TPTP Problem File: SLH0224^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Cotangent_PFD_Formula/0007_Cotangent_PFD_Formula/prob_00278_010775__14033972_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1417 ( 607 unt; 145 typ; 0 def)
% Number of atoms : 3560 (1067 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 9493 ( 366 ~; 86 |; 132 &;7443 @)
% ( 0 <=>;1466 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 6 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 437 ( 437 >; 0 *; 0 +; 0 <<)
% Number of symbols : 133 ( 132 usr; 32 con; 0-4 aty)
% Number of variables : 3149 ( 230 ^;2884 !; 35 ?;3149 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:58:03.711
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
thf(ty_n_t__Bounded____Continuous____Function__Obcontfun_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
bounde2223080031472583990omplex: $tType ).
thf(ty_n_t__Bounded____Continuous____Function__Obcontfun_It__Real__Oreal_Mt__Real__Oreal_J,type,
bounde2103802982846620082l_real: $tType ).
thf(ty_n_t__Set__Oset_It__Extended____Nat__Oenat_J,type,
set_Extended_enat: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $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__Extended____Nat__Oenat,type,
extended_enat: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (132)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
archim6058952711729229775r_real: real > int ).
thf(sy_c_Bounded__Continuous__Function_Obcontfun_Oapply__bcontfun_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
bounde4845209636850564516omplex: bounde2223080031472583990omplex > complex > complex ).
thf(sy_c_Bounded__Continuous__Function_Obcontfun_Oapply__bcontfun_001t__Real__Oreal_001t__Real__Oreal,type,
bounde974565180584300064l_real: bounde2103802982846620082l_real > real > real ).
thf(sy_c_Bounded__Continuous__Function_Oconst__bcontfun_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
bounde964246820246105844omplex: complex > bounde2223080031472583990omplex ).
thf(sy_c_Complex_Ocomplex_ORe,type,
re: complex > real ).
thf(sy_c_Complex__Analysis__Basics_Oholomorphic__on,type,
comple7700996537433184370hic_on: ( complex > complex ) > set_complex > $o ).
thf(sy_c_Cotangent__PFD__Formula_Ocot__pfd_001t__Complex__Ocomplex,type,
cotang8298477626502807258omplex: complex > complex ).
thf(sy_c_Cotangent__PFD__Formula_Ocot__pfd_001t__Real__Oreal,type,
cotang1502006655779026648d_real: real > real ).
thf(sy_c_Derivative_Opiecewise__differentiable__on_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
piecew5880207932299811686omplex: ( complex > complex ) > set_complex > $o ).
thf(sy_c_Derivative_Opiecewise__differentiable__on_001t__Real__Oreal_001t__Real__Oreal,type,
piecew745371121812451810l_real: ( real > real ) > set_real > $o ).
thf(sy_c_Factorial__Ring_Ofactorial__semiring__class_OGcd__factorial_001t__Int__Oint,type,
factor8536668470562680120al_int: set_int > int ).
thf(sy_c_Factorial__Ring_Ofactorial__semiring__class_OGcd__factorial_001t__Nat__Onat,type,
factor8539158941071730396al_nat: set_nat > nat ).
thf(sy_c_Gamma__Function_OGamma__class_OrGamma_001t__Complex__Ocomplex,type,
gamma_4773869415665495160omplex: complex > complex ).
thf(sy_c_Gamma__Function_OGamma__class_OrGamma_001t__Real__Oreal,type,
gamma_4599518313873207670a_real: real > real ).
thf(sy_c_Gamma__Function_Oln__Gamma__series_001t__Complex__Ocomplex,type,
gamma_3083380197916210424omplex: complex > nat > complex ).
thf(sy_c_Gamma__Function_Oln__Gamma__series_001t__Real__Oreal,type,
gamma_2780986569588390390s_real: real > nat > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Bounded____Continuous____Function__Obcontfun_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
minus_5896967405095338141omplex: bounde2223080031472583990omplex > bounde2223080031472583990omplex > bounde2223080031472583990omplex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
minus_minus_complex: complex > complex > complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
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_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
one_on7984719198319812577d_enat: extended_enat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Set__Oset_It__Complex__Ocomplex_J,type,
one_one_set_complex: set_complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Set__Oset_It__Int__Oint_J,type,
one_one_set_int: set_int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Set__Oset_It__Nat__Onat_J,type,
one_one_set_nat: set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Set__Oset_It__Real__Oreal_J,type,
one_one_set_real: set_real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Bounded____Continuous____Function__Obcontfun_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
uminus5413038955693695277omplex: bounde2223080031472583990omplex > bounde2223080031472583990omplex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
uminus1482373934393186551omplex: complex > complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
uminus8566677241136511917omplex: set_complex > set_complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
uminus612125837232591019t_real: set_real > set_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Bounded____Continuous____Function__Obcontfun_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
zero_z5184887426917238893omplex: bounde2223080031472583990omplex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
zero_z5237406670263579293d_enat: extended_enat ).
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_Ozero__class_Ozero_001t__Set__Oset_It__Complex__Ocomplex_J,type,
zero_z6614145512433583213omplex: set_complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
zero_z5219068577256268541d_enat: set_Extended_enat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Int__Oint_J,type,
zero_zero_set_int: set_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Nat__Onat_J,type,
zero_zero_set_nat: set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Real__Oreal_J,type,
zero_zero_set_real: set_real ).
thf(sy_c_Homeomorphism_Ocovering__space_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
coveri2851631837752942234omplex: set_complex > ( complex > complex ) > set_complex > $o ).
thf(sy_c_Homeomorphism_Ocovering__space_001t__Real__Oreal_001t__Real__Oreal,type,
coveri38709745440887062l_real: set_real > ( real > real ) > set_real > $o ).
thf(sy_c_Homotopy_ORetracts_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
retrac6640447365452120649omplex: set_complex > ( complex > complex ) > set_complex > ( complex > complex ) > $o ).
thf(sy_c_Homotopy_ORetracts_001t__Real__Oreal_001t__Real__Oreal,type,
retracts_real_real: set_real > ( real > real ) > set_real > ( real > real ) > $o ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Complex__Ocomplex,type,
ring_1_Ints_complex: set_complex ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
ring_1_Ints_int: set_int ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
ring_1_Ints_real: set_real ).
thf(sy_c_Lipschitz_Olocal__lipschitz_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
local_3408543445516732330omplex: set_complex > set_complex > ( complex > complex > complex ) > $o ).
thf(sy_c_Lipschitz_Olocal__lipschitz_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Real__Oreal,type,
local_3396545069758556710l_real: set_complex > set_real > ( complex > real > real ) > $o ).
thf(sy_c_Lipschitz_Olocal__lipschitz_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
local_3489851796131349160omplex: set_real > set_complex > ( real > complex > complex ) > $o ).
thf(sy_c_Lipschitz_Olocal__lipschitz_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
local_8704260266193821476l_real: set_real > set_real > ( real > real > real ) > $o ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Complex__Ocomplex_M_Eo_J,type,
bot_bot_complex_o: complex > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
bot_bot_real_o: real > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
bot_bo4199563552545308370d_enat: extended_enat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
bot_bot_set_complex: set_complex ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
bot_bo7653980558646680370d_enat: set_Extended_enat ).
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_Oord__class_Oless_001t__Complex__Ocomplex,type,
ord_less_complex: complex > complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
ord_le72135733267957522d_enat: extended_enat > extended_enat > $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__Num__Onum,type,
ord_less_num: num > num > $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__Complex__Ocomplex_J,type,
ord_less_set_complex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Complex__Ocomplex,type,
ord_less_eq_complex: complex > complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $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__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_le211207098394363844omplex: set_complex > set_complex > $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_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
real_V4546457046886955230omplex: real > complex ).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
real_V1803761363581548252l_real: real > real ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Bounded____Continuous____Function__Obcontfun_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
real_V9054712056520343207omplex: real > bounde2223080031472583990omplex > bounde2223080031472583990omplex ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
real_V2046097035970521341omplex: real > complex > complex ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
real_V1485227260804924795R_real: real > real > real ).
thf(sy_c_Real__Vector__Spaces_Ospan_001t__Complex__Ocomplex,type,
real_V8921647422947696435omplex: set_complex > set_complex ).
thf(sy_c_Real__Vector__Spaces_Ospan_001t__Real__Oreal,type,
real_V5325414057265605809n_real: set_real > set_real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
divide1717551699836669952omplex: complex > complex > complex ).
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_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
insert_complex: complex > set_complex > set_complex ).
thf(sy_c_Set_Oinsert_001t__Extended____Nat__Oenat,type,
insert_Extended_enat: extended_enat > set_Extended_enat > set_Extended_enat ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Ois__empty_001t__Complex__Ocomplex,type,
is_empty_complex: set_complex > $o ).
thf(sy_c_Set_Ois__singleton_001t__Complex__Ocomplex,type,
is_singleton_complex: set_complex > $o ).
thf(sy_c_Set_Ois__singleton_001t__Real__Oreal,type,
is_singleton_real: set_real > $o ).
thf(sy_c_Set_Oremove_001t__Complex__Ocomplex,type,
remove_complex: complex > set_complex > set_complex ).
thf(sy_c_Set_Oremove_001t__Real__Oreal,type,
remove_real: real > set_real > set_real ).
thf(sy_c_Set_Othe__elem_001t__Complex__Ocomplex,type,
the_elem_complex: set_complex > complex ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
topolo9015423870875150044omplex: set_complex > ( complex > complex ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Complex__Ocomplex,type,
topolo8620507378200602458omplex: set_real > ( real > complex ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_A,type,
a: set_complex ).
% Relevant facts (1271)
thf(fact_0_continuous__on__eq,axiom,
! [S: set_real,F: real > real,G: real > real] :
( ( topolo5044208981011980120l_real @ S @ F )
=> ( ! [X: real] :
( ( member_real @ X @ S )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( topolo5044208981011980120l_real @ S @ G ) ) ) ).
% continuous_on_eq
thf(fact_1_continuous__on__eq,axiom,
! [S: set_complex,F: complex > complex,G: complex > complex] :
( ( topolo9015423870875150044omplex @ S @ F )
=> ( ! [X: complex] :
( ( member_complex @ X @ S )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( topolo9015423870875150044omplex @ S @ G ) ) ) ).
% continuous_on_eq
thf(fact_2_continuous__on__cong,axiom,
! [S: set_real,T: set_real,F: real > real,G: real > real] :
( ( S = T )
=> ( ! [X: real] :
( ( member_real @ X @ T )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( topolo5044208981011980120l_real @ S @ F )
= ( topolo5044208981011980120l_real @ T @ G ) ) ) ) ).
% continuous_on_cong
thf(fact_3_continuous__on__cong,axiom,
! [S: set_complex,T: set_complex,F: complex > complex,G: complex > complex] :
( ( S = T )
=> ( ! [X: complex] :
( ( member_complex @ X @ T )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( topolo9015423870875150044omplex @ S @ F )
= ( topolo9015423870875150044omplex @ T @ G ) ) ) ) ).
% continuous_on_cong
thf(fact_4_cot__pfd__complex__of__real,axiom,
! [X2: real] :
( ( cotang8298477626502807258omplex @ ( real_V4546457046886955230omplex @ X2 ) )
= ( real_V4546457046886955230omplex @ ( cotang1502006655779026648d_real @ X2 ) ) ) ).
% cot_pfd_complex_of_real
thf(fact_5_Retracts_Oconth,axiom,
! [S: set_real,H: real > real,T: set_real,K: real > real] :
( ( retracts_real_real @ S @ H @ T @ K )
=> ( topolo5044208981011980120l_real @ S @ H ) ) ).
% Retracts.conth
thf(fact_6_Retracts_Oconth,axiom,
! [S: set_complex,H: complex > complex,T: set_complex,K: complex > complex] :
( ( retrac6640447365452120649omplex @ S @ H @ T @ K )
=> ( topolo9015423870875150044omplex @ S @ H ) ) ).
% Retracts.conth
thf(fact_7_Retracts_Ocontk,axiom,
! [S: set_real,H: real > real,T: set_real,K: real > real] :
( ( retracts_real_real @ S @ H @ T @ K )
=> ( topolo5044208981011980120l_real @ T @ K ) ) ).
% Retracts.contk
thf(fact_8_Retracts_Ocontk,axiom,
! [S: set_complex,H: complex > complex,T: set_complex,K: complex > complex] :
( ( retrac6640447365452120649omplex @ S @ H @ T @ K )
=> ( topolo9015423870875150044omplex @ T @ K ) ) ).
% Retracts.contk
thf(fact_9_local__lipschitz__continuous__on,axiom,
! [T2: set_complex,X3: set_real,F: complex > real > real,T: complex] :
( ( local_3396545069758556710l_real @ T2 @ X3 @ F )
=> ( ( member_complex @ T @ T2 )
=> ( topolo5044208981011980120l_real @ X3 @ ( F @ T ) ) ) ) ).
% local_lipschitz_continuous_on
thf(fact_10_local__lipschitz__continuous__on,axiom,
! [T2: set_real,X3: set_real,F: real > real > real,T: real] :
( ( local_8704260266193821476l_real @ T2 @ X3 @ F )
=> ( ( member_real @ T @ T2 )
=> ( topolo5044208981011980120l_real @ X3 @ ( F @ T ) ) ) ) ).
% local_lipschitz_continuous_on
thf(fact_11_local__lipschitz__continuous__on,axiom,
! [T2: set_complex,X3: set_complex,F: complex > complex > complex,T: complex] :
( ( local_3408543445516732330omplex @ T2 @ X3 @ F )
=> ( ( member_complex @ T @ T2 )
=> ( topolo9015423870875150044omplex @ X3 @ ( F @ T ) ) ) ) ).
% local_lipschitz_continuous_on
thf(fact_12_local__lipschitz__continuous__on,axiom,
! [T2: set_real,X3: set_complex,F: real > complex > complex,T: real] :
( ( local_3489851796131349160omplex @ T2 @ X3 @ F )
=> ( ( member_real @ T @ T2 )
=> ( topolo9015423870875150044omplex @ X3 @ ( F @ T ) ) ) ) ).
% local_lipschitz_continuous_on
thf(fact_13_piecewise__differentiable__on__imp__continuous__on,axiom,
! [F: real > real,S2: set_real] :
( ( piecew745371121812451810l_real @ F @ S2 )
=> ( topolo5044208981011980120l_real @ S2 @ F ) ) ).
% piecewise_differentiable_on_imp_continuous_on
thf(fact_14_piecewise__differentiable__on__imp__continuous__on,axiom,
! [F: complex > complex,S2: set_complex] :
( ( piecew5880207932299811686omplex @ F @ S2 )
=> ( topolo9015423870875150044omplex @ S2 @ F ) ) ).
% piecewise_differentiable_on_imp_continuous_on
thf(fact_15_continuous__on__apply__bcontfun,axiom,
! [T2: set_real,X2: bounde2103802982846620082l_real] : ( topolo5044208981011980120l_real @ T2 @ ( bounde974565180584300064l_real @ X2 ) ) ).
% continuous_on_apply_bcontfun
thf(fact_16_continuous__on__apply__bcontfun,axiom,
! [T2: set_complex,X2: bounde2223080031472583990omplex] : ( topolo9015423870875150044omplex @ T2 @ ( bounde4845209636850564516omplex @ X2 ) ) ).
% continuous_on_apply_bcontfun
thf(fact_17_continuous__on__rGamma,axiom,
! [A: set_complex] : ( topolo9015423870875150044omplex @ A @ gamma_4773869415665495160omplex ) ).
% continuous_on_rGamma
thf(fact_18_continuous__on__rGamma,axiom,
! [A: set_real] : ( topolo5044208981011980120l_real @ A @ gamma_4599518313873207670a_real ) ).
% continuous_on_rGamma
thf(fact_19_assms,axiom,
ord_le211207098394363844omplex @ a @ ( uminus8566677241136511917omplex @ ( minus_811609699411566653omplex @ ring_1_Ints_complex @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) ) ).
% assms
thf(fact_20_covering__space__imp__continuous,axiom,
! [C: set_complex,P: complex > complex,S2: set_complex] :
( ( coveri2851631837752942234omplex @ C @ P @ S2 )
=> ( topolo9015423870875150044omplex @ C @ P ) ) ).
% covering_space_imp_continuous
thf(fact_21_covering__space__imp__continuous,axiom,
! [C: set_real,P: real > real,S2: set_real] :
( ( coveri38709745440887062l_real @ C @ P @ S2 )
=> ( topolo5044208981011980120l_real @ C @ P ) ) ).
% covering_space_imp_continuous
thf(fact_22_zero__bcontfun_Orep__eq,axiom,
( ( bounde4845209636850564516omplex @ zero_z5184887426917238893omplex )
= ( ^ [Uu: complex] : zero_zero_complex ) ) ).
% zero_bcontfun.rep_eq
thf(fact_23_rGamma__0,axiom,
( ( gamma_4773869415665495160omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% rGamma_0
thf(fact_24_rGamma__0,axiom,
( ( gamma_4599518313873207670a_real @ zero_zero_real )
= zero_zero_real ) ).
% rGamma_0
thf(fact_25_minus__bcontfun_Orep__eq,axiom,
! [X2: bounde2223080031472583990omplex,Xa: bounde2223080031472583990omplex] :
( ( bounde4845209636850564516omplex @ ( minus_5896967405095338141omplex @ X2 @ Xa ) )
= ( ^ [X4: complex] : ( minus_minus_complex @ ( bounde4845209636850564516omplex @ X2 @ X4 ) @ ( bounde4845209636850564516omplex @ Xa @ X4 ) ) ) ) ).
% minus_bcontfun.rep_eq
thf(fact_26_uminus__bcontfun_Orep__eq,axiom,
! [X2: bounde2223080031472583990omplex] :
( ( bounde4845209636850564516omplex @ ( uminus5413038955693695277omplex @ X2 ) )
= ( ^ [X4: complex] : ( uminus1482373934393186551omplex @ ( bounde4845209636850564516omplex @ X2 @ X4 ) ) ) ) ).
% uminus_bcontfun.rep_eq
thf(fact_27_Retracts_Oidhk,axiom,
! [S: set_complex,H: complex > complex,T: set_complex,K: complex > complex,Y: complex] :
( ( retrac6640447365452120649omplex @ S @ H @ T @ K )
=> ( ( member_complex @ Y @ T )
=> ( ( H @ ( K @ Y ) )
= Y ) ) ) ).
% Retracts.idhk
thf(fact_28_apply__bcontfun__inject,axiom,
! [X2: bounde2223080031472583990omplex,Y: bounde2223080031472583990omplex] :
( ( ( bounde4845209636850564516omplex @ X2 )
= ( bounde4845209636850564516omplex @ Y ) )
= ( X2 = Y ) ) ).
% apply_bcontfun_inject
thf(fact_29_local__lipschitz__subset,axiom,
! [T2: set_real,X3: set_complex,F: real > complex > complex,S2: set_real,Y2: set_complex] :
( ( local_3489851796131349160omplex @ T2 @ X3 @ F )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ( ord_le211207098394363844omplex @ Y2 @ X3 )
=> ( local_3489851796131349160omplex @ S2 @ Y2 @ F ) ) ) ) ).
