TPTP Problem File: SLH0392^1.p
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%------------------------------------------------------------------------------
% 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_020375__14108330_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1426 ( 603 unt; 147 typ; 0 def)
% Number of atoms : 3919 (1537 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 14209 ( 471 ~; 41 |; 193 &;12057 @)
% ( 0 <=>;1447 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 1221 (1221 >; 0 *; 0 +; 0 <<)
% Number of symbols : 139 ( 136 usr; 30 con; 0-3 aty)
% Number of variables : 4269 ( 733 ^;3478 !; 58 ?;4269 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:59:54.479
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
thf(ty_n_t__Filter__Ofilter_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
filter_real_real: $tType ).
thf(ty_n_t__Filter__Ofilter_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
filter_nat_real: $tType ).
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 (136)
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__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).
thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
thf(sy_c_Elementary__Topology_Otopological__space__class_Oislimpt_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
elemen8049760308879840156t_real: ( nat > real ) > set_nat_real > $o ).
thf(sy_c_Elementary__Topology_Otopological__space__class_Oislimpt_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
elemen6793289397789523064l_real: ( real > real ) > set_real_real > $o ).
thf(sy_c_Elementary__Topology_Otopological__space__class_Oislimpt_001t__Nat__Onat,type,
elemen5607981409700034897pt_nat: nat > set_nat > $o ).
thf(sy_c_Elementary__Topology_Otopological__space__class_Oislimpt_001t__Real__Oreal,type,
elemen5683178629028408237t_real: real > set_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__bot_001t__Nat__Onat,type,
at_bot_nat: filter_nat ).
thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
at_bot_real: filter_real ).
thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
at_top_nat: filter_nat ).
thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
at_top_real: filter_real ).
thf(sy_c_Filter_Oeventually_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
eventually_nat_real: ( ( nat > real ) > $o ) > filter_nat_real > $o ).
thf(sy_c_Filter_Oeventually_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
eventually_real_real: ( ( real > real ) > $o ) > filter_real_real > $o ).
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_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).
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_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_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_Oone__class_Oone_001t__Set__Oset_It__Real__Oreal_J,type,
one_one_set_real: set_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__Real__Oreal_J,type,
plus_plus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Nat__Onat_M_Eo_J,type,
uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Real__Oreal_M_Eo_J,type,
uminus_uminus_real_o: ( real > $o ) > real > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
uminus612125837232591019t_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_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Real__Oreal_J,type,
zero_zero_set_real: set_real ).
thf(sy_c_HOL_ONO__MATCH_001t__Real__Oreal_001t__Nat__Onat,type,
nO_MATCH_real_nat: real > nat > $o ).
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__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
nO_MAT2475032472373502585et_nat: set_nat > set_nat > $o ).
thf(sy_c_HOL_ONO__MATCH_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Real__Oreal_J,type,
nO_MAT8790521740642007125t_real: set_nat > set_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_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
uniq_real_real: ( ( real > real ) > $o ) > $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_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Real__Oreal_J,type,
inf_inf_set_real: set_real > set_real > set_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_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
neg_nu6075765906172075777c_real: real > real ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
bot_bot_real_o: real > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
bot_bot_filter_nat: filter_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Real__Oreal_J,type,
bot_bot_filter_real: filter_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
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__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $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__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $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_001_Eo,type,
top_top_o: $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_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
top_top_set_nat_real: set_nat_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
top_to2071711978144146653l_real: set_real_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_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_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Real__Oreal,type,
is_singleton_real: set_real > $o ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
set_or5849166863359141190n_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
set_or5984915006950818249n_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_Ot2__space__class_OLim_001t__Nat__Onat_001t__Nat__Onat,type,
topolo4574594659991002850at_nat: filter_nat > ( nat > nat ) > nat ).
thf(sy_c_Topological__Spaces_Ot2__space__class_OLim_001t__Nat__Onat_001t__Real__Oreal,type,
topolo2843583510664976574t_real: filter_nat > ( nat > real ) > real ).
thf(sy_c_Topological__Spaces_Ot2__space__class_OLim_001t__Real__Oreal_001t__Real__Oreal,type,
topolo5976875806579547418l_real: filter_real > ( real > real ) > real ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
topolo3771493267791237467t_real: ( nat > real ) > set_nat_real > filter_nat_real ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
topolo4966363728887024311l_real: ( real > real ) > set_real_real > filter_real_real ).
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 (1273)
thf(fact_0_assms,axiom,
~ ( member_real @ x @ ring_1_Ints_real ) ).
% assms
thf(fact_1_islimpt__iff__eventually,axiom,
( elemen8049760308879840156t_real
= ( ^ [X: nat > real,S: set_nat_real] :
~ ( eventually_nat_real
@ ^ [Y: nat > real] :
~ ( member_nat_real @ Y @ S )
@ ( topolo3771493267791237467t_real @ X @ top_top_set_nat_real ) ) ) ) ).
% islimpt_iff_eventually
thf(fact_2_islimpt__iff__eventually,axiom,
( elemen6793289397789523064l_real
= ( ^ [X: real > real,S: set_real_real] :
~ ( eventually_real_real
@ ^ [Y: real > real] :
~ ( member_real_real @ Y @ S )
@ ( topolo4966363728887024311l_real @ X @ top_to2071711978144146653l_real ) ) ) ) ).
% islimpt_iff_eventually
thf(fact_3_islimpt__iff__eventually,axiom,
( elemen5607981409700034897pt_nat
= ( ^ [X: nat,S: set_nat] :
~ ( eventually_nat
@ ^ [Y: nat] :
~ ( member_nat @ Y @ S )
@ ( topolo4659099751122792720in_nat @ X @ top_top_set_nat ) ) ) ) ).
% islimpt_iff_eventually
thf(fact_4_islimpt__iff__eventually,axiom,
( elemen5683178629028408237t_real
= ( ^ [X: real,S: set_real] :
~ ( eventually_real
@ ^ [Y: real] :
~ ( member_real @ Y @ S )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% islimpt_iff_eventually
thf(fact_5__092_060open_062_092_060not_062_Ar_Ax_Aislimpt_A_092_060int_062_092_060close_062,axiom,
~ ( elemen5683178629028408237t_real @ ( r @ xa ) @ ring_1_Ints_real ) ).
% \<open>\<not> r x islimpt \<int>\<close>
thf(fact_6_that,axiom,
filterlim_real_real @ r @ ( topolo2177554685111907308n_real @ ( r @ xa ) @ top_top_set_real ) @ ( topolo2177554685111907308n_real @ xa @ top_top_set_real ) ).
% that
thf(fact_7_UNIV__I,axiom,
! [X2: nat > real] : ( member_nat_real @ X2 @ top_top_set_nat_real ) ).
% UNIV_I
thf(fact_8_UNIV__I,axiom,
! [X2: real > real] : ( member_real_real @ X2 @ top_to2071711978144146653l_real ) ).
% UNIV_I
thf(fact_9_UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_10_UNIV__I,axiom,
! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% UNIV_I
thf(fact_11_iso__tuple__UNIV__I,axiom,
! [X2: nat > real] : ( member_nat_real @ X2 @ top_top_set_nat_real ) ).
% iso_tuple_UNIV_I
thf(fact_12_iso__tuple__UNIV__I,axiom,
! [X2: real > real] : ( member_real_real @ X2 @ top_to2071711978144146653l_real ) ).
% iso_tuple_UNIV_I
thf(fact_13_iso__tuple__UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_14_iso__tuple__UNIV__I,axiom,
! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% iso_tuple_UNIV_I
thf(fact_15_top__apply,axiom,
( top_top_nat_o
= ( ^ [X: nat] : top_top_o ) ) ).
% top_apply
thf(fact_16_top__apply,axiom,
( top_top_real_o
= ( ^ [X: real] : top_top_o ) ) ).
% top_apply
thf(fact_17_Ints__not__limpt,axiom,
! [X2: real] :
~ ( elemen5683178629028408237t_real @ X2 @ ring_1_Ints_real ) ).
% Ints_not_limpt
thf(fact_18_eventually__frequently__const__simps_I6_J,axiom,
! [C: $o,P: real > $o,F: filter_real] :
( ( eventually_real
@ ^ [X: real] :
( C
=> ( P @ X ) )
@ F )
= ( C
=> ( eventually_real @ P @ F ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_19_eventually__frequently__const__simps_I6_J,axiom,
! [C: $o,P: nat > $o,F: filter_nat] :
( ( eventually_nat
@ ^ [X: nat] :
( C
=> ( P @ X ) )
@ F )
= ( C
=> ( eventually_nat @ P @ F ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_20_eventually__frequently__const__simps_I4_J,axiom,
! [C: $o,P: real > $o,F: filter_real] :
( ( eventually_real
@ ^ [X: real] :
( C
| ( P @ X ) )
@ F )
= ( C
| ( eventually_real @ P @ F ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_21_eventually__frequently__const__simps_I4_J,axiom,
! [C: $o,P: nat > $o,F: filter_nat] :
( ( eventually_nat
@ ^ [X: nat] :
( C
| ( P @ X ) )
@ F )
= ( C
| ( eventually_nat @ P @ F ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_22_eventually__frequently__const__simps_I3_J,axiom,
! [P: real > $o,C: $o,F: filter_real] :
( ( eventually_real
@ ^ [X: real] :
( ( P @ X )
| C )
@ F )
= ( ( eventually_real @ P @ F )
| C ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_23_eventually__frequently__const__simps_I3_J,axiom,
! [P: nat > $o,C: $o,F: filter_nat] :
( ( eventually_nat
@ ^ [X: nat] :
( ( P @ X )
| C )
@ F )
= ( ( eventually_nat @ P @ F )
| C ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_24_eventually__mp,axiom,
! [P: real > $o,Q: real > $o,F: filter_real] :
( ( eventually_real
@ ^ [X: real] :
( ( P @ X )
=> ( Q @ X ) )
@ F )
=> ( ( eventually_real @ P @ F )
=> ( eventually_real @ Q @ F ) ) ) ).
% eventually_mp
thf(fact_25_eventually__mp,axiom,
! [P: nat > $o,Q: nat > $o,F: filter_nat] :
( ( eventually_nat
@ ^ [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
@ F )
=> ( ( eventually_nat @ P @ F )
=> ( eventually_nat @ Q @ F ) ) ) ).
% eventually_mp
thf(fact_26_eventually__True,axiom,
! [F: filter_real] :
( eventually_real
@ ^ [X: real] : $true
@ F ) ).
% eventually_True
thf(fact_27_eventually__True,axiom,
! [F: filter_nat] :
( eventually_nat
@ ^ [X: nat] : $true
@ F ) ).
% eventually_True
thf(fact_28_eventually__conj,axiom,
! [P: real > $o,F: filter_real,Q: real > $o] :
( ( eventually_real @ P @ F )
=> ( ( eventually_real @ Q @ F )
=> ( eventually_real
@ ^ [X: real] :
( ( P @ X )
& ( Q @ X ) )
@ F ) ) ) ).
% eventually_conj
thf(fact_29_eventually__conj,axiom,
! [P: nat > $o,F: filter_nat,Q: nat > $o] :
( ( eventually_nat @ P @ F )
=> ( ( eventually_nat @ Q @ F )
=> ( eventually_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) )
@ F ) ) ) ).
% eventually_conj
thf(fact_30_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_31_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_32_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_33_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_34_eventually__top,axiom,
! [P: real > $o] :
( ( eventually_real @ P @ top_top_filter_real )
= ( ! [X3: real] : ( P @ X3 ) ) ) ).
% eventually_top
thf(fact_35_eventually__top,axiom,
! [P: nat > $o] :
( ( eventually_nat @ P @ top_top_filter_nat )
= ( ! [X3: nat] : ( P @ X3 ) ) ) ).
% eventually_top
thf(fact_36_islimpt__UNIV,axiom,
! [X2: real] : ( elemen5683178629028408237t_real @ X2 @ top_top_set_real ) ).
% islimpt_UNIV
thf(fact_37_filterlim__top,axiom,
! [F2: real > nat,F: filter_real] : ( filterlim_real_nat @ F2 @ top_top_filter_nat @ F ) ).
% filterlim_top
thf(fact_38_filterlim__top,axiom,
! [F2: nat > real,F: filter_nat] : ( filterlim_nat_real @ F2 @ top_top_filter_real @ F ) ).
% filterlim_top
thf(fact_39_filterlim__top,axiom,
! [F2: real > real,F: filter_real] : ( filterlim_real_real @ F2 @ top_top_filter_real @ F ) ).
% filterlim_top
thf(fact_40_filterlim__top,axiom,
! [F2: nat > nat,F: filter_nat] : ( filterlim_nat_nat @ F2 @ top_top_filter_nat @ F ) ).
% filterlim_top
thf(fact_41_filterlim__ident,axiom,
! [F: filter_real] :
( filterlim_real_real
@ ^ [X: real] : X
@ F
@ F ) ).
% filterlim_ident
thf(fact_42_filterlim__ident,axiom,
! [F: filter_nat] :
( filterlim_nat_nat
@ ^ [X: nat] : X
@ F
@ F ) ).
% filterlim_ident
thf(fact_43_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
@ ^ [X: real] : ( G @ ( F2 @ X ) )
@ F3
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_44_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
@ ^ [X: nat] : ( G @ ( F2 @ X ) )
@ F3
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_45_top__set__def,axiom,
( top_top_set_real
= ( collect_real @ top_top_real_o ) ) ).
% top_set_def
thf(fact_46_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_47_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
@ ^ [X: real] : ( P @ ( F2 @ X ) )
@ G2 ) ) ) ).
% eventually_compose_filterlim
thf(fact_48_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
@ ^ [X: nat] : ( P @ ( F2 @ X ) )
@ G2 ) ) ) ).
% eventually_compose_filterlim
thf(fact_49_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
@ ^ [X: real] : ( P @ ( F2 @ X ) )
@ G2 ) ) ) ).
% eventually_compose_filterlim
thf(fact_50_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
@ ^ [X: nat] : ( P @ ( F2 @ X ) )
@ G2 ) ) ) ).
% eventually_compose_filterlim
thf(fact_51_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
@ ^ [X: real] :
( ( F2 @ X )
= ( G @ X ) )
@ F22 )
=> ( ( filterlim_real_real @ F2 @ F1 @ F22 )
= ( filterlim_real_real @ G @ F12 @ F23 ) ) ) ) ) ).
% filterlim_cong
thf(fact_52_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
@ ^ [X: nat] :
( ( F2 @ X )
= ( G @ X ) )
@ F22 )
=> ( ( filterlim_nat_nat @ F2 @ F1 @ F22 )
= ( filterlim_nat_nat @ G @ F12 @ F23 ) ) ) ) ) ).
% filterlim_cong
thf(fact_53_filterlim__iff,axiom,
( filterlim_real_real
= ( ^ [F4: real > real,F24: filter_real,F13: filter_real] :
! [P2: real > $o] :
( ( eventually_real @ P2 @ F24 )
=> ( eventually_real
@ ^ [X: real] : ( P2 @ ( F4 @ X ) )
@ F13 ) ) ) ) ).
% filterlim_iff
thf(fact_54_filterlim__iff,axiom,
( filterlim_nat_real
= ( ^ [F4: nat > real,F24: filter_real,F13: filter_nat] :
! [P2: real > $o] :
( ( eventually_real @ P2 @ F24 )
=> ( eventually_nat
@ ^ [X: nat] : ( P2 @ ( F4 @ X ) )
@ F13 ) ) ) ) ).
% filterlim_iff
thf(fact_55_filterlim__iff,axiom,
( filterlim_real_nat
= ( ^ [F4: real > nat,F24: filter_nat,F13: filter_real] :
! [P2: nat > $o] :
( ( eventually_nat @ P2 @ F24 )
=> ( eventually_real
@ ^ [X: real] : ( P2 @ ( F4 @ X ) )
@ F13 ) ) ) ) ).
% filterlim_iff
thf(fact_56_filterlim__iff,axiom,
( filterlim_nat_nat
= ( ^ [F4: nat > nat,F24: filter_nat,F13: filter_nat] :
! [P2: nat > $o] :
( ( eventually_nat @ P2 @ F24 )
=> ( eventually_nat
@ ^ [X: nat] : ( P2 @ ( F4 @ X ) )
@ F13 ) ) ) ) ).
% filterlim_iff
thf(fact_57_UNIV__witness,axiom,
? [X4: real] : ( member_real @ X4 @ top_top_set_real ) ).
% UNIV_witness
thf(fact_58_UNIV__witness,axiom,
? [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_59_UNIV__eq__I,axiom,
! [A: set_real] :
( ! [X4: real] : ( member_real @ X4 @ A )
=> ( top_top_set_real = A ) ) ).
% UNIV_eq_I
thf(fact_60_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X4: nat] : ( member_nat @ X4 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_61_always__eventually,axiom,
! [P: real > $o,F: filter_real] :
( ! [X_1: real] : ( P @ X_1 )
=> ( eventually_real @ P @ F ) ) ).
% always_eventually
thf(fact_62_always__eventually,axiom,
! [P: nat > $o,F: filter_nat] :
( ! [X_1: nat] : ( P @ X_1 )
=> ( eventually_nat @ P @ F ) ) ).
% always_eventually
thf(fact_63_not__eventuallyD,axiom,
! [P: real > $o,F: filter_real] :
( ~ ( eventually_real @ P @ F )
=> ? [X4: real] :
~ ( P @ X4 ) ) ).
% not_eventuallyD
thf(fact_64_not__eventuallyD,axiom,
! [P: nat > $o,F: filter_nat] :
( ~ ( eventually_nat @ P @ F )
=> ? [X4: nat] :
~ ( P @ X4 ) ) ).
% not_eventuallyD
thf(fact_65_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_66_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_67_filter__eq__iff,axiom,
( ( ^ [Y2: filter_real,Z: filter_real] : ( Y2 = Z ) )
= ( ^ [F5: filter_real,F6: filter_real] :
! [P2: real > $o] :
( ( eventually_real @ P2 @ F5 )
= ( eventually_real @ P2 @ F6 ) ) ) ) ).
% filter_eq_iff
thf(fact_68_filter__eq__iff,axiom,
( ( ^ [Y2: filter_nat,Z: filter_nat] : ( Y2 = Z ) )
= ( ^ [F5: filter_nat,F6: filter_nat] :
! [P2: nat > $o] :
( ( eventually_nat @ P2 @ F5 )
= ( eventually_nat @ P2 @ F6 ) ) ) ) ).
% filter_eq_iff
thf(fact_69_eventuallyI,axiom,
! [P: real > $o,F: filter_real] :
( ! [X_1: real] : ( P @ X_1 )
=> ( eventually_real @ P @ F ) ) ).
% eventuallyI
thf(fact_70_eventuallyI,axiom,
! [P: nat > $o,F: filter_nat] :
( ! [X_1: nat] : ( P @ X_1 )
=> ( eventually_nat @ P @ F ) ) ).
% eventuallyI
thf(fact_71_UNIV__def,axiom,
( top_top_set_real
= ( collect_real
@ ^ [X: real] : $true ) ) ).
% UNIV_def
thf(fact_72_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X: nat] : $true ) ) ).
% UNIV_def
thf(fact_73_not__eventually__impI,axiom,
! [P: real > $o,F: filter_real,Q: real > $o] :
( ( eventually_real @ P @ F )
=> ( ~ ( eventually_real @ Q @ F )
=> ~ ( eventually_real
@ ^ [X: real] :
( ( P @ X )
=> ( Q @ X ) )
@ F ) ) ) ).
% not_eventually_impI
thf(fact_74_not__eventually__impI,axiom,
! [P: nat > $o,F: filter_nat,Q: nat > $o] :
( ( eventually_nat @ P @ F )
=> ( ~ ( eventually_nat @ Q @ F )
=> ~ ( eventually_nat
@ ^ [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
@ F ) ) ) ).
% not_eventually_impI
thf(fact_75_eventually__conj__iff,axiom,
! [P: real > $o,Q: real > $o,F: filter_real] :
( ( eventually_real
@ ^ [X: real] :
( ( P @ X )
& ( Q @ X ) )
@ F )
= ( ( eventually_real @ P @ F )
& ( eventually_real @ Q @ F ) ) ) ).
% eventually_conj_iff
thf(fact_76_eventually__conj__iff,axiom,
! [P: nat > $o,Q: nat > $o,F: filter_nat] :
( ( eventually_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) )
@ F )
= ( ( eventually_nat @ P @ F )
& ( eventually_nat @ Q @ F ) ) ) ).
% eventually_conj_iff
thf(fact_77_eventually__rev__mp,axiom,
! [P: real > $o,F: filter_real,Q: real > $o] :
( ( eventually_real @ P @ F )
=> ( ( eventually_real
@ ^ [X: real] :
( ( P @ X )
=> ( Q @ X ) )
@ F )
=> ( eventually_real @ Q @ F ) ) ) ).
% eventually_rev_mp
thf(fact_78_eventually__rev__mp,axiom,
! [P: nat > $o,F: filter_nat,Q: nat > $o] :
( ( eventually_nat @ P @ F )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
@ F )
=> ( eventually_nat @ Q @ F ) ) ) ).
% eventually_rev_mp
thf(fact_79_filterlim__at__within__not__equal,axiom,
! [F2: real > real,A2: real,S2: set_real,F: filter_real,B: real] :
( ( filterlim_real_real @ F2 @ ( topolo2177554685111907308n_real @ A2 @ S2 ) @ F )
=> ( eventually_real
@ ^ [W: real] :
( ( member_real @ ( F2 @ W ) @ S2 )
& ( ( F2 @ W )
!= B ) )
@ F ) ) ).
% filterlim_at_within_not_equal
thf(fact_80_filterlim__at__within__not__equal,axiom,
! [F2: nat > real,A2: real,S2: set_real,F: filter_nat,B: real] :
( ( filterlim_nat_real @ F2 @ ( topolo2177554685111907308n_real @ A2 @ S2 ) @ F )
=> ( eventually_nat
@ ^ [W: nat] :
( ( member_real @ ( F2 @ W ) @ S2 )
& ( ( F2 @ W )
!= B ) )
@ F ) ) ).
% filterlim_at_within_not_equal
thf(fact_81_filterlim__at__within__not__equal,axiom,
! [F2: real > nat,A2: nat,S2: set_nat,F: filter_real,B: nat] :
( ( filterlim_real_nat @ F2 @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) @ F )
=> ( eventually_real
@ ^ [W: real] :
( ( member_nat @ ( F2 @ W ) @ S2 )
& ( ( F2 @ W )
!= B ) )
@ F ) ) ).
% filterlim_at_within_not_equal
thf(fact_82_filterlim__at__within__not__equal,axiom,
! [F2: nat > nat,A2: nat,S2: set_nat,F: filter_nat,B: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) @ F )
=> ( eventually_nat
@ ^ [W: nat] :
( ( member_nat @ ( F2 @ W ) @ S2 )
& ( ( F2 @ W )
!= B ) )
@ F ) ) ).
% filterlim_at_within_not_equal
thf(fact_83_filterlim__at__If,axiom,
! [F2: real > real,G2: filter_real,X2: real,P: real > $o,G: real > real] :
( ( filterlim_real_real @ F2 @ G2 @ ( topolo2177554685111907308n_real @ X2 @ ( collect_real @ P ) ) )
=> ( ( filterlim_real_real @ G @ G2
@ ( topolo2177554685111907308n_real @ X2
@ ( collect_real
@ ^ [X: real] :
~ ( P @ X ) ) ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( if_real @ ( P @ X ) @ ( F2 @ X ) @ ( G @ X ) )
@ G2
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% filterlim_at_If
thf(fact_84_filterlim__at__If,axiom,
! [F2: nat > nat,G2: filter_nat,X2: nat,P: nat > $o,G: nat > nat] :
( ( filterlim_nat_nat @ F2 @ G2 @ ( topolo4659099751122792720in_nat @ X2 @ ( collect_nat @ P ) ) )
=> ( ( filterlim_nat_nat @ G @ G2
@ ( topolo4659099751122792720in_nat @ X2
@ ( collect_nat
@ ^ [X: nat] :
~ ( P @ X ) ) ) )
=> ( filterlim_nat_nat
@ ^ [X: nat] : ( if_nat @ ( P @ X ) @ ( F2 @ X ) @ ( G @ X ) )
@ G2
@ ( topolo4659099751122792720in_nat @ X2 @ top_top_set_nat ) ) ) ) ).
% filterlim_at_If
thf(fact_85_limpt__of__limpts,axiom,
! [X2: real,S3: set_real] :
( ( elemen5683178629028408237t_real @ X2
@ ( collect_real
@ ^ [Y: real] : ( elemen5683178629028408237t_real @ Y @ S3 ) ) )
=> ( elemen5683178629028408237t_real @ X2 @ S3 ) ) ).
% limpt_of_limpts
thf(fact_86_netlimit__at,axiom,
! [A2: real] :
( ( topolo5976875806579547418l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
@ ^ [X: real] : X )
= A2 ) ).
% netlimit_at
thf(fact_87_eventually__map__filter__on,axiom,
! [X5: set_real,F: filter_real,P: real > $o,F2: real > real] :
( ( eventually_real
@ ^ [X: real] : ( member_real @ X @ X5 )
@ F )
=> ( ( eventually_real @ P @ ( map_fi4827617581384206345l_real @ X5 @ F2 @ F ) )
= ( eventually_real
@ ^ [X: real] :
( ( P @ ( F2 @ X ) )
& ( member_real @ X @ X5 ) )
@ F ) ) ) ).
% eventually_map_filter_on
thf(fact_88_eventually__map__filter__on,axiom,
! [X5: set_real,F: filter_real,P: nat > $o,F2: real > nat] :
( ( eventually_real
@ ^ [X: real] : ( member_real @ X @ X5 )
@ F )
=> ( ( eventually_nat @ P @ ( map_fi4528196046924497837al_nat @ X5 @ F2 @ F ) )
= ( eventually_real
@ ^ [X: real] :
( ( P @ ( F2 @ X ) )
& ( member_real @ X @ X5 ) )
@ F ) ) ) ).
% eventually_map_filter_on
thf(fact_89_eventually__map__filter__on,axiom,
! [X5: set_nat,F: filter_nat,P: real > $o,F2: nat > real] :
( ( eventually_nat
@ ^ [X: nat] : ( member_nat @ X @ X5 )
@ F )
=> ( ( eventually_real @ P @ ( map_fi9184259510650374957t_real @ X5 @ F2 @ F ) )
= ( eventually_nat
@ ^ [X: nat] :
( ( P @ ( F2 @ X ) )
& ( member_nat @ X @ X5 ) )
@ F ) ) ) ).
% eventually_map_filter_on
thf(fact_90_eventually__map__filter__on,axiom,
! [X5: set_nat,F: filter_nat,P: nat > $o,F2: nat > nat] :
( ( eventually_nat
@ ^ [X: nat] : ( member_nat @ X @ X5 )
@ F )
=> ( ( eventually_nat @ P @ ( map_fi8816901555893508305at_nat @ X5 @ F2 @ F ) )
= ( eventually_nat
@ ^ [X: nat] :
( ( P @ ( F2 @ X ) )
& ( member_nat @ X @ X5 ) )
@ F ) ) ) ).
% eventually_map_filter_on
thf(fact_91_trivial__limit__at__iff,axiom,
! [A2: real] :
( ( ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
= bot_bot_filter_real )
= ( ~ ( elemen5683178629028408237t_real @ A2 @ top_top_set_real ) ) ) ).
% trivial_limit_at_iff
thf(fact_92_trivial__limit__at__iff,axiom,
! [A2: nat] :
( ( ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat )
= bot_bot_filter_nat )
= ( ~ ( elemen5607981409700034897pt_nat @ A2 @ top_top_set_nat ) ) ) ).
% trivial_limit_at_iff
thf(fact_93_mem__Collect__eq,axiom,
! [A2: real,P: real > $o] :
( ( member_real @ A2 @ ( collect_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_94_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_95_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X: real] : ( member_real @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_96_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_97_top__empty__eq,axiom,
( top_top_real_o
= ( ^ [X: real] : ( member_real @ X @ top_top_set_real ) ) ) ).
% top_empty_eq
thf(fact_98_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_99_Lim__transform__within__set__eq,axiom,
! [S3: set_real,T: set_real,A2: real,F2: real > real,L: real] :
( ( eventually_real
@ ^ [X: real] :
( ( member_real @ X @ S3 )
= ( member_real @ X @ T ) )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ S3 ) )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ T ) ) ) ) ).
% Lim_transform_within_set_eq
thf(fact_100_Lim__transform__within__set,axiom,
! [F2: real > real,L: real,A2: real,S3: set_real,T: set_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ S3 ) )
=> ( ( eventually_real
@ ^ [X: real] :
( ( member_real @ X @ S3 )
= ( member_real @ X @ 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_101_tendsto__const,axiom,
! [K: nat,F: filter_nat] :
( filterlim_nat_nat
@ ^ [X: nat] : K
@ ( topolo8926549440605965083ds_nat @ K )
@ F ) ).
% tendsto_const
thf(fact_102_tendsto__const,axiom,
! [K: real,F: filter_real] :
( filterlim_real_real
@ ^ [X: real] : K
@ ( topolo2815343760600316023s_real @ K )
@ F ) ).
% tendsto_const
thf(fact_103_eventually__const,axiom,
! [F: filter_real,P: $o] :
( ( F != bot_bot_filter_real )
=> ( ( eventually_real
@ ^ [X: real] : P
@ F )
= P ) ) ).
% eventually_const
thf(fact_104_eventually__const,axiom,
! [F: filter_nat,P: $o] :
( ( F != bot_bot_filter_nat )
=> ( ( eventually_nat
@ ^ [X: nat] : P
@ F )
= P ) ) ).
% eventually_const
thf(fact_105_tendsto__ident__at,axiom,
! [A2: real,S2: set_real] :
( filterlim_real_real
@ ^ [X: real] : X
@ ( topolo2815343760600316023s_real @ A2 )
@ ( topolo2177554685111907308n_real @ A2 @ S2 ) ) ).
% tendsto_ident_at
thf(fact_106_tendsto__ident__at,axiom,
! [A2: nat,S2: set_nat] :
( filterlim_nat_nat
@ ^ [X: nat] : X
@ ( topolo8926549440605965083ds_nat @ A2 )
@ ( topolo4659099751122792720in_nat @ A2 @ S2 ) ) ).
% tendsto_ident_at
thf(fact_107_tendsto__unique,axiom,
! [F: filter_nat,F2: nat > nat,A2: nat,B: nat] :
( ( F != bot_bot_filter_nat )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ A2 ) @ F )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ B ) @ F )
=> ( A2 = B ) ) ) ) ).
% tendsto_unique
thf(fact_108_tendsto__unique,axiom,
! [F: filter_real,F2: real > real,A2: real,B: real] :
( ( F != bot_bot_filter_real )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ B ) @ F )
=> ( A2 = B ) ) ) ) ).
% tendsto_unique
thf(fact_109_tendsto__unique,axiom,
! [F: filter_nat,F2: nat > real,A2: real,B: real] :
( ( F != bot_bot_filter_nat )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ B ) @ F )
=> ( A2 = B ) ) ) ) ).
% tendsto_unique
thf(fact_110_tendsto__bot,axiom,
! [F2: nat > nat,A2: nat] : ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ A2 ) @ bot_bot_filter_nat ) ).
% tendsto_bot
thf(fact_111_tendsto__bot,axiom,
! [F2: real > real,A2: real] : ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ bot_bot_filter_real ) ).
% tendsto_bot
thf(fact_112_tendsto__bot,axiom,
! [F2: nat > real,A2: real] : ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ bot_bot_filter_nat ) ).
% tendsto_bot
thf(fact_113_tendsto__Lim,axiom,
! [Net: filter_nat,F2: nat > nat,L: nat] :
( ( Net != bot_bot_filter_nat )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ Net )
=> ( ( topolo4574594659991002850at_nat @ Net @ F2 )
= L ) ) ) ).
% tendsto_Lim
thf(fact_114_tendsto__Lim,axiom,
! [Net: filter_real,F2: real > real,L: real] :
( ( Net != bot_bot_filter_real )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ Net )
=> ( ( topolo5976875806579547418l_real @ Net @ F2 )
= L ) ) ) ).
% tendsto_Lim
thf(fact_115_tendsto__Lim,axiom,
! [Net: filter_nat,F2: nat > real,L: real] :
( ( Net != bot_bot_filter_nat )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ Net )
=> ( ( topolo2843583510664976574t_real @ Net @ F2 )
= L ) ) ) ).
% tendsto_Lim
thf(fact_116_nhds__neq__bot,axiom,
! [A2: nat] :
( ( topolo8926549440605965083ds_nat @ A2 )
!= bot_bot_filter_nat ) ).
% nhds_neq_bot
thf(fact_117_nhds__neq__bot,axiom,
! [A2: real] :
( ( topolo2815343760600316023s_real @ A2 )
!= bot_bot_filter_real ) ).
% nhds_neq_bot
thf(fact_118_tendsto__const__iff,axiom,
! [F: filter_nat,A2: nat,B: nat] :
( ( F != bot_bot_filter_nat )
=> ( ( filterlim_nat_nat
@ ^ [X: nat] : A2
@ ( topolo8926549440605965083ds_nat @ B )
@ F )
= ( A2 = B ) ) ) ).
% tendsto_const_iff
thf(fact_119_tendsto__const__iff,axiom,
! [F: filter_real,A2: real,B: real] :
( ( F != bot_bot_filter_real )
=> ( ( filterlim_real_real
@ ^ [X: real] : A2
@ ( topolo2815343760600316023s_real @ B )
@ F )
= ( A2 = B ) ) ) ).
% tendsto_const_iff
thf(fact_120_tendsto__const__iff,axiom,
! [F: filter_nat,A2: real,B: real] :
( ( F != bot_bot_filter_nat )
=> ( ( filterlim_nat_real
@ ^ [X: nat] : A2
@ ( topolo2815343760600316023s_real @ B )
@ F )
= ( A2 = B ) ) ) ).
% tendsto_const_iff
thf(fact_121_Lim__ident__at,axiom,
! [X2: real,S2: set_real] :
( ( ( topolo2177554685111907308n_real @ X2 @ S2 )
!= bot_bot_filter_real )
=> ( ( topolo5976875806579547418l_real @ ( topolo2177554685111907308n_real @ X2 @ S2 )
@ ^ [X: real] : X )
= X2 ) ) ).
% Lim_ident_at
thf(fact_122_Lim__ident__at,axiom,
! [X2: nat,S2: set_nat] :
( ( ( topolo4659099751122792720in_nat @ X2 @ S2 )
!= bot_bot_filter_nat )
=> ( ( topolo4574594659991002850at_nat @ ( topolo4659099751122792720in_nat @ X2 @ S2 )
@ ^ [X: nat] : X )
= X2 ) ) ).
% Lim_ident_at
thf(fact_123_Lim__trivial__limit,axiom,
! [Net: filter_nat,F2: nat > nat,L: nat] :
( ( Net = bot_bot_filter_nat )
=> ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ Net ) ) ).
% Lim_trivial_limit
thf(fact_124_Lim__trivial__limit,axiom,
! [Net: filter_real,F2: real > real,L: real] :
( ( Net = bot_bot_filter_real )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ Net ) ) ).
% Lim_trivial_limit
thf(fact_125_Lim__trivial__limit,axiom,
! [Net: filter_nat,F2: nat > real,L: real] :
( ( Net = bot_bot_filter_nat )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ Net ) ) ).
% Lim_trivial_limit
thf(fact_126_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_127_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_128_tendsto__eq__rhs,axiom,
! [F2: nat > nat,X2: nat,F: filter_nat,Y3: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ X2 ) @ F )
=> ( ( X2 = Y3 )
=> ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ Y3 ) @ F ) ) ) ).
% tendsto_eq_rhs
thf(fact_129_tendsto__eq__rhs,axiom,
! [F2: real > real,X2: real,F: filter_real,Y3: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ X2 ) @ F )
=> ( ( X2 = Y3 )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ Y3 ) @ F ) ) ) ).
% tendsto_eq_rhs
thf(fact_130_at__discrete,axiom,
( topolo4659099751122792720in_nat
= ( ^ [X: nat,S: set_nat] : bot_bot_filter_nat ) ) ).
% at_discrete
thf(fact_131_eventually__nhds__x__imp__x,axiom,
! [P: nat > $o,X2: nat] :
( ( eventually_nat @ P @ ( topolo8926549440605965083ds_nat @ X2 ) )
=> ( P @ X2 ) ) ).
% eventually_nhds_x_imp_x
thf(fact_132_eventually__nhds__x__imp__x,axiom,
! [P: real > $o,X2: real] :
( ( eventually_real @ P @ ( topolo2815343760600316023s_real @ X2 ) )
=> ( P @ X2 ) ) ).
% eventually_nhds_x_imp_x
thf(fact_133_t1__space__nhds,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( eventually_nat
@ ^ [X: nat] : ( X != Y3 )
@ ( topolo8926549440605965083ds_nat @ X2 ) ) ) ).
% t1_space_nhds
thf(fact_134_t1__space__nhds,axiom,
! [X2: real,Y3: real] :
( ( X2 != Y3 )
=> ( eventually_real
@ ^ [X: real] : ( X != Y3 )
@ ( topolo2815343760600316023s_real @ X2 ) ) ) ).
% t1_space_nhds
thf(fact_135_eventually__eventually,axiom,
! [P: nat > $o,X2: nat] :
( ( eventually_nat
@ ^ [Y: nat] : ( eventually_nat @ P @ ( topolo8926549440605965083ds_nat @ Y ) )
@ ( topolo8926549440605965083ds_nat @ X2 ) )
= ( eventually_nat @ P @ ( topolo8926549440605965083ds_nat @ X2 ) ) ) ).
% eventually_eventually
thf(fact_136_eventually__eventually,axiom,
! [P: real > $o,X2: real] :
( ( eventually_real
@ ^ [Y: real] : ( eventually_real @ P @ ( topolo2815343760600316023s_real @ Y ) )
@ ( topolo2815343760600316023s_real @ X2 ) )
= ( eventually_real @ P @ ( topolo2815343760600316023s_real @ X2 ) ) ) ).
