TPTP Problem File: SLH0999^1.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 : Youngs_Inequality/0000_Youngs/prob_00643_027473__13185736_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1432 ( 473 unt; 156 typ; 0 def)
% Number of atoms : 4280 (1507 equ; 0 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 13125 ( 353 ~; 81 |; 314 &;10398 @)
% ( 0 <=>;1979 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 906 ( 906 >; 0 *; 0 +; 0 <<)
% Number of symbols : 149 ( 146 usr; 19 con; 0-3 aty)
% Number of variables : 3900 ( 193 ^;3604 !; 103 ?;3900 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:31:45.130
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J_J,type,
set_set_set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
set_set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_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__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (146)
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_Eo,type,
complete_Inf_Inf_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Int__Oint,type,
complete_Inf_Inf_int: set_int > int ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
comple4887499456419720421f_real: set_real > real ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Real__Oreal_J,type,
comple8289635161444856091t_real: set_set_real > set_real ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
complete_Sup_Sup_int: set_int > int ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
comple1385675409528146559p_real: set_real > real ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Real__Oreal_J,type,
comple3096694443085538997t_real: set_set_real > set_real ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
comple5917660045593844715t_real: set_set_set_real > set_set_real ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Int__Oint,type,
bij_betw_nat_int: ( nat > int ) > set_nat > set_int > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Real__Oreal,type,
bij_betw_nat_real: ( nat > real ) > set_nat > set_real > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Set__Oset_It__Real__Oreal_J,type,
bij_be7200859108919812778t_real: ( nat > set_real ) > set_nat > set_set_real > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Real__Oreal_J_001t__Nat__Onat,type,
bij_be7148990519944890154al_nat: ( set_real > nat ) > set_set_real > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J,type,
bij_be8850480603386830140t_real: ( set_real > set_real ) > set_set_real > set_set_real > $o ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
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_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
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_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
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__Big_Ocomm__monoid__add__class_Osum_001_Eo_001t__Int__Oint,type,
groups8505340233167759370_o_int: ( $o > int ) > set_o > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_Eo_001t__Nat__Onat,type,
groups8507830703676809646_o_nat: ( $o > nat ) > set_o > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_Eo_001t__Real__Oreal,type,
groups8691415230153176458o_real: ( $o > real ) > set_o > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
groups1932886352136224148al_int: ( real > int ) > set_real > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
groups8097168146408367636l_real: ( real > real ) > set_real > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Real__Oreal_J_001t__Int__Oint,type,
groups3009712052913938890al_int: ( set_real > int ) > set_set_real > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Real__Oreal_J_001t__Nat__Onat,type,
groups3012202523422989166al_nat: ( set_real > nat ) > set_set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Real__Oreal_J_001t__Real__Oreal,type,
groups8702937949983641418l_real: ( set_real > real ) > set_set_real > real ).
thf(sy_c_Henstock__Kurzweil__Integration_Ohas__integral_001t__Real__Oreal_001t__Real__Oreal,type,
hensto240673015341029504l_real: ( real > real ) > real > set_real > $o ).
thf(sy_c_Henstock__Kurzweil__Integration_Ointegral_001t__Real__Oreal_001t__Real__Oreal,type,
hensto2714581292692559302l_real: set_real > ( real > real ) > real ).
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_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
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_I_Eo_J,type,
bot_bot_set_o: set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
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_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
bot_bot_set_set_real: set_set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J_J,type,
bot_bo3378928929837779682t_real: set_set_set_real ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
ord_Least_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_Eo,type,
ord_less_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $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_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
ord_le3558479182127378552t_real: set_set_real > set_set_real > $o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
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_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
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_OCollect_001t__Set__Oset_It__Real__Oreal_J,type,
collect_set_real: ( set_real > $o ) > set_set_real ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Int__Oint,type,
image_o_int: ( $o > int ) > set_o > set_int ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Real__Oreal,type,
image_o_real: ( $o > real ) > set_o > set_real ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Real__Oreal_J,type,
image_o_set_real: ( $o > set_real ) > set_o > set_set_real ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Set__Oset_It__Real__Oreal_J,type,
image_int_set_real: ( int > set_real ) > set_int > set_set_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Real__Oreal_J,type,
image_nat_set_real: ( nat > set_real ) > set_nat > set_set_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
image_396256051147326063t_real: ( nat > set_set_real ) > set_nat > set_set_set_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001_Eo,type,
image_real_o: ( real > $o ) > set_real > set_o ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Int__Oint,type,
image_real_int: ( real > int ) > set_real > set_int ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Nat__Onat,type,
image_real_nat: ( real > nat ) > set_real > set_nat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Set__Oset_It__Nat__Onat_J,type,
image_real_set_nat: ( real > set_nat ) > set_real > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Set__Oset_It__Real__Oreal_J,type,
image_real_set_real: ( real > set_real ) > set_real > set_set_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
image_3243600997494576203t_real: ( real > set_set_real ) > set_real > set_set_set_real ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001_Eo,type,
image_set_real_o: ( set_real > $o ) > set_set_real > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Nat__Onat,type,
image_set_real_nat: ( set_real > nat ) > set_set_real > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Real__Oreal,type,
image_set_real_real: ( set_real > real ) > set_set_real > set_real ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7270232309134952815et_nat: ( set_real > set_nat ) > set_set_real > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J,type,
image_2436557299294012491t_real: ( set_real > set_real ) > set_set_real > set_set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001_Eo,type,
set_or8904488021354931149Most_o: $o > $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Real__Oreal_J,type,
set_or7743017856606604397t_real: set_real > set_real > set_set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001_Eo,type,
set_or7139685690850216873Than_o: $o > $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
set_or66887138388493659n_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Real__Oreal_J,type,
set_or5046967147999637905t_real: set_real > set_real > set_set_real ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_Eo,type,
set_ord_lessThan_o: $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
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_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Real__Oreal_J,type,
set_or3940062689191130623t_real: set_real > set_set_real ).
thf(sy_c_Youngs_Oregular__division,type,
regular_division: real > real > nat > set_set_real ).
thf(sy_c_Youngs_Osegment,type,
segment: nat > nat > set_real ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $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_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
member_set_real: set_real > set_set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
member_set_set_real: set_set_real > set_set_set_real > $o ).
thf(sy_v_DN____,type,
dn: set_real > nat ).
thf(sy_v__092_060delta_062____,type,
delta: real ).
thf(sy_v__092_060epsilon_062____,type,
epsilon: real ).
thf(sy_v_a,type,
a: real ).
thf(sy_v_a__seg____,type,
a_seg: real > real ).
thf(sy_v_b,type,
b: real ).
thf(sy_v_del____,type,
del: real > real ).
thf(sy_v_f,type,
f: real > real ).
thf(sy_v_f1____,type,
f1: real > real ).
thf(sy_v_f2____,type,
f2: real > real ).
thf(sy_v_g,type,
g: real > real ).
thf(sy_v_g1____,type,
g1: real > real ).
thf(sy_v_g2____,type,
g2: real > real ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_yidx____,type,
yidx: real > nat ).
% Relevant facts (1270)
thf(fact_0_False,axiom,
a != zero_zero_real ).
% False
thf(fact_1_f_I1_J,axiom,
( ( f @ zero_zero_real )
= zero_zero_real ) ).
% f(1)
thf(fact_2__092_060open_062_092_060And_062K_O_AK_A_092_060in_062_Aregular__division_A0_Aa_An_A_092_060Longrightarrow_062_Aa__seg_A_Ireal_A_IDN_AK_J_J_A_061_AInf_AK_092_060close_062,axiom,
! [K: set_real] :
( ( member_set_real @ K @ ( regular_division @ zero_zero_real @ a @ n ) )
=> ( ( a_seg @ ( semiri5074537144036343181t_real @ ( dn @ K ) ) )
= ( comple4887499456419720421f_real @ K ) ) ) ).
% \<open>\<And>K. K \<in> regular_division 0 a n \<Longrightarrow> a_seg (real (DN K)) = Inf K\<close>
thf(fact_3_a__seg__def,axiom,
( a_seg
= ( ^ [U: real] : ( divide_divide_real @ ( times_times_real @ U @ a ) @ ( semiri5074537144036343181t_real @ n ) ) ) ) ).
% a_seg_def
thf(fact_4_a__seg__eq__a__iff,axiom,
! [X: real] :
( ( ( a_seg @ X )
= a )
= ( X
= ( semiri5074537144036343181t_real @ n ) ) ) ).
% a_seg_eq_a_iff
thf(fact_5_a,axiom,
ord_less_eq_real @ zero_zero_real @ a ).
% a
thf(fact_6_f_I2_J,axiom,
( ( f @ a )
= b ) ).
% f(2)
thf(fact_7_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_8_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_9_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_10_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_11_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_12_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_13_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_14_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_15_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_16_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_17_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_18_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_19_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_20_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_21_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_22_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_23_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_24_g2__def,axiom,
( g2
= ( ^ [Y: real] : ( if_real @ ( Y = zero_zero_real ) @ zero_zero_real @ ( a_seg @ ( semiri5074537144036343181t_real @ ( yidx @ Y ) ) ) ) ) ) ).
% g2_def
thf(fact_25_of__nat__sum,axiom,
! [F: set_real > nat,A2: set_set_real] :
( ( semiri5074537144036343181t_real @ ( groups3012202523422989166al_nat @ F @ A2 ) )
= ( groups8702937949983641418l_real
@ ^ [X2: set_real] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_26_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3539618377306564664at_int
@ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_27_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [X2: nat] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_28_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( semiri1316708129612266289at_nat @ ( F @ X2 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_29__092_060open_0620_A_092_060le_062_Ab_092_060close_062,axiom,
ord_less_eq_real @ zero_zero_real @ b ).
% \<open>0 \<le> b\<close>
thf(fact_30__092_060open_062yidx_Ab_A_061_An_092_060close_062,axiom,
( ( yidx @ b )
= n ) ).
% \<open>yidx b = n\<close>
thf(fact_31_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_32_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_33_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_34_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_35_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_36_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_37_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_38_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_39_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_40_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_41_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_42_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_43_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_44_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_45_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_46_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_47_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_48_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_49_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_50_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_51_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_52_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_53_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_54_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_55_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_56_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_57_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_58_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_59_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_60_fa__eq__b,axiom,
( ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ n ) ) )
= b ) ).
% fa_eq_b
thf(fact_61_a__seg__le__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( a_seg @ X ) @ ( a_seg @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ).
% a_seg_le_iff
thf(fact_62_f__iff_I2_J,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ ( f @ X ) @ ( f @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ) ).
% f_iff(2)
thf(fact_63_a__seg__ge__0,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( a_seg @ X ) )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% a_seg_ge_0
thf(fact_64_a__seg__le__a,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( a_seg @ X ) @ a )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ n ) ) ) ).
% a_seg_le_a
thf(fact_65__092_060open_062_092_060delta_062_A_092_060le_062_Aa_092_060close_062,axiom,
ord_less_eq_real @ delta @ a ).
% \<open>\<delta> \<le> a\<close>
thf(fact_66_int__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3539618377306564664at_int
@ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
@ A2 ) ) ).
% int_sum
thf(fact_67_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_68_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_69_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_70_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_71_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_72_mem__Collect__eq,axiom,
! [A: set_real,P: set_real > $o] :
( ( member_set_real @ A @ ( collect_set_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_74_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_75_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_76_Collect__mem__eq,axiom,
! [A2: set_set_real] :
( ( collect_set_real
@ ^ [X2: set_real] : ( member_set_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_78_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_79_Collect__mem__eq,axiom,
! [A2: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_80_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_81_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_82_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_83_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_84_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_85_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_86_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_87_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_88_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_89_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_90_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_91_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_92_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_93_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_94_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_95_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_96_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_97_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_98_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_99_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_100_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_101_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_102_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_103_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_104_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_105_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_106_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_107_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_108_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_109_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_110_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_111_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_112_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_113_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_114_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_115_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_116_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_117_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_118_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_119_divide__right__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_120_divide__nonpos__nonpos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_121_divide__nonpos__nonneg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_122_divide__nonneg__nonpos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_123_divide__nonneg__nonneg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_124_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_125_divide__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_126_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_127_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_128_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_129_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_130_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_131_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_132_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_133_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_134_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_135_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_136_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_137_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_138_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_139_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_140_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_141_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_142_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_143_divide__divide__times__eq,axiom,
! [X: real,Y2: real,Z: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y2 ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y2 @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_144_times__divide__times__eq,axiom,
! [X: real,Y2: real,Z: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y2 ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y2 @ W ) ) ) ).
% times_divide_times_eq
thf(fact_145_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_146_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_147_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_148_nonzero__eq__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_149_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A )
= ( B
= ( times_times_real @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_150_eq__divide__imp,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= B )
=> ( A
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_151_divide__eq__imp,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_152_eq__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_153_divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( divide_divide_real @ B @ C )
= A )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_154_frac__eq__eq,axiom,
! [Y2: real,Z: real,X: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y2 )
= ( divide_divide_real @ W @ Z ) )
= ( ( times_times_real @ X @ Z )
= ( times_times_real @ W @ Y2 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_155_g,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ a )
=> ( ( g @ ( f @ X ) )
= X ) ) ) ).
% g
thf(fact_156_f2__upper,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ a )
=> ( ord_less_eq_real @ ( f @ X ) @ ( f2 @ X ) ) ) ) ).
% f2_upper
thf(fact_157_f1__lower,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ a )
=> ( ord_less_eq_real @ ( f1 @ X ) @ ( f @ X ) ) ) ) ).
% f1_lower
thf(fact_158_g1__def,axiom,
( g1
= ( ^ [Y: real] : ( if_real @ ( Y = b ) @ a @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ ( yidx @ Y ) ) ) ) ) ) ) ).
% g1_def
thf(fact_159_fa__yidx__le,axiom,
! [Y2: real] :
( ( member_real @ Y2 @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) )
=> ( ord_less_eq_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( yidx @ Y2 ) ) ) ) @ Y2 ) ) ).
% fa_yidx_le
thf(fact_160_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_161_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_162_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_163_sum_Oneutral__const,axiom,
! [A2: set_set_real] :
( ( groups8702937949983641418l_real
@ ^ [Uu: set_real] : zero_zero_real
@ A2 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_164_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu: nat] : zero_zero_real
@ A2 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_165_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu: nat] : zero_zero_nat
@ A2 )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_166_yidx__le__n,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ Y2 @ b )
=> ( ord_less_eq_nat @ ( yidx @ Y2 ) @ n ) ) ).
% yidx_le_n
thf(fact_167_g2,axiom,
! [Y2: real] :
( ( member_real @ Y2 @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) )
=> ( member_real @ ( g2 @ Y2 ) @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) ) ) ).
% g2
thf(fact_168_D__ne,axiom,
( ( regular_division @ zero_zero_real @ a @ n )
!= bot_bot_set_set_real ) ).
% D_ne
thf(fact_169_DN,axiom,
bij_be7148990519944890154al_nat @ dn @ ( regular_division @ zero_zero_real @ a @ n ) @ ( set_ord_lessThan_nat @ n ) ).
% DN
thf(fact_170_real__divide__square__eq,axiom,
! [R: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
= ( divide_divide_real @ A @ R ) ) ).
% real_divide_square_eq
thf(fact_171_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_172_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_173_mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_174_mult__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ( times_times_nat @ M @ K2 )
= ( times_times_nat @ N @ K2 ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_175_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_176_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_177_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_178_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_179_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_180_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_181_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_182_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_183_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_184_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_185_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_186_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_187_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_188_g1,axiom,
! [Y2: real] :
( ( member_real @ Y2 @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) )
=> ( member_real @ ( g1 @ Y2 ) @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) ) ) ).
% g1
thf(fact_189_g__le__g1,axiom,
! [Y2: real] :
( ( member_real @ Y2 @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) )
=> ( ord_less_eq_real @ ( g @ Y2 ) @ ( g1 @ Y2 ) ) ) ).
% g_le_g1
thf(fact_190_g2__le__g,axiom,
! [Y2: real] :
( ( member_real @ Y2 @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) )
=> ( ord_less_eq_real @ ( g2 @ Y2 ) @ ( g @ Y2 ) ) ) ).
% g2_le_g
thf(fact_191_sum_Oempty,axiom,
! [G: set_real > nat] :
( ( groups3012202523422989166al_nat @ G @ bot_bot_set_set_real )
= zero_zero_nat ) ).
% sum.empty
thf(fact_192_sum_Oempty,axiom,
! [G: set_real > int] :
( ( groups3009712052913938890al_int @ G @ bot_bot_set_set_real )
= zero_zero_int ) ).
% sum.empty
thf(fact_193_sum_Oempty,axiom,
! [G: nat > int] :
( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
= zero_zero_int ) ).
% sum.empty
thf(fact_194_sum_Oempty,axiom,
! [G: real > real] :
( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
= zero_zero_real ) ).
% sum.empty
thf(fact_195_sum_Oempty,axiom,
! [G: real > nat] :
( ( groups1935376822645274424al_nat @ G @ bot_bot_set_real )
= zero_zero_nat ) ).
% sum.empty
thf(fact_196_sum_Oempty,axiom,
! [G: real > int] :
( ( groups1932886352136224148al_int @ G @ bot_bot_set_real )
= zero_zero_int ) ).
% sum.empty
thf(fact_197_sum_Oempty,axiom,
! [G: set_real > real] :
( ( groups8702937949983641418l_real @ G @ bot_bot_set_set_real )
= zero_zero_real ) ).
% sum.empty
thf(fact_198_sum_Oempty,axiom,
! [G: nat > real] :
( ( groups6591440286371151544t_real @ G @ bot_bot_set_nat )
= zero_zero_real ) ).
% sum.empty
thf(fact_199_sum_Oempty,axiom,
! [G: nat > nat] :
( ( groups3542108847815614940at_nat @ G @ bot_bot_set_nat )
= zero_zero_nat ) ).
% sum.empty
thf(fact_200_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_201_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_202_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_203_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_204_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_205_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_206_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_207_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_208_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_209_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_210_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_211_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_212_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_213_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_214_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_215_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_216_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_217_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_218_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_219_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_220_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_221_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_222_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_223_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_224_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_225_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_226_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_227_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_228_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_229_Suc__le__D,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M2 )
=> ? [M3: nat] :
( M2
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_230_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_231_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_232_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_233_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_234_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_235_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_236_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_237_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_238_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_239_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_240_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_241_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_242_mult__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_243_mult__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_244_mult__le__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ).
% mult_le_mono2
thf(fact_245_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_246_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_247_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_248_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_249_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ~ ! [N2: nat] :
( K2
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_250_Suc__mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K2 ) @ M )
= ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_251_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_252_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ? [N2: nat] :
( K2
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_253_Suc__mult__le__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_254_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_255_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_256_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z2 )
=> ( R2 @ X3 @ Z2 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_257_zdiv__int,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zdiv_int
thf(fact_258_div__le__mono,axiom,
! [M: nat,N: nat,K2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K2 ) @ ( divide_divide_nat @ N @ K2 ) ) ) ).
% div_le_mono
thf(fact_259_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_260_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_261_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_262_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_263_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_264_Suc__inject,axiom,
! [X: nat,Y2: nat] :
( ( ( suc @ X )
= ( suc @ Y2 ) )
=> ( X = Y2 ) ) ).
% Suc_inject
thf(fact_265_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_266_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_267_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_268_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_269_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_270_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_271_sum__cong__Suc,axiom,
! [A2: set_nat,F: nat > real,G: nat > real] :
( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ ( suc @ X3 ) @ A2 )
=> ( ( F @ ( suc @ X3 ) )
= ( G @ ( suc @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ F @ A2 )
= ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).
% sum_cong_Suc
thf(fact_272_sum__cong__Suc,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ ( suc @ X3 ) @ A2 )
=> ( ( F @ ( suc @ X3 ) )
= ( G @ ( suc @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% sum_cong_Suc
thf(fact_273_sum_Oreindex__bij__betw,axiom,
! [H2: set_real > nat,S: set_set_real,T: set_nat,G: nat > nat] :
( ( bij_be7148990519944890154al_nat @ H2 @ S @ T )
=> ( ( groups3012202523422989166al_nat
@ ^ [X2: set_real] : ( G @ ( H2 @ X2 ) )
@ S )
= ( groups3542108847815614940at_nat @ G @ T ) ) ) ).
% sum.reindex_bij_betw
thf(fact_274_sum_Oreindex__bij__betw,axiom,
! [H2: set_real > set_real,S: set_set_real,T: set_set_real,G: set_real > real] :
( ( bij_be8850480603386830140t_real @ H2 @ S @ T )
=> ( ( groups8702937949983641418l_real
@ ^ [X2: set_real] : ( G @ ( H2 @ X2 ) )
@ S )
= ( groups8702937949983641418l_real @ G @ T ) ) ) ).
% sum.reindex_bij_betw
thf(fact_275_sum_Oreindex__bij__betw,axiom,
! [H2: set_real > nat,S: set_set_real,T: set_nat,G: nat > real] :
( ( bij_be7148990519944890154al_nat @ H2 @ S @ T )
=> ( ( groups8702937949983641418l_real
@ ^ [X2: set_real] : ( G @ ( H2 @ X2 ) )
@ S )
= ( groups6591440286371151544t_real @ G @ T ) ) ) ).
% sum.reindex_bij_betw
thf(fact_276_sum_Oreindex__bij__betw,axiom,
! [H2: nat > set_real,S: set_nat,T: set_set_real,G: set_real > real] :
( ( bij_be7200859108919812778t_real @ H2 @ S @ T )
=> ( ( groups6591440286371151544t_real
@ ^ [X2: nat] : ( G @ ( H2 @ X2 ) )
@ S )
= ( groups8702937949983641418l_real @ G @ T ) ) ) ).
% sum.reindex_bij_betw
thf(fact_277_sum_Oreindex__bij__betw,axiom,
! [H2: nat > nat,S: set_nat,T: set_nat,G: nat > real] :
( ( bij_betw_nat_nat @ H2 @ S @ T )
=> ( ( groups6591440286371151544t_real
@ ^ [X2: nat] : ( G @ ( H2 @ X2 ) )
@ S )
= ( groups6591440286371151544t_real @ G @ T ) ) ) ).
% sum.reindex_bij_betw
thf(fact_278_sum_Oreindex__bij__betw,axiom,
! [H2: nat > nat,S: set_nat,T: set_nat,G: nat > nat] :
( ( bij_betw_nat_nat @ H2 @ S @ T )
=> ( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( G @ ( H2 @ X2 ) )
@ S )
= ( groups3542108847815614940at_nat @ G @ T ) ) ) ).
