> ( topolo4422821103128117721l_real @ F
@ ^ [X2: real] : ( minus_minus_real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% continuous_diff
thf(fact_384_continuous__const,axiom,
! [F: filter_real,C2: real] :
( topolo4422821103128117721l_real @ F
@ ^ [X2: real] : C2 ) ).
% continuous_const
thf(fact_385_diff__left__imp__eq,axiom,
! [A2: real,B: real,C2: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ A2 @ C2 ) )
=> ( B = C2 ) ) ).
% diff_left_imp_eq
thf(fact_386_diff__eq__diff__eq,axiom,
! [A2: real,B: real,C2: real,D: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( A2 = B )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_387_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: real,C2: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A2 @ C2 ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A2 @ B ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_388_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_389_Multiseries__Expansion_Oreal__eqI,axiom,
! [A2: real,B: real] :
( ( ( minus_minus_real @ A2 @ B )
= zero_zero_real )
=> ( A2 = B ) ) ).
% Multiseries_Expansion.real_eqI
thf(fact_390_cot__pfd__plus__1__real,axiom,
! [X: real] :
( ~ ( member_real @ X @ ring_1_Ints_real )
=> ( ( cotang1502006655779026648d_real @ ( plus_plus_real @ X @ one_one_real ) )
= ( plus_plus_real @ ( minus_minus_real @ ( cotang1502006655779026648d_real @ X ) @ ( divide_divide_real @ one_one_real @ ( plus_plus_real @ X @ one_one_real ) ) ) @ ( divide_divide_real @ one_one_real @ X ) ) ) ) ).
% cot_pfd_plus_1_real
thf(fact_391_isCont__diff,axiom,
! [A2: real,F2: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F2 )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ G )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
@ ^ [X2: real] : ( minus_minus_real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% isCont_diff
thf(fact_392_tendsto__diff__smallo,axiom,
! [F14: real > real,A2: real,F: filter_real,F25: real > real] :
( ( filterlim_real_real @ F14 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( member_real_real @ F25 @ ( landau3007391416991288786l_real @ F @ F14 ) )
=> ( filterlim_real_real
@ ^ [X2: real] : ( minus_minus_real @ ( F14 @ X2 ) @ ( F25 @ X2 ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F ) ) ) ).
% tendsto_diff_smallo
thf(fact_393_tendsto__diff__smallo,axiom,
! [F14: nat > real,A2: real,F: filter_nat,F25: nat > real] :
( ( filterlim_nat_real @ F14 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( member_nat_real @ F25 @ ( landau997807338407142774t_real @ F @ F14 ) )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( minus_minus_real @ ( F14 @ X2 ) @ ( F25 @ X2 ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F ) ) ) ).
% tendsto_diff_smallo
thf(fact_394_tendsto__diff__smallo__iff,axiom,
! [F25: real > real,F: filter_real,F14: real > real,A2: real] :
( ( member_real_real @ F25 @ ( landau3007391416991288786l_real @ F @ F14 ) )
=> ( ( filterlim_real_real @ F14 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
= ( filterlim_real_real
@ ^ [X2: real] : ( minus_minus_real @ ( F14 @ X2 ) @ ( F25 @ X2 ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F ) ) ) ).
% tendsto_diff_smallo_iff
thf(fact_395_tendsto__diff__smallo__iff,axiom,
! [F25: nat > real,F: filter_nat,F14: nat > real,A2: real] :
( ( member_nat_real @ F25 @ ( landau997807338407142774t_real @ F @ F14 ) )
=> ( ( filterlim_nat_real @ F14 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
= ( filterlim_nat_real
@ ^ [X2: nat] : ( minus_minus_real @ ( F14 @ X2 ) @ ( F25 @ X2 ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F ) ) ) ).
% tendsto_diff_smallo_iff
thf(fact_396_eq__iff__diff__eq__0,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [A3: real,B2: real] :
( ( minus_minus_real @ A3 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_397_diff__diff__eq,axiom,
! [A2: real,B: real,C2: real] :
( ( minus_minus_real @ ( minus_minus_real @ A2 @ B ) @ C2 )
= ( minus_minus_real @ A2 @ ( plus_plus_real @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_398_diff__diff__eq,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C2 )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_399_add__implies__diff,axiom,
! [C2: real,B: real,A2: real] :
( ( ( plus_plus_real @ C2 @ B )
= A2 )
=> ( C2
= ( minus_minus_real @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_400_add__implies__diff,axiom,
! [C2: nat,B: nat,A2: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A2 )
=> ( C2
= ( minus_minus_nat @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_401_diff__add__eq__diff__diff__swap,axiom,
! [A2: real,B: real,C2: real] :
( ( minus_minus_real @ A2 @ ( plus_plus_real @ B @ C2 ) )
= ( minus_minus_real @ ( minus_minus_real @ A2 @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_402_diff__add__eq,axiom,
! [A2: real,B: real,C2: real] :
( ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ C2 )
= ( minus_minus_real @ ( plus_plus_real @ A2 @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_403_diff__diff__eq2,axiom,
! [A2: real,B: real,C2: real] :
( ( minus_minus_real @ A2 @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( plus_plus_real @ A2 @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_404_add__diff__eq,axiom,
! [A2: real,B: real,C2: real] :
( ( plus_plus_real @ A2 @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_405_eq__diff__eq,axiom,
! [A2: real,C2: real,B: real] :
( ( A2
= ( minus_minus_real @ C2 @ B ) )
= ( ( plus_plus_real @ A2 @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_406_diff__eq__eq,axiom,
! [A2: real,B: real,C2: real] :
( ( ( minus_minus_real @ A2 @ B )
= C2 )
= ( A2
= ( plus_plus_real @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_407_group__cancel_Osub1,axiom,
! [A: real,K: real,A2: real,B: real] :
( ( A
= ( plus_plus_real @ K @ A2 ) )
=> ( ( minus_minus_real @ A @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A2 @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_408_diff__divide__distrib,axiom,
! [A2: real,B: real,C2: real] :
( ( divide_divide_real @ ( minus_minus_real @ A2 @ B ) @ C2 )
= ( minus_minus_real @ ( divide_divide_real @ A2 @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ).
% diff_divide_distrib
thf(fact_409_Ints__diff,axiom,
! [A2: real,B: real] :
( ( member_real @ A2 @ ring_1_Ints_real )
=> ( ( member_real @ B @ ring_1_Ints_real )
=> ( member_real @ ( minus_minus_real @ A2 @ B ) @ ring_1_Ints_real ) ) ) ).
% Ints_diff
thf(fact_410_continuous__add,axiom,
! [F: filter_real,F2: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ F @ F2 )
=> ( ( topolo4422821103128117721l_real @ F @ G )
=> ( topolo4422821103128117721l_real @ F
@ ^ [X2: real] : ( plus_plus_real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% continuous_add
thf(fact_411_continuous__ident,axiom,
! [X: real,S2: set_real] :
( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ S2 )
@ ^ [X2: real] : X2 ) ).
% continuous_ident
thf(fact_412_continuous__at__imp__continuous__at__within,axiom,
! [X: real,F2: real > real,S: set_real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F2 )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ S ) @ F2 ) ) ).
% continuous_at_imp_continuous_at_within
thf(fact_413_continuous__within__topological,axiom,
! [X: real,S: set_real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ S ) @ F2 )
= ( ! [B5: set_real] :
( ( topolo4860482606490270245n_real @ B5 )
=> ( ( member_real @ ( F2 @ X ) @ B5 )
=> ? [A5: set_real] :
( ( topolo4860482606490270245n_real @ A5 )
& ( member_real @ X @ A5 )
& ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ( member_real @ X2 @ A5 )
=> ( member_real @ ( F2 @ X2 ) @ B5 ) ) ) ) ) ) ) ) ).
% continuous_within_topological
thf(fact_414_landau__o_Osmall__refl__iff,axiom,
! [F2: real > real,F: filter_real] :
( ( member_real_real @ F2 @ ( landau3007391416991288786l_real @ F @ F2 ) )
= ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
= zero_zero_real )
@ F ) ) ).
% landau_o.small_refl_iff
thf(fact_415_landau__o_Osmall__refl__iff,axiom,
! [F2: nat > real,F: filter_nat] :
( ( member_nat_real @ F2 @ ( landau997807338407142774t_real @ F @ F2 ) )
= ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
= zero_zero_real )
@ F ) ) ).
% landau_o.small_refl_iff
thf(fact_416_landau__o_Osmall__asymmetric,axiom,
! [F2: real > real,F: filter_real,G: real > real] :
( ( member_real_real @ F2 @ ( landau3007391416991288786l_real @ F @ G ) )
=> ( ( member_real_real @ G @ ( landau3007391416991288786l_real @ F @ F2 ) )
=> ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
= zero_zero_real )
@ F ) ) ) ).
% landau_o.small_asymmetric
thf(fact_417_landau__o_Osmall__asymmetric,axiom,
! [F2: nat > real,F: filter_nat,G: nat > real] :
( ( member_nat_real @ F2 @ ( landau997807338407142774t_real @ F @ G ) )
=> ( ( member_nat_real @ G @ ( landau997807338407142774t_real @ F @ F2 ) )
=> ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
= zero_zero_real )
@ F ) ) ) ).
% landau_o.small_asymmetric
thf(fact_418_isCont__o2,axiom,
! [A2: real,F2: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F2 )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F2 @ A2 ) @ top_top_set_real ) @ G )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
@ ^ [X2: real] : ( G @ ( F2 @ X2 ) ) ) ) ) ).
% isCont_o2
thf(fact_419_continuous__within__compose3,axiom,
! [F2: real > real,X: real,G: real > real,S: set_real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F2 @ X ) @ top_top_set_real ) @ G )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ S ) @ F2 )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ S )
@ ^ [X2: real] : ( G @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_within_compose3
thf(fact_420_tendsto__diff,axiom,
! [F2: real > real,A2: real,F: filter_real,G: real > real,B: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ B ) @ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( minus_minus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( minus_minus_real @ A2 @ B ) )
@ F ) ) ) ).
% tendsto_diff
thf(fact_421_tendsto__diff,axiom,
! [F2: nat > real,A2: real,F: filter_nat,G: nat > real,B: real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ B ) @ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( minus_minus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( minus_minus_real @ A2 @ B ) )
@ F ) ) ) ).
% tendsto_diff
thf(fact_422_continuous__within__open,axiom,
! [A2: real,A: set_real,F2: real > real] :
( ( member_real @ A2 @ A )
=> ( ( topolo4860482606490270245n_real @ A )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ A ) @ F2 )
= ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F2 ) ) ) ) ).
% continuous_within_open
thf(fact_423_continuous__at__open,axiom,
! [X: real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F2 )
= ( ! [T2: set_real] :
( ( ( topolo4860482606490270245n_real @ T2 )
& ( member_real @ ( F2 @ X ) @ T2 ) )
=> ? [S5: set_real] :
( ( topolo4860482606490270245n_real @ S5 )
& ( member_real @ X @ S5 )
& ! [X2: real] :
( ( member_real @ X2 @ S5 )
=> ( member_real @ ( F2 @ X2 ) @ T2 ) ) ) ) ) ) ).
% continuous_at_open
thf(fact_424_continuous__at__compose,axiom,
! [A2: real,F2: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F2 )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F2 @ A2 ) @ top_top_set_real ) @ G )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ ( comp_real_real_real @ G @ F2 ) ) ) ) ).
% continuous_at_compose
thf(fact_425_continuous__within,axiom,
! [X: nat,S: set_nat,F2: nat > nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ X @ S ) @ F2 )
= ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ ( F2 @ X ) ) @ ( topolo4659099751122792720in_nat @ X @ S ) ) ) ).
% continuous_within
thf(fact_426_continuous__within,axiom,
! [X: nat,S: set_nat,F2: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ X @ S ) @ F2 )
= ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ ( F2 @ X ) ) @ ( topolo4659099751122792720in_nat @ X @ S ) ) ) ).
% continuous_within
thf(fact_427_continuous__within,axiom,
! [X: real,S: set_real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ S ) @ F2 )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ ( F2 @ X ) ) @ ( topolo2177554685111907308n_real @ X @ S ) ) ) ).
% continuous_within
thf(fact_428_landau__o_Osmall_Odivide__right__iff,axiom,
! [H2: real > real,F: filter_real,F2: real > real,G: real > real] :
( ( eventually_real
@ ^ [X2: real] :
( ( H2 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( member_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F2 @ X2 ) @ ( H2 @ X2 ) )
@ ( landau3007391416991288786l_real @ F
@ ^ [X2: real] : ( divide_divide_real @ ( G @ X2 ) @ ( H2 @ X2 ) ) ) )
= ( member_real_real @ F2 @ ( landau3007391416991288786l_real @ F @ G ) ) ) ) ).
% landau_o.small.divide_right_iff
thf(fact_429_landau__o_Osmall_Odivide__right__iff,axiom,
! [H2: nat > real,F: filter_nat,F2: nat > real,G: nat > real] :
( ( eventually_nat
@ ^ [X2: nat] :
( ( H2 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real
@ ^ [X2: nat] : ( divide_divide_real @ ( F2 @ X2 ) @ ( H2 @ X2 ) )
@ ( landau997807338407142774t_real @ F
@ ^ [X2: nat] : ( divide_divide_real @ ( G @ X2 ) @ ( H2 @ X2 ) ) ) )
= ( member_nat_real @ F2 @ ( landau997807338407142774t_real @ F @ G ) ) ) ) ).
% landau_o.small.divide_right_iff
thf(fact_430_landau__o_Osmall_Odivide__left__iff,axiom,
! [F2: real > real,F: filter_real,G: real > real,H2: real > real] :
( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( G @ X2 )
!= zero_zero_real )
@ F )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( H2 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( member_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( H2 @ X2 ) @ ( F2 @ X2 ) )
@ ( landau3007391416991288786l_real @ F
@ ^ [X2: real] : ( divide_divide_real @ ( H2 @ X2 ) @ ( G @ X2 ) ) ) )
= ( member_real_real @ G @ ( landau3007391416991288786l_real @ F @ F2 ) ) ) ) ) ) ).
% landau_o.small.divide_left_iff
thf(fact_431_landau__o_Osmall_Odivide__left__iff,axiom,
! [F2: nat > real,F: filter_nat,G: nat > real,H2: nat > real] :
( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( G @ X2 )
!= zero_zero_real )
@ F )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( H2 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real
@ ^ [X2: nat] : ( divide_divide_real @ ( H2 @ X2 ) @ ( F2 @ X2 ) )
@ ( landau997807338407142774t_real @ F
@ ^ [X2: nat] : ( divide_divide_real @ ( H2 @ X2 ) @ ( G @ X2 ) ) ) )
= ( member_nat_real @ G @ ( landau997807338407142774t_real @ F @ F2 ) ) ) ) ) ) ).
% landau_o.small.divide_left_iff
thf(fact_432_landau__o_Osmall_Odivide__right,axiom,
! [H2: real > real,F: filter_real,F2: real > real,G: real > real] :
( ( eventually_real
@ ^ [X2: real] :
( ( H2 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( member_real_real @ F2 @ ( landau3007391416991288786l_real @ F @ G ) )
=> ( member_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F2 @ X2 ) @ ( H2 @ X2 ) )
@ ( landau3007391416991288786l_real @ F
@ ^ [X2: real] : ( divide_divide_real @ ( G @ X2 ) @ ( H2 @ X2 ) ) ) ) ) ) ).
% landau_o.small.divide_right
thf(fact_433_landau__o_Osmall_Odivide__right,axiom,
! [H2: nat > real,F: filter_nat,F2: nat > real,G: nat > real] :
( ( eventually_nat
@ ^ [X2: nat] :
( ( H2 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real @ F2 @ ( landau997807338407142774t_real @ F @ G ) )
=> ( member_nat_real
@ ^ [X2: nat] : ( divide_divide_real @ ( F2 @ X2 ) @ ( H2 @ X2 ) )
@ ( landau997807338407142774t_real @ F
@ ^ [X2: nat] : ( divide_divide_real @ ( G @ X2 ) @ ( H2 @ X2 ) ) ) ) ) ) ).
% landau_o.small.divide_right
thf(fact_434_landau__o_Osmall_Odivide__left,axiom,
! [F2: real > real,F: filter_real,G: real > real,H2: real > real] :
( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( G @ X2 )
!= zero_zero_real )
@ F )
=> ( ( member_real_real @ G @ ( landau3007391416991288786l_real @ F @ F2 ) )
=> ( member_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( H2 @ X2 ) @ ( F2 @ X2 ) )
@ ( landau3007391416991288786l_real @ F
@ ^ [X2: real] : ( divide_divide_real @ ( H2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% landau_o.small.divide_left
thf(fact_435_landau__o_Osmall_Odivide__left,axiom,
! [F2: nat > real,F: filter_nat,G: nat > real,H2: nat > real] :
( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( G @ X2 )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real @ G @ ( landau997807338407142774t_real @ F @ F2 ) )
=> ( member_nat_real
@ ^ [X2: nat] : ( divide_divide_real @ ( H2 @ X2 ) @ ( F2 @ X2 ) )
@ ( landau997807338407142774t_real @ F
@ ^ [X2: nat] : ( divide_divide_real @ ( H2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% landau_o.small.divide_left
thf(fact_436_landau__o_Osmall_Odivide,axiom,
! [G1: real > real,F: filter_real,G22: real > real,F14: real > real,F25: real > real] :
( ( eventually_real
@ ^ [X2: real] :
( ( G1 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( G22 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( member_real_real @ F14 @ ( landau3007391416991288786l_real @ F @ F25 ) )
=> ( ( member_real_real @ G22 @ ( landau3007391416991288786l_real @ F @ G1 ) )
=> ( member_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F14 @ X2 ) @ ( G1 @ X2 ) )
@ ( landau3007391416991288786l_real @ F
@ ^ [X2: real] : ( divide_divide_real @ ( F25 @ X2 ) @ ( G22 @ X2 ) ) ) ) ) ) ) ) ).
% landau_o.small.divide
thf(fact_437_landau__o_Osmall_Odivide,axiom,
! [G1: nat > real,F: filter_nat,G22: nat > real,F14: nat > real,F25: nat > real] :
( ( eventually_nat
@ ^ [X2: nat] :
( ( G1 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( G22 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real @ F14 @ ( landau997807338407142774t_real @ F @ F25 ) )
=> ( ( member_nat_real @ G22 @ ( landau997807338407142774t_real @ F @ G1 ) )
=> ( member_nat_real
@ ^ [X2: nat] : ( divide_divide_real @ ( F14 @ X2 ) @ ( G1 @ X2 ) )
@ ( landau997807338407142774t_real @ F
@ ^ [X2: nat] : ( divide_divide_real @ ( F25 @ X2 ) @ ( G22 @ X2 ) ) ) ) ) ) ) ) ).
% landau_o.small.divide
thf(fact_438_continuous__at__within__divide,axiom,
! [A2: real,S: set_real,F2: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S ) @ F2 )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S ) @ G )
=> ( ( ( G @ A2 )
!= zero_zero_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S )
@ ^ [X2: real] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_at_within_divide
thf(fact_439_isCont__add,axiom,
! [A2: real,F2: real > nat,G: real > nat] :
( ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F2 )
=> ( ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ G )
=> ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
@ ^ [X2: real] : ( plus_plus_nat @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% isCont_add
thf(fact_440_isCont__add,axiom,
! [A2: real,F2: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F2 )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ G )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
@ ^ [X2: real] : ( plus_plus_real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% isCont_add
thf(fact_441_landau__o_Osmall_Oinverse__cancel,axiom,
! [F2: real > real,F: filter_real,G: real > real] :
( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( G @ X2 )
!= zero_zero_real )
@ F )
=> ( ( member_real_real
@ ^ [X2: real] : ( inverse_inverse_real @ ( F2 @ X2 ) )
@ ( landau3007391416991288786l_real @ F
@ ^ [X2: real] : ( inverse_inverse_real @ ( G @ X2 ) ) ) )
= ( member_real_real @ G @ ( landau3007391416991288786l_real @ F @ F2 ) ) ) ) ) ).
% landau_o.small.inverse_cancel
thf(fact_442_landau__o_Osmall_Oinverse__cancel,axiom,
! [F2: nat > real,F: filter_nat,G: nat > real] :
( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( G @ X2 )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real
@ ^ [X2: nat] : ( inverse_inverse_real @ ( F2 @ X2 ) )
@ ( landau997807338407142774t_real @ F
@ ^ [X2: nat] : ( inverse_inverse_real @ ( G @ X2 ) ) ) )
= ( member_nat_real @ G @ ( landau997807338407142774t_real @ F @ F2 ) ) ) ) ) ).
% landau_o.small.inverse_cancel
thf(fact_443_landau__o_Osmall_Oinverse__flip,axiom,
! [G: real > real,F: filter_real,H2: real > real] :
( ( eventually_real
@ ^ [X2: real] :
( ( G @ X2 )
!= zero_zero_real )
@ F )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( H2 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( member_real_real
@ ^ [X2: real] : ( inverse_inverse_real @ ( G @ X2 ) )
@ ( landau3007391416991288786l_real @ F @ H2 ) )
=> ( member_real_real
@ ^ [X2: real] : ( inverse_inverse_real @ ( H2 @ X2 ) )
@ ( landau3007391416991288786l_real @ F @ G ) ) ) ) ) ).
% landau_o.small.inverse_flip
thf(fact_444_landau__o_Osmall_Oinverse__flip,axiom,
! [G: nat > real,F: filter_nat,H2: nat > real] :
( ( eventually_nat
@ ^ [X2: nat] :
( ( G @ X2 )
!= zero_zero_real )
@ F )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( H2 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real
@ ^ [X2: nat] : ( inverse_inverse_real @ ( G @ X2 ) )
@ ( landau997807338407142774t_real @ F @ H2 ) )
=> ( member_nat_real
@ ^ [X2: nat] : ( inverse_inverse_real @ ( H2 @ X2 ) )
@ ( landau997807338407142774t_real @ F @ G ) ) ) ) ) ).