% local_lipschitz_subset
thf(fact_30_local__lipschitz__subset,axiom,
! [T2: set_complex,X3: set_complex,F: complex > complex > complex,S2: set_complex,Y2: set_complex] :
( ( local_3408543445516732330omplex @ T2 @ X3 @ F )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ( ord_le211207098394363844omplex @ Y2 @ X3 )
=> ( local_3408543445516732330omplex @ S2 @ Y2 @ F ) ) ) ) ).
% local_lipschitz_subset
thf(fact_31_rGamma__complex__of__real,axiom,
! [X2: real] :
( ( gamma_4773869415665495160omplex @ ( real_V4546457046886955230omplex @ X2 ) )
= ( real_V4546457046886955230omplex @ ( gamma_4599518313873207670a_real @ X2 ) ) ) ).
% rGamma_complex_of_real
thf(fact_32_continuous__on__sing,axiom,
! [X2: complex,F: complex > complex] : ( topolo9015423870875150044omplex @ ( insert_complex @ X2 @ bot_bot_set_complex ) @ F ) ).
% continuous_on_sing
thf(fact_33_continuous__on__sing,axiom,
! [X2: real,F: real > real] : ( topolo5044208981011980120l_real @ ( insert_real @ X2 @ bot_bot_set_real ) @ F ) ).
% continuous_on_sing
thf(fact_34_bcontfun__eqI,axiom,
! [F: bounde2223080031472583990omplex,G: bounde2223080031472583990omplex] :
( ! [X: complex] :
( ( bounde4845209636850564516omplex @ F @ X )
= ( bounde4845209636850564516omplex @ G @ X ) )
=> ( F = G ) ) ).
% bcontfun_eqI
thf(fact_35_piecewise__differentiable__on__subset,axiom,
! [F: complex > complex,S2: set_complex,T2: set_complex] :
( ( piecew5880207932299811686omplex @ F @ S2 )
=> ( ( ord_le211207098394363844omplex @ T2 @ S2 )
=> ( piecew5880207932299811686omplex @ F @ T2 ) ) ) ).
% piecewise_differentiable_on_subset
thf(fact_36_continuous__on__subset,axiom,
! [S: set_complex,F: complex > complex,T: set_complex] :
( ( topolo9015423870875150044omplex @ S @ F )
=> ( ( ord_le211207098394363844omplex @ T @ S )
=> ( topolo9015423870875150044omplex @ T @ F ) ) ) ).
% continuous_on_subset
thf(fact_37_continuous__on__subset,axiom,
! [S: set_real,F: real > real,T: set_real] :
( ( topolo5044208981011980120l_real @ S @ F )
=> ( ( ord_less_eq_set_real @ T @ S )
=> ( topolo5044208981011980120l_real @ T @ F ) ) ) ).
% continuous_on_subset
thf(fact_38_continuous__on__empty,axiom,
! [F: complex > complex] : ( topolo9015423870875150044omplex @ bot_bot_set_complex @ F ) ).
% continuous_on_empty
thf(fact_39_continuous__on__empty,axiom,
! [F: real > real] : ( topolo5044208981011980120l_real @ bot_bot_set_real @ F ) ).
% continuous_on_empty
thf(fact_40_subset__Compl__singleton,axiom,
! [A: set_real,B: real] :
( ( ord_less_eq_set_real @ A @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
= ( ~ ( member_real @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_41_subset__Compl__singleton,axiom,
! [A: set_complex,B: complex] :
( ( ord_le211207098394363844omplex @ A @ ( uminus8566677241136511917omplex @ ( insert_complex @ B @ bot_bot_set_complex ) ) )
= ( ~ ( member_complex @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_42_insert__Diff__single,axiom,
! [A2: complex,A: set_complex] :
( ( insert_complex @ A2 @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
= ( insert_complex @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_43_Diff__eq__empty__iff,axiom,
! [A: set_complex,B2: set_complex] :
( ( ( minus_811609699411566653omplex @ A @ B2 )
= bot_bot_set_complex )
= ( ord_le211207098394363844omplex @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_44_singleton__insert__inj__eq,axiom,
! [B: complex,A2: complex,A: set_complex] :
( ( ( insert_complex @ B @ bot_bot_set_complex )
= ( insert_complex @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le211207098394363844omplex @ A @ ( insert_complex @ B @ bot_bot_set_complex ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_45_singleton__insert__inj__eq_H,axiom,
! [A2: complex,A: set_complex,B: complex] :
( ( ( insert_complex @ A2 @ A )
= ( insert_complex @ B @ bot_bot_set_complex ) )
= ( ( A2 = B )
& ( ord_le211207098394363844omplex @ A @ ( insert_complex @ B @ bot_bot_set_complex ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_46_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_47_verit__minus__simplify_I3_J,axiom,
! [B: complex] :
( ( minus_minus_complex @ zero_zero_complex @ B )
= ( uminus1482373934393186551omplex @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_48_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_49_diff__0,axiom,
! [A2: real] :
( ( minus_minus_real @ zero_zero_real @ A2 )
= ( uminus_uminus_real @ A2 ) ) ).
% diff_0
thf(fact_50_diff__0,axiom,
! [A2: complex] :
( ( minus_minus_complex @ zero_zero_complex @ A2 )
= ( uminus1482373934393186551omplex @ A2 ) ) ).
% diff_0
thf(fact_51_diff__0,axiom,
! [A2: int] :
( ( minus_minus_int @ zero_zero_int @ A2 )
= ( uminus_uminus_int @ A2 ) ) ).
% diff_0
thf(fact_52_neg__0__le__iff__le,axiom,
! [A2: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ ( uminus1482373934393186551omplex @ A2 ) )
= ( ord_less_eq_complex @ A2 @ zero_zero_complex ) ) ).
% neg_0_le_iff_le
thf(fact_53_neg__0__le__iff__le,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_54_neg__0__le__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_55_neg__le__0__iff__le,axiom,
! [A2: complex] :
( ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ A2 ) @ zero_zero_complex )
= ( ord_less_eq_complex @ zero_zero_complex @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_56_neg__le__0__iff__le,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_57_neg__le__0__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_58_less__eq__neg__nonpos,axiom,
! [A2: real] :
( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_59_less__eq__neg__nonpos,axiom,
! [A2: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_60_neg__less__eq__nonneg,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_61_neg__less__eq__nonneg,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_62_diff__ge__0__iff__ge,axiom,
! [A2: complex,B: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ ( minus_minus_complex @ A2 @ B ) )
= ( ord_less_eq_complex @ B @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_63_diff__ge__0__iff__ge,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B ) )
= ( ord_less_eq_real @ B @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_64_diff__ge__0__iff__ge,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
= ( ord_less_eq_int @ B @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_65_subset__antisym,axiom,
! [A: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ( ord_le211207098394363844omplex @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_66_subsetI,axiom,
! [A: set_real,B2: set_real] :
( ! [X: real] :
( ( member_real @ X @ A )
=> ( member_real @ X @ B2 ) )
=> ( ord_less_eq_set_real @ A @ B2 ) ) ).
% subsetI
thf(fact_67_subsetI,axiom,
! [A: set_complex,B2: set_complex] :
( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( member_complex @ X @ B2 ) )
=> ( ord_le211207098394363844omplex @ A @ B2 ) ) ).
% subsetI
thf(fact_68_empty__Collect__eq,axiom,
! [P2: complex > $o] :
( ( bot_bot_set_complex
= ( collect_complex @ P2 ) )
= ( ! [X4: complex] :
~ ( P2 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_69_Collect__empty__eq,axiom,
! [P2: complex > $o] :
( ( ( collect_complex @ P2 )
= bot_bot_set_complex )
= ( ! [X4: complex] :
~ ( P2 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_70_all__not__in__conv,axiom,
! [A: set_real] :
( ( ! [X4: real] :
~ ( member_real @ X4 @ A ) )
= ( A = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_71_all__not__in__conv,axiom,
! [A: set_complex] :
( ( ! [X4: complex] :
~ ( member_complex @ X4 @ A ) )
= ( A = bot_bot_set_complex ) ) ).
% all_not_in_conv
thf(fact_72_empty__iff,axiom,
! [C: real] :
~ ( member_real @ C @ bot_bot_set_real ) ).
% empty_iff
thf(fact_73_empty__iff,axiom,
! [C: complex] :
~ ( member_complex @ C @ bot_bot_set_complex ) ).
% empty_iff
thf(fact_74_neg__equal__iff__equal,axiom,
! [A2: real,B: real] :
( ( ( uminus_uminus_real @ A2 )
= ( uminus_uminus_real @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_75_neg__equal__iff__equal,axiom,
! [A2: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A2 )
= ( uminus1482373934393186551omplex @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_76_neg__equal__iff__equal,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_77_mem__Collect__eq,axiom,
! [A2: complex,P2: complex > $o] :
( ( member_complex @ A2 @ ( collect_complex @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
! [A2: real,P2: real > $o] :
( ( member_real @ A2 @ ( collect_real @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_79_Collect__mem__eq,axiom,
! [A: set_complex] :
( ( collect_complex
@ ^ [X4: complex] : ( member_complex @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_81_add_Oinverse__inverse,axiom,
! [A2: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_82_add_Oinverse__inverse,axiom,
! [A2: complex] :
( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_83_add_Oinverse__inverse,axiom,
! [A2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_84_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_85_verit__minus__simplify_I4_J,axiom,
! [B: complex] :
( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_86_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_87_insert__absorb2,axiom,
! [X2: complex,A: set_complex] :
( ( insert_complex @ X2 @ ( insert_complex @ X2 @ A ) )
= ( insert_complex @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_88_insert__iff,axiom,
! [A2: complex,B: complex,A: set_complex] :
( ( member_complex @ A2 @ ( insert_complex @ B @ A ) )
= ( ( A2 = B )
| ( member_complex @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_89_insert__iff,axiom,
! [A2: real,B: real,A: set_real] :
( ( member_real @ A2 @ ( insert_real @ B @ A ) )
= ( ( A2 = B )
| ( member_real @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_90_insertCI,axiom,
! [A2: complex,B2: set_complex,B: complex] :
( ( ~ ( member_complex @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_complex @ A2 @ ( insert_complex @ B @ B2 ) ) ) ).
% insertCI
thf(fact_91_insertCI,axiom,
! [A2: real,B2: set_real,B: real] :
( ( ~ ( member_real @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_real @ A2 @ ( insert_real @ B @ B2 ) ) ) ).
% insertCI
thf(fact_92_Diff__idemp,axiom,
! [A: set_complex,B2: set_complex] :
( ( minus_811609699411566653omplex @ ( minus_811609699411566653omplex @ A @ B2 ) @ B2 )
= ( minus_811609699411566653omplex @ A @ B2 ) ) ).
% Diff_idemp
thf(fact_93_Diff__iff,axiom,
! [C: real,A: set_real,B2: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A @ B2 ) )
= ( ( member_real @ C @ A )
& ~ ( member_real @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_94_Diff__iff,axiom,
! [C: complex,A: set_complex,B2: set_complex] :
( ( member_complex @ C @ ( minus_811609699411566653omplex @ A @ B2 ) )
= ( ( member_complex @ C @ A )
& ~ ( member_complex @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_95_DiffI,axiom,
! [C: real,A: set_real,B2: set_real] :
( ( member_real @ C @ A )
=> ( ~ ( member_real @ C @ B2 )
=> ( member_real @ C @ ( minus_minus_set_real @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_96_DiffI,axiom,
! [C: complex,A: set_complex,B2: set_complex] :
( ( member_complex @ C @ A )
=> ( ~ ( member_complex @ C @ B2 )
=> ( member_complex @ C @ ( minus_811609699411566653omplex @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_97_Compl__eq__Compl__iff,axiom,
! [A: set_complex,B2: set_complex] :
( ( ( uminus8566677241136511917omplex @ A )
= ( uminus8566677241136511917omplex @ B2 ) )
= ( A = B2 ) ) ).
% Compl_eq_Compl_iff
thf(fact_98_Compl__iff,axiom,
! [C: real,A: set_real] :
( ( member_real @ C @ ( uminus612125837232591019t_real @ A ) )
= ( ~ ( member_real @ C @ A ) ) ) ).
% Compl_iff
thf(fact_99_Compl__iff,axiom,
! [C: complex,A: set_complex] :
( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A ) )
= ( ~ ( member_complex @ C @ A ) ) ) ).
% Compl_iff
thf(fact_100_ComplI,axiom,
! [C: real,A: set_real] :
( ~ ( member_real @ C @ A )
=> ( member_real @ C @ ( uminus612125837232591019t_real @ A ) ) ) ).
% ComplI
thf(fact_101_ComplI,axiom,
! [C: complex,A: set_complex] :
( ~ ( member_complex @ C @ A )
=> ( member_complex @ C @ ( uminus8566677241136511917omplex @ A ) ) ) ).
% ComplI
thf(fact_102_le__zero__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% le_zero_eq
thf(fact_103_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_104_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: complex] :
( ( minus_minus_complex @ A2 @ A2 )
= zero_zero_complex ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_105_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_106_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_107_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_108_diff__zero,axiom,
! [A2: complex] :
( ( minus_minus_complex @ A2 @ zero_zero_complex )
= A2 ) ).
% diff_zero
thf(fact_109_diff__zero,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_zero
thf(fact_110_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_111_diff__zero,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_zero
thf(fact_112_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_113_diff__0__right,axiom,
! [A2: complex] :
( ( minus_minus_complex @ A2 @ zero_zero_complex )
= A2 ) ).
% diff_0_right
thf(fact_114_diff__0__right,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_0_right
thf(fact_115_diff__0__right,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_0_right
thf(fact_116_diff__self,axiom,
! [A2: complex] :
( ( minus_minus_complex @ A2 @ A2 )
= zero_zero_complex ) ).
% diff_self
thf(fact_117_diff__self,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% diff_self
thf(fact_118_diff__self,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% diff_self
thf(fact_119_neg__le__iff__le,axiom,
! [B: complex,A2: complex] :
( ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A2 ) )
= ( ord_less_eq_complex @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_120_neg__le__iff__le,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_121_neg__le__iff__le,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_122_neg__equal__zero,axiom,
! [A2: real] :
( ( ( uminus_uminus_real @ A2 )
= A2 )
= ( A2 = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_123_neg__equal__zero,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= A2 )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_124_equal__neg__zero,axiom,
! [A2: real] :
( ( A2
= ( uminus_uminus_real @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_125_equal__neg__zero,axiom,
! [A2: int] :
( ( A2
= ( uminus_uminus_int @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_126_neg__equal__0__iff__equal,axiom,
! [A2: real] :
( ( ( uminus_uminus_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_127_neg__equal__0__iff__equal,axiom,
! [A2: complex] :
( ( ( uminus1482373934393186551omplex @ A2 )
= zero_zero_complex )
= ( A2 = zero_zero_complex ) ) ).
% neg_equal_0_iff_equal
thf(fact_128_neg__equal__0__iff__equal,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_129_neg__0__equal__iff__equal,axiom,
! [A2: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A2 ) )
= ( zero_zero_real = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_130_neg__0__equal__iff__equal,axiom,
! [A2: complex] :
( ( zero_zero_complex
= ( uminus1482373934393186551omplex @ A2 ) )
= ( zero_zero_complex = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_131_neg__0__equal__iff__equal,axiom,
! [A2: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A2 ) )
= ( zero_zero_int = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_132_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_133_add_Oinverse__neutral,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% add.inverse_neutral
thf(fact_134_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_135_minus__diff__eq,axiom,
! [A2: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A2 @ B ) )
= ( minus_minus_real @ B @ A2 ) ) ).
% minus_diff_eq
thf(fact_136_minus__diff__eq,axiom,
! [A2: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A2 @ B ) )
= ( minus_minus_complex @ B @ A2 ) ) ).
% minus_diff_eq
thf(fact_137_minus__diff__eq,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B ) )
= ( minus_minus_int @ B @ A2 ) ) ).
% minus_diff_eq
thf(fact_138_empty__subsetI,axiom,
! [A: set_complex] : ( ord_le211207098394363844omplex @ bot_bot_set_complex @ A ) ).
% empty_subsetI
thf(fact_139_subset__empty,axiom,
! [A: set_complex] :
( ( ord_le211207098394363844omplex @ A @ bot_bot_set_complex )
= ( A = bot_bot_set_complex ) ) ).
% subset_empty
thf(fact_140_insert__subset,axiom,
! [X2: real,A: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ ( insert_real @ X2 @ A ) @ B2 )
= ( ( member_real @ X2 @ B2 )
& ( ord_less_eq_set_real @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_141_insert__subset,axiom,
! [X2: complex,A: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ ( insert_complex @ X2 @ A ) @ B2 )
= ( ( member_complex @ X2 @ B2 )
& ( ord_le211207098394363844omplex @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_142_singletonI,axiom,
! [A2: real] : ( member_real @ A2 @ ( insert_real @ A2 @ bot_bot_set_real ) ) ).
% singletonI
thf(fact_143_singletonI,axiom,
! [A2: complex] : ( member_complex @ A2 @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ).
% singletonI
thf(fact_144_Diff__cancel,axiom,
! [A: set_complex] :
( ( minus_811609699411566653omplex @ A @ A )
= bot_bot_set_complex ) ).
% Diff_cancel
thf(fact_145_empty__Diff,axiom,
! [A: set_complex] :
( ( minus_811609699411566653omplex @ bot_bot_set_complex @ A )
= bot_bot_set_complex ) ).
% empty_Diff
thf(fact_146_Diff__empty,axiom,
! [A: set_complex] :
( ( minus_811609699411566653omplex @ A @ bot_bot_set_complex )
= A ) ).
% Diff_empty
thf(fact_147_insert__Diff1,axiom,
! [X2: real,B2: set_real,A: set_real] :
( ( member_real @ X2 @ B2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X2 @ A ) @ B2 )
= ( minus_minus_set_real @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_148_insert__Diff1,axiom,
! [X2: complex,B2: set_complex,A: set_complex] :
( ( member_complex @ X2 @ B2 )
=> ( ( minus_811609699411566653omplex @ ( insert_complex @ X2 @ A ) @ B2 )
= ( minus_811609699411566653omplex @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_149_Diff__insert0,axiom,
! [X2: real,A: set_real,B2: set_real] :
( ~ ( member_real @ X2 @ A )
=> ( ( minus_minus_set_real @ A @ ( insert_real @ X2 @ B2 ) )
= ( minus_minus_set_real @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_150_Diff__insert0,axiom,
! [X2: complex,A: set_complex,B2: set_complex] :
( ~ ( member_complex @ X2 @ A )
=> ( ( minus_811609699411566653omplex @ A @ ( insert_complex @ X2 @ B2 ) )
= ( minus_811609699411566653omplex @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_151_Compl__subset__Compl__iff,axiom,
! [A: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ ( uminus8566677241136511917omplex @ A ) @ ( uminus8566677241136511917omplex @ B2 ) )
= ( ord_le211207098394363844omplex @ B2 @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_152_Compl__anti__mono,axiom,
! [A: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ord_le211207098394363844omplex @ ( uminus8566677241136511917omplex @ B2 ) @ ( uminus8566677241136511917omplex @ A ) ) ) ).