% sum.reindex_bij_betw
thf(fact_279_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_280_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_281_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_282_sum_Ocong,axiom,
! [A2: set_set_real,B2: set_set_real,G: set_real > real,H2: set_real > real] :
( ( A2 = B2 )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ B2 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups8702937949983641418l_real @ G @ A2 )
= ( groups8702937949983641418l_real @ H2 @ B2 ) ) ) ) ).
% sum.cong
thf(fact_283_sum_Ocong,axiom,
! [A2: set_nat,B2: set_nat,G: nat > real,H2: nat > real] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= ( groups6591440286371151544t_real @ H2 @ B2 ) ) ) ) ).
% sum.cong
thf(fact_284_sum_Ocong,axiom,
! [A2: set_nat,B2: set_nat,G: nat > nat,H2: nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= ( groups3542108847815614940at_nat @ H2 @ B2 ) ) ) ) ).
% sum.cong
thf(fact_285_sum_Oeq__general,axiom,
! [B2: set_nat,A2: set_real,H2: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ( H2 @ X4 )
= Y3 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_nat @ ( H2 @ X3 ) @ B2 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_286_sum_Oeq__general,axiom,
! [B2: set_nat,A2: set_o,H2: $o > nat,Gamma: nat > real,Phi: $o > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ( H2 @ X4 )
= Y3 )
& ! [Ya: $o] :
( ( ( member_o @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( member_nat @ ( H2 @ X3 ) @ B2 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8691415230153176458o_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_287_sum_Oeq__general,axiom,
! [B2: set_nat,A2: set_real,H2: real > nat,Gamma: nat > nat,Phi: real > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ( H2 @ X4 )
= Y3 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_nat @ ( H2 @ X3 ) @ B2 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups1935376822645274424al_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_288_sum_Oeq__general,axiom,
! [B2: set_nat,A2: set_o,H2: $o > nat,Gamma: nat > nat,Phi: $o > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ( H2 @ X4 )
= Y3 )
& ! [Ya: $o] :
( ( ( member_o @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( member_nat @ ( H2 @ X3 ) @ B2 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8507830703676809646_o_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_289_sum_Oeq__general,axiom,
! [B2: set_real,A2: set_nat,H2: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H2 @ X4 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_real @ ( H2 @ X3 ) @ B2 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8097168146408367636l_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_290_sum_Oeq__general,axiom,
! [B2: set_o,A2: set_nat,H2: nat > $o,Gamma: $o > real,Phi: nat > real] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H2 @ X4 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_o @ ( H2 @ X3 ) @ B2 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8691415230153176458o_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_291_sum_Oeq__general,axiom,
! [B2: set_nat,A2: set_nat,H2: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H2 @ X4 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ ( H2 @ X3 ) @ B2 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_292_sum_Oeq__general,axiom,
! [B2: set_real,A2: set_nat,H2: nat > real,Gamma: real > nat,Phi: nat > nat] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H2 @ X4 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_real @ ( H2 @ X3 ) @ B2 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups1935376822645274424al_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_293_sum_Oeq__general,axiom,
! [B2: set_o,A2: set_nat,H2: nat > $o,Gamma: $o > nat,Phi: nat > nat] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H2 @ X4 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_o @ ( H2 @ X3 ) @ B2 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups8507830703676809646_o_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_294_sum_Oeq__general,axiom,
! [B2: set_nat,A2: set_nat,H2: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H2 @ X4 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ ( H2 @ X3 ) @ B2 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_295_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K2: nat > real,A2: set_real,H2: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ( ( member_real @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_nat @ ( H2 @ X3 ) @ B2 )
& ( ( K2 @ ( H2 @ X3 ) )
= X3 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_296_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K2: nat > $o,A2: set_o,H2: $o > nat,Gamma: nat > real,Phi: $o > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ( ( member_o @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( member_nat @ ( H2 @ X3 ) @ B2 )
& ( ( K2 @ ( H2 @ X3 ) )
= X3 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8691415230153176458o_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_297_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K2: nat > real,A2: set_real,H2: real > nat,Gamma: nat > nat,Phi: real > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ( ( member_real @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_nat @ ( H2 @ X3 ) @ B2 )
& ( ( K2 @ ( H2 @ X3 ) )
= X3 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups1935376822645274424al_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_298_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K2: nat > $o,A2: set_o,H2: $o > nat,Gamma: nat > nat,Phi: $o > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ( ( member_o @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( member_nat @ ( H2 @ X3 ) @ B2 )
& ( ( K2 @ ( H2 @ X3 ) )
= X3 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8507830703676809646_o_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_299_sum_Oeq__general__inverses,axiom,
! [B2: set_real,K2: real > nat,A2: set_nat,H2: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B2 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_real @ ( H2 @ X3 ) @ B2 )
& ( ( K2 @ ( H2 @ X3 ) )
= X3 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8097168146408367636l_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_300_sum_Oeq__general__inverses,axiom,
! [B2: set_o,K2: $o > nat,A2: set_nat,H2: nat > $o,Gamma: $o > real,Phi: nat > real] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ B2 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_o @ ( H2 @ X3 ) @ B2 )
& ( ( K2 @ ( H2 @ X3 ) )
= X3 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8691415230153176458o_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_301_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K2: nat > nat,A2: set_nat,H2: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ ( H2 @ X3 ) @ B2 )
& ( ( K2 @ ( H2 @ X3 ) )
= X3 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_302_sum_Oeq__general__inverses,axiom,
! [B2: set_real,K2: real > nat,A2: set_nat,H2: nat > real,Gamma: real > nat,Phi: nat > nat] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B2 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_real @ ( H2 @ X3 ) @ B2 )
& ( ( K2 @ ( H2 @ X3 ) )
= X3 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups1935376822645274424al_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_303_sum_Oeq__general__inverses,axiom,
! [B2: set_o,K2: $o > nat,A2: set_nat,H2: nat > $o,Gamma: $o > nat,Phi: nat > nat] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ B2 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_o @ ( H2 @ X3 ) @ B2 )
& ( ( K2 @ ( H2 @ X3 ) )
= X3 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups8507830703676809646_o_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_304_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K2: nat > nat,A2: set_nat,H2: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ ( H2 @ X3 ) @ B2 )
& ( ( K2 @ ( H2 @ X3 ) )
= X3 )
& ( ( Gamma @ ( H2 @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_305_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I: nat > real,J: real > nat,T: set_nat,H2: nat > real,G: real > real] :
( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_real @ ( I @ B3 ) @ S ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups6591440286371151544t_real @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_306_sum_Oreindex__bij__witness,axiom,
! [S: set_o,I: nat > $o,J: $o > nat,T: set_nat,H2: nat > real,G: $o > real] :
( ! [A3: $o] :
( ( member_o @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: $o] :
( ( member_o @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_o @ ( I @ B3 ) @ S ) )
=> ( ! [A3: $o] :
( ( member_o @ A3 @ S )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8691415230153176458o_real @ G @ S )
= ( groups6591440286371151544t_real @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_307_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I: nat > real,J: real > nat,T: set_nat,H2: nat > nat,G: real > nat] :
( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_real @ ( I @ B3 ) @ S ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_308_sum_Oreindex__bij__witness,axiom,
! [S: set_o,I: nat > $o,J: $o > nat,T: set_nat,H2: nat > nat,G: $o > nat] :
( ! [A3: $o] :
( ( member_o @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: $o] :
( ( member_o @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_o @ ( I @ B3 ) @ S ) )
=> ( ! [A3: $o] :
( ( member_o @ A3 @ S )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8507830703676809646_o_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_309_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: real > nat,J: nat > real,T: set_real,H2: real > real,G: nat > real] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_real @ ( J @ A3 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups8097168146408367636l_real @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_310_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: $o > nat,J: nat > $o,T: set_o,H2: $o > real,G: nat > real] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_o @ ( J @ A3 ) @ T ) )
=> ( ! [B3: $o] :
( ( member_o @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: $o] :
( ( member_o @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups8691415230153176458o_real @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_311_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: nat > nat,J: nat > nat,T: set_nat,H2: nat > real,G: nat > real] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups6591440286371151544t_real @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_312_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: real > nat,J: nat > real,T: set_real,H2: real > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_real @ ( J @ A3 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups1935376822645274424al_nat @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_313_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: $o > nat,J: nat > $o,T: set_o,H2: $o > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_o @ ( J @ A3 ) @ T ) )
=> ( ! [B3: $o] :
( ( member_o @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: $o] :
( ( member_o @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups8507830703676809646_o_nat @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_314_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: nat > nat,J: nat > nat,T: set_nat,H2: nat > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_315_sum_Oswap,axiom,
! [G: set_real > set_real > real,B2: set_set_real,A2: set_set_real] :
( ( groups8702937949983641418l_real
@ ^ [I2: set_real] : ( groups8702937949983641418l_real @ ( G @ I2 ) @ B2 )
@ A2 )
= ( groups8702937949983641418l_real
@ ^ [J2: set_real] :
( groups8702937949983641418l_real
@ ^ [I2: set_real] : ( G @ I2 @ J2 )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_316_sum_Oswap,axiom,
! [G: set_real > nat > real,B2: set_nat,A2: set_set_real] :
( ( groups8702937949983641418l_real
@ ^ [I2: set_real] : ( groups6591440286371151544t_real @ ( G @ I2 ) @ B2 )
@ A2 )
= ( groups6591440286371151544t_real
@ ^ [J2: nat] :
( groups8702937949983641418l_real
@ ^ [I2: set_real] : ( G @ I2 @ J2 )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_317_sum_Oswap,axiom,
! [G: nat > set_real > real,B2: set_set_real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I2: nat] : ( groups8702937949983641418l_real @ ( G @ I2 ) @ B2 )
@ A2 )
= ( groups8702937949983641418l_real
@ ^ [J2: set_real] :
( groups6591440286371151544t_real
@ ^ [I2: nat] : ( G @ I2 @ J2 )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_318_sum_Oswap,axiom,
! [G: nat > nat > real,B2: set_nat,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I2: nat] : ( groups6591440286371151544t_real @ ( G @ I2 ) @ B2 )
@ A2 )
= ( groups6591440286371151544t_real
@ ^ [J2: nat] :
( groups6591440286371151544t_real
@ ^ [I2: nat] : ( G @ I2 @ J2 )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_319_sum_Oswap,axiom,
! [G: nat > nat > nat,B2: set_nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( groups3542108847815614940at_nat @ ( G @ I2 ) @ B2 )
@ A2 )
= ( groups3542108847815614940at_nat
@ ^ [J2: nat] :
( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ I2 @ J2 )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_320_sum_Oneutral,axiom,
! [A2: set_set_real,G: set_real > real] :
( ! [X3: set_real] :
( ( member_set_real @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups8702937949983641418l_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_321_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_322_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_323_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A2: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_324_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: $o > real,A2: set_o] :
( ( ( groups8691415230153176458o_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A3: $o] :
( ( member_o @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_325_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A2: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_326_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: $o > nat,A2: set_o] :
( ( ( groups8507830703676809646_o_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A3: $o] :
( ( member_o @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_327_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > int,A2: set_real] :
( ( ( groups1932886352136224148al_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_328_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > int,A2: set_nat] :
( ( ( groups3539618377306564664at_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_329_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: $o > int,A2: set_o] :
( ( ( groups8505340233167759370_o_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A3: $o] :
( ( member_o @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_330_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A2: set_nat] :
( ( ( groups6591440286371151544t_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_331_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A2: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_332_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: set_real > nat,A2: set_set_real] :
( ( ( groups3012202523422989166al_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A3: set_real] :
( ( member_set_real @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_333_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_334_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_335_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_336_mult__of__nat__commute,axiom,
! [X: nat,Y2: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y2 )
= ( times_times_nat @ Y2 @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_337_mult__of__nat__commute,axiom,
! [X: nat,Y2: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y2 )
= ( times_times_real @ Y2 @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_338_mult__of__nat__commute,axiom,
! [X: nat,Y2: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y2 )
= ( times_times_int @ Y2 @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_339_sum__mono,axiom,
! [K: set_real,F: real > real,G: real > real] :
( ! [I3: real] :
( ( member_real @ I3 @ K )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ K ) @ ( groups8097168146408367636l_real @ G @ K ) ) ) ).
% sum_mono
thf(fact_340_sum__mono,axiom,
! [K: set_o,F: $o > real,G: $o > real] :
( ! [I3: $o] :
( ( member_o @ I3 @ K )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ K ) @ ( groups8691415230153176458o_real @ G @ K ) ) ) ).
% sum_mono
thf(fact_341_sum__mono,axiom,
! [K: set_real,F: real > nat,G: real > nat] :
( ! [I3: real] :
( ( member_real @ I3 @ K )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K ) @ ( groups1935376822645274424al_nat @ G @ K ) ) ) ).
% sum_mono
thf(fact_342_sum__mono,axiom,
! [K: set_o,F: $o > nat,G: $o > nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ K )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ K ) @ ( groups8507830703676809646_o_nat @ G @ K ) ) ) ).
% sum_mono
thf(fact_343_sum__mono,axiom,
! [K: set_real,F: real > int,G: real > int] :
( ! [I3: real] :
( ( member_real @ I3 @ K )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K ) @ ( groups1932886352136224148al_int @ G @ K ) ) ) ).
% sum_mono
thf(fact_344_sum__mono,axiom,
! [K: set_nat,F: nat > int,G: nat > int] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K ) @ ( groups3539618377306564664at_int @ G @ K ) ) ) ).
% sum_mono
thf(fact_345_sum__mono,axiom,
! [K: set_o,F: $o > int,G: $o > int] :
( ! [I3: $o] :
( ( member_o @ I3 @ K )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ K ) @ ( groups8505340233167759370_o_int @ G @ K ) ) ) ).
% sum_mono
thf(fact_346_sum__mono,axiom,
! [K: set_nat,F: nat > real,G: nat > real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K ) @ ( groups6591440286371151544t_real @ G @ K ) ) ) ).
% sum_mono
thf(fact_347_sum__mono,axiom,
! [K: set_nat,F: nat > nat,G: nat > nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K ) @ ( groups3542108847815614940at_nat @ G @ K ) ) ) ).
% sum_mono
thf(fact_348_sum__mono,axiom,
! [K: set_set_real,F: set_real > nat,G: set_real > nat] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ K )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups3012202523422989166al_nat @ F @ K ) @ ( groups3012202523422989166al_nat @ G @ K ) ) ) ).
% sum_mono
thf(fact_349_sum__distrib__left,axiom,
! [R: real,F: set_real > real,A2: set_set_real] :
( ( times_times_real @ R @ ( groups8702937949983641418l_real @ F @ A2 ) )
= ( groups8702937949983641418l_real
@ ^ [N4: set_real] : ( times_times_real @ R @ ( F @ N4 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_350_sum__distrib__left,axiom,
! [R: real,F: nat > real,A2: set_nat] :
( ( times_times_real @ R @ ( groups6591440286371151544t_real @ F @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [N4: nat] : ( times_times_real @ R @ ( F @ N4 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_351_sum__distrib__left,axiom,
! [R: nat,F: nat > nat,A2: set_nat] :
( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3542108847815614940at_nat
@ ^ [N4: nat] : ( times_times_nat @ R @ ( F @ N4 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_352_sum__distrib__right,axiom,
! [F: set_real > real,A2: set_set_real,R: real] :
( ( times_times_real @ ( groups8702937949983641418l_real @ F @ A2 ) @ R )
= ( groups8702937949983641418l_real
@ ^ [N4: set_real] : ( times_times_real @ ( F @ N4 ) @ R )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_353_sum__distrib__right,axiom,
! [F: nat > real,A2: set_nat,R: real] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R )
= ( groups6591440286371151544t_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ R )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_354_sum__distrib__right,axiom,
! [F: nat > nat,A2: set_nat,R: nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R )
= ( groups3542108847815614940at_nat
@ ^ [N4: nat] : ( times_times_nat @ ( F @ N4 ) @ R )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_355_sum__product,axiom,
! [F: set_real > real,A2: set_set_real,G: set_real > real,B2: set_set_real] :
( ( times_times_real @ ( groups8702937949983641418l_real @ F @ A2 ) @ ( groups8702937949983641418l_real @ G @ B2 ) )
= ( groups8702937949983641418l_real
@ ^ [I2: set_real] :
( groups8702937949983641418l_real
@ ^ [J2: set_real] : ( times_times_real @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A2 ) ) ).
% sum_product
thf(fact_356_sum__product,axiom,
! [F: set_real > real,A2: set_set_real,G: nat > real,B2: set_nat] :
( ( times_times_real @ ( groups8702937949983641418l_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B2 ) )
= ( groups8702937949983641418l_real
@ ^ [I2: set_real] :
( groups6591440286371151544t_real
@ ^ [J2: nat] : ( times_times_real @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A2 ) ) ).
% sum_product
thf(fact_357_sum__product,axiom,
! [F: nat > real,A2: set_nat,G: set_real > real,B2: set_set_real] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups8702937949983641418l_real @ G @ B2 ) )
= ( groups6591440286371151544t_real
@ ^ [I2: nat] :
( groups8702937949983641418l_real
@ ^ [J2: set_real] : ( times_times_real @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A2 ) ) ).
% sum_product
thf(fact_358_sum__product,axiom,
! [F: nat > real,A2: set_nat,G: nat > real,B2: set_nat] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B2 ) )
= ( groups6591440286371151544t_real
@ ^ [I2: nat] :
( groups6591440286371151544t_real
@ ^ [J2: nat] : ( times_times_real @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A2 ) ) ).
% sum_product
thf(fact_359_sum__product,axiom,
! [F: nat > nat,A2: set_nat,G: nat > nat,B2: set_nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B2 ) )
= ( groups3542108847815614940at_nat
@ ^ [I2: nat] :
( groups3542108847815614940at_nat
@ ^ [J2: nat] : ( times_times_nat @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A2 ) ) ).
% sum_product
thf(fact_360_sum__divide__distrib,axiom,
! [F: set_real > real,A2: set_set_real,R: real] :
( ( divide_divide_real @ ( groups8702937949983641418l_real @ F @ A2 ) @ R )
= ( groups8702937949983641418l_real
@ ^ [N4: set_real] : ( divide_divide_real @ ( F @ N4 ) @ R )
@ A2 ) ) ).
% sum_divide_distrib
thf(fact_361_sum__divide__distrib,axiom,
! [F: nat > real,A2: set_nat,R: real] :
( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R )
= ( groups6591440286371151544t_real
@ ^ [N4: nat] : ( divide_divide_real @ ( F @ N4 ) @ R )
@ A2 ) ) ).
% sum_divide_distrib
thf(fact_362_sum__nonneg,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_363_sum__nonneg,axiom,
! [A2: set_o,F: $o > real] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_364_sum__nonneg,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_365_sum__nonneg,axiom,
! [A2: set_o,F: $o > nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_366_sum__nonneg,axiom,
! [A2: set_real,F: real > int] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_367_sum__nonneg,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_368_sum__nonneg,axiom,
! [A2: set_o,F: $o > int] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups8505340233167759370_o_int @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_369_sum__nonneg,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_370_sum__nonneg,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_371_sum__nonneg,axiom,
! [A2: set_set_real,F: set_real > nat] :
( ! [X3: set_real] :
( ( member_set_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3012202523422989166al_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_372_sum__nonpos,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_373_sum__nonpos,axiom,
! [A2: set_o,F: $o > real] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_374_sum__nonpos,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_375_sum__nonpos,axiom,
! [A2: set_o,F: $o > nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_376_sum__nonpos,axiom,
! [A2: set_real,F: real > int] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_377_sum__nonpos,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_378_sum__nonpos,axiom,
! [A2: set_o,F: $o > int] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_379_sum__nonpos,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_380_sum__nonpos,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_381_sum__nonpos,axiom,
! [A2: set_set_real,F: set_real > nat] :
( ! [X3: set_real] :
( ( member_set_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3012202523422989166al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_382_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_383_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_384_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_385_yidx__gt,axiom,
! [Y2: real] :
( ( member_real @ Y2 @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) )
=> ( ord_less_real @ Y2 @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ ( yidx @ Y2 ) ) ) ) ) ) ) ).
% yidx_gt
thf(fact_386_int__f1__D,axiom,
! [K: set_real] :
( ( member_set_real @ K @ ( regular_division @ zero_zero_real @ a @ n ) )
=> ( hensto240673015341029504l_real @ f1 @ ( times_times_real @ ( f @ ( comple4887499456419720421f_real @ K ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) ) @ K ) ) ).
% int_f1_D
thf(fact_387_cInf__atLeastAtMost,axiom,
! [Y2: $o,X: $o] :
( ( ord_less_eq_o @ Y2 @ X )
=> ( ( complete_Inf_Inf_o @ ( set_or8904488021354931149Most_o @ Y2 @ X ) )
= Y2 ) ) ).
% cInf_atLeastAtMost
thf(fact_388_cInf__atLeastAtMost,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ( ( comple4887499456419720421f_real @ ( set_or1222579329274155063t_real @ Y2 @ X ) )
= Y2 ) ) ).
% cInf_atLeastAtMost
thf(fact_389_cInf__atLeastAtMost,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( complete_Inf_Inf_nat @ ( set_or1269000886237332187st_nat @ Y2 @ X ) )
= Y2 ) ) ).
% cInf_atLeastAtMost
thf(fact_390_cInf__atLeastAtMost,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ( ( complete_Inf_Inf_int @ ( set_or1266510415728281911st_int @ Y2 @ X ) )
= Y2 ) ) ).
% cInf_atLeastAtMost
thf(fact_391_Inf__atLeastAtMost,axiom,
! [X: $o,Y2: $o] :
( ( ord_less_eq_o @ X @ Y2 )
=> ( ( complete_Inf_Inf_o @ ( set_or8904488021354931149Most_o @ X @ Y2 ) )
= X ) ) ).