% landau_o.small.inverse_flip
thf(fact_445_landau__o_Osmall_Oinverse,axiom,
! [F2: real > real,F: filter_real,G: real > real] :
( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( G @ X2 )
!= zero_zero_real )
@ F )
=> ( ( member_real_real @ F2 @ ( landau3007391416991288786l_real @ F @ G ) )
=> ( member_real_real
@ ^ [X2: real] : ( inverse_inverse_real @ ( G @ X2 ) )
@ ( landau3007391416991288786l_real @ F
@ ^ [X2: real] : ( inverse_inverse_real @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% landau_o.small.inverse
thf(fact_446_landau__o_Osmall_Oinverse,axiom,
! [F2: nat > real,F: filter_nat,G: nat > real] :
( ( eventually_nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
!= zero_zero_real )
@ F )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( G @ X2 )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real @ F2 @ ( landau997807338407142774t_real @ F @ G ) )
=> ( member_nat_real
@ ^ [X2: nat] : ( inverse_inverse_real @ ( G @ X2 ) )
@ ( landau997807338407142774t_real @ F
@ ^ [X2: nat] : ( inverse_inverse_real @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% landau_o.small.inverse
thf(fact_447_tendsto__add__smallo,axiom,
! [F14: real > real,A2: real,F: filter_real,F25: real > real] :
( ( filterlim_real_real @ F14 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( member_real_real @ F25 @ ( landau3007391416991288786l_real @ F @ F14 ) )
=> ( filterlim_real_real
@ ^ [X2: real] : ( plus_plus_real @ ( F14 @ X2 ) @ ( F25 @ X2 ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F ) ) ) ).
% tendsto_add_smallo
thf(fact_448_tendsto__add__smallo,axiom,
! [F14: nat > real,A2: real,F: filter_nat,F25: nat > real] :
( ( filterlim_nat_real @ F14 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( member_nat_real @ F25 @ ( landau997807338407142774t_real @ F @ F14 ) )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( plus_plus_real @ ( F14 @ X2 ) @ ( F25 @ X2 ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F ) ) ) ).
% tendsto_add_smallo
thf(fact_449_tendsto__add__smallo__iff,axiom,
! [F25: real > real,F: filter_real,F14: real > real,A2: real] :
( ( member_real_real @ F25 @ ( landau3007391416991288786l_real @ F @ F14 ) )
=> ( ( filterlim_real_real @ F14 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
= ( filterlim_real_real
@ ^ [X2: real] : ( plus_plus_real @ ( F14 @ X2 ) @ ( F25 @ X2 ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F ) ) ) ).
% tendsto_add_smallo_iff
thf(fact_450_tendsto__add__smallo__iff,axiom,
! [F25: nat > real,F: filter_nat,F14: nat > real,A2: real] :
( ( member_nat_real @ F25 @ ( landau997807338407142774t_real @ F @ F14 ) )
=> ( ( filterlim_nat_real @ F14 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
= ( filterlim_nat_real
@ ^ [X2: nat] : ( plus_plus_real @ ( F14 @ X2 ) @ ( F25 @ X2 ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F ) ) ) ).
% tendsto_add_smallo_iff
thf(fact_451_continuous__at__within__inverse,axiom,
! [A2: real,S: set_real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S ) @ F2 )
=> ( ( ( F2 @ A2 )
!= zero_zero_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S )
@ ^ [X2: real] : ( inverse_inverse_real @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_at_within_inverse
thf(fact_452_continuous__within__tendsto__compose_H,axiom,
! [A2: nat,S: set_nat,F2: nat > nat,X: nat > nat,F: filter_nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ A2 @ S ) @ F2 )
=> ( ! [N2: nat] : ( member_nat @ ( X @ N2 ) @ S )
=> ( ( filterlim_nat_nat @ X @ ( topolo8926549440605965083ds_nat @ A2 ) @ F )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F2 @ ( X @ N ) )
@ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose'
thf(fact_453_continuous__within__tendsto__compose_H,axiom,
! [A2: nat,S: set_nat,F2: nat > real,X: nat > nat,F: filter_nat] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ S ) @ F2 )
=> ( ! [N2: nat] : ( member_nat @ ( X @ N2 ) @ S )
=> ( ( filterlim_nat_nat @ X @ ( topolo8926549440605965083ds_nat @ A2 ) @ F )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F2 @ ( X @ N ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose'
thf(fact_454_continuous__within__tendsto__compose_H,axiom,
! [A2: real,S: set_real,F2: real > nat,X: nat > real,F: filter_nat] :
( ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ A2 @ S ) @ F2 )
=> ( ! [N2: nat] : ( member_real @ ( X @ N2 ) @ S )
=> ( ( filterlim_nat_real @ X @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F2 @ ( X @ N ) )
@ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose'
thf(fact_455_continuous__within__tendsto__compose_H,axiom,
! [A2: real,S: set_real,F2: real > real,X: real > real,F: filter_real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S ) @ F2 )
=> ( ! [N2: real] : ( member_real @ ( X @ N2 ) @ S )
=> ( ( filterlim_real_real @ X @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( filterlim_real_real
@ ^ [N: real] : ( F2 @ ( X @ N ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose'
thf(fact_456_continuous__within__tendsto__compose_H,axiom,
! [A2: real,S: set_real,F2: real > real,X: nat > real,F: filter_nat] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S ) @ F2 )
=> ( ! [N2: nat] : ( member_real @ ( X @ N2 ) @ S )
=> ( ( filterlim_nat_real @ X @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F2 @ ( X @ N ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose'
thf(fact_457_Lim__null,axiom,
! [F2: real > real,L: real,Net: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ Net )
= ( filterlim_real_real
@ ^ [X2: real] : ( minus_minus_real @ ( F2 @ X2 ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ Net ) ) ).
% Lim_null
thf(fact_458_Lim__null,axiom,
! [F2: nat > real,L: real,Net: filter_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ Net )
= ( filterlim_nat_real
@ ^ [X2: nat] : ( minus_minus_real @ ( F2 @ X2 ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ Net ) ) ).
% Lim_null
thf(fact_459_LIM__zero,axiom,
! [F2: real > real,L: real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( minus_minus_real @ ( F2 @ X2 ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ).
% LIM_zero
thf(fact_460_LIM__zero,axiom,
! [F2: nat > real,L: real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( minus_minus_real @ ( F2 @ X2 ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ).
% LIM_zero
thf(fact_461_LIM__zero__iff,axiom,
! [F2: real > real,L: real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X2: real] : ( minus_minus_real @ ( F2 @ X2 ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ).
% LIM_zero_iff
thf(fact_462_LIM__zero__iff,axiom,
! [F2: nat > real,L: real,F: filter_nat] :
( ( filterlim_nat_real
@ ^ [X2: nat] : ( minus_minus_real @ ( F2 @ X2 ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
= ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ).
% LIM_zero_iff
thf(fact_463_Lim__transform,axiom,
! [G: real > real,A2: real,F: filter_real,F2: real > real] :
( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( filterlim_real_real
@ ^ [X2: real] : ( minus_minus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F ) ) ) ).
% Lim_transform
thf(fact_464_Lim__transform,axiom,
! [G: nat > real,A2: real,F: filter_nat,F2: nat > real] :
( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( filterlim_nat_real
@ ^ [X2: nat] : ( minus_minus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F ) ) ) ).
% Lim_transform
thf(fact_465_Lim__transform2,axiom,
! [F2: real > real,A2: real,F: filter_real,G: real > real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( filterlim_real_real
@ ^ [X2: real] : ( minus_minus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
=> ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ A2 ) @ F ) ) ) ).
% Lim_transform2
thf(fact_466_Lim__transform2,axiom,
! [F2: nat > real,A2: real,F: filter_nat,G: nat > real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( filterlim_nat_real
@ ^ [X2: nat] : ( minus_minus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
=> ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ A2 ) @ F ) ) ) ).
% Lim_transform2
thf(fact_467_LIM__zero__cancel,axiom,
! [F2: real > real,L: real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X2: real] : ( minus_minus_real @ ( F2 @ X2 ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ).
% LIM_zero_cancel
thf(fact_468_LIM__zero__cancel,axiom,
! [F2: nat > real,L: real,F: filter_nat] :
( ( filterlim_nat_real
@ ^ [X2: nat] : ( minus_minus_real @ ( F2 @ X2 ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ).
% LIM_zero_cancel
thf(fact_469_Lim__transform__eq,axiom,
! [F2: real > real,G: real > real,F: filter_real,A2: real] :
( ( filterlim_real_real
@ ^ [X2: real] : ( minus_minus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
= ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ A2 ) @ F ) ) ) ).
% Lim_transform_eq
thf(fact_470_Lim__transform__eq,axiom,
! [F2: nat > real,G: nat > real,F: filter_nat,A2: real] :
( ( filterlim_nat_real
@ ^ [X2: nat] : ( minus_minus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
= ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ A2 ) @ F ) ) ) ).
% Lim_transform_eq
thf(fact_471_continuous__at,axiom,
! [X: nat,F2: nat > nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ X @ top_top_set_nat ) @ F2 )
= ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ ( F2 @ X ) ) @ ( topolo4659099751122792720in_nat @ X @ top_top_set_nat ) ) ) ).
% continuous_at
thf(fact_472_continuous__at,axiom,
! [X: nat,F2: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ X @ top_top_set_nat ) @ F2 )
= ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ ( F2 @ X ) ) @ ( topolo4659099751122792720in_nat @ X @ top_top_set_nat ) ) ) ).
% continuous_at
thf(fact_473_continuous__at,axiom,
! [X: real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F2 )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ ( F2 @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% continuous_at
thf(fact_474_isCont__def,axiom,
! [A2: nat,F2: nat > nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) @ F2 )
= ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ).
% isCont_def
thf(fact_475_isCont__def,axiom,
! [A2: nat,F2: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) @ F2 )
= ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ).
% isCont_def
thf(fact_476_isCont__def,axiom,
! [A2: real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F2 )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ).
% isCont_def
thf(fact_477_isContD,axiom,
! [X: nat,F2: nat > nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ X @ top_top_set_nat ) @ F2 )
=> ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ ( F2 @ X ) ) @ ( topolo4659099751122792720in_nat @ X @ top_top_set_nat ) ) ) ).
% isContD
thf(fact_478_isContD,axiom,
! [X: nat,F2: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ X @ top_top_set_nat ) @ F2 )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ ( F2 @ X ) ) @ ( topolo4659099751122792720in_nat @ X @ top_top_set_nat ) ) ) ).
% isContD
thf(fact_479_isContD,axiom,
! [X: real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F2 )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ ( F2 @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% isContD
thf(fact_480_filterlim__atI_H,axiom,
! [F2: real > real,C2: real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X2: real] : ( minus_minus_real @ ( F2 @ X2 ) @ C2 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real )
@ F )
=> ( filterlim_real_real @ F2 @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) @ F ) ) ).
% filterlim_atI'
thf(fact_481_filterlim__atI_H,axiom,
! [F2: nat > real,C2: real,F: filter_nat] :
( ( filterlim_nat_real
@ ^ [X2: nat] : ( minus_minus_real @ ( F2 @ X2 ) @ C2 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real )
@ F )
=> ( filterlim_nat_real @ F2 @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) @ F ) ) ).
% filterlim_atI'
thf(fact_482_isCont__divide,axiom,
! [A2: real,F2: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F2 )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ G )
=> ( ( ( G @ A2 )
!= zero_zero_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
@ ^ [X2: real] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% isCont_divide
thf(fact_483_smalloD__tendsto,axiom,
! [F2: real > real,F: filter_real,G: real > real] :
( ( member_real_real @ F2 @ ( landau3007391416991288786l_real @ F @ G ) )
=> ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ).
% smalloD_tendsto
thf(fact_484_smalloD__tendsto,axiom,
! [F2: nat > real,F: filter_nat,G: nat > real] :
( ( member_nat_real @ F2 @ ( landau997807338407142774t_real @ F @ G ) )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ).
% smalloD_tendsto
thf(fact_485_isCont__tendsto__compose,axiom,
! [L: nat,G: nat > nat,F2: nat > nat,F: filter_nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ L @ top_top_set_nat ) @ G )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
=> ( filterlim_nat_nat
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ ( topolo8926549440605965083ds_nat @ ( G @ L ) )
@ F ) ) ) ).
% isCont_tendsto_compose
thf(fact_486_isCont__tendsto__compose,axiom,
! [L: nat,G: nat > real,F2: nat > nat,F: filter_nat] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ L @ top_top_set_nat ) @ G )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( G @ L ) )
@ F ) ) ) ).
% isCont_tendsto_compose
thf(fact_487_isCont__tendsto__compose,axiom,
! [L: real,G: real > nat,F2: nat > real,F: filter_nat] :
( ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ L @ top_top_set_real ) @ G )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( filterlim_nat_nat
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ ( topolo8926549440605965083ds_nat @ ( G @ L ) )
@ F ) ) ) ).
% isCont_tendsto_compose
thf(fact_488_isCont__tendsto__compose,axiom,
! [L: real,G: real > real,F2: real > real,F: filter_real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ L @ top_top_set_real ) @ G )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( G @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( G @ L ) )
@ F ) ) ) ).
% isCont_tendsto_compose
thf(fact_489_isCont__tendsto__compose,axiom,
! [L: real,G: real > real,F2: nat > real,F: filter_nat] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ L @ top_top_set_real ) @ G )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( G @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( G @ L ) )
@ F ) ) ) ).
% isCont_tendsto_compose
thf(fact_490_isCont__cong,axiom,
! [F2: real > real,G: real > real,X: real] :
( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ X ) )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F2 )
= ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G ) ) ) ).
% isCont_cong
thf(fact_491_continuous__within__tendsto__compose,axiom,
! [A2: nat,S: set_nat,F2: nat > nat,X: nat > nat,F: filter_nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ A2 @ S ) @ F2 )
=> ( ( eventually_nat
@ ^ [N: nat] : ( member_nat @ ( X @ N ) @ S )
@ F )
=> ( ( filterlim_nat_nat @ X @ ( topolo8926549440605965083ds_nat @ A2 ) @ F )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F2 @ ( X @ N ) )
@ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose
thf(fact_492_continuous__within__tendsto__compose,axiom,
! [A2: nat,S: set_nat,F2: nat > real,X: nat > nat,F: filter_nat] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ S ) @ F2 )
=> ( ( eventually_nat
@ ^ [N: nat] : ( member_nat @ ( X @ N ) @ S )
@ F )
=> ( ( filterlim_nat_nat @ X @ ( topolo8926549440605965083ds_nat @ A2 ) @ F )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F2 @ ( X @ N ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose
thf(fact_493_continuous__within__tendsto__compose,axiom,
! [A2: real,S: set_real,F2: real > nat,X: nat > real,F: filter_nat] :
( ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ A2 @ S ) @ F2 )
=> ( ( eventually_nat
@ ^ [N: nat] : ( member_real @ ( X @ N ) @ S )
@ F )
=> ( ( filterlim_nat_real @ X @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F2 @ ( X @ N ) )
@ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose
thf(fact_494_continuous__within__tendsto__compose,axiom,
! [A2: real,S: set_real,F2: real > real,X: real > real,F: filter_real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S ) @ F2 )
=> ( ( eventually_real
@ ^ [N: real] : ( member_real @ ( X @ N ) @ S )
@ F )
=> ( ( filterlim_real_real @ X @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( filterlim_real_real
@ ^ [N: real] : ( F2 @ ( X @ N ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose
thf(fact_495_continuous__within__tendsto__compose,axiom,
! [A2: real,S: set_real,F2: real > real,X: nat > real,F: filter_nat] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S ) @ F2 )
=> ( ( eventually_nat
@ ^ [N: nat] : ( member_real @ ( X @ N ) @ S )
@ F )
=> ( ( filterlim_nat_real @ X @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F2 @ ( X @ N ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose
thf(fact_496_smalloI__tendsto,axiom,
! [F2: real > real,G: real > real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( G @ X2 )
!= zero_zero_real )
@ F )
=> ( member_real_real @ F2 @ ( landau3007391416991288786l_real @ F @ G ) ) ) ) ).
% smalloI_tendsto
thf(fact_497_smalloI__tendsto,axiom,
! [F2: nat > real,G: nat > real,F: filter_nat] :
( ( filterlim_nat_real
@ ^ [X2: nat] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
=> ( ( eventually_nat
@ ^ [X2: nat] :
( ( G @ X2 )
!= zero_zero_real )
@ F )
=> ( member_nat_real @ F2 @ ( landau997807338407142774t_real @ F @ G ) ) ) ) ).
% smalloI_tendsto
thf(fact_498_filterlim__shift__iff,axiom,
! [F2: real > real,D: real,F: filter_real,A2: real] :
( ( filterlim_real_real @ ( comp_real_real_real @ F2 @ ( plus_plus_real @ D ) ) @ F @ ( topolo2177554685111907308n_real @ ( minus_minus_real @ A2 @ D ) @ top_top_set_real ) )
= ( filterlim_real_real @ F2 @ F @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ).
% filterlim_shift_iff
thf(fact_499_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_500_tendsto__inverse__real,axiom,
! [U3: real > real,L: real,F: filter_real] :
( ( filterlim_real_real @ U3 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( ( L != zero_zero_real )
=> ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ one_one_real @ ( U3 @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( divide_divide_real @ one_one_real @ L ) )
@ F ) ) ) ).
% tendsto_inverse_real
thf(fact_501_tendsto__inverse__real,axiom,
! [U3: nat > real,L: real,F: filter_nat] :
( ( filterlim_nat_real @ U3 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( ( L != zero_zero_real )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( divide_divide_real @ one_one_real @ ( U3 @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( divide_divide_real @ one_one_real @ L ) )
@ F ) ) ) ).
% tendsto_inverse_real
thf(fact_502_DERIV__D,axiom,
! [F2: real > real,D2: real,X: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [H: real] : ( divide_divide_real @ ( minus_minus_real @ ( F2 @ ( plus_plus_real @ X @ H ) ) @ ( F2 @ X ) ) @ H )
@ ( topolo2815343760600316023s_real @ D2 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ).
% DERIV_D
thf(fact_503_DERIV__def,axiom,
! [F2: real > real,D2: real,X: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
= ( filterlim_real_real
@ ^ [H: real] : ( divide_divide_real @ ( minus_minus_real @ ( F2 @ ( plus_plus_real @ X @ H ) ) @ ( F2 @ X ) ) @ H )
@ ( topolo2815343760600316023s_real @ D2 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ).
% DERIV_def
thf(fact_504_add__diff__add,axiom,
! [A2: real,C2: real,B: real,D: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B @ D ) )
= ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ ( minus_minus_real @ C2 @ D ) ) ) ).
% add_diff_add
thf(fact_505_tendsto__norm__zero__iff,axiom,
! [F2: real > real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X2: real] : ( real_V7735802525324610683m_real @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F ) ) ).
% tendsto_norm_zero_iff
thf(fact_506_tendsto__norm__zero__iff,axiom,
! [F2: nat > real,F: filter_nat] :
( ( filterlim_nat_real
@ ^ [X2: nat] : ( real_V7735802525324610683m_real @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
= ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F ) ) ).
% tendsto_norm_zero_iff
thf(fact_507_tendsto__norm__zero__cancel,axiom,
! [F2: real > real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X2: real] : ( real_V7735802525324610683m_real @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F ) ) ).
% tendsto_norm_zero_cancel
thf(fact_508_tendsto__norm__zero__cancel,axiom,
! [F2: nat > real,F: filter_nat] :
( ( filterlim_nat_real
@ ^ [X2: nat] : ( real_V7735802525324610683m_real @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F ) ) ).
% tendsto_norm_zero_cancel
thf(fact_509_DERIV__const,axiom,
! [K: real,F: filter_real] :
( has_fi5821293074295781190e_real
@ ^ [X2: real] : K
@ zero_zero_real
@ F ) ).
% DERIV_const
thf(fact_510_DERIV__ident,axiom,
! [F: filter_real] :
( has_fi5821293074295781190e_real
@ ^ [X2: real] : X2
@ one_one_real
@ F ) ).
% DERIV_ident
thf(fact_511_Deriv_Ofield__differentiable__add,axiom,
! [F2: real > real,F7: real,F: filter_real,G: real > real,G3: real] :
( ( has_fi5821293074295781190e_real @ F2 @ F7 @ F )
=> ( ( has_fi5821293074295781190e_real @ G @ G3 @ F )
=> ( has_fi5821293074295781190e_real
@ ^ [Z2: real] : ( plus_plus_real @ ( F2 @ Z2 ) @ ( G @ Z2 ) )
@ ( plus_plus_real @ F7 @ G3 )
@ F ) ) ) ).
% Deriv.field_differentiable_add
thf(fact_512_Deriv_Ofield__differentiable__diff,axiom,
! [F2: real > real,F7: real,F: filter_real,G: real > real,G3: real] :
( ( has_fi5821293074295781190e_real @ F2 @ F7 @ F )
=> ( ( has_fi5821293074295781190e_real @ G @ G3 @ F )
=> ( has_fi5821293074295781190e_real
@ ^ [Z2: real] : ( minus_minus_real @ ( F2 @ Z2 ) @ ( G @ Z2 ) )
@ ( minus_minus_real @ F7 @ G3 )
@ F ) ) ) ).
% Deriv.field_differentiable_diff
thf(fact_513_continuous__norm,axiom,
! [F: filter_real,F2: real > real] :
( ( topolo4422821103128117721l_real @ F @ F2 )
=> ( topolo4422821103128117721l_real @ F
@ ^ [X2: real] : ( real_V7735802525324610683m_real @ ( F2 @ X2 ) ) ) ) ).
% continuous_norm
thf(fact_514_has__field__derivative__at__within,axiom,
! [F2: real > real,F7: real,X: real,S: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X @ S ) ) ) ).
% has_field_derivative_at_within
thf(fact_515_DERIV__unique,axiom,
! [F2: real > real,D2: real,X: real,E: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( has_fi5821293074295781190e_real @ F2 @ E @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( D2 = E ) ) ) ).
% DERIV_unique
thf(fact_516_DERIV__continuous,axiom,
! [F2: real > real,D2: real,X: real,S: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X @ S ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ S ) @ F2 ) ) ).
% DERIV_continuous
thf(fact_517_DERIV__const__ratio__const2,axiom,
! [A2: real,B: real,F2: real > real,K: real] :
( ( A2 != B )
=> ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F2 @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
=> ( ( divide_divide_real @ ( minus_minus_real @ ( F2 @ B ) @ ( F2 @ A2 ) ) @ ( minus_minus_real @ B @ A2 ) )
= K ) ) ) ).
% DERIV_const_ratio_const2
thf(fact_518_DERIV__add,axiom,
! [F2: real > real,D2: real,X: real,S: set_real,G: real > real,E: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X @ S ) )
=> ( ( has_fi5821293074295781190e_real @ G @ E @ ( topolo2177554685111907308n_real @ X @ S ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X2: real] : ( plus_plus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( plus_plus_real @ D2 @ E )
@ ( topolo2177554685111907308n_real @ X @ S ) ) ) ) ).