% Compl_anti_mono
thf(fact_153_verit__la__disequality,axiom,
! [A2: real,B: real] :
( ( A2 = B )
| ~ ( ord_less_eq_real @ A2 @ B )
| ~ ( ord_less_eq_real @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_154_verit__la__disequality,axiom,
! [A2: num,B: num] :
( ( A2 = B )
| ~ ( ord_less_eq_num @ A2 @ B )
| ~ ( ord_less_eq_num @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_155_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_156_verit__la__disequality,axiom,
! [A2: int,B: int] :
( ( A2 = B )
| ~ ( ord_less_eq_int @ A2 @ B )
| ~ ( ord_less_eq_int @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_157_verit__comp__simplify1_I2_J,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_158_verit__comp__simplify1_I2_J,axiom,
! [A2: set_complex] : ( ord_le211207098394363844omplex @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_159_verit__comp__simplify1_I2_J,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_160_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_161_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_162_zero__reorient,axiom,
! [X2: complex] :
( ( zero_zero_complex = X2 )
= ( X2 = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_163_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_164_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_165_zero__reorient,axiom,
! [X2: extended_enat] :
( ( zero_z5237406670263579293d_enat = X2 )
= ( X2 = zero_z5237406670263579293d_enat ) ) ).
% zero_reorient
thf(fact_166_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_167_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: complex,C: complex,B: complex] :
( ( minus_minus_complex @ ( minus_minus_complex @ A2 @ C ) @ B )
= ( minus_minus_complex @ ( minus_minus_complex @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_168_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_169_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_170_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_171_diff__eq__diff__eq,axiom,
! [A2: complex,B: complex,C: complex,D: complex] :
( ( ( minus_minus_complex @ A2 @ B )
= ( minus_minus_complex @ C @ D ) )
=> ( ( A2 = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_172_diff__eq__diff__eq,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A2 = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_173_diff__eq__diff__eq,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A2 = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_174_Collect__mono__iff,axiom,
! [P2: complex > $o,Q: complex > $o] :
( ( ord_le211207098394363844omplex @ ( collect_complex @ P2 ) @ ( collect_complex @ Q ) )
= ( ! [X4: complex] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_175_set__eq__subset,axiom,
( ( ^ [Y3: set_complex,Z: set_complex] : ( Y3 = Z ) )
= ( ^ [A3: set_complex,B3: set_complex] :
( ( ord_le211207098394363844omplex @ A3 @ B3 )
& ( ord_le211207098394363844omplex @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_176_subset__trans,axiom,
! [A: set_complex,B2: set_complex,C2: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ( ord_le211207098394363844omplex @ B2 @ C2 )
=> ( ord_le211207098394363844omplex @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_177_Collect__mono,axiom,
! [P2: complex > $o,Q: complex > $o] :
( ! [X: complex] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le211207098394363844omplex @ ( collect_complex @ P2 ) @ ( collect_complex @ Q ) ) ) ).
% Collect_mono
thf(fact_178_subset__refl,axiom,
! [A: set_complex] : ( ord_le211207098394363844omplex @ A @ A ) ).
% subset_refl
thf(fact_179_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
! [T3: real] :
( ( member_real @ T3 @ A3 )
=> ( member_real @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_180_subset__iff,axiom,
( ord_le211207098394363844omplex
= ( ^ [A3: set_complex,B3: set_complex] :
! [T3: complex] :
( ( member_complex @ T3 @ A3 )
=> ( member_complex @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_181_equalityD2,axiom,
! [A: set_complex,B2: set_complex] :
( ( A = B2 )
=> ( ord_le211207098394363844omplex @ B2 @ A ) ) ).
% equalityD2
thf(fact_182_equalityD1,axiom,
! [A: set_complex,B2: set_complex] :
( ( A = B2 )
=> ( ord_le211207098394363844omplex @ A @ B2 ) ) ).
% equalityD1
thf(fact_183_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
! [X4: real] :
( ( member_real @ X4 @ A3 )
=> ( member_real @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_184_subset__eq,axiom,
( ord_le211207098394363844omplex
= ( ^ [A3: set_complex,B3: set_complex] :
! [X4: complex] :
( ( member_complex @ X4 @ A3 )
=> ( member_complex @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_185_equalityE,axiom,
! [A: set_complex,B2: set_complex] :
( ( A = B2 )
=> ~ ( ( ord_le211207098394363844omplex @ A @ B2 )
=> ~ ( ord_le211207098394363844omplex @ B2 @ A ) ) ) ).
% equalityE
thf(fact_186_subsetD,axiom,
! [A: set_real,B2: set_real,C: real] :
( ( ord_less_eq_set_real @ A @ B2 )
=> ( ( member_real @ C @ A )
=> ( member_real @ C @ B2 ) ) ) ).
% subsetD
thf(fact_187_subsetD,axiom,
! [A: set_complex,B2: set_complex,C: complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ( member_complex @ C @ A )
=> ( member_complex @ C @ B2 ) ) ) ).
% subsetD
thf(fact_188_in__mono,axiom,
! [A: set_real,B2: set_real,X2: real] :
( ( ord_less_eq_set_real @ A @ B2 )
=> ( ( member_real @ X2 @ A )
=> ( member_real @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_189_in__mono,axiom,
! [A: set_complex,B2: set_complex,X2: complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ( member_complex @ X2 @ A )
=> ( member_complex @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_190_ex__in__conv,axiom,
! [A: set_real] :
( ( ? [X4: real] : ( member_real @ X4 @ A ) )
= ( A != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_191_ex__in__conv,axiom,
! [A: set_complex] :
( ( ? [X4: complex] : ( member_complex @ X4 @ A ) )
= ( A != bot_bot_set_complex ) ) ).
% ex_in_conv
thf(fact_192_equals0I,axiom,
! [A: set_real] :
( ! [Y4: real] :
~ ( member_real @ Y4 @ A )
=> ( A = bot_bot_set_real ) ) ).
% equals0I
thf(fact_193_equals0I,axiom,
! [A: set_complex] :
( ! [Y4: complex] :
~ ( member_complex @ Y4 @ A )
=> ( A = bot_bot_set_complex ) ) ).
% equals0I
thf(fact_194_equals0D,axiom,
! [A: set_real,A2: real] :
( ( A = bot_bot_set_real )
=> ~ ( member_real @ A2 @ A ) ) ).
% equals0D
thf(fact_195_equals0D,axiom,
! [A: set_complex,A2: complex] :
( ( A = bot_bot_set_complex )
=> ~ ( member_complex @ A2 @ A ) ) ).
% equals0D
thf(fact_196_emptyE,axiom,
! [A2: real] :
~ ( member_real @ A2 @ bot_bot_set_real ) ).
% emptyE
thf(fact_197_emptyE,axiom,
! [A2: complex] :
~ ( member_complex @ A2 @ bot_bot_set_complex ) ).
% emptyE
thf(fact_198_minus__equation__iff,axiom,
! [A2: real,B: real] :
( ( ( uminus_uminus_real @ A2 )
= B )
= ( ( uminus_uminus_real @ B )
= A2 ) ) ).
% minus_equation_iff
thf(fact_199_minus__equation__iff,axiom,
! [A2: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A2 )
= B )
= ( ( uminus1482373934393186551omplex @ B )
= A2 ) ) ).
% minus_equation_iff
thf(fact_200_minus__equation__iff,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= B )
= ( ( uminus_uminus_int @ B )
= A2 ) ) ).
% minus_equation_iff
thf(fact_201_equation__minus__iff,axiom,
! [A2: real,B: real] :
( ( A2
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_202_equation__minus__iff,axiom,
! [A2: complex,B: complex] :
( ( A2
= ( uminus1482373934393186551omplex @ B ) )
= ( B
= ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_203_equation__minus__iff,axiom,
! [A2: int,B: int] :
( ( A2
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_204_verit__negate__coefficient_I3_J,axiom,
! [A2: real,B: real] :
( ( A2 = B )
=> ( ( uminus_uminus_real @ A2 )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_205_verit__negate__coefficient_I3_J,axiom,
! [A2: complex,B: complex] :
( ( A2 = B )
=> ( ( uminus1482373934393186551omplex @ A2 )
= ( uminus1482373934393186551omplex @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_206_verit__negate__coefficient_I3_J,axiom,
! [A2: int,B: int] :
( ( A2 = B )
=> ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_207_mk__disjoint__insert,axiom,
! [A2: complex,A: set_complex] :
( ( member_complex @ A2 @ A )
=> ? [B4: set_complex] :
( ( A
= ( insert_complex @ A2 @ B4 ) )
& ~ ( member_complex @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_208_mk__disjoint__insert,axiom,
! [A2: real,A: set_real] :
( ( member_real @ A2 @ A )
=> ? [B4: set_real] :
( ( A
= ( insert_real @ A2 @ B4 ) )
& ~ ( member_real @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_209_insert__commute,axiom,
! [X2: complex,Y: complex,A: set_complex] :
( ( insert_complex @ X2 @ ( insert_complex @ Y @ A ) )
= ( insert_complex @ Y @ ( insert_complex @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_210_insert__eq__iff,axiom,
! [A2: complex,A: set_complex,B: complex,B2: set_complex] :
( ~ ( member_complex @ A2 @ A )
=> ( ~ ( member_complex @ B @ B2 )
=> ( ( ( insert_complex @ A2 @ A )
= ( insert_complex @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_complex] :
( ( A
= ( insert_complex @ B @ C3 ) )
& ~ ( member_complex @ B @ C3 )
& ( B2
= ( insert_complex @ A2 @ C3 ) )
& ~ ( member_complex @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_211_insert__eq__iff,axiom,
! [A2: real,A: set_real,B: real,B2: set_real] :
( ~ ( member_real @ A2 @ A )
=> ( ~ ( member_real @ B @ B2 )
=> ( ( ( insert_real @ A2 @ A )
= ( insert_real @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_real] :
( ( A
= ( insert_real @ B @ C3 ) )
& ~ ( member_real @ B @ C3 )
& ( B2
= ( insert_real @ A2 @ C3 ) )
& ~ ( member_real @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_212_insert__absorb,axiom,
! [A2: complex,A: set_complex] :
( ( member_complex @ A2 @ A )
=> ( ( insert_complex @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_213_insert__absorb,axiom,
! [A2: real,A: set_real] :
( ( member_real @ A2 @ A )
=> ( ( insert_real @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_214_insert__ident,axiom,
! [X2: complex,A: set_complex,B2: set_complex] :
( ~ ( member_complex @ X2 @ A )
=> ( ~ ( member_complex @ X2 @ B2 )
=> ( ( ( insert_complex @ X2 @ A )
= ( insert_complex @ X2 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_215_insert__ident,axiom,
! [X2: real,A: set_real,B2: set_real] :
( ~ ( member_real @ X2 @ A )
=> ( ~ ( member_real @ X2 @ B2 )
=> ( ( ( insert_real @ X2 @ A )
= ( insert_real @ X2 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_216_Set_Oset__insert,axiom,
! [X2: complex,A: set_complex] :
( ( member_complex @ X2 @ A )
=> ~ ! [B4: set_complex] :
( ( A
= ( insert_complex @ X2 @ B4 ) )
=> ( member_complex @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_217_Set_Oset__insert,axiom,
! [X2: real,A: set_real] :
( ( member_real @ X2 @ A )
=> ~ ! [B4: set_real] :
( ( A
= ( insert_real @ X2 @ B4 ) )
=> ( member_real @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_218_insertI2,axiom,
! [A2: complex,B2: set_complex,B: complex] :
( ( member_complex @ A2 @ B2 )
=> ( member_complex @ A2 @ ( insert_complex @ B @ B2 ) ) ) ).
% insertI2
thf(fact_219_insertI2,axiom,
! [A2: real,B2: set_real,B: real] :
( ( member_real @ A2 @ B2 )
=> ( member_real @ A2 @ ( insert_real @ B @ B2 ) ) ) ).
% insertI2
thf(fact_220_insertI1,axiom,
! [A2: complex,B2: set_complex] : ( member_complex @ A2 @ ( insert_complex @ A2 @ B2 ) ) ).
% insertI1
thf(fact_221_insertI1,axiom,
! [A2: real,B2: set_real] : ( member_real @ A2 @ ( insert_real @ A2 @ B2 ) ) ).
% insertI1
thf(fact_222_insertE,axiom,
! [A2: complex,B: complex,A: set_complex] :
( ( member_complex @ A2 @ ( insert_complex @ B @ A ) )
=> ( ( A2 != B )
=> ( member_complex @ A2 @ A ) ) ) ).
% insertE
thf(fact_223_insertE,axiom,
! [A2: real,B: real,A: set_real] :
( ( member_real @ A2 @ ( insert_real @ B @ A ) )
=> ( ( A2 != B )
=> ( member_real @ A2 @ A ) ) ) ).
% insertE
thf(fact_224_DiffD2,axiom,
! [C: real,A: set_real,B2: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A @ B2 ) )
=> ~ ( member_real @ C @ B2 ) ) ).
% DiffD2
thf(fact_225_DiffD2,axiom,
! [C: complex,A: set_complex,B2: set_complex] :
( ( member_complex @ C @ ( minus_811609699411566653omplex @ A @ B2 ) )
=> ~ ( member_complex @ C @ B2 ) ) ).
% DiffD2
thf(fact_226_DiffD1,axiom,
! [C: real,A: set_real,B2: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A @ B2 ) )
=> ( member_real @ C @ A ) ) ).
% DiffD1
thf(fact_227_DiffD1,axiom,
! [C: complex,A: set_complex,B2: set_complex] :
( ( member_complex @ C @ ( minus_811609699411566653omplex @ A @ B2 ) )
=> ( member_complex @ C @ A ) ) ).
% DiffD1
thf(fact_228_DiffE,axiom,
! [C: real,A: set_real,B2: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A @ B2 ) )
=> ~ ( ( member_real @ C @ A )
=> ( member_real @ C @ B2 ) ) ) ).
% DiffE
thf(fact_229_DiffE,axiom,
! [C: complex,A: set_complex,B2: set_complex] :
( ( member_complex @ C @ ( minus_811609699411566653omplex @ A @ B2 ) )
=> ~ ( ( member_complex @ C @ A )
=> ( member_complex @ C @ B2 ) ) ) ).
% DiffE
thf(fact_230_double__complement,axiom,
! [A: set_complex] :
( ( uminus8566677241136511917omplex @ ( uminus8566677241136511917omplex @ A ) )
= A ) ).
% double_complement
thf(fact_231_ComplD,axiom,
! [C: real,A: set_real] :
( ( member_real @ C @ ( uminus612125837232591019t_real @ A ) )
=> ~ ( member_real @ C @ A ) ) ).
% ComplD
thf(fact_232_ComplD,axiom,
! [C: complex,A: set_complex] :
( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A ) )
=> ~ ( member_complex @ C @ A ) ) ).
% ComplD
thf(fact_233_zero__le,axiom,
! [X2: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ X2 ) ).
% zero_le
thf(fact_234_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_235_diff__eq__diff__less__eq,axiom,
! [A2: complex,B: complex,C: complex,D: complex] :
( ( ( minus_minus_complex @ A2 @ B )
= ( minus_minus_complex @ C @ D ) )
=> ( ( ord_less_eq_complex @ A2 @ B )
= ( ord_less_eq_complex @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_236_diff__eq__diff__less__eq,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A2 @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_237_diff__eq__diff__less__eq,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A2 @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_238_diff__right__mono,axiom,
! [A2: complex,B: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B )
=> ( ord_less_eq_complex @ ( minus_minus_complex @ A2 @ C ) @ ( minus_minus_complex @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_239_diff__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_240_diff__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_241_diff__left__mono,axiom,
! [B: complex,A2: complex,C: complex] :
( ( ord_less_eq_complex @ B @ A2 )
=> ( ord_less_eq_complex @ ( minus_minus_complex @ C @ A2 ) @ ( minus_minus_complex @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_242_diff__left__mono,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_243_diff__left__mono,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_244_diff__mono,axiom,
! [A2: complex,B: complex,D: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B )
=> ( ( ord_less_eq_complex @ D @ C )
=> ( ord_less_eq_complex @ ( minus_minus_complex @ A2 @ C ) @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_245_diff__mono,axiom,
! [A2: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_246_diff__mono,axiom,
! [A2: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_247_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: complex,Z: complex] : ( Y3 = Z ) )
= ( ^ [A4: complex,B5: complex] :
( ( minus_minus_complex @ A4 @ B5 )
= zero_zero_complex ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_248_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A4: real,B5: real] :
( ( minus_minus_real @ A4 @ B5 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_249_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A4: int,B5: int] :
( ( minus_minus_int @ A4 @ B5 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_250_le__imp__neg__le,axiom,
! [A2: complex,B: complex] :
( ( ord_less_eq_complex @ A2 @ B )
=> ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_251_le__imp__neg__le,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_252_le__imp__neg__le,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_253_minus__le__iff,axiom,
! [A2: complex,B: complex] :
( ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B )
= ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ B ) @ A2 ) ) ).
% minus_le_iff
thf(fact_254_minus__le__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A2 ) ) ).
% minus_le_iff
thf(fact_255_minus__le__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% minus_le_iff
thf(fact_256_le__minus__iff,axiom,
! [A2: complex,B: complex] :
( ( ord_less_eq_complex @ A2 @ ( uminus1482373934393186551omplex @ B ) )
= ( ord_less_eq_complex @ B @ ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% le_minus_iff
thf(fact_257_le__minus__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A2 ) ) ) ).
% le_minus_iff
thf(fact_258_le__minus__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_minus_iff
thf(fact_259_minus__diff__commute,axiom,
! [B: real,A2: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A2 )
= ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ B ) ) ).
% minus_diff_commute
thf(fact_260_minus__diff__commute,axiom,
! [B: complex,A2: complex] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A2 )
= ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B ) ) ).
% minus_diff_commute
thf(fact_261_minus__diff__commute,axiom,
! [B: int,A2: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A2 )
= ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ B ) ) ).
% minus_diff_commute
thf(fact_262_subset__insertI2,axiom,
! [A: set_complex,B2: set_complex,B: complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ord_le211207098394363844omplex @ A @ ( insert_complex @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_263_subset__insertI,axiom,
! [B2: set_complex,A2: complex] : ( ord_le211207098394363844omplex @ B2 @ ( insert_complex @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_264_subset__insert,axiom,
! [X2: real,A: set_real,B2: set_real] :
( ~ ( member_real @ X2 @ A )
=> ( ( ord_less_eq_set_real @ A @ ( insert_real @ X2 @ B2 ) )
= ( ord_less_eq_set_real @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_265_subset__insert,axiom,
! [X2: complex,A: set_complex,B2: set_complex] :
( ~ ( member_complex @ X2 @ A )
=> ( ( ord_le211207098394363844omplex @ A @ ( insert_complex @ X2 @ B2 ) )
= ( ord_le211207098394363844omplex @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_266_insert__mono,axiom,
! [C2: set_complex,D2: set_complex,A2: complex] :
( ( ord_le211207098394363844omplex @ C2 @ D2 )
=> ( ord_le211207098394363844omplex @ ( insert_complex @ A2 @ C2 ) @ ( insert_complex @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_267_singleton__inject,axiom,
! [A2: complex,B: complex] :
( ( ( insert_complex @ A2 @ bot_bot_set_complex )
= ( insert_complex @ B @ bot_bot_set_complex ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_268_insert__not__empty,axiom,
! [A2: complex,A: set_complex] :
( ( insert_complex @ A2 @ A )
!= bot_bot_set_complex ) ).