% Inf_atLeastAtMost
thf(fact_392_atLeastatMost__empty__iff,axiom,
! [A: set_real,B: set_real] :
( ( ( set_or7743017856606604397t_real @ A @ B )
= bot_bot_set_set_real )
= ( ~ ( ord_less_eq_set_real @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_393_atLeastatMost__empty__iff,axiom,
! [A: real,B: real] :
( ( ( set_or1222579329274155063t_real @ A @ B )
= bot_bot_set_real )
= ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_394_atLeastatMost__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_395_atLeastatMost__empty__iff,axiom,
! [A: int,B: int] :
( ( ( set_or1266510415728281911st_int @ A @ B )
= bot_bot_set_int )
= ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_396_atLeastatMost__empty__iff2,axiom,
! [A: set_real,B: set_real] :
( ( bot_bot_set_set_real
= ( set_or7743017856606604397t_real @ A @ B ) )
= ( ~ ( ord_less_eq_set_real @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_397_atLeastatMost__empty__iff2,axiom,
! [A: real,B: real] :
( ( bot_bot_set_real
= ( set_or1222579329274155063t_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_398_atLeastatMost__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_399_atLeastatMost__empty__iff2,axiom,
! [A: int,B: int] :
( ( bot_bot_set_int
= ( set_or1266510415728281911st_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_400_atLeastatMost__empty_H,axiom,
! [A: set_real,B: set_real] :
( ~ ( ord_less_eq_set_real @ A @ B )
=> ( ( set_or7743017856606604397t_real @ A @ B )
= bot_bot_set_set_real ) ) ).
% atLeastatMost_empty'
thf(fact_401_atLeastatMost__empty_H,axiom,
! [A: real,B: real] :
( ~ ( ord_less_eq_real @ A @ B )
=> ( ( set_or1222579329274155063t_real @ A @ B )
= bot_bot_set_real ) ) ).
% atLeastatMost_empty'
thf(fact_402_atLeastatMost__empty_H,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty'
thf(fact_403_atLeastatMost__empty_H,axiom,
! [A: int,B: int] :
( ~ ( ord_less_eq_int @ A @ B )
=> ( ( set_or1266510415728281911st_int @ A @ B )
= bot_bot_set_int ) ) ).
% atLeastatMost_empty'
thf(fact_404_sum__clauses_I1_J,axiom,
! [F: set_real > nat] :
( ( groups3012202523422989166al_nat @ F @ bot_bot_set_set_real )
= zero_zero_nat ) ).
% sum_clauses(1)
thf(fact_405_sum__clauses_I1_J,axiom,
! [F: set_real > int] :
( ( groups3009712052913938890al_int @ F @ bot_bot_set_set_real )
= zero_zero_int ) ).
% sum_clauses(1)
thf(fact_406_sum__clauses_I1_J,axiom,
! [F: nat > int] :
( ( groups3539618377306564664at_int @ F @ bot_bot_set_nat )
= zero_zero_int ) ).
% sum_clauses(1)
thf(fact_407_sum__clauses_I1_J,axiom,
! [F: real > real] :
( ( groups8097168146408367636l_real @ F @ bot_bot_set_real )
= zero_zero_real ) ).
% sum_clauses(1)
thf(fact_408_sum__clauses_I1_J,axiom,
! [F: real > nat] :
( ( groups1935376822645274424al_nat @ F @ bot_bot_set_real )
= zero_zero_nat ) ).
% sum_clauses(1)
thf(fact_409_sum__clauses_I1_J,axiom,
! [F: real > int] :
( ( groups1932886352136224148al_int @ F @ bot_bot_set_real )
= zero_zero_int ) ).
% sum_clauses(1)
thf(fact_410_sum__clauses_I1_J,axiom,
! [F: set_real > real] :
( ( groups8702937949983641418l_real @ F @ bot_bot_set_set_real )
= zero_zero_real ) ).
% sum_clauses(1)
thf(fact_411_sum__clauses_I1_J,axiom,
! [F: nat > real] :
( ( groups6591440286371151544t_real @ F @ bot_bot_set_nat )
= zero_zero_real ) ).
% sum_clauses(1)
thf(fact_412_sum__clauses_I1_J,axiom,
! [F: nat > nat] :
( ( groups3542108847815614940at_nat @ F @ bot_bot_set_nat )
= zero_zero_nat ) ).
% sum_clauses(1)
thf(fact_413_yidx__equality,axiom,
! [Y2: real,K2: nat] :
( ( member_real @ Y2 @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) )
=> ( ( member_real @ Y2 @ ( set_or66887138388493659n_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K2 ) ) ) @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) ) ) ) )
=> ( ( yidx @ Y2 )
= K2 ) ) ) ).
% yidx_equality
thf(fact_414_fim,axiom,
( ( image_real_real @ f @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
= ( set_or1222579329274155063t_real @ zero_zero_real @ b ) ) ).
% fim
thf(fact_415_that,axiom,
ord_less_real @ zero_zero_real @ epsilon ).
% that
thf(fact_416__092_060open_0620_A_060_Aa_092_060close_062,axiom,
ord_less_real @ zero_zero_real @ a ).
% \<open>0 < a\<close>
thf(fact_417__092_060open_0620_A_060_A_092_060delta_062_092_060close_062,axiom,
ord_less_real @ zero_zero_real @ delta ).
% \<open>0 < \<delta>\<close>
thf(fact_418_lessThan__eq__iff,axiom,
! [X: nat,Y2: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y2 ) )
= ( X = Y2 ) ) ).
% lessThan_eq_iff
thf(fact_419_Icc__eq__Icc,axiom,
! [L: real,H2: real,L2: real,H3: real] :
( ( ( set_or1222579329274155063t_real @ L @ H2 )
= ( set_or1222579329274155063t_real @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_real @ L @ H2 )
& ~ ( ord_less_eq_real @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_420_Icc__eq__Icc,axiom,
! [L: nat,H2: nat,L2: nat,H3: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H2 )
= ( set_or1269000886237332187st_nat @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_nat @ L @ H2 )
& ~ ( ord_less_eq_nat @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_421_Icc__eq__Icc,axiom,
! [L: int,H2: int,L2: int,H3: int] :
( ( ( set_or1266510415728281911st_int @ L @ H2 )
= ( set_or1266510415728281911st_int @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_int @ L @ H2 )
& ~ ( ord_less_eq_int @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_422_atLeastAtMost__iff,axiom,
! [I: set_real,L: set_real,U2: set_real] :
( ( member_set_real @ I @ ( set_or7743017856606604397t_real @ L @ U2 ) )
= ( ( ord_less_eq_set_real @ L @ I )
& ( ord_less_eq_set_real @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_423_atLeastAtMost__iff,axiom,
! [I: $o,L: $o,U2: $o] :
( ( member_o @ I @ ( set_or8904488021354931149Most_o @ L @ U2 ) )
= ( ( ord_less_eq_o @ L @ I )
& ( ord_less_eq_o @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_424_atLeastAtMost__iff,axiom,
! [I: real,L: real,U2: real] :
( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U2 ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_eq_real @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_425_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U2: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U2 ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_426_atLeastAtMost__iff,axiom,
! [I: int,L: int,U2: int] :
( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U2 ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_eq_int @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_427_atLeastatMost__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_428_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_429_atLeastatMost__subset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_430_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_431_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_432_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_433_lessThan__subset__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_434_lessThan__subset__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y2 ) )
= ( ord_less_eq_int @ X @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_435_lessThan__subset__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_436_ivl__subset,axiom,
! [I: real,J: real,M: real,N: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ I @ J ) @ ( set_or66887138388493659n_real @ M @ N ) )
= ( ( ord_less_eq_real @ J @ I )
| ( ( ord_less_eq_real @ M @ I )
& ( ord_less_eq_real @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_437_ivl__subset,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_438_ivl__subset,axiom,
! [I: int,J: int,M: int,N: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J ) @ ( set_or4662586982721622107an_int @ M @ N ) )
= ( ( ord_less_eq_int @ J @ I )
| ( ( ord_less_eq_int @ M @ I )
& ( ord_less_eq_int @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_439_lessThan__iff,axiom,
! [I: set_real,K2: set_real] :
( ( member_set_real @ I @ ( set_or3940062689191130623t_real @ K2 ) )
= ( ord_less_set_real @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_440_lessThan__iff,axiom,
! [I: $o,K2: $o] :
( ( member_o @ I @ ( set_ord_lessThan_o @ K2 ) )
= ( ord_less_o @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_441_lessThan__iff,axiom,
! [I: real,K2: real] :
( ( member_real @ I @ ( set_or5984915006950818249n_real @ K2 ) )
= ( ord_less_real @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_442_lessThan__iff,axiom,
! [I: int,K2: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K2 ) )
= ( ord_less_int @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_443_lessThan__iff,axiom,
! [I: nat,K2: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K2 ) )
= ( ord_less_nat @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_444_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_445_a__seg__less__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ ( a_seg @ X ) @ ( a_seg @ Y2 ) )
= ( ord_less_real @ X @ Y2 ) ) ).
% a_seg_less_iff
thf(fact_446_atLeastatMost__empty,axiom,
! [B: set_real,A: set_real] :
( ( ord_less_set_real @ B @ A )
=> ( ( set_or7743017856606604397t_real @ A @ B )
= bot_bot_set_set_real ) ) ).
% atLeastatMost_empty
thf(fact_447_atLeastatMost__empty,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( set_or1222579329274155063t_real @ A @ B )
= bot_bot_set_real ) ) ).
% atLeastatMost_empty
thf(fact_448_atLeastatMost__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_449_atLeastatMost__empty,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( set_or1266510415728281911st_int @ A @ B )
= bot_bot_set_int ) ) ).
% atLeastatMost_empty
thf(fact_450_atLeastLessThan__iff,axiom,
! [I: set_real,L: set_real,U2: set_real] :
( ( member_set_real @ I @ ( set_or5046967147999637905t_real @ L @ U2 ) )
= ( ( ord_less_eq_set_real @ L @ I )
& ( ord_less_set_real @ I @ U2 ) ) ) ).
% atLeastLessThan_iff
thf(fact_451_atLeastLessThan__iff,axiom,
! [I: $o,L: $o,U2: $o] :
( ( member_o @ I @ ( set_or7139685690850216873Than_o @ L @ U2 ) )
= ( ( ord_less_eq_o @ L @ I )
& ( ord_less_o @ I @ U2 ) ) ) ).
% atLeastLessThan_iff
thf(fact_452_atLeastLessThan__iff,axiom,
! [I: real,L: real,U2: real] :
( ( member_real @ I @ ( set_or66887138388493659n_real @ L @ U2 ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_real @ I @ U2 ) ) ) ).
% atLeastLessThan_iff
thf(fact_453_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U2: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U2 ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U2 ) ) ) ).
% atLeastLessThan_iff
thf(fact_454_atLeastLessThan__iff,axiom,
! [I: int,L: int,U2: int] :
( ( member_int @ I @ ( set_or4662586982721622107an_int @ L @ U2 ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_int @ I @ U2 ) ) ) ).
% atLeastLessThan_iff
thf(fact_455_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_456_atLeastLessThan__empty,axiom,
! [B: set_real,A: set_real] :
( ( ord_less_eq_set_real @ B @ A )
=> ( ( set_or5046967147999637905t_real @ A @ B )
= bot_bot_set_set_real ) ) ).
% atLeastLessThan_empty
thf(fact_457_atLeastLessThan__empty,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( set_or66887138388493659n_real @ A @ B )
= bot_bot_set_real ) ) ).
% atLeastLessThan_empty
thf(fact_458_atLeastLessThan__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( set_or4665077453230672383an_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_459_atLeastLessThan__empty,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( set_or4662586982721622107an_int @ A @ B )
= bot_bot_set_int ) ) ).
% atLeastLessThan_empty
thf(fact_460_atLeastLessThan__empty__iff2,axiom,
! [A: set_real,B: set_real] :
( ( bot_bot_set_set_real
= ( set_or5046967147999637905t_real @ A @ B ) )
= ( ~ ( ord_less_set_real @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_461_atLeastLessThan__empty__iff2,axiom,
! [A: real,B: real] :
( ( bot_bot_set_real
= ( set_or66887138388493659n_real @ A @ B ) )
= ( ~ ( ord_less_real @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_462_atLeastLessThan__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or4665077453230672383an_nat @ A @ B ) )
= ( ~ ( ord_less_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_463_atLeastLessThan__empty__iff2,axiom,
! [A: int,B: int] :
( ( bot_bot_set_int
= ( set_or4662586982721622107an_int @ A @ B ) )
= ( ~ ( ord_less_int @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_464_atLeastLessThan__empty__iff,axiom,
! [A: set_real,B: set_real] :
( ( ( set_or5046967147999637905t_real @ A @ B )
= bot_bot_set_set_real )
= ( ~ ( ord_less_set_real @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_465_atLeastLessThan__empty__iff,axiom,
! [A: real,B: real] :
( ( ( set_or66887138388493659n_real @ A @ B )
= bot_bot_set_real )
= ( ~ ( ord_less_real @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_466_atLeastLessThan__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_467_atLeastLessThan__empty__iff,axiom,
! [A: int,B: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= bot_bot_set_int )
= ( ~ ( ord_less_int @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_468_Inf__atLeastLessThan,axiom,
! [X: $o,Y2: $o] :
( ( ord_less_o @ X @ Y2 )
=> ( ( complete_Inf_Inf_o @ ( set_or7139685690850216873Than_o @ X @ Y2 ) )
= X ) ) ).
% Inf_atLeastLessThan
thf(fact_469_cInf__atLeastLessThan,axiom,
! [Y2: $o,X: $o] :
( ( ord_less_o @ Y2 @ X )
=> ( ( complete_Inf_Inf_o @ ( set_or7139685690850216873Than_o @ Y2 @ X ) )
= Y2 ) ) ).
% cInf_atLeastLessThan
thf(fact_470_cInf__atLeastLessThan,axiom,
! [Y2: real,X: real] :
( ( ord_less_real @ Y2 @ X )
=> ( ( comple4887499456419720421f_real @ ( set_or66887138388493659n_real @ Y2 @ X ) )
= Y2 ) ) ).
% cInf_atLeastLessThan
thf(fact_471_cInf__atLeastLessThan,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_nat @ Y2 @ X )
=> ( ( complete_Inf_Inf_nat @ ( set_or4665077453230672383an_nat @ Y2 @ X ) )
= Y2 ) ) ).
% cInf_atLeastLessThan
thf(fact_472_cInf__atLeastLessThan,axiom,
! [Y2: int,X: int] :
( ( ord_less_int @ Y2 @ X )
=> ( ( complete_Inf_Inf_int @ ( set_or4662586982721622107an_int @ Y2 @ X ) )
= Y2 ) ) ).
% cInf_atLeastLessThan
thf(fact_473_cINF__const,axiom,
! [A2: set_nat,C: int] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_int
@ ( image_nat_int
@ ^ [X2: nat] : C
@ A2 ) )
= C ) ) ).
% cINF_const
thf(fact_474_cINF__const,axiom,
! [A2: set_nat,C: real] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple4887499456419720421f_real
@ ( image_nat_real
@ ^ [X2: nat] : C
@ A2 ) )
= C ) ) ).
% cINF_const
thf(fact_475_cINF__const,axiom,
! [A2: set_real,C: real] :
( ( A2 != bot_bot_set_real )
=> ( ( comple4887499456419720421f_real
@ ( image_real_real
@ ^ [X2: real] : C
@ A2 ) )
= C ) ) ).
% cINF_const
thf(fact_476_cINF__const,axiom,
! [A2: set_nat,C: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X2: nat] : C
@ A2 ) )
= C ) ) ).
% cINF_const
thf(fact_477_cINF__const,axiom,
! [A2: set_real,C: nat] :
( ( A2 != bot_bot_set_real )
=> ( ( complete_Inf_Inf_nat
@ ( image_real_nat
@ ^ [X2: real] : C
@ A2 ) )
= C ) ) ).
% cINF_const
thf(fact_478_cINF__const,axiom,
! [A2: set_nat,C: $o] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [X2: nat] : C
@ A2 ) )
= C ) ) ).
% cINF_const
thf(fact_479_cINF__const,axiom,
! [A2: set_real,C: $o] :
( ( A2 != bot_bot_set_real )
=> ( ( complete_Inf_Inf_o
@ ( image_real_o
@ ^ [X2: real] : C
@ A2 ) )
= C ) ) ).
% cINF_const
thf(fact_480_cINF__const,axiom,
! [A2: set_nat,C: set_real] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple8289635161444856091t_real
@ ( image_nat_set_real
@ ^ [X2: nat] : C
@ A2 ) )
= C ) ) ).
% cINF_const
thf(fact_481_cINF__const,axiom,
! [A2: set_set_real,C: real] :
( ( A2 != bot_bot_set_set_real )
=> ( ( comple4887499456419720421f_real
@ ( image_set_real_real
@ ^ [X2: set_real] : C
@ A2 ) )
= C ) ) ).
% cINF_const
thf(fact_482_cINF__const,axiom,
! [A2: set_set_real,C: nat] :
( ( A2 != bot_bot_set_set_real )
=> ( ( complete_Inf_Inf_nat
@ ( image_set_real_nat
@ ^ [X2: set_real] : C
@ A2 ) )
= C ) ) ).
% cINF_const
thf(fact_483_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_484_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_485_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_486_f__iff_I1_J,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ ( f @ X ) @ ( f @ Y2 ) )
= ( ord_less_real @ X @ Y2 ) ) ) ) ).
% f_iff(1)
thf(fact_487_image__mult__atLeastAtMost,axiom,
! [D: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ( image_real_real @ ( times_times_real @ D ) @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ ( times_times_real @ D @ A ) @ ( times_times_real @ D @ B ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_488_image__divide__atLeastAtMost,axiom,
! [D: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ( image_real_real
@ ^ [C2: real] : ( divide_divide_real @ C2 @ D )
@ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ ( divide_divide_real @ A @ D ) @ ( divide_divide_real @ B @ D ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_489_yidx__def,axiom,
( yidx
= ( ^ [Y: real] :
( ord_Least_nat
@ ^ [K3: nat] : ( ord_less_real @ Y @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K3 ) ) ) ) ) ) ) ) ).
% yidx_def
thf(fact_490_del__gt0,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_real @ zero_zero_real @ ( del @ E ) ) ) ).
% del_gt0
thf(fact_491_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_492_times__int__code_I1_J,axiom,
! [K2: int] :
( ( times_times_int @ K2 @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_493_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_494_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_495_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_496_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_497_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_498_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_499_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_500_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_501_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_502_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_503_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_504_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_505_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_506_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_507_atLeastLessThan__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A @ B ) @ ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_eq_real @ B @ A )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_508_atLeastLessThan__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_509_atLeastLessThan__subset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_eq_int @ B @ A )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_510_atLeastLessThan__inj_I2_J,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_511_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_512_atLeastLessThan__inj_I2_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_513_atLeastLessThan__inj_I1_J,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_514_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_515_atLeastLessThan__inj_I1_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_516_Ico__eq__Ico,axiom,
! [L: real,H2: real,L2: real,H3: real] :
( ( ( set_or66887138388493659n_real @ L @ H2 )
= ( set_or66887138388493659n_real @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_real @ L @ H2 )
& ~ ( ord_less_real @ L2 @ H3 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_517_Ico__eq__Ico,axiom,
! [L: nat,H2: nat,L2: nat,H3: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H2 )
= ( set_or4665077453230672383an_nat @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_nat @ L @ H2 )
& ~ ( ord_less_nat @ L2 @ H3 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_518_Ico__eq__Ico,axiom,
! [L: int,H2: int,L2: int,H3: int] :
( ( ( set_or4662586982721622107an_int @ L @ H2 )
= ( set_or4662586982721622107an_int @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_int @ L @ H2 )
& ~ ( ord_less_int @ L2 @ H3 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_519_ex__gt__or__lt,axiom,
! [A: real] :
? [B3: real] :
( ( ord_less_real @ A @ B3 )
| ( ord_less_real @ B3 @ A ) ) ).
% ex_gt_or_lt
thf(fact_520_atLeastLessThan__eq__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_521_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_522_atLeastLessThan__eq__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_523_lessThan__strict__subset__iff,axiom,
! [M: real,N: real] :
( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
= ( ord_less_real @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_524_lessThan__strict__subset__iff,axiom,
! [M: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_525_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_526_div__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) ).
% div_mult2_eq
thf(fact_527_linorder__neqE__linordered__idom,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_528_linorder__neqE__linordered__idom,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
=> ( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_529_linordered__field__no__ub,axiom,
! [X4: real] :
? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_530_linordered__field__no__lb,axiom,
! [X4: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X4 ) ).
% linordered_field_no_lb
thf(fact_531_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
& ~ ( ord_less_eq_real @ Y @ X2 ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_532_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: real] :
( ( ord_less_eq_real @ A @ C3 )
& ( ord_less_eq_real @ C3 @ B )
& ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_real @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D2: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_533_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D2: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_534_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: int] :
( ( ord_less_eq_int @ A @ C3 )
& ( ord_less_eq_int @ C3 @ B )
& ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D2: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_535_sum_Oivl__cong,axiom,
! [A: set_real,C: set_real,B: set_real,D: set_real,G: set_real > real,H2: set_real > real] :
( ( A = C )
=> ( ( B = D )
=> ( ! [X3: set_real] :
( ( ord_less_eq_set_real @ C @ X3 )
=> ( ( ord_less_set_real @ X3 @ D )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) ) )
=> ( ( groups8702937949983641418l_real @ G @ ( set_or5046967147999637905t_real @ A @ B ) )
= ( groups8702937949983641418l_real @ H2 @ ( set_or5046967147999637905t_real @ C @ D ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_536_sum_Oivl__cong,axiom,
! [A: nat,C: nat,B: nat,D: nat,G: nat > real,H2: nat > real] :
( ( A = C )
=> ( ( B = D )
=> ( ! [X3: nat] :
( ( ord_less_eq_nat @ C @ X3 )
=> ( ( ord_less_nat @ X3 @ D )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
= ( groups6591440286371151544t_real @ H2 @ ( set_or4665077453230672383an_nat @ C @ D ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_537_sum_Oivl__cong,axiom,
! [A: nat,C: nat,B: nat,D: nat,G: nat > nat,H2: nat > nat] :
( ( A = C )
=> ( ( B = D )
=> ( ! [X3: nat] :
( ( ord_less_eq_nat @ C @ X3 )
=> ( ( ord_less_nat @ X3 @ D )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
= ( groups3542108847815614940at_nat @ H2 @ ( set_or4665077453230672383an_nat @ C @ D ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_538_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_539_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or66887138388493659n_real @ C @ D ) )
= ( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_real @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_540_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_nat @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_541_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D ) )
= ( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ A )
& ( ord_less_int @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_542_atLeastatMost__psubset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D )
& ( ( ord_less_real @ C @ A )
| ( ord_less_real @ B @ D ) ) ) )
& ( ord_less_eq_real @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_543_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D )
& ( ( ord_less_nat @ C @ A )
| ( ord_less_nat @ B @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_544_atLeastatMost__psubset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
= ( ( ~ ( ord_less_eq_int @ A @ B )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D )
& ( ( ord_less_int @ C @ A )
| ( ord_less_int @ B @ D ) ) ) )
& ( ord_less_eq_int @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_545_lessThan__def,axiom,
( set_or5984915006950818249n_real
= ( ^ [U: real] :
( collect_real
@ ^ [X2: real] : ( ord_less_real @ X2 @ U ) ) ) ) ).