% DERIV_add
thf(fact_519_DERIV__diff,axiom,
! [F2: real > real,D2: real,X: real,S: set_real,G: real > real,E: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X @ S ) )
=> ( ( has_fi5821293074295781190e_real @ G @ E @ ( topolo2177554685111907308n_real @ X @ S ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X2: real] : ( minus_minus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( minus_minus_real @ D2 @ E )
@ ( topolo2177554685111907308n_real @ X @ S ) ) ) ) ).
% DERIV_diff
thf(fact_520_DERIV__cdivide,axiom,
! [F2: real > real,D2: real,X: real,S: set_real,C2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X @ S ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X2: real] : ( divide_divide_real @ ( F2 @ X2 ) @ C2 )
@ ( divide_divide_real @ D2 @ C2 )
@ ( topolo2177554685111907308n_real @ X @ S ) ) ) ).
% DERIV_cdivide
thf(fact_521_has__field__derivative__cong__eventually,axiom,
! [F2: real > real,G: real > real,X: real,S2: set_real,U3: real] :
( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( ( ( F2 @ X )
= ( G @ X ) )
=> ( ( has_fi5821293074295781190e_real @ F2 @ U3 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
= ( has_fi5821293074295781190e_real @ G @ U3 @ ( topolo2177554685111907308n_real @ X @ S2 ) ) ) ) ) ).
% has_field_derivative_cong_eventually
thf(fact_522_has__field__derivative__transform__within__open,axiom,
! [F2: real > real,F7: real,A2: real,S2: set_real,G: real > real] :
( ( has_fi5821293074295781190e_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( ( topolo4860482606490270245n_real @ S2 )
=> ( ( member_real @ A2 @ S2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) )
=> ( has_fi5821293074295781190e_real @ G @ F7 @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ) ).
% has_field_derivative_transform_within_open
thf(fact_523_DERIV__isCont,axiom,
! [F2: real > real,D2: real,X: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F2 ) ) ).
% DERIV_isCont
thf(fact_524_DERIV__isconst__all,axiom,
! [F2: real > real,X: real,Y2: real] :
( ! [X4: real] : ( has_fi5821293074295781190e_real @ F2 @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
=> ( ( F2 @ X )
= ( F2 @ Y2 ) ) ) ).
% DERIV_isconst_all
thf(fact_525_DERIV__shift,axiom,
! [F2: real > real,Y2: real,X: real,Z3: real] :
( ( has_fi5821293074295781190e_real @ F2 @ Y2 @ ( topolo2177554685111907308n_real @ ( plus_plus_real @ X @ Z3 ) @ top_top_set_real ) )
= ( has_fi5821293074295781190e_real
@ ^ [X2: real] : ( F2 @ ( plus_plus_real @ X2 @ Z3 ) )
@ Y2
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_shift
thf(fact_526_has__field__derivative__cong__ev,axiom,
! [X: real,Y2: real,S2: set_real,F2: real > real,G: real > real,U3: real,V: real,T3: set_real] :
( ( X = Y2 )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
@ ( topolo2815343760600316023s_real @ X ) )
=> ( ( U3 = V )
=> ( ( S2 = T3 )
=> ( ( member_real @ X @ S2 )
=> ( ( has_fi5821293074295781190e_real @ F2 @ U3 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
= ( has_fi5821293074295781190e_real @ G @ V @ ( topolo2177554685111907308n_real @ Y2 @ T3 ) ) ) ) ) ) ) ) ).
% has_field_derivative_cong_ev
thf(fact_527_is__num__normalize_I1_J,axiom,
! [A2: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A2 @ B ) @ C2 )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_528_DERIV__Uniq,axiom,
! [F2: real > real,X: real] :
( uniq_real
@ ^ [D3: real] : ( has_fi5821293074295781190e_real @ F2 @ D3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_Uniq
thf(fact_529_isCont__norm,axiom,
! [A2: real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F2 )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
@ ^ [X2: real] : ( real_V7735802525324610683m_real @ ( F2 @ X2 ) ) ) ) ).
% isCont_norm
thf(fact_530_DERIV__cong__ev,axiom,
! [X: real,Y2: real,F2: real > real,G: real > real,U3: real,V: real] :
( ( X = Y2 )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ X ) )
=> ( ( U3 = V )
=> ( ( has_fi5821293074295781190e_real @ F2 @ U3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
= ( has_fi5821293074295781190e_real @ G @ V @ ( topolo2177554685111907308n_real @ Y2 @ top_top_set_real ) ) ) ) ) ) ).
% DERIV_cong_ev
thf(fact_531_tendsto__norm,axiom,
! [F2: real > real,A2: real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( real_V7735802525324610683m_real @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( real_V7735802525324610683m_real @ A2 ) )
@ F ) ) ).
% tendsto_norm
thf(fact_532_tendsto__norm,axiom,
! [F2: nat > real,A2: real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( real_V7735802525324610683m_real @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ ( real_V7735802525324610683m_real @ A2 ) )
@ F ) ) ).
% tendsto_norm
thf(fact_533_has__field__derivative__iff,axiom,
! [F2: real > real,D2: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
= ( filterlim_real_real
@ ^ [Y3: real] : ( divide_divide_real @ ( minus_minus_real @ ( F2 @ Y3 ) @ ( F2 @ X ) ) @ ( minus_minus_real @ Y3 @ X ) )
@ ( topolo2815343760600316023s_real @ D2 )
@ ( topolo2177554685111907308n_real @ X @ S2 ) ) ) ).
% has_field_derivative_iff
thf(fact_534_has__field__derivativeD,axiom,
! [F2: real > real,D2: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( filterlim_real_real
@ ^ [Y3: real] : ( divide_divide_real @ ( minus_minus_real @ ( F2 @ Y3 ) @ ( F2 @ X ) ) @ ( minus_minus_real @ Y3 @ X ) )
@ ( topolo2815343760600316023s_real @ D2 )
@ ( topolo2177554685111907308n_real @ X @ S2 ) ) ) ).
% has_field_derivativeD
thf(fact_535_lhopital,axiom,
! [F2: real > real,X: real,G: real > real,G3: real > real,F7: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( G @ X2 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( G3 @ X2 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F7 @ X2 ) @ ( G3 @ X2 ) )
@ F
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).
% lhopital
thf(fact_536_tendsto__norm__zero,axiom,
! [F2: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( real_V7735802525324610683m_real @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ).
% tendsto_norm_zero
thf(fact_537_tendsto__norm__zero,axiom,
! [F2: nat > real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( real_V7735802525324610683m_real @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ).
% tendsto_norm_zero
thf(fact_538_lhopital__complex__simple,axiom,
! [F2: real > real,F7: real,Z3: real,G: real > real,G3: real,C2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) )
=> ( ( has_fi5821293074295781190e_real @ G @ G3 @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) )
=> ( ( ( F2 @ Z3 )
= zero_zero_real )
=> ( ( ( G @ Z3 )
= zero_zero_real )
=> ( ( G3 != zero_zero_real )
=> ( ( ( divide_divide_real @ F7 @ G3 )
= C2 )
=> ( filterlim_real_real
@ ^ [W: real] : ( divide_divide_real @ ( F2 @ W ) @ ( G @ W ) )
@ ( topolo2815343760600316023s_real @ C2 )
@ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% lhopital_complex_simple
thf(fact_539_norm__zero,axiom,
( ( real_V7735802525324610683m_real @ zero_zero_real )
= zero_zero_real ) ).
% norm_zero
thf(fact_540_norm__eq__zero,axiom,
! [X: real] :
( ( ( real_V7735802525324610683m_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_eq_zero
thf(fact_541_has__field__derivative__inverse__basic,axiom,
! [F2: real > real,F7: real,G: real > real,Y2: real,T3: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ ( G @ Y2 ) @ top_top_set_real ) )
=> ( ( F7 != zero_zero_real )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Y2 @ top_top_set_real ) @ G )
=> ( ( topolo4860482606490270245n_real @ T3 )
=> ( ( member_real @ Y2 @ T3 )
=> ( ! [Z4: real] :
( ( member_real @ Z4 @ T3 )
=> ( ( F2 @ ( G @ Z4 ) )
= Z4 ) )
=> ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ F7 ) @ ( topolo2177554685111907308n_real @ Y2 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% has_field_derivative_inverse_basic
thf(fact_542_nonzero__norm__divide,axiom,
! [B: real,A2: real] :
( ( B != zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A2 @ B ) )
= ( divide_divide_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% nonzero_norm_divide
thf(fact_543_norm__divide,axiom,
! [A2: real,B: real] :
( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A2 @ B ) )
= ( divide_divide_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% norm_divide
thf(fact_544_norm__inverse,axiom,
! [A2: real] :
( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A2 ) )
= ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A2 ) ) ) ).
% norm_inverse
thf(fact_545_nonzero__norm__inverse,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A2 ) )
= ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A2 ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_546_lhospital__at__top__at__top,axiom,
! [G: real > real,G3: real > real,F2: real > real,F7: real > real,X: real] :
( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( G3 @ X2 )
!= zero_zero_real )
@ at_top_real )
=> ( ( eventually_real
@ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
@ at_top_real )
=> ( ( eventually_real
@ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
@ at_top_real )
=> ( ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F7 @ X2 ) @ ( G3 @ X2 ) )
@ ( topolo2815343760600316023s_real @ X )
@ at_top_real )
=> ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ X )
@ at_top_real ) ) ) ) ) ) ).
% lhospital_at_top_at_top
thf(fact_547_lhopital__at__top,axiom,
! [G: real > real,X: real,G3: real > real,F2: real > real,F7: real > real,Y2: real] :
( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( G3 @ X2 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F7 @ X2 ) @ ( G3 @ X2 ) )
@ ( topolo2815343760600316023s_real @ Y2 )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ Y2 )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% lhopital_at_top
thf(fact_548_Lim__null__comparison,axiom,
! [F2: real > real,G: real > real,Net: filter_real] :
( ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F2 @ X2 ) ) @ ( G @ X2 ) )
@ Net )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ Net )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ Net ) ) ) ).
% Lim_null_comparison
thf(fact_549_Lim__null__comparison,axiom,
! [F2: nat > real,G: nat > real,Net: filter_nat] :
( ( eventually_nat
@ ^ [X2: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F2 @ X2 ) ) @ ( G @ X2 ) )
@ Net )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ Net )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ Net ) ) ) ).
% Lim_null_comparison
thf(fact_550_isCont__LIM__compose2,axiom,
! [A2: real,F2: real > nat,G: nat > nat,L: nat] :
( ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F2 )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ ( F2 @ A2 ) @ top_top_set_nat ) )
=> ( ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [X4: real] :
( ( ( X4 != A2 )
& ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ A2 ) ) @ D4 ) )
=> ( ( F2 @ X4 )
!= ( F2 @ A2 ) ) ) )
=> ( filterlim_real_nat
@ ^ [X2: real] : ( G @ ( F2 @ X2 ) )
@ ( topolo8926549440605965083ds_nat @ L )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ).
% isCont_LIM_compose2
thf(fact_551_isCont__LIM__compose2,axiom,
! [A2: real,F2: real > nat,G: nat > real,L: real] :
( ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F2 )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ ( F2 @ A2 ) @ top_top_set_nat ) )
=> ( ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [X4: real] :
( ( ( X4 != A2 )
& ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ A2 ) ) @ D4 ) )
=> ( ( F2 @ X4 )
!= ( F2 @ A2 ) ) ) )
=> ( filterlim_real_real
@ ^ [X2: real] : ( G @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ L )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ).
% isCont_LIM_compose2
thf(fact_552_isCont__LIM__compose2,axiom,
! [A2: real,F2: real > real,G: real > real,L: real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F2 )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ ( F2 @ A2 ) @ top_top_set_real ) )
=> ( ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [X4: real] :
( ( ( X4 != A2 )
& ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ A2 ) ) @ D4 ) )
=> ( ( F2 @ X4 )
!= ( F2 @ A2 ) ) ) )
=> ( filterlim_real_real
@ ^ [X2: real] : ( G @ ( F2 @ X2 ) )
@ ( topolo2815343760600316023s_real @ L )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ).
% isCont_LIM_compose2
thf(fact_553_field__derivative__lim__unique,axiom,
! [F2: real > real,Df: real,Z3: real,S: nat > real,A2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ Df @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) )
=> ( ( filterlim_nat_real @ S @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N2: nat] :
( ( S @ N2 )
!= zero_zero_real )
=> ( ( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ ( minus_minus_real @ ( F2 @ ( plus_plus_real @ Z3 @ ( S @ N ) ) ) @ ( F2 @ Z3 ) ) @ ( S @ N ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ at_top_nat )
=> ( Df = A2 ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_554_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_555_dual__order_Orefl,axiom,
! [A2: filter_nat] : ( ord_le2510731241096832064er_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_556_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_557_dual__order_Orefl,axiom,
! [A2: filter_real] : ( ord_le4104064031414453916r_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_558_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_559_order__refl,axiom,
! [X: filter_nat] : ( ord_le2510731241096832064er_nat @ X @ X ) ).
% order_refl
thf(fact_560_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_561_order__refl,axiom,
! [X: filter_real] : ( ord_le4104064031414453916r_real @ X @ X ) ).
% order_refl
thf(fact_562_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually_nat
@ ^ [N: nat] : ( P @ ( plus_plus_nat @ N @ K ) )
@ at_top_nat )
= ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_sequentially_seg
thf(fact_563_le__zero__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_564_add__le__cancel__left,axiom,
! [C2: real,A2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A2 ) @ ( plus_plus_real @ C2 @ B ) )
= ( ord_less_eq_real @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_565_add__le__cancel__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_566_add__le__cancel__right,axiom,
! [A2: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( ord_less_eq_real @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_567_add__le__cancel__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_568_add__less__cancel__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_569_add__less__cancel__left,axiom,
! [C2: real,A2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A2 ) @ ( plus_plus_real @ C2 @ B ) )
= ( ord_less_real @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_570_add__less__cancel__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_571_add__less__cancel__right,axiom,
! [A2: real,C2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( ord_less_real @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_572_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A2 @ A2 ) )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_573_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_574_le__add__same__cancel2,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ B @ A2 ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_575_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_576_le__add__same__cancel1,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ A2 @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_577_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_578_add__le__same__cancel2,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B ) @ B )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_579_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_580_add__le__same__cancel1,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A2 ) @ B )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_581_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_582_diff__ge__0__iff__ge,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B ) )
= ( ord_less_eq_real @ B @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_583_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ A2 ) )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_584_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ A2 ) @ zero_zero_real )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_585_less__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_586_less__add__same__cancel2,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ ( plus_plus_real @ B @ A2 ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_587_less__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_588_less__add__same__cancel1,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ ( plus_plus_real @ A2 @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_589_add__less__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_590_add__less__same__cancel2,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ B ) @ B )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_591_add__less__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_592_add__less__same__cancel1,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A2 ) @ B )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_593_diff__gt__0__iff__gt,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B ) )
= ( ord_less_real @ B @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_594_le__add__diff__inverse2,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_595_le__add__diff__inverse2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_596_le__add__diff__inverse,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_597_le__add__diff__inverse,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_598_inverse__nonpositive__iff__nonpositive,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_599_inverse__nonnegative__iff__nonnegative,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A2 ) )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_600_inverse__positive__iff__positive,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A2 ) )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% inverse_positive_iff_positive
thf(fact_601_inverse__negative__iff__negative,axiom,
! [A2: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ zero_zero_real )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% inverse_negative_iff_negative
thf(fact_602_inverse__less__iff__less__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A2 ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_603_inverse__less__iff__less,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A2 ) ) ) ) ).
% inverse_less_iff_less
thf(fact_604_zero__le__divide__1__iff,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A2 ) )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% zero_le_divide_1_iff
thf(fact_605_divide__le__0__1__iff,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_606_zero__less__divide__1__iff,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A2 ) )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% zero_less_divide_1_iff
thf(fact_607_less__divide__eq__1__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A2 ) )
= ( ord_less_real @ A2 @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_608_less__divide__eq__1__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A2 ) )
= ( ord_less_real @ B @ A2 ) ) ) ).
% less_divide_eq_1_neg
thf(fact_609_divide__less__eq__1__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A2 ) @ one_one_real )
= ( ord_less_real @ B @ A2 ) ) ) ).
% divide_less_eq_1_pos
thf(fact_610_divide__less__eq__1__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A2 ) @ one_one_real )
= ( ord_less_real @ A2 @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_611_divide__less__0__1__iff,axiom,
! [A2: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A2 ) @ zero_zero_real )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_612_inverse__le__iff__le,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_eq_real @ B @ A2 ) ) ) ) ).
% inverse_le_iff_le
thf(fact_613_inverse__le__iff__le__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_eq_real @ B @ A2 ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_614_norm__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_le_zero_iff
thf(fact_615_zero__less__norm__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
= ( X != zero_zero_real ) ) ).
% zero_less_norm_iff
thf(fact_616_le__divide__eq__1__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A2 ) )
= ( ord_less_eq_real @ A2 @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_617_le__divide__eq__1__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A2 ) )
= ( ord_less_eq_real @ B @ A2 ) ) ) ).
% le_divide_eq_1_neg
thf(fact_618_divide__le__eq__1__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A2 ) @ one_one_real )
= ( ord_less_eq_real @ B @ A2 ) ) ) ).
% divide_le_eq_1_pos
thf(fact_619_divide__le__eq__1__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A2 ) @ one_one_real )
= ( ord_less_eq_real @ A2 @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_620_tendsto__zero__divide__iff,axiom,
! [C2: real,A2: nat > real] :
( ( C2 != zero_zero_real )
=> ( ( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ ( A2 @ N ) @ C2 )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat )
= ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% tendsto_zero_divide_iff
thf(fact_621_isCont__bounded,axiom,
! [A2: real,B: real,F2: real > nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F2 ) )
=> ? [M3: nat] :
! [X6: real] :
( ( ( ord_less_eq_real @ A2 @ X6 )
& ( ord_less_eq_real @ X6 @ B ) )
=> ( ord_less_eq_nat @ ( F2 @ X6 ) @ M3 ) ) ) ) ).
% isCont_bounded
thf(fact_622_isCont__bounded,axiom,
! [A2: real,B: real,F2: real > real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F2 ) )
=> ? [M3: real] :
! [X6: real] :
( ( ( ord_less_eq_real @ A2 @ X6 )
& ( ord_less_eq_real @ X6 @ B ) )
=> ( ord_less_eq_real @ ( F2 @ X6 ) @ M3 ) ) ) ) ).
% isCont_bounded
thf(fact_623_isCont__eq__Ub,axiom,
! [A2: real,B: real,F2: real > nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F2 ) )
=> ? [M3: nat] :
( ! [X6: real] :
( ( ( ord_less_eq_real @ A2 @ X6 )
& ( ord_less_eq_real @ X6 @ B ) )
=> ( ord_less_eq_nat @ ( F2 @ X6 ) @ M3 ) )
& ? [X4: real] :
( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ( F2 @ X4 )
= M3 ) ) ) ) ) ).
% isCont_eq_Ub
thf(fact_624_isCont__eq__Ub,axiom,
! [A2: real,B: real,F2: real > real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F2 ) )
=> ? [M3: real] :
( ! [X6: real] :
( ( ( ord_less_eq_real @ A2 @ X6 )
& ( ord_less_eq_real @ X6 @ B ) )
=> ( ord_less_eq_real @ ( F2 @ X6 ) @ M3 ) )
& ? [X4: real] :
( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ( F2 @ X4 )
= M3 ) ) ) ) ) ).
% isCont_eq_Ub
thf(fact_625_isCont__eq__Lb,axiom,
! [A2: real,B: real,F2: real > nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F2 ) )
=> ? [M3: nat] :
( ! [X6: real] :
( ( ( ord_less_eq_real @ A2 @ X6 )
& ( ord_less_eq_real @ X6 @ B ) )
=> ( ord_less_eq_nat @ M3 @ ( F2 @ X6 ) ) )
& ? [X4: real] :
( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ( F2 @ X4 )
= M3 ) ) ) ) ) ).
% isCont_eq_Lb
thf(fact_626_isCont__eq__Lb,axiom,
! [A2: real,B: real,F2: real > real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F2 ) )
=> ? [M3: real] :
( ! [X6: real] :
( ( ( ord_less_eq_real @ A2 @ X6 )
& ( ord_less_eq_real @ X6 @ B ) )
=> ( ord_less_eq_real @ M3 @ ( F2 @ X6 ) ) )
& ? [X4: real] :
( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ( F2 @ X4 )
= M3 ) ) ) ) ) ).
% isCont_eq_Lb
thf(fact_627_isCont__inverse__function2,axiom,
! [A2: real,X: real,B: real,G: real > real,F2: real > real] :
( ( ord_less_real @ A2 @ X )
=> ( ( ord_less_real @ X @ B )
=> ( ! [Z4: real] :
( ( ord_less_eq_real @ A2 @ Z4 )
=> ( ( ord_less_eq_real @ Z4 @ B )
=> ( ( G @ ( F2 @ Z4 ) )
= Z4 ) ) )
=> ( ! [Z4: real] :
( ( ord_less_eq_real @ A2 @ Z4 )
=> ( ( ord_less_eq_real @ Z4 @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F2 ) ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F2 @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% isCont_inverse_function2
thf(fact_628_isCont__has__Ub,axiom,
! [A2: real,B: real,F2: real > nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F2 ) )
=> ? [M3: nat] :
( ! [X6: real] :
( ( ( ord_less_eq_real @ A2 @ X6 )
& ( ord_less_eq_real @ X6 @ B ) )
=> ( ord_less_eq_nat @ ( F2 @ X6 ) @ M3 ) )
& ! [N4: nat] :
( ( ord_less_nat @ N4 @ M3 )
=> ? [X4: real] :
( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ord_less_nat @ N4 @ ( F2 @ X4 ) ) ) ) ) ) ) ).
% isCont_has_Ub
thf(fact_629_isCont__has__Ub,axiom,
! [A2: real,B: real,F2: real > real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F2 ) )
=> ? [M3: real] :
( ! [X6: real] :
( ( ( ord_less_eq_real @ A2 @ X6 )
& ( ord_less_eq_real @ X6 @ B ) )
=> ( ord_less_eq_real @ ( F2 @ X6 ) @ M3 ) )
& ! [N4: real] :
( ( ord_less_real @ N4 @ M3 )
=> ? [X4: real] :
( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ord_less_real @ N4 @ ( F2 @ X4 ) ) ) ) ) ) ) ).