% insert_not_empty
thf(fact_269_doubleton__eq__iff,axiom,
! [A2: complex,B: complex,C: complex,D: complex] :
( ( ( insert_complex @ A2 @ ( insert_complex @ B @ bot_bot_set_complex ) )
= ( insert_complex @ C @ ( insert_complex @ D @ bot_bot_set_complex ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_270_singleton__iff,axiom,
! [B: real,A2: real] :
( ( member_real @ B @ ( insert_real @ A2 @ bot_bot_set_real ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_271_singleton__iff,axiom,
! [B: complex,A2: complex] :
( ( member_complex @ B @ ( insert_complex @ A2 @ bot_bot_set_complex ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_272_singletonD,axiom,
! [B: real,A2: real] :
( ( member_real @ B @ ( insert_real @ A2 @ bot_bot_set_real ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_273_singletonD,axiom,
! [B: complex,A2: complex] :
( ( member_complex @ B @ ( insert_complex @ A2 @ bot_bot_set_complex ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_274_double__diff,axiom,
! [A: set_complex,B2: set_complex,C2: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ( ord_le211207098394363844omplex @ B2 @ C2 )
=> ( ( minus_811609699411566653omplex @ B2 @ ( minus_811609699411566653omplex @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_275_Diff__subset,axiom,
! [A: set_complex,B2: set_complex] : ( ord_le211207098394363844omplex @ ( minus_811609699411566653omplex @ A @ B2 ) @ A ) ).
% Diff_subset
thf(fact_276_Diff__mono,axiom,
! [A: set_complex,C2: set_complex,D2: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ A @ C2 )
=> ( ( ord_le211207098394363844omplex @ D2 @ B2 )
=> ( ord_le211207098394363844omplex @ ( minus_811609699411566653omplex @ A @ B2 ) @ ( minus_811609699411566653omplex @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_277_insert__Diff__if,axiom,
! [X2: real,B2: set_real,A: set_real] :
( ( ( member_real @ X2 @ B2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X2 @ A ) @ B2 )
= ( minus_minus_set_real @ A @ B2 ) ) )
& ( ~ ( member_real @ X2 @ B2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X2 @ A ) @ B2 )
= ( insert_real @ X2 @ ( minus_minus_set_real @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_278_insert__Diff__if,axiom,
! [X2: complex,B2: set_complex,A: set_complex] :
( ( ( member_complex @ X2 @ B2 )
=> ( ( minus_811609699411566653omplex @ ( insert_complex @ X2 @ A ) @ B2 )
= ( minus_811609699411566653omplex @ A @ B2 ) ) )
& ( ~ ( member_complex @ X2 @ B2 )
=> ( ( minus_811609699411566653omplex @ ( insert_complex @ X2 @ A ) @ B2 )
= ( insert_complex @ X2 @ ( minus_811609699411566653omplex @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_279_le__iff__diff__le__0,axiom,
( ord_less_eq_complex
= ( ^ [A4: complex,B5: complex] : ( ord_less_eq_complex @ ( minus_minus_complex @ A4 @ B5 ) @ zero_zero_complex ) ) ) ).
% le_iff_diff_le_0
thf(fact_280_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B5: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B5 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_281_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B5: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B5 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_282_subset__singleton__iff,axiom,
! [X3: set_complex,A2: complex] :
( ( ord_le211207098394363844omplex @ X3 @ ( insert_complex @ A2 @ bot_bot_set_complex ) )
= ( ( X3 = bot_bot_set_complex )
| ( X3
= ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ).
% subset_singleton_iff
thf(fact_283_subset__singletonD,axiom,
! [A: set_complex,X2: complex] :
( ( ord_le211207098394363844omplex @ A @ ( insert_complex @ X2 @ bot_bot_set_complex ) )
=> ( ( A = bot_bot_set_complex )
| ( A
= ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ).
% subset_singletonD
thf(fact_284_subset__Diff__insert,axiom,
! [A: set_real,B2: set_real,X2: real,C2: set_real] :
( ( ord_less_eq_set_real @ A @ ( minus_minus_set_real @ B2 @ ( insert_real @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_real @ A @ ( minus_minus_set_real @ B2 @ C2 ) )
& ~ ( member_real @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_285_subset__Diff__insert,axiom,
! [A: set_complex,B2: set_complex,X2: complex,C2: set_complex] :
( ( ord_le211207098394363844omplex @ A @ ( minus_811609699411566653omplex @ B2 @ ( insert_complex @ X2 @ C2 ) ) )
= ( ( ord_le211207098394363844omplex @ A @ ( minus_811609699411566653omplex @ B2 @ C2 ) )
& ~ ( member_complex @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_286_Diff__insert__absorb,axiom,
! [X2: real,A: set_real] :
( ~ ( member_real @ X2 @ A )
=> ( ( minus_minus_set_real @ ( insert_real @ X2 @ A ) @ ( insert_real @ X2 @ bot_bot_set_real ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_287_Diff__insert__absorb,axiom,
! [X2: complex,A: set_complex] :
( ~ ( member_complex @ X2 @ A )
=> ( ( minus_811609699411566653omplex @ ( insert_complex @ X2 @ A ) @ ( insert_complex @ X2 @ bot_bot_set_complex ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_288_Diff__insert2,axiom,
! [A: set_complex,A2: complex,B2: set_complex] :
( ( minus_811609699411566653omplex @ A @ ( insert_complex @ A2 @ B2 ) )
= ( minus_811609699411566653omplex @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_289_insert__Diff,axiom,
! [A2: real,A: set_real] :
( ( member_real @ A2 @ A )
=> ( ( insert_real @ A2 @ ( minus_minus_set_real @ A @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
= A ) ) ).
% insert_Diff
thf(fact_290_insert__Diff,axiom,
! [A2: complex,A: set_complex] :
( ( member_complex @ A2 @ A )
=> ( ( insert_complex @ A2 @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
= A ) ) ).
% insert_Diff
thf(fact_291_Diff__insert,axiom,
! [A: set_complex,A2: complex,B2: set_complex] :
( ( minus_811609699411566653omplex @ A @ ( insert_complex @ A2 @ B2 ) )
= ( minus_811609699411566653omplex @ ( minus_811609699411566653omplex @ A @ B2 ) @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ).
% Diff_insert
thf(fact_292_subset__Compl__self__eq,axiom,
! [A: set_complex] :
( ( ord_le211207098394363844omplex @ A @ ( uminus8566677241136511917omplex @ A ) )
= ( A = bot_bot_set_complex ) ) ).
% subset_Compl_self_eq
thf(fact_293_Diff__single__insert,axiom,
! [A: set_complex,X2: complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) @ B2 )
=> ( ord_le211207098394363844omplex @ A @ ( insert_complex @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_294_subset__insert__iff,axiom,
! [A: set_real,X2: real,B2: set_real] :
( ( ord_less_eq_set_real @ A @ ( insert_real @ X2 @ B2 ) )
= ( ( ( member_real @ X2 @ A )
=> ( ord_less_eq_set_real @ ( minus_minus_set_real @ A @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B2 ) )
& ( ~ ( member_real @ X2 @ A )
=> ( ord_less_eq_set_real @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_295_subset__insert__iff,axiom,
! [A: set_complex,X2: complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ A @ ( insert_complex @ X2 @ B2 ) )
= ( ( ( member_complex @ X2 @ A )
=> ( ord_le211207098394363844omplex @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) @ B2 ) )
& ( ~ ( member_complex @ X2 @ A )
=> ( ord_le211207098394363844omplex @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_296_Compl__insert,axiom,
! [X2: complex,A: set_complex] :
( ( uminus8566677241136511917omplex @ ( insert_complex @ X2 @ A ) )
= ( minus_811609699411566653omplex @ ( uminus8566677241136511917omplex @ A ) @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ).
% Compl_insert
thf(fact_297_of__real__in__Ints__iff,axiom,
! [X2: real] :
( ( member_real @ ( real_V1803761363581548252l_real @ X2 ) @ ring_1_Ints_real )
= ( member_real @ X2 @ ring_1_Ints_real ) ) ).
% of_real_in_Ints_iff
thf(fact_298_of__real__in__Ints__iff,axiom,
! [X2: real] :
( ( member_complex @ ( real_V4546457046886955230omplex @ X2 ) @ ring_1_Ints_complex )
= ( member_real @ X2 @ ring_1_Ints_real ) ) ).
% of_real_in_Ints_iff
thf(fact_299_Ints__of__real,axiom,
! [X2: real] :
( ( member_real @ X2 @ ring_1_Ints_real )
=> ( member_real @ ( real_V1803761363581548252l_real @ X2 ) @ ring_1_Ints_real ) ) ).
% Ints_of_real
thf(fact_300_Ints__of__real,axiom,
! [X2: real] :
( ( member_real @ X2 @ ring_1_Ints_real )
=> ( member_complex @ ( real_V4546457046886955230omplex @ X2 ) @ ring_1_Ints_complex ) ) ).
% Ints_of_real
thf(fact_301_of__real__eq__minus__of__real__iff,axiom,
! [X2: real,Y: real] :
( ( ( real_V1803761363581548252l_real @ X2 )
= ( uminus_uminus_real @ ( real_V1803761363581548252l_real @ Y ) ) )
= ( X2
= ( uminus_uminus_real @ Y ) ) ) ).
% of_real_eq_minus_of_real_iff
thf(fact_302_of__real__eq__minus__of__real__iff,axiom,
! [X2: real,Y: real] :
( ( ( real_V4546457046886955230omplex @ X2 )
= ( uminus1482373934393186551omplex @ ( real_V4546457046886955230omplex @ Y ) ) )
= ( X2
= ( uminus_uminus_real @ Y ) ) ) ).
% of_real_eq_minus_of_real_iff
thf(fact_303_minus__of__real__eq__of__real__iff,axiom,
! [X2: real,Y: real] :
( ( ( uminus_uminus_real @ ( real_V1803761363581548252l_real @ X2 ) )
= ( real_V1803761363581548252l_real @ Y ) )
= ( ( uminus_uminus_real @ X2 )
= Y ) ) ).
% minus_of_real_eq_of_real_iff
thf(fact_304_minus__of__real__eq__of__real__iff,axiom,
! [X2: real,Y: real] :
( ( ( uminus1482373934393186551omplex @ ( real_V4546457046886955230omplex @ X2 ) )
= ( real_V4546457046886955230omplex @ Y ) )
= ( ( uminus_uminus_real @ X2 )
= Y ) ) ).
% minus_of_real_eq_of_real_iff
thf(fact_305_of__real__minus,axiom,
! [X2: real] :
( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ X2 ) )
= ( uminus_uminus_real @ ( real_V1803761363581548252l_real @ X2 ) ) ) ).
% of_real_minus
thf(fact_306_of__real__minus,axiom,
! [X2: real] :
( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ X2 ) )
= ( uminus1482373934393186551omplex @ ( real_V4546457046886955230omplex @ X2 ) ) ) ).
% of_real_minus
thf(fact_307_of__real__diff,axiom,
! [X2: real,Y: real] :
( ( real_V1803761363581548252l_real @ ( minus_minus_real @ X2 @ Y ) )
= ( minus_minus_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% of_real_diff
thf(fact_308_of__real__diff,axiom,
! [X2: real,Y: real] :
( ( real_V4546457046886955230omplex @ ( minus_minus_real @ X2 @ Y ) )
= ( minus_minus_complex @ ( real_V4546457046886955230omplex @ X2 ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% of_real_diff
thf(fact_309_of__real__eq__0__iff,axiom,
! [X2: real] :
( ( ( real_V1803761363581548252l_real @ X2 )
= zero_zero_real )
= ( X2 = zero_zero_real ) ) ).
% of_real_eq_0_iff
thf(fact_310_of__real__eq__0__iff,axiom,
! [X2: real] :
( ( ( real_V4546457046886955230omplex @ X2 )
= zero_zero_complex )
= ( X2 = zero_zero_real ) ) ).
% of_real_eq_0_iff
thf(fact_311_of__real__0,axiom,
( ( real_V1803761363581548252l_real @ zero_zero_real )
= zero_zero_real ) ).
% of_real_0
thf(fact_312_of__real__0,axiom,
( ( real_V4546457046886955230omplex @ zero_zero_real )
= zero_zero_complex ) ).
% of_real_0
thf(fact_313_compl__le__compl__iff,axiom,
! [X2: set_complex,Y: set_complex] :
( ( ord_le211207098394363844omplex @ ( uminus8566677241136511917omplex @ X2 ) @ ( uminus8566677241136511917omplex @ Y ) )
= ( ord_le211207098394363844omplex @ Y @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_314_set__zero,axiom,
( zero_zero_set_real
= ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) ).
% set_zero
thf(fact_315_set__zero,axiom,
( zero_zero_set_nat
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% set_zero
thf(fact_316_set__zero,axiom,
( zero_z5219068577256268541d_enat
= ( insert_Extended_enat @ zero_z5237406670263579293d_enat @ bot_bo7653980558646680370d_enat ) ) ).
% set_zero
thf(fact_317_set__zero,axiom,
( zero_zero_set_int
= ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ).
% set_zero
thf(fact_318_set__zero,axiom,
( zero_z6614145512433583213omplex
= ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) ).
% set_zero
thf(fact_319_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X2: set_complex] :
( ( uminus8566677241136511917omplex @ ( uminus8566677241136511917omplex @ X2 ) )
= X2 ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_320_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X2: set_complex,Y: set_complex] :
( ( ( uminus8566677241136511917omplex @ X2 )
= ( uminus8566677241136511917omplex @ Y ) )
= ( X2 = Y ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_321_of__real__eq__iff,axiom,
! [X2: real,Y: real] :
( ( ( real_V4546457046886955230omplex @ X2 )
= ( real_V4546457046886955230omplex @ Y ) )
= ( X2 = Y ) ) ).
% of_real_eq_iff
thf(fact_322_const__bcontfun_Orep__eq,axiom,
! [X2: complex] :
( ( bounde4845209636850564516omplex @ ( bounde964246820246105844omplex @ X2 ) )
= ( ^ [Uu: complex] : X2 ) ) ).
% const_bcontfun.rep_eq
thf(fact_323_compl__mono,axiom,
! [X2: set_complex,Y: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y )
=> ( ord_le211207098394363844omplex @ ( uminus8566677241136511917omplex @ Y ) @ ( uminus8566677241136511917omplex @ X2 ) ) ) ).
% compl_mono
thf(fact_324_compl__le__swap1,axiom,
! [Y: set_complex,X2: set_complex] :
( ( ord_le211207098394363844omplex @ Y @ ( uminus8566677241136511917omplex @ X2 ) )
=> ( ord_le211207098394363844omplex @ X2 @ ( uminus8566677241136511917omplex @ Y ) ) ) ).
% compl_le_swap1
thf(fact_325_compl__le__swap2,axiom,
! [Y: set_complex,X2: set_complex] :
( ( ord_le211207098394363844omplex @ ( uminus8566677241136511917omplex @ Y ) @ X2 )
=> ( ord_le211207098394363844omplex @ ( uminus8566677241136511917omplex @ X2 ) @ Y ) ) ).
% compl_le_swap2
thf(fact_326_diff__shunt__var,axiom,
! [X2: set_complex,Y: set_complex] :
( ( ( minus_811609699411566653omplex @ X2 @ Y )
= bot_bot_set_complex )
= ( ord_le211207098394363844omplex @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_327_ge__iff__diff__ge__0,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A4: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A4 @ B5 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_328_ge__iff__diff__ge__0,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A4: int] : ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A4 @ B5 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_329_holomorphic__on__cot__pfd,axiom,
! [A: set_complex] :
( ( ord_le211207098394363844omplex @ A @ ( uminus8566677241136511917omplex @ ( minus_811609699411566653omplex @ ring_1_Ints_complex @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) ) )
=> ( comple7700996537433184370hic_on @ cotang8298477626502807258omplex @ A ) ) ).
% holomorphic_on_cot_pfd
thf(fact_330_ball__insert,axiom,
! [A2: complex,B2: set_complex,P2: complex > $o] :
( ( ! [X4: complex] :
( ( member_complex @ X4 @ ( insert_complex @ A2 @ B2 ) )
=> ( P2 @ X4 ) ) )
= ( ( P2 @ A2 )
& ! [X4: complex] :
( ( member_complex @ X4 @ B2 )
=> ( P2 @ X4 ) ) ) ) ).
% ball_insert
thf(fact_331_the__elem__eq,axiom,
! [X2: complex] :
( ( the_elem_complex @ ( insert_complex @ X2 @ bot_bot_set_complex ) )
= X2 ) ).
% the_elem_eq
thf(fact_332_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_333_order__refl,axiom,
! [X2: set_complex] : ( ord_le211207098394363844omplex @ X2 @ X2 ) ).
% order_refl
thf(fact_334_order__refl,axiom,
! [X2: num] : ( ord_less_eq_num @ X2 @ X2 ) ).
% order_refl
thf(fact_335_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_336_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_337_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_338_dual__order_Orefl,axiom,
! [A2: set_complex] : ( ord_le211207098394363844omplex @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_339_dual__order_Orefl,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_340_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_341_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_342_bot__set__def,axiom,
( bot_bot_set_complex
= ( collect_complex @ bot_bot_complex_o ) ) ).
% bot_set_def
thf(fact_343_holomorphic__rGamma,axiom,
! [A: set_complex] : ( comple7700996537433184370hic_on @ gamma_4773869415665495160omplex @ A ) ).