% eventually_eventually
thf(fact_137_eventually__bot,axiom,
! [P: real > $o] : ( eventually_real @ P @ bot_bot_filter_real ) ).
% eventually_bot
thf(fact_138_eventually__bot,axiom,
! [P: nat > $o] : ( eventually_nat @ P @ bot_bot_filter_nat ) ).
% eventually_bot
thf(fact_139_eventually__happens,axiom,
! [P: real > $o,Net: filter_real] :
( ( eventually_real @ P @ Net )
=> ( ( Net = bot_bot_filter_real )
| ? [X_1: real] : ( P @ X_1 ) ) ) ).
% eventually_happens
thf(fact_140_eventually__happens,axiom,
! [P: nat > $o,Net: filter_nat] :
( ( eventually_nat @ P @ Net )
=> ( ( Net = bot_bot_filter_nat )
| ? [X_1: nat] : ( P @ X_1 ) ) ) ).
% eventually_happens
thf(fact_141_eventually__happens_H,axiom,
! [F: filter_real,P: real > $o] :
( ( F != bot_bot_filter_real )
=> ( ( eventually_real @ P @ F )
=> ? [X_1: real] : ( P @ X_1 ) ) ) ).
% eventually_happens'
thf(fact_142_eventually__happens_H,axiom,
! [F: filter_nat,P: nat > $o] :
( ( F != bot_bot_filter_nat )
=> ( ( eventually_nat @ P @ F )
=> ? [X_1: nat] : ( P @ X_1 ) ) ) ).
% eventually_happens'
thf(fact_143_trivial__limit__eq,axiom,
! [Net: filter_real] :
( ( Net = bot_bot_filter_real )
= ( ! [P2: real > $o] : ( eventually_real @ P2 @ Net ) ) ) ).
% trivial_limit_eq
thf(fact_144_trivial__limit__eq,axiom,
! [Net: filter_nat] :
( ( Net = bot_bot_filter_nat )
= ( ! [P2: nat > $o] : ( eventually_nat @ P2 @ Net ) ) ) ).
% trivial_limit_eq
thf(fact_145_trivial__limit__eventually,axiom,
! [Net: filter_real,P: real > $o] :
( ( Net = bot_bot_filter_real )
=> ( eventually_real @ P @ Net ) ) ).
% trivial_limit_eventually
thf(fact_146_trivial__limit__eventually,axiom,
! [Net: filter_nat,P: nat > $o] :
( ( Net = bot_bot_filter_nat )
=> ( eventually_nat @ P @ Net ) ) ).
% trivial_limit_eventually
thf(fact_147_trivial__limit__def,axiom,
! [F: filter_real] :
( ( F = bot_bot_filter_real )
= ( eventually_real
@ ^ [X: real] : $false
@ F ) ) ).
% trivial_limit_def
thf(fact_148_trivial__limit__def,axiom,
! [F: filter_nat] :
( ( F = bot_bot_filter_nat )
= ( eventually_nat
@ ^ [X: nat] : $false
@ F ) ) ).
% trivial_limit_def
thf(fact_149_eventually__const__iff,axiom,
! [P: $o,F: filter_real] :
( ( eventually_real
@ ^ [X: real] : P
@ F )
= ( P
| ( F = bot_bot_filter_real ) ) ) ).
% eventually_const_iff
thf(fact_150_eventually__const__iff,axiom,
! [P: $o,F: filter_nat] :
( ( eventually_nat
@ ^ [X: nat] : P
@ F )
= ( P
| ( F = bot_bot_filter_nat ) ) ) ).
% eventually_const_iff
thf(fact_151_False__imp__not__eventually,axiom,
! [P: real > $o,Net: filter_real] :
( ! [X4: real] :
~ ( P @ X4 )
=> ( ( Net != bot_bot_filter_real )
=> ~ ( eventually_real @ P @ Net ) ) ) ).
% False_imp_not_eventually
thf(fact_152_False__imp__not__eventually,axiom,
! [P: nat > $o,Net: filter_nat] :
( ! [X4: nat] :
~ ( P @ X4 )
=> ( ( Net != bot_bot_filter_nat )
=> ~ ( eventually_nat @ P @ Net ) ) ) ).
% False_imp_not_eventually
thf(fact_153_eventually__at__filter,axiom,
! [P: real > $o,A2: real,S2: set_real] :
( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A2 @ S2 ) )
= ( eventually_real
@ ^ [X: real] :
( ( X != A2 )
=> ( ( member_real @ X @ S2 )
=> ( P @ X ) ) )
@ ( topolo2815343760600316023s_real @ A2 ) ) ) ).
% eventually_at_filter
thf(fact_154_eventually__at__filter,axiom,
! [P: nat > $o,A2: nat,S2: set_nat] :
( ( eventually_nat @ P @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) )
= ( eventually_nat
@ ^ [X: nat] :
( ( X != A2 )
=> ( ( member_nat @ X @ S2 )
=> ( P @ X ) ) )
@ ( topolo8926549440605965083ds_nat @ A2 ) ) ) ).
% eventually_at_filter
thf(fact_155_tendsto__cong,axiom,
! [F2: nat > nat,G: nat > nat,F: filter_nat,C2: nat] :
( ( eventually_nat
@ ^ [X: nat] :
( ( F2 @ X )
= ( G @ X ) )
@ F )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ C2 ) @ F )
= ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ C2 ) @ F ) ) ) ).
% tendsto_cong
thf(fact_156_tendsto__cong,axiom,
! [F2: real > real,G: real > real,F: filter_real,C2: real] :
( ( eventually_real
@ ^ [X: real] :
( ( F2 @ X )
= ( G @ X ) )
@ F )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
= ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ C2 ) @ F ) ) ) ).
% tendsto_cong
thf(fact_157_tendsto__cong,axiom,
! [F2: nat > real,G: nat > real,F: filter_nat,C2: real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( F2 @ X )
= ( G @ X ) )
@ F )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
= ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ C2 ) @ F ) ) ) ).
% tendsto_cong
thf(fact_158_tendsto__discrete,axiom,
! [F2: nat > nat,Y3: nat,F: filter_nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ Y3 ) @ F )
= ( eventually_nat
@ ^ [X: nat] :
( ( F2 @ X )
= Y3 )
@ F ) ) ).
% tendsto_discrete
thf(fact_159_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_160_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_161_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_162_tendsto__eventually,axiom,
! [F2: nat > nat,L: nat,Net: filter_nat] :
( ( eventually_nat
@ ^ [X: nat] :
( ( F2 @ X )
= L )
@ Net )
=> ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ Net ) ) ).
% tendsto_eventually
thf(fact_163_tendsto__eventually,axiom,
! [F2: real > real,L: real,Net: filter_real] :
( ( eventually_real
@ ^ [X: real] :
( ( F2 @ X )
= L )
@ Net )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ Net ) ) ).
% tendsto_eventually
thf(fact_164_tendsto__eventually,axiom,
! [F2: nat > real,L: real,Net: filter_nat] :
( ( eventually_nat
@ ^ [X: nat] :
( ( F2 @ X )
= L )
@ Net )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ Net ) ) ).
% tendsto_eventually
thf(fact_165_netlimit__at__vector,axiom,
! [A2: real] :
( ( topolo5976875806579547418l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
@ ^ [X: real] : X )
= A2 ) ).
% netlimit_at_vector
thf(fact_166_eventually__Lim__ident__at,axiom,
! [P: real > real > $o,X2: real,X5: set_real] :
( ( eventually_real
@ ( P
@ ( topolo5976875806579547418l_real @ ( topolo2177554685111907308n_real @ X2 @ X5 )
@ ^ [X: real] : X ) )
@ ( topolo2177554685111907308n_real @ X2 @ X5 ) )
= ( eventually_real @ ( P @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ X5 ) ) ) ).
% eventually_Lim_ident_at
thf(fact_167_eventually__Lim__ident__at,axiom,
! [P: nat > nat > $o,X2: nat,X5: set_nat] :
( ( eventually_nat
@ ( P
@ ( topolo4574594659991002850at_nat @ ( topolo4659099751122792720in_nat @ X2 @ X5 )
@ ^ [X: nat] : X ) )
@ ( topolo4659099751122792720in_nat @ X2 @ X5 ) )
= ( eventually_nat @ ( P @ X2 ) @ ( topolo4659099751122792720in_nat @ X2 @ X5 ) ) ) ).
% eventually_Lim_ident_at
thf(fact_168_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_169_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_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_trivial__limit__at,axiom,
! [A2: real] :
( ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
!= bot_bot_filter_real ) ).
% trivial_limit_at
thf(fact_179_trivial__limit__within,axiom,
! [A2: real,S3: set_real] :
( ( ( topolo2177554685111907308n_real @ A2 @ S3 )
= bot_bot_filter_real )
= ( ~ ( elemen5683178629028408237t_real @ A2 @ S3 ) ) ) ).
% trivial_limit_within
thf(fact_180_trivial__limit__within,axiom,
! [A2: nat,S3: set_nat] :
( ( ( topolo4659099751122792720in_nat @ A2 @ S3 )
= bot_bot_filter_nat )
= ( ~ ( elemen5607981409700034897pt_nat @ A2 @ S3 ) ) ) ).
% trivial_limit_within
thf(fact_181_LIM__const__eq,axiom,
! [K: real,L2: real,A2: real] :
( ( filterlim_real_real
@ ^ [X: real] : K
@ ( topolo2815343760600316023s_real @ L2 )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( K = L2 ) ) ).
% LIM_const_eq
thf(fact_182_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
@ ^ [X: nat] : ( G @ ( F2 @ X ) )
@ ( topolo8926549440605965083ds_nat @ ( G @ L ) )
@ F ) ) ) ).
% tendsto_compose
thf(fact_183_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
@ ^ [X: real] : ( G @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ ( G @ L ) )
@ F ) ) ) ).
% tendsto_compose
thf(fact_184_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
@ ^ [X: nat] : ( G @ ( F2 @ X ) )
@ ( topolo8926549440605965083ds_nat @ ( G @ L ) )
@ F ) ) ) ).
% tendsto_compose
thf(fact_185_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
@ ^ [X: real] : ( G @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ ( G @ L ) )
@ F ) ) ) ).
% tendsto_compose
thf(fact_186_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
@ ^ [X: nat] : ( G @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ ( G @ L ) )
@ F ) ) ) ).
% tendsto_compose
thf(fact_187_LIM__const__not__eq,axiom,
! [K: real,L2: real,A2: real] :
( ( K != L2 )
=> ~ ( filterlim_real_real
@ ^ [X: real] : K
@ ( topolo2815343760600316023s_real @ L2 )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ).
% LIM_const_not_eq
thf(fact_188_filterlim__at,axiom,
! [F2: real > real,B: real,S2: set_real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2177554685111907308n_real @ B @ S2 ) @ F )
= ( ( eventually_real
@ ^ [X: real] :
( ( member_real @ ( F2 @ X ) @ S2 )
& ( ( F2 @ X )
!= B ) )
@ F )
& ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ B ) @ F ) ) ) ).
% filterlim_at
thf(fact_189_filterlim__at,axiom,
! [F2: nat > real,B: real,S2: set_real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2177554685111907308n_real @ B @ S2 ) @ F )
= ( ( eventually_nat
@ ^ [X: nat] :
( ( member_real @ ( F2 @ X ) @ S2 )
& ( ( F2 @ X )
!= B ) )
@ F )
& ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ B ) @ F ) ) ) ).
% filterlim_at
thf(fact_190_filterlim__at,axiom,
! [F2: real > nat,B: nat,S2: set_nat,F: filter_real] :
( ( filterlim_real_nat @ F2 @ ( topolo4659099751122792720in_nat @ B @ S2 ) @ F )
= ( ( eventually_real
@ ^ [X: real] :
( ( member_nat @ ( F2 @ X ) @ S2 )
& ( ( F2 @ X )
!= B ) )
@ F )
& ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ B ) @ F ) ) ) ).
% filterlim_at
thf(fact_191_filterlim__at,axiom,
! [F2: nat > nat,B: nat,S2: set_nat,F: filter_nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo4659099751122792720in_nat @ B @ S2 ) @ F )
= ( ( eventually_nat
@ ^ [X: nat] :
( ( member_nat @ ( F2 @ X ) @ S2 )
& ( ( F2 @ X )
!= B ) )
@ F )
& ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ B ) @ F ) ) ) ).
% filterlim_at
thf(fact_192_Lim__at__imp__Lim__at__within,axiom,
! [F2: real > real,L: real,X2: real,S3: set_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ X2 @ S3 ) ) ) ).
% Lim_at_imp_Lim_at_within
thf(fact_193_Lim__at__imp__Lim__at__within,axiom,
! [F2: nat > nat,L: nat,X2: nat,S3: set_nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ X2 @ top_top_set_nat ) )
=> ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ X2 @ S3 ) ) ) ).
% Lim_at_imp_Lim_at_within
thf(fact_194_Lim__at__imp__Lim__at__within,axiom,
! [F2: nat > real,L: real,X2: nat,S3: set_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ X2 @ top_top_set_nat ) )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ X2 @ S3 ) ) ) ).
% Lim_at_imp_Lim_at_within
thf(fact_195_filterlim__atI,axiom,
! [F2: real > real,C2: real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ( ( eventually_real
@ ^ [X: real] :
( ( F2 @ X )
!= C2 )
@ F )
=> ( filterlim_real_real @ F2 @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) @ F ) ) ) ).
% filterlim_atI
thf(fact_196_filterlim__atI,axiom,
! [F2: nat > real,C2: real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( F2 @ X )
!= C2 )
@ F )
=> ( filterlim_nat_real @ F2 @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) @ F ) ) ) ).
% filterlim_atI
thf(fact_197_filterlim__atI,axiom,
! [F2: real > nat,C2: nat,F: filter_real] :
( ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ C2 ) @ F )
=> ( ( eventually_real
@ ^ [X: real] :
( ( F2 @ X )
!= C2 )
@ F )
=> ( filterlim_real_nat @ F2 @ ( topolo4659099751122792720in_nat @ C2 @ top_top_set_nat ) @ F ) ) ) ).
% filterlim_atI
thf(fact_198_filterlim__atI,axiom,
! [F2: nat > nat,C2: nat,F: filter_nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ C2 ) @ F )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( F2 @ X )
!= C2 )
@ F )
=> ( filterlim_nat_nat @ F2 @ ( topolo4659099751122792720in_nat @ C2 @ top_top_set_nat ) @ F ) ) ) ).
% filterlim_atI
thf(fact_199_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
@ ^ [X: real] :
( ( F2 @ X )
!= B )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( G @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ C2 )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_200_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
@ ^ [X: real] :
( ( F2 @ X )
!= B )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( filterlim_real_nat
@ ^ [X: real] : ( G @ ( F2 @ X ) )
@ ( topolo8926549440605965083ds_nat @ C2 )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_201_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
@ ^ [X: real] :
( ( F2 @ X )
!= B )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( G @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ C2 )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_202_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
@ ^ [X: nat] :
( ( F2 @ X )
!= B )
@ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
=> ( filterlim_nat_nat
@ ^ [X: nat] : ( G @ ( F2 @ X ) )
@ ( topolo8926549440605965083ds_nat @ C2 )
@ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_203_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
@ ^ [X: nat] :
( ( F2 @ X )
!= B )
@ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
=> ( filterlim_nat_real
@ ^ [X: nat] : ( G @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ C2 )
@ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_204_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
@ ^ [X: nat] :
( ( F2 @ X )
!= B )
@ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
=> ( filterlim_nat_nat
@ ^ [X: nat] : ( G @ ( F2 @ X ) )
@ ( topolo8926549440605965083ds_nat @ C2 )
@ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_205_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
@ ^ [X: nat] :
( ( F2 @ X )
!= B )
@ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) )
=> ( filterlim_nat_real
@ ^ [X: nat] : ( G @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ C2 )
@ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_206_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
@ ^ [X: nat] :
( ( F2 @ X )
!= L )
@ F )
=> ( filterlim_nat_nat
@ ^ [X: nat] : ( G @ ( F2 @ X ) )
@ ( topolo8926549440605965083ds_nat @ M )
@ F ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_207_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
@ ^ [X: real] :
( ( F2 @ X )
!= L )
@ F )
=> ( filterlim_real_real
@ ^ [X: real] : ( G @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ M )
@ F ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_208_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
@ ^ [X: nat] :
( ( F2 @ X )
!= L )
@ F )
=> ( filterlim_nat_real
@ ^ [X: nat] : ( G @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ M )
@ F ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_209_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
@ ^ [X: real] :
( ( F2 @ X )
!= L )
@ F )
=> ( filterlim_real_nat
@ ^ [X: real] : ( G @ ( F2 @ X ) )
@ ( topolo8926549440605965083ds_nat @ M )
@ F ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_210_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
@ ^ [X: nat] :
( ( F2 @ X )
!= L )
@ F )
=> ( filterlim_nat_nat
@ ^ [X: nat] : ( G @ ( F2 @ X ) )
@ ( topolo8926549440605965083ds_nat @ M )
@ F ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_211_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
@ ^ [X: real] :
( ( F2 @ X )
!= L )
@ F )
=> ( filterlim_real_real
@ ^ [X: real] : ( G @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ M )
@ F ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_212_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
@ ^ [X: nat] :
( ( F2 @ X )
!= L )
@ F )
=> ( filterlim_nat_real
@ ^ [X: nat] : ( G @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ M )
@ F ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_213_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_214_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_215_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_216_tendsto__discrete__iff,axiom,
! [F2: nat > nat,C2: nat,F: filter_nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ C2 ) @ F )
= ( eventually_nat
@ ^ [X: nat] :
( ( F2 @ X )
= C2 )
@ F ) ) ).
% tendsto_discrete_iff
thf(fact_217_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
@ ^ [X: nat] :
( ( F2 @ X )
= ( G @ X ) )
@ F )
=> ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ L ) @ F ) ) ) ).
% Lim_transform_eventually
thf(fact_218_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
@ ^ [X: real] :
( ( F2 @ X )
= ( G @ X ) )
@ F )
=> ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ) ).
% Lim_transform_eventually
thf(fact_219_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
@ ^ [X: nat] :
( ( F2 @ X )
= ( G @ X ) )
@ F )
=> ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ) ).
% Lim_transform_eventually
thf(fact_220_Lim__transform__away__within,axiom,
! [A2: real,B: real,S3: set_real,F2: real > real,G: real > real,L: real] :
( ( A2 != B )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S3 )
=> ( ( ( X4 != A2 )
& ( X4 != B ) )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ S3 ) )
=> ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ S3 ) ) ) ) ) ).
% Lim_transform_away_within
thf(fact_221_Lim__transform__away__within,axiom,
! [A2: nat,B: nat,S3: set_nat,F2: nat > nat,G: nat > nat,L: nat] :
( ( A2 != B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S3 )
=> ( ( ( X4 != A2 )
& ( X4 != B ) )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S3 ) )
=> ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S3 ) ) ) ) ) ).
% Lim_transform_away_within
thf(fact_222_Lim__transform__away__within,axiom,
! [A2: nat,B: nat,S3: set_nat,F2: nat > real,G: nat > real,L: real] :
( ( A2 != B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S3 )
=> ( ( ( X4 != A2 )
& ( X4 != B ) )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S3 ) )
=> ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S3 ) ) ) ) ) ).
% Lim_transform_away_within
thf(fact_223_Lim__cong__within,axiom,
! [A2: real,B: real,X2: real,Y3: real,S3: set_real,T: set_real,F2: real > real,G: real > real] :
( ( A2 = B )
=> ( ( X2 = Y3 )
=> ( ( S3 = T )
=> ( ! [X4: real] :
( ( X4 != B )
=> ( ( member_real @ X4 @ T )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ X2 ) @ ( topolo2177554685111907308n_real @ A2 @ S3 ) )
= ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ Y3 ) @ ( topolo2177554685111907308n_real @ B @ T ) ) ) ) ) ) ) ).
% Lim_cong_within
thf(fact_224_Lim__cong__within,axiom,
! [A2: nat,B: nat,X2: nat,Y3: nat,S3: set_nat,T: set_nat,F2: nat > nat,G: nat > nat] :
( ( A2 = B )
=> ( ( X2 = Y3 )
=> ( ( S3 = T )
=> ( ! [X4: nat] :
( ( X4 != B )
=> ( ( member_nat @ X4 @ T )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ X2 ) @ ( topolo4659099751122792720in_nat @ A2 @ S3 ) )
= ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ Y3 ) @ ( topolo4659099751122792720in_nat @ B @ T ) ) ) ) ) ) ) ).
% Lim_cong_within
thf(fact_225_Lim__cong__within,axiom,
! [A2: nat,B: nat,X2: real,Y3: real,S3: set_nat,T: set_nat,F2: nat > real,G: nat > real] :
( ( A2 = B )
=> ( ( X2 = Y3 )
=> ( ( S3 = T )
=> ( ! [X4: nat] :
( ( X4 != B )
=> ( ( member_nat @ X4 @ T )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ X2 ) @ ( topolo4659099751122792720in_nat @ A2 @ S3 ) )
= ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ Y3 ) @ ( topolo4659099751122792720in_nat @ B @ T ) ) ) ) ) ) ) ).
% Lim_cong_within
thf(fact_226_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_227_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_228_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_229_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_230_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_231_tendsto__compose__at,axiom,
! [F2: nat > real,Y3: real,F: filter_nat,G: real > nat,Z3: nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ Y3 ) @ F )
=> ( ( filterlim_real_nat @ G @ ( topolo8926549440605965083ds_nat @ Z3 ) @ ( topolo2177554685111907308n_real @ Y3 @ top_top_set_real ) )
=> ( ( eventually_nat
@ ^ [W: nat] :
( ( ( F2 @ W )
= Y3 )
=> ( ( G @ Y3 )
= Z3 ) )
@ F )
=> ( filterlim_nat_nat @ ( comp_real_nat_nat @ G @ F2 ) @ ( topolo8926549440605965083ds_nat @ Z3 ) @ F ) ) ) ) ).
% tendsto_compose_at
thf(fact_232_tendsto__compose__at,axiom,
! [F2: real > real,Y3: real,F: filter_real,G: real > real,Z3: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ Y3 ) @ F )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ Z3 ) @ ( topolo2177554685111907308n_real @ Y3 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [W: real] :
( ( ( F2 @ W )
= Y3 )
=> ( ( G @ Y3 )
= Z3 ) )
@ F )
=> ( filterlim_real_real @ ( comp_real_real_real @ G @ F2 ) @ ( topolo2815343760600316023s_real @ Z3 ) @ F ) ) ) ) ).
% tendsto_compose_at
thf(fact_233_tendsto__compose__at,axiom,
! [F2: nat > real,Y3: real,F: filter_nat,G: real > real,Z3: real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ Y3 ) @ F )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ Z3 ) @ ( topolo2177554685111907308n_real @ Y3 @ top_top_set_real ) )
=> ( ( eventually_nat
@ ^ [W: nat] :
( ( ( F2 @ W )
= Y3 )
=> ( ( G @ Y3 )
= Z3 ) )
@ F )
=> ( filterlim_nat_real @ ( comp_real_real_nat @ G @ F2 ) @ ( topolo2815343760600316023s_real @ Z3 ) @ F ) ) ) ) ).
% tendsto_compose_at
thf(fact_234_tendsto__compose__at,axiom,
! [F2: real > nat,Y3: nat,F: filter_real,G: nat > nat,Z3: nat] :
( ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ Y3 ) @ F )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ Z3 ) @ ( topolo4659099751122792720in_nat @ Y3 @ top_top_set_nat ) )
=> ( ( eventually_real
@ ^ [W: real] :
( ( ( F2 @ W )
= Y3 )
=> ( ( G @ Y3 )
= Z3 ) )
@ F )
=> ( filterlim_real_nat @ ( comp_nat_nat_real @ G @ F2 ) @ ( topolo8926549440605965083ds_nat @ Z3 ) @ F ) ) ) ) ).
% tendsto_compose_at
thf(fact_235_tendsto__compose__at,axiom,
! [F2: nat > nat,Y3: nat,F: filter_nat,G: nat > nat,Z3: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ Y3 ) @ F )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ Z3 ) @ ( topolo4659099751122792720in_nat @ Y3 @ top_top_set_nat ) )
=> ( ( eventually_nat
@ ^ [W: nat] :
( ( ( F2 @ W )
= Y3 )
=> ( ( G @ Y3 )
= Z3 ) )
@ F )
=> ( filterlim_nat_nat @ ( comp_nat_nat_nat @ G @ F2 ) @ ( topolo8926549440605965083ds_nat @ Z3 ) @ F ) ) ) ) ).
% tendsto_compose_at
thf(fact_236_tendsto__compose__at,axiom,
! [F2: real > nat,Y3: nat,F: filter_real,G: nat > real,Z3: real] :
( ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ Y3 ) @ F )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ Z3 ) @ ( topolo4659099751122792720in_nat @ Y3 @ top_top_set_nat ) )
=> ( ( eventually_real
@ ^ [W: real] :
( ( ( F2 @ W )
= Y3 )
=> ( ( G @ Y3 )
= Z3 ) )
@ F )
=> ( filterlim_real_real @ ( comp_nat_real_real @ G @ F2 ) @ ( topolo2815343760600316023s_real @ Z3 ) @ F ) ) ) ) ).
% tendsto_compose_at
thf(fact_237_tendsto__compose__at,axiom,
! [F2: nat > nat,Y3: nat,F: filter_nat,G: nat > real,Z3: real] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ Y3 ) @ F )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ Z3 ) @ ( topolo4659099751122792720in_nat @ Y3 @ top_top_set_nat ) )
=> ( ( eventually_nat
@ ^ [W: nat] :
( ( ( F2 @ W )
= Y3 )
=> ( ( G @ Y3 )
= Z3 ) )
@ F )
=> ( filterlim_nat_real @ ( comp_nat_real_nat @ G @ F2 ) @ ( topolo2815343760600316023s_real @ Z3 ) @ F ) ) ) ) ).
% tendsto_compose_at
thf(fact_238_tendsto__unique_H,axiom,
! [F: filter_nat,F2: nat > nat] :
( ( F != bot_bot_filter_nat )
=> ( uniq_nat
@ ^ [L4: nat] : ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L4 ) @ F ) ) ) ).
% tendsto_unique'
thf(fact_239_tendsto__unique_H,axiom,
! [F: filter_real,F2: real > real] :
( ( F != bot_bot_filter_real )
=> ( uniq_real
@ ^ [L4: real] : ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L4 ) @ F ) ) ) ).
% tendsto_unique'
thf(fact_240_tendsto__unique_H,axiom,
! [F: filter_nat,F2: nat > real] :
( ( F != bot_bot_filter_nat )
=> ( uniq_real
@ ^ [L4: real] : ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L4 ) @ F ) ) ) ).
% tendsto_unique'
thf(fact_241_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_real
@ ^ [S4: real] : P )
= top_top_set_real ) )
& ( ~ P
=> ( ( collect_real
@ ^ [S4: real] : P )
= bot_bot_set_real ) ) ) ).
% Collect_const
thf(fact_242_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_nat
@ ^ [S4: nat] : P )
= top_top_set_nat ) )
& ( ~ P
=> ( ( collect_nat
@ ^ [S4: nat] : P )
= bot_bot_set_nat ) ) ) ).
% Collect_const
thf(fact_243_at__within__empty,axiom,
! [A2: real] :
( ( topolo2177554685111907308n_real @ A2 @ bot_bot_set_real )
= bot_bot_filter_real ) ).
% at_within_empty
thf(fact_244_at__within__empty,axiom,
! [A2: nat] :
( ( topolo4659099751122792720in_nat @ A2 @ bot_bot_set_nat )
= bot_bot_filter_nat ) ).
% at_within_empty
thf(fact_245_eventually__not__equal__at__infinity,axiom,
! [A2: real] :
( eventually_real
@ ^ [X: real] : ( X != A2 )
@ at_infinity_real ) ).
% eventually_not_equal_at_infinity
thf(fact_246_empty__not__UNIV,axiom,
bot_bot_set_real != top_top_set_real ).
% empty_not_UNIV
thf(fact_247_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_248_islimpt__EMPTY,axiom,
! [X2: real] :
~ ( elemen5683178629028408237t_real @ X2 @ bot_bot_set_real ) ).
% islimpt_EMPTY
thf(fact_249_islimpt__EMPTY,axiom,
! [X2: nat] :
~ ( elemen5607981409700034897pt_nat @ X2 @ bot_bot_set_nat ) ).
% islimpt_EMPTY
thf(fact_250_trivial__limit__at__infinity,axiom,
at_infinity_real != bot_bot_filter_real ).
% trivial_limit_at_infinity
thf(fact_251_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_252_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_253_not__tendsto__and__filterlim__at__infinity,axiom,
! [F: filter_real,F2: real > real,C2: real] :
( ( F != bot_bot_filter_real )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ~ ( filterlim_real_real @ F2 @ at_infinity_real @ F ) ) ) ).
% not_tendsto_and_filterlim_at_infinity
thf(fact_254_not__tendsto__and__filterlim__at__infinity,axiom,
! [F: filter_nat,F2: nat > real,C2: real] :
( ( F != bot_bot_filter_nat )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ~ ( filterlim_nat_real @ F2 @ at_infinity_real @ F ) ) ) ).
% not_tendsto_and_filterlim_at_infinity
thf(fact_255_LIM__not__zero,axiom,
! [K: real,A2: real] :
( ( K != zero_zero_real )
=> ~ ( filterlim_real_real
@ ^ [X: real] : K
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ).
% LIM_not_zero
thf(fact_256_Ints__0,axiom,
member_real @ zero_zero_real @ ring_1_Ints_real ).
% Ints_0
thf(fact_257_is__pole__tendsto,axiom,
! [F2: real > real,X2: real] :
( ( comple7683793008646357389l_real @ F2 @ X2 )
=> ( filterlim_real_real @ ( comp_real_real_real @ inverse_inverse_real @ F2 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% is_pole_tendsto
thf(fact_258_is__pole__tendsto,axiom,
! [F2: nat > real,X2: nat] :
( ( comple320020743214073521t_real @ F2 @ X2 )
=> ( filterlim_nat_real @ ( comp_real_real_nat @ inverse_inverse_real @ F2 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo4659099751122792720in_nat @ X2 @ top_top_set_nat ) ) ) ).
% is_pole_tendsto
thf(fact_259_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_260_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
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ at_infinity_real
@ F ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_261_empty__iff,axiom,
! [C2: real] :
~ ( member_real @ C2 @ bot_bot_set_real ) ).
% empty_iff
thf(fact_262_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_263_all__not__in__conv,axiom,
! [A: set_real] :
( ( ! [X: real] :
~ ( member_real @ X @ A ) )
= ( A = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_264_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_265_emptyE,axiom,
! [A2: real] :
~ ( member_real @ A2 @ bot_bot_set_real ) ).
% emptyE
thf(fact_266_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_267_equals0D,axiom,
! [A: set_real,A2: real] :
( ( A = bot_bot_set_real )
=> ~ ( member_real @ A2 @ A ) ) ).
% equals0D
thf(fact_268_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_269_equals0I,axiom,
! [A: set_real] :
( ! [Y4: real] :
~ ( member_real @ Y4 @ A )
=> ( A = bot_bot_set_real ) ) ).
% equals0I
thf(fact_270_equals0I,axiom,
! [A: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_271_ex__in__conv,axiom,
! [A: set_real] :
( ( ? [X: real] : ( member_real @ X @ A ) )
= ( A != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_272_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_273_bot__empty__eq,axiom,
( bot_bot_real_o
= ( ^ [X: real] : ( member_real @ X @ bot_bot_set_real ) ) ) ).
% bot_empty_eq
thf(fact_274_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_275_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_276_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
@ ^ [X: real] : ( inverse_inverse_real @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ ( inverse_inverse_real @ A2 ) )
@ F ) ) ) ).
% tendsto_inverse
thf(fact_277_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
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ ( divide_divide_real @ A2 @ B ) )
@ F ) ) ) ) ).
% tendsto_divide
thf(fact_278_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
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ C2 )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ).
% tendsto_divide_zero
thf(fact_279_tendsto__inverse__0,axiom,
filterlim_real_real @ inverse_inverse_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_infinity_real ).
% tendsto_inverse_0
thf(fact_280_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
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ) ).
% tendsto_divide_0
thf(fact_281_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_282_filterlim__inverse__at__iff,axiom,
! [G: real > real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X: real] : ( inverse_inverse_real @ ( G @ X ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real )
@ F )
= ( filterlim_real_real @ G @ at_infinity_real @ F ) ) ).
% filterlim_inverse_at_iff
thf(fact_283_inverse__divide,axiom,
! [A2: real,B: real] :
( ( inverse_inverse_real @ ( divide_divide_real @ A2 @ B ) )
= ( divide_divide_real @ B @ A2 ) ) ).
% inverse_divide
thf(fact_284_inverse__nonzero__iff__nonzero,axiom,
! [A2: real] :
( ( ( inverse_inverse_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_285_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_286_division__ring__divide__zero,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_287_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_288_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_289_div__by__0,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_290_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_291_inverse__inverse__eq,axiom,
! [A2: real] :
( ( inverse_inverse_real @ ( inverse_inverse_real @ A2 ) )
= A2 ) ).
% inverse_inverse_eq
thf(fact_292_div__0,axiom,
! [A2: real] :
( ( divide_divide_real @ zero_zero_real @ A2 )
= zero_zero_real ) ).
% div_0
thf(fact_293_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_294_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_295_nonzero__imp__inverse__nonzero,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( inverse_inverse_real @ A2 )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_296_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_297_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_298_inverse__zero__imp__zero,axiom,
! [A2: real] :
( ( ( inverse_inverse_real @ A2 )
= zero_zero_real )
=> ( A2 = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_299_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_300_lim__zero__infinity,axiom,
! [F2: real > real,L: real] :
( ( filterlim_real_real
@ ^ [X: real] : ( F2 @ ( divide_divide_real @ one_one_real @ X ) )
@ ( 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_301_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
@ ^ [X: real] : ( inverse_inverse_real @ ( F2 @ X ) )
@ F ) ) ) ).
% Bfun_inverse
thf(fact_302_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_303_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_304_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_305_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_306_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_307_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_308_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_309_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_310_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_311_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_312_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_313_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_314_add__0,axiom,
! [A2: real] :
( ( plus_plus_real @ zero_zero_real @ A2 )
= A2 ) ).
% add_0
thf(fact_315_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y3 ) )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_316_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_317_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_318_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_319_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_320_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_321_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_322_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_323_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_324_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_325_double__zero__sym,axiom,
! [A2: real] :
( ( zero_zero_real
= ( plus_plus_real @ A2 @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_326_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_327_add_Oright__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% add.right_neutral
thf(fact_328_div__by__1,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ one_one_real )
= A2 ) ).
% div_by_1
thf(fact_329_inverse__1,axiom,
( ( inverse_inverse_real @ one_one_real )
= one_one_real ) ).
% inverse_1
thf(fact_330_inverse__eq__1__iff,axiom,
! [X2: real] :
( ( ( inverse_inverse_real @ X2 )
= one_one_real )
= ( X2 = one_one_real ) ) ).
% inverse_eq_1_iff
thf(fact_331_Ints__add__iff1,axiom,
! [X2: real,Y3: real] :
( ( member_real @ X2 @ ring_1_Ints_real )
=> ( ( member_real @ ( plus_plus_real @ X2 @ Y3 ) @ ring_1_Ints_real )
= ( member_real @ Y3 @ ring_1_Ints_real ) ) ) ).
% Ints_add_iff1
thf(fact_332_Ints__add__iff2,axiom,
! [Y3: real,X2: real] :
( ( member_real @ Y3 @ ring_1_Ints_real )
=> ( ( member_real @ ( plus_plus_real @ X2 @ Y3 ) @ ring_1_Ints_real )
= ( member_real @ X2 @ ring_1_Ints_real ) ) ) ).
% Ints_add_iff2
thf(fact_333_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_334_div__self,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= one_one_real ) ) ).
% div_self
thf(fact_335_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_336_divide__self,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= one_one_real ) ) ).
% divide_self
thf(fact_337_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_338_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_339_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_340_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_341_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_342_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_343_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_344_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_345_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_346_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_347_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_348_group__cancel_Oadd2,axiom,
! [B2: real,K: real,B: real,A2: real] :
( ( B2
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A2 @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_349_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A2: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_350_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_351_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_352_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_353_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_354_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B3: real] : ( plus_plus_real @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_355_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_356_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_357_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_358_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_359_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_360_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_361_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_362_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_363_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_364_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_365_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_366_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 )
@ ^ [X: real] : ( plus_plus_real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% isCont_add
thf(fact_367_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 )
@ ^ [X: real] : ( plus_plus_nat @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% isCont_add
thf(fact_368_isCont__add,axiom,
! [A2: nat,F2: nat > real,G: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) @ F2 )
=> ( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) @ G )
=> ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat )
@ ^ [X: nat] : ( plus_plus_real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% isCont_add
thf(fact_369_isCont__add,axiom,
! [A2: nat,F2: nat > nat,G: nat > nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) @ F2 )
=> ( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) @ G )
=> ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat )
@ ^ [X: nat] : ( plus_plus_nat @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% isCont_add
thf(fact_370_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_371_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_372_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_373_add_Ocomm__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% add.comm_neutral
thf(fact_374_add_Ogroup__left__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ zero_zero_real @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_375_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_376_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_377_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_378_Ints__1,axiom,
member_real @ one_one_real @ ring_1_Ints_real ).
% Ints_1
thf(fact_379_continuous__ident,axiom,
! [X2: real,S3: set_real] :
( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ S3 )
@ ^ [X: real] : X ) ).
% continuous_ident
thf(fact_380_continuous__ident,axiom,
! [X2: nat,S3: set_nat] :
( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ X2 @ S3 )
@ ^ [X: nat] : X ) ).
% continuous_ident
thf(fact_381_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_382_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_383_inverse__eq__divide,axiom,
( inverse_inverse_real
= ( divide_divide_real @ one_one_real ) ) ).
% inverse_eq_divide
thf(fact_384_isCont__iff,axiom,
! [X2: real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ F2 )
= ( filterlim_real_real
@ ^ [H: real] : ( F2 @ ( plus_plus_real @ X2 @ H ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ X2 ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ).