% isCont_has_Ub
thf(fact_630_filterlim__at__top__dense,axiom,
! [F2: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ at_top_real @ F )
= ( ! [Z5: real] :
( eventually_real
@ ^ [X2: real] : ( ord_less_real @ Z5 @ ( F2 @ X2 ) )
@ F ) ) ) ).
% filterlim_at_top_dense
thf(fact_631_filterlim__at__top__dense,axiom,
! [F2: nat > real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ at_top_real @ F )
= ( ! [Z5: real] :
( eventually_nat
@ ^ [X2: nat] : ( ord_less_real @ Z5 @ ( F2 @ X2 ) )
@ F ) ) ) ).
% filterlim_at_top_dense
thf(fact_632_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_633_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_634_filterlim__at__top__mono,axiom,
! [F2: real > real,F: filter_real,G: real > real] :
( ( filterlim_real_real @ F2 @ at_top_real @ F )
=> ( ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F )
=> ( filterlim_real_real @ G @ at_top_real @ F ) ) ) ).
% filterlim_at_top_mono
thf(fact_635_filterlim__at__top__mono,axiom,
! [F2: nat > real,F: filter_nat,G: nat > real] :
( ( filterlim_nat_real @ F2 @ at_top_real @ F )
=> ( ( eventually_nat
@ ^ [X2: nat] : ( ord_less_eq_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F )
=> ( filterlim_nat_real @ G @ at_top_real @ F ) ) ) ).
% filterlim_at_top_mono
thf(fact_636_filterlim__at__top__mono,axiom,
! [F2: real > nat,F: filter_real,G: real > nat] :
( ( filterlim_real_nat @ F2 @ at_top_nat @ F )
=> ( ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F )
=> ( filterlim_real_nat @ G @ at_top_nat @ F ) ) ) ).
% filterlim_at_top_mono
thf(fact_637_filterlim__at__top__mono,axiom,
! [F2: nat > nat,F: filter_nat,G: nat > nat] :
( ( filterlim_nat_nat @ F2 @ at_top_nat @ F )
=> ( ( eventually_nat
@ ^ [X2: nat] : ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F )
=> ( filterlim_nat_nat @ G @ at_top_nat @ F ) ) ) ).
% filterlim_at_top_mono
thf(fact_638_filterlim__at__top__ge,axiom,
! [F2: real > real,F: filter_real,C2: real] :
( ( filterlim_real_real @ F2 @ at_top_real @ F )
= ( ! [Z5: real] :
( ( ord_less_eq_real @ C2 @ Z5 )
=> ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ Z5 @ ( F2 @ X2 ) )
@ F ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_639_filterlim__at__top__ge,axiom,
! [F2: nat > real,F: filter_nat,C2: real] :
( ( filterlim_nat_real @ F2 @ at_top_real @ F )
= ( ! [Z5: real] :
( ( ord_less_eq_real @ C2 @ Z5 )
=> ( eventually_nat
@ ^ [X2: nat] : ( ord_less_eq_real @ Z5 @ ( F2 @ X2 ) )
@ F ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_640_filterlim__at__top__ge,axiom,
! [F2: real > nat,F: filter_real,C2: nat] :
( ( filterlim_real_nat @ F2 @ at_top_nat @ F )
= ( ! [Z5: nat] :
( ( ord_less_eq_nat @ C2 @ Z5 )
=> ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_nat @ Z5 @ ( F2 @ X2 ) )
@ F ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_641_filterlim__at__top__ge,axiom,
! [F2: nat > nat,F: filter_nat,C2: nat] :
( ( filterlim_nat_nat @ F2 @ at_top_nat @ F )
= ( ! [Z5: nat] :
( ( ord_less_eq_nat @ C2 @ Z5 )
=> ( eventually_nat
@ ^ [X2: nat] : ( ord_less_eq_nat @ Z5 @ ( F2 @ X2 ) )
@ F ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_642_filterlim__at__top,axiom,
! [F2: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ at_top_real @ F )
= ( ! [Z5: real] :
( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ Z5 @ ( F2 @ X2 ) )
@ F ) ) ) ).
% filterlim_at_top
thf(fact_643_filterlim__at__top,axiom,
! [F2: nat > real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ at_top_real @ F )
= ( ! [Z5: real] :
( eventually_nat
@ ^ [X2: nat] : ( ord_less_eq_real @ Z5 @ ( F2 @ X2 ) )
@ F ) ) ) ).
% filterlim_at_top
thf(fact_644_filterlim__at__top,axiom,
! [F2: real > nat,F: filter_real] :
( ( filterlim_real_nat @ F2 @ at_top_nat @ F )
= ( ! [Z5: nat] :
( eventually_real
@ ^ [X2: real] : ( ord_less_eq_nat @ Z5 @ ( F2 @ X2 ) )
@ F ) ) ) ).
% filterlim_at_top
thf(fact_645_filterlim__at__top,axiom,
! [F2: nat > nat,F: filter_nat] :
( ( filterlim_nat_nat @ F2 @ at_top_nat @ F )
= ( ! [Z5: nat] :
( eventually_nat
@ ^ [X2: nat] : ( ord_less_eq_nat @ Z5 @ ( F2 @ X2 ) )
@ F ) ) ) ).
% filterlim_at_top
thf(fact_646_eventually__gt__at__top,axiom,
! [C2: nat] : ( eventually_nat @ ( ord_less_nat @ C2 ) @ at_top_nat ) ).
% eventually_gt_at_top
thf(fact_647_eventually__gt__at__top,axiom,
! [C2: real] : ( eventually_real @ ( ord_less_real @ C2 ) @ at_top_real ) ).
% eventually_gt_at_top
thf(fact_648_eventually__ge__at__top,axiom,
! [C2: real] : ( eventually_real @ ( ord_less_eq_real @ C2 ) @ at_top_real ) ).
% eventually_ge_at_top
thf(fact_649_eventually__ge__at__top,axiom,
! [C2: nat] : ( eventually_nat @ ( ord_less_eq_nat @ C2 ) @ at_top_nat ) ).
% eventually_ge_at_top
thf(fact_650_landau__omega_OR__linear,axiom,
! [Y2: real,X: real] :
( ~ ( ord_less_eq_real @ Y2 @ X )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% landau_omega.R_linear
thf(fact_651_landau__omega_OR__trans,axiom,
! [B: real,A2: real,C2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_eq_real @ C2 @ A2 ) ) ) ).
% landau_omega.R_trans
thf(fact_652_landau__omega_OR__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% landau_omega.R_refl
thf(fact_653_landau__o_OR__linear,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) ) ).
% landau_o.R_linear
thf(fact_654_landau__o_OR__trans,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ A2 @ C2 ) ) ) ).
% landau_o.R_trans
thf(fact_655_order__less__imp__not__less,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_656_order__le__imp__less__or__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_657_order__le__imp__less__or__eq,axiom,
! [X: filter_nat,Y2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y2 )
=> ( ( ord_less_filter_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_658_order__le__imp__less__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_659_order__le__imp__less__or__eq,axiom,
! [X: filter_real,Y2: filter_real] :
( ( ord_le4104064031414453916r_real @ X @ Y2 )
=> ( ( ord_less_filter_real @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_660_linorder__le__less__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
| ( ord_less_real @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_661_linorder__le__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_662_order__less__imp__not__eq2,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_663_order__less__imp__not__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_664_order__less__le__subst2,axiom,
! [A2: real,B: real,F2: real > real,C2: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F2 @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_665_order__less__le__subst2,axiom,
! [A2: real,B: real,F2: real > filter_nat,C2: filter_nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_le2510731241096832064er_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_filter_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_filter_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_666_order__less__le__subst2,axiom,
! [A2: real,B: real,F2: real > nat,C2: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_667_order__less__le__subst2,axiom,
! [A2: real,B: real,F2: real > filter_real,C2: filter_real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_le4104064031414453916r_real @ ( F2 @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_filter_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_filter_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_668_order__less__le__subst1,axiom,
! [A2: real,F2: real > real,B: real,C2: real] :
( ( ord_less_real @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_669_order__less__le__subst1,axiom,
! [A2: nat,F2: real > nat,B: real,C2: real] :
( ( ord_less_nat @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_670_order__less__le__subst1,axiom,
! [A2: real,F2: nat > real,B: nat,C2: nat] :
( ( ord_less_real @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_671_order__less__le__subst1,axiom,
! [A2: nat,F2: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_672_order__less__le__subst1,axiom,
! [A2: filter_nat,F2: real > filter_nat,B: real,C2: real] :
( ( ord_less_filter_nat @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_filter_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_673_order__less__le__subst1,axiom,
! [A2: filter_real,F2: real > filter_real,B: real,C2: real] :
( ( ord_less_filter_real @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_le4104064031414453916r_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_filter_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_674_order__less__le__subst1,axiom,
! [A2: real,F2: filter_nat > real,B: filter_nat,C2: filter_nat] :
( ( ord_less_real @ A2 @ ( F2 @ B ) )
=> ( ( ord_le2510731241096832064er_nat @ B @ C2 )
=> ( ! [X4: filter_nat,Y4: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_675_order__less__le__subst1,axiom,
! [A2: nat,F2: filter_nat > nat,B: filter_nat,C2: filter_nat] :
( ( ord_less_nat @ A2 @ ( F2 @ B ) )
=> ( ( ord_le2510731241096832064er_nat @ B @ C2 )
=> ( ! [X4: filter_nat,Y4: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_676_order__less__le__subst1,axiom,
! [A2: filter_nat,F2: nat > filter_nat,B: nat,C2: nat] :
( ( ord_less_filter_nat @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_filter_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_677_order__less__le__subst1,axiom,
! [A2: filter_real,F2: nat > filter_real,B: nat,C2: nat] :
( ( ord_less_filter_real @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le4104064031414453916r_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_filter_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_678_order__le__less__subst2,axiom,
! [A2: real,B: real,F2: real > real,C2: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_real @ ( F2 @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_679_order__le__less__subst2,axiom,
! [A2: real,B: real,F2: real > nat,C2: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_680_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F2: nat > real,C2: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_real @ ( F2 @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_681_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F2: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_682_order__le__less__subst2,axiom,
! [A2: real,B: real,F2: real > filter_nat,C2: filter_nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_filter_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_filter_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_683_order__le__less__subst2,axiom,
! [A2: real,B: real,F2: real > filter_real,C2: filter_real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_filter_real @ ( F2 @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_le4104064031414453916r_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_filter_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_684_order__le__less__subst2,axiom,
! [A2: filter_nat,B: filter_nat,F2: filter_nat > real,C2: real] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( ord_less_real @ ( F2 @ B ) @ C2 )
=> ( ! [X4: filter_nat,Y4: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_685_order__le__less__subst2,axiom,
! [A2: filter_nat,B: filter_nat,F2: filter_nat > nat,C2: nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X4: filter_nat,Y4: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_686_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F2: nat > filter_nat,C2: filter_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_filter_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_filter_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_687_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F2: nat > filter_real,C2: filter_real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_filter_real @ ( F2 @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le4104064031414453916r_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_filter_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_688_order__le__less__subst1,axiom,
! [A2: real,F2: real > real,B: real,C2: real] :
( ( ord_less_eq_real @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_689_order__le__less__subst1,axiom,
! [A2: filter_nat,F2: real > filter_nat,B: real,C2: real] :
( ( ord_le2510731241096832064er_nat @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_filter_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_filter_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_690_order__le__less__subst1,axiom,
! [A2: nat,F2: real > nat,B: real,C2: real] :
( ( ord_less_eq_nat @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_691_order__le__less__subst1,axiom,
! [A2: filter_real,F2: real > filter_real,B: real,C2: real] :
( ( ord_le4104064031414453916r_real @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_filter_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_filter_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_692_linorder__less__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_real @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_693_order__less__le__trans,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_694_order__less__le__trans,axiom,
! [X: filter_nat,Y2: filter_nat,Z3: filter_nat] :
( ( ord_less_filter_nat @ X @ Y2 )
=> ( ( ord_le2510731241096832064er_nat @ Y2 @ Z3 )
=> ( ord_less_filter_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_695_order__less__le__trans,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_696_order__less__le__trans,axiom,
! [X: filter_real,Y2: filter_real,Z3: filter_real] :
( ( ord_less_filter_real @ X @ Y2 )
=> ( ( ord_le4104064031414453916r_real @ Y2 @ Z3 )
=> ( ord_less_filter_real @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_697_order__less__imp__triv,axiom,
! [X: real,Y2: real,P: $o] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_698_order__le__less__trans,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_699_order__le__less__trans,axiom,
! [X: filter_nat,Y2: filter_nat,Z3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y2 )
=> ( ( ord_less_filter_nat @ Y2 @ Z3 )
=> ( ord_less_filter_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_700_order__le__less__trans,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_701_order__le__less__trans,axiom,
! [X: filter_real,Y2: filter_real,Z3: filter_real] :
( ( ord_le4104064031414453916r_real @ X @ Y2 )
=> ( ( ord_less_filter_real @ Y2 @ Z3 )
=> ( ord_less_filter_real @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_702_order__neq__le__trans,axiom,
! [A2: real,B: real] :
( ( A2 != B )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_real @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_703_order__neq__le__trans,axiom,
! [A2: filter_nat,B: filter_nat] :
( ( A2 != B )
=> ( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ord_less_filter_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_704_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_705_order__neq__le__trans,axiom,
! [A2: filter_real,B: filter_real] :
( ( A2 != B )
=> ( ( ord_le4104064031414453916r_real @ A2 @ B )
=> ( ord_less_filter_real @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_706_order__less__not__sym,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_707_order__le__neq__trans,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_real @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_708_order__le__neq__trans,axiom,
! [A2: filter_nat,B: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_filter_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_709_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_710_order__le__neq__trans,axiom,
! [A2: filter_real,B: filter_real] :
( ( ord_le4104064031414453916r_real @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_filter_real @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_711_order__antisym__conv,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_712_order__antisym__conv,axiom,
! [Y2: filter_nat,X: filter_nat] :
( ( ord_le2510731241096832064er_nat @ Y2 @ X )
=> ( ( ord_le2510731241096832064er_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_713_order__antisym__conv,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_714_order__antisym__conv,axiom,
! [Y2: filter_real,X: filter_real] :
( ( ord_le4104064031414453916r_real @ Y2 @ X )
=> ( ( ord_le4104064031414453916r_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_715_order__less__subst2,axiom,
! [A2: real,B: real,F2: real > real,C2: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ ( F2 @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_716_order__less__subst1,axiom,
! [A2: real,F2: real > real,B: real,C2: real] :
( ( ord_less_real @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_717_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_718_order__less__imp__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_719_order__less__imp__le,axiom,
! [X: filter_nat,Y2: filter_nat] :
( ( ord_less_filter_nat @ X @ Y2 )
=> ( ord_le2510731241096832064er_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_720_order__less__imp__le,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_721_order__less__imp__le,axiom,
! [X: filter_real,Y2: filter_real] :
( ( ord_less_filter_real @ X @ Y2 )
=> ( ord_le4104064031414453916r_real @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_722_ord__less__eq__subst,axiom,
! [A2: real,B: real,F2: real > real,C2: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_723_ord__eq__less__subst,axiom,
! [A2: real,F2: real > real,B: real,C2: real] :
( ( A2
= ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_724_linorder__not__less,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_725_linorder__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_726_linorder__le__cases,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_727_linorder__le__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_728_order__less__trans,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_729_order__less__asym_H,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ~ ( ord_less_real @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_730_linorder__neq__iff,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
= ( ( ord_less_real @ X @ Y2 )
| ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_731_order__less__asym,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_732_ord__le__eq__subst,axiom,
! [A2: real,B: real,F2: real > real,C2: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_733_ord__le__eq__subst,axiom,
! [A2: real,B: real,F2: real > nat,C2: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_734_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F2: nat > real,C2: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_735_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F2: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_736_ord__le__eq__subst,axiom,
! [A2: real,B: real,F2: real > filter_nat,C2: filter_nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_737_ord__le__eq__subst,axiom,
! [A2: real,B: real,F2: real > filter_real,C2: filter_real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_le4104064031414453916r_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le4104064031414453916r_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_738_ord__le__eq__subst,axiom,
! [A2: filter_nat,B: filter_nat,F2: filter_nat > real,C2: real] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X4: filter_nat,Y4: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_739_ord__le__eq__subst,axiom,
! [A2: filter_nat,B: filter_nat,F2: filter_nat > nat,C2: nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X4: filter_nat,Y4: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_740_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F2: nat > filter_nat,C2: filter_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_741_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F2: nat > filter_real,C2: filter_real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le4104064031414453916r_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le4104064031414453916r_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_742_ord__eq__le__subst,axiom,
! [A2: real,F2: real > real,B: real,C2: real] :
( ( A2
= ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_743_ord__eq__le__subst,axiom,
! [A2: nat,F2: real > nat,B: real,C2: real] :
( ( A2
= ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_744_ord__eq__le__subst,axiom,
! [A2: real,F2: nat > real,B: nat,C2: nat] :
( ( A2
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_745_ord__eq__le__subst,axiom,
! [A2: nat,F2: nat > nat,B: nat,C2: nat] :
( ( A2
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_746_ord__eq__le__subst,axiom,
! [A2: filter_nat,F2: real > filter_nat,B: real,C2: real] :
( ( A2
= ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le2510731241096832064er_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_747_ord__eq__le__subst,axiom,
! [A2: filter_real,F2: real > filter_real,B: real,C2: real] :
( ( A2
= ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_le4104064031414453916r_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le4104064031414453916r_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_748_ord__eq__le__subst,axiom,
! [A2: real,F2: filter_nat > real,B: filter_nat,C2: filter_nat] :
( ( A2
= ( F2 @ B ) )
=> ( ( ord_le2510731241096832064er_nat @ B @ C2 )
=> ( ! [X4: filter_nat,Y4: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_749_ord__eq__le__subst,axiom,
! [A2: nat,F2: filter_nat > nat,B: filter_nat,C2: filter_nat] :
( ( A2
= ( F2 @ B ) )
=> ( ( ord_le2510731241096832064er_nat @ B @ C2 )
=> ( ! [X4: filter_nat,Y4: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_750_ord__eq__le__subst,axiom,
! [A2: filter_nat,F2: nat > filter_nat,B: nat,C2: nat] :
( ( A2
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le2510731241096832064er_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_751_ord__eq__le__subst,axiom,
! [A2: filter_real,F2: nat > filter_real,B: nat,C2: nat] :
( ( A2
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le4104064031414453916r_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le4104064031414453916r_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_752_linorder__not__le,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X @ Y2 ) )
= ( ord_less_real @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_753_linorder__not__le,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y2 ) )
= ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_754_linorder__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
| ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_755_linorder__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_756_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_757_order__less__le,axiom,
( ord_less_filter_nat
= ( ^ [X2: filter_nat,Y3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_758_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_759_order__less__le,axiom,
( ord_less_filter_real
= ( ^ [X2: filter_real,Y3: filter_real] :
( ( ord_le4104064031414453916r_real @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_760_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_761_order__le__less,axiom,
( ord_le2510731241096832064er_nat
= ( ^ [X2: filter_nat,Y3: filter_nat] :
( ( ord_less_filter_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_762_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_763_order__le__less,axiom,
( ord_le4104064031414453916r_real
= ( ^ [X2: filter_real,Y3: filter_real] :
( ( ord_less_filter_real @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_764_order__eq__refl,axiom,
! [X: real,Y2: real] :
( ( X = Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_765_order__eq__refl,axiom,
! [X: filter_nat,Y2: filter_nat] :
( ( X = Y2 )
=> ( ord_le2510731241096832064er_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_766_order__eq__refl,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_767_order__eq__refl,axiom,
! [X: filter_real,Y2: filter_real] :
( ( X = Y2 )
=> ( ord_le4104064031414453916r_real @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_768_linorder__neqE,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_769_order__subst2,axiom,
! [A2: real,B: real,F2: real > real,C2: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F2 @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_770_order__subst2,axiom,
! [A2: real,B: real,F2: real > nat,C2: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_771_order__subst2,axiom,
! [A2: nat,B: nat,F2: nat > real,C2: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F2 @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_772_order__subst2,axiom,
! [A2: nat,B: nat,F2: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_773_order__subst2,axiom,
! [A2: real,B: real,F2: real > filter_nat,C2: filter_nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_le2510731241096832064er_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_774_order__subst2,axiom,
! [A2: real,B: real,F2: real > filter_real,C2: filter_real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_le4104064031414453916r_real @ ( F2 @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_le4104064031414453916r_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le4104064031414453916r_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_775_order__subst2,axiom,
! [A2: filter_nat,B: filter_nat,F2: filter_nat > real,C2: real] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F2 @ B ) @ C2 )
=> ( ! [X4: filter_nat,Y4: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_776_order__subst2,axiom,
! [A2: filter_nat,B: filter_nat,F2: filter_nat > nat,C2: nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X4: filter_nat,Y4: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_777_order__subst2,axiom,
! [A2: nat,B: nat,F2: nat > filter_nat,C2: filter_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le2510731241096832064er_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_778_order__subst2,axiom,
! [A2: nat,B: nat,F2: nat > filter_real,C2: filter_real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le4104064031414453916r_real @ ( F2 @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le4104064031414453916r_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le4104064031414453916r_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_779_order__subst1,axiom,
! [A2: real,F2: real > real,B: real,C2: real] :
( ( ord_less_eq_real @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_780_order__subst1,axiom,
! [A2: real,F2: nat > real,B: nat,C2: nat] :
( ( ord_less_eq_real @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_781_order__subst1,axiom,
! [A2: nat,F2: real > nat,B: real,C2: real] :
( ( ord_less_eq_nat @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_782_order__subst1,axiom,