% lessThan_def
thf(fact_546_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U: int] :
( collect_int
@ ^ [X2: int] : ( ord_less_int @ X2 @ U ) ) ) ) ).
% lessThan_def
thf(fact_547_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ X2 @ U ) ) ) ) ).
% lessThan_def
thf(fact_548_cInf__lessD,axiom,
! [X5: set_int,Z: int] :
( ( X5 != bot_bot_set_int )
=> ( ( ord_less_int @ ( complete_Inf_Inf_int @ X5 ) @ Z )
=> ? [X3: int] :
( ( member_int @ X3 @ X5 )
& ( ord_less_int @ X3 @ Z ) ) ) ) ).
% cInf_lessD
thf(fact_549_cInf__lessD,axiom,
! [X5: set_real,Z: real] :
( ( X5 != bot_bot_set_real )
=> ( ( ord_less_real @ ( comple4887499456419720421f_real @ X5 ) @ Z )
=> ? [X3: real] :
( ( member_real @ X3 @ X5 )
& ( ord_less_real @ X3 @ Z ) ) ) ) ).
% cInf_lessD
thf(fact_550_cInf__lessD,axiom,
! [X5: set_nat,Z: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( complete_Inf_Inf_nat @ X5 ) @ Z )
=> ? [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( ord_less_nat @ X3 @ Z ) ) ) ) ).
% cInf_lessD
thf(fact_551_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_552_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > real,M: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups6591440286371151544t_real
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_553_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_554_image__mult__atLeastAtMost__if,axiom,
! [C: real,X: real,Y2: real] :
( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( image_real_real @ ( times_times_real @ C ) @ ( set_or1222579329274155063t_real @ X @ Y2 ) )
= ( set_or1222579329274155063t_real @ ( times_times_real @ C @ X ) @ ( times_times_real @ C @ Y2 ) ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_eq_real @ X @ Y2 )
=> ( ( image_real_real @ ( times_times_real @ C ) @ ( set_or1222579329274155063t_real @ X @ Y2 ) )
= ( set_or1222579329274155063t_real @ ( times_times_real @ C @ Y2 ) @ ( times_times_real @ C @ X ) ) ) )
& ( ~ ( ord_less_eq_real @ X @ Y2 )
=> ( ( image_real_real @ ( times_times_real @ C ) @ ( set_or1222579329274155063t_real @ X @ Y2 ) )
= bot_bot_set_real ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_555_cINF__greatest,axiom,
! [A2: set_o,M: int,F: $o > int] :
( ( A2 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_int @ M @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ M @ ( complete_Inf_Inf_int @ ( image_o_int @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_556_cINF__greatest,axiom,
! [A2: set_nat,M: int,F: nat > int] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_int @ M @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ M @ ( complete_Inf_Inf_int @ ( image_nat_int @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_557_cINF__greatest,axiom,
! [A2: set_real,M: int,F: real > int] :
( ( A2 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_int @ M @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ M @ ( complete_Inf_Inf_int @ ( image_real_int @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_558_cINF__greatest,axiom,
! [A2: set_o,M: real,F: $o > real] :
( ( A2 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_real @ M @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ M @ ( comple4887499456419720421f_real @ ( image_o_real @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_559_cINF__greatest,axiom,
! [A2: set_nat,M: real,F: nat > real] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_real @ M @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ M @ ( comple4887499456419720421f_real @ ( image_nat_real @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_560_cINF__greatest,axiom,
! [A2: set_real,M: real,F: real > real] :
( ( A2 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ M @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ M @ ( comple4887499456419720421f_real @ ( image_real_real @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_561_cINF__greatest,axiom,
! [A2: set_o,M: nat,F: $o > nat] :
( ( A2 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_nat @ M @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ M @ ( complete_Inf_Inf_nat @ ( image_o_nat @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_562_cINF__greatest,axiom,
! [A2: set_nat,M: nat,F: nat > nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ M @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ M @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_563_cINF__greatest,axiom,
! [A2: set_real,M: nat,F: real > nat] :
( ( A2 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ M @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ M @ ( complete_Inf_Inf_nat @ ( image_real_nat @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_564_cINF__greatest,axiom,
! [A2: set_o,M: $o,F: $o > $o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ M @ ( F @ X3 ) ) )
=> ( ord_less_eq_o @ M @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_565_image__mult__atLeastAtMost__if_H,axiom,
! [X: real,Y2: real,C: real] :
( ( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( image_real_real
@ ^ [X2: real] : ( times_times_real @ X2 @ C )
@ ( set_or1222579329274155063t_real @ X @ Y2 ) )
= ( set_or1222579329274155063t_real @ ( times_times_real @ X @ C ) @ ( times_times_real @ Y2 @ C ) ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( image_real_real
@ ^ [X2: real] : ( times_times_real @ X2 @ C )
@ ( set_or1222579329274155063t_real @ X @ Y2 ) )
= ( set_or1222579329274155063t_real @ ( times_times_real @ Y2 @ C ) @ ( times_times_real @ X @ C ) ) ) ) ) )
& ( ~ ( ord_less_eq_real @ X @ Y2 )
=> ( ( image_real_real
@ ^ [X2: real] : ( times_times_real @ X2 @ C )
@ ( set_or1222579329274155063t_real @ X @ Y2 ) )
= bot_bot_set_real ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_566_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_567_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_568_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_569_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_570_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_571_mult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_572_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_573_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_574_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_575_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_576_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_577_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_578_mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_579_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_580_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_581_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_582_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_583_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_584_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_585_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_586_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_587_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_588_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_589_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_590_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_591_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_592_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_593_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_594_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_595_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_596_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_597_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_598_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_599_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_600_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_601_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_602_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_603_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_604_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_605_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_606_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_607_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_608_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_609_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_610_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_611_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_612_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_613_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_614_divide__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_615_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_616_divide__less__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_617_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_618_divide__pos__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_pos_pos
thf(fact_619_divide__pos__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_620_divide__neg__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_621_divide__neg__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_neg_neg
thf(fact_622_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_623_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_624_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_625_sum__shift__lb__Suc0__0,axiom,
! [F: nat > int,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_int )
=> ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_626_sum__shift__lb__Suc0__0,axiom,
! [F: nat > real,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_627_sum__shift__lb__Suc0__0,axiom,
! [F: nat > nat,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_628_mult__less__le__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_629_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_630_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_631_mult__le__less__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_632_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_633_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_634_mult__right__le__imp__le,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_635_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_636_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_637_mult__left__le__imp__le,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_638_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_639_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_640_mult__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_641_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_642_mult__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_643_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_644_mult__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_645_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_646_mult__strict__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_647_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_648_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_649_mult__right__less__imp__less,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_650_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_651_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_652_mult__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_653_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_654_mult__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_655_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_656_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_657_mult__left__less__imp__less,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_658_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_659_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_660_mult__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_661_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_662_mult__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_663_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_664_divide__nonpos__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_665_divide__nonpos__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonpos_neg
thf(fact_666_divide__nonneg__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonneg_pos
thf(fact_667_divide__nonneg__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_668_divide__le__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_669_frac__less2,axiom,
! [X: real,Y2: real,W: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_real @ W @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% frac_less2
thf(fact_670_frac__less,axiom,
! [X: real,Y2: real,W: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% frac_less
thf(fact_671_frac__le,axiom,
! [Y2: real,X: real,W: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% frac_le
thf(fact_672_divide__strict__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_673_divide__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_674_mult__imp__less__div__pos,axiom,
! [Y2: real,Z: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ ( times_times_real @ Z @ Y2 ) @ X )
=> ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_675_mult__imp__div__pos__less,axiom,
! [Y2: real,X: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y2 ) )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_676_pos__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_677_pos__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_678_neg__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_679_neg__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_680_less__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_681_divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_682_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_683_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_684_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_685_wellorder__InfI,axiom,
! [K2: nat,A2: set_nat] :
( ( member_nat @ K2 @ A2 )
=> ( member_nat @ ( complete_Inf_Inf_nat @ A2 ) @ A2 ) ) ).
% wellorder_InfI
thf(fact_686_divide__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_687_mult__imp__le__div__pos,axiom,
! [Y2: real,Z: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y2 ) @ X )
=> ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_688_mult__imp__div__pos__le,axiom,
! [Y2: real,X: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y2 ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ Z ) ) ) ).
% mult_imp_div_pos_le
thf(fact_689_pos__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_le_divide_eq
thf(fact_690_pos__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_le_eq
thf(fact_691_neg__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_le_divide_eq
thf(fact_692_neg__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_le_eq
thf(fact_693_divide__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono
thf(fact_694_le__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_695_divide__le__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_696_wellorder__Inf__le1,axiom,
! [K2: nat,A2: set_nat] :
( ( member_nat @ K2 @ A2 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A2 ) @ K2 ) ) ).
% wellorder_Inf_le1
thf(fact_697_cInf__eq,axiom,
! [X5: set_int,A: int] :
( ! [X3: int] :
( ( member_int @ X3 @ X5 )
=> ( ord_less_eq_int @ A @ X3 ) )
=> ( ! [Y3: int] :
( ! [X4: int] :
( ( member_int @ X4 @ X5 )
=> ( ord_less_eq_int @ Y3 @ X4 ) )
=> ( ord_less_eq_int @ Y3 @ A ) )
=> ( ( complete_Inf_Inf_int @ X5 )
= A ) ) ) ).
% cInf_eq
thf(fact_698_cInf__eq,axiom,
! [X5: set_real,A: real] :
( ! [X3: real] :
( ( member_real @ X3 @ X5 )
=> ( ord_less_eq_real @ A @ X3 ) )
=> ( ! [Y3: real] :
( ! [X4: real] :
( ( member_real @ X4 @ X5 )
=> ( ord_less_eq_real @ Y3 @ X4 ) )
=> ( ord_less_eq_real @ Y3 @ A ) )
=> ( ( comple4887499456419720421f_real @ X5 )
= A ) ) ) ).
% cInf_eq
thf(fact_699_cInf__eq,axiom,
! [X5: set_nat,A: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ A @ X3 ) )
=> ( ! [Y3: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ Y3 @ X4 ) )
=> ( ord_less_eq_nat @ Y3 @ A ) )
=> ( ( complete_Inf_Inf_nat @ X5 )
= A ) ) ) ).
% cInf_eq
thf(fact_700_cInf__eq__minimum,axiom,
! [Z: set_real,X5: set_set_real] :
( ( member_set_real @ Z @ X5 )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ X5 )
=> ( ord_less_eq_set_real @ Z @ X3 ) )
=> ( ( comple8289635161444856091t_real @ X5 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_701_cInf__eq__minimum,axiom,
! [Z: int,X5: set_int] :
( ( member_int @ Z @ X5 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X5 )
=> ( ord_less_eq_int @ Z @ X3 ) )
=> ( ( complete_Inf_Inf_int @ X5 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_702_cInf__eq__minimum,axiom,
! [Z: real,X5: set_real] :
( ( member_real @ Z @ X5 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X5 )
=> ( ord_less_eq_real @ Z @ X3 ) )
=> ( ( comple4887499456419720421f_real @ X5 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_703_cInf__eq__minimum,axiom,
! [Z: nat,X5: set_nat] :
( ( member_nat @ Z @ X5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ Z @ X3 ) )
=> ( ( complete_Inf_Inf_nat @ X5 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_704_cInf__eq__minimum,axiom,
! [Z: $o,X5: set_o] :
( ( member_o @ Z @ X5 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X5 )
=> ( ord_less_eq_o @ Z @ X3 ) )
=> ( ( complete_Inf_Inf_o @ X5 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_705_lessThan__non__empty,axiom,
! [X: real] :
( ( set_or5984915006950818249n_real @ X )
!= bot_bot_set_real ) ).
% lessThan_non_empty
thf(fact_706_cInf__greatest,axiom,
! [X5: set_set_real,Z: set_real] :
( ( X5 != bot_bot_set_set_real )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ X5 )
=> ( ord_less_eq_set_real @ Z @ X3 ) )
=> ( ord_less_eq_set_real @ Z @ ( comple8289635161444856091t_real @ X5 ) ) ) ) ).
% cInf_greatest
thf(fact_707_cInf__greatest,axiom,
! [X5: set_int,Z: int] :
( ( X5 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X5 )
=> ( ord_less_eq_int @ Z @ X3 ) )
=> ( ord_less_eq_int @ Z @ ( complete_Inf_Inf_int @ X5 ) ) ) ) ).
% cInf_greatest
thf(fact_708_cInf__greatest,axiom,
! [X5: set_real,Z: real] :
( ( X5 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X5 )
=> ( ord_less_eq_real @ Z @ X3 ) )
=> ( ord_less_eq_real @ Z @ ( comple4887499456419720421f_real @ X5 ) ) ) ) ).
% cInf_greatest
thf(fact_709_cInf__greatest,axiom,
! [X5: set_nat,Z: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ Z @ X3 ) )
=> ( ord_less_eq_nat @ Z @ ( complete_Inf_Inf_nat @ X5 ) ) ) ) ).
% cInf_greatest
thf(fact_710_cInf__greatest,axiom,
! [X5: set_o,Z: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X5 )
=> ( ord_less_eq_o @ Z @ X3 ) )
=> ( ord_less_eq_o @ Z @ ( complete_Inf_Inf_o @ X5 ) ) ) ) ).
% cInf_greatest
thf(fact_711_cInf__eq__non__empty,axiom,
! [X5: set_set_real,A: set_real] :
( ( X5 != bot_bot_set_set_real )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ X5 )
=> ( ord_less_eq_set_real @ A @ X3 ) )
=> ( ! [Y3: set_real] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ X5 )
=> ( ord_less_eq_set_real @ Y3 @ X4 ) )
=> ( ord_less_eq_set_real @ Y3 @ A ) )
=> ( ( comple8289635161444856091t_real @ X5 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_712_cInf__eq__non__empty,axiom,
! [X5: set_int,A: int] :
( ( X5 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X5 )
=> ( ord_less_eq_int @ A @ X3 ) )
=> ( ! [Y3: int] :
( ! [X4: int] :
( ( member_int @ X4 @ X5 )
=> ( ord_less_eq_int @ Y3 @ X4 ) )
=> ( ord_less_eq_int @ Y3 @ A ) )
=> ( ( complete_Inf_Inf_int @ X5 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_713_cInf__eq__non__empty,axiom,
! [X5: set_real,A: real] :
( ( X5 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X5 )
=> ( ord_less_eq_real @ A @ X3 ) )
=> ( ! [Y3: real] :
( ! [X4: real] :
( ( member_real @ X4 @ X5 )
=> ( ord_less_eq_real @ Y3 @ X4 ) )
=> ( ord_less_eq_real @ Y3 @ A ) )
=> ( ( comple4887499456419720421f_real @ X5 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_714_cInf__eq__non__empty,axiom,
! [X5: set_nat,A: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ A @ X3 ) )
=> ( ! [Y3: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ Y3 @ X4 ) )
=> ( ord_less_eq_nat @ Y3 @ A ) )
=> ( ( complete_Inf_Inf_nat @ X5 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_715_cInf__eq__non__empty,axiom,
! [X5: set_o,A: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X5 )
=> ( ord_less_eq_o @ A @ X3 ) )
=> ( ! [Y3: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ X5 )
=> ( ord_less_eq_o @ Y3 @ X4 ) )
=> ( ord_less_eq_o @ Y3 @ A ) )
=> ( ( complete_Inf_Inf_o @ X5 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_716_Iio__eq__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_717_int__f2__D,axiom,
! [K: set_real] :
( ( member_set_real @ K @ ( regular_division @ zero_zero_real @ a @ n ) )
=> ( hensto240673015341029504l_real @ f2 @ ( times_times_real @ ( f @ ( comple1385675409528146559p_real @ K ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) ) @ K ) ) ).
% int_f2_D
thf(fact_718_ccINF__const,axiom,
! [A2: set_nat,F: set_real] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple8289635161444856091t_real
@ ( image_nat_set_real
@ ^ [I2: nat] : F
@ A2 ) )
= F ) ) ).
% ccINF_const
thf(fact_719_ccINF__const,axiom,
! [A2: set_set_real,F: $o] :
( ( A2 != bot_bot_set_set_real )
=> ( ( complete_Inf_Inf_o
@ ( image_set_real_o
@ ^ [I2: set_real] : F
@ A2 ) )
= F ) ) ).
% ccINF_const
thf(fact_720_ccINF__const,axiom,
! [A2: set_nat,F: $o] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I2: nat] : F
@ A2 ) )
= F ) ) ).
% ccINF_const
thf(fact_721_ccINF__const,axiom,
! [A2: set_real,F: $o] :
( ( A2 != bot_bot_set_real )
=> ( ( complete_Inf_Inf_o
@ ( image_real_o
@ ^ [I2: real] : F
@ A2 ) )
= F ) ) ).
% ccINF_const
thf(fact_722_INF__const,axiom,
! [A2: set_nat,F: set_real] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple8289635161444856091t_real
@ ( image_nat_set_real
@ ^ [I2: nat] : F
@ A2 ) )
= F ) ) ).
% INF_const
thf(fact_723_INF__const,axiom,
! [A2: set_set_real,F: $o] :
( ( A2 != bot_bot_set_set_real )
=> ( ( complete_Inf_Inf_o
@ ( image_set_real_o
@ ^ [I2: set_real] : F
@ A2 ) )
= F ) ) ).
% INF_const
thf(fact_724_INF__const,axiom,
! [A2: set_nat,F: $o] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I2: nat] : F
@ A2 ) )
= F ) ) ).
% INF_const
thf(fact_725_INF__const,axiom,
! [A2: set_real,F: $o] :
( ( A2 != bot_bot_set_real )
=> ( ( complete_Inf_Inf_o
@ ( image_real_o
@ ^ [I2: real] : F
@ A2 ) )
= F ) ) ).
% INF_const
thf(fact_726_has__integral__empty__eq,axiom,
! [F: real > real,I: real] :
( ( hensto240673015341029504l_real @ F @ I @ bot_bot_set_real )
= ( I = zero_zero_real ) ) ).
% has_integral_empty_eq
thf(fact_727_has__integral__empty,axiom,
! [F: real > real] : ( hensto240673015341029504l_real @ F @ zero_zero_real @ bot_bot_set_real ) ).
% has_integral_empty
thf(fact_728_has__integral__0__eq,axiom,
! [I: real,S: set_real] :
( ( hensto240673015341029504l_real
@ ^ [X2: real] : zero_zero_real
@ I
@ S )
= ( I = zero_zero_real ) ) ).
% has_integral_0_eq
thf(fact_729__092_060open_0620_A_060_An_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% \<open>0 < n\<close>
thf(fact_730_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_731_bij__betw__Suc,axiom,
! [M5: set_nat,N5: set_nat] :
( ( bij_betw_nat_nat @ suc @ M5 @ N5 )
= ( ( image_nat_nat @ suc @ M5 )
= N5 ) ) ).
% bij_betw_Suc
thf(fact_732_bij__betw__of__nat,axiom,
! [N5: set_nat,A2: set_nat] :
( ( bij_betw_nat_nat @ semiri1316708129612266289at_nat @ N5 @ A2 )
= ( ( image_nat_nat @ semiri1316708129612266289at_nat @ N5 )
= A2 ) ) ).
% bij_betw_of_nat
thf(fact_733_bij__betw__of__nat,axiom,
! [N5: set_nat,A2: set_real] :
( ( bij_betw_nat_real @ semiri5074537144036343181t_real @ N5 @ A2 )
= ( ( image_nat_real @ semiri5074537144036343181t_real @ N5 )
= A2 ) ) ).
% bij_betw_of_nat
thf(fact_734_bij__betw__of__nat,axiom,
! [N5: set_nat,A2: set_int] :
( ( bij_betw_nat_int @ semiri1314217659103216013at_int @ N5 @ A2 )
= ( ( image_nat_int @ semiri1314217659103216013at_int @ N5 )
= A2 ) ) ).
% bij_betw_of_nat
thf(fact_735_yidx__less__n,axiom,
! [Y2: real] :
( ( ord_less_real @ Y2 @ b )
=> ( ord_less_nat @ ( yidx @ Y2 ) @ n ) ) ).
% yidx_less_n
thf(fact_736_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_set_real] :
( ( bot_bot_set_set_real
= ( comple5917660045593844715t_real @ A2 ) )
= ( ! [X2: set_set_real] :
( ( member_set_set_real @ X2 @ A2 )
=> ( X2 = bot_bot_set_set_real ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_737_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_738_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_real] :
( ( bot_bot_set_real
= ( comple3096694443085538997t_real @ A2 ) )
= ( ! [X2: set_real] :
( ( member_set_real @ X2 @ A2 )
=> ( X2 = bot_bot_set_real ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_739_Sup__bot__conv_I2_J,axiom,
! [A2: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ A2 ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( X2 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_740_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_set_real] :
( ( ( comple5917660045593844715t_real @ A2 )
= bot_bot_set_set_real )
= ( ! [X2: set_set_real] :
( ( member_set_set_real @ X2 @ A2 )
=> ( X2 = bot_bot_set_set_real ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_741_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_742_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_real] :
( ( ( comple3096694443085538997t_real @ A2 )
= bot_bot_set_real )
= ( ! [X2: set_real] :
( ( member_set_real @ X2 @ A2 )
=> ( X2 = bot_bot_set_real ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_743_Sup__bot__conv_I1_J,axiom,
! [A2: set_o] :
( ( ( complete_Sup_Sup_o @ A2 )
= bot_bot_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( X2 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_744_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_745_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_746_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_747_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_748_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_749_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_750_cSup__lessThan,axiom,
! [X: real] :
( ( comple1385675409528146559p_real @ ( set_or5984915006950818249n_real @ X ) )
= X ) ).
% cSup_lessThan
thf(fact_751_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
= ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_752_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= zero_zero_nat ) ) ).