% isCont_iff
thf(fact_385_tendsto__add__const__iff,axiom,
! [C2: real,F2: real > real,D: real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X: real] : ( plus_plus_real @ C2 @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ ( plus_plus_real @ C2 @ D ) )
@ F )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ D ) @ F ) ) ).
% tendsto_add_const_iff
thf(fact_386_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
@ ^ [X: nat] : ( plus_plus_nat @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo8926549440605965083ds_nat @ ( plus_plus_nat @ A2 @ B ) )
@ F ) ) ) ).
% tendsto_add
thf(fact_387_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
@ ^ [X: real] : ( plus_plus_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ ( plus_plus_real @ A2 @ B ) )
@ F ) ) ) ).
% tendsto_add
thf(fact_388_continuous__within,axiom,
! [X2: real,S2: set_real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ S2 ) @ F2 )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ ( F2 @ X2 ) ) @ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ).
% continuous_within
thf(fact_389_continuous__within,axiom,
! [X2: nat,S2: set_nat,F2: nat > nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ X2 @ S2 ) @ F2 )
= ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ ( F2 @ X2 ) ) @ ( topolo4659099751122792720in_nat @ X2 @ S2 ) ) ) ).
% continuous_within
thf(fact_390_continuous__within,axiom,
! [X2: nat,S2: set_nat,F2: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ X2 @ S2 ) @ F2 )
= ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ ( F2 @ X2 ) ) @ ( topolo4659099751122792720in_nat @ X2 @ S2 ) ) ) ).
% continuous_within
thf(fact_391_continuous__at__within__divide,axiom,
! [A2: real,S2: set_real,F2: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S2 ) @ F2 )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S2 ) @ G )
=> ( ( ( G @ A2 )
!= zero_zero_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S2 )
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_at_within_divide
thf(fact_392_continuous__at__within__divide,axiom,
! [A2: nat,S2: set_nat,F2: nat > real,G: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) @ F2 )
=> ( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) @ G )
=> ( ( ( G @ A2 )
!= zero_zero_real )
=> ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ S2 )
@ ^ [X: nat] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_at_within_divide
thf(fact_393_continuous__at__within__inverse,axiom,
! [A2: real,S2: set_real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S2 ) @ F2 )
=> ( ( ( F2 @ A2 )
!= zero_zero_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S2 )
@ ^ [X: real] : ( inverse_inverse_real @ ( F2 @ X ) ) ) ) ) ).
% continuous_at_within_inverse
thf(fact_394_continuous__at__within__inverse,axiom,
! [A2: nat,S2: set_nat,F2: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) @ F2 )
=> ( ( ( F2 @ A2 )
!= zero_zero_real )
=> ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ S2 )
@ ^ [X: nat] : ( inverse_inverse_real @ ( F2 @ X ) ) ) ) ) ).
% continuous_at_within_inverse
thf(fact_395_continuous__within__tendsto__compose_H,axiom,
! [A2: real,S2: set_real,F2: real > nat,X2: nat > real,F: filter_nat] :
( ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ A2 @ S2 ) @ F2 )
=> ( ! [N2: nat] : ( member_real @ ( X2 @ N2 ) @ S2 )
=> ( ( filterlim_nat_real @ X2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F2 @ ( X2 @ N ) )
@ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose'
thf(fact_396_continuous__within__tendsto__compose_H,axiom,
! [A2: real,S2: set_real,F2: real > real,X2: real > real,F: filter_real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S2 ) @ F2 )
=> ( ! [N2: real] : ( member_real @ ( X2 @ N2 ) @ S2 )
=> ( ( filterlim_real_real @ X2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( filterlim_real_real
@ ^ [N: real] : ( F2 @ ( X2 @ N ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose'
thf(fact_397_continuous__within__tendsto__compose_H,axiom,
! [A2: nat,S2: set_nat,F2: nat > nat,X2: nat > nat,F: filter_nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) @ F2 )
=> ( ! [N2: nat] : ( member_nat @ ( X2 @ N2 ) @ S2 )
=> ( ( filterlim_nat_nat @ X2 @ ( topolo8926549440605965083ds_nat @ A2 ) @ F )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F2 @ ( X2 @ N ) )
@ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose'
thf(fact_398_continuous__within__tendsto__compose_H,axiom,
! [A2: nat,S2: set_nat,F2: nat > real,X2: real > nat,F: filter_real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) @ F2 )
=> ( ! [N2: real] : ( member_nat @ ( X2 @ N2 ) @ S2 )
=> ( ( filterlim_real_nat @ X2 @ ( topolo8926549440605965083ds_nat @ A2 ) @ F )
=> ( filterlim_real_real
@ ^ [N: real] : ( F2 @ ( X2 @ N ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose'
thf(fact_399_continuous__within__tendsto__compose_H,axiom,
! [A2: nat,S2: set_nat,F2: nat > real,X2: nat > nat,F: filter_nat] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) @ F2 )
=> ( ! [N2: nat] : ( member_nat @ ( X2 @ N2 ) @ S2 )
=> ( ( filterlim_nat_nat @ X2 @ ( topolo8926549440605965083ds_nat @ A2 ) @ F )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F2 @ ( X2 @ N ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose'
thf(fact_400_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_401_continuous__def,axiom,
( topolo1306369304726495905at_nat
= ( ^ [F5: filter_nat,F4: nat > nat] :
( filterlim_nat_nat @ F4
@ ( topolo8926549440605965083ds_nat
@ ( F4
@ ( topolo4574594659991002850at_nat @ F5
@ ^ [X: nat] : X ) ) )
@ F5 ) ) ) ).
% continuous_def
thf(fact_402_continuous__def,axiom,
( topolo4422821103128117721l_real
= ( ^ [F5: filter_real,F4: real > real] :
( filterlim_real_real @ F4
@ ( topolo2815343760600316023s_real
@ ( F4
@ ( topolo5976875806579547418l_real @ F5
@ ^ [X: real] : X ) ) )
@ F5 ) ) ) ).
% continuous_def
thf(fact_403_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
@ ^ [X: nat] : ( plus_plus_nat @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo8926549440605965083ds_nat @ zero_zero_nat )
@ F ) ) ) ).
% tendsto_add_zero
thf(fact_404_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
@ ^ [X: real] : ( plus_plus_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ) ).
% tendsto_add_zero
thf(fact_405_continuous__at,axiom,
! [X2: real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ F2 )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ ( F2 @ X2 ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% continuous_at
thf(fact_406_continuous__at,axiom,
! [X2: nat,F2: nat > nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ X2 @ top_top_set_nat ) @ F2 )
= ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ ( F2 @ X2 ) ) @ ( topolo4659099751122792720in_nat @ X2 @ top_top_set_nat ) ) ) ).
% continuous_at
thf(fact_407_continuous__at,axiom,
! [X2: nat,F2: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ X2 @ top_top_set_nat ) @ F2 )
= ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ ( F2 @ X2 ) ) @ ( topolo4659099751122792720in_nat @ X2 @ top_top_set_nat ) ) ) ).
% continuous_at
thf(fact_408_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_409_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_410_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_411_isContD,axiom,
! [X2: real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ F2 )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ ( F2 @ X2 ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% isContD
thf(fact_412_isContD,axiom,
! [X2: nat,F2: nat > nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ X2 @ top_top_set_nat ) @ F2 )
=> ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ ( F2 @ X2 ) ) @ ( topolo4659099751122792720in_nat @ X2 @ top_top_set_nat ) ) ) ).
% isContD
thf(fact_413_isContD,axiom,
! [X2: nat,F2: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ X2 @ top_top_set_nat ) @ F2 )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ ( F2 @ X2 ) ) @ ( topolo4659099751122792720in_nat @ X2 @ top_top_set_nat ) ) ) ).
% isContD
thf(fact_414_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
@ ^ [X: real] : ( plus_plus_real @ ( F2 @ X ) @ ( G @ X ) )
@ at_infinity_real
@ F ) ) ) ).
% tendsto_add_filterlim_at_infinity
thf(fact_415_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
@ ^ [X: real] : ( plus_plus_real @ ( F2 @ X ) @ ( G @ X ) )
@ at_infinity_real
@ F ) ) ) ).
% tendsto_add_filterlim_at_infinity'
thf(fact_416_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 )
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% isCont_divide
thf(fact_417_isCont__divide,axiom,
! [A2: nat,F2: nat > real,G: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) @ F2 )
=> ( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) @ G )
=> ( ( ( G @ A2 )
!= zero_zero_real )
=> ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat )
@ ^ [X: nat] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% isCont_divide
thf(fact_418_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
@ ^ [X: nat] : ( G @ ( F2 @ X ) )
@ ( topolo8926549440605965083ds_nat @ ( G @ L ) )
@ F ) ) ) ).
% isCont_tendsto_compose
thf(fact_419_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
@ ^ [X: real] : ( G @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ ( G @ L ) )
@ F ) ) ) ).
% isCont_tendsto_compose
thf(fact_420_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
@ ^ [X: nat] : ( G @ ( F2 @ X ) )
@ ( topolo8926549440605965083ds_nat @ ( G @ L ) )
@ F ) ) ) ).
% isCont_tendsto_compose
thf(fact_421_isCont__tendsto__compose,axiom,
! [L: nat,G: nat > real,F2: real > nat,F: filter_real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ L @ top_top_set_nat ) @ G )
=> ( ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
=> ( filterlim_real_real
@ ^ [X: real] : ( G @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ ( G @ L ) )
@ F ) ) ) ).
% isCont_tendsto_compose
thf(fact_422_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
@ ^ [X: nat] : ( G @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ ( G @ L ) )
@ F ) ) ) ).
% isCont_tendsto_compose
thf(fact_423_continuous__within__tendsto__compose,axiom,
! [A2: real,S2: set_real,F2: real > nat,X2: nat > real,F: filter_nat] :
( ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ A2 @ S2 ) @ F2 )
=> ( ( eventually_nat
@ ^ [N: nat] : ( member_real @ ( X2 @ N ) @ S2 )
@ F )
=> ( ( filterlim_nat_real @ X2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F2 @ ( X2 @ N ) )
@ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose
thf(fact_424_continuous__within__tendsto__compose,axiom,
! [A2: real,S2: set_real,F2: real > real,X2: real > real,F: filter_real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S2 ) @ F2 )
=> ( ( eventually_real
@ ^ [N: real] : ( member_real @ ( X2 @ N ) @ S2 )
@ F )
=> ( ( filterlim_real_real @ X2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( filterlim_real_real
@ ^ [N: real] : ( F2 @ ( X2 @ N ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose
thf(fact_425_continuous__within__tendsto__compose,axiom,
! [A2: real,S2: set_real,F2: real > real,X2: nat > real,F: filter_nat] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S2 ) @ F2 )
=> ( ( eventually_nat
@ ^ [N: nat] : ( member_real @ ( X2 @ N ) @ S2 )
@ F )
=> ( ( filterlim_nat_real @ X2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F2 @ ( X2 @ N ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose
thf(fact_426_continuous__within__tendsto__compose,axiom,
! [A2: nat,S2: set_nat,F2: nat > nat,X2: nat > nat,F: filter_nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) @ F2 )
=> ( ( eventually_nat
@ ^ [N: nat] : ( member_nat @ ( X2 @ N ) @ S2 )
@ F )
=> ( ( filterlim_nat_nat @ X2 @ ( topolo8926549440605965083ds_nat @ A2 ) @ F )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F2 @ ( X2 @ N ) )
@ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose
thf(fact_427_continuous__within__tendsto__compose,axiom,
! [A2: nat,S2: set_nat,F2: nat > real,X2: real > nat,F: filter_real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) @ F2 )
=> ( ( eventually_real
@ ^ [N: real] : ( member_nat @ ( X2 @ N ) @ S2 )
@ F )
=> ( ( filterlim_real_nat @ X2 @ ( topolo8926549440605965083ds_nat @ A2 ) @ F )
=> ( filterlim_real_real
@ ^ [N: real] : ( F2 @ ( X2 @ N ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose
thf(fact_428_continuous__within__tendsto__compose,axiom,
! [A2: nat,S2: set_nat,F2: nat > real,X2: nat > nat,F: filter_nat] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) @ F2 )
=> ( ( eventually_nat
@ ^ [N: nat] : ( member_nat @ ( X2 @ N ) @ S2 )
@ F )
=> ( ( filterlim_nat_nat @ X2 @ ( topolo8926549440605965083ds_nat @ A2 ) @ F )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F2 @ ( X2 @ N ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ F ) ) ) ) ).
% continuous_within_tendsto_compose
thf(fact_429_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
@ ^ [X: real] : ( F2 @ ( plus_plus_real @ X @ A2 ) )
@ F
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ).
% filterlim_at_to_0
thf(fact_430_eventually__at__to__0,axiom,
! [P: real > $o,A2: real] :
( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
= ( eventually_real
@ ^ [X: real] : ( P @ ( plus_plus_real @ X @ A2 ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ).
% eventually_at_to_0
thf(fact_431_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
@ ^ [X: real] : ( F2 @ ( plus_plus_real @ A2 @ X ) )
@ ( topolo2815343760600316023s_real @ L )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ).
% Lim_at_zero
thf(fact_432_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_433_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_434_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_435_fps__tan__0,axiom,
( ( formal3683295897622742886n_real @ zero_zero_real )
= zero_z7760665558314615101s_real ) ).
% fps_tan_0
thf(fact_436_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_437_pth__7_I1_J,axiom,
! [X2: real] :
( ( plus_plus_real @ zero_zero_real @ X2 )
= X2 ) ).
% pth_7(1)
thf(fact_438_eq__add__iff,axiom,
! [X2: real,Y3: real] :
( ( X2
= ( plus_plus_real @ X2 @ Y3 ) )
= ( Y3 = zero_zero_real ) ) ).
% eq_add_iff
thf(fact_439_pth__d,axiom,
! [X2: real] :
( ( plus_plus_real @ X2 @ zero_zero_real )
= X2 ) ).
% pth_d
thf(fact_440_tendsto__offset__zero__iff,axiom,
! [A2: real,S3: set_real,F2: real > real,L2: real] :
( ( nO_MATCH_real_real @ zero_zero_real @ A2 )
=> ( ( member_real @ A2 @ S3 )
=> ( ( topolo4860482606490270245n_real @ S3 )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ A2 @ S3 ) )
= ( 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_441_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_442_verit__sum__simplify,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% verit_sum_simplify
thf(fact_443_add__0__iff,axiom,
! [B: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ B @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_444_add__0__iff,axiom,
! [B: real,A2: real] :
( ( B
= ( plus_plus_real @ B @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% add_0_iff
thf(fact_445_tendsto__divide__smallo,axiom,
! [F14: real > real,G1: real > real,A2: real,F: filter_real,F25: real > real,G22: real > real] :
( ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F14 @ X ) @ ( G1 @ X ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F )
=> ( ( member_real_real @ F25 @ ( landau3007391416991288786l_real @ F @ F14 ) )
=> ( ( member_real_real @ G22 @ ( landau3007391416991288786l_real @ F @ G1 ) )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G1 @ X )
!= zero_zero_real )
@ F )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( plus_plus_real @ ( F14 @ X ) @ ( F25 @ X ) ) @ ( plus_plus_real @ ( G1 @ X ) @ ( G22 @ X ) ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F ) ) ) ) ) ).
% tendsto_divide_smallo
thf(fact_446_tendsto__divide__smallo,axiom,
! [F14: nat > real,G1: nat > real,A2: real,F: filter_nat,F25: nat > real,G22: nat > real] :
( ( filterlim_nat_real
@ ^ [X: nat] : ( divide_divide_real @ ( F14 @ X ) @ ( G1 @ X ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F )
=> ( ( member_nat_real @ F25 @ ( landau997807338407142774t_real @ F @ F14 ) )
=> ( ( member_nat_real @ G22 @ ( landau997807338407142774t_real @ F @ G1 ) )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( G1 @ X )
!= zero_zero_real )
@ F )
=> ( filterlim_nat_real
@ ^ [X: nat] : ( divide_divide_real @ ( plus_plus_real @ ( F14 @ X ) @ ( F25 @ X ) ) @ ( plus_plus_real @ ( G1 @ X ) @ ( G22 @ X ) ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F ) ) ) ) ) ).
% tendsto_divide_smallo
thf(fact_447_open__UNIV,axiom,
topolo4860482606490270245n_real @ top_top_set_real ).
% open_UNIV
thf(fact_448_open__UNIV,axiom,
topolo4328251076210115529en_nat @ top_top_set_nat ).
% open_UNIV
thf(fact_449_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
@ ^ [X: real] :
( ( F2 @ X )
= zero_zero_real )
@ F ) ) ).
% in_smallo_zero_iff
thf(fact_450_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
@ ^ [X: nat] :
( ( F2 @ X )
= zero_zero_real )
@ F ) ) ).
% in_smallo_zero_iff
thf(fact_451_t0__space,axiom,
! [X2: real,Y3: real] :
( ( X2 != Y3 )
=> ? [U: set_real] :
( ( topolo4860482606490270245n_real @ U )
& ( ( member_real @ X2 @ U )
!= ( member_real @ Y3 @ U ) ) ) ) ).
% t0_space
thf(fact_452_t0__space,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ? [U: set_nat] :
( ( topolo4328251076210115529en_nat @ U )
& ( ( member_nat @ X2 @ U )
!= ( member_nat @ Y3 @ U ) ) ) ) ).
% t0_space
thf(fact_453_t1__space,axiom,
! [X2: real,Y3: real] :
( ( X2 != Y3 )
=> ? [U: set_real] :
( ( topolo4860482606490270245n_real @ U )
& ( member_real @ X2 @ U )
& ~ ( member_real @ Y3 @ U ) ) ) ).
% t1_space
thf(fact_454_t1__space,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ? [U: set_nat] :
( ( topolo4328251076210115529en_nat @ U )
& ( member_nat @ X2 @ U )
& ~ ( member_nat @ Y3 @ U ) ) ) ).
% t1_space
thf(fact_455_separation__t0,axiom,
! [X2: real,Y3: real] :
( ( X2 != Y3 )
= ( ? [U2: set_real] :
( ( topolo4860482606490270245n_real @ U2 )
& ( ( member_real @ X2 @ U2 )
!= ( member_real @ Y3 @ U2 ) ) ) ) ) ).
% separation_t0
thf(fact_456_separation__t0,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
= ( ? [U2: set_nat] :
( ( topolo4328251076210115529en_nat @ U2 )
& ( ( member_nat @ X2 @ U2 )
!= ( member_nat @ Y3 @ U2 ) ) ) ) ) ).
% separation_t0
thf(fact_457_separation__t1,axiom,
! [X2: real,Y3: real] :
( ( X2 != Y3 )
= ( ? [U2: set_real] :
( ( topolo4860482606490270245n_real @ U2 )
& ( member_real @ X2 @ U2 )
& ~ ( member_real @ Y3 @ U2 ) ) ) ) ).
% separation_t1
thf(fact_458_separation__t1,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
= ( ? [U2: set_nat] :
( ( topolo4328251076210115529en_nat @ U2 )
& ( member_nat @ X2 @ U2 )
& ~ ( member_nat @ Y3 @ U2 ) ) ) ) ).
% separation_t1
thf(fact_459_islimptI,axiom,
! [X2: real,S3: set_real] :
( ! [T2: set_real] :
( ( member_real @ X2 @ T2 )
=> ( ( topolo4860482606490270245n_real @ T2 )
=> ? [X6: real] :
( ( member_real @ X6 @ S3 )
& ( member_real @ X6 @ T2 )
& ( X6 != X2 ) ) ) )
=> ( elemen5683178629028408237t_real @ X2 @ S3 ) ) ).
% islimptI
thf(fact_460_islimptI,axiom,
! [X2: nat,S3: set_nat] :
( ! [T2: set_nat] :
( ( member_nat @ X2 @ T2 )
=> ( ( topolo4328251076210115529en_nat @ T2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ S3 )
& ( member_nat @ X6 @ T2 )
& ( X6 != X2 ) ) ) )
=> ( elemen5607981409700034897pt_nat @ X2 @ S3 ) ) ).
% islimptI
thf(fact_461_islimptE,axiom,
! [X2: real,S3: set_real,T: set_real] :
( ( elemen5683178629028408237t_real @ X2 @ S3 )
=> ( ( member_real @ X2 @ T )
=> ( ( topolo4860482606490270245n_real @ T )
=> ~ ! [Y4: real] :
( ( member_real @ Y4 @ S3 )
=> ( ( member_real @ Y4 @ T )
=> ( Y4 = X2 ) ) ) ) ) ) ).
% islimptE
thf(fact_462_islimptE,axiom,
! [X2: nat,S3: set_nat,T: set_nat] :
( ( elemen5607981409700034897pt_nat @ X2 @ S3 )
=> ( ( member_nat @ X2 @ T )
=> ( ( topolo4328251076210115529en_nat @ T )
=> ~ ! [Y4: nat] :
( ( member_nat @ Y4 @ S3 )
=> ( ( member_nat @ Y4 @ T )
=> ( Y4 = X2 ) ) ) ) ) ) ).
% islimptE
thf(fact_463_islimpt__def,axiom,
( elemen5683178629028408237t_real
= ( ^ [X: real,S: set_real] :
! [T3: set_real] :
( ( member_real @ X @ T3 )
=> ( ( topolo4860482606490270245n_real @ T3 )
=> ? [Y: real] :
( ( member_real @ Y @ S )
& ( member_real @ Y @ T3 )
& ( Y != X ) ) ) ) ) ) ).
% islimpt_def
thf(fact_464_islimpt__def,axiom,
( elemen5607981409700034897pt_nat
= ( ^ [X: nat,S: set_nat] :
! [T3: set_nat] :
( ( member_nat @ X @ T3 )
=> ( ( topolo4328251076210115529en_nat @ T3 )
=> ? [Y: nat] :
( ( member_nat @ Y @ S )
& ( member_nat @ Y @ T3 )
& ( Y != X ) ) ) ) ) ) ).
% islimpt_def
thf(fact_465_at__within__open,axiom,
! [A2: real,S3: set_real] :
( ( member_real @ A2 @ S3 )
=> ( ( topolo4860482606490270245n_real @ S3 )
=> ( ( topolo2177554685111907308n_real @ A2 @ S3 )
= ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ).
% at_within_open
thf(fact_466_at__within__open,axiom,
! [A2: nat,S3: set_nat] :
( ( member_nat @ A2 @ S3 )
=> ( ( topolo4328251076210115529en_nat @ S3 )
=> ( ( topolo4659099751122792720in_nat @ A2 @ S3 )
= ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ).
% at_within_open
thf(fact_467_at__within__open__NO__MATCH,axiom,
! [A2: real,S2: set_real] :
( ( member_real @ A2 @ S2 )
=> ( ( topolo4860482606490270245n_real @ S2 )
=> ( ( nO_MAT2855227906214470577t_real @ top_top_set_real @ S2 )
=> ( ( topolo2177554685111907308n_real @ A2 @ S2 )
= ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ).
% at_within_open_NO_MATCH
thf(fact_468_at__within__open__NO__MATCH,axiom,
! [A2: real,S2: set_real] :
( ( member_real @ A2 @ S2 )
=> ( ( topolo4860482606490270245n_real @ S2 )
=> ( ( nO_MAT8790521740642007125t_real @ top_top_set_nat @ S2 )
=> ( ( topolo2177554685111907308n_real @ A2 @ S2 )
= ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ).
% at_within_open_NO_MATCH
thf(fact_469_at__within__open__NO__MATCH,axiom,
! [A2: nat,S2: set_nat] :
( ( member_nat @ A2 @ S2 )
=> ( ( topolo4328251076210115529en_nat @ S2 )
=> ( ( nO_MAT504328087405689813et_nat @ top_top_set_real @ S2 )
=> ( ( topolo4659099751122792720in_nat @ A2 @ S2 )
= ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ).
% at_within_open_NO_MATCH
thf(fact_470_at__within__open__NO__MATCH,axiom,
! [A2: nat,S2: set_nat] :
( ( member_nat @ A2 @ S2 )
=> ( ( topolo4328251076210115529en_nat @ S2 )
=> ( ( nO_MAT2475032472373502585et_nat @ top_top_set_nat @ S2 )
=> ( ( topolo4659099751122792720in_nat @ A2 @ S2 )
= ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ).
% at_within_open_NO_MATCH
thf(fact_471_eventually__at__topological,axiom,
! [P: real > $o,A2: real,S2: set_real] :
( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A2 @ S2 ) )
= ( ? [S: set_real] :
( ( topolo4860482606490270245n_real @ S )
& ( member_real @ A2 @ S )
& ! [X: real] :
( ( member_real @ X @ S )
=> ( ( X != A2 )
=> ( ( member_real @ X @ S2 )
=> ( P @ X ) ) ) ) ) ) ) ).
% eventually_at_topological
thf(fact_472_eventually__at__topological,axiom,
! [P: nat > $o,A2: nat,S2: set_nat] :
( ( eventually_nat @ P @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) )
= ( ? [S: set_nat] :
( ( topolo4328251076210115529en_nat @ S )
& ( member_nat @ A2 @ S )
& ! [X: nat] :
( ( member_nat @ X @ S )
=> ( ( X != A2 )
=> ( ( member_nat @ X @ S2 )
=> ( P @ X ) ) ) ) ) ) ) ).
% eventually_at_topological
thf(fact_473_eventually__nhds,axiom,
! [P: nat > $o,A2: nat] :
( ( eventually_nat @ P @ ( topolo8926549440605965083ds_nat @ A2 ) )
= ( ? [S: set_nat] :
( ( topolo4328251076210115529en_nat @ S )
& ( member_nat @ A2 @ S )
& ! [X: nat] :
( ( member_nat @ X @ S )
=> ( P @ X ) ) ) ) ) ).
% eventually_nhds
thf(fact_474_eventually__nhds,axiom,
! [P: real > $o,A2: real] :
( ( eventually_real @ P @ ( topolo2815343760600316023s_real @ A2 ) )
= ( ? [S: set_real] :
( ( topolo4860482606490270245n_real @ S )
& ( member_real @ A2 @ S )
& ! [X: real] :
( ( member_real @ X @ S )
=> ( P @ X ) ) ) ) ) ).
% eventually_nhds
thf(fact_475_continuous__within__topological,axiom,
! [X2: real,S2: set_real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ S2 ) @ F2 )
= ( ! [B4: set_real] :
( ( topolo4860482606490270245n_real @ B4 )
=> ( ( member_real @ ( F2 @ X2 ) @ B4 )
=> ? [A4: set_real] :
( ( topolo4860482606490270245n_real @ A4 )
& ( member_real @ X2 @ A4 )
& ! [X: real] :
( ( member_real @ X @ S2 )
=> ( ( member_real @ X @ A4 )
=> ( member_real @ ( F2 @ X ) @ B4 ) ) ) ) ) ) ) ) ).
% continuous_within_topological
thf(fact_476_continuous__within__topological,axiom,
! [X2: real,S2: set_real,F2: real > nat] :
( ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ X2 @ S2 ) @ F2 )
= ( ! [B4: set_nat] :
( ( topolo4328251076210115529en_nat @ B4 )
=> ( ( member_nat @ ( F2 @ X2 ) @ B4 )
=> ? [A4: set_real] :
( ( topolo4860482606490270245n_real @ A4 )
& ( member_real @ X2 @ A4 )
& ! [X: real] :
( ( member_real @ X @ S2 )
=> ( ( member_real @ X @ A4 )
=> ( member_nat @ ( F2 @ X ) @ B4 ) ) ) ) ) ) ) ) ).
% continuous_within_topological
thf(fact_477_continuous__within__topological,axiom,
! [X2: nat,S2: set_nat,F2: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ X2 @ S2 ) @ F2 )
= ( ! [B4: set_real] :
( ( topolo4860482606490270245n_real @ B4 )
=> ( ( member_real @ ( F2 @ X2 ) @ B4 )
=> ? [A4: set_nat] :
( ( topolo4328251076210115529en_nat @ A4 )
& ( member_nat @ X2 @ A4 )
& ! [X: nat] :
( ( member_nat @ X @ S2 )
=> ( ( member_nat @ X @ A4 )
=> ( member_real @ ( F2 @ X ) @ B4 ) ) ) ) ) ) ) ) ).
% continuous_within_topological
thf(fact_478_continuous__within__topological,axiom,
! [X2: nat,S2: set_nat,F2: nat > nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ X2 @ S2 ) @ F2 )
= ( ! [B4: set_nat] :
( ( topolo4328251076210115529en_nat @ B4 )
=> ( ( member_nat @ ( F2 @ X2 ) @ B4 )
=> ? [A4: set_nat] :
( ( topolo4328251076210115529en_nat @ A4 )
& ( member_nat @ X2 @ A4 )
& ! [X: nat] :
( ( member_nat @ X @ S2 )
=> ( ( member_nat @ X @ A4 )
=> ( member_nat @ ( F2 @ X ) @ B4 ) ) ) ) ) ) ) ) ).
% continuous_within_topological
thf(fact_479_landau__o_Osmall__refl__iff,axiom,
! [F2: real > real,F: filter_real] :
( ( member_real_real @ F2 @ ( landau3007391416991288786l_real @ F @ F2 ) )
= ( eventually_real
@ ^ [X: real] :
( ( F2 @ X )
= zero_zero_real )
@ F ) ) ).
% landau_o.small_refl_iff
thf(fact_480_landau__o_Osmall__refl__iff,axiom,
! [F2: nat > real,F: filter_nat] :
( ( member_nat_real @ F2 @ ( landau997807338407142774t_real @ F @ F2 ) )
= ( eventually_nat
@ ^ [X: nat] :
( ( F2 @ X )
= zero_zero_real )
@ F ) ) ).
% landau_o.small_refl_iff
thf(fact_481_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
@ ^ [X: real] :
( ( F2 @ X )
= zero_zero_real )
@ F ) ) ) ).
% landau_o.small_asymmetric
thf(fact_482_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
@ ^ [X: nat] :
( ( F2 @ X )
= zero_zero_real )
@ F ) ) ) ).
% landau_o.small_asymmetric
thf(fact_483_eventually__nhds__in__open,axiom,
! [S2: set_nat,X2: nat] :
( ( topolo4328251076210115529en_nat @ S2 )
=> ( ( member_nat @ X2 @ S2 )
=> ( eventually_nat
@ ^ [Y: nat] : ( member_nat @ Y @ S2 )
@ ( topolo8926549440605965083ds_nat @ X2 ) ) ) ) ).
% eventually_nhds_in_open
thf(fact_484_eventually__nhds__in__open,axiom,
! [S2: set_real,X2: real] :
( ( topolo4860482606490270245n_real @ S2 )
=> ( ( member_real @ X2 @ S2 )
=> ( eventually_real
@ ^ [Y: real] : ( member_real @ Y @ S2 )
@ ( topolo2815343760600316023s_real @ X2 ) ) ) ) ).
% eventually_nhds_in_open
thf(fact_485_landau__o_Osmall_Odivide__right__iff,axiom,
! [H2: real > real,F: filter_real,F2: real > real,G: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( H2 @ X )
!= zero_zero_real )
@ F )
=> ( ( member_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( H2 @ X ) )
@ ( landau3007391416991288786l_real @ F
@ ^ [X: real] : ( divide_divide_real @ ( G @ X ) @ ( H2 @ X ) ) ) )
= ( member_real_real @ F2 @ ( landau3007391416991288786l_real @ F @ G ) ) ) ) ).
% landau_o.small.divide_right_iff
thf(fact_486_landau__o_Osmall_Odivide__right__iff,axiom,
! [H2: nat > real,F: filter_nat,F2: nat > real,G: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( H2 @ X )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real
@ ^ [X: nat] : ( divide_divide_real @ ( F2 @ X ) @ ( H2 @ X ) )
@ ( landau997807338407142774t_real @ F
@ ^ [X: nat] : ( divide_divide_real @ ( G @ X ) @ ( H2 @ X ) ) ) )
= ( member_nat_real @ F2 @ ( landau997807338407142774t_real @ F @ G ) ) ) ) ).
% landau_o.small.divide_right_iff
thf(fact_487_landau__o_Osmall_Odivide__left__iff,axiom,
! [F2: real > real,F: filter_real,G: real > real,H2: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( F2 @ X )
!= zero_zero_real )
@ F )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G @ X )
!= zero_zero_real )
@ F )
=> ( ( eventually_real
@ ^ [X: real] :
( ( H2 @ X )
!= zero_zero_real )
@ F )
=> ( ( member_real_real
@ ^ [X: real] : ( divide_divide_real @ ( H2 @ X ) @ ( F2 @ X ) )
@ ( landau3007391416991288786l_real @ F
@ ^ [X: real] : ( divide_divide_real @ ( H2 @ X ) @ ( G @ X ) ) ) )
= ( member_real_real @ G @ ( landau3007391416991288786l_real @ F @ F2 ) ) ) ) ) ) ).
% landau_o.small.divide_left_iff
thf(fact_488_landau__o_Osmall_Odivide__left__iff,axiom,
! [F2: nat > real,F: filter_nat,G: nat > real,H2: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( F2 @ X )
!= zero_zero_real )
@ F )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( G @ X )
!= zero_zero_real )
@ F )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( H2 @ X )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real
@ ^ [X: nat] : ( divide_divide_real @ ( H2 @ X ) @ ( F2 @ X ) )
@ ( landau997807338407142774t_real @ F
@ ^ [X: nat] : ( divide_divide_real @ ( H2 @ X ) @ ( G @ X ) ) ) )
= ( member_nat_real @ G @ ( landau997807338407142774t_real @ F @ F2 ) ) ) ) ) ) ).
% landau_o.small.divide_left_iff
thf(fact_489_landau__o_Osmall_Odivide__right,axiom,
! [H2: real > real,F: filter_real,F2: real > real,G: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( H2 @ X )
!= zero_zero_real )
@ F )
=> ( ( member_real_real @ F2 @ ( landau3007391416991288786l_real @ F @ G ) )
=> ( member_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( H2 @ X ) )
@ ( landau3007391416991288786l_real @ F
@ ^ [X: real] : ( divide_divide_real @ ( G @ X ) @ ( H2 @ X ) ) ) ) ) ) ).
% landau_o.small.divide_right
thf(fact_490_landau__o_Osmall_Odivide__right,axiom,
! [H2: nat > real,F: filter_nat,F2: nat > real,G: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( H2 @ X )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real @ F2 @ ( landau997807338407142774t_real @ F @ G ) )
=> ( member_nat_real
@ ^ [X: nat] : ( divide_divide_real @ ( F2 @ X ) @ ( H2 @ X ) )
@ ( landau997807338407142774t_real @ F
@ ^ [X: nat] : ( divide_divide_real @ ( G @ X ) @ ( H2 @ X ) ) ) ) ) ) ).
% landau_o.small.divide_right
thf(fact_491_landau__o_Osmall_Odivide__left,axiom,
! [F2: real > real,F: filter_real,G: real > real,H2: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( F2 @ X )
!= zero_zero_real )
@ F )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G @ X )
!= zero_zero_real )
@ F )
=> ( ( member_real_real @ G @ ( landau3007391416991288786l_real @ F @ F2 ) )
=> ( member_real_real
@ ^ [X: real] : ( divide_divide_real @ ( H2 @ X ) @ ( F2 @ X ) )
@ ( landau3007391416991288786l_real @ F
@ ^ [X: real] : ( divide_divide_real @ ( H2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% landau_o.small.divide_left
thf(fact_492_landau__o_Osmall_Odivide__left,axiom,
! [F2: nat > real,F: filter_nat,G: nat > real,H2: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( F2 @ X )
!= zero_zero_real )
@ F )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( G @ X )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real @ G @ ( landau997807338407142774t_real @ F @ F2 ) )
=> ( member_nat_real
@ ^ [X: nat] : ( divide_divide_real @ ( H2 @ X ) @ ( F2 @ X ) )
@ ( landau997807338407142774t_real @ F
@ ^ [X: nat] : ( divide_divide_real @ ( H2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% landau_o.small.divide_left
thf(fact_493_landau__o_Osmall_Odivide,axiom,
! [G1: real > real,F: filter_real,G22: real > real,F14: real > real,F25: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( G1 @ X )
!= zero_zero_real )
@ F )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G22 @ X )
!= zero_zero_real )
@ F )
=> ( ( member_real_real @ F14 @ ( landau3007391416991288786l_real @ F @ F25 ) )
=> ( ( member_real_real @ G22 @ ( landau3007391416991288786l_real @ F @ G1 ) )
=> ( member_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F14 @ X ) @ ( G1 @ X ) )
@ ( landau3007391416991288786l_real @ F
@ ^ [X: real] : ( divide_divide_real @ ( F25 @ X ) @ ( G22 @ X ) ) ) ) ) ) ) ) ).
% landau_o.small.divide
thf(fact_494_landau__o_Osmall_Odivide,axiom,
! [G1: nat > real,F: filter_nat,G22: nat > real,F14: nat > real,F25: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( G1 @ X )
!= zero_zero_real )
@ F )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( G22 @ X )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real @ F14 @ ( landau997807338407142774t_real @ F @ F25 ) )
=> ( ( member_nat_real @ G22 @ ( landau997807338407142774t_real @ F @ G1 ) )
=> ( member_nat_real
@ ^ [X: nat] : ( divide_divide_real @ ( F14 @ X ) @ ( G1 @ X ) )
@ ( landau997807338407142774t_real @ F
@ ^ [X: nat] : ( divide_divide_real @ ( F25 @ X ) @ ( G22 @ X ) ) ) ) ) ) ) ) ).
% landau_o.small.divide
thf(fact_495_landau__o_Osmall_Oinverse__cancel,axiom,
! [F2: real > real,F: filter_real,G: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( F2 @ X )
!= zero_zero_real )
@ F )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G @ X )
!= zero_zero_real )
@ F )
=> ( ( member_real_real
@ ^ [X: real] : ( inverse_inverse_real @ ( F2 @ X ) )
@ ( landau3007391416991288786l_real @ F
@ ^ [X: real] : ( inverse_inverse_real @ ( G @ X ) ) ) )
= ( member_real_real @ G @ ( landau3007391416991288786l_real @ F @ F2 ) ) ) ) ) ).