! [A2: nat,F2: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_783_order__subst1,axiom,
! [A2: real,F2: filter_nat > real,B: filter_nat,C2: filter_nat] :
( ( ord_less_eq_real @ A2 @ ( F2 @ B ) )
=> ( ( ord_le2510731241096832064er_nat @ B @ C2 )
=> ( ! [X4: filter_nat,Y4: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_784_order__subst1,axiom,
! [A2: real,F2: filter_real > real,B: filter_real,C2: filter_real] :
( ( ord_less_eq_real @ A2 @ ( F2 @ B ) )
=> ( ( ord_le4104064031414453916r_real @ B @ C2 )
=> ( ! [X4: filter_real,Y4: filter_real] :
( ( ord_le4104064031414453916r_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_785_order__subst1,axiom,
! [A2: filter_nat,F2: real > filter_nat,B: real,C2: real] :
( ( ord_le2510731241096832064er_nat @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le2510731241096832064er_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_786_order__subst1,axiom,
! [A2: filter_nat,F2: nat > filter_nat,B: nat,C2: nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le2510731241096832064er_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_787_order__subst1,axiom,
! [A2: nat,F2: filter_nat > nat,B: filter_nat,C2: filter_nat] :
( ( ord_less_eq_nat @ A2 @ ( F2 @ B ) )
=> ( ( ord_le2510731241096832064er_nat @ B @ C2 )
=> ( ! [X4: filter_nat,Y4: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_788_order__subst1,axiom,
! [A2: nat,F2: filter_real > nat,B: filter_real,C2: filter_real] :
( ( ord_less_eq_nat @ A2 @ ( F2 @ B ) )
=> ( ( ord_le4104064031414453916r_real @ B @ C2 )
=> ( ! [X4: filter_real,Y4: filter_real] :
( ( ord_le4104064031414453916r_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_789_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_790_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: filter_nat,Z: filter_nat] : ( Y = Z ) )
= ( ^ [A3: filter_nat,B2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A3 @ B2 )
& ( ord_le2510731241096832064er_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_791_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_792_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: filter_real,Z: filter_real] : ( Y = Z ) )
= ( ^ [A3: filter_real,B2: filter_real] :
( ( ord_le4104064031414453916r_real @ A3 @ B2 )
& ( ord_le4104064031414453916r_real @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_793_antisym,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_794_antisym,axiom,
! [A2: filter_nat,B: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( ord_le2510731241096832064er_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_795_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_796_antisym,axiom,
! [A2: filter_real,B: filter_real] :
( ( ord_le4104064031414453916r_real @ A2 @ B )
=> ( ( ord_le4104064031414453916r_real @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_797_dual__order_Ostrict__implies__order,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ( ord_less_eq_real @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_798_dual__order_Ostrict__implies__order,axiom,
! [B: filter_nat,A2: filter_nat] :
( ( ord_less_filter_nat @ B @ A2 )
=> ( ord_le2510731241096832064er_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_799_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_800_dual__order_Ostrict__implies__order,axiom,
! [B: filter_real,A2: filter_real] :
( ( ord_less_filter_real @ B @ A2 )
=> ( ord_le4104064031414453916r_real @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_801_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_802_order_Ostrict__implies__order,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_eq_real @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_803_order_Ostrict__implies__order,axiom,
! [A2: filter_nat,B: filter_nat] :
( ( ord_less_filter_nat @ A2 @ B )
=> ( ord_le2510731241096832064er_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_804_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_805_order_Ostrict__implies__order,axiom,
! [A2: filter_real,B: filter_real] :
( ( ord_less_filter_real @ A2 @ B )
=> ( ord_le4104064031414453916r_real @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_806_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_807_dual__order_Ostrict__iff__not,axiom,
( ord_less_filter_nat
= ( ^ [B2: filter_nat,A3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B2 @ A3 )
& ~ ( ord_le2510731241096832064er_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_808_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_809_dual__order_Ostrict__iff__not,axiom,
( ord_less_filter_real
= ( ^ [B2: filter_real,A3: filter_real] :
( ( ord_le4104064031414453916r_real @ B2 @ A3 )
& ~ ( ord_le4104064031414453916r_real @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_810_dual__order_Ostrict__trans2,axiom,
! [B: real,A2: real,C2: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_811_dual__order_Ostrict__trans2,axiom,
! [B: filter_nat,A2: filter_nat,C2: filter_nat] :
( ( ord_less_filter_nat @ B @ A2 )
=> ( ( ord_le2510731241096832064er_nat @ C2 @ B )
=> ( ord_less_filter_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_812_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_813_dual__order_Ostrict__trans2,axiom,
! [B: filter_real,A2: filter_real,C2: filter_real] :
( ( ord_less_filter_real @ B @ A2 )
=> ( ( ord_le4104064031414453916r_real @ C2 @ B )
=> ( ord_less_filter_real @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_814_dual__order_Ostrict__trans1,axiom,
! [B: real,A2: real,C2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_815_dual__order_Ostrict__trans1,axiom,
! [B: filter_nat,A2: filter_nat,C2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B @ A2 )
=> ( ( ord_less_filter_nat @ C2 @ B )
=> ( ord_less_filter_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_816_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_817_dual__order_Ostrict__trans1,axiom,
! [B: filter_real,A2: filter_real,C2: filter_real] :
( ( ord_le4104064031414453916r_real @ B @ A2 )
=> ( ( ord_less_filter_real @ C2 @ B )
=> ( ord_less_filter_real @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_818_order_Ostrict__implies__not__eq,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_819_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_820_dual__order_Ostrict__iff__order,axiom,
( ord_less_filter_nat
= ( ^ [B2: filter_nat,A3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_821_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_822_dual__order_Ostrict__iff__order,axiom,
( ord_less_filter_real
= ( ^ [B2: filter_real,A3: filter_real] :
( ( ord_le4104064031414453916r_real @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_823_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_real @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_824_dual__order_Oorder__iff__strict,axiom,
( ord_le2510731241096832064er_nat
= ( ^ [B2: filter_nat,A3: filter_nat] :
( ( ord_less_filter_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_825_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_826_dual__order_Oorder__iff__strict,axiom,
( ord_le4104064031414453916r_real
= ( ^ [B2: filter_real,A3: filter_real] :
( ( ord_less_filter_real @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_827_dual__order_Ostrict__trans,axiom,
! [B: real,A2: real,C2: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_828_dense__le__bounded,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ! [W2: real] :
( ( ord_less_real @ X @ W2 )
=> ( ( ord_less_real @ W2 @ Y2 )
=> ( ord_less_eq_real @ W2 @ Z3 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_829_dense__ge__bounded,axiom,
! [Z3: real,X: real,Y2: real] :
( ( ord_less_real @ Z3 @ X )
=> ( ! [W2: real] :
( ( ord_less_real @ Z3 @ W2 )
=> ( ( ord_less_real @ W2 @ X )
=> ( ord_less_eq_real @ Y2 @ W2 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_830_not__less__iff__gr__or__eq,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_831_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_832_order_Ostrict__iff__not,axiom,
( ord_less_filter_nat
= ( ^ [A3: filter_nat,B2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A3 @ B2 )
& ~ ( ord_le2510731241096832064er_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_833_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_834_order_Ostrict__iff__not,axiom,
( ord_less_filter_real
= ( ^ [A3: filter_real,B2: filter_real] :
( ( ord_le4104064031414453916r_real @ A3 @ B2 )
& ~ ( ord_le4104064031414453916r_real @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_835_order_Ostrict__trans2,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_836_order_Ostrict__trans2,axiom,
! [A2: filter_nat,B: filter_nat,C2: filter_nat] :
( ( ord_less_filter_nat @ A2 @ B )
=> ( ( ord_le2510731241096832064er_nat @ B @ C2 )
=> ( ord_less_filter_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_837_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_838_order_Ostrict__trans2,axiom,
! [A2: filter_real,B: filter_real,C2: filter_real] :
( ( ord_less_filter_real @ A2 @ B )
=> ( ( ord_le4104064031414453916r_real @ B @ C2 )
=> ( ord_less_filter_real @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_839_order_Ostrict__trans1,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_840_order_Ostrict__trans1,axiom,
! [A2: filter_nat,B: filter_nat,C2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( ord_less_filter_nat @ B @ C2 )
=> ( ord_less_filter_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_841_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_842_order_Ostrict__trans1,axiom,
! [A2: filter_real,B: filter_real,C2: filter_real] :
( ( ord_le4104064031414453916r_real @ A2 @ B )
=> ( ( ord_less_filter_real @ B @ C2 )
=> ( ord_less_filter_real @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_843_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_844_order_Ostrict__iff__order,axiom,
( ord_less_filter_nat
= ( ^ [A3: filter_nat,B2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_845_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_846_order_Ostrict__iff__order,axiom,
( ord_less_filter_real
= ( ^ [A3: filter_real,B2: filter_real] :
( ( ord_le4104064031414453916r_real @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_847_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_848_order_Oorder__iff__strict,axiom,
( ord_le2510731241096832064er_nat
= ( ^ [A3: filter_nat,B2: filter_nat] :
( ( ord_less_filter_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_849_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_850_order_Oorder__iff__strict,axiom,
( ord_le4104064031414453916r_real
= ( ^ [A3: filter_real,B2: filter_real] :
( ( ord_less_filter_real @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_851_order_Ostrict__trans,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_852_linorder__less__wlog,axiom,
! [P: real > real > $o,A2: real,B: real] :
( ! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_853_dual__order_Oirrefl,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_854_dual__order_Otrans,axiom,
! [B: real,A2: real,C2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_eq_real @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_855_dual__order_Otrans,axiom,
! [B: filter_nat,A2: filter_nat,C2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B @ A2 )
=> ( ( ord_le2510731241096832064er_nat @ C2 @ B )
=> ( ord_le2510731241096832064er_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_856_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_857_dual__order_Otrans,axiom,
! [B: filter_real,A2: filter_real,C2: filter_real] :
( ( ord_le4104064031414453916r_real @ B @ A2 )
=> ( ( ord_le4104064031414453916r_real @ C2 @ B )
=> ( ord_le4104064031414453916r_real @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_858_dual__order_Oasym,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ~ ( ord_less_real @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_859_dual__order_Oantisym,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_860_dual__order_Oantisym,axiom,
! [B: filter_nat,A2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B @ A2 )
=> ( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_861_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_862_dual__order_Oantisym,axiom,
! [B: filter_real,A2: filter_real] :
( ( ord_le4104064031414453916r_real @ B @ A2 )
=> ( ( ord_le4104064031414453916r_real @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_863_not__le__imp__less,axiom,
! [Y2: real,X: real] :
( ~ ( ord_less_eq_real @ Y2 @ X )
=> ( ord_less_real @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_864_not__le__imp__less,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X )
=> ( ord_less_nat @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_865_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_866_less__le__not__le,axiom,
( ord_less_filter_nat
= ( ^ [X2: filter_nat,Y3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X2 @ Y3 )
& ~ ( ord_le2510731241096832064er_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_867_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_868_less__le__not__le,axiom,
( ord_less_filter_real
= ( ^ [X2: filter_real,Y3: filter_real] :
( ( ord_le4104064031414453916r_real @ X2 @ Y3 )
& ~ ( ord_le4104064031414453916r_real @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_869_dual__order_Oeq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_870_dual__order_Oeq__iff,axiom,
( ( ^ [Y: filter_nat,Z: filter_nat] : ( Y = Z ) )
= ( ^ [A3: filter_nat,B2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B2 @ A3 )
& ( ord_le2510731241096832064er_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_871_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_872_dual__order_Oeq__iff,axiom,
( ( ^ [Y: filter_real,Z: filter_real] : ( Y = Z ) )
= ( ^ [A3: filter_real,B2: filter_real] :
( ( ord_le4104064031414453916r_real @ B2 @ A3 )
& ( ord_le4104064031414453916r_real @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_873_linorder__cases,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_874_dense__le,axiom,
! [Y2: real,Z3: real] :
( ! [X4: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_eq_real @ X4 @ Z3 ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ).
% dense_le
thf(fact_875_dense__ge,axiom,
! [Z3: real,Y2: real] :
( ! [X4: real] :
( ( ord_less_real @ Z3 @ X4 )
=> ( ord_less_eq_real @ Y2 @ X4 ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ).
% dense_ge
thf(fact_876_linorder__wlog,axiom,
! [P: real > real > $o,A2: real,B: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_877_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_878_antisym__conv3,axiom,
! [Y2: real,X: real] :
( ~ ( ord_less_real @ Y2 @ X )
=> ( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_879_ord__less__eq__trans,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_880_ord__eq__less__trans,axiom,
! [A2: real,B: real,C2: real] :
( ( A2 = B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_881_order__trans,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ( ord_less_eq_real @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_882_order__trans,axiom,
! [X: filter_nat,Y2: filter_nat,Z3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y2 )
=> ( ( ord_le2510731241096832064er_nat @ Y2 @ Z3 )
=> ( ord_le2510731241096832064er_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_883_order__trans,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_884_order__trans,axiom,
! [X: filter_real,Y2: filter_real,Z3: filter_real] :
( ( ord_le4104064031414453916r_real @ X @ Y2 )
=> ( ( ord_le4104064031414453916r_real @ Y2 @ Z3 )
=> ( ord_le4104064031414453916r_real @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_885_order_Otrans,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_886_order_Otrans,axiom,
! [A2: filter_nat,B: filter_nat,C2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( ord_le2510731241096832064er_nat @ B @ C2 )
=> ( ord_le2510731241096832064er_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_887_order_Otrans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_888_order_Otrans,axiom,
! [A2: filter_real,B: filter_real,C2: filter_real] :
( ( ord_le4104064031414453916r_real @ A2 @ B )
=> ( ( ord_le4104064031414453916r_real @ B @ C2 )
=> ( ord_le4104064031414453916r_real @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_889_order_Oasym,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ~ ( ord_less_real @ B @ A2 ) ) ).
% order.asym
thf(fact_890_order__antisym,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_891_order__antisym,axiom,
! [X: filter_nat,Y2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y2 )
=> ( ( ord_le2510731241096832064er_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_892_order__antisym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_893_order__antisym,axiom,
! [X: filter_real,Y2: filter_real] :
( ( ord_le4104064031414453916r_real @ X @ Y2 )
=> ( ( ord_le4104064031414453916r_real @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_894_antisym__conv2,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_895_antisym__conv2,axiom,
! [X: filter_nat,Y2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_filter_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_896_antisym__conv2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_897_antisym__conv2,axiom,
! [X: filter_real,Y2: filter_real] :
( ( ord_le4104064031414453916r_real @ X @ Y2 )
=> ( ( ~ ( ord_less_filter_real @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_898_antisym__conv1,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_899_antisym__conv1,axiom,
! [X: filter_nat,Y2: filter_nat] :
( ~ ( ord_less_filter_nat @ X @ Y2 )
=> ( ( ord_le2510731241096832064er_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_900_antisym__conv1,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_901_antisym__conv1,axiom,
! [X: filter_real,Y2: filter_real] :
( ~ ( ord_less_filter_real @ X @ Y2 )
=> ( ( ord_le4104064031414453916r_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_902_ord__le__eq__trans,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_real @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_903_ord__le__eq__trans,axiom,
! [A2: filter_nat,B: filter_nat,C2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_le2510731241096832064er_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_904_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_905_ord__le__eq__trans,axiom,
! [A2: filter_real,B: filter_real,C2: filter_real] :
( ( ord_le4104064031414453916r_real @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_le4104064031414453916r_real @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_906_ord__eq__le__trans,axiom,
! [A2: real,B: real,C2: real] :
( ( A2 = B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_907_ord__eq__le__trans,axiom,
! [A2: filter_nat,B: filter_nat,C2: filter_nat] :
( ( A2 = B )
=> ( ( ord_le2510731241096832064er_nat @ B @ C2 )
=> ( ord_le2510731241096832064er_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_908_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_909_ord__eq__le__trans,axiom,
! [A2: filter_real,B: filter_real,C2: filter_real] :
( ( A2 = B )
=> ( ( ord_le4104064031414453916r_real @ B @ C2 )
=> ( ord_le4104064031414453916r_real @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_910_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
& ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_911_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: filter_nat,Z: filter_nat] : ( Y = Z ) )
= ( ^ [X2: filter_nat,Y3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X2 @ Y3 )
& ( ord_le2510731241096832064er_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_912_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_913_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: filter_real,Z: filter_real] : ( Y = Z ) )
= ( ^ [X2: filter_real,Y3: filter_real] :
( ( ord_le4104064031414453916r_real @ X2 @ Y3 )
& ( ord_le4104064031414453916r_real @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_914_less__imp__neq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_915_le__cases3,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ( ord_less_eq_real @ X @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X )
=> ~ ( ord_less_eq_real @ X @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X )
=> ~ ( ord_less_eq_real @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_916_le__cases3,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_917_dense,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ? [Z4: real] :
( ( ord_less_real @ X @ Z4 )
& ( ord_less_real @ Z4 @ Y2 ) ) ) ).
% dense
thf(fact_918_nle__le,axiom,
! [A2: real,B: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B ) )
= ( ( ord_less_eq_real @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_919_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_920_nless__le,axiom,
! [A2: real,B: real] :
( ( ~ ( ord_less_real @ A2 @ B ) )
= ( ~ ( ord_less_eq_real @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_921_nless__le,axiom,
! [A2: filter_nat,B: filter_nat] :
( ( ~ ( ord_less_filter_nat @ A2 @ B ) )
= ( ~ ( ord_le2510731241096832064er_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_922_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_923_nless__le,axiom,
! [A2: filter_real,B: filter_real] :
( ( ~ ( ord_less_filter_real @ A2 @ B ) )
= ( ~ ( ord_le4104064031414453916r_real @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_924_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_925_lt__ex,axiom,
! [X: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X ) ).
% lt_ex
thf(fact_926_leI,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) ) ).
% leI
thf(fact_927_leI,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% leI
thf(fact_928_leD,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ~ ( ord_less_real @ X @ Y2 ) ) ).
% leD
thf(fact_929_leD,axiom,
! [Y2: filter_nat,X: filter_nat] :
( ( ord_le2510731241096832064er_nat @ Y2 @ X )
=> ~ ( ord_less_filter_nat @ X @ Y2 ) ) ).
% leD
thf(fact_930_leD,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_nat @ X @ Y2 ) ) ).
% leD
thf(fact_931_leD,axiom,
! [Y2: filter_real,X: filter_real] :
( ( ord_le4104064031414453916r_real @ Y2 @ X )
=> ~ ( ord_less_filter_real @ X @ Y2 ) ) ).
% leD
thf(fact_932_eventually__at__top__linorderI,axiom,
! [C2: real,P: real > $o] :
( ! [X4: real] :
( ( ord_less_eq_real @ C2 @ X4 )
=> ( P @ X4 ) )
=> ( eventually_real @ P @ at_top_real ) ) ).
% eventually_at_top_linorderI
thf(fact_933_eventually__at__top__linorderI,axiom,
! [C2: nat,P: nat > $o] :
( ! [X4: nat] :
( ( ord_less_eq_nat @ C2 @ X4 )
=> ( P @ X4 ) )
=> ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_at_top_linorderI
thf(fact_934_eventually__at__top__linorder,axiom,
! [P: real > $o] :
( ( eventually_real @ P @ at_top_real )
= ( ? [N5: real] :
! [N: real] :
( ( ord_less_eq_real @ N5 @ N )
=> ( P @ N ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_935_eventually__at__top__linorder,axiom,
! [P: nat > $o] :
( ( eventually_nat @ P @ at_top_nat )
= ( ? [N5: nat] :
! [N: nat] :
( ( ord_less_eq_nat @ N5 @ N )
=> ( P @ N ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_936_eventually__at__top__dense,axiom,
! [P: nat > $o] :
( ( eventually_nat @ P @ at_top_nat )
= ( ? [N5: nat] :
! [N: nat] :
( ( ord_less_nat @ N5 @ N )
=> ( P @ N ) ) ) ) ).
% eventually_at_top_dense
thf(fact_937_eventually__at__top__dense,axiom,
! [P: real > $o] :
( ( eventually_real @ P @ at_top_real )
= ( ? [N5: real] :
! [N: real] :
( ( ord_less_real @ N5 @ N )
=> ( P @ N ) ) ) ) ).
% eventually_at_top_dense
thf(fact_938_landau__omega_OR,axiom,
( ( ( ^ [X2: real,Y3: real] : ( ord_less_eq_real @ Y3 @ X2 ) )
= ord_less_eq_real )
| ( ( ^ [X2: real,Y3: real] : ( ord_less_eq_real @ Y3 @ X2 ) )
= ( ^ [X2: real,Y3: real] : ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% landau_omega.R
thf(fact_939_landau__o_OR,axiom,
( ( ord_less_eq_real = ord_less_eq_real )
| ( ord_less_eq_real
= ( ^ [X2: real,Y3: real] : ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% landau_o.R
thf(fact_940_eventually__False__sequentially,axiom,
~ ( eventually_nat
@ ^ [N: nat] : $false
@ at_top_nat ) ).
% eventually_False_sequentially
thf(fact_941_linorder__neqE__linordered__idom,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_942_linordered__field__no__ub,axiom,
! [X6: real] :
? [X_1: real] : ( ord_less_real @ X6 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_943_linordered__field__no__lb,axiom,
! [X6: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X6 ) ).
% linordered_field_no_lb
thf(fact_944_Multiseries__Expansion_Oeventually__lt__imp__eventually__le,axiom,
! [F2: real > real,G: real > real,F: filter_real] :
( ( eventually_real
@ ^ [X2: real] : ( ord_less_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F )
=> ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F ) ) ).
% Multiseries_Expansion.eventually_lt_imp_eventually_le
thf(fact_945_Multiseries__Expansion_Oeventually__lt__imp__eventually__le,axiom,
! [F2: nat > real,G: nat > real,F: filter_nat] :
( ( eventually_nat
@ ^ [X2: nat] : ( ord_less_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F )
=> ( eventually_nat
@ ^ [X2: nat] : ( ord_less_eq_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F ) ) ).
% Multiseries_Expansion.eventually_lt_imp_eventually_le
thf(fact_946_inverse__le__imp__le,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_eq_real @ B @ A2 ) ) ) ).
% inverse_le_imp_le
thf(fact_947_le__imp__inverse__le,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A2 ) ) ) ) ).
% le_imp_inverse_le
thf(fact_948_inverse__le__imp__le__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A2 ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_949_le__imp__inverse__le__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A2 ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_950_frac__le,axiom,
! [Y2: real,X: real,W3: real,Z3: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W3 )
=> ( ( ord_less_eq_real @ W3 @ Z3 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z3 ) @ ( divide_divide_real @ Y2 @ W3 ) ) ) ) ) ) ).
% frac_le
thf(fact_951_frac__less,axiom,
! [X: real,Y2: real,W3: real,Z3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W3 )
=> ( ( ord_less_eq_real @ W3 @ Z3 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z3 ) @ ( divide_divide_real @ Y2 @ W3 ) ) ) ) ) ) ).
% frac_less
thf(fact_952_frac__less2,axiom,
! [X: real,Y2: real,W3: real,Z3: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W3 )
=> ( ( ord_less_real @ W3 @ Z3 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z3 ) @ ( divide_divide_real @ Y2 @ W3 ) ) ) ) ) ) ).
% frac_less2
thf(fact_953_divide__le__cancel,axiom,
! [A2: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ C2 ) @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A2 @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A2 ) ) ) ) ).