% Least_eq_0
thf(fact_753_SUP__identity__eq,axiom,
! [A2: set_real] :
( ( comple1385675409528146559p_real
@ ( image_real_real
@ ^ [X2: real] : X2
@ A2 ) )
= ( comple1385675409528146559p_real @ A2 ) ) ).
% SUP_identity_eq
thf(fact_754_SUP__identity__eq,axiom,
! [A2: set_set_real] :
( ( comple3096694443085538997t_real
@ ( image_2436557299294012491t_real
@ ^ [X2: set_real] : X2
@ A2 ) )
= ( comple3096694443085538997t_real @ A2 ) ) ).
% SUP_identity_eq
thf(fact_755_SUP__identity__eq,axiom,
! [A2: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( complete_Sup_Sup_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_756_SUP__identity__eq,axiom,
! [A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X2: $o] : X2
@ A2 ) )
= ( complete_Sup_Sup_o @ A2 ) ) ).
% SUP_identity_eq
thf(fact_757_INF__identity__eq,axiom,
! [A2: set_real] :
( ( comple4887499456419720421f_real
@ ( image_real_real
@ ^ [X2: real] : X2
@ A2 ) )
= ( comple4887499456419720421f_real @ A2 ) ) ).
% INF_identity_eq
thf(fact_758_INF__identity__eq,axiom,
! [A2: set_nat] :
( ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( complete_Inf_Inf_nat @ A2 ) ) ).
% INF_identity_eq
thf(fact_759_INF__identity__eq,axiom,
! [A2: set_o] :
( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [X2: $o] : X2
@ A2 ) )
= ( complete_Inf_Inf_o @ A2 ) ) ).
% INF_identity_eq
thf(fact_760_an__less__del,axiom,
ord_less_real @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) @ ( del @ ( divide_divide_real @ epsilon @ a ) ) ).
% an_less_del
thf(fact_761_ccSup__empty,axiom,
( ( comple5917660045593844715t_real @ bot_bo3378928929837779682t_real )
= bot_bot_set_set_real ) ).
% ccSup_empty
thf(fact_762_ccSup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% ccSup_empty
thf(fact_763_ccSup__empty,axiom,
( ( comple3096694443085538997t_real @ bot_bot_set_set_real )
= bot_bot_set_real ) ).
% ccSup_empty
thf(fact_764_ccSup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% ccSup_empty
thf(fact_765_Sup__empty,axiom,
( ( comple5917660045593844715t_real @ bot_bo3378928929837779682t_real )
= bot_bot_set_set_real ) ).
% Sup_empty
thf(fact_766_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_767_Sup__empty,axiom,
( ( comple3096694443085538997t_real @ bot_bot_set_set_real )
= bot_bot_set_real ) ).
% Sup_empty
thf(fact_768_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_769_Sup__atLeastAtMost,axiom,
! [X: set_real,Y2: set_real] :
( ( ord_less_eq_set_real @ X @ Y2 )
=> ( ( comple3096694443085538997t_real @ ( set_or7743017856606604397t_real @ X @ Y2 ) )
= Y2 ) ) ).
% Sup_atLeastAtMost
thf(fact_770_Sup__atLeastAtMost,axiom,
! [X: $o,Y2: $o] :
( ( ord_less_eq_o @ X @ Y2 )
=> ( ( complete_Sup_Sup_o @ ( set_or8904488021354931149Most_o @ X @ Y2 ) )
= Y2 ) ) ).
% Sup_atLeastAtMost
thf(fact_771_cSup__atLeastAtMost,axiom,
! [Y2: set_real,X: set_real] :
( ( ord_less_eq_set_real @ Y2 @ X )
=> ( ( comple3096694443085538997t_real @ ( set_or7743017856606604397t_real @ Y2 @ X ) )
= X ) ) ).
% cSup_atLeastAtMost
thf(fact_772_cSup__atLeastAtMost,axiom,
! [Y2: $o,X: $o] :
( ( ord_less_eq_o @ Y2 @ X )
=> ( ( complete_Sup_Sup_o @ ( set_or8904488021354931149Most_o @ Y2 @ X ) )
= X ) ) ).
% cSup_atLeastAtMost
thf(fact_773_cSup__atLeastAtMost,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ( ( comple1385675409528146559p_real @ ( set_or1222579329274155063t_real @ Y2 @ X ) )
= X ) ) ).
% cSup_atLeastAtMost
thf(fact_774_cSup__atLeastAtMost,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( complete_Sup_Sup_nat @ ( set_or1269000886237332187st_nat @ Y2 @ X ) )
= X ) ) ).
% cSup_atLeastAtMost
thf(fact_775_cSup__atLeastAtMost,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ( ( complete_Sup_Sup_int @ ( set_or1266510415728281911st_int @ Y2 @ X ) )
= X ) ) ).
% cSup_atLeastAtMost
thf(fact_776_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_777_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_778_cSup__atLeastLessThan,axiom,
! [Y2: real,X: real] :
( ( ord_less_real @ Y2 @ X )
=> ( ( comple1385675409528146559p_real @ ( set_or66887138388493659n_real @ Y2 @ X ) )
= X ) ) ).
% cSup_atLeastLessThan
thf(fact_779_mult__less__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_780_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_781_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_782_div__pos__pos__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ K2 @ L )
=> ( ( divide_divide_int @ K2 @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_783_div__neg__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ( ( ord_less_int @ L @ K2 )
=> ( ( divide_divide_int @ K2 @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_784_SUP__bot__conv_I2_J,axiom,
! [B2: nat > set_real,A2: set_nat] :
( ( bot_bot_set_real
= ( comple3096694443085538997t_real @ ( image_nat_set_real @ B2 @ A2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_real ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_785_SUP__bot__conv_I1_J,axiom,
! [B2: nat > set_real,A2: set_nat] :
( ( ( comple3096694443085538997t_real @ ( image_nat_set_real @ B2 @ A2 ) )
= bot_bot_set_real )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_real ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_786_SUP__bot,axiom,
! [A2: set_nat] :
( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [X2: nat] : bot_bot_set_real
@ A2 ) )
= bot_bot_set_real ) ).
% SUP_bot
thf(fact_787_ccSUP__bot,axiom,
! [A2: set_nat] :
( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [X2: nat] : bot_bot_set_real
@ A2 ) )
= bot_bot_set_real ) ).
% ccSUP_bot
thf(fact_788_SUP__const,axiom,
! [A2: set_set_real,F: set_real] :
( ( A2 != bot_bot_set_set_real )
=> ( ( comple3096694443085538997t_real
@ ( image_2436557299294012491t_real
@ ^ [I2: set_real] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_789_SUP__const,axiom,
! [A2: set_nat,F: set_real] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [I2: nat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_790_SUP__const,axiom,
! [A2: set_real,F: set_real] :
( ( A2 != bot_bot_set_real )
=> ( ( comple3096694443085538997t_real
@ ( image_real_set_real
@ ^ [I2: real] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_791_SUP__const,axiom,
! [A2: set_set_real,F: $o] :
( ( A2 != bot_bot_set_set_real )
=> ( ( complete_Sup_Sup_o
@ ( image_set_real_o
@ ^ [I2: set_real] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_792_SUP__const,axiom,
! [A2: set_nat,F: $o] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [I2: nat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_793_SUP__const,axiom,
! [A2: set_real,F: $o] :
( ( A2 != bot_bot_set_real )
=> ( ( complete_Sup_Sup_o
@ ( image_real_o
@ ^ [I2: real] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_794_ccSUP__const,axiom,
! [A2: set_set_real,F: set_real] :
( ( A2 != bot_bot_set_set_real )
=> ( ( comple3096694443085538997t_real
@ ( image_2436557299294012491t_real
@ ^ [I2: set_real] : F
@ A2 ) )
= F ) ) ).
% ccSUP_const
thf(fact_795_ccSUP__const,axiom,
! [A2: set_nat,F: set_real] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [I2: nat] : F
@ A2 ) )
= F ) ) ).
% ccSUP_const
thf(fact_796_ccSUP__const,axiom,
! [A2: set_real,F: set_real] :
( ( A2 != bot_bot_set_real )
=> ( ( comple3096694443085538997t_real
@ ( image_real_set_real
@ ^ [I2: real] : F
@ A2 ) )
= F ) ) ).
% ccSUP_const
thf(fact_797_ccSUP__const,axiom,
! [A2: set_set_real,F: $o] :
( ( A2 != bot_bot_set_set_real )
=> ( ( complete_Sup_Sup_o
@ ( image_set_real_o
@ ^ [I2: set_real] : F
@ A2 ) )
= F ) ) ).
% ccSUP_const
thf(fact_798_ccSUP__const,axiom,
! [A2: set_nat,F: $o] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [I2: nat] : F
@ A2 ) )
= F ) ) ).
% ccSUP_const
thf(fact_799_ccSUP__const,axiom,
! [A2: set_real,F: $o] :
( ( A2 != bot_bot_set_real )
=> ( ( complete_Sup_Sup_o
@ ( image_real_o
@ ^ [I2: real] : F
@ A2 ) )
= F ) ) ).
% ccSUP_const
thf(fact_800_cSUP__const,axiom,
! [A2: set_nat,C: int] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_int
@ ( image_nat_int
@ ^ [X2: nat] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_801_cSUP__const,axiom,
! [A2: set_nat,C: real] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple1385675409528146559p_real
@ ( image_nat_real
@ ^ [X2: nat] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_802_cSUP__const,axiom,
! [A2: set_real,C: real] :
( ( A2 != bot_bot_set_real )
=> ( ( comple1385675409528146559p_real
@ ( image_real_real
@ ^ [X2: real] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_803_cSUP__const,axiom,
! [A2: set_nat,C: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_804_cSUP__const,axiom,
! [A2: set_real,C: nat] :
( ( A2 != bot_bot_set_real )
=> ( ( complete_Sup_Sup_nat
@ ( image_real_nat
@ ^ [X2: real] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_805_cSUP__const,axiom,
! [A2: set_nat,C: $o] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X2: nat] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_806_cSUP__const,axiom,
! [A2: set_real,C: $o] :
( ( A2 != bot_bot_set_real )
=> ( ( complete_Sup_Sup_o
@ ( image_real_o
@ ^ [X2: real] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_807_cSUP__const,axiom,
! [A2: set_set_real,C: real] :
( ( A2 != bot_bot_set_set_real )
=> ( ( comple1385675409528146559p_real
@ ( image_set_real_real
@ ^ [X2: set_real] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_808_cSUP__const,axiom,
! [A2: set_nat,C: set_real] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [X2: nat] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_809_cSUP__const,axiom,
! [A2: set_real,C: set_real] :
( ( A2 != bot_bot_set_real )
=> ( ( comple3096694443085538997t_real
@ ( image_real_set_real
@ ^ [X2: real] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_810_has__integral__restrict,axiom,
! [S: set_real,T: set_real,F: real > real,I: real] :
( ( ord_less_eq_set_real @ S @ T )
=> ( ( hensto240673015341029504l_real
@ ^ [X2: real] : ( if_real @ ( member_real @ X2 @ S ) @ ( F @ X2 ) @ zero_zero_real )
@ I
@ T )
= ( hensto240673015341029504l_real @ F @ I @ S ) ) ) ).
% has_integral_restrict
thf(fact_811_mult__le__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_812_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_813_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_814_ccSUP__empty,axiom,
! [F: nat > $o] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ bot_bot_set_nat ) )
= bot_bot_o ) ).
% ccSUP_empty
thf(fact_815_ccSUP__empty,axiom,
! [F: real > $o] :
( ( complete_Sup_Sup_o @ ( image_real_o @ F @ bot_bot_set_real ) )
= bot_bot_o ) ).
% ccSUP_empty
thf(fact_816_ccSUP__empty,axiom,
! [F: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% ccSUP_empty
thf(fact_817_ccSUP__empty,axiom,
! [F: real > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ bot_bot_set_real ) )
= bot_bot_set_nat ) ).
% ccSUP_empty
thf(fact_818_ccSUP__empty,axiom,
! [F: nat > set_real] :
( ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ bot_bot_set_nat ) )
= bot_bot_set_real ) ).
% ccSUP_empty
thf(fact_819_ccSUP__empty,axiom,
! [F: real > set_real] :
( ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ bot_bot_set_real ) )
= bot_bot_set_real ) ).
% ccSUP_empty
thf(fact_820_ccSUP__empty,axiom,
! [F: set_real > $o] :
( ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ bot_bot_set_set_real ) )
= bot_bot_o ) ).
% ccSUP_empty
thf(fact_821_ccSUP__empty,axiom,
! [F: set_real > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_7270232309134952815et_nat @ F @ bot_bot_set_set_real ) )
= bot_bot_set_nat ) ).
% ccSUP_empty
thf(fact_822_ccSUP__empty,axiom,
! [F: nat > set_set_real] :
( ( comple5917660045593844715t_real @ ( image_396256051147326063t_real @ F @ bot_bot_set_nat ) )
= bot_bot_set_set_real ) ).
% ccSUP_empty
thf(fact_823_ccSUP__empty,axiom,
! [F: real > set_set_real] :
( ( comple5917660045593844715t_real @ ( image_3243600997494576203t_real @ F @ bot_bot_set_real ) )
= bot_bot_set_set_real ) ).
% ccSUP_empty
thf(fact_824_INT__lower,axiom,
! [A: nat,A2: set_nat,B2: nat > set_real] :
( ( member_nat @ A @ A2 )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_nat_set_real @ B2 @ A2 ) ) @ ( B2 @ A ) ) ) ).
% INT_lower
thf(fact_825_Inter__lower,axiom,
! [B2: set_real,A2: set_set_real] :
( ( member_set_real @ B2 @ A2 )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ A2 ) @ B2 ) ) ).
% Inter_lower
thf(fact_826_INT__greatest,axiom,
! [A2: set_nat,C4: set_real,B2: nat > set_real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_real @ C4 @ ( B2 @ X3 ) ) )
=> ( ord_less_eq_set_real @ C4 @ ( comple8289635161444856091t_real @ ( image_nat_set_real @ B2 @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_827_INT__anti__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_real,G: nat > set_real] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_nat_set_real @ F @ B2 ) ) @ ( comple8289635161444856091t_real @ ( image_nat_set_real @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_828_INT__subset__iff,axiom,
! [B2: set_real,A2: nat > set_real,I4: set_nat] :
( ( ord_less_eq_set_real @ B2 @ ( comple8289635161444856091t_real @ ( image_nat_set_real @ A2 @ I4 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I4 )
=> ( ord_less_eq_set_real @ B2 @ ( A2 @ X2 ) ) ) ) ) ).
% INT_subset_iff
thf(fact_829_Inter__greatest,axiom,
! [A2: set_set_real,C4: set_real] :
( ! [X6: set_real] :
( ( member_set_real @ X6 @ A2 )
=> ( ord_less_eq_set_real @ C4 @ X6 ) )
=> ( ord_less_eq_set_real @ C4 @ ( comple8289635161444856091t_real @ A2 ) ) ) ).
% Inter_greatest
thf(fact_830_image__int__atLeastLessThan,axiom,
! [A: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
= ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastLessThan
thf(fact_831_Inf__nat__def1,axiom,
! [K: set_nat] :
( ( K != bot_bot_set_nat )
=> ( member_nat @ ( complete_Inf_Inf_nat @ K ) @ K ) ) ).
% Inf_nat_def1
thf(fact_832_linorder__neqE__nat,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_833_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_834_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_835_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_836_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_837_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_838_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_839_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_840_Sup__subset__mono,axiom,
! [A2: set_set_real,B2: set_set_real] :
( ( ord_le3558479182127378552t_real @ A2 @ B2 )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A2 ) @ ( comple3096694443085538997t_real @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_841_Sup__subset__mono,axiom,
! [A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_842_Sup__upper2,axiom,
! [U2: set_real,A2: set_set_real,V: set_real] :
( ( member_set_real @ U2 @ A2 )
=> ( ( ord_less_eq_set_real @ V @ U2 )
=> ( ord_less_eq_set_real @ V @ ( comple3096694443085538997t_real @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_843_Sup__upper2,axiom,
! [U2: $o,A2: set_o,V: $o] :
( ( member_o @ U2 @ A2 )
=> ( ( ord_less_eq_o @ V @ U2 )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_844_Sup__le__iff,axiom,
! [A2: set_set_real,B: set_real] :
( ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A2 ) @ B )
= ( ! [X2: set_real] :
( ( member_set_real @ X2 @ A2 )
=> ( ord_less_eq_set_real @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_845_Sup__le__iff,axiom,
! [A2: set_o,B: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ B )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_o @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_846_Sup__upper,axiom,
! [X: set_real,A2: set_set_real] :
( ( member_set_real @ X @ A2 )
=> ( ord_less_eq_set_real @ X @ ( comple3096694443085538997t_real @ A2 ) ) ) ).
% Sup_upper
thf(fact_847_Sup__upper,axiom,
! [X: $o,A2: set_o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ X @ ( complete_Sup_Sup_o @ A2 ) ) ) ).
% Sup_upper
thf(fact_848_Sup__least,axiom,
! [A2: set_set_real,Z: set_real] :
( ! [X3: set_real] :
( ( member_set_real @ X3 @ A2 )
=> ( ord_less_eq_set_real @ X3 @ Z ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_849_Sup__least,axiom,
! [A2: set_o,Z: $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ X3 @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_850_Sup__mono,axiom,
! [A2: set_set_real,B2: set_set_real] :
( ! [A3: set_real] :
( ( member_set_real @ A3 @ A2 )
=> ? [X4: set_real] :
( ( member_set_real @ X4 @ B2 )
& ( ord_less_eq_set_real @ A3 @ X4 ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A2 ) @ ( comple3096694443085538997t_real @ B2 ) ) ) ).
% Sup_mono
thf(fact_851_Sup__mono,axiom,
! [A2: set_o,B2: set_o] :
( ! [A3: $o] :
( ( member_o @ A3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_o @ A3 @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_mono
thf(fact_852_Sup__eqI,axiom,
! [A2: set_set_real,X: set_real] :
( ! [Y3: set_real] :
( ( member_set_real @ Y3 @ A2 )
=> ( ord_less_eq_set_real @ Y3 @ X ) )
=> ( ! [Y3: set_real] :
( ! [Z3: set_real] :
( ( member_set_real @ Z3 @ A2 )
=> ( ord_less_eq_set_real @ Z3 @ Y3 ) )
=> ( ord_less_eq_set_real @ X @ Y3 ) )
=> ( ( comple3096694443085538997t_real @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_853_Sup__eqI,axiom,
! [A2: set_o,X: $o] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ A2 )
=> ( ord_less_eq_o @ Y3 @ X ) )
=> ( ! [Y3: $o] :
( ! [Z3: $o] :
( ( member_o @ Z3 @ A2 )
=> ( ord_less_eq_o @ Z3 @ Y3 ) )
=> ( ord_less_eq_o @ X @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_854_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > int,D3: nat > int] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_nat_int @ C4 @ A2 ) )
= ( complete_Sup_Sup_int @ ( image_nat_int @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_855_SUP__cong,axiom,
! [A2: set_real,B2: set_real,C4: real > real,D3: real > real] :
( ( A2 = B2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( comple1385675409528146559p_real @ ( image_real_real @ C4 @ A2 ) )
= ( comple1385675409528146559p_real @ ( image_real_real @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_856_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > real,D3: nat > real] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( comple1385675409528146559p_real @ ( image_nat_real @ C4 @ A2 ) )
= ( comple1385675409528146559p_real @ ( image_nat_real @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_857_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C4: $o > real,D3: $o > real] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( comple1385675409528146559p_real @ ( image_o_real @ C4 @ A2 ) )
= ( comple1385675409528146559p_real @ ( image_o_real @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_858_SUP__cong,axiom,
! [A2: set_real,B2: set_real,C4: real > nat,D3: real > nat] :
( ( A2 = B2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_real_nat @ C4 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_real_nat @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_859_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > nat,D3: nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C4 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_860_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C4: $o > nat,D3: $o > nat] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat @ C4 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_o_nat @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_861_SUP__cong,axiom,
! [A2: set_real,B2: set_real,C4: real > $o,D3: real > $o] :
( ( A2 = B2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ C4 @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_real_o @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_862_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > $o,D3: nat > $o] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ C4 @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_863_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C4: $o > $o,D3: $o > $o] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C4 @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_864_SUP__UNION,axiom,
! [F: nat > set_real,G: nat > set_nat,A2: set_nat] :
( ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) )
= ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [Y: nat] : ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_865_SUP__UNION,axiom,
! [F: real > set_real,G: nat > set_real,A2: set_nat] :
( ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ A2 ) ) ) )
= ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [Y: nat] : ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_866_SUP__UNION,axiom,
! [F: real > $o,G: nat > set_real,A2: set_nat] :
( ( complete_Sup_Sup_o @ ( image_real_o @ F @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y: nat] : ( complete_Sup_Sup_o @ ( image_real_o @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_867_Inf__nat__def,axiom,
( complete_Inf_Inf_nat
= ( ^ [X7: set_nat] :
( ord_Least_nat
@ ^ [N4: nat] : ( member_nat @ N4 @ X7 ) ) ) ) ).
% Inf_nat_def
thf(fact_868_SUP__commute,axiom,
! [F: nat > nat > set_real,B2: set_nat,A2: set_nat] :
( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [I2: nat] : ( comple3096694443085538997t_real @ ( image_nat_set_real @ ( F @ I2 ) @ B2 ) )
@ A2 ) )
= ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [J2: nat] :
( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [I2: nat] : ( F @ I2 @ J2 )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_869_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K2: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_870_cSup__eq__maximum,axiom,
! [Z: int,X5: set_int] :
( ( member_int @ Z @ X5 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X5 )
=> ( ord_less_eq_int @ X3 @ Z ) )
=> ( ( complete_Sup_Sup_int @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_871_cSup__eq__maximum,axiom,
! [Z: real,X5: set_real] :
( ( member_real @ Z @ X5 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X5 )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ( ( comple1385675409528146559p_real @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_872_cSup__eq__maximum,axiom,
! [Z: set_real,X5: set_set_real] :
( ( member_set_real @ Z @ X5 )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ X5 )
=> ( ord_less_eq_set_real @ X3 @ Z ) )
=> ( ( comple3096694443085538997t_real @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_873_cSup__eq__maximum,axiom,
! [Z: nat,X5: set_nat] :
( ( member_nat @ Z @ X5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_874_cSup__eq__maximum,axiom,
! [Z: $o,X5: set_o] :
( ( member_o @ Z @ X5 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X5 )
=> ( ord_less_eq_o @ X3 @ Z ) )
=> ( ( complete_Sup_Sup_o @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_875_cSup__eq,axiom,
! [X5: set_int,A: int] :
( ! [X3: int] :
( ( member_int @ X3 @ X5 )
=> ( ord_less_eq_int @ X3 @ A ) )
=> ( ! [Y3: int] :
( ! [X4: int] :
( ( member_int @ X4 @ X5 )
=> ( ord_less_eq_int @ X4 @ Y3 ) )
=> ( ord_less_eq_int @ A @ Y3 ) )
=> ( ( complete_Sup_Sup_int @ X5 )
= A ) ) ) ).