% landau_o.small.inverse_cancel
thf(fact_496_landau__o_Osmall_Oinverse__cancel,axiom,
! [F2: nat > real,F: filter_nat,G: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( F2 @ X )
!= zero_zero_real )
@ F )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( G @ X )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real
@ ^ [X: nat] : ( inverse_inverse_real @ ( F2 @ X ) )
@ ( landau997807338407142774t_real @ F
@ ^ [X: nat] : ( inverse_inverse_real @ ( G @ X ) ) ) )
= ( member_nat_real @ G @ ( landau997807338407142774t_real @ F @ F2 ) ) ) ) ) ).
% landau_o.small.inverse_cancel
thf(fact_497_landau__o_Osmall_Oinverse__flip,axiom,
! [G: real > real,F: filter_real,H2: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( G @ X )
!= zero_zero_real )
@ F )
=> ( ( eventually_real
@ ^ [X: real] :
( ( H2 @ X )
!= zero_zero_real )
@ F )
=> ( ( member_real_real
@ ^ [X: real] : ( inverse_inverse_real @ ( G @ X ) )
@ ( landau3007391416991288786l_real @ F @ H2 ) )
=> ( member_real_real
@ ^ [X: real] : ( inverse_inverse_real @ ( H2 @ X ) )
@ ( landau3007391416991288786l_real @ F @ G ) ) ) ) ) ).
% landau_o.small.inverse_flip
thf(fact_498_landau__o_Osmall_Oinverse__flip,axiom,
! [G: nat > real,F: filter_nat,H2: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( G @ X )
!= zero_zero_real )
@ F )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( H2 @ X )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real
@ ^ [X: nat] : ( inverse_inverse_real @ ( G @ X ) )
@ ( landau997807338407142774t_real @ F @ H2 ) )
=> ( member_nat_real
@ ^ [X: nat] : ( inverse_inverse_real @ ( H2 @ X ) )
@ ( landau997807338407142774t_real @ F @ G ) ) ) ) ) ).
% landau_o.small.inverse_flip
thf(fact_499_landau__o_Osmall_Oinverse,axiom,
! [F2: real > real,F: filter_real,G: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( F2 @ X )
!= zero_zero_real )
@ F )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G @ X )
!= zero_zero_real )
@ F )
=> ( ( member_real_real @ F2 @ ( landau3007391416991288786l_real @ F @ G ) )
=> ( member_real_real
@ ^ [X: real] : ( inverse_inverse_real @ ( G @ X ) )
@ ( landau3007391416991288786l_real @ F
@ ^ [X: real] : ( inverse_inverse_real @ ( F2 @ X ) ) ) ) ) ) ) ).
% landau_o.small.inverse
thf(fact_500_landau__o_Osmall_Oinverse,axiom,
! [F2: nat > real,F: filter_nat,G: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( F2 @ X )
!= zero_zero_real )
@ F )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( G @ X )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real @ F2 @ ( landau997807338407142774t_real @ F @ G ) )
=> ( member_nat_real
@ ^ [X: nat] : ( inverse_inverse_real @ ( G @ X ) )
@ ( landau997807338407142774t_real @ F
@ ^ [X: nat] : ( inverse_inverse_real @ ( F2 @ X ) ) ) ) ) ) ) ).
% landau_o.small.inverse
thf(fact_501_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
@ ^ [X: real] : ( plus_plus_real @ ( F14 @ X ) @ ( F25 @ X ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F ) ) ) ).
% tendsto_add_smallo
thf(fact_502_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
@ ^ [X: real] : ( plus_plus_real @ ( F14 @ X ) @ ( F25 @ X ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F ) ) ) ).
% tendsto_add_smallo_iff
thf(fact_503_Lim__transform__within__open,axiom,
! [F2: real > real,L: real,A2: real,T: set_real,S2: set_real,G: real > real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ T ) )
=> ( ( topolo4860482606490270245n_real @ S2 )
=> ( ( member_real @ A2 @ S2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( X4 != A2 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ T ) ) ) ) ) ) ).
% Lim_transform_within_open
thf(fact_504_Lim__transform__within__open,axiom,
! [F2: nat > nat,L: nat,A2: nat,T: set_nat,S2: set_nat,G: nat > nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ T ) )
=> ( ( topolo4328251076210115529en_nat @ S2 )
=> ( ( member_nat @ A2 @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( ( X4 != A2 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ T ) ) ) ) ) ) ).
% Lim_transform_within_open
thf(fact_505_Lim__transform__within__open,axiom,
! [F2: nat > real,L: real,A2: nat,T: set_nat,S2: set_nat,G: nat > real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ T ) )
=> ( ( topolo4328251076210115529en_nat @ S2 )
=> ( ( member_nat @ A2 @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( ( X4 != A2 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ T ) ) ) ) ) ) ).
% Lim_transform_within_open
thf(fact_506_continuous__at__open,axiom,
! [X2: real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ F2 )
= ( ! [T4: set_real] :
( ( ( topolo4860482606490270245n_real @ T4 )
& ( member_real @ ( F2 @ X2 ) @ T4 ) )
=> ? [S4: set_real] :
( ( topolo4860482606490270245n_real @ S4 )
& ( member_real @ X2 @ S4 )
& ! [X: real] :
( ( member_real @ X @ S4 )
=> ( member_real @ ( F2 @ X ) @ T4 ) ) ) ) ) ) ).
% continuous_at_open
thf(fact_507_continuous__at__open,axiom,
! [X2: real,F2: real > nat] :
( ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ F2 )
= ( ! [T4: set_nat] :
( ( ( topolo4328251076210115529en_nat @ T4 )
& ( member_nat @ ( F2 @ X2 ) @ T4 ) )
=> ? [S4: set_real] :
( ( topolo4860482606490270245n_real @ S4 )
& ( member_real @ X2 @ S4 )
& ! [X: real] :
( ( member_real @ X @ S4 )
=> ( member_nat @ ( F2 @ X ) @ T4 ) ) ) ) ) ) ).
% continuous_at_open
thf(fact_508_continuous__at__open,axiom,
! [X2: nat,F2: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ X2 @ top_top_set_nat ) @ F2 )
= ( ! [T4: set_real] :
( ( ( topolo4860482606490270245n_real @ T4 )
& ( member_real @ ( F2 @ X2 ) @ T4 ) )
=> ? [S4: set_nat] :
( ( topolo4328251076210115529en_nat @ S4 )
& ( member_nat @ X2 @ S4 )
& ! [X: nat] :
( ( member_nat @ X @ S4 )
=> ( member_real @ ( F2 @ X ) @ T4 ) ) ) ) ) ) ).
% continuous_at_open
thf(fact_509_continuous__at__open,axiom,
! [X2: nat,F2: nat > nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ X2 @ top_top_set_nat ) @ F2 )
= ( ! [T4: set_nat] :
( ( ( topolo4328251076210115529en_nat @ T4 )
& ( member_nat @ ( F2 @ X2 ) @ T4 ) )
=> ? [S4: set_nat] :
( ( topolo4328251076210115529en_nat @ S4 )
& ( member_nat @ X2 @ S4 )
& ! [X: nat] :
( ( member_nat @ X @ S4 )
=> ( member_nat @ ( F2 @ X ) @ T4 ) ) ) ) ) ) ).
% continuous_at_open
thf(fact_510_topological__tendstoI,axiom,
! [L: nat,F2: real > nat,F: filter_real] :
( ! [S5: set_nat] :
( ( topolo4328251076210115529en_nat @ S5 )
=> ( ( member_nat @ L @ S5 )
=> ( eventually_real
@ ^ [X: real] : ( member_nat @ ( F2 @ X ) @ S5 )
@ F ) ) )
=> ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F ) ) ).
% topological_tendstoI
thf(fact_511_topological__tendstoI,axiom,
! [L: nat,F2: nat > nat,F: filter_nat] :
( ! [S5: set_nat] :
( ( topolo4328251076210115529en_nat @ S5 )
=> ( ( member_nat @ L @ S5 )
=> ( eventually_nat
@ ^ [X: nat] : ( member_nat @ ( F2 @ X ) @ S5 )
@ F ) ) )
=> ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F ) ) ).
% topological_tendstoI
thf(fact_512_topological__tendstoI,axiom,
! [L: real,F2: real > real,F: filter_real] :
( ! [S5: set_real] :
( ( topolo4860482606490270245n_real @ S5 )
=> ( ( member_real @ L @ S5 )
=> ( eventually_real
@ ^ [X: real] : ( member_real @ ( F2 @ X ) @ S5 )
@ F ) ) )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ).
% topological_tendstoI
thf(fact_513_topological__tendstoI,axiom,
! [L: real,F2: nat > real,F: filter_nat] :
( ! [S5: set_real] :
( ( topolo4860482606490270245n_real @ S5 )
=> ( ( member_real @ L @ S5 )
=> ( eventually_nat
@ ^ [X: nat] : ( member_real @ ( F2 @ X ) @ S5 )
@ F ) ) )
=> ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ).
% topological_tendstoI
thf(fact_514_topological__tendstoD,axiom,
! [F2: real > nat,L: nat,F: filter_real,S3: set_nat] :
( ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
=> ( ( topolo4328251076210115529en_nat @ S3 )
=> ( ( member_nat @ L @ S3 )
=> ( eventually_real
@ ^ [X: real] : ( member_nat @ ( F2 @ X ) @ S3 )
@ F ) ) ) ) ).
% topological_tendstoD
thf(fact_515_topological__tendstoD,axiom,
! [F2: nat > nat,L: nat,F: filter_nat,S3: set_nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
=> ( ( topolo4328251076210115529en_nat @ S3 )
=> ( ( member_nat @ L @ S3 )
=> ( eventually_nat
@ ^ [X: nat] : ( member_nat @ ( F2 @ X ) @ S3 )
@ F ) ) ) ) ).
% topological_tendstoD
thf(fact_516_topological__tendstoD,axiom,
! [F2: real > real,L: real,F: filter_real,S3: set_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( ( topolo4860482606490270245n_real @ S3 )
=> ( ( member_real @ L @ S3 )
=> ( eventually_real
@ ^ [X: real] : ( member_real @ ( F2 @ X ) @ S3 )
@ F ) ) ) ) ).
% topological_tendstoD
thf(fact_517_topological__tendstoD,axiom,
! [F2: nat > real,L: real,F: filter_nat,S3: set_real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( ( topolo4860482606490270245n_real @ S3 )
=> ( ( member_real @ L @ S3 )
=> ( eventually_nat
@ ^ [X: nat] : ( member_real @ ( F2 @ X ) @ S3 )
@ F ) ) ) ) ).
% topological_tendstoD
thf(fact_518_tendsto__def,axiom,
! [F2: real > nat,L: nat,F: filter_real] :
( ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
= ( ! [S: set_nat] :
( ( topolo4328251076210115529en_nat @ S )
=> ( ( member_nat @ L @ S )
=> ( eventually_real
@ ^ [X: real] : ( member_nat @ ( F2 @ X ) @ S )
@ F ) ) ) ) ) ).
% tendsto_def
thf(fact_519_tendsto__def,axiom,
! [F2: nat > nat,L: nat,F: filter_nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
= ( ! [S: set_nat] :
( ( topolo4328251076210115529en_nat @ S )
=> ( ( member_nat @ L @ S )
=> ( eventually_nat
@ ^ [X: nat] : ( member_nat @ ( F2 @ X ) @ S )
@ F ) ) ) ) ) ).
% tendsto_def
thf(fact_520_tendsto__def,axiom,
! [F2: real > real,L: real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
= ( ! [S: set_real] :
( ( topolo4860482606490270245n_real @ S )
=> ( ( member_real @ L @ S )
=> ( eventually_real
@ ^ [X: real] : ( member_real @ ( F2 @ X ) @ S )
@ F ) ) ) ) ) ).
% tendsto_def
thf(fact_521_tendsto__def,axiom,
! [F2: nat > real,L: real,F: filter_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
= ( ! [S: set_real] :
( ( topolo4860482606490270245n_real @ S )
=> ( ( member_real @ L @ S )
=> ( eventually_nat
@ ^ [X: nat] : ( member_real @ ( F2 @ X ) @ S )
@ F ) ) ) ) ) ).
% tendsto_def
thf(fact_522_smalloD__tendsto,axiom,
! [F2: real > real,F: filter_real,G: real > real] :
( ( member_real_real @ F2 @ ( landau3007391416991288786l_real @ F @ G ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ).
% smalloD_tendsto
thf(fact_523_tendsto__within__open,axiom,
! [A2: real,S3: set_real,F2: real > real,L: real] :
( ( member_real @ A2 @ S3 )
=> ( ( topolo4860482606490270245n_real @ S3 )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ S3 ) )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ).
% tendsto_within_open
thf(fact_524_tendsto__within__open,axiom,
! [A2: nat,S3: set_nat,F2: nat > nat,L: nat] :
( ( member_nat @ A2 @ S3 )
=> ( ( topolo4328251076210115529en_nat @ S3 )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S3 ) )
= ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ).
% tendsto_within_open
thf(fact_525_tendsto__within__open,axiom,
! [A2: nat,S3: set_nat,F2: nat > real,L: real] :
( ( member_nat @ A2 @ S3 )
=> ( ( topolo4328251076210115529en_nat @ S3 )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S3 ) )
= ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ).
% tendsto_within_open
thf(fact_526_tendsto__within__open__NO__MATCH,axiom,
! [A2: real,S3: set_real,F2: real > real,L: real] :
( ( member_real @ A2 @ S3 )
=> ( ( nO_MAT2855227906214470577t_real @ top_top_set_real @ S3 )
=> ( ( topolo4860482606490270245n_real @ S3 )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ S3 ) )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ) ).
% tendsto_within_open_NO_MATCH
thf(fact_527_tendsto__within__open__NO__MATCH,axiom,
! [A2: real,S3: set_real,F2: real > real,L: real] :
( ( member_real @ A2 @ S3 )
=> ( ( nO_MAT8790521740642007125t_real @ top_top_set_nat @ S3 )
=> ( ( topolo4860482606490270245n_real @ S3 )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ S3 ) )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ) ).
% tendsto_within_open_NO_MATCH
thf(fact_528_tendsto__within__open__NO__MATCH,axiom,
! [A2: nat,S3: set_nat,F2: nat > nat,L: nat] :
( ( member_nat @ A2 @ S3 )
=> ( ( nO_MAT504328087405689813et_nat @ top_top_set_real @ S3 )
=> ( ( topolo4328251076210115529en_nat @ S3 )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S3 ) )
= ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ) ).
% tendsto_within_open_NO_MATCH
thf(fact_529_tendsto__within__open__NO__MATCH,axiom,
! [A2: nat,S3: set_nat,F2: nat > nat,L: nat] :
( ( member_nat @ A2 @ S3 )
=> ( ( nO_MAT2475032472373502585et_nat @ top_top_set_nat @ S3 )
=> ( ( topolo4328251076210115529en_nat @ S3 )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S3 ) )
= ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ) ).
% tendsto_within_open_NO_MATCH
thf(fact_530_tendsto__within__open__NO__MATCH,axiom,
! [A2: nat,S3: set_nat,F2: nat > real,L: real] :
( ( member_nat @ A2 @ S3 )
=> ( ( nO_MAT504328087405689813et_nat @ top_top_set_real @ S3 )
=> ( ( topolo4328251076210115529en_nat @ S3 )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S3 ) )
= ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ) ).
% tendsto_within_open_NO_MATCH
thf(fact_531_tendsto__within__open__NO__MATCH,axiom,
! [A2: nat,S3: set_nat,F2: nat > real,L: real] :
( ( member_nat @ A2 @ S3 )
=> ( ( nO_MAT2475032472373502585et_nat @ top_top_set_nat @ S3 )
=> ( ( topolo4328251076210115529en_nat @ S3 )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ S3 ) )
= ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ) ) ) ).
% tendsto_within_open_NO_MATCH
thf(fact_532_Lim__topological,axiom,
! [F2: real > nat,L: nat,Net: filter_real] :
( ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ Net )
= ( ( Net = bot_bot_filter_real )
| ! [S: set_nat] :
( ( topolo4328251076210115529en_nat @ S )
=> ( ( member_nat @ L @ S )
=> ( eventually_real
@ ^ [X: real] : ( member_nat @ ( F2 @ X ) @ S )
@ Net ) ) ) ) ) ).
% Lim_topological
thf(fact_533_Lim__topological,axiom,
! [F2: nat > nat,L: nat,Net: filter_nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ Net )
= ( ( Net = bot_bot_filter_nat )
| ! [S: set_nat] :
( ( topolo4328251076210115529en_nat @ S )
=> ( ( member_nat @ L @ S )
=> ( eventually_nat
@ ^ [X: nat] : ( member_nat @ ( F2 @ X ) @ S )
@ Net ) ) ) ) ) ).
% Lim_topological
thf(fact_534_Lim__topological,axiom,
! [F2: real > real,L: real,Net: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ Net )
= ( ( Net = bot_bot_filter_real )
| ! [S: set_real] :
( ( topolo4860482606490270245n_real @ S )
=> ( ( member_real @ L @ S )
=> ( eventually_real
@ ^ [X: real] : ( member_real @ ( F2 @ X ) @ S )
@ Net ) ) ) ) ) ).
% Lim_topological
thf(fact_535_Lim__topological,axiom,
! [F2: nat > real,L: real,Net: filter_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ Net )
= ( ( Net = bot_bot_filter_nat )
| ! [S: set_real] :
( ( topolo4860482606490270245n_real @ S )
=> ( ( member_real @ L @ S )
=> ( eventually_nat
@ ^ [X: nat] : ( member_real @ ( F2 @ X ) @ S )
@ Net ) ) ) ) ) ).
% Lim_topological
thf(fact_536_smalloI__tendsto,axiom,
! [F2: real > real,G: real > real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G @ X )
!= zero_zero_real )
@ F )
=> ( member_real_real @ F2 @ ( landau3007391416991288786l_real @ F @ G ) ) ) ) ).
% smalloI_tendsto
thf(fact_537_smalloI__tendsto,axiom,
! [F2: nat > real,G: nat > real,F: filter_nat] :
( ( filterlim_nat_real
@ ^ [X: nat] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( G @ X )
!= zero_zero_real )
@ F )
=> ( member_nat_real @ F2 @ ( landau997807338407142774t_real @ F @ G ) ) ) ) ).
% smalloI_tendsto
thf(fact_538_landau__o_Osmall_Oinverse__eq1,axiom,
! [G: real > real,F: filter_real,F2: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( G @ X )
!= zero_zero_real )
@ F )
=> ( ( member_real_real @ F2
@ ( landau3007391416991288786l_real @ F
@ ^ [X: real] : ( inverse_inverse_real @ ( G @ X ) ) ) )
= ( member_real_real
@ ^ [X: real] : ( times_times_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( landau3007391416991288786l_real @ F
@ ^ [Uu: real] : one_one_real ) ) ) ) ).
% landau_o.small.inverse_eq1
thf(fact_539_landau__o_Osmall_Oinverse__eq1,axiom,
! [G: nat > real,F: filter_nat,F2: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( G @ X )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real @ F2
@ ( landau997807338407142774t_real @ F
@ ^ [X: nat] : ( inverse_inverse_real @ ( G @ X ) ) ) )
= ( member_nat_real
@ ^ [X: nat] : ( times_times_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( landau997807338407142774t_real @ F
@ ^ [Uu: nat] : one_one_real ) ) ) ) ).
% landau_o.small.inverse_eq1
thf(fact_540_landau__o_Osmall_Oinverse__eq2,axiom,
! [F2: real > real,F: filter_real,G: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( F2 @ X )
!= zero_zero_real )
@ F )
=> ( ( member_real_real
@ ^ [X: real] : ( inverse_inverse_real @ ( F2 @ X ) )
@ ( landau3007391416991288786l_real @ F @ G ) )
= ( member_real_real
@ ^ [X: real] : one_one_real
@ ( landau3007391416991288786l_real @ F
@ ^ [X: real] : ( times_times_real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% landau_o.small.inverse_eq2
thf(fact_541_landau__o_Osmall_Oinverse__eq2,axiom,
! [F2: nat > real,F: filter_nat,G: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( F2 @ X )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real
@ ^ [X: nat] : ( inverse_inverse_real @ ( F2 @ X ) )
@ ( landau997807338407142774t_real @ F @ G ) )
= ( member_nat_real
@ ^ [X: nat] : one_one_real
@ ( landau997807338407142774t_real @ F
@ ^ [X: nat] : ( times_times_real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% landau_o.small.inverse_eq2
thf(fact_542_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
@ ^ [X: real] : ( F2 @ ( plus_plus_real @ X @ K ) )
@ ( topolo2815343760600316023s_real @ L2 )
@ ( topolo2177554685111907308n_real @ ( minus_minus_real @ A2 @ K ) @ top_top_set_real ) ) ) ).
% LIM_offset
thf(fact_543_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
@ ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ ( F2 @ X ) @ ( F2 @ A2 ) ) @ ( minus_minus_real @ X @ A2 ) )
@ ( topolo2815343760600316023s_real @ D2 )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ).
% DERIV_LIM_iff
thf(fact_544_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_545_at__eq__bot__iff,axiom,
! [A2: real] :
( ( ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
= bot_bot_filter_real )
= ( topolo4860482606490270245n_real @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ).
% at_eq_bot_iff
thf(fact_546_at__eq__bot__iff,axiom,
! [A2: nat] :
( ( ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat )
= bot_bot_filter_nat )
= ( topolo4328251076210115529en_nat @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% at_eq_bot_iff
thf(fact_547_insert__Diff1,axiom,
! [X2: real,B2: set_real,A: set_real] :
( ( member_real @ X2 @ B2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X2 @ A ) @ B2 )
= ( minus_minus_set_real @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_548_insert__Diff1,axiom,
! [X2: nat,B2: set_nat,A: set_nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_549_Diff__insert0,axiom,
! [X2: real,A: set_real,B2: set_real] :
( ~ ( member_real @ X2 @ A )
=> ( ( minus_minus_set_real @ A @ ( insert_real @ X2 @ B2 ) )
= ( minus_minus_set_real @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_550_Diff__insert0,axiom,
! [X2: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X2 @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ B2 ) )
= ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_551_insert__iff,axiom,
! [A2: real,B: real,A: set_real] :
( ( member_real @ A2 @ ( insert_real @ B @ A ) )
= ( ( A2 = B )
| ( member_real @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_552_insert__iff,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
= ( ( A2 = B )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_553_insertCI,axiom,
! [A2: real,B2: set_real,B: real] :
( ( ~ ( member_real @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_real @ A2 @ ( insert_real @ B @ B2 ) ) ) ).
% insertCI
thf(fact_554_insertCI,axiom,
! [A2: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_555_mult__zero__left,axiom,
! [A2: real] :
( ( times_times_real @ zero_zero_real @ A2 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_556_mult__zero__right,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_557_mult__eq__0__iff,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ B )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_558_mult__cancel__left,axiom,
! [C2: real,A2: real,B: real] :
( ( ( times_times_real @ C2 @ A2 )
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_559_mult__cancel__right,axiom,
! [A2: real,C2: real,B: real] :
( ( ( times_times_real @ A2 @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_560_diff__self,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% diff_self
thf(fact_561_diff__0__right,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_0_right
thf(fact_562_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_563_diff__zero,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_zero
thf(fact_564_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_565_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_566_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_567_mult__1,axiom,
! [A2: real] :
( ( times_times_real @ one_one_real @ A2 )
= A2 ) ).
% mult_1
thf(fact_568_mult_Oright__neutral,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% mult.right_neutral
thf(fact_569_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_570_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_571_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_572_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_573_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_574_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_575_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_576_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_577_diff__add__cancel,axiom,
! [A2: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ B )
= A2 ) ).
% diff_add_cancel
thf(fact_578_add__diff__cancel,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel
thf(fact_579_times__divide__eq__left,axiom,
! [B: real,C2: real,A2: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C2 ) @ A2 )
= ( divide_divide_real @ ( times_times_real @ B @ A2 ) @ C2 ) ) ).
% times_divide_eq_left
thf(fact_580_divide__divide__eq__left,axiom,
! [A2: real,B: real,C2: real] :
( ( divide_divide_real @ ( divide_divide_real @ A2 @ B ) @ C2 )
= ( divide_divide_real @ A2 @ ( times_times_real @ B @ C2 ) ) ) ).
% divide_divide_eq_left
thf(fact_581_divide__divide__eq__right,axiom,
! [A2: real,B: real,C2: real] :
( ( divide_divide_real @ A2 @ ( divide_divide_real @ B @ C2 ) )
= ( divide_divide_real @ ( times_times_real @ A2 @ C2 ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_582_times__divide__eq__right,axiom,
! [A2: real,B: real,C2: real] :
( ( times_times_real @ A2 @ ( divide_divide_real @ B @ C2 ) )
= ( divide_divide_real @ ( times_times_real @ A2 @ B ) @ C2 ) ) ).
% times_divide_eq_right
thf(fact_583_Diff__UNIV,axiom,
! [A: set_real] :
( ( minus_minus_set_real @ A @ top_top_set_real )
= bot_bot_set_real ) ).
% Diff_UNIV
thf(fact_584_Diff__UNIV,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_585_singletonI,axiom,
! [A2: real] : ( member_real @ A2 @ ( insert_real @ A2 @ bot_bot_set_real ) ) ).
% singletonI
thf(fact_586_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_587_inverse__mult__distrib,axiom,
! [A2: real,B: real] :
( ( inverse_inverse_real @ ( times_times_real @ A2 @ B ) )
= ( times_times_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B ) ) ) ).
% inverse_mult_distrib
thf(fact_588_mult__cancel__left1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_589_mult__cancel__left2,axiom,
! [C2: real,A2: real] :
( ( ( times_times_real @ C2 @ A2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A2 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_590_mult__cancel__right1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_591_mult__cancel__right2,axiom,
! [A2: real,C2: real] :
( ( ( times_times_real @ A2 @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A2 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_592_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_593_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C2: real,A2: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ C2 @ B ) )
= ( divide_divide_real @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_594_nonzero__mult__div__cancel__right,axiom,
! [B: real,A2: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ B ) @ B )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_595_nonzero__mult__divide__mult__cancel__right,axiom,
! [C2: real,A2: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( divide_divide_real @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_596_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C2: real,A2: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ B @ C2 ) )
= ( divide_divide_real @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_597_nonzero__mult__div__cancel__left,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ B ) @ A2 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_598_nonzero__mult__divide__mult__cancel__left,axiom,
! [C2: real,A2: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B ) )
= ( divide_divide_real @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_599_mult__divide__mult__cancel__left__if,axiom,
! [C2: real,A2: real,B: real] :
( ( ( C2 = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B ) )
= zero_zero_real ) )
& ( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B ) )
= ( divide_divide_real @ A2 @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_600_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A2: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A2 @ B ) )
= ( divide_divide_real @ one_one_real @ A2 ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_601_nonzero__divide__mult__cancel__left,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ ( times_times_real @ A2 @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_602_div__mult__self4,axiom,
! [B: nat,C2: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C2 ) @ A2 ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self4
thf(fact_603_div__mult__self3,axiom,
! [B: nat,C2: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B ) @ A2 ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self3
thf(fact_604_div__mult__self2,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ B @ C2 ) ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self2
thf(fact_605_div__mult__self1,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ C2 @ B ) ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self1
thf(fact_606_right__inverse,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( times_times_real @ A2 @ ( inverse_inverse_real @ A2 ) )
= one_one_real ) ) ).
% right_inverse
thf(fact_607_left__inverse,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A2 ) @ A2 )
= one_one_real ) ) ).
% left_inverse
thf(fact_608_set__add__0,axiom,
! [A: set_real] :
( ( plus_plus_set_real @ ( insert_real @ zero_zero_real @ bot_bot_set_real ) @ A )
= A ) ).
% set_add_0
thf(fact_609_set__add__0__right,axiom,
! [A: set_real] :
( ( plus_plus_set_real @ A @ ( insert_real @ zero_zero_real @ bot_bot_set_real ) )
= A ) ).
% set_add_0_right
thf(fact_610_tendsto__mult__right__iff,axiom,
! [C2: real,F2: real > real,L: real,F: filter_real] :
( ( C2 != zero_zero_real )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( times_times_real @ ( F2 @ X ) @ C2 )
@ ( topolo2815343760600316023s_real @ ( times_times_real @ L @ C2 ) )
@ F )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ) ).
% tendsto_mult_right_iff
thf(fact_611_tendsto__mult__left__iff,axiom,
! [C2: real,F2: real > real,L: real,F: filter_real] :
( ( C2 != zero_zero_real )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( times_times_real @ C2 @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ ( times_times_real @ C2 @ L ) )
@ F )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ) ).
% tendsto_mult_left_iff
thf(fact_612_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_613_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_614_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_615_mult_Oleft__commute,axiom,
! [B: real,A2: real,C2: real] :
( ( times_times_real @ B @ ( times_times_real @ A2 @ C2 ) )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_616_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_617_mult_Oassoc,axiom,
! [A2: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B ) @ C2 )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_618_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_619_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_620_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B ) @ C2 )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_621_insert__compr,axiom,
( insert_real
= ( ^ [A3: real,B4: set_real] :
( collect_real
@ ^ [X: real] :
( ( X = A3 )
| ( member_real @ X @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_622_insert__compr,axiom,
( insert_nat
= ( ^ [A3: nat,B4: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( X = A3 )
| ( member_nat @ X @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_623_mult__commute__abs,axiom,
! [C2: real] :
( ( ^ [X: real] : ( times_times_real @ X @ C2 ) )
= ( times_times_real @ C2 ) ) ).
% mult_commute_abs
thf(fact_624_mk__disjoint__insert,axiom,
! [A2: real,A: set_real] :
( ( member_real @ A2 @ A )
=> ? [B5: set_real] :
( ( A
= ( insert_real @ A2 @ B5 ) )
& ~ ( member_real @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_625_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B5: set_nat] :
( ( A
= ( insert_nat @ A2 @ B5 ) )
& ~ ( member_nat @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_626_insert__Diff__if,axiom,
! [X2: real,B2: set_real,A: set_real] :
( ( ( member_real @ X2 @ B2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X2 @ A ) @ B2 )
= ( minus_minus_set_real @ A @ B2 ) ) )
& ( ~ ( member_real @ X2 @ B2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X2 @ A ) @ B2 )
= ( insert_real @ X2 @ ( minus_minus_set_real @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_627_insert__Diff__if,axiom,
! [X2: nat,B2: set_nat,A: set_nat] :
( ( ( member_nat @ X2 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) )
& ( ~ ( member_nat @ X2 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A ) @ B2 )
= ( insert_nat @ X2 @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_628_insert__eq__iff,axiom,
! [A2: real,A: set_real,B: real,B2: set_real] :
( ~ ( member_real @ A2 @ A )
=> ( ~ ( member_real @ B @ B2 )
=> ( ( ( insert_real @ A2 @ A )
= ( insert_real @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C4: set_real] :
( ( A
= ( insert_real @ B @ C4 ) )
& ~ ( member_real @ B @ C4 )
& ( B2
= ( insert_real @ A2 @ C4 ) )
& ~ ( member_real @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_629_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C4: set_nat] :
( ( A
= ( insert_nat @ B @ C4 ) )
& ~ ( member_nat @ B @ C4 )
& ( B2
= ( insert_nat @ A2 @ C4 ) )
& ~ ( member_nat @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_630_insert__absorb,axiom,
! [A2: real,A: set_real] :
( ( member_real @ A2 @ A )
=> ( ( insert_real @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_631_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_632_insert__ident,axiom,
! [X2: real,A: set_real,B2: set_real] :
( ~ ( member_real @ X2 @ A )
=> ( ~ ( member_real @ X2 @ B2 )
=> ( ( ( insert_real @ X2 @ A )
= ( insert_real @ X2 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_633_insert__ident,axiom,
! [X2: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X2 @ A )
=> ( ~ ( member_nat @ X2 @ B2 )
=> ( ( ( insert_nat @ X2 @ A )
= ( insert_nat @ X2 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_634_Set_Oset__insert,axiom,
! [X2: real,A: set_real] :
( ( member_real @ X2 @ A )
=> ~ ! [B5: set_real] :
( ( A
= ( insert_real @ X2 @ B5 ) )
=> ( member_real @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_635_Set_Oset__insert,axiom,
! [X2: nat,A: set_nat] :
( ( member_nat @ X2 @ A )
=> ~ ! [B5: set_nat] :
( ( A
= ( insert_nat @ X2 @ B5 ) )
=> ( member_nat @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_636_insertI2,axiom,
! [A2: real,B2: set_real,B: real] :
( ( member_real @ A2 @ B2 )
=> ( member_real @ A2 @ ( insert_real @ B @ B2 ) ) ) ).
% insertI2
thf(fact_637_insertI2,axiom,
! [A2: nat,B2: set_nat,B: nat] :
( ( member_nat @ A2 @ B2 )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_638_insertI1,axiom,
! [A2: real,B2: set_real] : ( member_real @ A2 @ ( insert_real @ A2 @ B2 ) ) ).
% insertI1
thf(fact_639_insertI1,axiom,
! [A2: nat,B2: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_640_insertE,axiom,
! [A2: real,B: real,A: set_real] :
( ( member_real @ A2 @ ( insert_real @ B @ A ) )
=> ( ( A2 != B )
=> ( member_real @ A2 @ A ) ) ) ).
% insertE
thf(fact_641_insertE,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_642_right__diff__distrib_H,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( times_times_nat @ A2 @ ( minus_minus_nat @ B @ C2 ) )
= ( minus_minus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_643_right__diff__distrib_H,axiom,
! [A2: real,B: real,C2: real] :
( ( times_times_real @ A2 @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_644_left__diff__distrib_H,axiom,
! [B: nat,C2: nat,A2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C2 ) @ A2 )
= ( minus_minus_nat @ ( times_times_nat @ B @ A2 ) @ ( times_times_nat @ C2 @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_645_left__diff__distrib_H,axiom,
! [B: real,C2: real,A2: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C2 ) @ A2 )
= ( minus_minus_real @ ( times_times_real @ B @ A2 ) @ ( times_times_real @ C2 @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_646_right__diff__distrib,axiom,
! [A2: real,B: real,C2: real] :
( ( times_times_real @ A2 @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_647_left__diff__distrib,axiom,
! [A2: real,B: real,C2: real] :
( ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ C2 )
= ( minus_minus_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_648_Diff__insert__absorb,axiom,
! [X2: real,A: set_real] :
( ~ ( member_real @ X2 @ A )
=> ( ( minus_minus_set_real @ ( insert_real @ X2 @ A ) @ ( insert_real @ X2 @ bot_bot_set_real ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_649_Diff__insert__absorb,axiom,
! [X2: nat,A: set_nat] :
( ~ ( member_nat @ X2 @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_650_insert__Diff,axiom,
! [A2: real,A: set_real] :
( ( member_real @ A2 @ A )
=> ( ( insert_real @ A2 @ ( minus_minus_set_real @ A @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
= A ) ) ).
% insert_Diff
thf(fact_651_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_652_cot__pfd__plus__1__real,axiom,
! [X2: real] :
( ~ ( member_real @ X2 @ ring_1_Ints_real )
=> ( ( cotang1502006655779026648d_real @ ( plus_plus_real @ X2 @ one_one_real ) )
= ( plus_plus_real @ ( minus_minus_real @ ( cotang1502006655779026648d_real @ X2 ) @ ( divide_divide_real @ one_one_real @ ( plus_plus_real @ X2 @ one_one_real ) ) ) @ ( divide_divide_real @ one_one_real @ X2 ) ) ) ) ).
% cot_pfd_plus_1_real
thf(fact_653_square__diff__square__factored,axiom,
! [X2: real,Y3: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) )
= ( times_times_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( minus_minus_real @ X2 @ Y3 ) ) ) ).
% square_diff_square_factored
thf(fact_654_eq__add__iff2,axiom,
! [A2: real,E: real,C2: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( C2
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A2 ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_655_eq__add__iff1,axiom,
! [A2: real,E: real,C2: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ E ) @ C2 )
= D ) ) ).
% eq_add_iff1
thf(fact_656_divide__diff__eq__iff,axiom,
! [Z3: real,X2: real,Y3: real] :
( ( Z3 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Z3 ) @ Y3 )
= ( divide_divide_real @ ( minus_minus_real @ X2 @ ( times_times_real @ Y3 @ Z3 ) ) @ Z3 ) ) ) ).
% divide_diff_eq_iff
thf(fact_657_diff__divide__eq__iff,axiom,
! [Z3: real,X2: real,Y3: real] :
( ( Z3 != zero_zero_real )
=> ( ( minus_minus_real @ X2 @ ( divide_divide_real @ Y3 @ Z3 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z3 ) @ Y3 ) @ Z3 ) ) ) ).
% diff_divide_eq_iff
thf(fact_658_diff__frac__eq,axiom,
! [Y3: real,Z3: real,X2: real,W2: real] :
( ( Y3 != zero_zero_real )
=> ( ( Z3 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W2 @ Z3 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z3 ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z3 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_659_add__divide__eq__if__simps_I4_J,axiom,
! [Z3: real,A2: real,B: real] :
( ( ( Z3 = zero_zero_real )
=> ( ( minus_minus_real @ A2 @ ( divide_divide_real @ B @ Z3 ) )
= A2 ) )
& ( ( Z3 != zero_zero_real )
=> ( ( minus_minus_real @ A2 @ ( divide_divide_real @ B @ Z3 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A2 @ Z3 ) @ B ) @ Z3 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_660_square__diff__one__factored,axiom,
! [X2: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X2 @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_661_division__ring__inverse__diff,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( minus_minus_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B ) )
= ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A2 ) @ ( minus_minus_real @ B @ A2 ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_662_islimpt__punctured,axiom,
( elemen5683178629028408237t_real
= ( ^ [X: real,S: set_real] : ( elemen5683178629028408237t_real @ X @ ( minus_minus_set_real @ S @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ).