% divide_le_cancel
thf(fact_954_divide__nonneg__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_955_divide__nonneg__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonneg_pos
thf(fact_956_divide__nonpos__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonpos_neg
thf(fact_957_divide__nonpos__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_958_add__neg__nonpos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_959_add__neg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_960_add__nonneg__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_961_add__nonneg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_962_add__nonpos__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_963_add__nonpos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_964_add__pos__nonneg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_965_add__pos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_966_add__strict__increasing,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_real @ B @ ( plus_plus_real @ A2 @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_967_add__strict__increasing,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_968_add__strict__increasing2,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ B @ ( plus_plus_real @ A2 @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_969_add__strict__increasing2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_970_field__le__epsilon,axiom,
! [X: real,Y2: real] :
( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y2 @ E2 ) ) )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% field_le_epsilon
thf(fact_971_filterlim__at__top__at__top,axiom,
! [Q: real > $o,F2: real > real,P: real > $o,G: real > real] :
( ! [X4: real,Y4: real] :
( ( Q @ X4 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) )
=> ( ! [X4: real] :
( ( P @ X4 )
=> ( ( F2 @ ( G @ X4 ) )
= X4 ) )
=> ( ! [X4: real] :
( ( P @ X4 )
=> ( Q @ ( G @ X4 ) ) )
=> ( ( eventually_real @ Q @ at_top_real )
=> ( ( eventually_real @ P @ at_top_real )
=> ( filterlim_real_real @ F2 @ at_top_real @ at_top_real ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_972_filterlim__at__top__at__top,axiom,
! [Q: real > $o,F2: real > nat,P: nat > $o,G: nat > real] :
( ! [X4: real,Y4: real] :
( ( Q @ X4 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ( F2 @ ( G @ X4 ) )
= X4 ) )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ ( G @ X4 ) ) )
=> ( ( eventually_real @ Q @ at_top_real )
=> ( ( eventually_nat @ P @ at_top_nat )
=> ( filterlim_real_nat @ F2 @ at_top_nat @ at_top_real ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_973_filterlim__at__top__at__top,axiom,
! [Q: nat > $o,F2: nat > real,P: real > $o,G: real > nat] :
( ! [X4: nat,Y4: nat] :
( ( Q @ X4 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) )
=> ( ! [X4: real] :
( ( P @ X4 )
=> ( ( F2 @ ( G @ X4 ) )
= X4 ) )
=> ( ! [X4: real] :
( ( P @ X4 )
=> ( Q @ ( G @ X4 ) ) )
=> ( ( eventually_nat @ Q @ at_top_nat )
=> ( ( eventually_real @ P @ at_top_real )
=> ( filterlim_nat_real @ F2 @ at_top_real @ at_top_nat ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_974_filterlim__at__top__at__top,axiom,
! [Q: nat > $o,F2: nat > nat,P: nat > $o,G: nat > nat] :
( ! [X4: nat,Y4: nat] :
( ( Q @ X4 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ( F2 @ ( G @ X4 ) )
= X4 ) )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ ( G @ X4 ) ) )
=> ( ( eventually_nat @ Q @ at_top_nat )
=> ( ( eventually_nat @ P @ at_top_nat )
=> ( filterlim_nat_nat @ F2 @ at_top_nat @ at_top_nat ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_975_filterlim__at__top__gt,axiom,
! [F2: real > real,F: filter_real,C2: real] :
( ( filterlim_real_real @ F2 @ at_top_real @ F )
= ( ! [Z5: real] :
( ( ord_less_real @ C2 @ Z5 )
=> ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ Z5 @ ( F2 @ X2 ) )
@ F ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_976_filterlim__at__top__gt,axiom,
! [F2: nat > real,F: filter_nat,C2: real] :
( ( filterlim_nat_real @ F2 @ at_top_real @ F )
= ( ! [Z5: real] :
( ( ord_less_real @ C2 @ Z5 )
=> ( eventually_nat
@ ^ [X2: nat] : ( ord_less_eq_real @ Z5 @ ( F2 @ X2 ) )
@ F ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_977_add__less__le__mono,axiom,
! [A2: real,B: real,C2: real,D: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_978_add__less__le__mono,axiom,
! [A2: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_979_add__le__less__mono,axiom,
! [A2: real,B: real,C2: real,D: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_980_add__le__less__mono,axiom,
! [A2: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_981_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I2 @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_982_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_983_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_984_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_985_sequentially__offset,axiom,
! [P: nat > $o,K: nat] :
( ( eventually_nat @ P @ at_top_nat )
=> ( eventually_nat
@ ^ [I3: nat] : ( P @ ( plus_plus_nat @ I3 @ K ) )
@ at_top_nat ) ) ).
% sequentially_offset
thf(fact_986_filterlim__add__const__nat__at__top,axiom,
! [C2: nat] :
( filterlim_nat_nat
@ ^ [N: nat] : ( plus_plus_nat @ N @ C2 )
@ at_top_nat
@ at_top_nat ) ).
% filterlim_add_const_nat_at_top
thf(fact_987_Bolzano,axiom,
! [A2: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [A4: real,B4: real,C4: real] :
( ( P @ A4 @ B4 )
=> ( ( P @ B4 @ C4 )
=> ( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ B4 @ C4 )
=> ( P @ A4 @ C4 ) ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A2 @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [A4: real,B4: real] :
( ( ( ord_less_eq_real @ A4 @ X4 )
& ( ord_less_eq_real @ X4 @ B4 )
& ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D4 ) )
=> ( P @ A4 @ B4 ) ) ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Bolzano
thf(fact_988_approx__from__above__dense__linorder,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ? [U4: nat > real] :
( ! [N6: nat] : ( ord_less_real @ X @ ( U4 @ N6 ) )
& ( filterlim_nat_real @ U4 @ ( topolo2815343760600316023s_real @ X ) @ at_top_nat ) ) ) ).
% approx_from_above_dense_linorder
thf(fact_989_approx__from__below__dense__linorder,axiom,
! [Y2: real,X: real] :
( ( ord_less_real @ Y2 @ X )
=> ? [U4: nat > real] :
( ! [N6: nat] : ( ord_less_real @ ( U4 @ N6 ) @ X )
& ( filterlim_nat_real @ U4 @ ( topolo2815343760600316023s_real @ X ) @ at_top_nat ) ) ) ).
% approx_from_below_dense_linorder
thf(fact_990_lim__mono,axiom,
! [N7: nat,X5: nat > nat,Y5: nat > nat,X: nat,Y2: nat] :
( ! [N2: nat] :
( ( ord_less_eq_nat @ N7 @ N2 )
=> ( ord_less_eq_nat @ ( X5 @ N2 ) @ ( Y5 @ N2 ) ) )
=> ( ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ X ) @ at_top_nat )
=> ( ( filterlim_nat_nat @ Y5 @ ( topolo8926549440605965083ds_nat @ Y2 ) @ at_top_nat )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ).
% lim_mono
thf(fact_991_lim__mono,axiom,
! [N7: nat,X5: nat > real,Y5: nat > real,X: real,Y2: real] :
( ! [N2: nat] :
( ( ord_less_eq_nat @ N7 @ N2 )
=> ( ord_less_eq_real @ ( X5 @ N2 ) @ ( Y5 @ N2 ) ) )
=> ( ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ X ) @ at_top_nat )
=> ( ( filterlim_nat_real @ Y5 @ ( topolo2815343760600316023s_real @ Y2 ) @ at_top_nat )
=> ( ord_less_eq_real @ X @ Y2 ) ) ) ) ).
% lim_mono
thf(fact_992_LIMSEQ__le,axiom,
! [X5: nat > nat,X: nat,Y5: nat > nat,Y2: nat] :
( ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ X ) @ at_top_nat )
=> ( ( filterlim_nat_nat @ Y5 @ ( topolo8926549440605965083ds_nat @ Y2 ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_less_eq_nat @ ( X5 @ N2 ) @ ( Y5 @ N2 ) ) )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ).
% LIMSEQ_le
thf(fact_993_LIMSEQ__le,axiom,
! [X5: nat > real,X: real,Y5: nat > real,Y2: real] :
( ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ X ) @ at_top_nat )
=> ( ( filterlim_nat_real @ Y5 @ ( topolo2815343760600316023s_real @ Y2 ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_less_eq_real @ ( X5 @ N2 ) @ ( Y5 @ N2 ) ) )
=> ( ord_less_eq_real @ X @ Y2 ) ) ) ) ).
% LIMSEQ_le
thf(fact_994_Lim__bounded,axiom,
! [F2: nat > nat,L: nat,M2: nat,C: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ at_top_nat )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( F2 @ N2 ) @ C ) )
=> ( ord_less_eq_nat @ L @ C ) ) ) ).
% Lim_bounded
thf(fact_995_Lim__bounded,axiom,
! [F2: nat > real,L: real,M2: nat,C: real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_real @ ( F2 @ N2 ) @ C ) )
=> ( ord_less_eq_real @ L @ C ) ) ) ).
% Lim_bounded
thf(fact_996_Lim__bounded2,axiom,
! [F2: nat > nat,L: nat,N7: nat,C: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ at_top_nat )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ N7 @ N2 )
=> ( ord_less_eq_nat @ C @ ( F2 @ N2 ) ) )
=> ( ord_less_eq_nat @ C @ L ) ) ) ).
% Lim_bounded2
thf(fact_997_Lim__bounded2,axiom,
! [F2: nat > real,L: real,N7: nat,C: real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ N7 @ N2 )
=> ( ord_less_eq_real @ C @ ( F2 @ N2 ) ) )
=> ( ord_less_eq_real @ C @ L ) ) ) ).
% Lim_bounded2
thf(fact_998_LIMSEQ__le__const,axiom,
! [X5: nat > nat,X: nat,A2: nat] :
( ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ X ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_less_eq_nat @ A2 @ ( X5 @ N2 ) ) )
=> ( ord_less_eq_nat @ A2 @ X ) ) ) ).
% LIMSEQ_le_const
thf(fact_999_LIMSEQ__le__const,axiom,
! [X5: nat > real,X: real,A2: real] :
( ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ X ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_less_eq_real @ A2 @ ( X5 @ N2 ) ) )
=> ( ord_less_eq_real @ A2 @ X ) ) ) ).
% LIMSEQ_le_const
thf(fact_1000_LIMSEQ__le__const2,axiom,
! [X5: nat > nat,X: nat,A2: nat] :
( ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ X ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_less_eq_nat @ ( X5 @ N2 ) @ A2 ) )
=> ( ord_less_eq_nat @ X @ A2 ) ) ) ).
% LIMSEQ_le_const2
thf(fact_1001_LIMSEQ__le__const2,axiom,
! [X5: nat > real,X: real,A2: real] :
( ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ X ) @ at_top_nat )
=> ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_less_eq_real @ ( X5 @ N2 ) @ A2 ) )
=> ( ord_less_eq_real @ X @ A2 ) ) ) ).
% LIMSEQ_le_const2
thf(fact_1002_filterlim__minus__const__nat__at__top,axiom,
! [C2: nat] :
( filterlim_nat_nat
@ ^ [N: nat] : ( minus_minus_nat @ N @ C2 )
@ at_top_nat
@ at_top_nat ) ).
% filterlim_minus_const_nat_at_top
thf(fact_1003_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_1004_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1005_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_1006_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1007_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I2 @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1008_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1009_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: real,J: real,K: real,L: real] :
( ( ( I2 = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1010_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1011_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I2 @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1012_add__strict__mono,axiom,
! [A2: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1013_add__strict__mono,axiom,
! [A2: real,B: real,C2: real,D: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1014_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1015_add__strict__left__mono,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( plus_plus_real @ C2 @ A2 ) @ ( plus_plus_real @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1016_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1017_add__strict__right__mono,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1018_add__less__imp__less__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_1019_add__less__imp__less__left,axiom,
! [C2: real,A2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A2 ) @ ( plus_plus_real @ C2 @ B ) )
=> ( ord_less_real @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_1020_add__less__imp__less__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_1021_add__less__imp__less__right,axiom,
! [A2: real,C2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
=> ( ord_less_real @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_1022_top_Onot__eq__extremum,axiom,
! [A2: set_real] :
( ( A2 != top_top_set_real )
= ( ord_less_set_real @ A2 @ top_top_set_real ) ) ).
% top.not_eq_extremum
thf(fact_1023_top_Oextremum__strict,axiom,
! [A2: set_real] :
~ ( ord_less_set_real @ top_top_set_real @ A2 ) ).
% top.extremum_strict
thf(fact_1024_diff__strict__right__mono,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( minus_minus_real @ A2 @ C2 ) @ ( minus_minus_real @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_1025_diff__strict__left__mono,axiom,
! [B: real,A2: real,C2: real] :
( ( ord_less_real @ B @ A2 )
=> ( ord_less_real @ ( minus_minus_real @ C2 @ A2 ) @ ( minus_minus_real @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1026_diff__eq__diff__less,axiom,
! [A2: real,B: real,C2: real,D: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( ord_less_real @ A2 @ B )
= ( ord_less_real @ C2 @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1027_diff__strict__mono,axiom,
! [A2: real,B: real,D: real,C2: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ D @ C2 )
=> ( ord_less_real @ ( minus_minus_real @ A2 @ C2 ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1028_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1029_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1030_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: real,J: real,K: real,L: real] :
( ( ( I2 = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1031_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1032_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1033_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1034_add__mono,axiom,
! [A2: real,B: real,C2: real,D: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1035_add__mono,axiom,
! [A2: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1036_add__left__mono,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A2 ) @ ( plus_plus_real @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_1037_add__left__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_1038_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C4: nat] :
( B
!= ( plus_plus_nat @ A2 @ C4 ) ) ) ).
% less_eqE
thf(fact_1039_add__right__mono,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_1040_add__right__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_1041_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C5: nat] :
( B2
= ( plus_plus_nat @ A3 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_1042_add__le__imp__le__left,axiom,
! [C2: real,A2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A2 ) @ ( plus_plus_real @ C2 @ B ) )
=> ( ord_less_eq_real @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_1043_add__le__imp__le__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_1044_add__le__imp__le__right,axiom,
! [A2: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
=> ( ord_less_eq_real @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_1045_add__le__imp__le__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_1046_top_Oextremum__uniqueI,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A2 )
=> ( A2 = top_top_set_real ) ) ).
% top.extremum_uniqueI
thf(fact_1047_top_Oextremum__uniqueI,axiom,
! [A2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ top_top_filter_nat @ A2 )
=> ( A2 = top_top_filter_nat ) ) ).
% top.extremum_uniqueI
thf(fact_1048_top_Oextremum__uniqueI,axiom,
! [A2: filter_real] :
( ( ord_le4104064031414453916r_real @ top_top_filter_real @ A2 )
=> ( A2 = top_top_filter_real ) ) ).
% top.extremum_uniqueI
thf(fact_1049_top_Oextremum__unique,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A2 )
= ( A2 = top_top_set_real ) ) ).
% top.extremum_unique
thf(fact_1050_top_Oextremum__unique,axiom,
! [A2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ top_top_filter_nat @ A2 )
= ( A2 = top_top_filter_nat ) ) ).
% top.extremum_unique
thf(fact_1051_top_Oextremum__unique,axiom,
! [A2: filter_real] :
( ( ord_le4104064031414453916r_real @ top_top_filter_real @ A2 )
= ( A2 = top_top_filter_real ) ) ).
% top.extremum_unique
thf(fact_1052_top__greatest,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ top_top_set_real ) ).
% top_greatest
thf(fact_1053_top__greatest,axiom,
! [A2: filter_nat] : ( ord_le2510731241096832064er_nat @ A2 @ top_top_filter_nat ) ).
% top_greatest
thf(fact_1054_top__greatest,axiom,
! [A2: filter_real] : ( ord_le4104064031414453916r_real @ A2 @ top_top_filter_real ) ).
% top_greatest
thf(fact_1055_diff__eq__diff__less__eq,axiom,
! [A2: real,B: real,C2: real,D: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( ord_less_eq_real @ A2 @ B )
= ( ord_less_eq_real @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1056_diff__right__mono,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C2 ) @ ( minus_minus_real @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_1057_diff__left__mono,axiom,
! [B: real,A2: real,C2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ C2 @ A2 ) @ ( minus_minus_real @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_1058_diff__mono,axiom,
! [A2: real,B: real,D: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ D @ C2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C2 ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_1059_LIMSEQ__unique,axiom,
! [X5: nat > nat,A2: nat,B: nat] :
( ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat )
=> ( ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ B ) @ at_top_nat )
=> ( A2 = B ) ) ) ).
% LIMSEQ_unique
thf(fact_1060_LIMSEQ__unique,axiom,
! [X5: nat > real,A2: real,B: real] :
( ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat )
=> ( ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ B ) @ at_top_nat )
=> ( A2 = B ) ) ) ).
% LIMSEQ_unique
thf(fact_1061_filterlim__at__top__imp__at__infinity,axiom,
! [F2: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ at_top_real @ F )
=> ( filterlim_real_real @ F2 @ at_infinity_real @ F ) ) ).
% filterlim_at_top_imp_at_infinity
thf(fact_1062_filterlim__at__top__imp__at__infinity,axiom,
! [F2: nat > real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ at_top_real @ F )
=> ( filterlim_nat_real @ F2 @ at_infinity_real @ F ) ) ).
% filterlim_at_top_imp_at_infinity
thf(fact_1063_tendsto__at__topI__sequentially,axiom,
! [F2: real > nat,Y2: nat] :
( ! [X7: nat > real] :
( ( filterlim_nat_real @ X7 @ at_top_real @ at_top_nat )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F2 @ ( X7 @ N ) )
@ ( topolo8926549440605965083ds_nat @ Y2 )
@ at_top_nat ) )
=> ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ Y2 ) @ at_top_real ) ) ).
% tendsto_at_topI_sequentially
thf(fact_1064_tendsto__at__topI__sequentially,axiom,
! [F2: real > real,Y2: real] :
( ! [X7: nat > real] :
( ( filterlim_nat_real @ X7 @ at_top_real @ at_top_nat )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F2 @ ( X7 @ N ) )
@ ( topolo2815343760600316023s_real @ Y2 )
@ at_top_nat ) )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ Y2 ) @ at_top_real ) ) ).
% tendsto_at_topI_sequentially
thf(fact_1065_le__divide__eq__1,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ A2 @ B ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B @ A2 ) ) ) ) ).
% le_divide_eq_1
thf(fact_1066_divide__le__eq__1,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ A2 ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ B @ A2 ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ A2 @ B ) )
| ( A2 = zero_zero_real ) ) ) ).
% divide_le_eq_1
thf(fact_1067_eventually__at__top__not__equal,axiom,
! [C2: real] :
( eventually_real
@ ^ [X2: real] : ( X2 != C2 )
@ at_top_real ) ).
% eventually_at_top_not_equal
thf(fact_1068_eventually__at__top__not__equal,axiom,
! [C2: nat] :
( eventually_nat
@ ^ [X2: nat] : ( X2 != C2 )
@ at_top_nat ) ).
% eventually_at_top_not_equal
thf(fact_1069_one__le__inverse__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X ) )
= ( ( ord_less_real @ zero_zero_real @ X )
& ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% one_le_inverse_iff
thf(fact_1070_inverse__less__1__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( inverse_inverse_real @ X ) @ one_one_real )
= ( ( ord_less_eq_real @ X @ zero_zero_real )
| ( ord_less_real @ one_one_real @ X ) ) ) ).
% inverse_less_1_iff
thf(fact_1071_one__le__inverse,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ A2 @ one_one_real )
=> ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A2 ) ) ) ) ).
% one_le_inverse
thf(fact_1072_norm__inverse__le__norm,axiom,
! [R2: real,X: real] :
( ( ord_less_eq_real @ R2 @ ( real_V7735802525324610683m_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_1073_Bseq__add__iff,axiom,
! [F2: nat > real,C2: real] :
( ( bfun_nat_real
@ ^ [X2: nat] : ( plus_plus_real @ ( F2 @ X2 ) @ C2 )
@ at_top_nat )
= ( bfun_nat_real @ F2 @ at_top_nat ) ) ).
% Bseq_add_iff
thf(fact_1074_Bseq__add,axiom,
! [F2: nat > real,C2: real] :
( ( bfun_nat_real @ F2 @ at_top_nat )
=> ( bfun_nat_real
@ ^ [X2: nat] : ( plus_plus_real @ ( F2 @ X2 ) @ C2 )
@ at_top_nat ) ) ).
% Bseq_add
thf(fact_1075_LIMSEQ__const__iff,axiom,
! [K: nat,L: nat] :
( ( filterlim_nat_nat
@ ^ [N: nat] : K
@ ( topolo8926549440605965083ds_nat @ L )
@ at_top_nat )
= ( K = L ) ) ).
% LIMSEQ_const_iff
thf(fact_1076_LIMSEQ__const__iff,axiom,
! [K: real,L: real] :
( ( filterlim_nat_real
@ ^ [N: nat] : K
@ ( topolo2815343760600316023s_real @ L )
@ at_top_nat )
= ( K = L ) ) ).
% LIMSEQ_const_iff
thf(fact_1077_LIMSEQ__ignore__initial__segment,axiom,
! [F2: nat > nat,A2: nat,K: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F2 @ ( plus_plus_nat @ N @ K ) )
@ ( topolo8926549440605965083ds_nat @ A2 )
@ at_top_nat ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_1078_LIMSEQ__ignore__initial__segment,axiom,
! [F2: nat > real,A2: real,K: nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F2 @ ( plus_plus_nat @ N @ K ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ at_top_nat ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_1079_LIMSEQ__offset,axiom,
! [F2: nat > nat,K: nat,A2: nat] :
( ( filterlim_nat_nat
@ ^ [N: nat] : ( F2 @ ( plus_plus_nat @ N @ K ) )
@ ( topolo8926549440605965083ds_nat @ A2 )
@ at_top_nat )
=> ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat ) ) ).
% LIMSEQ_offset
thf(fact_1080_LIMSEQ__offset,axiom,
! [F2: nat > real,K: nat,A2: real] :
( ( filterlim_nat_real
@ ^ [N: nat] : ( F2 @ ( plus_plus_nat @ N @ K ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ at_top_nat )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat ) ) ).
% LIMSEQ_offset
thf(fact_1081_seq__offset__neg,axiom,
! [F2: nat > nat,L: nat,K: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ at_top_nat )
=> ( filterlim_nat_nat
@ ^ [I3: nat] : ( F2 @ ( minus_minus_nat @ I3 @ K ) )
@ ( topolo8926549440605965083ds_nat @ L )
@ at_top_nat ) ) ).