% holomorphic_rGamma
thf(fact_344_order__antisym__conv,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_345_order__antisym__conv,axiom,
! [Y: set_complex,X2: set_complex] :
( ( ord_le211207098394363844omplex @ Y @ X2 )
=> ( ( ord_le211207098394363844omplex @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_346_order__antisym__conv,axiom,
! [Y: num,X2: num] :
( ( ord_less_eq_num @ Y @ X2 )
=> ( ( ord_less_eq_num @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_347_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_348_order__antisym__conv,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_349_linorder__le__cases,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_350_linorder__le__cases,axiom,
! [X2: num,Y: num] :
( ~ ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_num @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_351_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_352_linorder__le__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_353_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_354_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > num,C: num] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_355_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_356_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_357_ord__le__eq__subst,axiom,
! [A2: num,B: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_358_ord__le__eq__subst,axiom,
! [A2: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_359_ord__le__eq__subst,axiom,
! [A2: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_360_ord__le__eq__subst,axiom,
! [A2: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_361_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_362_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_363_ord__eq__le__subst,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_364_ord__eq__le__subst,axiom,
! [A2: num,F: real > num,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_365_ord__eq__le__subst,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_366_ord__eq__le__subst,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_367_ord__eq__le__subst,axiom,
! [A2: real,F: num > real,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_368_ord__eq__le__subst,axiom,
! [A2: num,F: num > num,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_369_ord__eq__le__subst,axiom,
! [A2: nat,F: num > nat,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_370_ord__eq__le__subst,axiom,
! [A2: int,F: num > int,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_371_ord__eq__le__subst,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_372_ord__eq__le__subst,axiom,
! [A2: num,F: nat > num,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_373_linorder__linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
| ( ord_less_eq_real @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_374_linorder__linear,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
| ( ord_less_eq_num @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_375_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_376_linorder__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_377_order__eq__refl,axiom,
! [X2: real,Y: real] :
( ( X2 = Y )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_378_order__eq__refl,axiom,
! [X2: set_complex,Y: set_complex] :
( ( X2 = Y )
=> ( ord_le211207098394363844omplex @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_379_order__eq__refl,axiom,
! [X2: num,Y: num] :
( ( X2 = Y )
=> ( ord_less_eq_num @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_380_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_381_order__eq__refl,axiom,
! [X2: int,Y: int] :
( ( X2 = Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_382_order__subst2,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_383_order__subst2,axiom,
! [A2: real,B: real,F: real > num,C: num] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_384_order__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_385_order__subst2,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_386_order__subst2,axiom,
! [A2: num,B: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_387_order__subst2,axiom,
! [A2: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_388_order__subst2,axiom,
! [A2: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_389_order__subst2,axiom,
! [A2: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_390_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_391_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_392_order__subst1,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_393_order__subst1,axiom,
! [A2: real,F: num > real,B: num,C: num] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_394_order__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_395_order__subst1,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_396_order__subst1,axiom,
! [A2: num,F: real > num,B: real,C: real] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_397_order__subst1,axiom,
! [A2: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_398_order__subst1,axiom,
! [A2: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_399_order__subst1,axiom,
! [A2: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_400_order__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_401_order__subst1,axiom,
! [A2: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_402_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A4: real,B5: real] :
( ( ord_less_eq_real @ A4 @ B5 )
& ( ord_less_eq_real @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_403_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_complex,Z: set_complex] : ( Y3 = Z ) )
= ( ^ [A4: set_complex,B5: set_complex] :
( ( ord_le211207098394363844omplex @ A4 @ B5 )
& ( ord_le211207098394363844omplex @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_404_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: num,Z: num] : ( Y3 = Z ) )
= ( ^ [A4: num,B5: num] :
( ( ord_less_eq_num @ A4 @ B5 )
& ( ord_less_eq_num @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_405_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ( ord_less_eq_nat @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_406_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A4: int,B5: int] :
( ( ord_less_eq_int @ A4 @ B5 )
& ( ord_less_eq_int @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_407_antisym,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_408_antisym,axiom,
! [A2: set_complex,B: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ B )
=> ( ( ord_le211207098394363844omplex @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_409_antisym,axiom,
! [A2: num,B: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_num @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_410_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_411_antisym,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_412_dual__order_Otrans,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_413_dual__order_Otrans,axiom,
! [B: set_complex,A2: set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ B @ A2 )
=> ( ( ord_le211207098394363844omplex @ C @ B )
=> ( ord_le211207098394363844omplex @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_414_dual__order_Otrans,axiom,
! [B: num,A2: num,C: num] :
( ( ord_less_eq_num @ B @ A2 )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_415_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_416_dual__order_Otrans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_417_dual__order_Oantisym,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_418_dual__order_Oantisym,axiom,
! [B: set_complex,A2: set_complex] :
( ( ord_le211207098394363844omplex @ B @ A2 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_419_dual__order_Oantisym,axiom,
! [B: num,A2: num] :
( ( ord_less_eq_num @ B @ A2 )
=> ( ( ord_less_eq_num @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_420_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_421_dual__order_Oantisym,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_422_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A4: real,B5: real] :
( ( ord_less_eq_real @ B5 @ A4 )
& ( ord_less_eq_real @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_423_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_complex,Z: set_complex] : ( Y3 = Z ) )
= ( ^ [A4: set_complex,B5: set_complex] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
& ( ord_le211207098394363844omplex @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_424_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: num,Z: num] : ( Y3 = Z ) )
= ( ^ [A4: num,B5: num] :
( ( ord_less_eq_num @ B5 @ A4 )
& ( ord_less_eq_num @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_425_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ( ord_less_eq_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_426_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A4: int,B5: int] :
( ( ord_less_eq_int @ B5 @ A4 )
& ( ord_less_eq_int @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_427_linorder__wlog,axiom,
! [P2: real > real > $o,A2: real,B: real] :
( ! [A5: real,B6: real] :
( ( ord_less_eq_real @ A5 @ B6 )
=> ( P2 @ A5 @ B6 ) )
=> ( ! [A5: real,B6: real] :
( ( P2 @ B6 @ A5 )
=> ( P2 @ A5 @ B6 ) )
=> ( P2 @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_428_linorder__wlog,axiom,
! [P2: num > num > $o,A2: num,B: num] :
( ! [A5: num,B6: num] :
( ( ord_less_eq_num @ A5 @ B6 )
=> ( P2 @ A5 @ B6 ) )
=> ( ! [A5: num,B6: num] :
( ( P2 @ B6 @ A5 )
=> ( P2 @ A5 @ B6 ) )
=> ( P2 @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_429_linorder__wlog,axiom,
! [P2: nat > nat > $o,A2: nat,B: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
=> ( P2 @ A5 @ B6 ) )
=> ( ! [A5: nat,B6: nat] :
( ( P2 @ B6 @ A5 )
=> ( P2 @ A5 @ B6 ) )
=> ( P2 @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_430_linorder__wlog,axiom,
! [P2: int > int > $o,A2: int,B: int] :
( ! [A5: int,B6: int] :
( ( ord_less_eq_int @ A5 @ B6 )
=> ( P2 @ A5 @ B6 ) )
=> ( ! [A5: int,B6: int] :
( ( P2 @ B6 @ A5 )
=> ( P2 @ A5 @ B6 ) )
=> ( P2 @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_431_order__trans,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_eq_real @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_432_order__trans,axiom,
! [X2: set_complex,Y: set_complex,Z2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y )
=> ( ( ord_le211207098394363844omplex @ Y @ Z2 )
=> ( ord_le211207098394363844omplex @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_433_order__trans,axiom,
! [X2: num,Y: num,Z2: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ( ord_less_eq_num @ Y @ Z2 )
=> ( ord_less_eq_num @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_434_order__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_435_order__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_436_order_Otrans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_437_order_Otrans,axiom,
! [A2: set_complex,B: set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ B )
=> ( ( ord_le211207098394363844omplex @ B @ C )
=> ( ord_le211207098394363844omplex @ A2 @ C ) ) ) ).
% order.trans
thf(fact_438_order_Otrans,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% order.trans
thf(fact_439_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_440_order_Otrans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_441_order__antisym,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_442_order__antisym,axiom,
! [X2: set_complex,Y: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y )
=> ( ( ord_le211207098394363844omplex @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_443_order__antisym,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ( ord_less_eq_num @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_444_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_445_order__antisym,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_446_ord__le__eq__trans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_447_ord__le__eq__trans,axiom,
! [A2: set_complex,B: set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ B )
=> ( ( B = C )
=> ( ord_le211207098394363844omplex @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_448_ord__le__eq__trans,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_449_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_450_ord__le__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_451_ord__eq__le__trans,axiom,
! [A2: real,B: real,C: real] :
( ( A2 = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_452_ord__eq__le__trans,axiom,
! [A2: set_complex,B: set_complex,C: set_complex] :
( ( A2 = B )
=> ( ( ord_le211207098394363844omplex @ B @ C )
=> ( ord_le211207098394363844omplex @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_453_ord__eq__le__trans,axiom,
! [A2: num,B: num,C: num] :
( ( A2 = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_454_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_455_ord__eq__le__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_456_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
& ( ord_less_eq_real @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_457_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_complex,Z: set_complex] : ( Y3 = Z ) )
= ( ^ [X4: set_complex,Y5: set_complex] :
( ( ord_le211207098394363844omplex @ X4 @ Y5 )
& ( ord_le211207098394363844omplex @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_458_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: num,Z: num] : ( Y3 = Z ) )
= ( ^ [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
& ( ord_less_eq_num @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_459_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_460_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [X4: int,Y5: int] :
( ( ord_less_eq_int @ X4 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_461_le__cases3,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ( ord_less_eq_real @ X2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_462_le__cases3,axiom,
! [X2: num,Y: num,Z2: num] :
( ( ( ord_less_eq_num @ X2 @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_num @ Y @ X2 )
=> ~ ( ord_less_eq_num @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_num @ X2 @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_num @ Z2 @ Y )
=> ~ ( ord_less_eq_num @ Y @ X2 ) )
=> ( ( ( ord_less_eq_num @ Y @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_num @ Z2 @ X2 )
=> ~ ( ord_less_eq_num @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_463_le__cases3,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_464_le__cases3,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_465_nle__le,axiom,
! [A2: real,B: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B ) )
= ( ( ord_less_eq_real @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_466_nle__le,axiom,
! [A2: num,B: num] :
( ( ~ ( ord_less_eq_num @ A2 @ B ) )
= ( ( ord_less_eq_num @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_467_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_468_nle__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B ) )
= ( ( ord_less_eq_int @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_469_bot_Oextremum__uniqueI,axiom,
! [A2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ bot_bo4199563552545308370d_enat )
=> ( A2 = bot_bo4199563552545308370d_enat ) ) ).
% bot.extremum_uniqueI
thf(fact_470_bot_Oextremum__uniqueI,axiom,
! [A2: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ bot_bot_set_complex )
=> ( A2 = bot_bot_set_complex ) ) ).
% bot.extremum_uniqueI
thf(fact_471_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_472_bot_Oextremum__unique,axiom,
! [A2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ bot_bo4199563552545308370d_enat )
= ( A2 = bot_bo4199563552545308370d_enat ) ) ).
% bot.extremum_unique
thf(fact_473_bot_Oextremum__unique,axiom,
! [A2: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ bot_bot_set_complex )
= ( A2 = bot_bot_set_complex ) ) ).
% bot.extremum_unique
thf(fact_474_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_475_bot_Oextremum,axiom,
! [A2: extended_enat] : ( ord_le2932123472753598470d_enat @ bot_bo4199563552545308370d_enat @ A2 ) ).
% bot.extremum
thf(fact_476_bot_Oextremum,axiom,
! [A2: set_complex] : ( ord_le211207098394363844omplex @ bot_bot_set_complex @ A2 ) ).
% bot.extremum
thf(fact_477_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_478_real__eq__0__iff__le__ge__0,axiom,
! [X2: real] :
( ( X2 = zero_zero_real )
= ( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X2 ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_479_is__singleton__the__elem,axiom,
( is_singleton_complex
= ( ^ [A3: set_complex] :
( A3
= ( insert_complex @ ( the_elem_complex @ A3 ) @ bot_bot_set_complex ) ) ) ) ).
% is_singleton_the_elem
thf(fact_480_holomorphic__on__imp__continuous__on,axiom,
! [F: complex > complex,S: set_complex] :
( ( comple7700996537433184370hic_on @ F @ S )
=> ( topolo9015423870875150044omplex @ S @ F ) ) ).
% holomorphic_on_imp_continuous_on
thf(fact_481_is__singletonI,axiom,
! [X2: complex] : ( is_singleton_complex @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ).
% is_singletonI
thf(fact_482_holomorphic__on__subset,axiom,
! [F: complex > complex,S: set_complex,T: set_complex] :
( ( comple7700996537433184370hic_on @ F @ S )
=> ( ( ord_le211207098394363844omplex @ T @ S )
=> ( comple7700996537433184370hic_on @ F @ T ) ) ) ).
% holomorphic_on_subset
thf(fact_483_holomorphic__on__empty,axiom,
! [F: complex > complex] : ( comple7700996537433184370hic_on @ F @ bot_bot_set_complex ) ).
% holomorphic_on_empty
thf(fact_484_minus__in__Ints__iff,axiom,
! [X2: real] :
( ( member_real @ ( uminus_uminus_real @ X2 ) @ ring_1_Ints_real )
= ( member_real @ X2 @ ring_1_Ints_real ) ) ).
% minus_in_Ints_iff
thf(fact_485_minus__in__Ints__iff,axiom,
! [X2: complex] :
( ( member_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ring_1_Ints_complex )
= ( member_complex @ X2 @ ring_1_Ints_complex ) ) ).
% minus_in_Ints_iff
thf(fact_486_minus__in__Ints__iff,axiom,
! [X2: int] :
( ( member_int @ ( uminus_uminus_int @ X2 ) @ ring_1_Ints_int )
= ( member_int @ X2 @ ring_1_Ints_int ) ) ).
% minus_in_Ints_iff
thf(fact_487_Ints__minus,axiom,
! [A2: real] :
( ( member_real @ A2 @ ring_1_Ints_real )
=> ( member_real @ ( uminus_uminus_real @ A2 ) @ ring_1_Ints_real ) ) ).
% Ints_minus
thf(fact_488_Ints__minus,axiom,
! [A2: complex] :
( ( member_complex @ A2 @ ring_1_Ints_complex )
=> ( member_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ring_1_Ints_complex ) ) ).
% Ints_minus
thf(fact_489_Ints__minus,axiom,
! [A2: int] :
( ( member_int @ A2 @ ring_1_Ints_int )
=> ( member_int @ ( uminus_uminus_int @ A2 ) @ ring_1_Ints_int ) ) ).
% Ints_minus
thf(fact_490_Ints__diff,axiom,
! [A2: complex,B: complex] :
( ( member_complex @ A2 @ ring_1_Ints_complex )
=> ( ( member_complex @ B @ ring_1_Ints_complex )
=> ( member_complex @ ( minus_minus_complex @ A2 @ B ) @ ring_1_Ints_complex ) ) ) ).
% Ints_diff
thf(fact_491_Ints__diff,axiom,
! [A2: real,B: real] :
( ( member_real @ A2 @ ring_1_Ints_real )
=> ( ( member_real @ B @ ring_1_Ints_real )
=> ( member_real @ ( minus_minus_real @ A2 @ B ) @ ring_1_Ints_real ) ) ) ).
% Ints_diff
thf(fact_492_Ints__diff,axiom,
! [A2: int,B: int] :
( ( member_int @ A2 @ ring_1_Ints_int )
=> ( ( member_int @ B @ ring_1_Ints_int )
=> ( member_int @ ( minus_minus_int @ A2 @ B ) @ ring_1_Ints_int ) ) ) ).
% Ints_diff
thf(fact_493_is__singletonI_H,axiom,
! [A: set_real] :
( ( A != bot_bot_set_real )
=> ( ! [X: real,Y4: real] :
( ( member_real @ X @ A )
=> ( ( member_real @ Y4 @ A )
=> ( X = Y4 ) ) )
=> ( is_singleton_real @ A ) ) ) ).
% is_singletonI'
thf(fact_494_is__singletonI_H,axiom,
! [A: set_complex] :
( ( A != bot_bot_set_complex )
=> ( ! [X: complex,Y4: complex] :
( ( member_complex @ X @ A )
=> ( ( member_complex @ Y4 @ A )
=> ( X = Y4 ) ) )
=> ( is_singleton_complex @ A ) ) ) ).
% is_singletonI'
thf(fact_495_holomorphic__cong,axiom,
! [S: set_complex,T: set_complex,F: complex > complex,G: complex > complex] :
( ( S = T )
=> ( ! [X: complex] :
( ( member_complex @ X @ S )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( comple7700996537433184370hic_on @ F @ S )
= ( comple7700996537433184370hic_on @ G @ T ) ) ) ) ).
% holomorphic_cong
thf(fact_496_holomorphic__transform,axiom,
! [F: complex > complex,S: set_complex,G: complex > complex] :
( ( comple7700996537433184370hic_on @ F @ S )
=> ( ! [X: complex] :
( ( member_complex @ X @ S )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( comple7700996537433184370hic_on @ G @ S ) ) ) ).
% holomorphic_transform
thf(fact_497_is__singletonE,axiom,
! [A: set_complex] :
( ( is_singleton_complex @ A )
=> ~ ! [X: complex] :
( A
!= ( insert_complex @ X @ bot_bot_set_complex ) ) ) ).
% is_singletonE
thf(fact_498_is__singleton__def,axiom,
( is_singleton_complex
= ( ^ [A3: set_complex] :
? [X4: complex] :
( A3
= ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) ) ).
% is_singleton_def
thf(fact_499_Ints__0,axiom,
member_complex @ zero_zero_complex @ ring_1_Ints_complex ).
% Ints_0
thf(fact_500_Ints__0,axiom,
member_real @ zero_zero_real @ ring_1_Ints_real ).
% Ints_0
thf(fact_501_Ints__0,axiom,
member_int @ zero_zero_int @ ring_1_Ints_int ).
% Ints_0
thf(fact_502_Collect__empty__eq__bot,axiom,
! [P2: complex > $o] :
( ( ( collect_complex @ P2 )
= bot_bot_set_complex )
= ( P2 = bot_bot_complex_o ) ) ).
% Collect_empty_eq_bot
thf(fact_503_bot__empty__eq,axiom,
( bot_bot_real_o
= ( ^ [X4: real] : ( member_real @ X4 @ bot_bot_set_real ) ) ) ).
% bot_empty_eq
thf(fact_504_bot__empty__eq,axiom,
( bot_bot_complex_o
= ( ^ [X4: complex] : ( member_complex @ X4 @ bot_bot_set_complex ) ) ) ).
% bot_empty_eq
thf(fact_505_Multiseries__Expansion_Oreal__eqI,axiom,
! [A2: real,B: real] :
( ( ( minus_minus_real @ A2 @ B )
= zero_zero_real )
=> ( A2 = B ) ) ).
% Multiseries_Expansion.real_eqI
thf(fact_506_continuous__on__op__minus,axiom,
! [S: set_complex,X2: complex] : ( topolo9015423870875150044omplex @ S @ ( minus_minus_complex @ X2 ) ) ).
% continuous_on_op_minus
thf(fact_507_continuous__on__op__minus,axiom,
! [S: set_real,X2: real] : ( topolo5044208981011980120l_real @ S @ ( minus_minus_real @ X2 ) ) ).
% continuous_on_op_minus
thf(fact_508_insert__subsetI,axiom,
! [X2: real,A: set_real,X3: set_real] :
( ( member_real @ X2 @ A )
=> ( ( ord_less_eq_set_real @ X3 @ A )
=> ( ord_less_eq_set_real @ ( insert_real @ X2 @ X3 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_509_insert__subsetI,axiom,
! [X2: complex,A: set_complex,X3: set_complex] :
( ( member_complex @ X2 @ A )
=> ( ( ord_le211207098394363844omplex @ X3 @ A )
=> ( ord_le211207098394363844omplex @ ( insert_complex @ X2 @ X3 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_510_Multiseries__Expansion__Bounds_Oeq__zero__imp__nonneg,axiom,
! [X2: real] :
( ( X2 = zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% Multiseries_Expansion_Bounds.eq_zero_imp_nonneg
thf(fact_511_subset__emptyI,axiom,
! [A: set_real] :
( ! [X: real] :
~ ( member_real @ X @ A )
=> ( ord_less_eq_set_real @ A @ bot_bot_set_real ) ) ).
% subset_emptyI
thf(fact_512_subset__emptyI,axiom,
! [A: set_complex] :
( ! [X: complex] :
~ ( member_complex @ X @ A )
=> ( ord_le211207098394363844omplex @ A @ bot_bot_set_complex ) ) ).
% subset_emptyI
thf(fact_513_gbinomial__series__aux_Oexhaust,axiom,
! [Abort: $o,Acc: real] :
( ( Abort
=> ( Acc != zero_zero_real ) )
=> ( ~ Abort
| ( Acc != zero_zero_real ) ) ) ).
% gbinomial_series_aux.exhaust
thf(fact_514_minus__diff__minus,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( minus_minus_real @ A2 @ B ) ) ) ).
% minus_diff_minus
thf(fact_515_minus__diff__minus,axiom,
! [A2: complex,B: complex] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( uminus1482373934393186551omplex @ B ) )
= ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A2 @ B ) ) ) ).
% minus_diff_minus
thf(fact_516_minus__diff__minus,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B ) ) ) ).