% islimpt_punctured
thf(fact_663_islimpt__punctured,axiom,
( elemen5607981409700034897pt_nat
= ( ^ [X: nat,S: set_nat] : ( elemen5607981409700034897pt_nat @ X @ ( minus_minus_set_nat @ S @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% islimpt_punctured
thf(fact_664_right__diff__distrib__NO__MATCH,axiom,
! [X2: real,Y3: real,A2: real,B: real,C2: real] :
( ( nO_MATCH_real_real @ ( divide_divide_real @ X2 @ Y3 ) @ A2 )
=> ( ( times_times_real @ A2 @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C2 ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_665_left__diff__distrib__NO__MATCH,axiom,
! [X2: real,Y3: real,C2: real,A2: real,B: real] :
( ( nO_MATCH_real_real @ ( divide_divide_real @ X2 @ Y3 ) @ C2 )
=> ( ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ C2 )
= ( minus_minus_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_666_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A3: real,B3: real] :
( ( minus_minus_real @ A3 @ B3 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_667_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_668_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_669_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_670_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_671_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_672_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_673_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_674_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_675_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_676_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_677_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_678_insert__UNIV,axiom,
! [X2: real] :
( ( insert_real @ X2 @ top_top_set_real )
= top_top_set_real ) ).
% insert_UNIV
thf(fact_679_insert__UNIV,axiom,
! [X2: nat] :
( ( insert_nat @ X2 @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_680_singleton__iff,axiom,
! [B: real,A2: real] :
( ( member_real @ B @ ( insert_real @ A2 @ bot_bot_set_real ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_681_singleton__iff,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_682_singletonD,axiom,
! [B: real,A2: real] :
( ( member_real @ B @ ( insert_real @ A2 @ bot_bot_set_real ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_683_singletonD,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_684_mult__right__cancel,axiom,
! [C2: real,A2: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_685_mult__left__cancel,axiom,
! [C2: real,A2: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A2 )
= ( times_times_real @ C2 @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_686_no__zero__divisors,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A2 @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_687_divisors__zero,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ B )
= zero_zero_real )
=> ( ( A2 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_688_mult__not__zero,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ B )
!= zero_zero_real )
=> ( ( A2 != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_689_mult__delta__right,axiom,
! [B: $o,X2: real,Y3: real] :
( ( B
=> ( ( times_times_real @ X2 @ ( if_real @ B @ Y3 @ zero_zero_real ) )
= ( times_times_real @ X2 @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_real @ X2 @ ( if_real @ B @ Y3 @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_690_mult__delta__left,axiom,
! [B: $o,X2: real,Y3: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X2 @ zero_zero_real ) @ Y3 )
= ( times_times_real @ X2 @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X2 @ zero_zero_real ) @ Y3 )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_691_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_692_mult_Ocomm__neutral,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% mult.comm_neutral
thf(fact_693_comm__monoid__mult__class_Omult__1,axiom,
! [A2: real] :
( ( times_times_real @ one_one_real @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_694_combine__common__factor,axiom,
! [A2: nat,E: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A2 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_695_combine__common__factor,axiom,
! [A2: real,E: real,B: real,C2: real] :
( ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_696_distrib__right,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_697_distrib__right,axiom,
! [A2: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_698_distrib__left,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( times_times_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C2 ) ) ) ).
% distrib_left
thf(fact_699_distrib__left,axiom,
! [A2: real,B: real,C2: real] :
( ( times_times_real @ A2 @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C2 ) ) ) ).
% distrib_left
thf(fact_700_comm__semiring__class_Odistrib,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_701_comm__semiring__class_Odistrib,axiom,
! [A2: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_702_ring__class_Oring__distribs_I1_J,axiom,
! [A2: real,B: real,C2: real] :
( ( times_times_real @ A2 @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_703_ring__class_Oring__distribs_I2_J,axiom,
! [A2: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_704_divide__divide__eq__left_H,axiom,
! [A2: real,B: real,C2: real] :
( ( divide_divide_real @ ( divide_divide_real @ A2 @ B ) @ C2 )
= ( divide_divide_real @ A2 @ ( times_times_real @ C2 @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_705_divide__divide__times__eq,axiom,
! [X2: real,Y3: real,Z3: real,W2: real] :
( ( divide_divide_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ Z3 @ W2 ) )
= ( divide_divide_real @ ( times_times_real @ X2 @ W2 ) @ ( times_times_real @ Y3 @ Z3 ) ) ) ).
% divide_divide_times_eq
thf(fact_706_times__divide__times__eq,axiom,
! [X2: real,Y3: real,Z3: real,W2: real] :
( ( times_times_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ Z3 @ W2 ) )
= ( divide_divide_real @ ( times_times_real @ X2 @ Z3 ) @ ( times_times_real @ Y3 @ W2 ) ) ) ).
% times_divide_times_eq
thf(fact_707_mult__commute__imp__mult__inverse__commute,axiom,
! [Y3: real,X2: real] :
( ( ( times_times_real @ Y3 @ X2 )
= ( times_times_real @ X2 @ Y3 ) )
=> ( ( times_times_real @ ( inverse_inverse_real @ Y3 ) @ X2 )
= ( times_times_real @ X2 @ ( inverse_inverse_real @ Y3 ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_708_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_709_Ints__mult,axiom,
! [A2: real,B: real] :
( ( member_real @ A2 @ ring_1_Ints_real )
=> ( ( member_real @ B @ ring_1_Ints_real )
=> ( member_real @ ( times_times_real @ A2 @ B ) @ ring_1_Ints_real ) ) ) ).
% Ints_mult
thf(fact_710_islimpt__insert,axiom,
! [X2: real,A2: real,S2: set_real] :
( ( elemen5683178629028408237t_real @ X2 @ ( insert_real @ A2 @ S2 ) )
= ( elemen5683178629028408237t_real @ X2 @ S2 ) ) ).
% islimpt_insert
thf(fact_711_islimpt__insert,axiom,
! [X2: nat,A2: nat,S2: set_nat] :
( ( elemen5607981409700034897pt_nat @ X2 @ ( insert_nat @ A2 @ S2 ) )
= ( elemen5607981409700034897pt_nat @ X2 @ S2 ) ) ).
% islimpt_insert
thf(fact_712_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_713_lambda__one,axiom,
( ( ^ [X: real] : X )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_714_set__zero,axiom,
( zero_zero_set_real
= ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) ).
% set_zero
thf(fact_715_set__one,axiom,
( one_one_set_real
= ( insert_real @ one_one_real @ bot_bot_set_real ) ) ).
% set_one
thf(fact_716_perfect__space__class_OUNIV__not__singleton,axiom,
! [X2: real] :
( top_top_set_real
!= ( insert_real @ X2 @ bot_bot_set_real ) ) ).
% perfect_space_class.UNIV_not_singleton
thf(fact_717_add__scale__eq__noteq,axiom,
! [R2: nat,A2: nat,B: nat,C2: nat,D: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A2 = B )
& ( C2 != D ) )
=> ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R2 @ C2 ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_718_add__scale__eq__noteq,axiom,
! [R2: real,A2: real,B: real,C2: real,D: real] :
( ( R2 != zero_zero_real )
=> ( ( ( A2 = B )
& ( C2 != D ) )
=> ( ( plus_plus_real @ A2 @ ( times_times_real @ R2 @ C2 ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_719_nonzero__eq__divide__eq,axiom,
! [C2: real,A2: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( A2
= ( divide_divide_real @ B @ C2 ) )
= ( ( times_times_real @ A2 @ C2 )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_720_nonzero__divide__eq__eq,axiom,
! [C2: real,B: real,A2: real] :
( ( C2 != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C2 )
= A2 )
= ( B
= ( times_times_real @ A2 @ C2 ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_721_eq__divide__imp,axiom,
! [C2: real,A2: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ C2 )
= B )
=> ( A2
= ( divide_divide_real @ B @ C2 ) ) ) ) ).
% eq_divide_imp
thf(fact_722_divide__eq__imp,axiom,
! [C2: real,B: real,A2: real] :
( ( C2 != zero_zero_real )
=> ( ( B
= ( times_times_real @ A2 @ C2 ) )
=> ( ( divide_divide_real @ B @ C2 )
= A2 ) ) ) ).
% divide_eq_imp
thf(fact_723_eq__divide__eq,axiom,
! [A2: real,B: real,C2: real] :
( ( A2
= ( divide_divide_real @ B @ C2 ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ A2 @ C2 )
= B ) )
& ( ( C2 = zero_zero_real )
=> ( A2 = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_724_divide__eq__eq,axiom,
! [B: real,C2: real,A2: real] :
( ( ( divide_divide_real @ B @ C2 )
= A2 )
= ( ( ( C2 != zero_zero_real )
=> ( B
= ( times_times_real @ A2 @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( A2 = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_725_frac__eq__eq,axiom,
! [Y3: real,Z3: real,X2: real,W2: real] :
( ( Y3 != zero_zero_real )
=> ( ( Z3 != zero_zero_real )
=> ( ( ( divide_divide_real @ X2 @ Y3 )
= ( divide_divide_real @ W2 @ Z3 ) )
= ( ( times_times_real @ X2 @ Z3 )
= ( times_times_real @ W2 @ Y3 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_726_nonzero__inverse__mult__distrib,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( inverse_inverse_real @ ( times_times_real @ A2 @ B ) )
= ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A2 ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_727_inverse__unique,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ B )
= one_one_real )
=> ( ( inverse_inverse_real @ A2 )
= B ) ) ).
% inverse_unique
thf(fact_728_field__class_Ofield__divide__inverse,axiom,
( divide_divide_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B3 ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_729_divide__inverse,axiom,
( divide_divide_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B3 ) ) ) ) ).
% divide_inverse
thf(fact_730_divide__inverse__commute,axiom,
( divide_divide_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ ( inverse_inverse_real @ B3 ) @ A3 ) ) ) ).
% divide_inverse_commute
thf(fact_731_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
@ ^ [X: real] : ( minus_minus_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ ( minus_minus_real @ A2 @ B ) )
@ F ) ) ) ).
% tendsto_diff
thf(fact_732_tendsto__mult,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
@ ^ [X: nat] : ( times_times_nat @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo8926549440605965083ds_nat @ ( times_times_nat @ A2 @ B ) )
@ F ) ) ) ).
% tendsto_mult
thf(fact_733_tendsto__mult,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
@ ^ [X: real] : ( times_times_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ ( times_times_real @ A2 @ B ) )
@ F ) ) ) ).
% tendsto_mult
thf(fact_734_tendsto__mult__left,axiom,
! [F2: nat > nat,L: nat,F: filter_nat,C2: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
=> ( filterlim_nat_nat
@ ^ [X: nat] : ( times_times_nat @ C2 @ ( F2 @ X ) )
@ ( topolo8926549440605965083ds_nat @ ( times_times_nat @ C2 @ L ) )
@ F ) ) ).
% tendsto_mult_left
thf(fact_735_tendsto__mult__left,axiom,
! [F2: real > real,L: real,F: filter_real,C2: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( filterlim_real_real
@ ^ [X: real] : ( times_times_real @ C2 @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ ( times_times_real @ C2 @ L ) )
@ F ) ) ).
% tendsto_mult_left
thf(fact_736_tendsto__mult__right,axiom,
! [F2: nat > nat,L: nat,F: filter_nat,C2: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F )
=> ( filterlim_nat_nat
@ ^ [X: nat] : ( times_times_nat @ ( F2 @ X ) @ C2 )
@ ( topolo8926549440605965083ds_nat @ ( times_times_nat @ L @ C2 ) )
@ F ) ) ).
% tendsto_mult_right
thf(fact_737_tendsto__mult__right,axiom,
! [F2: real > real,L: real,F: filter_real,C2: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( filterlim_real_real
@ ^ [X: real] : ( times_times_real @ ( F2 @ X ) @ C2 )
@ ( topolo2815343760600316023s_real @ ( times_times_real @ L @ C2 ) )
@ F ) ) ).
% tendsto_mult_right
thf(fact_738_filterlim__at__withinI,axiom,
! [F2: real > real,C2: real,F: filter_real,A: set_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ( ( eventually_real
@ ^ [X: real] : ( member_real @ ( F2 @ X ) @ ( minus_minus_set_real @ A @ ( insert_real @ C2 @ bot_bot_set_real ) ) )
@ F )
=> ( filterlim_real_real @ F2 @ ( topolo2177554685111907308n_real @ C2 @ A ) @ F ) ) ) ).
% filterlim_at_withinI
thf(fact_739_filterlim__at__withinI,axiom,
! [F2: nat > real,C2: real,F: filter_nat,A: set_real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ( ( eventually_nat
@ ^ [X: nat] : ( member_real @ ( F2 @ X ) @ ( minus_minus_set_real @ A @ ( insert_real @ C2 @ bot_bot_set_real ) ) )
@ F )
=> ( filterlim_nat_real @ F2 @ ( topolo2177554685111907308n_real @ C2 @ A ) @ F ) ) ) ).
% filterlim_at_withinI
thf(fact_740_filterlim__at__withinI,axiom,
! [F2: real > nat,C2: nat,F: filter_real,A: set_nat] :
( ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ C2 ) @ F )
=> ( ( eventually_real
@ ^ [X: real] : ( member_nat @ ( F2 @ X ) @ ( minus_minus_set_nat @ A @ ( insert_nat @ C2 @ bot_bot_set_nat ) ) )
@ F )
=> ( filterlim_real_nat @ F2 @ ( topolo4659099751122792720in_nat @ C2 @ A ) @ F ) ) ) ).
% filterlim_at_withinI
thf(fact_741_filterlim__at__withinI,axiom,
! [F2: nat > nat,C2: nat,F: filter_nat,A: set_nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ C2 ) @ F )
=> ( ( eventually_nat
@ ^ [X: nat] : ( member_nat @ ( F2 @ X ) @ ( minus_minus_set_nat @ A @ ( insert_nat @ C2 @ bot_bot_set_nat ) ) )
@ F )
=> ( filterlim_nat_nat @ F2 @ ( topolo4659099751122792720in_nat @ C2 @ A ) @ F ) ) ) ).
% filterlim_at_withinI
thf(fact_742_divide__add__eq__iff,axiom,
! [Z3: real,X2: real,Y3: real] :
( ( Z3 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Z3 ) @ Y3 )
= ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Y3 @ Z3 ) ) @ Z3 ) ) ) ).
% divide_add_eq_iff
thf(fact_743_add__divide__eq__iff,axiom,
! [Z3: real,X2: real,Y3: real] :
( ( Z3 != zero_zero_real )
=> ( ( plus_plus_real @ X2 @ ( divide_divide_real @ Y3 @ Z3 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z3 ) @ Y3 ) @ Z3 ) ) ) ).
% add_divide_eq_iff
thf(fact_744_add__num__frac,axiom,
! [Y3: real,Z3: real,X2: real] :
( ( Y3 != zero_zero_real )
=> ( ( plus_plus_real @ Z3 @ ( divide_divide_real @ X2 @ Y3 ) )
= ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z3 @ Y3 ) ) @ Y3 ) ) ) ).
% add_num_frac
thf(fact_745_add__frac__num,axiom,
! [Y3: real,X2: real,Z3: real] :
( ( Y3 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y3 ) @ Z3 )
= ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z3 @ Y3 ) ) @ Y3 ) ) ) ).
% add_frac_num
thf(fact_746_add__frac__eq,axiom,
! [Y3: real,Z3: real,X2: real,W2: real] :
( ( Y3 != zero_zero_real )
=> ( ( Z3 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W2 @ Z3 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z3 ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z3 ) ) ) ) ) ).
% add_frac_eq
thf(fact_747_add__divide__eq__if__simps_I1_J,axiom,
! [Z3: real,A2: real,B: real] :
( ( ( Z3 = zero_zero_real )
=> ( ( plus_plus_real @ A2 @ ( divide_divide_real @ B @ Z3 ) )
= A2 ) )
& ( ( Z3 != zero_zero_real )
=> ( ( plus_plus_real @ A2 @ ( divide_divide_real @ B @ Z3 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A2 @ Z3 ) @ B ) @ Z3 ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_748_add__divide__eq__if__simps_I2_J,axiom,
! [Z3: real,A2: real,B: real] :
( ( ( Z3 = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A2 @ Z3 ) @ B )
= B ) )
& ( ( Z3 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A2 @ Z3 ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A2 @ ( times_times_real @ B @ Z3 ) ) @ Z3 ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_749_field__class_Ofield__inverse,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A2 ) @ A2 )
= one_one_real ) ) ).
% field_class.field_inverse
thf(fact_750_division__ring__inverse__add,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( plus_plus_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B ) )
= ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A2 ) @ ( plus_plus_real @ A2 @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_751_inverse__add,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( plus_plus_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B ) )
= ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ ( inverse_inverse_real @ A2 ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% inverse_add
thf(fact_752_distrib__left__NO__MATCH,axiom,
! [X2: real,Y3: real,A2: nat,B: nat,C2: nat] :
( ( nO_MATCH_real_nat @ ( divide_divide_real @ X2 @ Y3 ) @ A2 )
=> ( ( times_times_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C2 ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_753_distrib__left__NO__MATCH,axiom,
! [X2: real,Y3: real,A2: real,B: real,C2: real] :
( ( nO_MATCH_real_real @ ( divide_divide_real @ X2 @ Y3 ) @ A2 )
=> ( ( times_times_real @ A2 @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C2 ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_754_distrib__right__NO__MATCH,axiom,
! [X2: real,Y3: real,C2: nat,A2: nat,B: nat] :
( ( nO_MATCH_real_nat @ ( divide_divide_real @ X2 @ Y3 ) @ C2 )
=> ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_755_distrib__right__NO__MATCH,axiom,
! [X2: real,Y3: real,C2: real,A2: real,B: real] :
( ( nO_MATCH_real_real @ ( divide_divide_real @ X2 @ Y3 ) @ C2 )
=> ( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_756_Lim__transform__eq,axiom,
! [F2: real > real,G: real > real,F: filter_real,A2: real] :
( ( filterlim_real_real
@ ^ [X: real] : ( minus_minus_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( 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_757_LIM__zero__cancel,axiom,
! [F2: real > real,L: real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X: real] : ( minus_minus_real @ ( F2 @ X ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ).
% LIM_zero_cancel
thf(fact_758_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
@ ^ [X: real] : ( minus_minus_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
=> ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ A2 ) @ F ) ) ) ).
% Lim_transform2
thf(fact_759_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
@ ^ [X: real] : ( minus_minus_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F ) ) ) ).
% Lim_transform
thf(fact_760_LIM__zero__iff,axiom,
! [F2: real > real,L: real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X: real] : ( minus_minus_real @ ( F2 @ X ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F )
= ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ).
% LIM_zero_iff
thf(fact_761_LIM__zero,axiom,
! [F2: real > real,L: real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F )
=> ( filterlim_real_real
@ ^ [X: real] : ( minus_minus_real @ ( F2 @ X ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ).
% LIM_zero
thf(fact_762_Lim__null,axiom,
! [F2: real > real,L: real,Net: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ Net )
= ( filterlim_real_real
@ ^ [X: real] : ( minus_minus_real @ ( F2 @ X ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ Net ) ) ).
% Lim_null
thf(fact_763_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 )
@ ^ [X: real] : ( minus_minus_real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% isCont_diff
thf(fact_764_isCont__diff,axiom,
! [A2: nat,F2: nat > real,G: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) @ F2 )
=> ( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) @ G )
=> ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat )
@ ^ [X: nat] : ( minus_minus_real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% isCont_diff
thf(fact_765_tendsto__mult__right__zero,axiom,
! [F2: real > real,F: filter_real,C2: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F )
=> ( filterlim_real_real
@ ^ [X: real] : ( times_times_real @ C2 @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ).
% tendsto_mult_right_zero
thf(fact_766_tendsto__mult__left__zero,axiom,
! [F2: real > real,F: filter_real,C2: real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F )
=> ( filterlim_real_real
@ ^ [X: real] : ( times_times_real @ ( F2 @ X ) @ C2 )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ).
% tendsto_mult_left_zero
thf(fact_767_tendsto__mult__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
@ ^ [X: real] : ( times_times_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ) ).
% tendsto_mult_zero
thf(fact_768_tendsto__mult__one,axiom,
! [F2: nat > nat,F: filter_nat,G: nat > nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ one_one_nat ) @ F )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ one_one_nat ) @ F )
=> ( filterlim_nat_nat
@ ^ [X: nat] : ( times_times_nat @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo8926549440605965083ds_nat @ one_one_nat )
@ F ) ) ) ).
% tendsto_mult_one
thf(fact_769_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
@ ^ [X: real] : ( minus_minus_real @ ( F14 @ X ) @ ( F25 @ X ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F ) ) ) ).
% tendsto_diff_smallo_iff
thf(fact_770_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
@ ^ [X: real] : ( minus_minus_real @ ( F14 @ X ) @ ( F25 @ X ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ F ) ) ) ).
% tendsto_diff_smallo
thf(fact_771_isCont__mult,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 )
@ ^ [X: real] : ( times_times_real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% isCont_mult
thf(fact_772_isCont__mult,axiom,
! [A2: nat,F2: nat > real,G: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) @ F2 )
=> ( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) @ G )
=> ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat )
@ ^ [X: nat] : ( times_times_real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% isCont_mult
thf(fact_773_filterlim__atI_H,axiom,
! [F2: real > real,C2: real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X: real] : ( minus_minus_real @ ( F2 @ X ) @ 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_774_islimpt__UNIV__iff,axiom,
! [X2: real] :
( ( elemen5683178629028408237t_real @ X2 @ top_top_set_real )
= ( ~ ( topolo4860482606490270245n_real @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ).
% islimpt_UNIV_iff
thf(fact_775_islimpt__UNIV__iff,axiom,
! [X2: nat] :
( ( elemen5607981409700034897pt_nat @ X2 @ top_top_set_nat )
= ( ~ ( topolo4328251076210115529en_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% islimpt_UNIV_iff
thf(fact_776_landau__o_Osmall_Odivide__eq1,axiom,
! [H2: real > real,F: filter_real,F2: real > real,G: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( H2 @ X )
!= zero_zero_real )
@ F )
=> ( ( member_real_real @ F2
@ ( landau3007391416991288786l_real @ F
@ ^ [X: real] : ( divide_divide_real @ ( G @ X ) @ ( H2 @ X ) ) ) )
= ( member_real_real
@ ^ [X: real] : ( times_times_real @ ( F2 @ X ) @ ( H2 @ X ) )
@ ( landau3007391416991288786l_real @ F @ G ) ) ) ) ).
% landau_o.small.divide_eq1
thf(fact_777_landau__o_Osmall_Odivide__eq1,axiom,
! [H2: nat > real,F: filter_nat,F2: nat > real,G: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( H2 @ X )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real @ F2
@ ( landau997807338407142774t_real @ F
@ ^ [X: nat] : ( divide_divide_real @ ( G @ X ) @ ( H2 @ X ) ) ) )
= ( member_nat_real
@ ^ [X: nat] : ( times_times_real @ ( F2 @ X ) @ ( H2 @ X ) )
@ ( landau997807338407142774t_real @ F @ G ) ) ) ) ).
% landau_o.small.divide_eq1
thf(fact_778_landau__o_Osmall_Odivide__eq2,axiom,
! [H2: real > real,F: filter_real,F2: real > real,G: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( H2 @ X )
!= zero_zero_real )
@ F )
=> ( ( member_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( H2 @ X ) )
@ ( landau3007391416991288786l_real @ F @ G ) )
= ( member_real_real @ F2
@ ( landau3007391416991288786l_real @ F
@ ^ [X: real] : ( times_times_real @ ( G @ X ) @ ( H2 @ X ) ) ) ) ) ) ).
% landau_o.small.divide_eq2
thf(fact_779_landau__o_Osmall_Odivide__eq2,axiom,
! [H2: nat > real,F: filter_nat,F2: nat > real,G: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( H2 @ X )
!= zero_zero_real )
@ F )
=> ( ( member_nat_real
@ ^ [X: nat] : ( divide_divide_real @ ( F2 @ X ) @ ( H2 @ X ) )
@ ( landau997807338407142774t_real @ F @ G ) )
= ( member_nat_real @ F2
@ ( landau997807338407142774t_real @ F
@ ^ [X: nat] : ( times_times_real @ ( G @ X ) @ ( H2 @ X ) ) ) ) ) ) ).
% landau_o.small.divide_eq2
thf(fact_780_tendsto__mult__filterlim__at__infinity,axiom,
! [F2: real > real,C2: real,F: filter_real,G: real > real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ C2 ) @ F )
=> ( ( C2 != zero_zero_real )
=> ( ( filterlim_real_real @ G @ at_infinity_real @ F )
=> ( filterlim_real_real
@ ^ [X: real] : ( times_times_real @ ( F2 @ X ) @ ( G @ X ) )
@ at_infinity_real
@ F ) ) ) ) ).
% tendsto_mult_filterlim_at_infinity
thf(fact_781_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_782_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_783_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_784_sum__squares__eq__zero__iff,axiom,
! [X2: real,Y3: real] :
( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y3 = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_785_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A2: real,X2: real,B: real] :
( ( ( times_times_real @ A2 @ X2 )
= ( times_times_real @ B @ X2 ) )
= ( ( A2 = B )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_786_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A2: real,X2: real,Y3: real] :
( ( ( times_times_real @ A2 @ X2 )
= ( times_times_real @ A2 @ Y3 ) )
= ( ( X2 = Y3 )
| ( A2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_787_vector__space__over__itself_Oscale__zero__right,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_788_Diff__iff,axiom,
! [C2: real,A: set_real,B2: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A @ B2 ) )
= ( ( member_real @ C2 @ A )
& ~ ( member_real @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_789_Diff__iff,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
= ( ( member_nat @ C2 @ A )
& ~ ( member_nat @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_790_DiffI,axiom,
! [C2: real,A: set_real,B2: set_real] :
( ( member_real @ C2 @ A )
=> ( ~ ( member_real @ C2 @ B2 )
=> ( member_real @ C2 @ ( minus_minus_set_real @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_791_DiffI,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ A )
=> ( ~ ( member_nat @ C2 @ B2 )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_792_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A2: real,X2: real] :
( ( ( times_times_real @ A2 @ X2 )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_793_vector__space__over__itself_Oscale__zero__left,axiom,
! [X2: real] :
( ( times_times_real @ zero_zero_real @ X2 )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_794_set__diff__eq,axiom,
( minus_minus_set_real
= ( ^ [A4: set_real,B4: set_real] :
( collect_real
@ ^ [X: real] :
( ( member_real @ X @ A4 )
& ~ ( member_real @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_795_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A4 )
& ~ ( member_nat @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_796_minus__set__def,axiom,
( minus_minus_set_real
= ( ^ [A4: set_real,B4: set_real] :
( collect_real
@ ( minus_minus_real_o
@ ^ [X: real] : ( member_real @ X @ A4 )
@ ^ [X: real] : ( member_real @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_797_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A4 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_798_DiffD2,axiom,
! [C2: real,A: set_real,B2: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A @ B2 ) )
=> ~ ( member_real @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_799_DiffD2,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( member_nat @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_800_DiffD1,axiom,
! [C2: real,A: set_real,B2: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A @ B2 ) )
=> ( member_real @ C2 @ A ) ) ).
% DiffD1
thf(fact_801_DiffD1,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
=> ( member_nat @ C2 @ A ) ) ).
% DiffD1
thf(fact_802_DiffE,axiom,
! [C2: real,A: set_real,B2: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A @ B2 ) )
=> ~ ( ( member_real @ C2 @ A )
=> ( member_real @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_803_DiffE,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_804_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X2: real,A2: real,B: real] :
( ( X2 != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ X2 )
= ( times_times_real @ B @ X2 ) )
=> ( A2 = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_805_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A2: real,X2: real,Y3: real] :
( ( A2 != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ X2 )
= ( times_times_real @ A2 @ Y3 ) )
=> ( X2 = Y3 ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_806_vector__space__over__itself_Ozero__not__in__Basis,axiom,
~ ( member_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) ).
% vector_space_over_itself.zero_not_in_Basis
thf(fact_807_vector__space__over__itself_Oscale__right__distrib__NO__MATCH,axiom,
! [X2: real,Y3: real,A2: real] :
( ( nO_MATCH_real_real @ ( divide_divide_real @ X2 @ Y3 ) @ A2 )
=> ( ( times_times_real @ A2 @ ( plus_plus_real @ X2 @ Y3 ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ X2 ) @ ( times_times_real @ A2 @ Y3 ) ) ) ) ).
% vector_space_over_itself.scale_right_distrib_NO_MATCH
thf(fact_808_vector__space__over__itself_Oscale__right__diff__distrib__NO__MATCH,axiom,
! [X2: real,Y3: real,A2: real] :
( ( nO_MATCH_real_real @ ( divide_divide_real @ X2 @ Y3 ) @ A2 )
=> ( ( times_times_real @ A2 @ ( minus_minus_real @ X2 @ Y3 ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ X2 ) @ ( times_times_real @ A2 @ Y3 ) ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib_NO_MATCH
thf(fact_809_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
@ ^ [X: real] : ( divide_divide_real @ one_one_real @ ( U3 @ X ) )
@ ( topolo2815343760600316023s_real @ ( divide_divide_real @ one_one_real @ L ) )
@ F ) ) ) ).
% tendsto_inverse_real
thf(fact_810_mult__if__delta,axiom,
! [P: $o,Q2: real] :
( ( P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q2 )
= Q2 ) )
& ( ~ P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q2 )
= zero_zero_real ) ) ) ).
% mult_if_delta
thf(fact_811_field__has__derivative__at,axiom,
! [F2: real > real,D2: real,X2: real] :
( ( has_de1759254742604945161l_real @ F2 @ ( times_times_real @ D2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
= ( filterlim_real_real
@ ^ [H: real] : ( divide_divide_real @ ( minus_minus_real @ ( F2 @ ( plus_plus_real @ X2 @ H ) ) @ ( F2 @ X2 ) ) @ H )
@ ( topolo2815343760600316023s_real @ D2 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ).
% field_has_derivative_at
thf(fact_812_has__derivative__diff,axiom,
! [F2: real > real,F7: real > real,F: filter_real,G: real > real,G3: real > real] :
( ( has_de1759254742604945161l_real @ F2 @ F7 @ F )
=> ( ( has_de1759254742604945161l_real @ G @ G3 @ F )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( minus_minus_real @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [X: real] : ( minus_minus_real @ ( F7 @ X ) @ ( G3 @ X ) )
@ F ) ) ) ).
% has_derivative_diff
thf(fact_813_has__derivative__ident,axiom,
! [F: filter_real] :
( has_de1759254742604945161l_real
@ ^ [X: real] : X
@ ^ [X: real] : X
@ F ) ).
% has_derivative_ident
thf(fact_814_has__derivative__transform,axiom,
! [X2: real,S2: set_real,G: real > real,F2: real > real,F7: real > real] :
( ( member_real @ X2 @ S2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( F2 @ X4 ) ) )
=> ( ( has_de1759254742604945161l_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( has_de1759254742604945161l_real @ G @ F7 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ) ) ).
% has_derivative_transform
thf(fact_815_has__derivative__add,axiom,
! [F2: real > real,F7: real > real,F: filter_real,G: real > real,G3: real > real] :
( ( has_de1759254742604945161l_real @ F2 @ F7 @ F )
=> ( ( has_de1759254742604945161l_real @ G @ G3 @ F )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( plus_plus_real @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [X: real] : ( plus_plus_real @ ( F7 @ X ) @ ( G3 @ X ) )
@ F ) ) ) ).
% has_derivative_add
thf(fact_816_has__derivative__mult__left,axiom,
! [G: real > real,G3: real > real,F: filter_real,Y3: real] :
( ( has_de1759254742604945161l_real @ G @ G3 @ F )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( times_times_real @ ( G @ X ) @ Y3 )
@ ^ [X: real] : ( times_times_real @ ( G3 @ X ) @ Y3 )
@ F ) ) ).
% has_derivative_mult_left
thf(fact_817_has__derivative__mult__right,axiom,
! [G: real > real,G3: real > real,F: filter_real,X2: real] :
( ( has_de1759254742604945161l_real @ G @ G3 @ F )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( times_times_real @ X2 @ ( G @ X ) )
@ ^ [X: real] : ( times_times_real @ X2 @ ( G3 @ X ) )
@ F ) ) ).
% has_derivative_mult_right
thf(fact_818_has__derivative__const,axiom,
! [C2: real,F: filter_real] :
( has_de1759254742604945161l_real
@ ^ [X: real] : C2
@ ^ [X: real] : zero_zero_real
@ F ) ).
% has_derivative_const
thf(fact_819_has__derivative__unique,axiom,
! [F2: real > real,F: real > real,X2: real,F8: real > real] :
( ( has_de1759254742604945161l_real @ F2 @ F @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( has_de1759254742604945161l_real @ F2 @ F8 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( F = F8 ) ) ) ).
% has_derivative_unique
thf(fact_820_has__derivative__transform__within__open,axiom,
! [F2: real > real,F7: real > real,X2: real,T5: set_real,S2: set_real,G: real > real] :
( ( has_de1759254742604945161l_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X2 @ T5 ) )
=> ( ( topolo4860482606490270245n_real @ S2 )
=> ( ( member_real @ X2 @ S2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) )
=> ( has_de1759254742604945161l_real @ G @ F7 @ ( topolo2177554685111907308n_real @ X2 @ T5 ) ) ) ) ) ) ).
% has_derivative_transform_within_open
thf(fact_821_has__derivative__continuous,axiom,
! [F2: real > real,F7: real > real,X2: real,S2: set_real] :
( ( has_de1759254742604945161l_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ S2 ) @ F2 ) ) ).
% has_derivative_continuous
thf(fact_822_has__derivative__compose,axiom,
! [F2: real > real,F7: real > real,X2: real,S2: set_real,G: real > real,G3: real > real] :
( ( has_de1759254742604945161l_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( has_de1759254742604945161l_real @ G @ G3 @ ( topolo2177554685111907308n_real @ ( F2 @ X2 ) @ top_top_set_real ) )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( G @ ( F2 @ X ) )
@ ^ [X: real] : ( G3 @ ( F7 @ X ) )
@ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ) ).
% has_derivative_compose
thf(fact_823_has__derivative__Uniq,axiom,
! [F2: real > real,X2: real] :
( uniq_real_real
@ ^ [F5: real > real] : ( has_de1759254742604945161l_real @ F2 @ F5 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% has_derivative_Uniq
thf(fact_824_has__derivative__transform__eventually,axiom,
! [F2: real > real,F7: real > real,X2: real,S2: set_real,G: real > real] :
( ( has_de1759254742604945161l_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( eventually_real
@ ^ [X7: real] :
( ( F2 @ X7 )
= ( G @ X7 ) )
@ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( ( F2 @ X2 )
= ( G @ X2 ) )
=> ( ( member_real @ X2 @ S2 )
=> ( has_de1759254742604945161l_real @ G @ F7 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ) ) ) ).
% has_derivative_transform_eventually
thf(fact_825_is__singletonI_H,axiom,
! [A: set_real] :
( ( A != bot_bot_set_real )
=> ( ! [X4: real,Y4: real] :
( ( member_real @ X4 @ A )
=> ( ( member_real @ Y4 @ A )
=> ( X4 = Y4 ) ) )
=> ( is_singleton_real @ A ) ) ) ).
% is_singletonI'
thf(fact_826_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X4: nat,Y4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( X4 = Y4 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_827_Deriv_Ohas__derivative__zero__unique,axiom,
! [F: real > real,X2: real] :
( ( has_de1759254742604945161l_real
@ ^ [X: real] : zero_zero_real
@ F
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( F
= ( ^ [H: real] : zero_zero_real ) ) ) ).
% Deriv.has_derivative_zero_unique
thf(fact_828_has__derivative__mult,axiom,
! [F2: real > real,F7: real > real,X2: real,S2: set_real,G: real > real,G3: real > real] :
( ( has_de1759254742604945161l_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( has_de1759254742604945161l_real @ G @ G3 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( times_times_real @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [H: real] : ( plus_plus_real @ ( times_times_real @ ( F2 @ X2 ) @ ( G3 @ H ) ) @ ( times_times_real @ ( F7 @ H ) @ ( G @ X2 ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ) ).
% has_derivative_mult
thf(fact_829_has__derivative__divide_H,axiom,
! [F2: real > real,F7: real > real,X2: real,S3: set_real,G: real > real,G3: real > real] :
( ( has_de1759254742604945161l_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X2 @ S3 ) )
=> ( ( has_de1759254742604945161l_real @ G @ G3 @ ( topolo2177554685111907308n_real @ X2 @ S3 ) )
=> ( ( ( G @ X2 )
!= zero_zero_real )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [H: real] : ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( F7 @ H ) @ ( G @ X2 ) ) @ ( times_times_real @ ( F2 @ X2 ) @ ( G3 @ H ) ) ) @ ( times_times_real @ ( G @ X2 ) @ ( G @ X2 ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ S3 ) ) ) ) ) ).
% has_derivative_divide'
thf(fact_830_diff__chain__at,axiom,
! [F2: real > real,F7: real > real,X2: real,G: real > real,G3: real > real] :
( ( has_de1759254742604945161l_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( has_de1759254742604945161l_real @ G @ G3 @ ( topolo2177554685111907308n_real @ ( F2 @ X2 ) @ top_top_set_real ) )
=> ( has_de1759254742604945161l_real @ ( comp_real_real_real @ G @ F2 ) @ ( comp_real_real_real @ G3 @ F7 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% diff_chain_at
thf(fact_831_has__derivative__at__withinI,axiom,
! [F2: real > real,F7: real > real,X2: real,S2: set_real] :
( ( has_de1759254742604945161l_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( has_de1759254742604945161l_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ).
% has_derivative_at_withinI
thf(fact_832_has__derivative__subst,axiom,
! [F2: real > real,Df: real > real,X2: real,D: real > real] :
( ( has_de1759254742604945161l_real @ F2 @ Df @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( Df = D )
=> ( has_de1759254742604945161l_real @ F2 @ D @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% has_derivative_subst
thf(fact_833_has__derivative__add__const,axiom,
! [F2: real > real,F7: real > real,Net: filter_real,C2: real] :
( ( has_de1759254742604945161l_real @ F2 @ F7 @ Net )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( plus_plus_real @ ( F2 @ X ) @ C2 )
@ F7
@ Net ) ) ).
% has_derivative_add_const
thf(fact_834_has__derivative__divide,axiom,
! [F2: real > real,F7: real > real,X2: real,S3: set_real,G: real > real,G3: real > real] :
( ( has_de1759254742604945161l_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X2 @ S3 ) )
=> ( ( has_de1759254742604945161l_real @ G @ G3 @ ( topolo2177554685111907308n_real @ X2 @ S3 ) )
=> ( ( ( G @ X2 )
!= zero_zero_real )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [H: real] : ( plus_plus_real @ ( times_times_real @ ( uminus_uminus_real @ ( F2 @ X2 ) ) @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ ( G @ X2 ) ) @ ( G3 @ H ) ) @ ( inverse_inverse_real @ ( G @ X2 ) ) ) ) @ ( divide_divide_real @ ( F7 @ H ) @ ( G @ X2 ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ S3 ) ) ) ) ) ).