% seq_offset_neg
thf(fact_1082_seq__offset__neg,axiom,
! [F2: nat > real,L: real,K: nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat )
=> ( filterlim_nat_real
@ ^ [I3: nat] : ( F2 @ ( minus_minus_nat @ I3 @ K ) )
@ ( topolo2815343760600316023s_real @ L )
@ at_top_nat ) ) ).
% seq_offset_neg
thf(fact_1083_decreasing__tendsto,axiom,
! [L: nat,F2: real > nat,F: filter_real] :
( ( eventually_real
@ ^ [N: real] : ( ord_less_eq_nat @ L @ ( F2 @ N ) )
@ F )
=> ( ! [X4: nat] :
( ( ord_less_nat @ L @ X4 )
=> ( eventually_real
@ ^ [N: real] : ( ord_less_nat @ ( F2 @ N ) @ X4 )
@ F ) )
=> ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F ) ) ) ).
% decreasing_tendsto
thf(fact_1084_decreasing__tendsto,axiom,
! [L: nat,F2: nat > nat,F: filter_nat] :
( ( eventually_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ L @ ( F2 @ N ) )
@ F )
=> ( ! [X4: nat] :
( ( ord_less_nat @ L @ X4 )
=> ( eventually_nat
@ ^ [N: nat] : ( ord_less_nat @ ( F2 @ N ) @ X4 )
@ F ) )
=> ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F ) ) ) ).
% decreasing_tendsto
thf(fact_1085_decreasing__tendsto,axiom,
! [L: real,F2: real > real,F: filter_real] :
( ( eventually_real
@ ^ [N: real] : ( ord_less_eq_real @ L @ ( F2 @ N ) )
@ F )
=> ( ! [X4: real] :
( ( ord_less_real @ L @ X4 )
=> ( eventually_real
@ ^ [N: real] : ( ord_less_real @ ( F2 @ N ) @ X4 )
@ F ) )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ) ).
% decreasing_tendsto
thf(fact_1086_decreasing__tendsto,axiom,
! [L: real,F2: nat > real,F: filter_nat] :
( ( eventually_nat
@ ^ [N: nat] : ( ord_less_eq_real @ L @ ( F2 @ N ) )
@ F )
=> ( ! [X4: real] :
( ( ord_less_real @ L @ X4 )
=> ( eventually_nat
@ ^ [N: nat] : ( ord_less_real @ ( F2 @ N ) @ X4 )
@ F ) )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ) ).
% decreasing_tendsto
thf(fact_1087_increasing__tendsto,axiom,
! [F2: real > nat,L: nat,F: filter_real] :
( ( eventually_real
@ ^ [N: real] : ( ord_less_eq_nat @ ( F2 @ N ) @ L )
@ F )
=> ( ! [X4: nat] :
( ( ord_less_nat @ X4 @ L )
=> ( eventually_real
@ ^ [N: real] : ( ord_less_nat @ X4 @ ( F2 @ N ) )
@ F ) )
=> ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F ) ) ) ).
% increasing_tendsto
thf(fact_1088_increasing__tendsto,axiom,
! [F2: nat > nat,L: nat,F: filter_nat] :
( ( eventually_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ ( F2 @ N ) @ L )
@ F )
=> ( ! [X4: nat] :
( ( ord_less_nat @ X4 @ L )
=> ( eventually_nat
@ ^ [N: nat] : ( ord_less_nat @ X4 @ ( F2 @ N ) )
@ F ) )
=> ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F ) ) ) ).
% increasing_tendsto
thf(fact_1089_increasing__tendsto,axiom,
! [F2: real > real,L: real,F: filter_real] :
( ( eventually_real
@ ^ [N: real] : ( ord_less_eq_real @ ( F2 @ N ) @ L )
@ F )
=> ( ! [X4: real] :
( ( ord_less_real @ X4 @ L )
=> ( eventually_real
@ ^ [N: real] : ( ord_less_real @ X4 @ ( F2 @ N ) )
@ F ) )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ) ).
% increasing_tendsto
thf(fact_1090_increasing__tendsto,axiom,
! [F2: nat > real,L: real,F: filter_nat] :
( ( eventually_nat
@ ^ [N: nat] : ( ord_less_eq_real @ ( F2 @ N ) @ L )
@ F )
=> ( ! [X4: real] :
( ( ord_less_real @ X4 @ L )
=> ( eventually_nat
@ ^ [N: nat] : ( ord_less_real @ X4 @ ( F2 @ N ) )
@ F ) )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ) ).
% increasing_tendsto
thf(fact_1091_filterlim__at__infinity__imp__filterlim__at__top,axiom,
! [F2: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ at_infinity_real @ F )
=> ( ( eventually_real
@ ^ [X2: real] : ( ord_less_real @ zero_zero_real @ ( F2 @ X2 ) )
@ F )
=> ( filterlim_real_real @ F2 @ at_top_real @ F ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_top
thf(fact_1092_filterlim__at__infinity__imp__filterlim__at__top,axiom,
! [F2: nat > real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ at_infinity_real @ F )
=> ( ( eventually_nat
@ ^ [X2: nat] : ( ord_less_real @ zero_zero_real @ ( F2 @ X2 ) )
@ F )
=> ( filterlim_nat_real @ F2 @ at_top_real @ F ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_top
thf(fact_1093_DERIV__neg__imp__decreasing__at__top,axiom,
! [B: real,F2: real > real,Flim: real] :
( ! [X4: real] :
( ( ord_less_eq_real @ B @ X4 )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F2 @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ Y6 @ zero_zero_real ) ) )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
=> ( ord_less_real @ Flim @ ( F2 @ B ) ) ) ) ).
% DERIV_neg_imp_decreasing_at_top
thf(fact_1094_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_1095_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_1096_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_1097_add__less__zeroD,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y2 ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y2 @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_1098_pos__add__strict,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1099_pos__add__strict,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ B @ ( plus_plus_real @ A2 @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1100_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ! [C4: nat] :
( ( B
= ( plus_plus_nat @ A2 @ C4 ) )
=> ( C4 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1101_add__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1102_add__pos__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1103_add__neg__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1104_add__neg__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_1105_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_1106_divide__strict__right__mono__neg,axiom,
! [B: real,A2: real,C2: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A2 @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_1107_divide__strict__right__mono,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( divide_divide_real @ A2 @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ) ).
% divide_strict_right_mono
thf(fact_1108_zero__less__divide__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_1109_divide__less__cancel,axiom,
! [A2: real,C2: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A2 @ C2 ) @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A2 @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ A2 ) )
& ( C2 != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_1110_divide__less__0__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A2 @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_1111_divide__pos__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_pos_pos
thf(fact_1112_divide__pos__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_1113_divide__neg__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_1114_divide__neg__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_neg_neg
thf(fact_1115_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_1116_less__add__one,axiom,
! [A2: real] : ( ord_less_real @ A2 @ ( plus_plus_real @ A2 @ one_one_real ) ) ).
% less_add_one
thf(fact_1117_add__mono1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1118_add__mono1,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_1119_DERIV__neg__imp__decreasing,axiom,
! [A2: real,B: real,F2: real > real] :
( ( ord_less_real @ A2 @ B )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A2 @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F2 @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ Y6 @ zero_zero_real ) ) ) )
=> ( ord_less_real @ ( F2 @ B ) @ ( F2 @ A2 ) ) ) ) ).
% DERIV_neg_imp_decreasing
thf(fact_1120_DERIV__pos__imp__increasing,axiom,
! [A2: real,B: real,F2: real > real] :
( ( ord_less_real @ A2 @ B )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A2 @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F2 @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ zero_zero_real @ Y6 ) ) ) )
=> ( ord_less_real @ ( F2 @ A2 ) @ ( F2 @ B ) ) ) ) ).
% DERIV_pos_imp_increasing
thf(fact_1121_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ~ ( ord_less_nat @ A2 @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1122_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: real,B: real] :
( ~ ( ord_less_real @ A2 @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A2 @ B ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1123_diff__less__eq,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_real @ ( minus_minus_real @ A2 @ B ) @ C2 )
= ( ord_less_real @ A2 @ ( plus_plus_real @ C2 @ B ) ) ) ).
% diff_less_eq
thf(fact_1124_less__diff__eq,axiom,
! [A2: real,C2: real,B: real] :
( ( ord_less_real @ A2 @ ( minus_minus_real @ C2 @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A2 @ B ) @ C2 ) ) ).
% less_diff_eq
thf(fact_1125_eventually__at__infinity__pos,axiom,
! [P3: real > $o] :
( ( eventually_real @ P3 @ at_infinity_real )
= ( ? [B2: real] :
( ( ord_less_real @ zero_zero_real @ B2 )
& ! [X2: real] :
( ( ord_less_eq_real @ B2 @ ( real_V7735802525324610683m_real @ X2 ) )
=> ( P3 @ X2 ) ) ) ) ) ).
% eventually_at_infinity_pos
thf(fact_1126_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_1127_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1128_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1129_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1130_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_1131_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1132_add__nonpos__eq__0__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y2 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1133_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1134_add__nonneg__eq__0__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( plus_plus_real @ X @ Y2 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1135_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1136_add__nonpos__nonpos,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_1137_add__nonpos__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1138_add__nonneg__nonneg,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1139_add__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1140_add__increasing2,axiom,
! [C2: real,B: real,A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ B @ A2 )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A2 @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1141_add__increasing2,axiom,
! [C2: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1142_add__decreasing2,axiom,
! [C2: real,A2: real,B: real] :
( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1143_add__decreasing2,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1144_add__increasing,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A2 @ C2 ) ) ) ) ).
% add_increasing
thf(fact_1145_add__increasing,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% add_increasing
thf(fact_1146_add__decreasing,axiom,
! [A2: real,C2: real,B: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_1147_add__decreasing,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_1148_ge__iff__diff__ge__0,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B2 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_1149_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_1150_divide__right__mono__neg,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ ( divide_divide_real @ A2 @ C2 ) ) ) ) ).
% divide_right_mono_neg
thf(fact_1151_divide__nonpos__nonpos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1152_divide__nonpos__nonneg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1153_divide__nonneg__nonpos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1154_divide__nonneg__nonneg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1155_zero__le__divide__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A2 @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_1156_divide__right__mono,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ A2 @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ) ).
% divide_right_mono
thf(fact_1157_divide__le__0__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_1158_positive__imp__inverse__positive,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A2 ) ) ) ).
% positive_imp_inverse_positive
thf(fact_1159_negative__imp__inverse__negative,axiom,
! [A2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ zero_zero_real ) ) ).
% negative_imp_inverse_negative
thf(fact_1160_inverse__positive__imp__positive,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A2 ) )
=> ( ( A2 != zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A2 ) ) ) ).
% inverse_positive_imp_positive
thf(fact_1161_inverse__negative__imp__negative,axiom,
! [A2: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ zero_zero_real )
=> ( ( A2 != zero_zero_real )
=> ( ord_less_real @ A2 @ zero_zero_real ) ) ) ).
% inverse_negative_imp_negative
thf(fact_1162_less__imp__inverse__less__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A2 ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_1163_inverse__less__imp__less__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ B @ A2 ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_1164_less__imp__inverse__less,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A2 ) ) ) ) ).
% less_imp_inverse_less
thf(fact_1165_inverse__less__imp__less,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ B @ A2 ) ) ) ).
% inverse_less_imp_less
thf(fact_1166_add__le__add__imp__diff__le,axiom,
! [I2: real,K: real,N3: real,J: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N3 )
=> ( ( ord_less_eq_real @ N3 @ ( plus_plus_real @ J @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N3 )
=> ( ( ord_less_eq_real @ N3 @ ( plus_plus_real @ J @ K ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N3 @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1167_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N3: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N3 @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1168_add__le__imp__le__diff,axiom,
! [I2: real,K: real,N3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N3 )
=> ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N3 @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1169_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N3 )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N3 @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1170_diff__le__eq,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B ) @ C2 )
= ( ord_less_eq_real @ A2 @ ( plus_plus_real @ C2 @ B ) ) ) ).
% diff_le_eq
thf(fact_1171_le__diff__eq,axiom,
! [A2: real,C2: real,B: real] :
( ( ord_less_eq_real @ A2 @ ( minus_minus_real @ C2 @ B ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B ) @ C2 ) ) ).
% le_diff_eq
thf(fact_1172_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1173_le__add__diff,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_1174_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1175_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1176_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A2 )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1177_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1178_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1179_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1180_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1181_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( minus_minus_nat @ B @ A2 )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1182_Multiseries__Expansion_Ocompare__reals__diff__sgnD_I3_J,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B ) )
=> ( ord_less_real @ B @ A2 ) ) ).
% Multiseries_Expansion.compare_reals_diff_sgnD(3)
thf(fact_1183_Multiseries__Expansion_Ocompare__reals__diff__sgnD_I1_J,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( minus_minus_real @ A2 @ B ) @ zero_zero_real )
=> ( ord_less_real @ A2 @ B ) ) ).
% Multiseries_Expansion.compare_reals_diff_sgnD(1)
thf(fact_1184_first__countable__topology__class_Ocountable__basis,axiom,
! [X: nat] :
~ ! [A6: nat > set_nat] :
( ! [I4: nat] : ( topolo4328251076210115529en_nat @ ( A6 @ I4 ) )
=> ( ! [I4: nat] : ( member_nat @ X @ ( A6 @ I4 ) )
=> ~ ! [F8: nat > nat] :
( ! [N2: nat] : ( member_nat @ ( F8 @ N2 ) @ ( A6 @ N2 ) )
=> ( filterlim_nat_nat @ F8 @ ( topolo8926549440605965083ds_nat @ X ) @ at_top_nat ) ) ) ) ).
% first_countable_topology_class.countable_basis
thf(fact_1185_first__countable__topology__class_Ocountable__basis,axiom,
! [X: real] :
~ ! [A6: nat > set_real] :
( ! [I4: nat] : ( topolo4860482606490270245n_real @ ( A6 @ I4 ) )
=> ( ! [I4: nat] : ( member_real @ X @ ( A6 @ I4 ) )
=> ~ ! [F8: nat > real] :
( ! [N2: nat] : ( member_real @ ( F8 @ N2 ) @ ( A6 @ N2 ) )
=> ( filterlim_nat_real @ F8 @ ( topolo2815343760600316023s_real @ X ) @ at_top_nat ) ) ) ) ).
% first_countable_topology_class.countable_basis
thf(fact_1186_filterlim__inverse__at__top,axiom,
! [F2: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F )
=> ( ( eventually_real
@ ^ [X2: real] : ( ord_less_real @ zero_zero_real @ ( F2 @ X2 ) )
@ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( inverse_inverse_real @ ( F2 @ X2 ) )
@ at_top_real
@ F ) ) ) ).
% filterlim_inverse_at_top
thf(fact_1187_filterlim__inverse__at__top,axiom,
! [F2: nat > real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F )
=> ( ( eventually_nat
@ ^ [X2: nat] : ( ord_less_real @ zero_zero_real @ ( F2 @ X2 ) )
@ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( inverse_inverse_real @ ( F2 @ X2 ) )
@ at_top_real
@ F ) ) ) ).
% filterlim_inverse_at_top
thf(fact_1188_filterlim__inverse__at__top__iff,axiom,
! [F2: real > real,F: filter_real] :
( ( eventually_real
@ ^ [X2: real] : ( ord_less_real @ zero_zero_real @ ( F2 @ X2 ) )
@ F )
=> ( ( filterlim_real_real
@ ^ [X2: real] : ( inverse_inverse_real @ ( F2 @ X2 ) )
@ at_top_real
@ F )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F ) ) ) ).
% filterlim_inverse_at_top_iff
thf(fact_1189_filterlim__inverse__at__top__iff,axiom,
! [F2: nat > real,F: filter_nat] :
( ( eventually_nat
@ ^ [X2: nat] : ( ord_less_real @ zero_zero_real @ ( F2 @ X2 ) )
@ F )
=> ( ( filterlim_nat_real
@ ^ [X2: nat] : ( inverse_inverse_real @ ( F2 @ X2 ) )
@ at_top_real
@ F )
= ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F ) ) ) ).
% filterlim_inverse_at_top_iff
thf(fact_1190_filterlim__at__top__iff__inverse__0,axiom,
! [F2: real > real,F: filter_real] :
( ( eventually_real
@ ^ [X2: real] : ( ord_less_real @ zero_zero_real @ ( F2 @ X2 ) )
@ F )
=> ( ( filterlim_real_real @ F2 @ at_top_real @ F )
= ( filterlim_real_real @ ( comp_real_real_real @ inverse_inverse_real @ F2 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F ) ) ) ).
% filterlim_at_top_iff_inverse_0
thf(fact_1191_filterlim__at__top__iff__inverse__0,axiom,
! [F2: nat > real,F: filter_nat] :
( ( eventually_nat
@ ^ [X2: nat] : ( ord_less_real @ zero_zero_real @ ( F2 @ X2 ) )
@ F )
=> ( ( filterlim_nat_real @ F2 @ at_top_real @ F )
= ( filterlim_nat_real @ ( comp_real_real_nat @ inverse_inverse_real @ F2 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F ) ) ) ).
% filterlim_at_top_iff_inverse_0
thf(fact_1192_isCont__Lb__Ub,axiom,
! [A2: real,B: real,F2: real > real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F2 ) )
=> ? [L4: real,M3: real] :
( ! [X6: real] :
( ( ( ord_less_eq_real @ A2 @ X6 )
& ( ord_less_eq_real @ X6 @ B ) )
=> ( ( ord_less_eq_real @ L4 @ ( F2 @ X6 ) )
& ( ord_less_eq_real @ ( F2 @ X6 ) @ M3 ) ) )
& ! [Y6: real] :
( ( ( ord_less_eq_real @ L4 @ Y6 )
& ( ord_less_eq_real @ Y6 @ M3 ) )
=> ? [X4: real] :
( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ( F2 @ X4 )
= Y6 ) ) ) ) ) ) ).
% isCont_Lb_Ub
thf(fact_1193_eventually__nhds__iff__sequentially,axiom,
! [P: nat > $o,A2: nat] :
( ( eventually_nat @ P @ ( topolo8926549440605965083ds_nat @ A2 ) )
= ( ! [F4: nat > nat] :
( ( filterlim_nat_nat @ F4 @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat )
=> ( eventually_nat
@ ^ [N: nat] : ( P @ ( F4 @ N ) )
@ at_top_nat ) ) ) ) ).
% eventually_nhds_iff_sequentially
thf(fact_1194_eventually__nhds__iff__sequentially,axiom,
! [P: real > $o,A2: real] :
( ( eventually_real @ P @ ( topolo2815343760600316023s_real @ A2 ) )
= ( ! [F4: nat > real] :
( ( filterlim_nat_real @ F4 @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat )
=> ( eventually_nat
@ ^ [N: nat] : ( P @ ( F4 @ N ) )
@ at_top_nat ) ) ) ) ).
% eventually_nhds_iff_sequentially
thf(fact_1195_filterlim__at__top__add__at__top,axiom,
! [F2: real > real,F: filter_real,G: real > real] :
( ( filterlim_real_real @ F2 @ at_top_real @ F )
=> ( ( filterlim_real_real @ G @ at_top_real @ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( plus_plus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ at_top_real
@ F ) ) ) ).
% filterlim_at_top_add_at_top
thf(fact_1196_filterlim__at__top__add__at__top,axiom,
! [F2: nat > real,F: filter_nat,G: nat > real] :
( ( filterlim_nat_real @ F2 @ at_top_real @ F )
=> ( ( filterlim_nat_real @ G @ at_top_real @ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( plus_plus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ at_top_real
@ F ) ) ) ).
% filterlim_at_top_add_at_top
thf(fact_1197_LIM__at__top__divide,axiom,
! [F2: real > real,A2: real,F: filter_real,G: real > real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F )
=> ( ( eventually_real
@ ^ [X2: real] : ( ord_less_real @ zero_zero_real @ ( G @ X2 ) )
@ F )
=> ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ at_top_real
@ F ) ) ) ) ) ).
% LIM_at_top_divide
thf(fact_1198_LIM__at__top__divide,axiom,
! [F2: nat > real,A2: real,F: filter_nat,G: nat > real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F )
=> ( ( eventually_nat
@ ^ [X2: nat] : ( ord_less_real @ zero_zero_real @ ( G @ X2 ) )
@ F )
=> ( filterlim_nat_real
@ ^ [X2: nat] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ at_top_real
@ F ) ) ) ) ) ).
% LIM_at_top_divide
thf(fact_1199_filterlim__at__infinity,axiom,
! [C2: real,F2: real > real,F: filter_real] :
( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( filterlim_real_real @ F2 @ at_infinity_real @ F )
= ( ! [R3: real] :
( ( ord_less_real @ C2 @ R3 )
=> ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ R3 @ ( real_V7735802525324610683m_real @ ( F2 @ X2 ) ) )
@ F ) ) ) ) ) ).
% filterlim_at_infinity
thf(fact_1200_filterlim__at__infinity,axiom,
! [C2: real,F2: nat > real,F: filter_nat] :
( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( filterlim_nat_real @ F2 @ at_infinity_real @ F )
= ( ! [R3: real] :
( ( ord_less_real @ C2 @ R3 )
=> ( eventually_nat
@ ^ [X2: nat] : ( ord_less_eq_real @ R3 @ ( real_V7735802525324610683m_real @ ( F2 @ X2 ) ) )
@ F ) ) ) ) ) ).
% filterlim_at_infinity
thf(fact_1201_LIMSEQ__Uniq,axiom,
! [X5: nat > nat] :
( uniq_nat
@ ^ [L5: nat] : ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ L5 ) @ at_top_nat ) ) ).
% LIMSEQ_Uniq
thf(fact_1202_LIMSEQ__Uniq,axiom,
! [X5: nat > real] :
( uniq_real
@ ^ [L5: real] : ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ L5 ) @ at_top_nat ) ) ).
% LIMSEQ_Uniq
thf(fact_1203_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1204_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_1205_less__divide__eq__1,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ A2 @ B ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ B @ A2 ) ) ) ) ).
% less_divide_eq_1
thf(fact_1206_divide__less__eq__1,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ A2 ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ B @ A2 ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ A2 @ B ) )
| ( A2 = zero_zero_real ) ) ) ).
% divide_less_eq_1
thf(fact_1207_gt__half__sum,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A2 @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% gt_half_sum
thf(fact_1208_less__half__sum,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ A2 @ ( divide_divide_real @ ( plus_plus_real @ A2 @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% less_half_sum
thf(fact_1209_one__less__inverse__iff,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X ) )
= ( ( ord_less_real @ zero_zero_real @ X )
& ( ord_less_real @ X @ one_one_real ) ) ) ).