% minus_diff_minus
thf(fact_517_le__numeral__extra_I3_J,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ).
% le_numeral_extra(3)
thf(fact_518_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_519_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_520_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_521_remove__def,axiom,
( remove_complex
= ( ^ [X4: complex,A3: set_complex] : ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) ) ).
% remove_def
thf(fact_522_Set_Ois__empty__def,axiom,
( is_empty_complex
= ( ^ [A3: set_complex] : ( A3 = bot_bot_set_complex ) ) ) ).
% Set.is_empty_def
thf(fact_523_span__delete__0,axiom,
! [S2: set_real] :
( ( real_V5325414057265605809n_real @ ( minus_minus_set_real @ S2 @ ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
= ( real_V5325414057265605809n_real @ S2 ) ) ).
% span_delete_0
thf(fact_524_span__delete__0,axiom,
! [S2: set_complex] :
( ( real_V8921647422947696435omplex @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) )
= ( real_V8921647422947696435omplex @ S2 ) ) ).
% span_delete_0
thf(fact_525_Gcd__factorial__eq__0__iff,axiom,
! [A: set_nat] :
( ( ( factor8539158941071730396al_nat @ A )
= zero_zero_nat )
= ( ord_less_eq_set_nat @ A @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% Gcd_factorial_eq_0_iff
thf(fact_526_Gcd__factorial__eq__0__iff,axiom,
! [A: set_int] :
( ( ( factor8536668470562680120al_int @ A )
= zero_zero_int )
= ( ord_less_eq_set_int @ A @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ) ).
% Gcd_factorial_eq_0_iff
thf(fact_527_rGamma__real__def,axiom,
( gamma_4599518313873207670a_real
= ( ^ [X4: real] : ( re @ ( gamma_4773869415665495160omplex @ ( real_V4546457046886955230omplex @ X4 ) ) ) ) ) ).
% rGamma_real_def
thf(fact_528_member__remove,axiom,
! [X2: complex,Y: complex,A: set_complex] :
( ( member_complex @ X2 @ ( remove_complex @ Y @ A ) )
= ( ( member_complex @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_529_member__remove,axiom,
! [X2: real,Y: real,A: set_real] :
( ( member_real @ X2 @ ( remove_real @ Y @ A ) )
= ( ( member_real @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_530_span__insert__0,axiom,
! [S2: set_complex] :
( ( real_V8921647422947696435omplex @ ( insert_complex @ zero_zero_complex @ S2 ) )
= ( real_V8921647422947696435omplex @ S2 ) ) ).
% span_insert_0
thf(fact_531_span__insert__0,axiom,
! [S2: set_real] :
( ( real_V5325414057265605809n_real @ ( insert_real @ zero_zero_real @ S2 ) )
= ( real_V5325414057265605809n_real @ S2 ) ) ).
% span_insert_0
thf(fact_532_span__empty,axiom,
( ( real_V5325414057265605809n_real @ bot_bot_set_real )
= ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) ).
% span_empty
thf(fact_533_span__empty,axiom,
( ( real_V8921647422947696435omplex @ bot_bot_set_complex )
= ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) ).
% span_empty
thf(fact_534_span__base,axiom,
! [A2: complex,S2: set_complex] :
( ( member_complex @ A2 @ S2 )
=> ( member_complex @ A2 @ ( real_V8921647422947696435omplex @ S2 ) ) ) ).
% span_base
thf(fact_535_span__base,axiom,
! [A2: real,S2: set_real] :
( ( member_real @ A2 @ S2 )
=> ( member_real @ A2 @ ( real_V5325414057265605809n_real @ S2 ) ) ) ).
% span_base
thf(fact_536_span__0,axiom,
! [S2: set_complex] : ( member_complex @ zero_zero_complex @ ( real_V8921647422947696435omplex @ S2 ) ) ).
% span_0
thf(fact_537_span__0,axiom,
! [S2: set_real] : ( member_real @ zero_zero_real @ ( real_V5325414057265605809n_real @ S2 ) ) ).
% span_0
thf(fact_538_span__diff,axiom,
! [X2: complex,S2: set_complex,Y: complex] :
( ( member_complex @ X2 @ ( real_V8921647422947696435omplex @ S2 ) )
=> ( ( member_complex @ Y @ ( real_V8921647422947696435omplex @ S2 ) )
=> ( member_complex @ ( minus_minus_complex @ X2 @ Y ) @ ( real_V8921647422947696435omplex @ S2 ) ) ) ) ).
% span_diff
thf(fact_539_span__diff,axiom,
! [X2: real,S2: set_real,Y: real] :
( ( member_real @ X2 @ ( real_V5325414057265605809n_real @ S2 ) )
=> ( ( member_real @ Y @ ( real_V5325414057265605809n_real @ S2 ) )
=> ( member_real @ ( minus_minus_real @ X2 @ Y ) @ ( real_V5325414057265605809n_real @ S2 ) ) ) ) ).
% span_diff
thf(fact_540_span__eq,axiom,
! [S2: set_complex,T2: set_complex] :
( ( ( real_V8921647422947696435omplex @ S2 )
= ( real_V8921647422947696435omplex @ T2 ) )
= ( ( ord_le211207098394363844omplex @ S2 @ ( real_V8921647422947696435omplex @ T2 ) )
& ( ord_le211207098394363844omplex @ T2 @ ( real_V8921647422947696435omplex @ S2 ) ) ) ) ).
% span_eq
thf(fact_541_span__mono,axiom,
! [A: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ord_le211207098394363844omplex @ ( real_V8921647422947696435omplex @ A ) @ ( real_V8921647422947696435omplex @ B2 ) ) ) ).
% span_mono
thf(fact_542_span__superset,axiom,
! [S2: set_complex] : ( ord_le211207098394363844omplex @ S2 @ ( real_V8921647422947696435omplex @ S2 ) ) ).
% span_superset
thf(fact_543_span__neg,axiom,
! [X2: real,S2: set_real] :
( ( member_real @ X2 @ ( real_V5325414057265605809n_real @ S2 ) )
=> ( member_real @ ( uminus_uminus_real @ X2 ) @ ( real_V5325414057265605809n_real @ S2 ) ) ) ).
% span_neg
thf(fact_544_span__neg,axiom,
! [X2: complex,S2: set_complex] :
( ( member_complex @ X2 @ ( real_V8921647422947696435omplex @ S2 ) )
=> ( member_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( real_V8921647422947696435omplex @ S2 ) ) ) ).
% span_neg
thf(fact_545_span__trans,axiom,
! [X2: complex,S2: set_complex,Y: complex] :
( ( member_complex @ X2 @ ( real_V8921647422947696435omplex @ S2 ) )
=> ( ( member_complex @ Y @ ( real_V8921647422947696435omplex @ ( insert_complex @ X2 @ S2 ) ) )
=> ( member_complex @ Y @ ( real_V8921647422947696435omplex @ S2 ) ) ) ) ).
% span_trans
thf(fact_546_span__trans,axiom,
! [X2: real,S2: set_real,Y: real] :
( ( member_real @ X2 @ ( real_V5325414057265605809n_real @ S2 ) )
=> ( ( member_real @ Y @ ( real_V5325414057265605809n_real @ ( insert_real @ X2 @ S2 ) ) )
=> ( member_real @ Y @ ( real_V5325414057265605809n_real @ S2 ) ) ) ) ).
% span_trans
thf(fact_547_in__span__insert,axiom,
! [A2: complex,B: complex,S2: set_complex] :
( ( member_complex @ A2 @ ( real_V8921647422947696435omplex @ ( insert_complex @ B @ S2 ) ) )
=> ( ~ ( member_complex @ A2 @ ( real_V8921647422947696435omplex @ S2 ) )
=> ( member_complex @ B @ ( real_V8921647422947696435omplex @ ( insert_complex @ A2 @ S2 ) ) ) ) ) ).
% in_span_insert
thf(fact_548_in__span__insert,axiom,
! [A2: real,B: real,S2: set_real] :
( ( member_real @ A2 @ ( real_V5325414057265605809n_real @ ( insert_real @ B @ S2 ) ) )
=> ( ~ ( member_real @ A2 @ ( real_V5325414057265605809n_real @ S2 ) )
=> ( member_real @ B @ ( real_V5325414057265605809n_real @ ( insert_real @ A2 @ S2 ) ) ) ) ) ).
% in_span_insert
thf(fact_549_span__redundant,axiom,
! [X2: complex,S2: set_complex] :
( ( member_complex @ X2 @ ( real_V8921647422947696435omplex @ S2 ) )
=> ( ( real_V8921647422947696435omplex @ ( insert_complex @ X2 @ S2 ) )
= ( real_V8921647422947696435omplex @ S2 ) ) ) ).
% span_redundant
thf(fact_550_span__redundant,axiom,
! [X2: real,S2: set_real] :
( ( member_real @ X2 @ ( real_V5325414057265605809n_real @ S2 ) )
=> ( ( real_V5325414057265605809n_real @ ( insert_real @ X2 @ S2 ) )
= ( real_V5325414057265605809n_real @ S2 ) ) ) ).
% span_redundant
thf(fact_551_eq__span__insert__eq,axiom,
! [X2: complex,Y: complex,S2: set_complex] :
( ( member_complex @ ( minus_minus_complex @ X2 @ Y ) @ ( real_V8921647422947696435omplex @ S2 ) )
=> ( ( real_V8921647422947696435omplex @ ( insert_complex @ X2 @ S2 ) )
= ( real_V8921647422947696435omplex @ ( insert_complex @ Y @ S2 ) ) ) ) ).
% eq_span_insert_eq
thf(fact_552_eq__span__insert__eq,axiom,
! [X2: real,Y: real,S2: set_real] :
( ( member_real @ ( minus_minus_real @ X2 @ Y ) @ ( real_V5325414057265605809n_real @ S2 ) )
=> ( ( real_V5325414057265605809n_real @ ( insert_real @ X2 @ S2 ) )
= ( real_V5325414057265605809n_real @ ( insert_real @ Y @ S2 ) ) ) ) ).
% eq_span_insert_eq
thf(fact_553_in__span__delete,axiom,
! [A2: real,S2: set_real,B: real] :
( ( member_real @ A2 @ ( real_V5325414057265605809n_real @ S2 ) )
=> ( ~ ( member_real @ A2 @ ( real_V5325414057265605809n_real @ ( minus_minus_set_real @ S2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) )
=> ( member_real @ B @ ( real_V5325414057265605809n_real @ ( insert_real @ A2 @ ( minus_minus_set_real @ S2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ) ) ) ).
% in_span_delete
thf(fact_554_in__span__delete,axiom,
! [A2: complex,S2: set_complex,B: complex] :
( ( member_complex @ A2 @ ( real_V8921647422947696435omplex @ S2 ) )
=> ( ~ ( member_complex @ A2 @ ( real_V8921647422947696435omplex @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ B @ bot_bot_set_complex ) ) ) )
=> ( member_complex @ B @ ( real_V8921647422947696435omplex @ ( insert_complex @ A2 @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ B @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% in_span_delete
thf(fact_555_complete__real,axiom,
! [S2: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S2 )
=> ( ? [Z3: real] :
! [X: real] :
( ( member_real @ X @ S2 )
=> ( ord_less_eq_real @ X @ Z3 ) )
=> ? [Y4: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S2 )
=> ( ord_less_eq_real @ X5 @ Y4 ) )
& ! [Z3: real] :
( ! [X: real] :
( ( member_real @ X @ S2 )
=> ( ord_less_eq_real @ X @ Z3 ) )
=> ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ) ) ).
% complete_real
thf(fact_556_Re__complex__of__real,axiom,
! [Z2: real] :
( ( re @ ( real_V4546457046886955230omplex @ Z2 ) )
= Z2 ) ).
% Re_complex_of_real
thf(fact_557_zero__complex_Osimps_I1_J,axiom,
( ( re @ zero_zero_complex )
= zero_zero_real ) ).
% zero_complex.simps(1)
thf(fact_558_uminus__complex_Osimps_I1_J,axiom,
! [X2: complex] :
( ( re @ ( uminus1482373934393186551omplex @ X2 ) )
= ( uminus_uminus_real @ ( re @ X2 ) ) ) ).
% uminus_complex.simps(1)
thf(fact_559_minus__complex_Osimps_I1_J,axiom,
! [X2: complex,Y: complex] :
( ( re @ ( minus_minus_complex @ X2 @ Y ) )
= ( minus_minus_real @ ( re @ X2 ) @ ( re @ Y ) ) ) ).
% minus_complex.simps(1)
thf(fact_560_span__breakdown,axiom,
! [B: complex,S2: set_complex,A2: complex] :
( ( member_complex @ B @ S2 )
=> ( ( member_complex @ A2 @ ( real_V8921647422947696435omplex @ S2 ) )
=> ? [K2: real] : ( member_complex @ ( minus_minus_complex @ A2 @ ( real_V2046097035970521341omplex @ K2 @ B ) ) @ ( real_V8921647422947696435omplex @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ B @ bot_bot_set_complex ) ) ) ) ) ) ).
% span_breakdown
thf(fact_561_span__breakdown,axiom,
! [B: real,S2: set_real,A2: real] :
( ( member_real @ B @ S2 )
=> ( ( member_real @ A2 @ ( real_V5325414057265605809n_real @ S2 ) )
=> ? [K2: real] : ( member_real @ ( minus_minus_real @ A2 @ ( real_V1485227260804924795R_real @ K2 @ B ) ) @ ( real_V5325414057265605809n_real @ ( minus_minus_set_real @ S2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ) ) ).
% span_breakdown
thf(fact_562_rGamma__neg__1,axiom,
( ( gamma_4773869415665495160omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% rGamma_neg_1
thf(fact_563_rGamma__neg__1,axiom,
( ( gamma_4599518313873207670a_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% rGamma_neg_1
thf(fact_564_psubset__insert__iff,axiom,
! [A: set_real,X2: real,B2: set_real] :
( ( ord_less_set_real @ A @ ( insert_real @ X2 @ B2 ) )
= ( ( ( member_real @ X2 @ B2 )
=> ( ord_less_set_real @ A @ B2 ) )
& ( ~ ( member_real @ X2 @ B2 )
=> ( ( ( member_real @ X2 @ A )
=> ( ord_less_set_real @ ( minus_minus_set_real @ A @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B2 ) )
& ( ~ ( member_real @ X2 @ A )
=> ( ord_less_eq_set_real @ A @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_565_psubset__insert__iff,axiom,
! [A: set_complex,X2: complex,B2: set_complex] :
( ( ord_less_set_complex @ A @ ( insert_complex @ X2 @ B2 ) )
= ( ( ( member_complex @ X2 @ B2 )
=> ( ord_less_set_complex @ A @ B2 ) )
& ( ~ ( member_complex @ X2 @ B2 )
=> ( ( ( member_complex @ X2 @ A )
=> ( ord_less_set_complex @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) @ B2 ) )
& ( ~ ( member_complex @ X2 @ A )
=> ( ord_le211207098394363844omplex @ A @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_566_not__gr__zero,axiom,
! [N: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% not_gr_zero
thf(fact_567_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_568_neg__less__iff__less,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_real @ A2 @ B ) ) ).
% neg_less_iff_less
thf(fact_569_neg__less__iff__less,axiom,
! [B: complex,A2: complex] :
( ( ord_less_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A2 ) )
= ( ord_less_complex @ A2 @ B ) ) ).
% neg_less_iff_less
thf(fact_570_neg__less__iff__less,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ B ) ) ).
% neg_less_iff_less
thf(fact_571_compl__less__compl__iff,axiom,
! [X2: set_complex,Y: set_complex] :
( ( ord_less_set_complex @ ( uminus8566677241136511917omplex @ X2 ) @ ( uminus8566677241136511917omplex @ Y ) )
= ( ord_less_set_complex @ Y @ X2 ) ) ).
% compl_less_compl_iff
thf(fact_572_scaleR__cancel__right,axiom,
! [A2: real,X2: complex,B: real] :
( ( ( real_V2046097035970521341omplex @ A2 @ X2 )
= ( real_V2046097035970521341omplex @ B @ X2 ) )
= ( ( A2 = B )
| ( X2 = zero_zero_complex ) ) ) ).
% scaleR_cancel_right
thf(fact_573_scaleR__cancel__right,axiom,
! [A2: real,X2: real,B: real] :
( ( ( real_V1485227260804924795R_real @ A2 @ X2 )
= ( real_V1485227260804924795R_real @ B @ X2 ) )
= ( ( A2 = B )
| ( X2 = zero_zero_real ) ) ) ).
% scaleR_cancel_right
thf(fact_574_scaleR__zero__right,axiom,
! [A2: real] :
( ( real_V2046097035970521341omplex @ A2 @ zero_zero_complex )
= zero_zero_complex ) ).
% scaleR_zero_right
thf(fact_575_scaleR__zero__right,axiom,
! [A2: real] :
( ( real_V1485227260804924795R_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% scaleR_zero_right
thf(fact_576_scaleR__minus__right,axiom,
! [A2: real,X2: real] :
( ( real_V1485227260804924795R_real @ A2 @ ( uminus_uminus_real @ X2 ) )
= ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ A2 @ X2 ) ) ) ).
% scaleR_minus_right
thf(fact_577_scaleR__minus__right,axiom,
! [A2: real,X2: complex] :
( ( real_V2046097035970521341omplex @ A2 @ ( uminus1482373934393186551omplex @ X2 ) )
= ( uminus1482373934393186551omplex @ ( real_V2046097035970521341omplex @ A2 @ X2 ) ) ) ).
% scaleR_minus_right
thf(fact_578_of__real__eq__1__iff,axiom,
! [X2: real] :
( ( ( real_V1803761363581548252l_real @ X2 )
= one_one_real )
= ( X2 = one_one_real ) ) ).
% of_real_eq_1_iff
thf(fact_579_of__real__eq__1__iff,axiom,
! [X2: real] :
( ( ( real_V4546457046886955230omplex @ X2 )
= one_one_complex )
= ( X2 = one_one_real ) ) ).
% of_real_eq_1_iff
thf(fact_580_of__real__1,axiom,
( ( real_V1803761363581548252l_real @ one_one_real )
= one_one_real ) ).
% of_real_1
thf(fact_581_of__real__1,axiom,
( ( real_V4546457046886955230omplex @ one_one_real )
= one_one_complex ) ).
% of_real_1
thf(fact_582_psubsetI,axiom,
! [A: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_complex @ A @ B2 ) ) ) ).
% psubsetI
thf(fact_583_rGamma__1,axiom,
( ( gamma_4773869415665495160omplex @ one_one_complex )
= one_one_complex ) ).
% rGamma_1
thf(fact_584_rGamma__1,axiom,
( ( gamma_4599518313873207670a_real @ one_one_real )
= one_one_real ) ).
% rGamma_1
thf(fact_585_scaleR__bcontfun_Orep__eq,axiom,
! [X2: real,Xa: bounde2223080031472583990omplex] :
( ( bounde4845209636850564516omplex @ ( real_V9054712056520343207omplex @ X2 @ Xa ) )
= ( ^ [X4: complex] : ( real_V2046097035970521341omplex @ X2 @ ( bounde4845209636850564516omplex @ Xa @ X4 ) ) ) ) ).