% has_derivative_divide
thf(fact_835_DERIV__def,axiom,
! [F2: real > real,D2: real,X2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
= ( filterlim_real_real
@ ^ [H: real] : ( divide_divide_real @ ( minus_minus_real @ ( F2 @ ( plus_plus_real @ X2 @ H ) ) @ ( F2 @ X2 ) ) @ H )
@ ( topolo2815343760600316023s_real @ D2 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ).
% DERIV_def
thf(fact_836_DERIV__D,axiom,
! [F2: real > real,D2: real,X2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [H: real] : ( divide_divide_real @ ( minus_minus_real @ ( F2 @ ( plus_plus_real @ X2 @ H ) ) @ ( F2 @ X2 ) ) @ H )
@ ( topolo2815343760600316023s_real @ D2 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ) ).
% DERIV_D
thf(fact_837_add_Oinverse__inverse,axiom,
! [A2: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_838_neg__equal__iff__equal,axiom,
! [A2: real,B: real] :
( ( ( uminus_uminus_real @ A2 )
= ( uminus_uminus_real @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_839_cot__pfd__real__minus,axiom,
! [X2: real] :
( ( cotang1502006655779026648d_real @ ( uminus_uminus_real @ X2 ) )
= ( uminus_uminus_real @ ( cotang1502006655779026648d_real @ X2 ) ) ) ).
% cot_pfd_real_minus
thf(fact_840_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_841_neg__0__equal__iff__equal,axiom,
! [A2: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A2 ) )
= ( zero_zero_real = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_842_neg__equal__0__iff__equal,axiom,
! [A2: real] :
( ( ( uminus_uminus_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_843_equal__neg__zero,axiom,
! [A2: real] :
( ( A2
= ( uminus_uminus_real @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_844_neg__equal__zero,axiom,
! [A2: real] :
( ( ( uminus_uminus_real @ A2 )
= A2 )
= ( A2 = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_845_mult__minus__right,axiom,
! [A2: real,B: real] :
( ( times_times_real @ A2 @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ A2 @ B ) ) ) ).
% mult_minus_right
thf(fact_846_minus__mult__minus,axiom,
! [A2: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B ) )
= ( times_times_real @ A2 @ B ) ) ).
% minus_mult_minus
thf(fact_847_mult__minus__left,axiom,
! [A2: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( uminus_uminus_real @ ( times_times_real @ A2 @ B ) ) ) ).
% mult_minus_left
thf(fact_848_minus__add__distrib,axiom,
! [A2: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_849_minus__add__cancel,axiom,
! [A2: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( plus_plus_real @ A2 @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_850_add__minus__cancel,axiom,
! [A2: real,B: real] :
( ( plus_plus_real @ A2 @ ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_851_minus__diff__eq,axiom,
! [A2: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A2 @ B ) )
= ( minus_minus_real @ B @ A2 ) ) ).
% minus_diff_eq
thf(fact_852_inverse__minus__eq,axiom,
! [A2: real] :
( ( inverse_inverse_real @ ( uminus_uminus_real @ A2 ) )
= ( uminus_uminus_real @ ( inverse_inverse_real @ A2 ) ) ) ).
% inverse_minus_eq
thf(fact_853_ab__left__minus,axiom,
! [A2: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_854_add_Oright__inverse,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ ( uminus_uminus_real @ A2 ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_855_diff__0,axiom,
! [A2: real] :
( ( minus_minus_real @ zero_zero_real @ A2 )
= ( uminus_uminus_real @ A2 ) ) ).
% diff_0
thf(fact_856_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_857_diff__minus__eq__add,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ A2 @ ( uminus_uminus_real @ B ) )
= ( plus_plus_real @ A2 @ B ) ) ).
% diff_minus_eq_add
thf(fact_858_uminus__add__conv__diff,axiom,
! [A2: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( minus_minus_real @ B @ A2 ) ) ).
% uminus_add_conv_diff
thf(fact_859_divide__minus1,axiom,
! [X2: real] :
( ( divide_divide_real @ X2 @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ X2 ) ) ).
% divide_minus1
thf(fact_860_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_861_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_862_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_863_has__derivative__minus,axiom,
! [F2: real > real,F7: real > real,F: filter_real] :
( ( has_de1759254742604945161l_real @ F2 @ F7 @ F )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( uminus_uminus_real @ ( F2 @ X ) )
@ ^ [X: real] : ( uminus_uminus_real @ ( F7 @ X ) )
@ F ) ) ).
% has_derivative_minus
thf(fact_864_DERIV__mirror,axiom,
! [F2: real > real,Y3: real,X2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ Y3 @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ X2 ) @ top_top_set_real ) )
= ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( F2 @ ( uminus_uminus_real @ X ) )
@ ( uminus_uminus_real @ Y3 )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% DERIV_mirror
thf(fact_865_DERIV__const,axiom,
! [K: real,F: filter_real] :
( has_fi5821293074295781190e_real
@ ^ [X: real] : K
@ zero_zero_real
@ F ) ).
% DERIV_const
thf(fact_866_DERIV__ident,axiom,
! [F: filter_real] :
( has_fi5821293074295781190e_real
@ ^ [X: real] : X
@ one_one_real
@ F ) ).
% DERIV_ident
thf(fact_867_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_868_DERIV__minus,axiom,
! [F2: real > real,D2: real,X2: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( uminus_uminus_real @ ( F2 @ X ) )
@ ( uminus_uminus_real @ D2 )
@ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ).
% DERIV_minus
thf(fact_869_Deriv_Ofield__differentiable__minus,axiom,
! [F2: real > real,F7: real,F: filter_real] :
( ( has_fi5821293074295781190e_real @ F2 @ F7 @ F )
=> ( has_fi5821293074295781190e_real
@ ^ [Z2: real] : ( uminus_uminus_real @ ( F2 @ Z2 ) )
@ ( uminus_uminus_real @ F7 )
@ F ) ) ).
% Deriv.field_differentiable_minus
thf(fact_870_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_871_DERIV__cmult__Id,axiom,
! [C2: real,X2: real,S2: set_real] : ( has_fi5821293074295781190e_real @ ( times_times_real @ C2 ) @ C2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ).
% DERIV_cmult_Id
thf(fact_872_has__field__derivative__at__within,axiom,
! [F2: real > real,F7: real,X2: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ).
% has_field_derivative_at_within
thf(fact_873_DERIV__unique,axiom,
! [F2: real > real,D2: real,X2: real,E2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( has_fi5821293074295781190e_real @ F2 @ E2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( D2 = E2 ) ) ) ).
% DERIV_unique
thf(fact_874_minus__diff__commute,axiom,
! [B: real,A2: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A2 )
= ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ B ) ) ).
% minus_diff_commute
thf(fact_875_equation__minus__iff,axiom,
! [A2: real,B: real] :
( ( A2
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_876_minus__equation__iff,axiom,
! [A2: real,B: real] :
( ( ( uminus_uminus_real @ A2 )
= B )
= ( ( uminus_uminus_real @ B )
= A2 ) ) ).
% minus_equation_iff
thf(fact_877_minus__in__Ints__iff,axiom,
! [X2: real] :
( ( member_real @ ( uminus_uminus_real @ X2 ) @ ring_1_Ints_real )
= ( member_real @ X2 @ ring_1_Ints_real ) ) ).
% minus_in_Ints_iff
thf(fact_878_Ints__minus,axiom,
! [A2: real] :
( ( member_real @ A2 @ ring_1_Ints_real )
=> ( member_real @ ( uminus_uminus_real @ A2 ) @ ring_1_Ints_real ) ) ).
% Ints_minus
thf(fact_879_minus__divide__left,axiom,
! [A2: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B ) )
= ( divide_divide_real @ ( uminus_uminus_real @ A2 ) @ B ) ) ).
% minus_divide_left
thf(fact_880_minus__divide__divide,axiom,
! [A2: real,B: real] :
( ( divide_divide_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A2 @ B ) ) ).
% minus_divide_divide
thf(fact_881_minus__divide__right,axiom,
! [A2: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B ) )
= ( divide_divide_real @ A2 @ ( uminus_uminus_real @ B ) ) ) ).
% minus_divide_right
thf(fact_882_add_Oinverse__distrib__swap,axiom,
! [A2: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_883_group__cancel_Oneg1,axiom,
! [A: real,K: real,A2: real] :
( ( A
= ( plus_plus_real @ K @ A2 ) )
=> ( ( uminus_uminus_real @ A )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A2 ) ) ) ) ).
% group_cancel.neg1
thf(fact_884_minus__mult__commute,axiom,
! [A2: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( times_times_real @ A2 @ ( uminus_uminus_real @ B ) ) ) ).
% minus_mult_commute
thf(fact_885_square__eq__iff,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ A2 )
= ( times_times_real @ B @ B ) )
= ( ( A2 = B )
| ( A2
= ( uminus_uminus_real @ B ) ) ) ) ).
% square_eq_iff
thf(fact_886_DERIV__continuous,axiom,
! [F2: real > real,D2: real,X2: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ S2 ) @ F2 ) ) ).
% DERIV_continuous
thf(fact_887_DERIV__inverse_H,axiom,
! [F2: real > real,D2: real,X2: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( ( F2 @ X2 )
!= zero_zero_real )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( inverse_inverse_real @ ( F2 @ X ) )
@ ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ ( F2 @ X2 ) ) @ D2 ) @ ( inverse_inverse_real @ ( F2 @ X2 ) ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ) ).
% DERIV_inverse'
thf(fact_888_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_889_neg__eq__iff__add__eq__0,axiom,
! [A2: real,B: real] :
( ( ( uminus_uminus_real @ A2 )
= B )
= ( ( plus_plus_real @ A2 @ B )
= zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_890_eq__neg__iff__add__eq__0,axiom,
! [A2: real,B: real] :
( ( A2
= ( uminus_uminus_real @ B ) )
= ( ( plus_plus_real @ A2 @ B )
= zero_zero_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_891_add_Oinverse__unique,axiom,
! [A2: real,B: real] :
( ( ( plus_plus_real @ A2 @ B )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A2 )
= B ) ) ).
% add.inverse_unique
thf(fact_892_ab__group__add__class_Oab__left__minus,axiom,
! [A2: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= zero_zero_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_893_add__eq__0__iff,axiom,
! [A2: real,B: real] :
( ( ( plus_plus_real @ A2 @ B )
= zero_zero_real )
= ( B
= ( uminus_uminus_real @ A2 ) ) ) ).
% add_eq_0_iff
thf(fact_894_square__eq__1__iff,axiom,
! [X2: real] :
( ( ( times_times_real @ X2 @ X2 )
= one_one_real )
= ( ( X2 = one_one_real )
| ( X2
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% square_eq_1_iff
thf(fact_895_nonzero__minus__divide__right,axiom,
! [B: real,A2: real] :
( ( B != zero_zero_real )
=> ( ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B ) )
= ( divide_divide_real @ A2 @ ( uminus_uminus_real @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_896_nonzero__minus__divide__divide,axiom,
! [B: real,A2: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A2 @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_897_minus__eq__plus__uminus,axiom,
( minus_minus_real
= ( ^ [F5: real,G4: real] : ( plus_plus_real @ F5 @ ( uminus_uminus_real @ G4 ) ) ) ) ).
% minus_eq_plus_uminus
thf(fact_898_group__cancel_Osub2,axiom,
! [B2: real,K: real,B: real,A2: real] :
( ( B2
= ( plus_plus_real @ K @ B ) )
=> ( ( minus_minus_real @ A2 @ B2 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A2 @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_899_diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A3: real,B3: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_900_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A3: real,B3: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_901_Compl__empty__eq,axiom,
( ( uminus612125837232591019t_real @ bot_bot_set_real )
= top_top_set_real ) ).
% Compl_empty_eq
thf(fact_902_Compl__empty__eq,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% Compl_empty_eq
thf(fact_903_Compl__UNIV__eq,axiom,
( ( uminus612125837232591019t_real @ top_top_set_real )
= bot_bot_set_real ) ).
% Compl_UNIV_eq
thf(fact_904_Compl__UNIV__eq,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Compl_UNIV_eq
thf(fact_905_nonzero__inverse__minus__eq,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( inverse_inverse_real @ ( uminus_uminus_real @ A2 ) )
= ( uminus_uminus_real @ ( inverse_inverse_real @ A2 ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_906_Compl__eq__Diff__UNIV,axiom,
( uminus612125837232591019t_real
= ( minus_minus_set_real @ top_top_set_real ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_907_Compl__eq__Diff__UNIV,axiom,
( uminus5710092332889474511et_nat
= ( minus_minus_set_nat @ top_top_set_nat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_908_tendsto__uminus__nhds,axiom,
! [A2: real] : ( filterlim_real_real @ uminus_uminus_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ A2 ) ) @ ( topolo2815343760600316023s_real @ A2 ) ) ).
% tendsto_uminus_nhds
thf(fact_909_DERIV__cmult__right,axiom,
! [F2: real > real,D2: real,X2: real,S2: set_real,C2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( times_times_real @ ( F2 @ X ) @ C2 )
@ ( times_times_real @ D2 @ C2 )
@ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ).
% DERIV_cmult_right
thf(fact_910_DERIV__cmult,axiom,
! [F2: real > real,D2: real,X2: real,S2: set_real,C2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( times_times_real @ C2 @ ( F2 @ X ) )
@ ( times_times_real @ C2 @ D2 )
@ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ).
% DERIV_cmult
thf(fact_911_DERIV__add,axiom,
! [F2: real > real,D2: real,X2: real,S2: set_real,G: real > real,E2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( has_fi5821293074295781190e_real @ G @ E2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( plus_plus_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( plus_plus_real @ D2 @ E2 )
@ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ) ).
% DERIV_add
thf(fact_912_DERIV__diff,axiom,
! [F2: real > real,D2: real,X2: real,S2: set_real,G: real > real,E2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( has_fi5821293074295781190e_real @ G @ E2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( minus_minus_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( minus_minus_real @ D2 @ E2 )
@ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ) ).
% DERIV_diff
thf(fact_913_DERIV__cdivide,axiom,
! [F2: real > real,D2: real,X2: real,S2: set_real,C2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ C2 )
@ ( divide_divide_real @ D2 @ C2 )
@ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ).
% DERIV_cdivide
thf(fact_914_has__field__derivative__cong__eventually,axiom,
! [F2: real > real,G: real > real,X2: real,S3: set_real,U3: real] :
( ( eventually_real
@ ^ [X: real] :
( ( F2 @ X )
= ( G @ X ) )
@ ( topolo2177554685111907308n_real @ X2 @ S3 ) )
=> ( ( ( F2 @ X2 )
= ( G @ X2 ) )
=> ( ( has_fi5821293074295781190e_real @ F2 @ U3 @ ( topolo2177554685111907308n_real @ X2 @ S3 ) )
= ( has_fi5821293074295781190e_real @ G @ U3 @ ( topolo2177554685111907308n_real @ X2 @ S3 ) ) ) ) ) ).
% has_field_derivative_cong_eventually
thf(fact_915_tendsto__minus__cancel__left,axiom,
! [F2: real > real,Y3: real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ Y3 ) ) @ F )
= ( filterlim_real_real
@ ^ [X: real] : ( uminus_uminus_real @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ Y3 )
@ F ) ) ).
% tendsto_minus_cancel_left
thf(fact_916_tendsto__minus__cancel,axiom,
! [F2: real > real,A2: real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X: real] : ( uminus_uminus_real @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ A2 ) )
@ F )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F ) ) ).
% tendsto_minus_cancel
thf(fact_917_tendsto__minus,axiom,
! [F2: real > real,A2: real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ F )
=> ( filterlim_real_real
@ ^ [X: real] : ( uminus_uminus_real @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ A2 ) )
@ F ) ) ).
% tendsto_minus
thf(fact_918_has__field__derivative__transform__within__open,axiom,
! [F2: real > real,F7: real,A2: real,S3: set_real,G: real > real] :
( ( has_fi5821293074295781190e_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( ( topolo4860482606490270245n_real @ S3 )
=> ( ( member_real @ A2 @ S3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S3 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) )
=> ( has_fi5821293074295781190e_real @ G @ F7 @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ) ).
% has_field_derivative_transform_within_open
thf(fact_919_DERIV__isCont,axiom,
! [F2: real > real,D2: real,X2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ F2 ) ) ).
% DERIV_isCont
thf(fact_920_DERIV__isconst__all,axiom,
! [F2: real > real,X2: real,Y3: real] :
( ! [X4: real] : ( has_fi5821293074295781190e_real @ F2 @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
=> ( ( F2 @ X2 )
= ( F2 @ Y3 ) ) ) ).
% DERIV_isconst_all
thf(fact_921_DERIV__const__ratio__const,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 ) )
=> ( ( minus_minus_real @ ( F2 @ B ) @ ( F2 @ A2 ) )
= ( times_times_real @ ( minus_minus_real @ B @ A2 ) @ K ) ) ) ) ).
% DERIV_const_ratio_const
thf(fact_922_DERIV__mult_H,axiom,
! [F2: real > real,D2: real,X2: real,S2: set_real,G: real > real,E2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( has_fi5821293074295781190e_real @ G @ E2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( times_times_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( plus_plus_real @ ( times_times_real @ ( F2 @ X2 ) @ E2 ) @ ( times_times_real @ D2 @ ( G @ X2 ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ) ).
% DERIV_mult'
thf(fact_923_DERIV__mult,axiom,
! [F2: real > real,Da: real,X2: real,S2: set_real,G: real > real,Db: real] :
( ( has_fi5821293074295781190e_real @ F2 @ Da @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( has_fi5821293074295781190e_real @ G @ Db @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( times_times_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( plus_plus_real @ ( times_times_real @ Da @ ( G @ X2 ) ) @ ( times_times_real @ Db @ ( F2 @ X2 ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ) ).
% DERIV_mult
thf(fact_924_DERIV__chain__s,axiom,
! [S2: set_real,G: real > real,G3: real > real,F2: real > real,F7: real,X2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( has_fi5821293074295781190e_real @ G @ ( G3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
=> ( ( has_fi5821293074295781190e_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( member_real @ ( F2 @ X2 ) @ S2 )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( G @ ( F2 @ X ) )
@ ( times_times_real @ F7 @ ( G3 @ ( F2 @ X2 ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).
% DERIV_chain_s
thf(fact_925_DERIV__chain3,axiom,
! [G: real > real,G3: real > real,F2: real > real,F7: real,X2: real] :
( ! [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
=> ( ( has_fi5821293074295781190e_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( G @ ( F2 @ X ) )
@ ( times_times_real @ F7 @ ( G3 @ ( F2 @ X2 ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% DERIV_chain3
thf(fact_926_DERIV__chain2,axiom,
! [F2: real > real,Da: real,G: real > real,X2: real,Db: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ Da @ ( topolo2177554685111907308n_real @ ( G @ X2 ) @ top_top_set_real ) )
=> ( ( has_fi5821293074295781190e_real @ G @ Db @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( F2 @ ( G @ X ) )
@ ( times_times_real @ Da @ Db )
@ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ) ).
% DERIV_chain2
thf(fact_927_DERIV__chain_H,axiom,
! [F2: real > real,D2: real,X2: real,S2: set_real,G: real > real,E2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( has_fi5821293074295781190e_real @ G @ E2 @ ( topolo2177554685111907308n_real @ ( F2 @ X2 ) @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( G @ ( F2 @ X ) )
@ ( times_times_real @ E2 @ D2 )
@ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ) ).
% DERIV_chain'
thf(fact_928_DERIV__shift,axiom,
! [F2: real > real,Y3: real,X2: real,Z3: real] :
( ( has_fi5821293074295781190e_real @ F2 @ Y3 @ ( topolo2177554685111907308n_real @ ( plus_plus_real @ X2 @ Z3 ) @ top_top_set_real ) )
= ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( F2 @ ( plus_plus_real @ X @ Z3 ) )
@ Y3
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% DERIV_shift
thf(fact_929_eq__minus__divide__eq,axiom,
! [A2: real,B: real,C2: real] :
( ( A2
= ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ A2 @ C2 )
= ( uminus_uminus_real @ B ) ) )
& ( ( C2 = zero_zero_real )
=> ( A2 = zero_zero_real ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_930_minus__divide__eq__eq,axiom,
! [B: real,C2: real,A2: real] :
( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) )
= A2 )
= ( ( ( C2 != zero_zero_real )
=> ( ( uminus_uminus_real @ B )
= ( times_times_real @ A2 @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( A2 = zero_zero_real ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_931_nonzero__neg__divide__eq__eq,axiom,
! [B: real,A2: real,C2: real] :
( ( B != zero_zero_real )
=> ( ( ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B ) )
= C2 )
= ( ( uminus_uminus_real @ A2 )
= ( times_times_real @ C2 @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_932_nonzero__neg__divide__eq__eq2,axiom,
! [B: real,C2: real,A2: real] :
( ( B != zero_zero_real )
=> ( ( C2
= ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B ) ) )
= ( ( times_times_real @ C2 @ B )
= ( uminus_uminus_real @ A2 ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_933_divide__eq__minus__1__iff,axiom,
! [A2: real,B: real] :
( ( ( divide_divide_real @ A2 @ B )
= ( uminus_uminus_real @ one_one_real ) )
= ( ( B != zero_zero_real )
& ( A2
= ( uminus_uminus_real @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_934_has__field__derivative__cong__ev,axiom,
! [X2: real,Y3: real,S3: set_real,F2: real > real,G: real > real,U3: real,V: real,T5: set_real] :
( ( X2 = Y3 )
=> ( ( eventually_real
@ ^ [X: real] :
( ( member_real @ X @ S3 )
=> ( ( F2 @ X )
= ( G @ X ) ) )
@ ( topolo2815343760600316023s_real @ X2 ) )
=> ( ( U3 = V )
=> ( ( S3 = T5 )
=> ( ( member_real @ X2 @ S3 )
=> ( ( has_fi5821293074295781190e_real @ F2 @ U3 @ ( topolo2177554685111907308n_real @ X2 @ S3 ) )
= ( has_fi5821293074295781190e_real @ G @ V @ ( topolo2177554685111907308n_real @ Y3 @ T5 ) ) ) ) ) ) ) ) ).
% has_field_derivative_cong_ev
thf(fact_935_DERIV__compose__FDERIV,axiom,
! [F2: real > real,F7: real,G: real > real,X2: real,G3: real > real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ ( G @ X2 ) @ top_top_set_real ) )
=> ( ( has_de1759254742604945161l_real @ G @ G3 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( F2 @ ( G @ X ) )
@ ^ [X: real] : ( times_times_real @ ( G3 @ X ) @ F7 )
@ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ) ).
% DERIV_compose_FDERIV
thf(fact_936_DERIV__Uniq,axiom,
! [F2: real > real,X2: real] :
( uniq_real
@ ^ [D3: real] : ( has_fi5821293074295781190e_real @ F2 @ D3 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% DERIV_Uniq
thf(fact_937_isCont__minus,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 )
@ ^ [X: real] : ( uminus_uminus_real @ ( F2 @ X ) ) ) ) ).
% isCont_minus
thf(fact_938_isCont__minus,axiom,
! [A2: nat,F2: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) @ F2 )
=> ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat )
@ ^ [X: nat] : ( uminus_uminus_real @ ( F2 @ X ) ) ) ) ).
% isCont_minus
thf(fact_939_DERIV__chain,axiom,
! [F2: real > real,Da: real,G: real > real,X2: real,Db: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ Da @ ( topolo2177554685111907308n_real @ ( G @ X2 ) @ top_top_set_real ) )
=> ( ( has_fi5821293074295781190e_real @ G @ Db @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( has_fi5821293074295781190e_real @ ( comp_real_real_real @ F2 @ G ) @ ( times_times_real @ Da @ Db ) @ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ) ).
% DERIV_chain
thf(fact_940_DERIV__caratheodory__within,axiom,
! [F2: real > real,L: real,X2: real,S3: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ L @ ( topolo2177554685111907308n_real @ X2 @ S3 ) )
= ( ? [G5: real > real] :
( ! [Z2: real] :
( ( minus_minus_real @ ( F2 @ Z2 ) @ ( F2 @ X2 ) )
= ( times_times_real @ ( G5 @ Z2 ) @ ( minus_minus_real @ Z2 @ X2 ) ) )
& ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ S3 ) @ G5 )
& ( ( G5 @ X2 )
= L ) ) ) ) ).
% DERIV_caratheodory_within
thf(fact_941_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_942_DERIV__cong__ev,axiom,
! [X2: real,Y3: real,F2: real > real,G: real > real,U3: real,V: real] :
( ( X2 = Y3 )
=> ( ( eventually_real
@ ^ [X: real] :
( ( F2 @ X )
= ( G @ X ) )
@ ( topolo2815343760600316023s_real @ X2 ) )
=> ( ( U3 = V )
=> ( ( has_fi5821293074295781190e_real @ F2 @ U3 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
= ( has_fi5821293074295781190e_real @ G @ V @ ( topolo2177554685111907308n_real @ Y3 @ top_top_set_real ) ) ) ) ) ) ).
% DERIV_cong_ev
thf(fact_943_add__divide__eq__if__simps_I3_J,axiom,
! [Z3: real,A2: real,B: real] :
( ( ( Z3 = zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z3 ) ) @ B )
= B ) )
& ( ( Z3 != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z3 ) ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( times_times_real @ B @ Z3 ) ) @ Z3 ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_944_minus__divide__add__eq__iff,axiom,
! [Z3: real,X2: real,Y3: real] :
( ( Z3 != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z3 ) ) @ Y3 )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y3 @ Z3 ) ) @ Z3 ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_945_minus__divide__diff__eq__iff,axiom,
! [Z3: real,X2: real,Y3: real] :
( ( Z3 != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z3 ) ) @ Y3 )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y3 @ Z3 ) ) @ Z3 ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_946_add__divide__eq__if__simps_I5_J,axiom,
! [Z3: real,A2: real,B: real] :
( ( ( Z3 = zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A2 @ Z3 ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z3 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A2 @ Z3 ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ A2 @ ( times_times_real @ B @ Z3 ) ) @ Z3 ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_947_add__divide__eq__if__simps_I6_J,axiom,
! [Z3: real,A2: real,B: real] :
( ( ( Z3 = zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z3 ) ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z3 != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z3 ) ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ ( times_times_real @ B @ Z3 ) ) @ Z3 ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_948_CARAT__DERIV,axiom,
! [F2: real > real,L: real,X2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
= ( ? [G5: real > real] :
( ! [Z2: real] :
( ( minus_minus_real @ ( F2 @ Z2 ) @ ( F2 @ X2 ) )
= ( times_times_real @ ( G5 @ Z2 ) @ ( minus_minus_real @ Z2 @ X2 ) ) )
& ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ G5 )
& ( ( G5 @ X2 )
= L ) ) ) ) ).
% CARAT_DERIV
thf(fact_949_DERIV__divide,axiom,
! [F2: real > real,D2: real,X2: real,S2: set_real,G: real > real,E2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( has_fi5821293074295781190e_real @ G @ E2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( ( G @ X2 )
!= zero_zero_real )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ D2 @ ( G @ X2 ) ) @ ( times_times_real @ ( F2 @ X2 ) @ E2 ) ) @ ( times_times_real @ ( G @ X2 ) @ ( G @ X2 ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ) ) ).
% DERIV_divide
thf(fact_950_has__field__derivativeD,axiom,
! [F2: real > real,D2: real,X2: real,S3: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X2 @ S3 ) )
=> ( filterlim_real_real
@ ^ [Y: real] : ( divide_divide_real @ ( minus_minus_real @ ( F2 @ Y ) @ ( F2 @ X2 ) ) @ ( minus_minus_real @ Y @ X2 ) )
@ ( topolo2815343760600316023s_real @ D2 )
@ ( topolo2177554685111907308n_real @ X2 @ S3 ) ) ) ).
% has_field_derivativeD
thf(fact_951_has__field__derivative__iff,axiom,
! [F2: real > real,D2: real,X2: real,S3: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ D2 @ ( topolo2177554685111907308n_real @ X2 @ S3 ) )
= ( filterlim_real_real
@ ^ [Y: real] : ( divide_divide_real @ ( minus_minus_real @ ( F2 @ Y ) @ ( F2 @ X2 ) ) @ ( minus_minus_real @ Y @ X2 ) )
@ ( topolo2815343760600316023s_real @ D2 )
@ ( topolo2177554685111907308n_real @ X2 @ S3 ) ) ) ).
% has_field_derivative_iff
thf(fact_952_real__eq__affinity,axiom,
! [M: real,Y3: real,X2: real,C2: real] :
( ( M != zero_zero_real )
=> ( ( Y3
= ( plus_plus_real @ ( times_times_real @ M @ X2 ) @ C2 ) )
= ( ( plus_plus_real @ ( times_times_real @ ( inverse_inverse_real @ M ) @ Y3 ) @ ( uminus_uminus_real @ ( divide_divide_real @ C2 @ M ) ) )
= X2 ) ) ) ).
% real_eq_affinity
thf(fact_953_real__affinity__eq,axiom,
! [M: real,X2: real,C2: real,Y3: real] :
( ( M != zero_zero_real )
=> ( ( ( plus_plus_real @ ( times_times_real @ M @ X2 ) @ C2 )
= Y3 )
= ( X2
= ( plus_plus_real @ ( times_times_real @ ( inverse_inverse_real @ M ) @ Y3 ) @ ( uminus_uminus_real @ ( divide_divide_real @ C2 @ M ) ) ) ) ) ) ).
% real_affinity_eq
thf(fact_954_has__field__derivative__inverse__basic,axiom,
! [F2: real > real,F7: real,G: real > real,Y3: real,T5: set_real] :
( ( has_fi5821293074295781190e_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ ( G @ Y3 ) @ top_top_set_real ) )
=> ( ( F7 != zero_zero_real )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Y3 @ top_top_set_real ) @ G )
=> ( ( topolo4860482606490270245n_real @ T5 )
=> ( ( member_real @ Y3 @ T5 )
=> ( ! [Z4: real] :
( ( member_real @ Z4 @ T5 )
=> ( ( F2 @ ( G @ Z4 ) )
= Z4 ) )
=> ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ F7 ) @ ( topolo2177554685111907308n_real @ Y3 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% has_field_derivative_inverse_basic
thf(fact_955_Deriv_Ohas__derivative__inverse,axiom,
! [F2: real > real,X2: real,F7: real > real,S3: set_real] :
( ( ( F2 @ X2 )
!= zero_zero_real )
=> ( ( has_de1759254742604945161l_real @ F2 @ F7 @ ( topolo2177554685111907308n_real @ X2 @ S3 ) )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( inverse_inverse_real @ ( F2 @ X ) )
@ ^ [H: real] : ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ ( F2 @ X2 ) ) @ ( F7 @ H ) ) @ ( inverse_inverse_real @ ( F2 @ X2 ) ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ S3 ) ) ) ) ).
% Deriv.has_derivative_inverse
thf(fact_956_has__derivative__inverse_H,axiom,
! [X2: real,S3: set_real] :
( ( X2 != zero_zero_real )
=> ( has_de1759254742604945161l_real @ inverse_inverse_real
@ ^ [H: real] : ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ X2 ) @ H ) @ ( inverse_inverse_real @ X2 ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ S3 ) ) ) ).
% has_derivative_inverse'
thf(fact_957_lhopital,axiom,
! [F2: real > real,X2: real,G: real > real,G3: real > real,F7: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G @ X )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G3 @ X )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F7 @ X ) @ ( G3 @ X ) )
@ F
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ F
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ) ) ).
% lhopital
thf(fact_958_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_959_boolean__algebra_Ocompl__zero,axiom,
( ( uminus612125837232591019t_real @ bot_bot_set_real )
= top_top_set_real ) ).
% boolean_algebra.compl_zero
thf(fact_960_boolean__algebra_Ocompl__zero,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.compl_zero
thf(fact_961_boolean__algebra_Ocompl__one,axiom,
( ( uminus612125837232591019t_real @ top_top_set_real )
= bot_bot_set_real ) ).
% boolean_algebra.compl_one
thf(fact_962_boolean__algebra_Ocompl__one,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.compl_one
thf(fact_963_ComplI,axiom,
! [C2: real,A: set_real] :
( ~ ( member_real @ C2 @ A )
=> ( member_real @ C2 @ ( uminus612125837232591019t_real @ A ) ) ) ).
% ComplI
thf(fact_964_ComplI,axiom,
! [C2: nat,A: set_nat] :
( ~ ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_965_Compl__iff,axiom,
! [C2: real,A: set_real] :
( ( member_real @ C2 @ ( uminus612125837232591019t_real @ A ) )
= ( ~ ( member_real @ C2 @ A ) ) ) ).
% Compl_iff
thf(fact_966_Compl__iff,axiom,
! [C2: nat,A: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat @ C2 @ A ) ) ) ).
% Compl_iff
thf(fact_967_Compl__eq,axiom,
( uminus612125837232591019t_real
= ( ^ [A4: set_real] :
( collect_real
@ ^ [X: real] :
~ ( member_real @ X @ A4 ) ) ) ) ).
% Compl_eq
thf(fact_968_Compl__eq,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A4: set_nat] :
( collect_nat
@ ^ [X: nat] :
~ ( member_nat @ X @ A4 ) ) ) ) ).
% Compl_eq
thf(fact_969_uminus__set__def,axiom,
( uminus612125837232591019t_real
= ( ^ [A4: set_real] :
( collect_real
@ ( uminus_uminus_real_o
@ ^ [X: real] : ( member_real @ X @ A4 ) ) ) ) ) ).
% uminus_set_def
thf(fact_970_uminus__set__def,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A4: set_nat] :
( collect_nat
@ ( uminus_uminus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A4 ) ) ) ) ) ).
% uminus_set_def
thf(fact_971_ComplD,axiom,
! [C2: real,A: set_real] :
( ( member_real @ C2 @ ( uminus612125837232591019t_real @ A ) )
=> ~ ( member_real @ C2 @ A ) ) ).
% ComplD
thf(fact_972_ComplD,axiom,
! [C2: nat,A: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat @ C2 @ A ) ) ).
% ComplD
thf(fact_973_inverse__diff__inverse,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( minus_minus_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A2 ) @ ( minus_minus_real @ A2 @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_974_lhopital__at__top,axiom,
! [G: real > real,X2: real,G3: real > real,F2: real > real,F7: real > real,Y3: real] :
( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G3 @ X )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F7 @ X ) @ ( G3 @ X ) )
@ ( topolo2815343760600316023s_real @ Y3 )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ Y3 )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ).
% lhopital_at_top
thf(fact_975_eventually__at__top__not__equal,axiom,
! [C2: real] :
( eventually_real
@ ^ [X: real] : ( X != C2 )
@ at_top_real ) ).
% eventually_at_top_not_equal
thf(fact_976_eventually__at__top__not__equal,axiom,
! [C2: nat] :
( eventually_nat
@ ^ [X: nat] : ( X != C2 )
@ at_top_nat ) ).
% eventually_at_top_not_equal
thf(fact_977_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_978_trivial__limit__at__top__linorder,axiom,
at_top_real != bot_bot_filter_real ).
% trivial_limit_at_top_linorder
thf(fact_979_trivial__limit__at__top__linorder,axiom,
at_top_nat != bot_bot_filter_nat ).
% trivial_limit_at_top_linorder
thf(fact_980_filterlim__at__top__mult__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
@ ^ [X: real] : ( times_times_real @ ( F2 @ X ) @ ( G @ X ) )
@ at_top_real
@ F ) ) ) ).
% filterlim_at_top_mult_at_top
thf(fact_981_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
@ ^ [X: real] : ( plus_plus_real @ ( F2 @ X ) @ ( G @ X ) )
@ at_top_real
@ F ) ) ) ).
% filterlim_at_top_add_at_top
thf(fact_982_eventually__diff__zero__imp__eq,axiom,
! [F2: real > real,G: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( minus_minus_real @ ( F2 @ X ) @ ( G @ X ) )
= zero_zero_real )
@ at_top_real )
=> ( eventually_real
@ ^ [X: real] :
( ( F2 @ X )
= ( G @ X ) )
@ at_top_real ) ) ).
% eventually_diff_zero_imp_eq
thf(fact_983_tendsto__inverse__0__at__top,axiom,
! [F2: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ at_top_real @ F )
=> ( filterlim_real_real
@ ^ [X: real] : ( inverse_inverse_real @ ( F2 @ X ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ).
% tendsto_inverse_0_at_top
thf(fact_984_filterlim__tendsto__add__at__top,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_top_real @ F )
=> ( filterlim_real_real
@ ^ [X: real] : ( plus_plus_real @ ( F2 @ X ) @ ( G @ X ) )
@ at_top_real
@ F ) ) ) ).
% filterlim_tendsto_add_at_top
thf(fact_985_real__tendsto__divide__at__top,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_top_real @ F )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F ) ) ) ).
% real_tendsto_divide_at_top
thf(fact_986_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
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F7 @ X ) @ ( G3 @ X ) )
@ at_top_real
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ at_top_real
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ) ) ).
% lhopital_at_top_at_top
thf(fact_987_lhospital__at__top__at__top,axiom,
! [G: real > real,G3: real > real,F2: real > real,F7: real > real,X2: real] :
( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G3 @ X )
!= zero_zero_real )
@ at_top_real )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ at_top_real )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ at_top_real )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F7 @ X ) @ ( G3 @ X ) )
@ ( topolo2815343760600316023s_real @ X2 )
@ at_top_real )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ X2 )
@ at_top_real ) ) ) ) ) ) ).
% lhospital_at_top_at_top
thf(fact_988_Multiseries__Expansion__Bounds_Oeventually__0__imp__prod__zero_I2_J,axiom,
! [F2: real > real,G: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( F2 @ X )
= zero_zero_real )
@ at_top_real )
=> ( eventually_real
@ ^ [X: real] :
( ( times_times_real @ ( G @ X ) @ ( F2 @ X ) )
= zero_zero_real )
@ at_top_real ) ) ).