% cSup_eq
thf(fact_876_cSup__eq,axiom,
! [X5: set_real,A: real] :
( ! [X3: real] :
( ( member_real @ X3 @ X5 )
=> ( ord_less_eq_real @ X3 @ A ) )
=> ( ! [Y3: real] :
( ! [X4: real] :
( ( member_real @ X4 @ X5 )
=> ( ord_less_eq_real @ X4 @ Y3 ) )
=> ( ord_less_eq_real @ A @ Y3 ) )
=> ( ( comple1385675409528146559p_real @ X5 )
= A ) ) ) ).
% cSup_eq
thf(fact_877_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_878_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_879_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U2: nat] :
( ( set_or4665077453230672383an_nat @ L @ ( suc @ U2 ) )
= ( set_or1269000886237332187st_nat @ L @ U2 ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_880_image__int__atLeastAtMost,axiom,
! [A: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastAtMost
thf(fact_881_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_882_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_883_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_884_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_885_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_886_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_887_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_888_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_889_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_890_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K4: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K4 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K4 )
=> ( P @ I3 @ K4 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_891_less__trans__Suc,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_892_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_893_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_894_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_895_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_896_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_897_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_898_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_899_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_900_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_901_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_902_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K2 )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_903_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_904_Nat_OlessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_905_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M7: nat,N4: nat] :
( ( ord_less_eq_nat @ M7 @ N4 )
& ( M7 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_906_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_907_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M7: nat,N4: nat] :
( ( ord_less_nat @ M7 @ N4 )
| ( M7 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_908_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_909_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_910_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_911_zmult__zless__mono2,axiom,
! [I: int,J: int,K2: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ord_less_int @ ( times_times_int @ K2 @ I ) @ ( times_times_int @ K2 @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_912_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_913_SUP__eq,axiom,
! [A2: set_real,B2: set_real,F: real > $o,G: real > $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ? [X4: real] :
( ( member_real @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B2 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_real_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_914_SUP__eq,axiom,
! [A2: set_real,B2: set_nat,F: real > $o,G: nat > $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_915_SUP__eq,axiom,
! [A2: set_real,B2: set_o,F: real > $o,G: $o > $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B2 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_916_SUP__eq,axiom,
! [A2: set_nat,B2: set_real,F: nat > $o,G: real > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: real] :
( ( member_real @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_real_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_917_SUP__eq,axiom,
! [A2: set_nat,B2: set_nat,F: nat > $o,G: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_918_SUP__eq,axiom,
! [A2: set_nat,B2: set_o,F: nat > $o,G: $o > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_919_SUP__eq,axiom,
! [A2: set_o,B2: set_real,F: $o > $o,G: real > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: real] :
( ( member_real @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_real_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_920_SUP__eq,axiom,
! [A2: set_o,B2: set_nat,F: $o > $o,G: nat > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_921_SUP__eq,axiom,
! [A2: set_o,B2: set_o,F: $o > $o,G: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_922_SUP__eq,axiom,
! [A2: set_real,B2: set_real,F: real > set_real,G: real > set_real] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ? [X4: real] :
( ( member_real @ X4 @ B2 )
& ( ord_less_eq_set_real @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B2 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ord_less_eq_set_real @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A2 ) )
= ( comple3096694443085538997t_real @ ( image_real_set_real @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_923_less__eq__Sup,axiom,
! [A2: set_set_real,U2: set_real] :
( ! [V2: set_real] :
( ( member_set_real @ V2 @ A2 )
=> ( ord_less_eq_set_real @ U2 @ V2 ) )
=> ( ( A2 != bot_bot_set_set_real )
=> ( ord_less_eq_set_real @ U2 @ ( comple3096694443085538997t_real @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_924_less__eq__Sup,axiom,
! [A2: set_o,U2: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A2 )
=> ( ord_less_eq_o @ U2 @ V2 ) )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_925_SUP__eq__const,axiom,
! [I4: set_o,F: $o > set_real,X: set_real] :
( ( I4 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ I4 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_926_SUP__eq__const,axiom,
! [I4: set_set_real,F: set_real > set_real,X: set_real] :
( ( I4 != bot_bot_set_set_real )
=> ( ! [I3: set_real] :
( ( member_set_real @ I3 @ I4 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ I4 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_927_SUP__eq__const,axiom,
! [I4: set_nat,F: nat > set_real,X: set_real] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ I4 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_928_SUP__eq__const,axiom,
! [I4: set_real,F: real > set_real,X: set_real] :
( ( I4 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I4 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ I4 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_929_SUP__eq__const,axiom,
! [I4: set_o,F: $o > $o,X: $o] :
( ( I4 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I4 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_930_SUP__eq__const,axiom,
! [I4: set_set_real,F: set_real > $o,X: $o] :
( ( I4 != bot_bot_set_set_real )
=> ( ! [I3: set_real] :
( ( member_set_real @ I3 @ I4 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ I4 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_931_SUP__eq__const,axiom,
! [I4: set_nat,F: nat > $o,X: $o] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I4 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_932_SUP__eq__const,axiom,
! [I4: set_real,F: real > $o,X: $o] :
( ( I4 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I4 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ F @ I4 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_933_interval__bounds__real_I1_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( comple1385675409528146559p_real @ ( set_or1222579329274155063t_real @ A @ B ) )
= B ) ) ).
% interval_bounds_real(1)
thf(fact_934_SUP__subset__mono,axiom,
! [A2: set_set_real,B2: set_set_real,F: set_real > set_real,G: set_real > set_real] :
( ( ord_le3558479182127378552t_real @ A2 @ B2 )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_935_SUP__subset__mono,axiom,
! [A2: set_real,B2: set_real,F: real > set_real,G: real > set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_real_set_real @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_936_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_real,G: nat > set_real] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_937_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_real,G: $o > set_real] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_o_set_real @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_938_SUP__subset__mono,axiom,
! [A2: set_set_real,B2: set_set_real,F: set_real > $o,G: set_real > $o] :
( ( ord_le3558479182127378552t_real @ A2 @ B2 )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ A2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_set_real_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_939_SUP__subset__mono,axiom,
! [A2: set_real,B2: set_real,F: real > $o,G: real > $o] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_real_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_940_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > $o,G: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_941_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > $o,G: $o > $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_942_SUP__upper2,axiom,
! [I: set_real,A2: set_set_real,U2: set_real,F: set_real > set_real] :
( ( member_set_real @ I @ A2 )
=> ( ( ord_less_eq_set_real @ U2 @ ( F @ I ) )
=> ( ord_less_eq_set_real @ U2 @ ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_943_SUP__upper2,axiom,
! [I: real,A2: set_real,U2: set_real,F: real > set_real] :
( ( member_real @ I @ A2 )
=> ( ( ord_less_eq_set_real @ U2 @ ( F @ I ) )
=> ( ord_less_eq_set_real @ U2 @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_944_SUP__upper2,axiom,
! [I: nat,A2: set_nat,U2: set_real,F: nat > set_real] :
( ( member_nat @ I @ A2 )
=> ( ( ord_less_eq_set_real @ U2 @ ( F @ I ) )
=> ( ord_less_eq_set_real @ U2 @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_945_SUP__upper2,axiom,
! [I: $o,A2: set_o,U2: set_real,F: $o > set_real] :
( ( member_o @ I @ A2 )
=> ( ( ord_less_eq_set_real @ U2 @ ( F @ I ) )
=> ( ord_less_eq_set_real @ U2 @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_946_SUP__upper2,axiom,
! [I: set_real,A2: set_set_real,U2: $o,F: set_real > $o] :
( ( member_set_real @ I @ A2 )
=> ( ( ord_less_eq_o @ U2 @ ( F @ I ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_947_SUP__upper2,axiom,
! [I: real,A2: set_real,U2: $o,F: real > $o] :
( ( member_real @ I @ A2 )
=> ( ( ord_less_eq_o @ U2 @ ( F @ I ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_948_SUP__upper2,axiom,
! [I: nat,A2: set_nat,U2: $o,F: nat > $o] :
( ( member_nat @ I @ A2 )
=> ( ( ord_less_eq_o @ U2 @ ( F @ I ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_949_SUP__upper2,axiom,
! [I: $o,A2: set_o,U2: $o,F: $o > $o] :
( ( member_o @ I @ A2 )
=> ( ( ord_less_eq_o @ U2 @ ( F @ I ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_950_SUP__le__iff,axiom,
! [F: nat > set_real,A2: set_nat,U2: set_real] :
( ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) ) @ U2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ X2 ) @ U2 ) ) ) ) ).
% SUP_le_iff
thf(fact_951_SUP__upper,axiom,
! [I: set_real,A2: set_set_real,F: set_real > set_real] :
( ( member_set_real @ I @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I ) @ ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_952_SUP__upper,axiom,
! [I: real,A2: set_real,F: real > set_real] :
( ( member_real @ I @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I ) @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_953_SUP__upper,axiom,
! [I: nat,A2: set_nat,F: nat > set_real] :
( ( member_nat @ I @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_954_SUP__upper,axiom,
! [I: $o,A2: set_o,F: $o > set_real] :
( ( member_o @ I @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I ) @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_955_SUP__upper,axiom,
! [I: set_real,A2: set_set_real,F: set_real > $o] :
( ( member_set_real @ I @ A2 )
=> ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_956_SUP__upper,axiom,
! [I: real,A2: set_real,F: real > $o] :
( ( member_real @ I @ A2 )
=> ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_957_SUP__upper,axiom,
! [I: nat,A2: set_nat,F: nat > $o] :
( ( member_nat @ I @ A2 )
=> ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_958_SUP__upper,axiom,
! [I: $o,A2: set_o,F: $o > $o] :
( ( member_o @ I @ A2 )
=> ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_959_SUP__mono_H,axiom,
! [F: nat > set_real,G: nat > set_real,A2: set_nat] :
( ! [X3: nat] : ( ord_less_eq_set_real @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_960_SUP__least,axiom,
! [A2: set_set_real,F: set_real > set_real,U2: set_real] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_961_SUP__least,axiom,
! [A2: set_real,F: real > set_real,U2: set_real] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_962_SUP__least,axiom,
! [A2: set_nat,F: nat > set_real,U2: set_real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_963_SUP__least,axiom,
! [A2: set_o,F: $o > set_real,U2: set_real] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_964_SUP__least,axiom,
! [A2: set_set_real,F: set_real > $o,U2: $o] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_965_SUP__least,axiom,
! [A2: set_real,F: real > $o,U2: $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_966_SUP__least,axiom,
! [A2: set_nat,F: nat > $o,U2: $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_967_SUP__least,axiom,
! [A2: set_o,F: $o > $o,U2: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_968_SUP__mono,axiom,
! [A2: set_set_real,B2: set_nat,F: set_real > set_real,G: nat > set_real] :
( ! [N2: set_real] :
( ( member_set_real @ N2 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_set_real @ ( F @ N2 ) @ ( G @ X4 ) ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_969_SUP__mono,axiom,
! [A2: set_real,B2: set_nat,F: real > set_real,G: nat > set_real] :
( ! [N2: real] :
( ( member_real @ N2 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_set_real @ ( F @ N2 ) @ ( G @ X4 ) ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_970_SUP__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_real,G: nat > set_real] :
( ! [N2: nat] :
( ( member_nat @ N2 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_set_real @ ( F @ N2 ) @ ( G @ X4 ) ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_971_SUP__mono,axiom,
! [A2: set_o,B2: set_nat,F: $o > set_real,G: nat > set_real] :
( ! [N2: $o] :
( ( member_o @ N2 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_set_real @ ( F @ N2 ) @ ( G @ X4 ) ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_972_SUP__eqI,axiom,
! [A2: set_set_real,F: set_real > set_real,X: set_real] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ X ) )
=> ( ! [Y3: set_real] :
( ! [I5: set_real] :
( ( member_set_real @ I5 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I5 ) @ Y3 ) )
=> ( ord_less_eq_set_real @ X @ Y3 ) )
=> ( ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_973_SUP__eqI,axiom,
! [A2: set_real,F: real > set_real,X: set_real] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ X ) )
=> ( ! [Y3: set_real] :
( ! [I5: real] :
( ( member_real @ I5 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I5 ) @ Y3 ) )
=> ( ord_less_eq_set_real @ X @ Y3 ) )
=> ( ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_974_SUP__eqI,axiom,
! [A2: set_nat,F: nat > set_real,X: set_real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ X ) )
=> ( ! [Y3: set_real] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I5 ) @ Y3 ) )
=> ( ord_less_eq_set_real @ X @ Y3 ) )
=> ( ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_975_SUP__eqI,axiom,
! [A2: set_o,F: $o > set_real,X: set_real] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ X ) )
=> ( ! [Y3: set_real] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I5 ) @ Y3 ) )
=> ( ord_less_eq_set_real @ X @ Y3 ) )
=> ( ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_976_SUP__eqI,axiom,
! [A2: set_set_real,F: set_real > $o,X: $o] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ X ) )
=> ( ! [Y3: $o] :
( ! [I5: set_real] :
( ( member_set_real @ I5 @ A2 )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y3 ) )
=> ( ord_less_eq_o @ X @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_977_SUP__eqI,axiom,
! [A2: set_real,F: real > $o,X: $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ X ) )
=> ( ! [Y3: $o] :
( ! [I5: real] :
( ( member_real @ I5 @ A2 )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y3 ) )
=> ( ord_less_eq_o @ X @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_978_SUP__eqI,axiom,
! [A2: set_nat,F: nat > $o,X: $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ X ) )
=> ( ! [Y3: $o] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A2 )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y3 ) )
=> ( ord_less_eq_o @ X @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_979_SUP__eqI,axiom,
! [A2: set_o,F: $o > $o,X: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ X ) )
=> ( ! [Y3: $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y3 ) )
=> ( ord_less_eq_o @ X @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_980_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N2: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_981_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K2
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_982_SUP__lessD,axiom,
! [F: set_real > set_real,A2: set_set_real,Y2: set_real,I: set_real] :
( ( ord_less_set_real @ ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A2 ) ) @ Y2 )
=> ( ( member_set_real @ I @ A2 )
=> ( ord_less_set_real @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_983_SUP__lessD,axiom,
! [F: real > set_real,A2: set_real,Y2: set_real,I: real] :
( ( ord_less_set_real @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A2 ) ) @ Y2 )
=> ( ( member_real @ I @ A2 )
=> ( ord_less_set_real @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_984_SUP__lessD,axiom,
! [F: nat > set_real,A2: set_nat,Y2: set_real,I: nat] :
( ( ord_less_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) ) @ Y2 )
=> ( ( member_nat @ I @ A2 )
=> ( ord_less_set_real @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_985_SUP__lessD,axiom,
! [F: $o > set_real,A2: set_o,Y2: set_real,I: $o] :
( ( ord_less_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A2 ) ) @ Y2 )
=> ( ( member_o @ I @ A2 )
=> ( ord_less_set_real @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_986_SUP__lessD,axiom,
! [F: set_real > $o,A2: set_set_real,Y2: $o,I: set_real] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ A2 ) ) @ Y2 )
=> ( ( member_set_real @ I @ A2 )
=> ( ord_less_o @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_987_SUP__lessD,axiom,
! [F: real > $o,A2: set_real,Y2: $o,I: real] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) ) @ Y2 )
=> ( ( member_real @ I @ A2 )
=> ( ord_less_o @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_988_SUP__lessD,axiom,
! [F: nat > $o,A2: set_nat,Y2: $o,I: nat] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) @ Y2 )
=> ( ( member_nat @ I @ A2 )
=> ( ord_less_o @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_989_SUP__lessD,axiom,
! [F: $o > $o,A2: set_o,Y2: $o,I: $o] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ Y2 )
=> ( ( member_o @ I @ A2 )
=> ( ord_less_o @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_990_SUP__eq__iff,axiom,
! [I4: set_o,C: set_real,F: $o > set_real] :
( ( I4 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ord_less_eq_set_real @ C @ ( F @ I3 ) ) )
=> ( ( ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ I4 ) )
= C )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I4 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_991_SUP__eq__iff,axiom,
! [I4: set_set_real,C: set_real,F: set_real > set_real] :
( ( I4 != bot_bot_set_set_real )
=> ( ! [I3: set_real] :
( ( member_set_real @ I3 @ I4 )
=> ( ord_less_eq_set_real @ C @ ( F @ I3 ) ) )
=> ( ( ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ I4 ) )
= C )
= ( ! [X2: set_real] :
( ( member_set_real @ X2 @ I4 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_992_SUP__eq__iff,axiom,
! [I4: set_nat,C: set_real,F: nat > set_real] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_less_eq_set_real @ C @ ( F @ I3 ) ) )
=> ( ( ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ I4 ) )
= C )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I4 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_993_SUP__eq__iff,axiom,
! [I4: set_real,C: set_real,F: real > set_real] :
( ( I4 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I4 )
=> ( ord_less_eq_set_real @ C @ ( F @ I3 ) ) )
=> ( ( ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ I4 ) )
= C )
= ( ! [X2: real] :
( ( member_real @ X2 @ I4 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_994_SUP__eq__iff,axiom,
! [I4: set_o,C: $o,F: $o > $o] :
( ( I4 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I4 ) )
= C )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I4 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_995_SUP__eq__iff,axiom,
! [I4: set_set_real,C: $o,F: set_real > $o] :
( ( I4 != bot_bot_set_set_real )
=> ( ! [I3: set_real] :
( ( member_set_real @ I3 @ I4 )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ I4 ) )
= C )
= ( ! [X2: set_real] :
( ( member_set_real @ X2 @ I4 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_996_SUP__eq__iff,axiom,
! [I4: set_nat,C: $o,F: nat > $o] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I4 ) )
= C )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I4 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_997_SUP__eq__iff,axiom,
! [I4: set_real,C: $o,F: real > $o] :
( ( I4 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I4 )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_real_o @ F @ I4 ) )
= C )
= ( ! [X2: real] :
( ( member_real @ X2 @ I4 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_998_Inf__le__Sup,axiom,
! [A2: set_set_real] :
( ( A2 != bot_bot_set_set_real )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ A2 ) @ ( comple3096694443085538997t_real @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_999_Inf__le__Sup,axiom,
! [A2: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Sup_Sup_o @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_1000_cSup__least,axiom,
! [X5: set_int,Z: int] :
( ( X5 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X5 )
=> ( ord_less_eq_int @ X3 @ Z ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1001_cSup__least,axiom,
! [X5: set_real,Z: real] :
( ( X5 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X5 )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ( ord_less_eq_real @ ( comple1385675409528146559p_real @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1002_cSup__least,axiom,
! [X5: set_set_real,Z: set_real] :
( ( X5 != bot_bot_set_set_real )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ X5 )
=> ( ord_less_eq_set_real @ X3 @ Z ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1003_cSup__least,axiom,
! [X5: set_nat,Z: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1004_cSup__least,axiom,
! [X5: set_o,Z: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X5 )
=> ( ord_less_eq_o @ X3 @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1005_cSup__eq__non__empty,axiom,
! [X5: set_int,A: int] :
( ( X5 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X5 )
=> ( ord_less_eq_int @ X3 @ A ) )
=> ( ! [Y3: int] :
( ! [X4: int] :
( ( member_int @ X4 @ X5 )
=> ( ord_less_eq_int @ X4 @ Y3 ) )
=> ( ord_less_eq_int @ A @ Y3 ) )
=> ( ( complete_Sup_Sup_int @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1006_cSup__eq__non__empty,axiom,
! [X5: set_real,A: real] :
( ( X5 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X5 )
=> ( ord_less_eq_real @ X3 @ A ) )
=> ( ! [Y3: real] :
( ! [X4: real] :
( ( member_real @ X4 @ X5 )
=> ( ord_less_eq_real @ X4 @ Y3 ) )
=> ( ord_less_eq_real @ A @ Y3 ) )
=> ( ( comple1385675409528146559p_real @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1007_cSup__eq__non__empty,axiom,
! [X5: set_set_real,A: set_real] :
( ( X5 != bot_bot_set_set_real )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ X5 )
=> ( ord_less_eq_set_real @ X3 @ A ) )
=> ( ! [Y3: set_real] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ X5 )
=> ( ord_less_eq_set_real @ X4 @ Y3 ) )
=> ( ord_less_eq_set_real @ A @ Y3 ) )
=> ( ( comple3096694443085538997t_real @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1008_cSup__eq__non__empty,axiom,
! [X5: set_nat,A: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ A ) )
=> ( ! [Y3: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ Y3 ) )
=> ( ord_less_eq_nat @ A @ Y3 ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1009_cSup__eq__non__empty,axiom,
! [X5: set_o,A: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X5 )
=> ( ord_less_eq_o @ X3 @ A ) )
=> ( ! [Y3: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ X5 )
=> ( ord_less_eq_o @ X4 @ Y3 ) )
=> ( ord_less_eq_o @ A @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1010_less__cSupD,axiom,
! [X5: set_int,Z: int] :
( ( X5 != bot_bot_set_int )
=> ( ( ord_less_int @ Z @ ( complete_Sup_Sup_int @ X5 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ X5 )
& ( ord_less_int @ Z @ X3 ) ) ) ) ).
% less_cSupD
thf(fact_1011_less__cSupD,axiom,
! [X5: set_real,Z: real] :
( ( X5 != bot_bot_set_real )
=> ( ( ord_less_real @ Z @ ( comple1385675409528146559p_real @ X5 ) )
=> ? [X3: real] :
( ( member_real @ X3 @ X5 )
& ( ord_less_real @ Z @ X3 ) ) ) ) ).
% less_cSupD
thf(fact_1012_less__cSupD,axiom,
! [X5: set_nat,Z: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ( ord_less_nat @ Z @ ( complete_Sup_Sup_nat @ X5 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( ord_less_nat @ Z @ X3 ) ) ) ) ).