% scaleR_bcontfun.rep_eq
thf(fact_586_diff__gt__0__iff__gt,axiom,
! [A2: complex,B: complex] :
( ( ord_less_complex @ zero_zero_complex @ ( minus_minus_complex @ A2 @ B ) )
= ( ord_less_complex @ B @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_587_diff__gt__0__iff__gt,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B ) )
= ( ord_less_real @ B @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_588_diff__gt__0__iff__gt,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
= ( ord_less_int @ B @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_589_diff__numeral__special_I9_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% diff_numeral_special(9)
thf(fact_590_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_591_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_592_neg__less__0__iff__less,axiom,
! [A2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_593_neg__less__0__iff__less,axiom,
! [A2: complex] :
( ( ord_less_complex @ ( uminus1482373934393186551omplex @ A2 ) @ zero_zero_complex )
= ( ord_less_complex @ zero_zero_complex @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_594_neg__less__0__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_595_neg__0__less__iff__less,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_596_neg__0__less__iff__less,axiom,
! [A2: complex] :
( ( ord_less_complex @ zero_zero_complex @ ( uminus1482373934393186551omplex @ A2 ) )
= ( ord_less_complex @ A2 @ zero_zero_complex ) ) ).
% neg_0_less_iff_less
thf(fact_597_neg__0__less__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_598_neg__less__pos,axiom,
! [A2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% neg_less_pos
thf(fact_599_neg__less__pos,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_pos
thf(fact_600_less__neg__neg,axiom,
! [A2: real] :
( ( ord_less_real @ A2 @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_601_less__neg__neg,axiom,
! [A2: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_602_scaleR__zero__left,axiom,
! [X2: complex] :
( ( real_V2046097035970521341omplex @ zero_zero_real @ X2 )
= zero_zero_complex ) ).
% scaleR_zero_left
thf(fact_603_scaleR__zero__left,axiom,
! [X2: real] :
( ( real_V1485227260804924795R_real @ zero_zero_real @ X2 )
= zero_zero_real ) ).
% scaleR_zero_left
thf(fact_604_scaleR__eq__0__iff,axiom,
! [A2: real,X2: complex] :
( ( ( real_V2046097035970521341omplex @ A2 @ X2 )
= zero_zero_complex )
= ( ( A2 = zero_zero_real )
| ( X2 = zero_zero_complex ) ) ) ).
% scaleR_eq_0_iff
thf(fact_605_scaleR__eq__0__iff,axiom,
! [A2: real,X2: real] :
( ( ( real_V1485227260804924795R_real @ A2 @ X2 )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( X2 = zero_zero_real ) ) ) ).
% scaleR_eq_0_iff
thf(fact_606_scaleR__minus1__left,axiom,
! [X2: real] :
( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
= ( uminus_uminus_real @ X2 ) ) ).
% scaleR_minus1_left
thf(fact_607_scaleR__minus1__left,axiom,
! [X2: complex] :
( ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ one_one_real ) @ X2 )
= ( uminus1482373934393186551omplex @ X2 ) ) ).
% scaleR_minus1_left
thf(fact_608_scaleR__minus__left,axiom,
! [A2: real,X2: real] :
( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ A2 ) @ X2 )
= ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ A2 @ X2 ) ) ) ).
% scaleR_minus_left
thf(fact_609_scaleR__minus__left,axiom,
! [A2: real,X2: complex] :
( ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ A2 ) @ X2 )
= ( uminus1482373934393186551omplex @ ( real_V2046097035970521341omplex @ A2 @ X2 ) ) ) ).
% scaleR_minus_left
thf(fact_610_scaleR__left_Ominus,axiom,
! [X2: real,Xa: real] :
( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ X2 ) @ Xa )
= ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ X2 @ Xa ) ) ) ).
% scaleR_left.minus
thf(fact_611_scaleR__left_Ominus,axiom,
! [X2: real,Xa: complex] :
( ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ X2 ) @ Xa )
= ( uminus1482373934393186551omplex @ ( real_V2046097035970521341omplex @ X2 @ Xa ) ) ) ).
% scaleR_left.minus
thf(fact_612_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_613_diff__numeral__special_I12_J,axiom,
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% diff_numeral_special(12)
thf(fact_614_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_615_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(4)
thf(fact_616_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_617_less__minus__one__simps_I2_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% less_minus_one_simps(2)
thf(fact_618_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_619_less__numeral__extra_I1_J,axiom,
ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% less_numeral_extra(1)
thf(fact_620_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_621_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_622_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_623_of__real__def,axiom,
( real_V1803761363581548252l_real
= ( ^ [R: real] : ( real_V1485227260804924795R_real @ R @ one_one_real ) ) ) ).
% of_real_def
thf(fact_624_of__real__def,axiom,
( real_V4546457046886955230omplex
= ( ^ [R: real] : ( real_V2046097035970521341omplex @ R @ one_one_complex ) ) ) ).
% of_real_def
thf(fact_625_verit__comp__simplify1_I1_J,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_626_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_627_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_628_verit__comp__simplify1_I1_J,axiom,
! [A2: num] :
~ ( ord_less_num @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_629_psubsetD,axiom,
! [A: set_complex,B2: set_complex,C: complex] :
( ( ord_less_set_complex @ A @ B2 )
=> ( ( member_complex @ C @ A )
=> ( member_complex @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_630_psubsetD,axiom,
! [A: set_real,B2: set_real,C: real] :
( ( ord_less_set_real @ A @ B2 )
=> ( ( member_real @ C @ A )
=> ( member_real @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_631_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_632_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_633_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_634_lt__ex,axiom,
! [X2: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X2 ) ).
% lt_ex
thf(fact_635_lt__ex,axiom,
! [X2: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X2 ) ).
% lt_ex
thf(fact_636_gt__ex,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% gt_ex
thf(fact_637_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_638_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_639_dense,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ? [Z4: real] :
( ( ord_less_real @ X2 @ Z4 )
& ( ord_less_real @ Z4 @ Y ) ) ) ).
% dense
thf(fact_640_less__imp__neq,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_641_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_642_less__imp__neq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_643_less__imp__neq,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_644_order_Oasym,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ~ ( ord_less_real @ B @ A2 ) ) ).
% order.asym
thf(fact_645_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_646_order_Oasym,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order.asym
thf(fact_647_order_Oasym,axiom,
! [A2: num,B: num] :
( ( ord_less_num @ A2 @ B )
=> ~ ( ord_less_num @ B @ A2 ) ) ).
% order.asym
thf(fact_648_ord__eq__less__trans,axiom,
! [A2: real,B: real,C: real] :
( ( A2 = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_649_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_650_ord__eq__less__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_651_ord__eq__less__trans,axiom,
! [A2: num,B: num,C: num] :
( ( A2 = B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_652_ord__less__eq__trans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_653_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_654_ord__less__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_655_ord__less__eq__trans,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_656_less__induct,axiom,
! [P2: nat > $o,A2: nat] :
( ! [X: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X )
=> ( P2 @ Y6 ) )
=> ( P2 @ X ) )
=> ( P2 @ A2 ) ) ).
% less_induct
thf(fact_657_antisym__conv3,axiom,
! [Y: real,X2: real] :
( ~ ( ord_less_real @ Y @ X2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_658_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_659_antisym__conv3,axiom,
! [Y: int,X2: int] :
( ~ ( ord_less_int @ Y @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_660_antisym__conv3,axiom,
! [Y: num,X2: num] :
( ~ ( ord_less_num @ Y @ X2 )
=> ( ( ~ ( ord_less_num @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_661_linorder__cases,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_real @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_662_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_663_linorder__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_664_linorder__cases,axiom,
! [X2: num,Y: num] :
( ~ ( ord_less_num @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_num @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_665_dual__order_Oasym,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ~ ( ord_less_real @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_666_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_667_dual__order_Oasym,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ~ ( ord_less_int @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_668_dual__order_Oasym,axiom,
! [B: num,A2: num] :
( ( ord_less_num @ B @ A2 )
=> ~ ( ord_less_num @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_669_dual__order_Oirrefl,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_670_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_671_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_672_dual__order_Oirrefl,axiom,
! [A2: num] :
~ ( ord_less_num @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_673_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P4 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_674_linorder__less__wlog,axiom,
! [P2: real > real > $o,A2: real,B: real] :
( ! [A5: real,B6: real] :
( ( ord_less_real @ A5 @ B6 )
=> ( P2 @ A5 @ B6 ) )
=> ( ! [A5: real] : ( P2 @ A5 @ A5 )
=> ( ! [A5: real,B6: real] :
( ( P2 @ B6 @ A5 )
=> ( P2 @ A5 @ B6 ) )
=> ( P2 @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_675_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A2: nat,B: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_nat @ A5 @ B6 )
=> ( P2 @ A5 @ B6 ) )
=> ( ! [A5: nat] : ( P2 @ A5 @ A5 )
=> ( ! [A5: nat,B6: nat] :
( ( P2 @ B6 @ A5 )
=> ( P2 @ A5 @ B6 ) )
=> ( P2 @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_676_linorder__less__wlog,axiom,
! [P2: int > int > $o,A2: int,B: int] :
( ! [A5: int,B6: int] :
( ( ord_less_int @ A5 @ B6 )
=> ( P2 @ A5 @ B6 ) )
=> ( ! [A5: int] : ( P2 @ A5 @ A5 )
=> ( ! [A5: int,B6: int] :
( ( P2 @ B6 @ A5 )
=> ( P2 @ A5 @ B6 ) )
=> ( P2 @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_677_linorder__less__wlog,axiom,
! [P2: num > num > $o,A2: num,B: num] :
( ! [A5: num,B6: num] :
( ( ord_less_num @ A5 @ B6 )
=> ( P2 @ A5 @ B6 ) )
=> ( ! [A5: num] : ( P2 @ A5 @ A5 )
=> ( ! [A5: num,B6: num] :
( ( P2 @ B6 @ A5 )
=> ( P2 @ A5 @ B6 ) )
=> ( P2 @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_678_order_Ostrict__trans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_679_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_680_order_Ostrict__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_681_order_Ostrict__trans,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_682_not__less__iff__gr__or__eq,axiom,
! [X2: real,Y: real] :
( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( ( ord_less_real @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_683_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_684_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( ( ord_less_int @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_685_not__less__iff__gr__or__eq,axiom,
! [X2: num,Y: num] :
( ( ~ ( ord_less_num @ X2 @ Y ) )
= ( ( ord_less_num @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_686_dual__order_Ostrict__trans,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_687_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_688_dual__order_Ostrict__trans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_689_dual__order_Ostrict__trans,axiom,
! [B: num,A2: num,C: num] :
( ( ord_less_num @ B @ A2 )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_690_order_Ostrict__implies__not__eq,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_691_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_692_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_693_order_Ostrict__implies__not__eq,axiom,
! [A2: num,B: num] :
( ( ord_less_num @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_694_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_695_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_696_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_697_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A2: num] :
( ( ord_less_num @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_698_linorder__neqE,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
=> ( ~ ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_699_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_700_linorder__neqE,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
=> ( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_701_linorder__neqE,axiom,
! [X2: num,Y: num] :
( ( X2 != Y )
=> ( ~ ( ord_less_num @ X2 @ Y )
=> ( ord_less_num @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_702_order__less__asym,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_703_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_704_order__less__asym,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_705_order__less__asym,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ~ ( ord_less_num @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_706_linorder__neq__iff,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
= ( ( ord_less_real @ X2 @ Y )
| ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_707_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_708_linorder__neq__iff,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
= ( ( ord_less_int @ X2 @ Y )
| ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_709_linorder__neq__iff,axiom,
! [X2: num,Y: num] :
( ( X2 != Y )
= ( ( ord_less_num @ X2 @ Y )
| ( ord_less_num @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_710_order__less__asym_H,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ~ ( ord_less_real @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_711_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_712_order__less__asym_H,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_713_order__less__asym_H,axiom,
! [A2: num,B: num] :
( ( ord_less_num @ A2 @ B )
=> ~ ( ord_less_num @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_714_order__less__trans,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_715_order__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_716_order__less__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_717_order__less__trans,axiom,
! [X2: num,Y: num,Z2: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ( ord_less_num @ Y @ Z2 )
=> ( ord_less_num @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_718_ord__eq__less__subst,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_719_ord__eq__less__subst,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_720_ord__eq__less__subst,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_721_ord__eq__less__subst,axiom,
! [A2: num,F: real > num,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_722_ord__eq__less__subst,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_723_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_724_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_725_ord__eq__less__subst,axiom,
! [A2: num,F: nat > num,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_726_ord__eq__less__subst,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_727_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_728_ord__less__eq__subst,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_729_ord__less__eq__subst,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_730_ord__less__eq__subst,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_731_ord__less__eq__subst,axiom,
! [A2: real,B: real,F: real > num,C: num] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_732_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_733_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_734_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_735_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_736_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_737_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_738_order__less__irrefl,axiom,
! [X2: real] :
~ ( ord_less_real @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_739_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_740_order__less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_741_order__less__irrefl,axiom,
! [X2: num] :
~ ( ord_less_num @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_742_order__less__subst1,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_743_order__less__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_744_order__less__subst1,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_745_order__less__subst1,axiom,
! [A2: real,F: num > real,B: num,C: num] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_746_order__less__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_747_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_748_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_749_order__less__subst1,axiom,
! [A2: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_750_order__less__subst1,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_751_order__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_752_order__less__subst2,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_753_order__less__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_754_order__less__subst2,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_755_order__less__subst2,axiom,
! [A2: real,B: real,F: real > num,C: num] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_756_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_757_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_758_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_759_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_760_order__less__subst2,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_761_order__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_762_order__less__not__sym,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_763_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_764_order__less__not__sym,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_765_order__less__not__sym,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ~ ( ord_less_num @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_766_order__less__imp__triv,axiom,
! [X2: real,Y: real,P2: $o] :
( ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_real @ Y @ X2 )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_767_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P2: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_768_order__less__imp__triv,axiom,
! [X2: int,Y: int,P2: $o] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ X2 )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_769_order__less__imp__triv,axiom,
! [X2: num,Y: num,P2: $o] :
( ( ord_less_num @ X2 @ Y )
=> ( ( ord_less_num @ Y @ X2 )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_770_linorder__less__linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_real @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_771_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_772_linorder__less__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_int @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_773_linorder__less__linear,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_num @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_774_order__less__imp__not__eq,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_775_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_776_order__less__imp__not__eq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_777_order__less__imp__not__eq,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_778_order__less__imp__not__eq2,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_779_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_780_order__less__imp__not__eq2,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_781_order__less__imp__not__eq2,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_782_order__less__imp__not__less,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_783_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_784_order__less__imp__not__less,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_785_order__less__imp__not__less,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ~ ( ord_less_num @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_786_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(3)
thf(fact_787_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_788_less__minus__one__simps_I1_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% less_minus_one_simps(1)
thf(fact_789_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_790_pth__3,axiom,
( uminus_uminus_real
= ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% pth_3
thf(fact_791_pth__3,axiom,
( uminus1482373934393186551omplex
= ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% pth_3
thf(fact_792_set__one,axiom,
( one_one_set_real
= ( insert_real @ one_one_real @ bot_bot_set_real ) ) ).
% set_one
thf(fact_793_set__one,axiom,
( one_one_set_nat
= ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ).
% set_one
thf(fact_794_set__one,axiom,
( one_one_set_int
= ( insert_int @ one_one_int @ bot_bot_set_int ) ) ).
% set_one
thf(fact_795_set__one,axiom,
( one_one_set_complex
= ( insert_complex @ one_one_complex @ bot_bot_set_complex ) ) ).
% set_one
thf(fact_796_pth__4_I2_J,axiom,
! [C: real] :
( ( real_V2046097035970521341omplex @ C @ zero_zero_complex )
= zero_zero_complex ) ).
% pth_4(2)
thf(fact_797_pth__4_I2_J,axiom,
! [C: real] :
( ( real_V1485227260804924795R_real @ C @ zero_zero_real )
= zero_zero_real ) ).
% pth_4(2)
thf(fact_798_scaleR__right__imp__eq,axiom,
! [X2: complex,A2: real,B: real] :
( ( X2 != zero_zero_complex )
=> ( ( ( real_V2046097035970521341omplex @ A2 @ X2 )
= ( real_V2046097035970521341omplex @ B @ X2 ) )
=> ( A2 = B ) ) ) ).
% scaleR_right_imp_eq
thf(fact_799_scaleR__right__imp__eq,axiom,
! [X2: real,A2: real,B: real] :
( ( X2 != zero_zero_real )
=> ( ( ( real_V1485227260804924795R_real @ A2 @ X2 )
= ( real_V1485227260804924795R_real @ B @ X2 ) )
=> ( A2 = B ) ) ) ).
% scaleR_right_imp_eq
thf(fact_800_leD,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ~ ( ord_less_real @ X2 @ Y ) ) ).
% leD
thf(fact_801_leD,axiom,
! [Y: set_complex,X2: set_complex] :
( ( ord_le211207098394363844omplex @ Y @ X2 )
=> ~ ( ord_less_set_complex @ X2 @ Y ) ) ).
% leD
thf(fact_802_leD,axiom,
! [Y: num,X2: num] :
( ( ord_less_eq_num @ Y @ X2 )
=> ~ ( ord_less_num @ X2 @ Y ) ) ).
% leD
thf(fact_803_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_804_leD,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_int @ X2 @ Y ) ) ).
% leD
thf(fact_805_leI,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% leI
thf(fact_806_leI,axiom,
! [X2: num,Y: num] :
( ~ ( ord_less_num @ X2 @ Y )
=> ( ord_less_eq_num @ Y @ X2 ) ) ).