% Multiseries_Expansion_Bounds.eventually_0_imp_prod_zero(2)
thf(fact_989_Multiseries__Expansion__Bounds_Oeventually__0__imp__prod__zero_I1_J,axiom,
! [F2: real > real,G: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( F2 @ X )
= zero_zero_real )
@ at_top_real )
=> ( eventually_real
@ ^ [X: real] :
( ( times_times_real @ ( F2 @ X ) @ ( G @ X ) )
= zero_zero_real )
@ at_top_real ) ) ).
% Multiseries_Expansion_Bounds.eventually_0_imp_prod_zero(1)
thf(fact_990_tendsto__zero__mult__left__iff,axiom,
! [C2: real,A2: nat > real] :
( ( C2 != zero_zero_real )
=> ( ( filterlim_nat_real
@ ^ [N: nat] : ( times_times_real @ C2 @ ( A2 @ N ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat )
= ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% tendsto_zero_mult_left_iff
thf(fact_991_tendsto__zero__mult__right__iff,axiom,
! [C2: real,A2: nat > real] :
( ( C2 != zero_zero_real )
=> ( ( filterlim_nat_real
@ ^ [N: nat] : ( times_times_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_mult_right_iff
thf(fact_992_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_993_Bseq__minus__iff,axiom,
! [X5: nat > real] :
( ( bfun_nat_real
@ ^ [N: nat] : ( uminus_uminus_real @ ( X5 @ N ) )
@ at_top_nat )
= ( bfun_nat_real @ X5 @ at_top_nat ) ) ).
% Bseq_minus_iff
thf(fact_994_Bseq__mult,axiom,
! [F2: nat > real,G: nat > real] :
( ( bfun_nat_real @ F2 @ at_top_nat )
=> ( ( bfun_nat_real @ G @ at_top_nat )
=> ( bfun_nat_real
@ ^ [X: nat] : ( times_times_real @ ( F2 @ X ) @ ( G @ X ) )
@ at_top_nat ) ) ) ).
% Bseq_mult
thf(fact_995_first__countable__topology__class_Ocountable__basis,axiom,
! [X2: nat] :
~ ! [A5: nat > set_nat] :
( ! [I3: nat] : ( topolo4328251076210115529en_nat @ ( A5 @ I3 ) )
=> ( ! [I3: nat] : ( member_nat @ X2 @ ( A5 @ I3 ) )
=> ~ ! [F9: nat > nat] :
( ! [N2: nat] : ( member_nat @ ( F9 @ N2 ) @ ( A5 @ N2 ) )
=> ( filterlim_nat_nat @ F9 @ ( topolo8926549440605965083ds_nat @ X2 ) @ at_top_nat ) ) ) ) ).
% first_countable_topology_class.countable_basis
thf(fact_996_first__countable__topology__class_Ocountable__basis,axiom,
! [X2: real] :
~ ! [A5: nat > set_real] :
( ! [I3: nat] : ( topolo4860482606490270245n_real @ ( A5 @ I3 ) )
=> ( ! [I3: nat] : ( member_real @ X2 @ ( A5 @ I3 ) )
=> ~ ! [F9: nat > real] :
( ! [N2: nat] : ( member_real @ ( F9 @ N2 ) @ ( A5 @ N2 ) )
=> ( filterlim_nat_real @ F9 @ ( topolo2815343760600316023s_real @ X2 ) @ at_top_nat ) ) ) ) ).
% first_countable_topology_class.countable_basis
thf(fact_997_Bseq__add,axiom,
! [F2: nat > real,C2: real] :
( ( bfun_nat_real @ F2 @ at_top_nat )
=> ( bfun_nat_real
@ ^ [X: nat] : ( plus_plus_real @ ( F2 @ X ) @ C2 )
@ at_top_nat ) ) ).
% Bseq_add
thf(fact_998_Bseq__add__iff,axiom,
! [F2: nat > real,C2: real] :
( ( bfun_nat_real
@ ^ [X: nat] : ( plus_plus_real @ ( F2 @ X ) @ C2 )
@ at_top_nat )
= ( bfun_nat_real @ F2 @ at_top_nat ) ) ).
% Bseq_add_iff
thf(fact_999_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_1000_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_1001_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_1002_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_1003_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_1004_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_1005_seq__offset__neg,axiom,
! [F2: nat > nat,L: nat,K: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ at_top_nat )
=> ( filterlim_nat_nat
@ ^ [I4: nat] : ( F2 @ ( minus_minus_nat @ I4 @ K ) )
@ ( topolo8926549440605965083ds_nat @ L )
@ at_top_nat ) ) ).
% seq_offset_neg
thf(fact_1006_seq__offset__neg,axiom,
! [F2: nat > real,L: real,K: nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat )
=> ( filterlim_nat_real
@ ^ [I4: nat] : ( F2 @ ( minus_minus_nat @ I4 @ K ) )
@ ( topolo2815343760600316023s_real @ L )
@ at_top_nat ) ) ).
% seq_offset_neg
thf(fact_1007_limI,axiom,
! [X5: nat > nat,L2: nat] :
( ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ L2 ) @ at_top_nat )
=> ( ( topolo4574594659991002850at_nat @ at_top_nat @ X5 )
= L2 ) ) ).
% limI
thf(fact_1008_limI,axiom,
! [X5: nat > real,L2: real] :
( ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat )
=> ( ( topolo2843583510664976574t_real @ at_top_nat @ X5 )
= L2 ) ) ).
% limI
thf(fact_1009_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_1010_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_1011_Bseq__cmult__iff,axiom,
! [C2: real,F2: nat > real] :
( ( C2 != zero_zero_real )
=> ( ( bfun_nat_real
@ ^ [X: nat] : ( times_times_real @ C2 @ ( F2 @ X ) )
@ at_top_nat )
= ( bfun_nat_real @ F2 @ at_top_nat ) ) ) ).
% Bseq_cmult_iff
thf(fact_1012_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_1013_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_1014_LIMSEQ__Uniq,axiom,
! [X5: nat > nat] :
( uniq_nat
@ ^ [L4: nat] : ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ L4 ) @ at_top_nat ) ) ).
% LIMSEQ_Uniq
thf(fact_1015_LIMSEQ__Uniq,axiom,
! [X5: nat > real] :
( uniq_real
@ ^ [L4: real] : ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ).
% LIMSEQ_Uniq
thf(fact_1016_continuous__within__sequentially,axiom,
! [A2: real,S2: set_real,F2: real > nat] :
( ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ A2 @ S2 ) @ F2 )
= ( ! [X: nat > real] :
( ( ! [N: nat] : ( member_real @ ( X @ N ) @ S2 )
& ( filterlim_nat_real @ X @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat ) )
=> ( filterlim_nat_nat @ ( comp_real_nat_nat @ F2 @ X ) @ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) ) @ at_top_nat ) ) ) ) ).
% continuous_within_sequentially
thf(fact_1017_continuous__within__sequentially,axiom,
! [A2: real,S2: set_real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S2 ) @ F2 )
= ( ! [X: nat > real] :
( ( ! [N: nat] : ( member_real @ ( X @ N ) @ S2 )
& ( filterlim_nat_real @ X @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat ) )
=> ( filterlim_nat_real @ ( comp_real_real_nat @ F2 @ X ) @ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) ) @ at_top_nat ) ) ) ) ).
% continuous_within_sequentially
thf(fact_1018_continuous__within__sequentially,axiom,
! [A2: nat,S2: set_nat,F2: nat > nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) @ F2 )
= ( ! [X: nat > nat] :
( ( ! [N: nat] : ( member_nat @ ( X @ N ) @ S2 )
& ( filterlim_nat_nat @ X @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat ) )
=> ( filterlim_nat_nat @ ( comp_nat_nat_nat @ F2 @ X ) @ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) ) @ at_top_nat ) ) ) ) ).
% continuous_within_sequentially
thf(fact_1019_continuous__within__sequentially,axiom,
! [A2: nat,S2: set_nat,F2: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) @ F2 )
= ( ! [X: nat > nat] :
( ( ! [N: nat] : ( member_nat @ ( X @ N ) @ S2 )
& ( filterlim_nat_nat @ X @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat ) )
=> ( filterlim_nat_real @ ( comp_nat_real_nat @ F2 @ X ) @ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) ) @ at_top_nat ) ) ) ) ).
% continuous_within_sequentially
thf(fact_1020_Lim__within__LIMSEQ,axiom,
! [A2: real,T: set_real,X5: real > nat,L2: nat] :
( ! [S5: nat > real] :
( ( ! [N3: nat] :
( ( ( S5 @ N3 )
!= A2 )
& ( member_real @ ( S5 @ N3 ) @ T ) )
& ( filterlim_nat_real @ S5 @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat ) )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( X5 @ ( S5 @ N ) )
@ ( topolo8926549440605965083ds_nat @ L2 )
@ at_top_nat ) )
=> ( filterlim_real_nat @ X5 @ ( topolo8926549440605965083ds_nat @ L2 ) @ ( topolo2177554685111907308n_real @ A2 @ T ) ) ) ).
% Lim_within_LIMSEQ
thf(fact_1021_Lim__within__LIMSEQ,axiom,
! [A2: real,T: set_real,X5: real > real,L2: real] :
( ! [S5: nat > real] :
( ( ! [N3: nat] :
( ( ( S5 @ N3 )
!= A2 )
& ( member_real @ ( S5 @ N3 ) @ T ) )
& ( filterlim_nat_real @ S5 @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat ) )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( X5 @ ( S5 @ N ) )
@ ( topolo2815343760600316023s_real @ L2 )
@ at_top_nat ) )
=> ( filterlim_real_real @ X5 @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ A2 @ T ) ) ) ).
% Lim_within_LIMSEQ
thf(fact_1022_Lim__within__LIMSEQ,axiom,
! [A2: nat,T: set_nat,X5: nat > nat,L2: nat] :
( ! [S5: nat > nat] :
( ( ! [N3: nat] :
( ( ( S5 @ N3 )
!= A2 )
& ( member_nat @ ( S5 @ N3 ) @ T ) )
& ( filterlim_nat_nat @ S5 @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat ) )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( X5 @ ( S5 @ N ) )
@ ( topolo8926549440605965083ds_nat @ L2 )
@ at_top_nat ) )
=> ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ L2 ) @ ( topolo4659099751122792720in_nat @ A2 @ T ) ) ) ).
% Lim_within_LIMSEQ
thf(fact_1023_Lim__within__LIMSEQ,axiom,
! [A2: nat,T: set_nat,X5: nat > real,L2: real] :
( ! [S5: nat > nat] :
( ( ! [N3: nat] :
( ( ( S5 @ N3 )
!= A2 )
& ( member_nat @ ( S5 @ N3 ) @ T ) )
& ( filterlim_nat_nat @ S5 @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat ) )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( X5 @ ( S5 @ N ) )
@ ( topolo2815343760600316023s_real @ L2 )
@ at_top_nat ) )
=> ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo4659099751122792720in_nat @ A2 @ T ) ) ) ).
% Lim_within_LIMSEQ
thf(fact_1024_sequentially__imp__eventually__within,axiom,
! [S2: set_real,A2: real,P: real > $o] :
( ! [F10: nat > real] :
( ( ! [N3: nat] :
( ( member_real @ ( F10 @ N3 ) @ S2 )
& ( ( F10 @ N3 )
!= A2 ) )
& ( filterlim_nat_real @ F10 @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat ) )
=> ( eventually_nat
@ ^ [N: nat] : ( P @ ( F10 @ N ) )
@ at_top_nat ) )
=> ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A2 @ S2 ) ) ) ).
% sequentially_imp_eventually_within
thf(fact_1025_sequentially__imp__eventually__within,axiom,
! [S2: set_nat,A2: nat,P: nat > $o] :
( ! [F10: nat > nat] :
( ( ! [N3: nat] :
( ( member_nat @ ( F10 @ N3 ) @ S2 )
& ( ( F10 @ N3 )
!= A2 ) )
& ( filterlim_nat_nat @ F10 @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat ) )
=> ( eventually_nat
@ ^ [N: nat] : ( P @ ( F10 @ N ) )
@ at_top_nat ) )
=> ( eventually_nat @ P @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) ) ) ).
% sequentially_imp_eventually_within
thf(fact_1026_continuous__within__sequentiallyI,axiom,
! [A2: real,S2: set_real,F2: real > nat] :
( ! [U4: nat > real] :
( ( filterlim_nat_real @ U4 @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat )
=> ( ! [N3: nat] : ( member_real @ ( U4 @ N3 ) @ S2 )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F2 @ ( U4 @ N ) )
@ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) )
@ at_top_nat ) ) )
=> ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ A2 @ S2 ) @ F2 ) ) ).
% continuous_within_sequentiallyI
thf(fact_1027_continuous__within__sequentiallyI,axiom,
! [A2: real,S2: set_real,F2: real > real] :
( ! [U4: nat > real] :
( ( filterlim_nat_real @ U4 @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat )
=> ( ! [N3: nat] : ( member_real @ ( U4 @ N3 ) @ S2 )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F2 @ ( U4 @ N ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ at_top_nat ) ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S2 ) @ F2 ) ) ).
% continuous_within_sequentiallyI
thf(fact_1028_continuous__within__sequentiallyI,axiom,
! [A2: nat,S2: set_nat,F2: nat > nat] :
( ! [U4: nat > nat] :
( ( filterlim_nat_nat @ U4 @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat )
=> ( ! [N3: nat] : ( member_nat @ ( U4 @ N3 ) @ S2 )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F2 @ ( U4 @ N ) )
@ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) )
@ at_top_nat ) ) )
=> ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) @ F2 ) ) ).
% continuous_within_sequentiallyI
thf(fact_1029_continuous__within__sequentiallyI,axiom,
! [A2: nat,S2: set_nat,F2: nat > real] :
( ! [U4: nat > nat] :
( ( filterlim_nat_nat @ U4 @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat )
=> ( ! [N3: nat] : ( member_nat @ ( U4 @ N3 ) @ S2 )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F2 @ ( U4 @ N ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ at_top_nat ) ) )
=> ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ S2 ) @ F2 ) ) ).
% continuous_within_sequentiallyI
thf(fact_1030_continuous__at__sequentially,axiom,
! [A2: real,F2: real > nat] :
( ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F2 )
= ( ! [X: nat > real] :
( ( filterlim_nat_real @ X @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat )
=> ( filterlim_nat_nat @ ( comp_real_nat_nat @ F2 @ X ) @ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) ) @ at_top_nat ) ) ) ) ).
% continuous_at_sequentially
thf(fact_1031_continuous__at__sequentially,axiom,
! [A2: real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F2 )
= ( ! [X: nat > real] :
( ( filterlim_nat_real @ X @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat )
=> ( filterlim_nat_real @ ( comp_real_real_nat @ F2 @ X ) @ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) ) @ at_top_nat ) ) ) ) ).
% continuous_at_sequentially
thf(fact_1032_continuous__imp__tendsto,axiom,
! [X0: real,F2: real > nat,X2: nat > real] :
( ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) @ F2 )
=> ( ( filterlim_nat_real @ X2 @ ( topolo2815343760600316023s_real @ X0 ) @ at_top_nat )
=> ( filterlim_nat_nat @ ( comp_real_nat_nat @ F2 @ X2 ) @ ( topolo8926549440605965083ds_nat @ ( F2 @ X0 ) ) @ at_top_nat ) ) ) ).
% continuous_imp_tendsto
thf(fact_1033_continuous__imp__tendsto,axiom,
! [X0: real,F2: real > real,X2: nat > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) @ F2 )
=> ( ( filterlim_nat_real @ X2 @ ( topolo2815343760600316023s_real @ X0 ) @ at_top_nat )
=> ( filterlim_nat_real @ ( comp_real_real_nat @ F2 @ X2 ) @ ( topolo2815343760600316023s_real @ ( F2 @ X0 ) ) @ at_top_nat ) ) ) ).
% continuous_imp_tendsto
thf(fact_1034_continuous__imp__tendsto,axiom,
! [X0: nat,F2: nat > nat,X2: nat > nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ X0 @ top_top_set_nat ) @ F2 )
=> ( ( filterlim_nat_nat @ X2 @ ( topolo8926549440605965083ds_nat @ X0 ) @ at_top_nat )
=> ( filterlim_nat_nat @ ( comp_nat_nat_nat @ F2 @ X2 ) @ ( topolo8926549440605965083ds_nat @ ( F2 @ X0 ) ) @ at_top_nat ) ) ) ).
% continuous_imp_tendsto
thf(fact_1035_continuous__imp__tendsto,axiom,
! [X0: nat,F2: nat > real,X2: nat > nat] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ X0 @ top_top_set_nat ) @ F2 )
=> ( ( filterlim_nat_nat @ X2 @ ( topolo8926549440605965083ds_nat @ X0 ) @ at_top_nat )
=> ( filterlim_nat_real @ ( comp_nat_real_nat @ F2 @ X2 ) @ ( topolo2815343760600316023s_real @ ( F2 @ X0 ) ) @ at_top_nat ) ) ) ).
% continuous_imp_tendsto
thf(fact_1036_LIMSEQ__SEQ__conv,axiom,
! [A2: real,X5: real > nat,L2: nat] :
( ( ! [S: nat > real] :
( ( ! [N: nat] :
( ( S @ N )
!= A2 )
& ( filterlim_nat_real @ S @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat ) )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( X5 @ ( S @ N ) )
@ ( topolo8926549440605965083ds_nat @ L2 )
@ at_top_nat ) ) )
= ( filterlim_real_nat @ X5 @ ( topolo8926549440605965083ds_nat @ L2 ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ).
% LIMSEQ_SEQ_conv
thf(fact_1037_LIMSEQ__SEQ__conv,axiom,
! [A2: real,X5: real > real,L2: real] :
( ( ! [S: nat > real] :
( ( ! [N: nat] :
( ( S @ N )
!= A2 )
& ( filterlim_nat_real @ S @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat ) )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( X5 @ ( S @ N ) )
@ ( topolo2815343760600316023s_real @ L2 )
@ at_top_nat ) ) )
= ( filterlim_real_real @ X5 @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ).
% LIMSEQ_SEQ_conv
thf(fact_1038_LIMSEQ__SEQ__conv,axiom,
! [A2: nat,X5: nat > nat,L2: nat] :
( ( ! [S: nat > nat] :
( ( ! [N: nat] :
( ( S @ N )
!= A2 )
& ( filterlim_nat_nat @ S @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat ) )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( X5 @ ( S @ N ) )
@ ( topolo8926549440605965083ds_nat @ L2 )
@ at_top_nat ) ) )
= ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ L2 ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ).
% LIMSEQ_SEQ_conv
thf(fact_1039_LIMSEQ__SEQ__conv,axiom,
! [A2: nat,X5: nat > real,L2: real] :
( ( ! [S: nat > nat] :
( ( ! [N: nat] :
( ( S @ N )
!= A2 )
& ( filterlim_nat_nat @ S @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat ) )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( X5 @ ( S @ N ) )
@ ( topolo2815343760600316023s_real @ L2 )
@ at_top_nat ) ) )
= ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ).
% LIMSEQ_SEQ_conv
thf(fact_1040_sequentially__imp__eventually__at,axiom,
! [A2: real,P: real > $o] :
( ! [F10: nat > real] :
( ( ! [N3: nat] :
( ( F10 @ N3 )
!= A2 )
& ( filterlim_nat_real @ F10 @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat ) )
=> ( eventually_nat
@ ^ [N: nat] : ( P @ ( F10 @ N ) )
@ at_top_nat ) )
=> ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ).
% sequentially_imp_eventually_at
thf(fact_1041_sequentially__imp__eventually__at,axiom,
! [A2: nat,P: nat > $o] :
( ! [F10: nat > nat] :
( ( ! [N3: nat] :
( ( F10 @ N3 )
!= A2 )
& ( filterlim_nat_nat @ F10 @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat ) )
=> ( eventually_nat
@ ^ [N: nat] : ( P @ ( F10 @ N ) )
@ at_top_nat ) )
=> ( eventually_nat @ P @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) ) ) ).
% sequentially_imp_eventually_at
thf(fact_1042_continuous__at__sequentiallyI,axiom,
! [A2: real,F2: real > nat] :
( ! [U4: nat > real] :
( ( filterlim_nat_real @ U4 @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F2 @ ( U4 @ N ) )
@ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) )
@ at_top_nat ) )
=> ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F2 ) ) ).
% continuous_at_sequentiallyI
thf(fact_1043_continuous__at__sequentiallyI,axiom,
! [A2: real,F2: real > real] :
( ! [U4: nat > real] :
( ( filterlim_nat_real @ U4 @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F2 @ ( U4 @ N ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ at_top_nat ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F2 ) ) ).
% continuous_at_sequentiallyI
thf(fact_1044_continuous__at__sequentiallyI,axiom,
! [A2: nat,F2: nat > nat] :
( ! [U4: nat > nat] :
( ( filterlim_nat_nat @ U4 @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F2 @ ( U4 @ N ) )
@ ( topolo8926549440605965083ds_nat @ ( F2 @ A2 ) )
@ at_top_nat ) )
=> ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) @ F2 ) ) ).
% continuous_at_sequentiallyI
thf(fact_1045_continuous__at__sequentiallyI,axiom,
! [A2: nat,F2: nat > real] :
( ! [U4: nat > nat] :
( ( filterlim_nat_nat @ U4 @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F2 @ ( U4 @ N ) )
@ ( topolo2815343760600316023s_real @ ( F2 @ A2 ) )
@ at_top_nat ) )
=> ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat ) @ F2 ) ) ).
% continuous_at_sequentiallyI
thf(fact_1046_islimpt__sequential,axiom,
( elemen5683178629028408237t_real
= ( ^ [X: real,S: set_real] :
? [F4: nat > real] :
( ! [N: nat] : ( member_real @ ( F4 @ N ) @ ( minus_minus_set_real @ S @ ( insert_real @ X @ bot_bot_set_real ) ) )
& ( filterlim_nat_real @ F4 @ ( topolo2815343760600316023s_real @ X ) @ at_top_nat ) ) ) ) ).
% islimpt_sequential
thf(fact_1047_islimpt__sequential,axiom,
( elemen5607981409700034897pt_nat
= ( ^ [X: nat,S: set_nat] :
? [F4: nat > nat] :
( ! [N: nat] : ( member_nat @ ( F4 @ N ) @ ( minus_minus_set_nat @ S @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
& ( filterlim_nat_nat @ F4 @ ( topolo8926549440605965083ds_nat @ X ) @ at_top_nat ) ) ) ) ).
% islimpt_sequential
thf(fact_1048_tendsto__at__iff__sequentially,axiom,
! [F2: real > nat,A2: nat,X2: real,S2: set_real] :
( ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ A2 ) @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
= ( ! [X3: nat > real] :
( ! [I4: nat] : ( member_real @ ( X3 @ I4 ) @ ( minus_minus_set_real @ S2 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
=> ( ( filterlim_nat_real @ X3 @ ( topolo2815343760600316023s_real @ X2 ) @ at_top_nat )
=> ( filterlim_nat_nat @ ( comp_real_nat_nat @ F2 @ X3 ) @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat ) ) ) ) ) ).
% tendsto_at_iff_sequentially
thf(fact_1049_tendsto__at__iff__sequentially,axiom,
! [F2: real > real,A2: real,X2: real,S2: set_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
= ( ! [X3: nat > real] :
( ! [I4: nat] : ( member_real @ ( X3 @ I4 ) @ ( minus_minus_set_real @ S2 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
=> ( ( filterlim_nat_real @ X3 @ ( topolo2815343760600316023s_real @ X2 ) @ at_top_nat )
=> ( filterlim_nat_real @ ( comp_real_real_nat @ F2 @ X3 ) @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat ) ) ) ) ) ).
% tendsto_at_iff_sequentially
thf(fact_1050_tendsto__at__iff__sequentially,axiom,
! [F2: nat > nat,A2: nat,X2: nat,S2: set_nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ A2 ) @ ( topolo4659099751122792720in_nat @ X2 @ S2 ) )
= ( ! [X3: nat > nat] :
( ! [I4: nat] : ( member_nat @ ( X3 @ I4 ) @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
=> ( ( filterlim_nat_nat @ X3 @ ( topolo8926549440605965083ds_nat @ X2 ) @ at_top_nat )
=> ( filterlim_nat_nat @ ( comp_nat_nat_nat @ F2 @ X3 ) @ ( topolo8926549440605965083ds_nat @ A2 ) @ at_top_nat ) ) ) ) ) ).
% tendsto_at_iff_sequentially
thf(fact_1051_tendsto__at__iff__sequentially,axiom,
! [F2: nat > real,A2: real,X2: nat,S2: set_nat] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ A2 ) @ ( topolo4659099751122792720in_nat @ X2 @ S2 ) )
= ( ! [X3: nat > nat] :
( ! [I4: nat] : ( member_nat @ ( X3 @ I4 ) @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
=> ( ( filterlim_nat_nat @ X3 @ ( topolo8926549440605965083ds_nat @ X2 ) @ at_top_nat )
=> ( filterlim_nat_real @ ( comp_nat_real_nat @ F2 @ X3 ) @ ( topolo2815343760600316023s_real @ A2 ) @ at_top_nat ) ) ) ) ) ).
% tendsto_at_iff_sequentially
thf(fact_1052_field__derivative__lim__unique,axiom,
! [F2: real > real,Df: real,Z3: real,S2: nat > real,A2: real] :
( ( has_fi5821293074295781190e_real @ F2 @ Df @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) )
=> ( ( filterlim_nat_real @ S2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N2: nat] :
( ( S2 @ N2 )
!= zero_zero_real )
=> ( ( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ ( minus_minus_real @ ( F2 @ ( plus_plus_real @ Z3 @ ( S2 @ N ) ) ) @ ( F2 @ Z3 ) ) @ ( S2 @ N ) )
@ ( topolo2815343760600316023s_real @ A2 )
@ at_top_nat )
=> ( Df = A2 ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_1053_tendsto__at__topI__sequentially,axiom,
! [F2: real > nat,Y3: nat] :
( ! [X8: nat > real] :
( ( filterlim_nat_real @ X8 @ at_top_real @ at_top_nat )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F2 @ ( X8 @ N ) )
@ ( topolo8926549440605965083ds_nat @ Y3 )
@ at_top_nat ) )
=> ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ Y3 ) @ at_top_real ) ) ).
% tendsto_at_topI_sequentially
thf(fact_1054_tendsto__at__topI__sequentially,axiom,
! [F2: real > real,Y3: real] :
( ! [X8: nat > real] :
( ( filterlim_nat_real @ X8 @ at_top_real @ at_top_nat )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F2 @ ( X8 @ N ) )
@ ( topolo2815343760600316023s_real @ Y3 )
@ at_top_nat ) )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ Y3 ) @ at_top_real ) ) ).
% tendsto_at_topI_sequentially
thf(fact_1055_lhopital__right__at__top,axiom,
! [G: real > real,X2: real,G3: real > real,F2: real > real,F7: real > real,Y3: real] :
( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G3 @ X )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F7 @ X ) @ ( G3 @ X ) )
@ ( topolo2815343760600316023s_real @ Y3 )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ Y3 )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top
thf(fact_1056_trivial__limit__sequentially,axiom,
at_top_nat != bot_bot_filter_nat ).
% trivial_limit_sequentially
thf(fact_1057_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_1058_sequentially__offset,axiom,
! [P: nat > $o,K: nat] :
( ( eventually_nat @ P @ at_top_nat )
=> ( eventually_nat
@ ^ [I4: nat] : ( P @ ( plus_plus_nat @ I4 @ K ) )
@ at_top_nat ) ) ).
% sequentially_offset
thf(fact_1059_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_1060_eventually__False__sequentially,axiom,
~ ( eventually_nat
@ ^ [N: nat] : $false
@ at_top_nat ) ).
% eventually_False_sequentially
thf(fact_1061_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_1062_trivial__limit__at__right__real,axiom,
! [X2: real] :
( ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) )
!= bot_bot_filter_real ) ).
% trivial_limit_at_right_real
thf(fact_1063_open__greaterThan,axiom,
! [A2: real] : ( topolo4860482606490270245n_real @ ( set_or5849166863359141190n_real @ A2 ) ) ).
% open_greaterThan
thf(fact_1064_filterlim__at__rightI,axiom,
! [F2: real > real,C2: real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X: real] : ( minus_minus_real @ ( F2 @ X ) @ C2 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) )
@ F )
=> ( filterlim_real_real @ F2 @ ( topolo2177554685111907308n_real @ C2 @ ( set_or5849166863359141190n_real @ C2 ) ) @ F ) ) ).
% filterlim_at_rightI
thf(fact_1065_eventually__at__right__to__0,axiom,
! [P: real > $o,A2: real] :
( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
= ( eventually_real
@ ^ [X: real] : ( P @ ( plus_plus_real @ X @ A2 ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% eventually_at_right_to_0
thf(fact_1066_filterlim__at__right__to__0,axiom,
! [F2: real > real,F: filter_real,A2: real] :
( ( filterlim_real_real @ F2 @ F @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
= ( filterlim_real_real
@ ^ [X: real] : ( F2 @ ( plus_plus_real @ X @ A2 ) )
@ F
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% filterlim_at_right_to_0
thf(fact_1067_eventually__at__top__to__right,axiom,
! [P: real > $o] :
( ( eventually_real @ P @ at_top_real )
= ( eventually_real
@ ^ [X: real] : ( P @ ( inverse_inverse_real @ X ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% eventually_at_top_to_right
thf(fact_1068_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
@ ^ [X: real] : ( P @ ( inverse_inverse_real @ X ) )
@ at_top_real ) ) ).
% eventually_at_right_to_top
thf(fact_1069_filterlim__at__top__to__right,axiom,
! [F2: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ F @ at_top_real )
= ( filterlim_real_real
@ ^ [X: real] : ( F2 @ ( inverse_inverse_real @ X ) )
@ F
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% filterlim_at_top_to_right
thf(fact_1070_filterlim__at__right__to__top,axiom,
! [F2: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ F @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
= ( filterlim_real_real
@ ^ [X: real] : ( F2 @ ( inverse_inverse_real @ X ) )
@ F
@ at_top_real ) ) ).
% filterlim_at_right_to_top
thf(fact_1071_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_1072_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_1073_lhopital__right__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 @ ( set_or5849166863359141190n_real @ A2 ) ) )
=> ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F7 @ X ) @ ( G3 @ X ) )
@ at_top_real
@ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ at_top_real
@ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_top
thf(fact_1074_lhopital__right__0,axiom,
! [F0: real > real,G0: real > real,G3: real > real,F7: real > real,F: filter_real] :
( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G0 @ X )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G3 @ X )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F7 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F7 @ X ) @ ( G3 @ X ) )
@ F
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F0 @ X ) @ ( G0 @ X ) )
@ F
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0
thf(fact_1075_lhopital__right,axiom,
! [F2: real > real,X2: real,G: real > real,G3: real > real,F7: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G @ X )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G3 @ X )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F7 @ X ) @ ( G3 @ X ) )
@ F
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ F
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right
thf(fact_1076_lhopital__right__0__at__top,axiom,
! [G: real > real,G3: real > real,F2: real > real,F7: real > real,X2: real] :
( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G3 @ X )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F7 @ X ) @ ( G3 @ X ) )
@ ( topolo2815343760600316023s_real @ X2 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ X2 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).
% lhopital_right_0_at_top
thf(fact_1077_lhopital__left__at__top,axiom,
! [G: real > real,X2: real,G3: real > real,F2: real > real,F7: real > real,Y3: real] :
( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G3 @ X )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F7 @ X ) @ ( G3 @ X ) )
@ ( topolo2815343760600316023s_real @ Y3 )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ Y3 )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top
thf(fact_1078_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_1079_open__lessThan,axiom,
! [A2: real] : ( topolo4860482606490270245n_real @ ( set_or5984915006950818249n_real @ A2 ) ) ).
% open_lessThan
thf(fact_1080_trivial__limit__at__left__real,axiom,
! [X2: real] :
( ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) )
!= bot_bot_filter_real ) ).
% trivial_limit_at_left_real
thf(fact_1081_trivial__limit__at__left__bot,axiom,
( ( topolo4659099751122792720in_nat @ bot_bot_nat @ ( set_ord_lessThan_nat @ bot_bot_nat ) )
= bot_bot_filter_nat ) ).
% trivial_limit_at_left_bot
thf(fact_1082_filterlim__split__at,axiom,
! [F2: real > real,F: filter_real,X2: real] :
( ( filterlim_real_real @ F2 @ F @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
=> ( ( filterlim_real_real @ F2 @ F @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
=> ( filterlim_real_real @ F2 @ F @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% filterlim_split_at
thf(fact_1083_filterlim__split__at,axiom,
! [F2: nat > nat,F: filter_nat,X2: nat] :
( ( filterlim_nat_nat @ F2 @ F @ ( topolo4659099751122792720in_nat @ X2 @ ( set_ord_lessThan_nat @ X2 ) ) )
=> ( ( filterlim_nat_nat @ F2 @ F @ ( topolo4659099751122792720in_nat @ X2 @ ( set_or1210151606488870762an_nat @ X2 ) ) )
=> ( filterlim_nat_nat @ F2 @ F @ ( topolo4659099751122792720in_nat @ X2 @ top_top_set_nat ) ) ) ) ).
% filterlim_split_at
thf(fact_1084_filterlim__at__split,axiom,
! [F2: real > real,F: filter_real,X2: real] :
( ( filterlim_real_real @ F2 @ F @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
= ( ( filterlim_real_real @ F2 @ F @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
& ( filterlim_real_real @ F2 @ F @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) ) ) ) ).
% filterlim_at_split
thf(fact_1085_filterlim__at__split,axiom,
! [F2: nat > nat,F: filter_nat,X2: nat] :
( ( filterlim_nat_nat @ F2 @ F @ ( topolo4659099751122792720in_nat @ X2 @ top_top_set_nat ) )
= ( ( filterlim_nat_nat @ F2 @ F @ ( topolo4659099751122792720in_nat @ X2 @ ( set_ord_lessThan_nat @ X2 ) ) )
& ( filterlim_nat_nat @ F2 @ F @ ( topolo4659099751122792720in_nat @ X2 @ ( set_or1210151606488870762an_nat @ X2 ) ) ) ) ) ).
% filterlim_at_split
thf(fact_1086_eventually__at__split,axiom,
! [P: real > $o,X2: real] :
( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
= ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
& ( eventually_real @ P @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) ) ) ) ).
% eventually_at_split
thf(fact_1087_eventually__at__split,axiom,
! [P: nat > $o,X2: nat] :
( ( eventually_nat @ P @ ( topolo4659099751122792720in_nat @ X2 @ top_top_set_nat ) )
= ( ( eventually_nat @ P @ ( topolo4659099751122792720in_nat @ X2 @ ( set_ord_lessThan_nat @ X2 ) ) )
& ( eventually_nat @ P @ ( topolo4659099751122792720in_nat @ X2 @ ( set_or1210151606488870762an_nat @ X2 ) ) ) ) ) ).
% eventually_at_split
thf(fact_1088_eventually__at__left__at__right__imp__at,axiom,
! [P: real > $o,A2: real] :
( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
=> ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
=> ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ).
% eventually_at_left_at_right_imp_at
thf(fact_1089_filterlim__split__at__real,axiom,
! [F2: real > real,F: filter_real,X2: real] :
( ( filterlim_real_real @ F2 @ F @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
=> ( ( filterlim_real_real @ F2 @ F @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
=> ( filterlim_real_real @ F2 @ F @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% filterlim_split_at_real
thf(fact_1090_filterlim__at__leftI,axiom,
! [F2: real > real,C2: real,F: filter_real] :
( ( filterlim_real_real
@ ^ [X: real] : ( minus_minus_real @ ( F2 @ X ) @ C2 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5984915006950818249n_real @ zero_zero_real ) )
@ F )
=> ( filterlim_real_real @ F2 @ ( topolo2177554685111907308n_real @ C2 @ ( set_or5984915006950818249n_real @ C2 ) ) @ F ) ) ).
% filterlim_at_leftI
thf(fact_1091_eventually__at__left__to__right,axiom,
! [P: real > $o,A2: real] :
( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
= ( eventually_real
@ ^ [X: real] : ( P @ ( uminus_uminus_real @ X ) )
@ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ A2 ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ A2 ) ) ) ) ) ).
% eventually_at_left_to_right
thf(fact_1092_filterlim__at__left__to__right,axiom,
! [F2: real > real,F: filter_real,A2: real] :
( ( filterlim_real_real @ F2 @ F @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
= ( filterlim_real_real
@ ^ [X: real] : ( F2 @ ( uminus_uminus_real @ X ) )
@ F
@ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ A2 ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ A2 ) ) ) ) ) ).
% filterlim_at_left_to_right
thf(fact_1093_lhopital__left__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 @ ( set_or5984915006950818249n_real @ A2 ) ) )
=> ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F7 @ X ) @ ( G3 @ X ) )
@ at_top_real
@ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ at_top_real
@ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_top
thf(fact_1094_lhopital__left,axiom,
! [F2: real > real,X2: real,G: real > real,G3: real > real,F7: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G @ X )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G3 @ X )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F7 @ X ) @ ( G3 @ X ) )
@ F
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ F
@ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) ) ) ) ) ) ) ) ) ).
% lhopital_left
thf(fact_1095_single__Diff__lessThan,axiom,
! [K: real] :
( ( minus_minus_set_real @ ( insert_real @ K @ bot_bot_set_real ) @ ( set_or5984915006950818249n_real @ K ) )
= ( insert_real @ K @ bot_bot_set_real ) ) ).
% single_Diff_lessThan
thf(fact_1096_lessThan__non__empty,axiom,
! [X2: real] :
( ( set_or5984915006950818249n_real @ X2 )
!= bot_bot_set_real ) ).
% lessThan_non_empty
thf(fact_1097_greaterThan__non__empty,axiom,
! [X2: real] :
( ( set_or5849166863359141190n_real @ X2 )
!= bot_bot_set_real ) ).
% greaterThan_non_empty
thf(fact_1098_lhopital__left__at__top__at__bot,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 @ ( set_or5984915006950818249n_real @ A2 ) ) )
=> ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F7 @ X ) @ ( G3 @ X ) )
@ at_bot_real
@ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ at_bot_real
@ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_bot
thf(fact_1099_lhopital__right__at__top__at__bot,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 @ ( set_or5849166863359141190n_real @ A2 ) ) )
=> ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F7 @ X ) @ ( G3 @ X ) )
@ at_bot_real
@ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ at_bot_real
@ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_bot
thf(fact_1100_eventually__at__bot__not__equal,axiom,
! [C2: real] :
( eventually_real
@ ^ [X: real] : ( X != C2 )
@ at_bot_real ) ).