% less_cSupD
thf(fact_1013_less__cSupE,axiom,
! [Y2: int,X5: set_int] :
( ( ord_less_int @ Y2 @ ( complete_Sup_Sup_int @ X5 ) )
=> ( ( X5 != bot_bot_set_int )
=> ~ ! [X3: int] :
( ( member_int @ X3 @ X5 )
=> ~ ( ord_less_int @ Y2 @ X3 ) ) ) ) ).
% less_cSupE
thf(fact_1014_less__cSupE,axiom,
! [Y2: real,X5: set_real] :
( ( ord_less_real @ Y2 @ ( comple1385675409528146559p_real @ X5 ) )
=> ( ( X5 != bot_bot_set_real )
=> ~ ! [X3: real] :
( ( member_real @ X3 @ X5 )
=> ~ ( ord_less_real @ Y2 @ X3 ) ) ) ) ).
% less_cSupE
thf(fact_1015_less__cSupE,axiom,
! [Y2: nat,X5: set_nat] :
( ( ord_less_nat @ Y2 @ ( complete_Sup_Sup_nat @ X5 ) )
=> ( ( X5 != bot_bot_set_nat )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ~ ( ord_less_nat @ Y2 @ X3 ) ) ) ) ).
% less_cSupE
thf(fact_1016_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_1017_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_1018_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1019_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M7: nat] :
( N
= ( suc @ M7 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1020_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1021_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_1022_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1023_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1024_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1025_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1026_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1027_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1028_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1029_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1030_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1031_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1032_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1033_mult__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_1034_mult__less__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1035_Suc__mult__less__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1036_SUP__constant,axiom,
! [C: $o,A2: set_nat] :
( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y: nat] : C
@ A2 ) )
= ( ( ( A2 = bot_bot_set_nat )
=> bot_bot_o )
& ( ( A2 != bot_bot_set_nat )
=> C ) ) ) ).
% SUP_constant
thf(fact_1037_SUP__constant,axiom,
! [C: $o,A2: set_real] :
( ( complete_Sup_Sup_o
@ ( image_real_o
@ ^ [Y: real] : C
@ A2 ) )
= ( ( ( A2 = bot_bot_set_real )
=> bot_bot_o )
& ( ( A2 != bot_bot_set_real )
=> C ) ) ) ).
% SUP_constant
thf(fact_1038_SUP__constant,axiom,
! [A2: set_nat,C: set_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] : C
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] : C
@ A2 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1039_SUP__constant,axiom,
! [A2: set_real,C: set_nat] :
( ( ( A2 = bot_bot_set_real )
=> ( ( comple7399068483239264473et_nat
@ ( image_real_set_nat
@ ^ [Y: real] : C
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_real )
=> ( ( comple7399068483239264473et_nat
@ ( image_real_set_nat
@ ^ [Y: real] : C
@ A2 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1040_SUP__constant,axiom,
! [A2: set_nat,C: set_real] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [Y: nat] : C
@ A2 ) )
= bot_bot_set_real ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [Y: nat] : C
@ A2 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1041_SUP__constant,axiom,
! [A2: set_real,C: set_real] :
( ( ( A2 = bot_bot_set_real )
=> ( ( comple3096694443085538997t_real
@ ( image_real_set_real
@ ^ [Y: real] : C
@ A2 ) )
= bot_bot_set_real ) )
& ( ( A2 != bot_bot_set_real )
=> ( ( comple3096694443085538997t_real
@ ( image_real_set_real
@ ^ [Y: real] : C
@ A2 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1042_SUP__constant,axiom,
! [C: $o,A2: set_set_real] :
( ( complete_Sup_Sup_o
@ ( image_set_real_o
@ ^ [Y: set_real] : C
@ A2 ) )
= ( ( ( A2 = bot_bot_set_set_real )
=> bot_bot_o )
& ( ( A2 != bot_bot_set_set_real )
=> C ) ) ) ).
% SUP_constant
thf(fact_1043_SUP__constant,axiom,
! [A2: set_set_real,C: set_nat] :
( ( ( A2 = bot_bot_set_set_real )
=> ( ( comple7399068483239264473et_nat
@ ( image_7270232309134952815et_nat
@ ^ [Y: set_real] : C
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_set_real )
=> ( ( comple7399068483239264473et_nat
@ ( image_7270232309134952815et_nat
@ ^ [Y: set_real] : C
@ A2 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1044_SUP__constant,axiom,
! [A2: set_nat,C: set_set_real] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple5917660045593844715t_real
@ ( image_396256051147326063t_real
@ ^ [Y: nat] : C
@ A2 ) )
= bot_bot_set_set_real ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple5917660045593844715t_real
@ ( image_396256051147326063t_real
@ ^ [Y: nat] : C
@ A2 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1045_SUP__constant,axiom,
! [A2: set_real,C: set_set_real] :
( ( ( A2 = bot_bot_set_real )
=> ( ( comple5917660045593844715t_real
@ ( image_3243600997494576203t_real
@ ^ [Y: real] : C
@ A2 ) )
= bot_bot_set_set_real ) )
& ( ( A2 != bot_bot_set_real )
=> ( ( comple5917660045593844715t_real
@ ( image_3243600997494576203t_real
@ ^ [Y: real] : C
@ A2 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1046_SUP__empty,axiom,
! [F: nat > $o] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ bot_bot_set_nat ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_1047_SUP__empty,axiom,
! [F: real > $o] :
( ( complete_Sup_Sup_o @ ( image_real_o @ F @ bot_bot_set_real ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_1048_SUP__empty,axiom,
! [F: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_1049_SUP__empty,axiom,
! [F: real > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ bot_bot_set_real ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_1050_SUP__empty,axiom,
! [F: nat > set_real] :
( ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ bot_bot_set_nat ) )
= bot_bot_set_real ) ).
% SUP_empty
thf(fact_1051_SUP__empty,axiom,
! [F: real > set_real] :
( ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ bot_bot_set_real ) )
= bot_bot_set_real ) ).
% SUP_empty
thf(fact_1052_SUP__empty,axiom,
! [F: set_real > $o] :
( ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ bot_bot_set_set_real ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_1053_SUP__empty,axiom,
! [F: set_real > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_7270232309134952815et_nat @ F @ bot_bot_set_set_real ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_1054_SUP__empty,axiom,
! [F: nat > set_set_real] :
( ( comple5917660045593844715t_real @ ( image_396256051147326063t_real @ F @ bot_bot_set_nat ) )
= bot_bot_set_set_real ) ).
% SUP_empty
thf(fact_1055_SUP__empty,axiom,
! [F: real > set_set_real] :
( ( comple5917660045593844715t_real @ ( image_3243600997494576203t_real @ F @ bot_bot_set_real ) )
= bot_bot_set_set_real ) ).
% SUP_empty
thf(fact_1056_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1057_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1058_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1059_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1060_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1061_Least__Suc2,axiom,
! [P: nat > $o,N: nat,Q2: nat > $o,M: nat] :
( ( P @ N )
=> ( ( Q2 @ M )
=> ( ~ ( P @ zero_zero_nat )
=> ( ! [K4: nat] :
( ( P @ ( suc @ K4 ) )
= ( Q2 @ K4 ) )
=> ( ( ord_Least_nat @ P )
= ( suc @ ( ord_Least_nat @ Q2 ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_1062_sum_Oshift__bounds__Suc__ivl,axiom,
! [G: nat > real,M: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups6591440286371151544t_real
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_1063_sum_Oshift__bounds__Suc__ivl,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_1064_INF__le__SUP,axiom,
! [A2: set_set_real,F: set_real > set_real] :
( ( A2 != bot_bot_set_set_real )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_2436557299294012491t_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1065_INF__le__SUP,axiom,
! [A2: set_nat,F: nat > set_real] :
( ( A2 != bot_bot_set_nat )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_nat_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1066_INF__le__SUP,axiom,
! [A2: set_real,F: real > set_real] :
( ( A2 != bot_bot_set_real )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_real_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1067_INF__le__SUP,axiom,
! [A2: set_set_real,F: set_real > $o] :
( ( A2 != bot_bot_set_set_real )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_set_real_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1068_INF__le__SUP,axiom,
! [A2: set_nat,F: nat > $o] :
( ( A2 != bot_bot_set_nat )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1069_INF__le__SUP,axiom,
! [A2: set_real,F: real > $o] :
( ( A2 != bot_bot_set_real )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_real_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1070_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= ( suc
@ ( ord_Least_nat
@ ^ [M7: nat] : ( P @ ( suc @ M7 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_1071_cSUP__least,axiom,
! [A2: set_o,F: $o > int,M5: int] :
( ( A2 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ M5 ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_o_int @ F @ A2 ) ) @ M5 ) ) ) ).
% cSUP_least
thf(fact_1072_cSUP__least,axiom,
! [A2: set_nat,F: nat > int,M5: int] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ M5 ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_nat_int @ F @ A2 ) ) @ M5 ) ) ) ).
% cSUP_least
thf(fact_1073_cSUP__least,axiom,
! [A2: set_real,F: real > int,M5: int] :
( ( A2 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ M5 ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_real_int @ F @ A2 ) ) @ M5 ) ) ) ).
% cSUP_least
thf(fact_1074_cSUP__least,axiom,
! [A2: set_o,F: $o > real,M5: real] :
( ( A2 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ M5 ) )
=> ( ord_less_eq_real @ ( comple1385675409528146559p_real @ ( image_o_real @ F @ A2 ) ) @ M5 ) ) ) ).
% cSUP_least
thf(fact_1075_cSUP__least,axiom,
! [A2: set_nat,F: nat > real,M5: real] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ M5 ) )
=> ( ord_less_eq_real @ ( comple1385675409528146559p_real @ ( image_nat_real @ F @ A2 ) ) @ M5 ) ) ) ).
% cSUP_least
thf(fact_1076_cSUP__least,axiom,
! [A2: set_real,F: real > real,M5: real] :
( ( A2 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ M5 ) )
=> ( ord_less_eq_real @ ( comple1385675409528146559p_real @ ( image_real_real @ F @ A2 ) ) @ M5 ) ) ) ).
% cSUP_least
thf(fact_1077_cSUP__least,axiom,
! [A2: set_o,F: $o > nat,M5: nat] :
( ( A2 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ M5 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A2 ) ) @ M5 ) ) ) ).
% cSUP_least
thf(fact_1078_cSUP__least,axiom,
! [A2: set_nat,F: nat > nat,M5: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ M5 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A2 ) ) @ M5 ) ) ) ).
% cSUP_least
thf(fact_1079_cSUP__least,axiom,
! [A2: set_real,F: real > nat,M5: nat] :
( ( A2 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ M5 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_real_nat @ F @ A2 ) ) @ M5 ) ) ) ).
% cSUP_least
thf(fact_1080_cSUP__least,axiom,
! [A2: set_o,F: $o > $o,M5: $o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ M5 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ M5 ) ) ) ).
% cSUP_least
thf(fact_1081_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K4 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1082_Sup_OSUP__cong,axiom,
! [A2: set_real,B2: set_real,C4: real > real,D3: real > real,Sup: set_real > real] :
( ( A2 = B2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image_real_real @ C4 @ A2 ) )
= ( Sup @ ( image_real_real @ D3 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1083_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > nat,D3: nat > nat,Sup: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C4 @ A2 ) )
= ( Sup @ ( image_nat_nat @ D3 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1084_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > int,D3: nat > int,Sup: set_int > int] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image_nat_int @ C4 @ A2 ) )
= ( Sup @ ( image_nat_int @ D3 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1085_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > set_real,D3: nat > set_real,Sup: set_set_real > set_real] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image_nat_set_real @ C4 @ A2 ) )
= ( Sup @ ( image_nat_set_real @ D3 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1086_Inf_OINF__cong,axiom,
! [A2: set_real,B2: set_real,C4: real > real,D3: real > real,Inf: set_real > real] :
( ( A2 = B2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image_real_real @ C4 @ A2 ) )
= ( Inf @ ( image_real_real @ D3 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1087_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > nat,D3: nat > nat,Inf: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C4 @ A2 ) )
= ( Inf @ ( image_nat_nat @ D3 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1088_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > int,D3: nat > int,Inf: set_int > int] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image_nat_int @ C4 @ A2 ) )
= ( Inf @ ( image_nat_int @ D3 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1089_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > set_real,D3: nat > set_real,Inf: set_set_real > set_real] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image_nat_set_real @ C4 @ A2 ) )
= ( Inf @ ( image_nat_set_real @ D3 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1090_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1091_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1092_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1093_div__le__mono2,axiom,
! [M: nat,N: nat,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K2 @ N ) @ ( divide_divide_nat @ K2 @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1094_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1095_div__less__iff__less__mult,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1096_sum__SucD,axiom,
! [F: nat > nat,A2: set_nat,N: nat] :
( ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( suc @ N ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ).
% sum_SucD
thf(fact_1097_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1098_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1099_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1100_pos__imp__zdiv__pos__iff,axiom,
! [K2: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K2 ) )
= ( ord_less_eq_int @ K2 @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1101_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1102_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1103_div__int__pos__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K2 @ L ) )
= ( ( K2 = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K2 )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K2 @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1104_zdiv__mono2__neg,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1105_zdiv__mono1__neg,axiom,
! [A: int,A4: int,B: int] :
( ( ord_less_eq_int @ A @ A4 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1106_zdiv__eq__0__iff,axiom,
! [I: int,K2: int] :
( ( ( divide_divide_int @ I @ K2 )
= zero_zero_int )
= ( ( K2 = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K2 ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K2 @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1107_zdiv__mono2,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1108_zdiv__mono1,axiom,
! [A: int,A4: int,B: int] :
( ( ord_less_eq_int @ A @ A4 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A4 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1109_sum_Onested__swap,axiom,
! [A: nat > nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I2: nat] : ( groups6591440286371151544t_real @ ( A @ I2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( groups6591440286371151544t_real
@ ^ [J2: nat] :
( groups6591440286371151544t_real
@ ^ [I2: nat] : ( A @ I2 @ J2 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J2 ) @ N ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% sum.nested_swap
thf(fact_1110_sum_Onested__swap,axiom,
! [A: nat > nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( groups3542108847815614940at_nat @ ( A @ I2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [J2: nat] :
( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( A @ I2 @ J2 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J2 ) @ N ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% sum.nested_swap
thf(fact_1111_has__integral__unique,axiom,
! [F: real > real,K1: real,I: set_real,K22: real] :
( ( hensto240673015341029504l_real @ F @ K1 @ I )
=> ( ( hensto240673015341029504l_real @ F @ K22 @ I )
=> ( K1 = K22 ) ) ) ).
% has_integral_unique
thf(fact_1112_has__integral__eq__rhs,axiom,
! [F: real > real,J: real,S: set_real,I: real] :
( ( hensto240673015341029504l_real @ F @ J @ S )
=> ( ( I = J )
=> ( hensto240673015341029504l_real @ F @ I @ S ) ) ) ).
% has_integral_eq_rhs
thf(fact_1113_has__integral__cong,axiom,
! [S2: set_real,F: real > real,G: real > real,I: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( hensto240673015341029504l_real @ F @ I @ S2 )
= ( hensto240673015341029504l_real @ G @ I @ S2 ) ) ) ).
% has_integral_cong
thf(fact_1114_has__integral__eq,axiom,
! [S2: set_real,F: real > real,G: real > real,K2: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( hensto240673015341029504l_real @ F @ K2 @ S2 )
=> ( hensto240673015341029504l_real @ G @ K2 @ S2 ) ) ) ).
% has_integral_eq
thf(fact_1115_Inter__subset,axiom,
! [A2: set_set_real,B2: set_real] :
( ! [X6: set_real] :
( ( member_set_real @ X6 @ A2 )
=> ( ord_less_eq_set_real @ X6 @ B2 ) )
=> ( ( A2 != bot_bot_set_set_real )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ A2 ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_1116_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > int,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_int )
=> ( ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_1117_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > real,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_1118_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > nat,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_1119_less__eq__div__iff__mult__less__eq,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1120_div__nat__eqI,axiom,
! [N: nat,Q: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q ) ) ) ).
% div_nat_eqI
thf(fact_1121_Sup_OSUP__identity__eq,axiom,
! [Sup: set_real > real,A2: set_real] :
( ( Sup
@ ( image_real_real
@ ^ [X2: real] : X2
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_1122_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A2: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_1123_Inf_OINF__identity__eq,axiom,
! [Inf: set_real > real,A2: set_real] :
( ( Inf
@ ( image_real_real
@ ^ [X2: real] : X2
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_1124_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A2: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_1125_INT__extend__simps_I10_J,axiom,
! [B2: int > set_real,F: nat > int,A2: set_nat] :
( ( comple8289635161444856091t_real
@ ( image_nat_set_real
@ ^ [A5: nat] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple8289635161444856091t_real @ ( image_int_set_real @ B2 @ ( image_nat_int @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1126_INT__extend__simps_I10_J,axiom,
! [B2: set_real > set_real,F: nat > set_real,A2: set_nat] :
( ( comple8289635161444856091t_real
@ ( image_nat_set_real
@ ^ [A5: nat] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple8289635161444856091t_real @ ( image_2436557299294012491t_real @ B2 @ ( image_nat_set_real @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1127_INT__extend__simps_I10_J,axiom,
! [B2: nat > set_real,F: nat > nat,A2: set_nat] :
( ( comple8289635161444856091t_real
@ ( image_nat_set_real
@ ^ [A5: nat] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple8289635161444856091t_real @ ( image_nat_set_real @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1128_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
& ( P @ Q3 ) ) ) ) ).
% split_div'
thf(fact_1129_Inf__eqI,axiom,
! [A2: set_set_real,X: set_real] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ A2 )
=> ( ord_less_eq_set_real @ X @ I3 ) )
=> ( ! [Y3: set_real] :
( ! [I5: set_real] :
( ( member_set_real @ I5 @ A2 )
=> ( ord_less_eq_set_real @ Y3 @ I5 ) )
=> ( ord_less_eq_set_real @ Y3 @ X ) )
=> ( ( comple8289635161444856091t_real @ A2 )
= X ) ) ) ).
% Inf_eqI
thf(fact_1130_Inf__eqI,axiom,
! [A2: set_o,X: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ( ord_less_eq_o @ X @ I3 ) )
=> ( ! [Y3: $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_o @ Y3 @ I5 ) )
=> ( ord_less_eq_o @ Y3 @ X ) )
=> ( ( complete_Inf_Inf_o @ A2 )
= X ) ) ) ).
% Inf_eqI
thf(fact_1131_Inf__mono,axiom,
! [B2: set_set_real,A2: set_set_real] :
( ! [B3: set_real] :
( ( member_set_real @ B3 @ B2 )
=> ? [X4: set_real] :
( ( member_set_real @ X4 @ A2 )
& ( ord_less_eq_set_real @ X4 @ B3 ) ) )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ A2 ) @ ( comple8289635161444856091t_real @ B2 ) ) ) ).
% Inf_mono
thf(fact_1132_Inf__mono,axiom,
! [B2: set_o,A2: set_o] :
( ! [B3: $o] :
( ( member_o @ B3 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ X4 @ B3 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Inf_Inf_o @ B2 ) ) ) ).
% Inf_mono
thf(fact_1133_Inf__lower,axiom,
! [X: set_real,A2: set_set_real] :
( ( member_set_real @ X @ A2 )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ A2 ) @ X ) ) ).
% Inf_lower
thf(fact_1134_Inf__lower,axiom,
! [X: $o,A2: set_o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ X ) ) ).
% Inf_lower
thf(fact_1135_Inf__lower2,axiom,
! [U2: set_real,A2: set_set_real,V: set_real] :
( ( member_set_real @ U2 @ A2 )
=> ( ( ord_less_eq_set_real @ U2 @ V )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_1136_Inf__lower2,axiom,
! [U2: $o,A2: set_o,V: $o] :
( ( member_o @ U2 @ A2 )
=> ( ( ord_less_eq_o @ U2 @ V )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_1137_le__Inf__iff,axiom,
! [B: $o,A2: set_o] :
( ( ord_less_eq_o @ B @ ( complete_Inf_Inf_o @ A2 ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_o @ B @ X2 ) ) ) ) ).
% le_Inf_iff
thf(fact_1138_Inf__greatest,axiom,
! [A2: set_set_real,Z: set_real] :
( ! [X3: set_real] :
( ( member_set_real @ X3 @ A2 )
=> ( ord_less_eq_set_real @ Z @ X3 ) )
=> ( ord_less_eq_set_real @ Z @ ( comple8289635161444856091t_real @ A2 ) ) ) ).
% Inf_greatest
thf(fact_1139_Inf__greatest,axiom,
! [A2: set_o,Z: $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ Z @ X3 ) )
=> ( ord_less_eq_o @ Z @ ( complete_Inf_Inf_o @ A2 ) ) ) ).
% Inf_greatest
thf(fact_1140_Inf__superset__mono,axiom,
! [B2: set_o,A2: set_o] :
( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Inf_Inf_o @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_1141_INF__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > int,D3: nat > int] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_nat_int @ C4 @ A2 ) )
= ( complete_Inf_Inf_int @ ( image_nat_int @ D3 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1142_INF__cong,axiom,
! [A2: set_real,B2: set_real,C4: real > real,D3: real > real] :
( ( A2 = B2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( comple4887499456419720421f_real @ ( image_real_real @ C4 @ A2 ) )
= ( comple4887499456419720421f_real @ ( image_real_real @ D3 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1143_INF__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > real,D3: nat > real] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( comple4887499456419720421f_real @ ( image_nat_real @ C4 @ A2 ) )
= ( comple4887499456419720421f_real @ ( image_nat_real @ D3 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1144_INF__cong,axiom,
! [A2: set_o,B2: set_o,C4: $o > real,D3: $o > real] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( comple4887499456419720421f_real @ ( image_o_real @ C4 @ A2 ) )
= ( comple4887499456419720421f_real @ ( image_o_real @ D3 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1145_INF__cong,axiom,
! [A2: set_real,B2: set_real,C4: real > nat,D3: real > nat] :
( ( A2 = B2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_real_nat @ C4 @ A2 ) )
= ( complete_Inf_Inf_nat @ ( image_real_nat @ D3 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1146_INF__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > nat,D3: nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ C4 @ A2 ) )
= ( complete_Inf_Inf_nat @ ( image_nat_nat @ D3 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1147_INF__cong,axiom,
! [A2: set_o,B2: set_o,C4: $o > nat,D3: $o > nat] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_o_nat @ C4 @ A2 ) )
= ( complete_Inf_Inf_nat @ ( image_o_nat @ D3 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1148_INF__cong,axiom,
! [A2: set_real,B2: set_real,C4: real > $o,D3: real > $o] :
( ( A2 = B2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_real_o @ C4 @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_real_o @ D3 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1149_INF__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > $o,D3: nat > $o] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ C4 @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ D3 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1150_INF__cong,axiom,
! [A2: set_o,B2: set_o,C4: $o > $o,D3: $o > $o] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ C4 @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ D3 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1151_has__integral__is__0,axiom,
! [S: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ( F @ X3 )
= zero_zero_real ) )
=> ( hensto240673015341029504l_real @ F @ zero_zero_real @ S ) ) ).