% one_less_inverse_iff
thf(fact_1210_one__less__inverse,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ A2 @ one_one_real )
=> ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A2 ) ) ) ) ).
% one_less_inverse
thf(fact_1211_inverse__le__1__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ X ) @ one_one_real )
= ( ( ord_less_eq_real @ X @ zero_zero_real )
| ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% inverse_le_1_iff
thf(fact_1212_IVT2,axiom,
! [F2: real > nat,B: real,Y2: nat,A2: real] :
( ( ord_less_eq_nat @ ( F2 @ B ) @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ ( F2 @ A2 ) )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F2 ) )
=> ? [X4: real] :
( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ( F2 @ X4 )
= Y2 ) ) ) ) ) ) ).
% IVT2
thf(fact_1213_IVT2,axiom,
! [F2: real > real,B: real,Y2: real,A2: real] :
( ( ord_less_eq_real @ ( F2 @ B ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ ( F2 @ A2 ) )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F2 ) )
=> ? [X4: real] :
( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ( F2 @ X4 )
= Y2 ) ) ) ) ) ) ).
% IVT2
thf(fact_1214_IVT,axiom,
! [F2: real > nat,A2: real,Y2: nat,B: real] :
( ( ord_less_eq_nat @ ( F2 @ A2 ) @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ ( F2 @ B ) )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F2 ) )
=> ? [X4: real] :
( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ( F2 @ X4 )
= Y2 ) ) ) ) ) ) ).
% IVT
thf(fact_1215_IVT,axiom,
! [F2: real > real,A2: real,Y2: real,B: real] :
( ( ord_less_eq_real @ ( F2 @ A2 ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ ( F2 @ B ) )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F2 ) )
=> ? [X4: real] :
( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ( F2 @ X4 )
= Y2 ) ) ) ) ) ) ).
% IVT
thf(fact_1216_norm__add__less,axiom,
! [X: real,R2: real,Y2: real,S: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y2 ) @ S )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y2 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% norm_add_less
thf(fact_1217_norm__triangle__lt,axiom,
! [X: real,Y2: real,E3: real] :
( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y2 ) ) @ E3 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y2 ) ) @ E3 ) ) ).
% norm_triangle_lt
thf(fact_1218_norm__add__rule__thm,axiom,
! [X1: real,B1: real,X22: real,B22: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X1 ) @ B1 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X22 ) @ B22 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X1 @ X22 ) ) @ ( plus_plus_real @ B1 @ B22 ) ) ) ) ).
% norm_add_rule_thm
thf(fact_1219_norm__add__leD,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A2 @ B ) ) @ C2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A2 ) @ C2 ) ) ) ).
% norm_add_leD
thf(fact_1220_norm__triangle__le,axiom,
! [X: real,Y2: real,E3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y2 ) ) @ E3 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y2 ) ) @ E3 ) ) ).
% norm_triangle_le
thf(fact_1221_norm__triangle__ineq,axiom,
! [X: real,Y2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y2 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y2 ) ) ) ).
% norm_triangle_ineq
thf(fact_1222_norm__triangle__mono,axiom,
! [A2: real,R2: real,B: real,S: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A2 ) @ R2 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A2 @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% norm_triangle_mono
thf(fact_1223_norm__diff__ineq,axiom,
! [A2: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A2 @ B ) ) ) ).
% norm_diff_ineq
thf(fact_1224_continuous__within__sequentially,axiom,
! [A2: nat,S: set_nat,F2: nat > nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ A2 @ S ) @ F2 )
= ( ! [X2: nat > nat] :
( ( ! [N: nat] : ( member_nat @ ( X2 @ N ) @ S )
& ( filterlim_nat_nat @ X2 @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat ) )
=> ( filterlim_nat_nat @ ( comp_nat_nat_nat @ F2 @ X2 ) @ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) ) @ at_top_nat ) ) ) ) ).
% continuous_within_sequentially
thf(fact_1225_continuous__within__sequentially,axiom,
! [A2: nat,S: set_nat,F2: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ S ) @ F2 )
= ( ! [X2: nat > nat] :
( ( ! [N: nat] : ( member_nat @ ( X2 @ N ) @ S )
& ( filterlim_nat_nat @ X2 @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat ) )
=> ( filterlim_nat_real @ ( comp_nat_real_nat @ F2 @ X2 ) @ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) ) @ at_top_nat ) ) ) ) ).
% continuous_within_sequentially
thf(fact_1226_continuous__within__sequentially,axiom,
! [A2: real,S: set_real,F2: real > nat] :
( ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ A2 @ S ) @ F2 )
= ( ! [X2: nat > real] :
( ( ! [N: nat] : ( member_real @ ( X2 @ N ) @ S )
& ( filterlim_nat_real @ X2 @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat ) )
=> ( filterlim_nat_nat @ ( comp_real_nat_nat @ F2 @ X2 ) @ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) ) @ at_top_nat ) ) ) ) ).
% continuous_within_sequentially
thf(fact_1227_continuous__within__sequentially,axiom,
! [A2: real,S: set_real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S ) @ F2 )
= ( ! [X2: nat > real] :
( ( ! [N: nat] : ( member_real @ ( X2 @ N ) @ S )
& ( filterlim_nat_real @ X2 @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat ) )
=> ( filterlim_nat_real @ ( comp_real_real_nat @ F2 @ X2 ) @ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) ) @ at_top_nat ) ) ) ) ).
% continuous_within_sequentially
thf(fact_1228_filterlim__norm__at__top,axiom,
filterlim_real_real @ real_V7735802525324610683m_real @ at_top_real @ at_infinity_real ).
% filterlim_norm_at_top
thf(fact_1229_eventually__at__infinity,axiom,
! [P: real > $o] :
( ( eventually_real @ P @ at_infinity_real )
= ( ? [B2: real] :
! [X2: real] :
( ( ord_less_eq_real @ B2 @ ( real_V7735802525324610683m_real @ X2 ) )
=> ( P @ X2 ) ) ) ) ).
% eventually_at_infinity
thf(fact_1230_order__tendstoD_I2_J,axiom,
! [F2: nat > nat,Y2: nat,F: filter_nat,A2: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ Y2 ) @ F )
=> ( ( ord_less_nat @ Y2 @ A2 )
=> ( eventually_nat
@ ^ [X2: nat] : ( ord_less_nat @ ( F2 @ X2 ) @ A2 )
@ F ) ) ) ).
% order_tendstoD(2)
thf(fact_1231_order__tendstoD_I2_J,axiom,
! [F2: real > real,Y2: real,F: filter_real,A2: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ Y2 ) @ F )
=> ( ( ord_less_real @ Y2 @ A2 )
=> ( eventually_real
@ ^ [X2: real] : ( ord_less_real @ ( F2 @ X2 ) @ A2 )
@ F ) ) ) ).
% order_tendstoD(2)
thf(fact_1232_order__tendstoD_I2_J,axiom,
! [F2: nat > real,Y2: real,F: filter_nat,A2: real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ Y2 ) @ F )
=> ( ( ord_less_real @ Y2 @ A2 )
=> ( eventually_nat
@ ^ [X2: nat] : ( ord_less_real @ ( F2 @ X2 ) @ A2 )
@ F ) ) ) ).
% order_tendstoD(2)
thf(fact_1233_order__tendstoD_I1_J,axiom,
! [F2: nat > nat,Y2: nat,F: filter_nat,A2: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ Y2 ) @ F )
=> ( ( ord_less_nat @ A2 @ Y2 )
=> ( eventually_nat
@ ^ [X2: nat] : ( ord_less_nat @ A2 @ ( F2 @ X2 ) )
@ F ) ) ) ).
% order_tendstoD(1)
thf(fact_1234_order__tendstoD_I1_J,axiom,
! [F2: real > real,Y2: real,F: filter_real,A2: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ Y2 ) @ F )
=> ( ( ord_less_real @ A2 @ Y2 )
=> ( eventually_real
@ ^ [X2: real] : ( ord_less_real @ A2 @ ( F2 @ X2 ) )
@ F ) ) ) ).
% order_tendstoD(1)
thf(fact_1235_order__tendstoD_I1_J,axiom,
! [F2: nat > real,Y2: real,F: filter_nat,A2: real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ Y2 ) @ F )
=> ( ( ord_less_real @ A2 @ Y2 )
=> ( eventually_nat
@ ^ [X2: nat] : ( ord_less_real @ A2 @ ( F2 @ X2 ) )
@ F ) ) ) ).
% order_tendstoD(1)
thf(fact_1236_order__tendstoI,axiom,
! [Y2: real,F2: nat > real,F: filter_nat] :
( ! [A4: real] :
( ( ord_less_real @ A4 @ Y2 )
=> ( eventually_nat
@ ^ [X2: nat] : ( ord_less_real @ A4 @ ( F2 @ X2 ) )
@ F ) )
=> ( ! [A4: real] :
( ( ord_less_real @ Y2 @ A4 )
=> ( eventually_nat
@ ^ [X2: nat] : ( ord_less_real @ ( F2 @ X2 ) @ A4 )
@ F ) )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ Y2 ) @ F ) ) ) ).
% order_tendstoI
thf(fact_1237_eventually__diff__zero__imp__eq,axiom,
! [F2: real > real,G: real > real] :
( ( eventually_real
@ ^ [X2: real] :
( ( minus_minus_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
= zero_zero_real )
@ at_top_real )
=> ( eventually_real
@ ^ [X2: real] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ at_top_real ) ) ).
% eventually_diff_zero_imp_eq
thf(fact_1238_has__real__derivative__neg__dec__right,axiom,
! [F2: real > real,L: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [H3: real] :
( ( ord_less_real @ zero_zero_real @ H3 )
=> ( ( member_real @ ( plus_plus_real @ X @ H3 ) @ S2 )
=> ( ( ord_less_real @ H3 @ D5 )
=> ( ord_less_real @ ( F2 @ ( plus_plus_real @ X @ H3 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_1239_has__real__derivative__pos__inc__right,axiom,
! [F2: real > real,L: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [H3: real] :
( ( ord_less_real @ zero_zero_real @ H3 )
=> ( ( member_real @ ( plus_plus_real @ X @ H3 ) @ S2 )
=> ( ( ord_less_real @ H3 @ D5 )
=> ( ord_less_real @ ( F2 @ X ) @ ( F2 @ ( plus_plus_real @ X @ H3 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_1240_has__real__derivative__pos__inc__left,axiom,
! [F2: real > real,L: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [H3: real] :
( ( ord_less_real @ zero_zero_real @ H3 )
=> ( ( member_real @ ( minus_minus_real @ X @ H3 ) @ S2 )
=> ( ( ord_less_real @ H3 @ D5 )
=> ( ord_less_real @ ( F2 @ ( minus_minus_real @ X @ H3 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_1241_has__real__derivative__neg__dec__left,axiom,
! [F2: real > real,L: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [H3: real] :
( ( ord_less_real @ zero_zero_real @ H3 )
=> ( ( member_real @ ( minus_minus_real @ X @ H3 ) @ S2 )
=> ( ( ord_less_real @ H3 @ D5 )
=> ( ord_less_real @ ( F2 @ X ) @ ( F2 @ ( minus_minus_real @ X @ H3 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_1242_DERIV__inverse__function,axiom,
! [F2: real > real,D2: real,G: real > real,X: real,A2: real,B: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
=> ( ( D2 != zero_zero_real )
=> ( ( ord_less_real @ A2 @ X )
=> ( ( ord_less_real @ X @ B )
=> ( ! [Y4: real] :
( ( ord_less_real @ A2 @ Y4 )
=> ( ( ord_less_real @ Y4 @ B )
=> ( ( F2 @ ( G @ Y4 ) )
= Y4 ) ) )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
=> ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D2 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).
% DERIV_inverse_function
thf(fact_1243_DERIV__nonpos__imp__nonincreasing,axiom,
! [A2: real,B: real,F2: real > real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A2 @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F2 @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_eq_real @ Y6 @ zero_zero_real ) ) ) )
=> ( ord_less_eq_real @ ( F2 @ B ) @ ( F2 @ A2 ) ) ) ) ).
% DERIV_nonpos_imp_nonincreasing
thf(fact_1244_DERIV__nonneg__imp__nondecreasing,axiom,
! [A2: real,B: real,F2: real > real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A2 @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F2 @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_eq_real @ zero_zero_real @ Y6 ) ) ) )
=> ( ord_less_eq_real @ ( F2 @ A2 ) @ ( F2 @ B ) ) ) ) ).
% DERIV_nonneg_imp_nondecreasing
thf(fact_1245_DERIV__neg__dec__right,axiom,
! [F2: real > real,L: real,X: real] :
( ( has_fi5821293074295781190e_real @ F2 @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [H3: real] :
( ( ord_less_real @ zero_zero_real @ H3 )
=> ( ( ord_less_real @ H3 @ D5 )
=> ( ord_less_real @ ( F2 @ ( plus_plus_real @ X @ H3 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_1246_DERIV__pos__inc__right,axiom,
! [F2: real > real,L: real,X: real] :
( ( has_fi5821293074295781190e_real @ F2 @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [H3: real] :
( ( ord_less_real @ zero_zero_real @ H3 )
=> ( ( ord_less_real @ H3 @ D5 )
=> ( ord_less_real @ ( F2 @ X ) @ ( F2 @ ( plus_plus_real @ X @ H3 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_1247_DERIV__neg__dec__left,axiom,
! [F2: real > real,L: real,X: real] :
( ( has_fi5821293074295781190e_real @ F2 @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [H3: real] :
( ( ord_less_real @ zero_zero_real @ H3 )
=> ( ( ord_less_real @ H3 @ D5 )
=> ( ord_less_real @ ( F2 @ X ) @ ( F2 @ ( minus_minus_real @ X @ H3 ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_1248_DERIV__pos__inc__left,axiom,
! [F2: real > real,L: real,X: real] :
( ( has_fi5821293074295781190e_real @ F2 @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [H3: real] :
( ( ord_less_real @ zero_zero_real @ H3 )
=> ( ( ord_less_real @ H3 @ D5 )
=> ( ord_less_real @ ( F2 @ ( minus_minus_real @ X @ H3 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_1249_lhopital__at__top__at__top,axiom,
! [F2: real > real,A2: real,G: real > real,F7: real > real,G3: real > real] :
( ( filterlim_real_real @ F2 @ at_top_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F7 @ X2 ) @ ( G3 @ X2 ) )
@ at_top_real
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ at_top_real
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ) ) ).
% lhopital_at_top_at_top
thf(fact_1250_Multiseries__Expansion__Bounds_Ocombine__bounds__diff,axiom,
! [A2: real > real,B: real > real,C2: real > real,D: real > real] :
( ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( A2 @ X2 ) @ ( B @ X2 ) )
@ at_top_real )
=> ( ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( C2 @ X2 ) @ ( D @ X2 ) )
@ at_top_real )
=> ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( A2 @ X2 ) @ ( D @ X2 ) ) @ ( minus_minus_real @ ( B @ X2 ) @ ( C2 @ X2 ) ) )
@ at_top_real ) ) ) ).
% Multiseries_Expansion_Bounds.combine_bounds_diff
thf(fact_1251_Multiseries__Expansion__Bounds_Ocombine__bounds__add,axiom,
! [A2: real > real,B: real > real,C2: real > real,D: real > real] :
( ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( A2 @ X2 ) @ ( B @ X2 ) )
@ at_top_real )
=> ( ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( C2 @ X2 ) @ ( D @ X2 ) )
@ at_top_real )
=> ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( plus_plus_real @ ( A2 @ X2 ) @ ( C2 @ X2 ) ) @ ( plus_plus_real @ ( B @ X2 ) @ ( D @ X2 ) ) )
@ at_top_real ) ) ) ).
% Multiseries_Expansion_Bounds.combine_bounds_add
thf(fact_1252_le__sequentially,axiom,
! [F: filter_nat] :
( ( ord_le2510731241096832064er_nat @ F @ at_top_nat )
= ( ! [N5: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N5 ) @ F ) ) ) ).
% le_sequentially
thf(fact_1253_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually_nat @ P @ at_top_nat )
= ( ? [N5: nat] :
! [N: nat] :
( ( ord_less_eq_nat @ N5 @ N )
=> ( P @ N ) ) ) ) ).
% eventually_sequentially
thf(fact_1254_eventually__sequentiallyI,axiom,
! [C2: nat,P: nat > $o] :
( ! [X4: nat] :
( ( ord_less_eq_nat @ C2 @ X4 )
=> ( P @ X4 ) )
=> ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_sequentiallyI
thf(fact_1255_at__top__le__at__infinity,axiom,
ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).
% at_top_le_at_infinity
thf(fact_1256_LIMSEQ__inverse__zero,axiom,
! [X5: nat > real] :
( ! [R4: real] :
? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( ord_less_real @ R4 @ ( X5 @ N2 ) ) )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( inverse_inverse_real @ ( X5 @ N ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_inverse_zero
thf(fact_1257_Multiseries__Expansion__Bounds_Otransfer__upper__bound,axiom,
! [G: real > real,U3: real > real,F2: real > real] :
( ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( G @ X2 ) @ ( U3 @ X2 ) )
@ at_top_real )
=> ( ( F2 = G )
=> ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( F2 @ X2 ) @ ( U3 @ X2 ) )
@ at_top_real ) ) ) ).
% Multiseries_Expansion_Bounds.transfer_upper_bound
thf(fact_1258_Multiseries__Expansion__Bounds_Otransfer__lower__bound,axiom,
! [L: real > real,G: real > real,F2: real > real] :
( ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( L @ X2 ) @ ( G @ X2 ) )
@ at_top_real )
=> ( ( F2 = G )
=> ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( L @ X2 ) @ ( F2 @ X2 ) )
@ at_top_real ) ) ) ).
% Multiseries_Expansion_Bounds.transfer_lower_bound
thf(fact_1259_Multiseries__Expansion__Bounds_Oeventually__le__self,axiom,
! [F2: real > real] :
( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( F2 @ X2 ) @ ( F2 @ X2 ) )
@ at_top_real ) ).
% Multiseries_Expansion_Bounds.eventually_le_self
thf(fact_1260_Multiseries__Expansion__Bounds_Onatmod__upper__bound,axiom,
! [F2: real > real,L22: real > real,G: real > real,U22: real > real] :
( ( F2 = F2 )
=> ( ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( L22 @ X2 ) @ ( G @ X2 ) )
@ at_top_real )
=> ( ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( G @ X2 ) @ ( U22 @ X2 ) )
@ at_top_real )
=> ( ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( L22 @ X2 ) @ one_one_real ) )
@ at_top_real )
=> ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( multiseries_rnatmod @ ( F2 @ X2 ) @ ( G @ X2 ) ) @ ( minus_minus_real @ ( U22 @ X2 ) @ one_one_real ) )
@ at_top_real ) ) ) ) ) ).
% Multiseries_Expansion_Bounds.natmod_upper_bound
thf(fact_1261_Multiseries__Expansion__Bounds_Onatmod__trivial__lower__bound,axiom,
! [F2: real > real,G: real > real] :
( ( F2 = F2 )
=> ( ( G = G )
=> ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( multiseries_rnatmod @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
@ at_top_real ) ) ) ).
% Multiseries_Expansion_Bounds.natmod_trivial_lower_bound
thf(fact_1262_Multiseries__Expansion__Bounds_Onatmod__upper__bound_H,axiom,
! [G: real > real,U1: real > real,F2: real > real] :
( ( G = G )
=> ( ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( U1 @ X2 ) )
@ at_top_real )
=> ( ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( F2 @ X2 ) @ ( U1 @ X2 ) )
@ at_top_real )
=> ( eventually_real
@ ^ [X2: real] : ( ord_less_eq_real @ ( multiseries_rnatmod @ ( F2 @ X2 ) @ ( G @ X2 ) ) @ ( U1 @ X2 ) )
@ at_top_real ) ) ) ) ).
% Multiseries_Expansion_Bounds.natmod_upper_bound'
thf(fact_1263_lhopital__right__0__at__top,axiom,
! [G: real > real,G3: real > real,F2: real > real,F7: real > real,X: real] :
( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X2: real] :
( ( G3 @ X2 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F7 @ X2 ) @ ( G3 @ X2 ) )
@ ( topolo2815343760600316023s_real @ X )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo2815343760600316023s_real @ X )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).
% lhopital_right_0_at_top
thf(fact_1264_interval__neqs_I5_J,axiom,
! [R2: real] :
( ( set_or5849166863359141190n_real @ R2 )
!= top_top_set_real ) ).
% interval_neqs(5)
thf(fact_1265_eventually__at__right__to__0,axiom,
! [P: real > $o,A2: real] :
( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
= ( eventually_real
@ ^ [X2: real] : ( P @ ( plus_plus_real @ X2 @ A2 ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% eventually_at_right_to_0
thf(fact_1266_eventually__at__top__to__right,axiom,
! [P: real > $o] :
( ( eventually_real @ P @ at_top_real )
= ( eventually_real
@ ^ [X2: real] : ( P @ ( inverse_inverse_real @ X2 ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% eventually_at_top_to_right
thf(fact_1267_eventually__at__right__to__top,axiom,
! [P: real > $o] :
( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
= ( eventually_real
@ ^ [X2: real] : ( P @ ( inverse_inverse_real @ X2 ) )
@ at_top_real ) ) ).
% eventually_at_right_to_top
thf(fact_1268_filterlim__inverse__at__right__top,axiom,
filterlim_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) @ at_top_real ).
% filterlim_inverse_at_right_top
thf(fact_1269_filterlim__inverse__at__top__right,axiom,
filterlim_real_real @ inverse_inverse_real @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).
% filterlim_inverse_at_top_right
thf(fact_1270_continuous__at__right__real__increasing,axiom,
! [F2: real > real,A2: real] :
( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) @ F2 )
= ( ! [E4: real] :
( ( ord_less_real @ zero_zero_real @ E4 )
=> ? [D6: real] :
( ( ord_less_real @ zero_zero_real @ D6 )
& ( ord_less_real @ ( minus_minus_real @ ( F2 @ ( plus_plus_real @ A2 @ D6 ) ) @ ( F2 @ A2 ) ) @ E4 ) ) ) ) ) ) ).
% continuous_at_right_real_increasing
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( eventually_real
@ ^ [X2: real] :
~ ( member_real @ X2 @ ring_1_Ints_real )
@ ( topolo2177554685111907308n_real @ ( r @ xa ) @ top_top_set_real ) ) ).
%------------------------------------------------------------------------------