% eventually_at_bot_not_equal
thf(fact_1101_trivial__limit__at__bot__linorder,axiom,
at_bot_real != bot_bot_filter_real ).
% trivial_limit_at_bot_linorder
thf(fact_1102_trivial__limit__at__bot__linorder,axiom,
at_bot_nat != bot_bot_filter_nat ).
% trivial_limit_at_bot_linorder
thf(fact_1103_filterlim__at__bot__imp__at__infinity,axiom,
! [F2: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ at_bot_real @ F )
=> ( filterlim_real_real @ F2 @ at_infinity_real @ F ) ) ).
% filterlim_at_bot_imp_at_infinity
thf(fact_1104_filterlim__uminus__at__top,axiom,
! [F2: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ at_top_real @ F )
= ( filterlim_real_real
@ ^ [X: real] : ( uminus_uminus_real @ ( F2 @ X ) )
@ at_bot_real
@ F ) ) ).
% filterlim_uminus_at_top
thf(fact_1105_filterlim__uminus__at__bot,axiom,
! [F2: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ at_bot_real @ F )
= ( filterlim_real_real
@ ^ [X: real] : ( uminus_uminus_real @ ( F2 @ X ) )
@ at_top_real
@ F ) ) ).
% filterlim_uminus_at_bot
thf(fact_1106_filterlim__at__top__mirror,axiom,
! [F2: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ F @ at_top_real )
= ( filterlim_real_real
@ ^ [X: real] : ( F2 @ ( uminus_uminus_real @ X ) )
@ F
@ at_bot_real ) ) ).
% filterlim_at_top_mirror
thf(fact_1107_filterlim__at__bot__mirror,axiom,
! [F2: real > real,F: filter_real] :
( ( filterlim_real_real @ F2 @ F @ at_bot_real )
= ( filterlim_real_real
@ ^ [X: real] : ( F2 @ ( uminus_uminus_real @ X ) )
@ F
@ at_top_real ) ) ).
% filterlim_at_bot_mirror
thf(fact_1108_filterlim__uminus__at__top__at__bot,axiom,
filterlim_real_real @ uminus_uminus_real @ at_top_real @ at_bot_real ).
% filterlim_uminus_at_top_at_bot
thf(fact_1109_filterlim__uminus__at__bot__at__top,axiom,
filterlim_real_real @ uminus_uminus_real @ at_bot_real @ at_top_real ).
% filterlim_uminus_at_bot_at_top
thf(fact_1110_tendsto__at__botI__sequentially,axiom,
! [F2: real > nat,Y3: nat] :
( ! [X8: nat > real] :
( ( filterlim_nat_real @ X8 @ at_bot_real @ at_top_nat )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F2 @ ( X8 @ N ) )
@ ( topolo8926549440605965083ds_nat @ Y3 )
@ at_top_nat ) )
=> ( filterlim_real_nat @ F2 @ ( topolo8926549440605965083ds_nat @ Y3 ) @ at_bot_real ) ) ).
% tendsto_at_botI_sequentially
thf(fact_1111_tendsto__at__botI__sequentially,axiom,
! [F2: real > real,Y3: real] :
( ! [X8: nat > real] :
( ( filterlim_nat_real @ X8 @ at_bot_real @ at_top_nat )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F2 @ ( X8 @ N ) )
@ ( topolo2815343760600316023s_real @ Y3 )
@ at_top_nat ) )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ Y3 ) @ at_bot_real ) ) ).
% tendsto_at_botI_sequentially
thf(fact_1112_filterlim__inverse__at__bot__neg,axiom,
filterlim_real_real @ inverse_inverse_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5984915006950818249n_real @ zero_zero_real ) ) ).
% filterlim_inverse_at_bot_neg
thf(fact_1113_lhopital__at__top__at__bot,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_bot_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F7 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F7 @ X ) @ ( G3 @ X ) )
@ at_bot_real
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F2 @ X ) @ ( G @ X ) )
@ at_bot_real
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ) ) ).
% lhopital_at_top_at_bot
thf(fact_1114_has__derivative__right,axiom,
! [F2: real > real,Y3: real,X2: real,I5: set_real] :
( ( has_de1759254742604945161l_real @ F2 @ ( times_times_real @ Y3 ) @ ( topolo2177554685111907308n_real @ X2 @ ( inf_inf_set_real @ ( set_or5849166863359141190n_real @ X2 ) @ I5 ) ) )
= ( filterlim_real_real
@ ^ [T4: real] : ( divide_divide_real @ ( minus_minus_real @ ( F2 @ X2 ) @ ( F2 @ T4 ) ) @ ( minus_minus_real @ X2 @ T4 ) )
@ ( topolo2815343760600316023s_real @ Y3 )
@ ( topolo2177554685111907308n_real @ X2 @ ( inf_inf_set_real @ ( set_or5849166863359141190n_real @ X2 ) @ I5 ) ) ) ) ).
% has_derivative_right
thf(fact_1115_isCont__If__ge,axiom,
! [A2: real,G: real > real,F2: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) @ G )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ ( G @ A2 ) ) @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
@ ^ [X: real] : ( if_real @ ( ord_less_eq_real @ X @ A2 ) @ ( G @ X ) @ ( F2 @ X ) ) ) ) ) ).
% isCont_If_ge
thf(fact_1116_isCont__If__ge,axiom,
! [A2: nat,G: nat > nat,F2: nat > nat] :
( ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ A2 @ ( set_ord_lessThan_nat @ A2 ) ) @ G )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ ( G @ A2 ) ) @ ( topolo4659099751122792720in_nat @ A2 @ ( set_or1210151606488870762an_nat @ A2 ) ) )
=> ( topolo1306369304726495905at_nat @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat )
@ ^ [X: nat] : ( if_nat @ ( ord_less_eq_nat @ X @ A2 ) @ ( G @ X ) @ ( F2 @ X ) ) ) ) ) ).
% isCont_If_ge
thf(fact_1117_isCont__If__ge,axiom,
! [A2: nat,G: nat > real,F2: nat > real] :
( ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ ( set_ord_lessThan_nat @ A2 ) ) @ G )
=> ( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ ( G @ A2 ) ) @ ( topolo4659099751122792720in_nat @ A2 @ ( set_or1210151606488870762an_nat @ A2 ) ) )
=> ( topolo3806541068715748605t_real @ ( topolo4659099751122792720in_nat @ A2 @ top_top_set_nat )
@ ^ [X: nat] : ( if_real @ ( ord_less_eq_nat @ X @ A2 ) @ ( G @ X ) @ ( F2 @ X ) ) ) ) ) ).
% isCont_If_ge
thf(fact_1118_order__refl,axiom,
! [X2: filter_real] : ( ord_le4104064031414453916r_real @ X2 @ X2 ) ).
% order_refl
thf(fact_1119_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_1120_dual__order_Orefl,axiom,
! [A2: filter_real] : ( ord_le4104064031414453916r_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_1121_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_1122_insert__subset,axiom,
! [X2: real,A: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ ( insert_real @ X2 @ A ) @ B2 )
= ( ( member_real @ X2 @ B2 )
& ( ord_less_eq_set_real @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_1123_insert__subset,axiom,
! [X2: nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A ) @ B2 )
= ( ( member_nat @ X2 @ B2 )
& ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_1124_IntI,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ A )
=> ( ( member_nat @ C2 @ B2 )
=> ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_1125_IntI,axiom,
! [C2: real,A: set_real,B2: set_real] :
( ( member_real @ C2 @ A )
=> ( ( member_real @ C2 @ B2 )
=> ( member_real @ C2 @ ( inf_inf_set_real @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_1126_Int__iff,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) )
= ( ( member_nat @ C2 @ A )
& ( member_nat @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_1127_Int__iff,axiom,
! [C2: real,A: set_real,B2: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A @ B2 ) )
= ( ( member_real @ C2 @ A )
& ( member_real @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_1128_Int__subset__iff,axiom,
! [C: set_real,A: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ C @ ( inf_inf_set_real @ A @ B2 ) )
= ( ( ord_less_eq_set_real @ C @ A )
& ( ord_less_eq_set_real @ C @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_1129_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_1130_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_1131_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_1132_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_1133_neg__le__iff__le,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_1134_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_real] :
( ( inf_inf_set_real @ bot_bot_set_real @ X2 )
= bot_bot_set_real ) ).
% boolean_algebra.conj_zero_left
thf(fact_1135_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_real] :
( ( inf_inf_set_real @ X2 @ bot_bot_set_real )
= bot_bot_set_real ) ).
% boolean_algebra.conj_zero_right
thf(fact_1136_Int__UNIV,axiom,
! [A: set_real,B2: set_real] :
( ( ( inf_inf_set_real @ A @ B2 )
= top_top_set_real )
= ( ( A = top_top_set_real )
& ( B2 = top_top_set_real ) ) ) ).
% Int_UNIV
thf(fact_1137_Int__UNIV,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_1138_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B2 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1139_Int__insert__right__if1,axiom,
! [A2: real,A: set_real,B2: set_real] :
( ( member_real @ A2 @ A )
=> ( ( inf_inf_set_real @ A @ ( insert_real @ A2 @ B2 ) )
= ( insert_real @ A2 @ ( inf_inf_set_real @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1140_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_1141_Int__insert__right__if0,axiom,
! [A2: real,A: set_real,B2: set_real] :
( ~ ( member_real @ A2 @ A )
=> ( ( inf_inf_set_real @ A @ ( insert_real @ A2 @ B2 ) )
= ( inf_inf_set_real @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_1142_insert__inter__insert,axiom,
! [A2: real,A: set_real,B2: set_real] :
( ( inf_inf_set_real @ ( insert_real @ A2 @ A ) @ ( insert_real @ A2 @ B2 ) )
= ( insert_real @ A2 @ ( inf_inf_set_real @ A @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_1143_Int__insert__left__if1,axiom,
! [A2: nat,C: set_nat,B2: set_nat] :
( ( member_nat @ A2 @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1144_Int__insert__left__if1,axiom,
! [A2: real,C: set_real,B2: set_real] :
( ( member_real @ A2 @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A2 @ B2 ) @ C )
= ( insert_real @ A2 @ ( inf_inf_set_real @ B2 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1145_Int__insert__left__if0,axiom,
! [A2: nat,C: set_nat,B2: set_nat] :
( ~ ( member_nat @ A2 @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C )
= ( inf_inf_set_nat @ B2 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_1146_Int__insert__left__if0,axiom,
! [A2: real,C: set_real,B2: set_real] :
( ~ ( member_real @ A2 @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A2 @ B2 ) @ C )
= ( inf_inf_set_real @ B2 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_1147_open__Int,axiom,
! [S3: set_real,T: set_real] :
( ( topolo4860482606490270245n_real @ S3 )
=> ( ( topolo4860482606490270245n_real @ T )
=> ( topolo4860482606490270245n_real @ ( inf_inf_set_real @ S3 @ T ) ) ) ) ).
% open_Int
thf(fact_1148_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_1149_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_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_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_1157_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_1158_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_1159_neg__less__eq__nonneg,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_1160_less__eq__neg__nonpos,axiom,
! [A2: real] :
( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_1161_neg__le__0__iff__le,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_1162_neg__0__le__iff__le,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_1163_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_1164_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_1165_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_1166_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_1167_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_1168_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_1169_inf__compl__bot__left1,axiom,
! [X2: set_real,Y3: set_real] :
( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ ( inf_inf_set_real @ X2 @ Y3 ) )
= bot_bot_set_real ) ).
% inf_compl_bot_left1
thf(fact_1170_inf__compl__bot__left2,axiom,
! [X2: set_real,Y3: set_real] :
( ( inf_inf_set_real @ X2 @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ Y3 ) )
= bot_bot_set_real ) ).
% inf_compl_bot_left2
thf(fact_1171_inf__compl__bot__right,axiom,
! [X2: set_real,Y3: set_real] :
( ( inf_inf_set_real @ X2 @ ( inf_inf_set_real @ Y3 @ ( uminus612125837232591019t_real @ X2 ) ) )
= bot_bot_set_real ) ).
% inf_compl_bot_right
thf(fact_1172_boolean__algebra_Oconj__cancel__left,axiom,
! [X2: set_real] :
( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ X2 )
= bot_bot_set_real ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1173_boolean__algebra_Oconj__cancel__right,axiom,
! [X2: set_real] :
( ( inf_inf_set_real @ X2 @ ( uminus612125837232591019t_real @ X2 ) )
= bot_bot_set_real ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1174_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B2 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B2 )
& ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_1175_insert__disjoint_I1_J,axiom,
! [A2: real,A: set_real,B2: set_real] :
( ( ( inf_inf_set_real @ ( insert_real @ A2 @ A ) @ B2 )
= bot_bot_set_real )
= ( ~ ( member_real @ A2 @ B2 )
& ( ( inf_inf_set_real @ A @ B2 )
= bot_bot_set_real ) ) ) ).
% insert_disjoint(1)
thf(fact_1176_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B2 ) )
= ( ~ ( member_nat @ A2 @ B2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1177_insert__disjoint_I2_J,axiom,
! [A2: real,A: set_real,B2: set_real] :
( ( bot_bot_set_real
= ( inf_inf_set_real @ ( insert_real @ A2 @ A ) @ B2 ) )
= ( ~ ( member_real @ A2 @ B2 )
& ( bot_bot_set_real
= ( inf_inf_set_real @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1178_disjoint__insert_I1_J,axiom,
! [B2: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B2 @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B2 )
& ( ( inf_inf_set_nat @ B2 @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_1179_disjoint__insert_I1_J,axiom,
! [B2: set_real,A2: real,A: set_real] :
( ( ( inf_inf_set_real @ B2 @ ( insert_real @ A2 @ A ) )
= bot_bot_set_real )
= ( ~ ( member_real @ A2 @ B2 )
& ( ( inf_inf_set_real @ B2 @ A )
= bot_bot_set_real ) ) ) ).
% disjoint_insert(1)
thf(fact_1180_disjoint__insert_I2_J,axiom,
! [A: set_nat,B: nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B @ B2 ) ) )
= ( ~ ( member_nat @ B @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1181_disjoint__insert_I2_J,axiom,
! [A: set_real,B: real,B2: set_real] :
( ( bot_bot_set_real
= ( inf_inf_set_real @ A @ ( insert_real @ B @ B2 ) ) )
= ( ~ ( member_real @ B @ A )
& ( bot_bot_set_real
= ( inf_inf_set_real @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1182_Diff__disjoint,axiom,
! [A: set_real,B2: set_real] :
( ( inf_inf_set_real @ A @ ( minus_minus_set_real @ B2 @ A ) )
= bot_bot_set_real ) ).
% Diff_disjoint
thf(fact_1183_subset__Compl__singleton,axiom,
! [A: set_real,B: real] :
( ( ord_less_eq_set_real @ A @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
= ( ~ ( member_real @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_1184_subset__Compl__singleton,axiom,
! [A: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_1185_Compl__disjoint,axiom,
! [A: set_real] :
( ( inf_inf_set_real @ A @ ( uminus612125837232591019t_real @ A ) )
= bot_bot_set_real ) ).
% Compl_disjoint
thf(fact_1186_Compl__disjoint2,axiom,
! [A: set_real] :
( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ A ) @ A )
= bot_bot_set_real ) ).
% Compl_disjoint2
thf(fact_1187_Diff__Compl,axiom,
! [A: set_real,B2: set_real] :
( ( minus_minus_set_real @ A @ ( uminus612125837232591019t_real @ B2 ) )
= ( inf_inf_set_real @ A @ B2 ) ) ).
% Diff_Compl
thf(fact_1188_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_1189_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_1190_eventually__le__at__bot,axiom,
! [C2: nat] :
( eventually_nat
@ ^ [X: nat] : ( ord_less_eq_nat @ X @ C2 )
@ at_bot_nat ) ).
% eventually_le_at_bot
thf(fact_1191_eventually__le__at__bot,axiom,
! [C2: real] :
( eventually_real
@ ^ [X: real] : ( ord_less_eq_real @ X @ C2 )
@ at_bot_real ) ).
% eventually_le_at_bot
thf(fact_1192_eventually__at__bot__linorder,axiom,
! [P: nat > $o] :
( ( eventually_nat @ P @ at_bot_nat )
= ( ? [N4: nat] :
! [N: nat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( P @ N ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_1193_eventually__at__bot__linorder,axiom,
! [P: real > $o] :
( ( eventually_real @ P @ at_bot_real )
= ( ? [N4: real] :
! [N: real] :
( ( ord_less_eq_real @ N @ N4 )
=> ( P @ N ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_1194_at__bot__le__at__infinity,axiom,
ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).
% at_bot_le_at_infinity
thf(fact_1195_LIMSEQ__le__const2,axiom,
! [X5: nat > nat,X2: nat,A2: nat] :
( ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ X2 ) @ at_top_nat )
=> ( ? [N5: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N5 @ N2 )
=> ( ord_less_eq_nat @ ( X5 @ N2 ) @ A2 ) )
=> ( ord_less_eq_nat @ X2 @ A2 ) ) ) ).
% LIMSEQ_le_const2
thf(fact_1196_LIMSEQ__le__const2,axiom,
! [X5: nat > real,X2: real,A2: real] :
( ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ X2 ) @ at_top_nat )
=> ( ? [N5: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N5 @ N2 )
=> ( ord_less_eq_real @ ( X5 @ N2 ) @ A2 ) )
=> ( ord_less_eq_real @ X2 @ A2 ) ) ) ).
% LIMSEQ_le_const2
thf(fact_1197_LIMSEQ__le__const,axiom,
! [X5: nat > nat,X2: nat,A2: nat] :
( ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ X2 ) @ at_top_nat )
=> ( ? [N5: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N5 @ N2 )
=> ( ord_less_eq_nat @ A2 @ ( X5 @ N2 ) ) )
=> ( ord_less_eq_nat @ A2 @ X2 ) ) ) ).
% LIMSEQ_le_const
thf(fact_1198_LIMSEQ__le__const,axiom,
! [X5: nat > real,X2: real,A2: real] :
( ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ X2 ) @ at_top_nat )
=> ( ? [N5: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N5 @ N2 )
=> ( ord_less_eq_real @ A2 @ ( X5 @ N2 ) ) )
=> ( ord_less_eq_real @ A2 @ X2 ) ) ) ).
% LIMSEQ_le_const
thf(fact_1199_Lim__bounded2,axiom,
! [F2: nat > nat,L: nat,N6: nat,C: nat] :
( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ at_top_nat )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ N6 @ N2 )
=> ( ord_less_eq_nat @ C @ ( F2 @ N2 ) ) )
=> ( ord_less_eq_nat @ C @ L ) ) ) ).
% Lim_bounded2
thf(fact_1200_Lim__bounded2,axiom,
! [F2: nat > real,L: real,N6: nat,C: real] :
( ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ N6 @ N2 )
=> ( ord_less_eq_real @ C @ ( F2 @ N2 ) ) )
=> ( ord_less_eq_real @ C @ L ) ) ) ).
% Lim_bounded2
thf(fact_1201_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_1202_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_1203_LIMSEQ__le,axiom,
! [X5: nat > nat,X2: nat,Y5: nat > nat,Y3: nat] :
( ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ X2 ) @ at_top_nat )
=> ( ( filterlim_nat_nat @ Y5 @ ( topolo8926549440605965083ds_nat @ Y3 ) @ at_top_nat )
=> ( ? [N5: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N5 @ N2 )
=> ( ord_less_eq_nat @ ( X5 @ N2 ) @ ( Y5 @ N2 ) ) )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ).
% LIMSEQ_le
thf(fact_1204_LIMSEQ__le,axiom,
! [X5: nat > real,X2: real,Y5: nat > real,Y3: real] :
( ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ X2 ) @ at_top_nat )
=> ( ( filterlim_nat_real @ Y5 @ ( topolo2815343760600316023s_real @ Y3 ) @ at_top_nat )
=> ( ? [N5: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N5 @ N2 )
=> ( ord_less_eq_real @ ( X5 @ N2 ) @ ( Y5 @ N2 ) ) )
=> ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ).
% LIMSEQ_le
thf(fact_1205_lim__mono,axiom,
! [N6: nat,X5: nat > nat,Y5: nat > nat,X2: nat,Y3: nat] :
( ! [N2: nat] :
( ( ord_less_eq_nat @ N6 @ N2 )
=> ( ord_less_eq_nat @ ( X5 @ N2 ) @ ( Y5 @ N2 ) ) )
=> ( ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ X2 ) @ at_top_nat )
=> ( ( filterlim_nat_nat @ Y5 @ ( topolo8926549440605965083ds_nat @ Y3 ) @ at_top_nat )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ).
% lim_mono
thf(fact_1206_lim__mono,axiom,
! [N6: nat,X5: nat > real,Y5: nat > real,X2: real,Y3: real] :
( ! [N2: nat] :
( ( ord_less_eq_nat @ N6 @ N2 )
=> ( ord_less_eq_real @ ( X5 @ N2 ) @ ( Y5 @ N2 ) ) )
=> ( ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ X2 ) @ at_top_nat )
=> ( ( filterlim_nat_real @ Y5 @ ( topolo2815343760600316023s_real @ Y3 ) @ at_top_nat )
=> ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ).
% lim_mono
thf(fact_1207_Int__Diff__disjoint,axiom,
! [A: set_real,B2: set_real] :
( ( inf_inf_set_real @ ( inf_inf_set_real @ A @ B2 ) @ ( minus_minus_set_real @ A @ B2 ) )
= bot_bot_set_real ) ).
% Int_Diff_disjoint
thf(fact_1208_Diff__triv,axiom,
! [A: set_real,B2: set_real] :
( ( ( inf_inf_set_real @ A @ B2 )
= bot_bot_set_real )
=> ( ( minus_minus_set_real @ A @ B2 )
= A ) ) ).
% Diff_triv
thf(fact_1209_tendsto__mono,axiom,
! [F: filter_nat,F8: filter_nat,F2: nat > nat,L: nat] :
( ( ord_le2510731241096832064er_nat @ F @ F8 )
=> ( ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F8 )
=> ( filterlim_nat_nat @ F2 @ ( topolo8926549440605965083ds_nat @ L ) @ F ) ) ) ).
% tendsto_mono
thf(fact_1210_tendsto__mono,axiom,
! [F: filter_real,F8: filter_real,F2: real > real,L: real] :
( ( ord_le4104064031414453916r_real @ F @ F8 )
=> ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F8 )
=> ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ F ) ) ) ).
% tendsto_mono
thf(fact_1211_tendsto__within__subset,axiom,
! [F2: real > real,L: filter_real,X2: real,S3: set_real,T: set_real] :
( ( filterlim_real_real @ F2 @ L @ ( topolo2177554685111907308n_real @ X2 @ S3 ) )
=> ( ( ord_less_eq_set_real @ T @ S3 )
=> ( filterlim_real_real @ F2 @ L @ ( topolo2177554685111907308n_real @ X2 @ T ) ) ) ) ).
% tendsto_within_subset
thf(fact_1212_tendsto__within__subset,axiom,
! [F2: nat > nat,L: filter_nat,X2: nat,S3: set_nat,T: set_nat] :
( ( filterlim_nat_nat @ F2 @ L @ ( topolo4659099751122792720in_nat @ X2 @ S3 ) )
=> ( ( ord_less_eq_set_nat @ T @ S3 )
=> ( filterlim_nat_nat @ F2 @ L @ ( topolo4659099751122792720in_nat @ X2 @ T ) ) ) ) ).
% tendsto_within_subset
thf(fact_1213_filterlim__mono_H,axiom,
! [F2: nat > nat,F: filter_nat,G2: filter_nat,F8: filter_nat] :
( ( filterlim_nat_nat @ F2 @ F @ G2 )
=> ( ( ord_le2510731241096832064er_nat @ F @ F8 )
=> ( filterlim_nat_nat @ F2 @ F8 @ G2 ) ) ) ).
% filterlim_mono'
thf(fact_1214_filterlim__mono_H,axiom,
! [F2: real > real,F: filter_real,G2: filter_real,F8: filter_real] :
( ( filterlim_real_real @ F2 @ F @ G2 )
=> ( ( ord_le4104064031414453916r_real @ F @ F8 )
=> ( filterlim_real_real @ F2 @ F8 @ G2 ) ) ) ).
% filterlim_mono'
thf(fact_1215_filterlim__mono,axiom,
! [F2: nat > nat,F22: filter_nat,F1: filter_nat,F23: filter_nat,F12: filter_nat] :
( ( filterlim_nat_nat @ F2 @ F22 @ F1 )
=> ( ( ord_le2510731241096832064er_nat @ F22 @ F23 )
=> ( ( ord_le2510731241096832064er_nat @ F12 @ F1 )
=> ( filterlim_nat_nat @ F2 @ F23 @ F12 ) ) ) ) ).
% filterlim_mono
thf(fact_1216_filterlim__mono,axiom,
! [F2: real > real,F22: filter_real,F1: filter_real,F23: filter_real,F12: filter_real] :
( ( filterlim_real_real @ F2 @ F22 @ F1 )
=> ( ( ord_le4104064031414453916r_real @ F22 @ F23 )
=> ( ( ord_le4104064031414453916r_real @ F12 @ F1 )
=> ( filterlim_real_real @ F2 @ F23 @ F12 ) ) ) ) ).
% filterlim_mono
thf(fact_1217_filterlim__at__within__If,axiom,
! [F2: real > real,G2: filter_real,X2: real,A: set_real,P: real > $o,G: real > real] :
( ( filterlim_real_real @ F2 @ G2 @ ( topolo2177554685111907308n_real @ X2 @ ( inf_inf_set_real @ A @ ( collect_real @ P ) ) ) )
=> ( ( filterlim_real_real @ G @ G2
@ ( topolo2177554685111907308n_real @ X2
@ ( inf_inf_set_real @ A
@ ( collect_real
@ ^ [X: real] :
~ ( P @ X ) ) ) ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( if_real @ ( P @ X ) @ ( F2 @ X ) @ ( G @ X ) )
@ G2
@ ( topolo2177554685111907308n_real @ X2 @ A ) ) ) ) ).
% filterlim_at_within_If
thf(fact_1218_filterlim__at__within__If,axiom,
! [F2: nat > nat,G2: filter_nat,X2: nat,A: set_nat,P: nat > $o,G: nat > nat] :
( ( filterlim_nat_nat @ F2 @ G2 @ ( topolo4659099751122792720in_nat @ X2 @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) ) )
=> ( ( filterlim_nat_nat @ G @ G2
@ ( topolo4659099751122792720in_nat @ X2
@ ( inf_inf_set_nat @ A
@ ( collect_nat
@ ^ [X: nat] :
~ ( P @ X ) ) ) ) )
=> ( filterlim_nat_nat
@ ^ [X: nat] : ( if_nat @ ( P @ X ) @ ( F2 @ X ) @ ( G @ X ) )
@ G2
@ ( topolo4659099751122792720in_nat @ X2 @ A ) ) ) ) ).
% filterlim_at_within_If
thf(fact_1219_filterlim__mono__eventually,axiom,
! [F2: nat > nat,F: filter_nat,G2: filter_nat,F8: filter_nat,G6: filter_nat,F7: nat > nat] :
( ( filterlim_nat_nat @ F2 @ F @ G2 )
=> ( ( ord_le2510731241096832064er_nat @ F @ F8 )
=> ( ( ord_le2510731241096832064er_nat @ G6 @ G2 )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( F2 @ X )
= ( F7 @ X ) )
@ G6 )
=> ( filterlim_nat_nat @ F7 @ F8 @ G6 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_1220_filterlim__mono__eventually,axiom,
! [F2: nat > real,F: filter_real,G2: filter_nat,F8: filter_real,G6: filter_nat,F7: nat > real] :
( ( filterlim_nat_real @ F2 @ F @ G2 )
=> ( ( ord_le4104064031414453916r_real @ F @ F8 )
=> ( ( ord_le2510731241096832064er_nat @ G6 @ G2 )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( F2 @ X )
= ( F7 @ X ) )
@ G6 )
=> ( filterlim_nat_real @ F7 @ F8 @ G6 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_1221_filterlim__mono__eventually,axiom,
! [F2: real > real,F: filter_real,G2: filter_real,F8: filter_real,G6: filter_real,F7: real > real] :
( ( filterlim_real_real @ F2 @ F @ G2 )
=> ( ( ord_le4104064031414453916r_real @ F @ F8 )
=> ( ( ord_le4104064031414453916r_real @ G6 @ G2 )
=> ( ( eventually_real
@ ^ [X: real] :
( ( F2 @ X )
= ( F7 @ X ) )
@ G6 )
=> ( filterlim_real_real @ F7 @ F8 @ G6 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_1222_disjoint__eq__subset__Compl,axiom,
! [A: set_real,B2: set_real] :
( ( ( inf_inf_set_real @ A @ B2 )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ A @ ( uminus612125837232591019t_real @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1223_inf__cancel__left2,axiom,
! [X2: set_real,A2: set_real,B: set_real] :
( ( inf_inf_set_real @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ A2 ) @ ( inf_inf_set_real @ X2 @ B ) )
= bot_bot_set_real ) ).
% inf_cancel_left2
thf(fact_1224_inf__cancel__left1,axiom,
! [X2: set_real,A2: set_real,B: set_real] :
( ( inf_inf_set_real @ ( inf_inf_set_real @ X2 @ A2 ) @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ B ) )
= bot_bot_set_real ) ).
% inf_cancel_left1
thf(fact_1225_inf__shunt,axiom,
! [X2: set_real,Y3: set_real] :
( ( ( inf_inf_set_real @ X2 @ Y3 )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ X2 @ ( uminus612125837232591019t_real @ Y3 ) ) ) ).
% inf_shunt
thf(fact_1226_Int__emptyI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ~ ( member_nat @ X4 @ B2 ) )
=> ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_1227_Int__emptyI,axiom,
! [A: set_real,B2: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A )
=> ~ ( member_real @ X4 @ B2 ) )
=> ( ( inf_inf_set_real @ A @ B2 )
= bot_bot_set_real ) ) ).
% Int_emptyI
thf(fact_1228_disjoint__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_nat @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1229_disjoint__iff,axiom,
! [A: set_real,B2: set_real] :
( ( ( inf_inf_set_real @ A @ B2 )
= bot_bot_set_real )
= ( ! [X: real] :
( ( member_real @ X @ A )
=> ~ ( member_real @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1230_Int__empty__left,axiom,
! [B2: set_real] :
( ( inf_inf_set_real @ bot_bot_set_real @ B2 )
= bot_bot_set_real ) ).
% Int_empty_left
thf(fact_1231_Int__empty__right,axiom,
! [A: set_real] :
( ( inf_inf_set_real @ A @ bot_bot_set_real )
= bot_bot_set_real ) ).
% Int_empty_right
thf(fact_1232_disjoint__iff__not__equal,axiom,
! [A: set_real,B2: set_real] :
( ( ( inf_inf_set_real @ A @ B2 )
= bot_bot_set_real )
= ( ! [X: real] :
( ( member_real @ X @ A )
=> ! [Y: real] :
( ( member_real @ Y @ B2 )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1233_bot_Oextremum,axiom,
! [A2: filter_nat] : ( ord_le2510731241096832064er_nat @ bot_bot_filter_nat @ A2 ) ).
% bot.extremum
thf(fact_1234_bot_Oextremum,axiom,
! [A2: filter_real] : ( ord_le4104064031414453916r_real @ bot_bot_filter_real @ A2 ) ).
% bot.extremum
thf(fact_1235_bot_Oextremum__unique,axiom,
! [A2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ bot_bot_filter_nat )
= ( A2 = bot_bot_filter_nat ) ) ).
% bot.extremum_unique
thf(fact_1236_bot_Oextremum__unique,axiom,
! [A2: filter_real] :
( ( ord_le4104064031414453916r_real @ A2 @ bot_bot_filter_real )
= ( A2 = bot_bot_filter_real ) ) ).
% bot.extremum_unique
thf(fact_1237_bot_Oextremum__uniqueI,axiom,
! [A2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ bot_bot_filter_nat )
=> ( A2 = bot_bot_filter_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1238_bot_Oextremum__uniqueI,axiom,
! [A2: filter_real] :
( ( ord_le4104064031414453916r_real @ A2 @ bot_bot_filter_real )
=> ( A2 = bot_bot_filter_real ) ) ).
% bot.extremum_uniqueI
thf(fact_1239_separation__t2,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
= ( ? [U2: set_nat,V2: set_nat] :
( ( topolo4328251076210115529en_nat @ U2 )
& ( topolo4328251076210115529en_nat @ V2 )
& ( member_nat @ X2 @ U2 )
& ( member_nat @ Y3 @ V2 )
& ( ( inf_inf_set_nat @ U2 @ V2 )
= bot_bot_set_nat ) ) ) ) ).
% separation_t2
thf(fact_1240_separation__t2,axiom,
! [X2: real,Y3: real] :
( ( X2 != Y3 )
= ( ? [U2: set_real,V2: set_real] :
( ( topolo4860482606490270245n_real @ U2 )
& ( topolo4860482606490270245n_real @ V2 )
& ( member_real @ X2 @ U2 )
& ( member_real @ Y3 @ V2 )
& ( ( inf_inf_set_real @ U2 @ V2 )
= bot_bot_set_real ) ) ) ) ).
% separation_t2
thf(fact_1241_hausdorff,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ? [U: set_nat,V3: set_nat] :
( ( topolo4328251076210115529en_nat @ U )
& ( topolo4328251076210115529en_nat @ V3 )
& ( member_nat @ X2 @ U )
& ( member_nat @ Y3 @ V3 )
& ( ( inf_inf_set_nat @ U @ V3 )
= bot_bot_set_nat ) ) ) ).
% hausdorff
thf(fact_1242_hausdorff,axiom,
! [X2: real,Y3: real] :
( ( X2 != Y3 )
=> ? [U: set_real,V3: set_real] :
( ( topolo4860482606490270245n_real @ U )
& ( topolo4860482606490270245n_real @ V3 )
& ( member_real @ X2 @ U )
& ( member_real @ Y3 @ V3 )
& ( ( inf_inf_set_real @ U @ V3 )
= bot_bot_set_real ) ) ) ).
% hausdorff
thf(fact_1243_at__top__le__at__infinity,axiom,
ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).
% at_top_le_at_infinity
thf(fact_1244_Multiseries__Expansion__Bounds_Omult__mono__nonneg__nonpos,axiom,
! [C2: real,A2: real,B: real,D: real] :
( ( ord_less_eq_real @ C2 @ A2 )
=> ( ( ord_less_eq_real @ B @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ D @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ C2 @ D ) ) ) ) ) ) ).
% Multiseries_Expansion_Bounds.mult_mono_nonneg_nonpos
thf(fact_1245_Multiseries__Expansion__Bounds_Omult__mono__nonpos__nonneg,axiom,
! [A2: real,C2: real,D: real,B: real] :
( ( ord_less_eq_real @ A2 @ C2 )
=> ( ( ord_less_eq_real @ D @ B )
=> ( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ D )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ C2 @ D ) ) ) ) ) ) ).
% Multiseries_Expansion_Bounds.mult_mono_nonpos_nonneg
thf(fact_1246_Multiseries__Expansion__Bounds_Omult__mono__nonpos__nonpos,axiom,
! [C2: real,A2: real,D: real,B: real] :
( ( ord_less_eq_real @ C2 @ A2 )
=> ( ( ord_less_eq_real @ D @ B )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ C2 @ D ) ) ) ) ) ) ).
% Multiseries_Expansion_Bounds.mult_mono_nonpos_nonpos
thf(fact_1247_landau__o_OR__mult__left__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 @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% landau_o.R_mult_left_mono
thf(fact_1248_landau__o_OR__mult__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 @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% landau_o.R_mult_right_mono
thf(fact_1249_landau__omega_OR__mult__left__mono,axiom,
! [B: real,A2: real,C2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ B ) @ ( times_times_real @ C2 @ A2 ) ) ) ) ).
% landau_omega.R_mult_left_mono
thf(fact_1250_landau__omega_OR__mult__right__mono,axiom,
! [B: real,A2: real,C2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ B @ C2 ) @ ( times_times_real @ A2 @ C2 ) ) ) ) ).
% landau_omega.R_mult_right_mono
thf(fact_1251_boolean__algebra_Oconj__one__right,axiom,
! [X2: set_real] :
( ( inf_inf_set_real @ X2 @ top_top_set_real )
= X2 ) ).
% boolean_algebra.conj_one_right
thf(fact_1252_boolean__algebra_Oconj__one__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ top_top_set_nat )
= X2 ) ).
% boolean_algebra.conj_one_right
thf(fact_1253_ordered__comm__semiring__class_Ocomm__mult__left__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 @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1254_zero__le__mult__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_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_mult_iff
thf(fact_1255_mult__nonneg__nonpos2,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A2 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1256_mult__nonpos__nonneg,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1257_mult__nonneg__nonpos,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1258_mult__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 @ ( times_times_real @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1259_split__mult__neg__le,axiom,
! [A2: real,B: 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 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_1260_mult__le__0__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ ( times_times_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 ) ) ) ) ).
% mult_le_0_iff
thf(fact_1261_mult__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 @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_1262_mult__right__mono__neg,axiom,
! [B: real,A2: real,C2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1263_mult__left__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 @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1264_mult__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 @ zero_zero_real @ ( times_times_real @ A2 @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1265_mult__left__mono__neg,axiom,
! [B: real,A2: real,C2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1266_split__mult__pos__le,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 @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) ) ) ).
% split_mult_pos_le
thf(fact_1267_zero__le__square,axiom,
! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_1268_mult__mono_H,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 @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1269_mult__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 @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1270_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_1271_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_1272_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $true @ X2 @ Y3 )
= X2 ) ).
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,
! [X2: real,Y3: real] :
( ( if_real @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y3: real] :
( ( if_real @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( eventually_real
@ ^ [X: real] :
~ ( member_real @ X @ ring_1_Ints_real )
@ ( topolo2177554685111907308n_real @ ( r @ xa ) @ top_top_set_real ) ) ).
%------------------------------------------------------------------------------