% has_integral_is_0
thf(fact_1152_has__integral__on__superset,axiom,
! [F: real > real,I: real,S: set_real,T: set_real] :
( ( hensto240673015341029504l_real @ F @ I @ S )
=> ( ! [X3: real] :
( ~ ( member_real @ X3 @ S )
=> ( ( F @ X3 )
= zero_zero_real ) )
=> ( ( ord_less_eq_set_real @ S @ T )
=> ( hensto240673015341029504l_real @ F @ I @ T ) ) ) ) ).
% has_integral_on_superset
thf(fact_1153_has__integral__le,axiom,
! [F: real > real,I: real,S: set_real,G: real > real,J: real] :
( ( hensto240673015341029504l_real @ F @ I @ S )
=> ( ( hensto240673015341029504l_real @ G @ J @ S )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_real @ I @ J ) ) ) ) ).
% has_integral_le
thf(fact_1154_INF__commute,axiom,
! [F: nat > nat > set_real,B2: set_nat,A2: set_nat] :
( ( comple8289635161444856091t_real
@ ( image_nat_set_real
@ ^ [I2: nat] : ( comple8289635161444856091t_real @ ( image_nat_set_real @ ( F @ I2 ) @ B2 ) )
@ A2 ) )
= ( comple8289635161444856091t_real
@ ( image_nat_set_real
@ ^ [J2: nat] :
( comple8289635161444856091t_real
@ ( image_nat_set_real
@ ^ [I2: nat] : ( F @ I2 @ J2 )
@ A2 ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_1155_has__integral__0,axiom,
! [S: set_real] :
( hensto240673015341029504l_real
@ ^ [X2: real] : zero_zero_real
@ zero_zero_real
@ S ) ).
% has_integral_0
thf(fact_1156_has__integral__mult__left,axiom,
! [F: real > real,Y2: real,S: set_real,C: real] :
( ( hensto240673015341029504l_real @ F @ Y2 @ S )
=> ( hensto240673015341029504l_real
@ ^ [X2: real] : ( times_times_real @ ( F @ X2 ) @ C )
@ ( times_times_real @ Y2 @ C )
@ S ) ) ).
% has_integral_mult_left
thf(fact_1157_has__integral__mult__right,axiom,
! [F: real > real,Y2: real,I: set_real,C: real] :
( ( hensto240673015341029504l_real @ F @ Y2 @ I )
=> ( hensto240673015341029504l_real
@ ^ [X2: real] : ( times_times_real @ C @ ( F @ X2 ) )
@ ( times_times_real @ C @ Y2 )
@ I ) ) ).
% has_integral_mult_right
thf(fact_1158_has__integral__divide,axiom,
! [F: real > real,Y2: real,S: set_real,C: real] :
( ( hensto240673015341029504l_real @ F @ Y2 @ S )
=> ( hensto240673015341029504l_real
@ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ C )
@ ( divide_divide_real @ Y2 @ C )
@ S ) ) ).
% has_integral_divide
thf(fact_1159_INF__eq,axiom,
! [A2: set_real,B2: set_real,G: real > $o,F: real > $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ? [X4: real] :
( ( member_real @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B2 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_real_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_real_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1160_INF__eq,axiom,
! [A2: set_real,B2: set_nat,G: nat > $o,F: real > $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_real_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1161_INF__eq,axiom,
! [A2: set_real,B2: set_o,G: $o > $o,F: real > $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B2 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_real_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1162_INF__eq,axiom,
! [A2: set_nat,B2: set_real,G: real > $o,F: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: real] :
( ( member_real @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_real_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1163_INF__eq,axiom,
! [A2: set_nat,B2: set_nat,G: nat > $o,F: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1164_INF__eq,axiom,
! [A2: set_nat,B2: set_o,G: $o > $o,F: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1165_INF__eq,axiom,
! [A2: set_o,B2: set_real,G: real > $o,F: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: real] :
( ( member_real @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_real_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1166_INF__eq,axiom,
! [A2: set_o,B2: set_nat,G: nat > $o,F: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1167_INF__eq,axiom,
! [A2: set_o,B2: set_o,G: $o > $o,F: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1168_INF__eq,axiom,
! [A2: set_real,B2: set_nat,G: nat > set_real,F: real > set_real] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_set_real @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ord_less_eq_set_real @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( comple8289635161444856091t_real @ ( image_real_set_real @ F @ A2 ) )
= ( comple8289635161444856091t_real @ ( image_nat_set_real @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1169_Inf__less__eq,axiom,
! [A2: set_set_real,U2: set_real] :
( ! [V2: set_real] :
( ( member_set_real @ V2 @ A2 )
=> ( ord_less_eq_set_real @ V2 @ U2 ) )
=> ( ( A2 != bot_bot_set_set_real )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ A2 ) @ U2 ) ) ) ).
% Inf_less_eq
thf(fact_1170_Inf__less__eq,axiom,
! [A2: set_o,U2: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A2 )
=> ( ord_less_eq_o @ V2 @ U2 ) )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ U2 ) ) ) ).
% Inf_less_eq
thf(fact_1171_INF__eq__const,axiom,
! [I4: set_nat,F: nat > set_real,X: set_real] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple8289635161444856091t_real @ ( image_nat_set_real @ F @ I4 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_1172_INF__eq__const,axiom,
! [I4: set_o,F: $o > $o,X: $o] :
( ( I4 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ I4 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_1173_INF__eq__const,axiom,
! [I4: set_set_real,F: set_real > $o,X: $o] :
( ( I4 != bot_bot_set_set_real )
=> ( ! [I3: set_real] :
( ( member_set_real @ I3 @ I4 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Inf_Inf_o @ ( image_set_real_o @ F @ I4 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_1174_INF__eq__const,axiom,
! [I4: set_nat,F: nat > $o,X: $o] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ I4 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_1175_INF__eq__const,axiom,
! [I4: set_real,F: real > $o,X: $o] :
( ( I4 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I4 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Inf_Inf_o @ ( image_real_o @ F @ I4 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_1176_has__integral__subset__le,axiom,
! [S2: set_real,T2: set_real,F: real > real,I: real,J: real] :
( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ( hensto240673015341029504l_real @ F @ I @ S2 )
=> ( ( hensto240673015341029504l_real @ F @ J @ T2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ T2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ I @ J ) ) ) ) ) ).
% has_integral_subset_le
thf(fact_1177_has__integral__nonneg,axiom,
! [F: real > real,I: real,S: set_real] :
( ( hensto240673015341029504l_real @ F @ I @ S )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ I ) ) ) ).
% has_integral_nonneg
thf(fact_1178_interval__bounds__real_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( comple4887499456419720421f_real @ ( set_or1222579329274155063t_real @ A @ B ) )
= A ) ) ).
% interval_bounds_real(2)
thf(fact_1179_INF__eqI,axiom,
! [A2: set_nat,X: set_real,F: nat > set_real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ord_less_eq_set_real @ X @ ( F @ I3 ) ) )
=> ( ! [Y3: set_real] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A2 )
=> ( ord_less_eq_set_real @ Y3 @ ( F @ I5 ) ) )
=> ( ord_less_eq_set_real @ Y3 @ X ) )
=> ( ( comple8289635161444856091t_real @ ( image_nat_set_real @ F @ A2 ) )
= X ) ) ) ).
% INF_eqI
thf(fact_1180_INF__eqI,axiom,
! [A2: set_set_real,X: $o,F: set_real > $o] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ A2 )
=> ( ord_less_eq_o @ X @ ( F @ I3 ) ) )
=> ( ! [Y3: $o] :
( ! [I5: set_real] :
( ( member_set_real @ I5 @ A2 )
=> ( ord_less_eq_o @ Y3 @ ( F @ I5 ) ) )
=> ( ord_less_eq_o @ Y3 @ X ) )
=> ( ( complete_Inf_Inf_o @ ( image_set_real_o @ F @ A2 ) )
= X ) ) ) ).
% INF_eqI
thf(fact_1181_INF__eqI,axiom,
! [A2: set_real,X: $o,F: real > $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ( ord_less_eq_o @ X @ ( F @ I3 ) ) )
=> ( ! [Y3: $o] :
( ! [I5: real] :
( ( member_real @ I5 @ A2 )
=> ( ord_less_eq_o @ Y3 @ ( F @ I5 ) ) )
=> ( ord_less_eq_o @ Y3 @ X ) )
=> ( ( complete_Inf_Inf_o @ ( image_real_o @ F @ A2 ) )
= X ) ) ) ).
% INF_eqI
thf(fact_1182_INF__eqI,axiom,
! [A2: set_nat,X: $o,F: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ord_less_eq_o @ X @ ( F @ I3 ) ) )
=> ( ! [Y3: $o] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A2 )
=> ( ord_less_eq_o @ Y3 @ ( F @ I5 ) ) )
=> ( ord_less_eq_o @ Y3 @ X ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A2 ) )
= X ) ) ) ).
% INF_eqI
thf(fact_1183_INF__eqI,axiom,
! [A2: set_o,X: $o,F: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ( ord_less_eq_o @ X @ ( F @ I3 ) ) )
=> ( ! [Y3: $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_o @ Y3 @ ( F @ I5 ) ) )
=> ( ord_less_eq_o @ Y3 @ X ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= X ) ) ) ).
% INF_eqI
thf(fact_1184_INF__mono,axiom,
! [B2: set_set_real,A2: set_nat,F: nat > set_real,G: set_real > set_real] :
( ! [M3: set_real] :
( ( member_set_real @ M3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_set_real @ ( F @ X4 ) @ ( G @ M3 ) ) ) )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_nat_set_real @ F @ A2 ) ) @ ( comple8289635161444856091t_real @ ( image_2436557299294012491t_real @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_1185_INF__mono,axiom,
! [B2: set_real,A2: set_nat,F: nat > set_real,G: real > set_real] :
( ! [M3: real] :
( ( member_real @ M3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_set_real @ ( F @ X4 ) @ ( G @ M3 ) ) ) )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_nat_set_real @ F @ A2 ) ) @ ( comple8289635161444856091t_real @ ( image_real_set_real @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_1186_INF__mono,axiom,
! [B2: set_nat,A2: set_nat,F: nat > set_real,G: nat > set_real] :
( ! [M3: nat] :
( ( member_nat @ M3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_set_real @ ( F @ X4 ) @ ( G @ M3 ) ) ) )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_nat_set_real @ F @ A2 ) ) @ ( comple8289635161444856091t_real @ ( image_nat_set_real @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_1187_INF__mono,axiom,
! [B2: set_o,A2: set_nat,F: nat > set_real,G: $o > set_real] :
( ! [M3: $o] :
( ( member_o @ M3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_set_real @ ( F @ X4 ) @ ( G @ M3 ) ) ) )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_nat_set_real @ F @ A2 ) ) @ ( comple8289635161444856091t_real @ ( image_o_set_real @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_1188_INF__lower,axiom,
! [I: nat,A2: set_nat,F: nat > set_real] :
( ( member_nat @ I @ A2 )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_nat_set_real @ F @ A2 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_1189_INF__lower,axiom,
! [I: set_real,A2: set_set_real,F: set_real > $o] :
( ( member_set_real @ I @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_set_real_o @ F @ A2 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_1190_INF__lower,axiom,
! [I: real,A2: set_real,F: real > $o] :
( ( member_real @ I @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_real_o @ F @ A2 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_1191_INF__lower,axiom,
! [I: nat,A2: set_nat,F: nat > $o] :
( ( member_nat @ I @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A2 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_1192_INF__lower,axiom,
! [I: $o,A2: set_o,F: $o > $o] :
( ( member_o @ I @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_1193_INF__mono_H,axiom,
! [F: nat > set_real,G: nat > set_real,A2: set_nat] :
( ! [X3: nat] : ( ord_less_eq_set_real @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_nat_set_real @ F @ A2 ) ) @ ( comple8289635161444856091t_real @ ( image_nat_set_real @ G @ A2 ) ) ) ) ).
% INF_mono'
thf(fact_1194_INF__lower2,axiom,
! [I: set_real,A2: set_set_real,F: set_real > $o,U2: $o] :
( ( member_set_real @ I @ A2 )
=> ( ( ord_less_eq_o @ ( F @ I ) @ U2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_set_real_o @ F @ A2 ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1195_INF__lower2,axiom,
! [I: real,A2: set_real,F: real > $o,U2: $o] :
( ( member_real @ I @ A2 )
=> ( ( ord_less_eq_o @ ( F @ I ) @ U2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_real_o @ F @ A2 ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1196_INF__lower2,axiom,
! [I: nat,A2: set_nat,F: nat > $o,U2: $o] :
( ( member_nat @ I @ A2 )
=> ( ( ord_less_eq_o @ ( F @ I ) @ U2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A2 ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1197_INF__lower2,axiom,
! [I: $o,A2: set_o,F: $o > $o,U2: $o] :
( ( member_o @ I @ A2 )
=> ( ( ord_less_eq_o @ ( F @ I ) @ U2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1198_f12,axiom,
( hensto240673015341029504l_real
@ ^ [X2: real] : ( minus_minus_real @ ( f2 @ X2 ) @ ( f1 @ X2 ) )
@ ( groups8702937949983641418l_real
@ ^ [K5: set_real] : ( times_times_real @ ( minus_minus_real @ ( f @ ( comple1385675409528146559p_real @ K5 ) ) @ ( f @ ( comple4887499456419720421f_real @ K5 ) ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) )
@ ( regular_division @ zero_zero_real @ a @ n ) )
@ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) ) ).
% f12
thf(fact_1199_int__21__D,axiom,
! [K: set_real] :
( ( member_set_real @ K @ ( regular_division @ zero_zero_real @ a @ n ) )
=> ( hensto240673015341029504l_real
@ ^ [X2: real] : ( minus_minus_real @ ( f2 @ X2 ) @ ( f1 @ X2 ) )
@ ( times_times_real @ ( minus_minus_real @ ( f @ ( comple1385675409528146559p_real @ K ) ) @ ( f @ ( comple4887499456419720421f_real @ K ) ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) )
@ K ) ) ).
% int_21_D
thf(fact_1200__092_060open_062_I_092_060Sum_062K_092_060in_062regular__division_A0_Aa_An_O_A_If_A_ISup_AK_J_A_N_Af_A_IInf_AK_J_J_A_K_A_Ia_A_P_Areal_An_J_J_A_060_A_092_060epsilon_062_092_060close_062,axiom,
( ord_less_real
@ ( groups8702937949983641418l_real
@ ^ [K5: set_real] : ( times_times_real @ ( minus_minus_real @ ( f @ ( comple1385675409528146559p_real @ K5 ) ) @ ( f @ ( comple4887499456419720421f_real @ K5 ) ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) )
@ ( regular_division @ zero_zero_real @ a @ n ) )
@ epsilon ) ).
% \<open>(\<Sum>K\<in>regular_division 0 a n. (f (Sup K) - f (Inf K)) * (a / real n)) < \<epsilon>\<close>
thf(fact_1201_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1202_zero__to__b__eq,axiom,
( ( set_or1222579329274155063t_real @ zero_zero_real @ b )
= ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [K3: nat] : ( set_or1222579329274155063t_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K3 ) ) ) @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K3 ) ) ) ) )
@ ( set_ord_lessThan_nat @ n ) ) ) ) ).
% zero_to_b_eq
thf(fact_1203_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1204_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1205_Union__regular__division,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( N = zero_zero_nat )
=> ( ( comple3096694443085538997t_real @ ( regular_division @ A @ B @ N ) )
= bot_bot_set_real ) )
& ( ( N != zero_zero_nat )
=> ( ( comple3096694443085538997t_real @ ( regular_division @ A @ B @ N ) )
= ( set_or1222579329274155063t_real @ A @ B ) ) ) ) ) ).
% Union_regular_division
thf(fact_1206_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( ( K2 = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1207_nat__mult__less__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1208_nat__mult__eq__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1209_nat__mult__div__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( K2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= zero_zero_nat ) )
& ( ( K2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1210_nat__mult__le__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1211_nat__mult__div__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1212_less,axiom,
! [K: set_real] :
( ( member_set_real @ K @ ( regular_division @ zero_zero_real @ a @ n ) )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( f @ ( comple1385675409528146559p_real @ K ) ) @ ( f @ ( comple4887499456419720421f_real @ K ) ) ) ) @ ( divide_divide_real @ epsilon @ a ) ) ) ).
% less
thf(fact_1213_f2__near__f1,axiom,
( ord_less_real
@ ( hensto2714581292692559302l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ a )
@ ^ [X2: real] : ( minus_minus_real @ ( f2 @ X2 ) @ ( f1 @ X2 ) ) )
@ epsilon ) ).
% f2_near_f1
thf(fact_1214_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1215_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1216_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1217_Suc__diff__diff,axiom,
! [M: nat,N: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_1218_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1219__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062del_O_A_092_060lbrakk_062_092_060And_062e_O_A0_A_060_Ae_A_092_060Longrightarrow_062_A0_A_060_Adel_Ae_059_A_092_060And_062e_Ax_Ax_H_O_A_092_060lbrakk_062_092_060bar_062x_H_A_N_Ax_092_060bar_062_A_060_Adel_Ae_059_A0_A_060_Ae_059_Ax_A_092_060in_062_A_1230_O_Oa_125_059_Ax_H_A_092_060in_062_A_1230_O_Oa_125_092_060rbrakk_062_A_092_060Longrightarrow_062_A_092_060bar_062f_Ax_H_A_N_Af_Ax_092_060bar_062_A_060_Ae_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Del: real > real] :
( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_less_real @ zero_zero_real @ ( Del @ E2 ) ) )
=> ~ ! [E2: real,X4: real,X8: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X8 @ X4 ) ) @ ( Del @ E2 ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
=> ( ( member_real @ X8 @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( f @ X8 ) @ ( f @ X4 ) ) ) @ E2 ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>del. \<lbrakk>\<And>e. 0 < e \<Longrightarrow> 0 < del e; \<And>e x x'. \<lbrakk>\<bar>x' - x\<bar> < del e; 0 < e; x \<in> {0..a}; x' \<in> {0..a}\<rbrakk> \<Longrightarrow> \<bar>f x' - f x\<bar> < e\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1220_del,axiom,
! [X9: real,X: real,E: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X9 @ X ) ) @ ( del @ E ) )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ( ( member_real @ X @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
=> ( ( member_real @ X9 @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( f @ X9 ) @ ( f @ X ) ) ) @ E ) ) ) ) ) ).
% del
thf(fact_1221_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1222_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1223_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1224_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1225_Inf__bool__def,axiom,
( complete_Inf_Inf_o
= ( ^ [A6: set_o] :
~ ( member_o @ $false @ A6 ) ) ) ).
% Inf_bool_def
thf(fact_1226_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1227_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_1228_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1229_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1230_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1231_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1232_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1233_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1234_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1235_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1236_le__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1237_eq__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1238_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1239_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_1240_minus__int__code_I1_J,axiom,
! [K2: int] :
( ( minus_minus_int @ K2 @ zero_zero_int )
= K2 ) ).
% minus_int_code(1)
thf(fact_1241_diff__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_1242_diff__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1243_int__diff__cases,axiom,
! [Z: int] :
~ ! [M3: nat,N2: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_1244_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1245_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1246_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1247_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1248_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1249_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1250_less__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1251_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1252_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1253_div__if,axiom,
( divide_divide_nat
= ( ^ [M7: nat,N4: nat] :
( if_nat
@ ( ( ord_less_nat @ M7 @ N4 )
| ( N4 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M7 @ N4 ) @ N4 ) ) ) ) ) ).
% div_if
thf(fact_1254_le__div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ M @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_1255_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A: nat > nat,B: nat > nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J3 ) ) ) )
=> ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ord_less_eq_nat @ ( B @ J3 ) @ ( B @ I3 ) ) ) )
=> ( ord_less_eq_nat
@ ( times_times_nat @ N
@ ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( B @ I2 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
@ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_1256_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y2: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y2 ) ) )
= ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) )
& ( ( ord_less_nat @ X @ Y2 )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1257_conj__le__cong,axiom,
! [X: int,X9: int,P: $o,P2: $o] :
( ( X = X9 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X9 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X9 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_1258_imp__le__cong,axiom,
! [X: int,X9: int,P: $o,P2: $o] :
( ( X = X9 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X9 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X9 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_1259_plusinfinity,axiom,
! [D: int,P2: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K4: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D ) ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [X_12: int] : ( P2 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1260_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K4: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D ) ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1261_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K2: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1262_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_1263_Union__segment__image,axiom,
! [K2: nat,N: nat] :
( ( ( K2 = zero_zero_nat )
=> ( ( comple3096694443085538997t_real @ ( image_nat_set_real @ ( segment @ N ) @ ( set_ord_lessThan_nat @ K2 ) ) )
= bot_bot_set_real ) )
& ( ( K2 != zero_zero_nat )
=> ( ( comple3096694443085538997t_real @ ( image_nat_set_real @ ( segment @ N ) @ ( set_ord_lessThan_nat @ K2 ) ) )
= ( set_or1222579329274155063t_real @ zero_zero_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% Union_segment_image
thf(fact_1264_segment__nonempty,axiom,
! [N: nat,K2: nat] :
( ( segment @ N @ K2 )
!= bot_bot_set_real ) ).
% segment_nonempty
thf(fact_1265_complete__real,axiom,
! [S: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S )
=> ( ? [Z3: real] :
! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ? [Y3: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Y3 ) )
& ! [Z3: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ord_less_eq_real @ Y3 @ Z3 ) ) ) ) ) ).
% complete_real
thf(fact_1266_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% less_eq_real_def
thf(fact_1267_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y4: real] :
? [N2: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1268_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1269_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( groups8702937949983641418l_real
@ ^ [K5: set_real] : ( times_times_real @ ( f @ ( comple4887499456419720421f_real @ K5 ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) )
@ ( regular_division @ zero_zero_real @ a @ n ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K3 ) ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) )
@ ( set_ord_lessThan_nat @ n ) ) ) ).
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