% leI
thf(fact_807_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_808_leI,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% leI
thf(fact_809_nless__le,axiom,
! [A2: real,B: real] :
( ( ~ ( ord_less_real @ A2 @ B ) )
= ( ~ ( ord_less_eq_real @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_810_nless__le,axiom,
! [A2: set_complex,B: set_complex] :
( ( ~ ( ord_less_set_complex @ A2 @ B ) )
= ( ~ ( ord_le211207098394363844omplex @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_811_nless__le,axiom,
! [A2: num,B: num] :
( ( ~ ( ord_less_num @ A2 @ B ) )
= ( ~ ( ord_less_eq_num @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_812_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_813_nless__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_int @ A2 @ B ) )
= ( ~ ( ord_less_eq_int @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_814_antisym__conv1,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_815_antisym__conv1,axiom,
! [X2: set_complex,Y: set_complex] :
( ~ ( ord_less_set_complex @ X2 @ Y )
=> ( ( ord_le211207098394363844omplex @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_816_antisym__conv1,axiom,
! [X2: num,Y: num] :
( ~ ( ord_less_num @ X2 @ Y )
=> ( ( ord_less_eq_num @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_817_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_818_antisym__conv1,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_819_antisym__conv2,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_820_antisym__conv2,axiom,
! [X2: set_complex,Y: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y )
=> ( ( ~ ( ord_less_set_complex @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_821_antisym__conv2,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ( ~ ( ord_less_num @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_822_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_823_antisym__conv2,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_824_dense__ge,axiom,
! [Z2: real,Y: real] :
( ! [X: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ord_less_eq_real @ Y @ X ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_825_dense__le,axiom,
! [Y: real,Z2: real] :
( ! [X: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Z2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_le
thf(fact_826_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_827_less__le__not__le,axiom,
( ord_less_set_complex
= ( ^ [X4: set_complex,Y5: set_complex] :
( ( ord_le211207098394363844omplex @ X4 @ Y5 )
& ~ ( ord_le211207098394363844omplex @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_828_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
& ~ ( ord_less_eq_num @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_829_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_830_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y5: int] :
( ( ord_less_eq_int @ X4 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_831_not__le__imp__less,axiom,
! [Y: real,X2: real] :
( ~ ( ord_less_eq_real @ Y @ X2 )
=> ( ord_less_real @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_832_not__le__imp__less,axiom,
! [Y: num,X2: num] :
( ~ ( ord_less_eq_num @ Y @ X2 )
=> ( ord_less_num @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_833_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_834_not__le__imp__less,axiom,
! [Y: int,X2: int] :
( ~ ( ord_less_eq_int @ Y @ X2 )
=> ( ord_less_int @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_835_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B5: real] :
( ( ord_less_real @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_836_order_Oorder__iff__strict,axiom,
( ord_le211207098394363844omplex
= ( ^ [A4: set_complex,B5: set_complex] :
( ( ord_less_set_complex @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_837_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A4: num,B5: num] :
( ( ord_less_num @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_838_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_nat @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_839_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B5: int] :
( ( ord_less_int @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_840_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B5: real] :
( ( ord_less_eq_real @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_841_order_Ostrict__iff__order,axiom,
( ord_less_set_complex
= ( ^ [A4: set_complex,B5: set_complex] :
( ( ord_le211207098394363844omplex @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_842_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A4: num,B5: num] :
( ( ord_less_eq_num @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_843_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_844_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B5: int] :
( ( ord_less_eq_int @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_845_order_Ostrict__trans1,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_846_order_Ostrict__trans1,axiom,
! [A2: set_complex,B: set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ B )
=> ( ( ord_less_set_complex @ B @ C )
=> ( ord_less_set_complex @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_847_order_Ostrict__trans1,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_848_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_849_order_Ostrict__trans1,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_850_order_Ostrict__trans2,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_851_order_Ostrict__trans2,axiom,
! [A2: set_complex,B: set_complex,C: set_complex] :
( ( ord_less_set_complex @ A2 @ B )
=> ( ( ord_le211207098394363844omplex @ B @ C )
=> ( ord_less_set_complex @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_852_order_Ostrict__trans2,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_853_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_854_order_Ostrict__trans2,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_855_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B5: real] :
( ( ord_less_eq_real @ A4 @ B5 )
& ~ ( ord_less_eq_real @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_856_order_Ostrict__iff__not,axiom,
( ord_less_set_complex
= ( ^ [A4: set_complex,B5: set_complex] :
( ( ord_le211207098394363844omplex @ A4 @ B5 )
& ~ ( ord_le211207098394363844omplex @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_857_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A4: num,B5: num] :
( ( ord_less_eq_num @ A4 @ B5 )
& ~ ( ord_less_eq_num @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_858_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_859_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B5: int] :
( ( ord_less_eq_int @ A4 @ B5 )
& ~ ( ord_less_eq_int @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_860_dense__ge__bounded,axiom,
! [Z2: real,X2: real,Y: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X2 )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_861_dense__le__bounded,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X2 @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_862_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A4: real] :
( ( ord_less_real @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_863_dual__order_Oorder__iff__strict,axiom,
( ord_le211207098394363844omplex
= ( ^ [B5: set_complex,A4: set_complex] :
( ( ord_less_set_complex @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_864_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B5: num,A4: num] :
( ( ord_less_num @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_865_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_nat @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_866_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A4: int] :
( ( ord_less_int @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_867_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B5: real,A4: real] :
( ( ord_less_eq_real @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_868_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_complex
= ( ^ [B5: set_complex,A4: set_complex] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_869_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B5: num,A4: num] :
( ( ord_less_eq_num @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_870_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_871_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B5: int,A4: int] :
( ( ord_less_eq_int @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_872_dual__order_Ostrict__trans1,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_873_dual__order_Ostrict__trans1,axiom,
! [B: set_complex,A2: set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ B @ A2 )
=> ( ( ord_less_set_complex @ C @ B )
=> ( ord_less_set_complex @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_874_dual__order_Ostrict__trans1,axiom,
! [B: num,A2: num,C: num] :
( ( ord_less_eq_num @ B @ A2 )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_875_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_876_dual__order_Ostrict__trans1,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_877_dual__order_Ostrict__trans2,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_878_dual__order_Ostrict__trans2,axiom,
! [B: set_complex,A2: set_complex,C: set_complex] :
( ( ord_less_set_complex @ B @ A2 )
=> ( ( ord_le211207098394363844omplex @ C @ B )
=> ( ord_less_set_complex @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_879_dual__order_Ostrict__trans2,axiom,
! [B: num,A2: num,C: num] :
( ( ord_less_num @ B @ A2 )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_880_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_881_dual__order_Ostrict__trans2,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_882_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B5: real,A4: real] :
( ( ord_less_eq_real @ B5 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_883_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_complex
= ( ^ [B5: set_complex,A4: set_complex] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
& ~ ( ord_le211207098394363844omplex @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_884_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B5: num,A4: num] :
( ( ord_less_eq_num @ B5 @ A4 )
& ~ ( ord_less_eq_num @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_885_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_886_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B5: int,A4: int] :
( ( ord_less_eq_int @ B5 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_887_order_Ostrict__implies__order,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_eq_real @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_888_order_Ostrict__implies__order,axiom,
! [A2: set_complex,B: set_complex] :
( ( ord_less_set_complex @ A2 @ B )
=> ( ord_le211207098394363844omplex @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_889_order_Ostrict__implies__order,axiom,
! [A2: num,B: num] :
( ( ord_less_num @ A2 @ B )
=> ( ord_less_eq_num @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_890_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_891_order_Ostrict__implies__order,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_892_dual__order_Ostrict__implies__order,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ( ord_less_eq_real @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_893_dual__order_Ostrict__implies__order,axiom,
! [B: set_complex,A2: set_complex] :
( ( ord_less_set_complex @ B @ A2 )
=> ( ord_le211207098394363844omplex @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_894_dual__order_Ostrict__implies__order,axiom,
! [B: num,A2: num] :
( ( ord_less_num @ B @ A2 )
=> ( ord_less_eq_num @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_895_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_896_dual__order_Ostrict__implies__order,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_eq_int @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_897_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_real @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_898_order__le__less,axiom,
( ord_le211207098394363844omplex
= ( ^ [X4: set_complex,Y5: set_complex] :
( ( ord_less_set_complex @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_899_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_900_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_901_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y5: int] :
( ( ord_less_int @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_902_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_903_order__less__le,axiom,
( ord_less_set_complex
= ( ^ [X4: set_complex,Y5: set_complex] :
( ( ord_le211207098394363844omplex @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_904_order__less__le,axiom,
( ord_less_num
= ( ^ [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_905_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_906_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y5: int] :
( ( ord_less_eq_int @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_907_linorder__not__le,axiom,
! [X2: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X2 @ Y ) )
= ( ord_less_real @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_908_linorder__not__le,axiom,
! [X2: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X2 @ Y ) )
= ( ord_less_num @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_909_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_910_linorder__not__le,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y ) )
= ( ord_less_int @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_911_linorder__not__less,axiom,
! [X2: real,Y: real] :
( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( ord_less_eq_real @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_912_linorder__not__less,axiom,
! [X2: num,Y: num] :
( ( ~ ( ord_less_num @ X2 @ Y ) )
= ( ord_less_eq_num @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_913_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_914_linorder__not__less,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_915_order__less__imp__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_916_order__less__imp__le,axiom,
! [X2: set_complex,Y: set_complex] :
( ( ord_less_set_complex @ X2 @ Y )
=> ( ord_le211207098394363844omplex @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_917_order__less__imp__le,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_eq_num @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_918_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_919_order__less__imp__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_920_order__le__neq__trans,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_real @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_921_order__le__neq__trans,axiom,
! [A2: set_complex,B: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_complex @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_922_order__le__neq__trans,axiom,
! [A2: num,B: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_num @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_923_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_924_order__le__neq__trans,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_925_order__neq__le__trans,axiom,
! [A2: real,B: real] :
( ( A2 != B )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_real @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_926_order__neq__le__trans,axiom,
! [A2: set_complex,B: set_complex] :
( ( A2 != B )
=> ( ( ord_le211207098394363844omplex @ A2 @ B )
=> ( ord_less_set_complex @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_927_order__neq__le__trans,axiom,
! [A2: num,B: num] :
( ( A2 != B )
=> ( ( ord_less_eq_num @ A2 @ B )
=> ( ord_less_num @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_928_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_929_order__neq__le__trans,axiom,
! [A2: int,B: int] :
( ( A2 != B )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_930_order__le__less__trans,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_931_order__le__less__trans,axiom,
! [X2: set_complex,Y: set_complex,Z2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y )
=> ( ( ord_less_set_complex @ Y @ Z2 )
=> ( ord_less_set_complex @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_932_order__le__less__trans,axiom,
! [X2: num,Y: num,Z2: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ( ord_less_num @ Y @ Z2 )
=> ( ord_less_num @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_933_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_934_order__le__less__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_935_order__less__le__trans,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_936_order__less__le__trans,axiom,
! [X2: set_complex,Y: set_complex,Z2: set_complex] :
( ( ord_less_set_complex @ X2 @ Y )
=> ( ( ord_le211207098394363844omplex @ Y @ Z2 )
=> ( ord_less_set_complex @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_937_order__less__le__trans,axiom,
! [X2: num,Y: num,Z2: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ( ord_less_eq_num @ Y @ Z2 )
=> ( ord_less_num @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_938_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_939_order__less__le__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_940_order__le__less__subst1,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_941_order__le__less__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_942_order__le__less__subst1,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_943_order__le__less__subst1,axiom,
! [A2: real,F: num > real,B: num,C: num] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_944_order__le__less__subst1,axiom,
! [A2: num,F: real > num,B: real,C: real] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_945_order__le__less__subst1,axiom,
! [A2: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_946_order__le__less__subst1,axiom,
! [A2: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_947_order__le__less__subst1,axiom,
! [A2: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_948_order__le__less__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_949_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_950_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_951_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > num,C: num] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_952_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_953_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_954_order__le__less__subst2,axiom,
! [A2: num,B: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_955_order__le__less__subst2,axiom,
! [A2: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_956_order__le__less__subst2,axiom,
! [A2: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_957_order__le__less__subst2,axiom,
! [A2: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_958_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_959_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_960_order__less__le__subst1,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_961_order__less__le__subst1,axiom,
! [A2: num,F: real > num,B: real,C: real] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_962_order__less__le__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_963_order__less__le__subst1,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_964_order__less__le__subst1,axiom,
! [A2: real,F: num > real,B: num,C: num] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_965_order__less__le__subst1,axiom,
! [A2: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_966_order__less__le__subst1,axiom,
! [A2: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_967_order__less__le__subst1,axiom,
! [A2: int,F: num > int,B: num,C: num] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_968_order__less__le__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_969_order__less__le__subst1,axiom,
! [A2: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_970_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_971_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_972_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_973_order__less__le__subst2,axiom,
! [A2: num,B: num,F: num > real,C: real] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_974_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > num,C: num] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_975_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_976_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > num,C: num] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_977_order__less__le__subst2,axiom,
! [A2: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_978_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_979_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_980_linorder__le__less__linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
| ( ord_less_real @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_981_linorder__le__less__linear,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
| ( ord_less_num @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_982_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_983_linorder__le__less__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_int @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_984_order__le__imp__less__or__eq,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_real @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_985_order__le__imp__less__or__eq,axiom,
! [X2: set_complex,Y: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y )
=> ( ( ord_less_set_complex @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_986_order__le__imp__less__or__eq,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ( ord_less_num @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_987_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_988_order__le__imp__less__or__eq,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_int @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_989_verit__comp__simplify1_I3_J,axiom,
! [B7: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B7 @ A6 ) )
= ( ord_less_real @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_990_verit__comp__simplify1_I3_J,axiom,
! [B7: num,A6: num] :
( ( ~ ( ord_less_eq_num @ B7 @ A6 ) )
= ( ord_less_num @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_991_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
= ( ord_less_nat @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_992_verit__comp__simplify1_I3_J,axiom,
! [B7: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B7 @ A6 ) )
= ( ord_less_int @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_993_zero__less__iff__neq__zero,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% zero_less_iff_neq_zero
thf(fact_994_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_995_gr__implies__not__zero,axiom,
! [M2: extended_enat,N: extended_enat] :
( ( ord_le72135733267957522d_enat @ M2 @ N )
=> ( N != zero_z5237406670263579293d_enat ) ) ).
% gr_implies_not_zero
thf(fact_996_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_997_not__less__zero,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_less_zero
thf(fact_998_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_999_gr__zeroI,axiom,
! [N: extended_enat] :
( ( N != zero_z5237406670263579293d_enat )
=> ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ).
% gr_zeroI
thf(fact_1000_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_1001_less__numeral__extra_I3_J,axiom,
~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ) ).
% less_numeral_extra(3)
thf(fact_1002_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_1003_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1004_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_1005_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1006_scaleR__right__diff__distrib,axiom,
! [A2: real,X2: complex,Y: complex] :
( ( real_V2046097035970521341omplex @ A2 @ ( minus_minus_complex @ X2 @ Y ) )
= ( minus_minus_complex @ ( real_V2046097035970521341omplex @ A2 @ X2 ) @ ( real_V2046097035970521341omplex @ A2 @ Y ) ) ) ).
% scaleR_right_diff_distrib
thf(fact_1007_scaleR__right__diff__distrib,axiom,
! [A2: real,X2: real,Y: real] :
( ( real_V1485227260804924795R_real @ A2 @ ( minus_minus_real @ X2 @ Y ) )
= ( minus_minus_real @ ( real_V1485227260804924795R_real @ A2 @ X2 ) @ ( real_V1485227260804924795R_real @ A2 @ Y ) ) ) ).
% scaleR_right_diff_distrib
thf(fact_1008_diff__strict__right__mono,axiom,
! [A2: complex,B: complex,C: complex] :
( ( ord_less_complex @ A2 @ B )
=> ( ord_less_complex @ ( minus_minus_complex @ A2 @ C ) @ ( minus_minus_complex @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1009_diff__strict__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1010_diff__strict__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1011_diff__strict__left__mono,axiom,
! [B: complex,A2: complex,C: complex] :
( ( ord_less_complex @ B @ A2 )
=> ( ord_less_complex @ ( minus_minus_complex @ C @ A2 ) @ ( minus_minus_complex @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1012_diff__strict__left__mono,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ord_less_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1013_diff__strict__left__mono,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1014_diff__eq__diff__less,axiom,
! [A2: complex,B: complex,C: complex,D: complex] :
( ( ( minus_minus_complex @ A2 @ B )
= ( minus_minus_complex @ C @ D ) )
=> ( ( ord_less_complex @ A2 @ B )
= ( ord_less_complex @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1015_diff__eq__diff__less,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A2 @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1016_diff__eq__diff__less,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A2 @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1017_diff__strict__mono,axiom,
! [A2: complex,B: complex,D: complex,C: complex] :
( ( ord_less_complex @ A2 @ B )
=> ( ( ord_less_complex @ D @ C )
=> ( ord_less_complex @ ( minus_minus_complex @ A2 @ C ) @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1018_diff__strict__mono,axiom,
! [A2: real,B: real,D: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1019_diff__strict__mono,axiom,
! [A2: int,B: int,D: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1020_bot_Oextremum__strict,axiom,
! [A2: set_complex] :
~ ( ord_less_set_complex @ A2 @ bot_bot_set_complex ) ).
% bot.extremum_strict
thf(fact_1021_bot_Oextremum__strict,axiom,
! [A2: extended_enat] :
~ ( ord_le72135733267957522d_enat @ A2 @ bot_bo4199563552545308370d_enat ) ).
% bot.extremum_strict
thf(fact_1022_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1023_bot_Onot__eq__extremum,axiom,
! [A2: set_complex] :
( ( A2 != bot_bot_set_complex )
= ( ord_less_set_complex @ bot_bot_set_complex @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1024_bot_Onot__eq__extremum,axiom,
! [A2: extended_enat] :
( ( A2 != bot_bo4199563552545308370d_enat )
= ( ord_le72135733267957522d_enat @ bot_bo4199563552545308370d_enat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1025_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1026_minus__less__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A2 ) ) ).
% minus_less_iff
thf(fact_1027_minus__less__iff,axiom,
! [A2: complex,B: complex] :
( ( ord_less_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B )
= ( ord_less_complex @ ( uminus1482373934393186551omplex @ B ) @ A2 ) ) ).
% minus_less_iff
thf(fact_1028_minus__less__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% minus_less_iff
thf(fact_1029_less__minus__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A2 ) ) ) ).
% less_minus_iff
thf(fact_1030_less__minus__iff,axiom,
! [A2: complex,B: complex] :
( ( ord_less_complex @ A2 @ ( uminus1482373934393186551omplex @ B ) )
= ( ord_less_complex @ B @ ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% less_minus_iff
thf(fact_1031_less__minus__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% less_minus_iff
thf(fact_1032_verit__negate__coefficient_I2_J,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1033_verit__negate__coefficient_I2_J,axiom,
! [A2: complex,B: complex] :
( ( ord_less_complex @ A2 @ B )
=> ( ord_less_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1034_verit__negate__coefficient_I2_J,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1035_compl__less__swap2,axiom,
! [Y: set_complex,X2: set_complex] :
( ( ord_less_set_complex @ ( uminus8566677241136511917omplex @ Y ) @ X2 )
=> ( ord_less_set_complex @ ( uminus8566677241136511917omplex @ X2 ) @ Y ) ) ).
% compl_less_swap2
thf(fact_1036_compl__less__swap1,axiom,
! [Y: set_complex,X2: set_complex] :
( ( ord_less_set_complex @ Y @ ( uminus8566677241136511917omplex @ X2 ) )
=> ( ord_less_set_complex @ X2 @ ( uminus8566677241136511917omplex @ Y ) ) ) ).
% compl_less_swap1
thf(fact_1037_subset__iff__psubset__eq,axiom,
( ord_le211207098394363844omplex
= ( ^ [A3: set_complex,B3: set_complex] :
( ( ord_less_set_complex @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1038_subset__psubset__trans,axiom,
! [A: set_complex,B2: set_complex,C2: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ( ord_less_set_complex @ B2 @ C2 )
=> ( ord_less_set_complex @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1039_subset__not__subset__eq,axiom,
( ord_less_set_complex
= ( ^ [A3: set_complex,B3: set_complex] :
( ( ord_le211207098394363844omplex @ A3 @ B3 )
& ~ ( ord_le211207098394363844omplex @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1040_psubset__subset__trans,axiom,
! [A: set_complex,B2: set_complex,C2: set_complex] :
( ( ord_less_set_complex @ A @ B2 )
=> ( ( ord_le211207098394363844omplex @ B2 @ C2 )
=> ( ord_less_set_complex @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1041_psubset__imp__subset,axiom,
! [A: set_complex,B2: set_complex] :
( ( ord_less_set_complex @ A @ B2 )
=> ( ord_le211207098394363844omplex @ A @ B2 ) ) ).
% psubset_imp_subset
thf(fact_1042_psubset__eq,axiom,
( ord_less_set_complex
= ( ^ [A3: set_complex,B3: set_complex] :
( ( ord_le211207098394363844omplex @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_1043_psubsetE,axiom,
! [A: set_complex,B2: set_complex] :
( ( ord_less_set_complex @ A @ B2 )
=> ~ ( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ord_le211207098394363844omplex @ B2 @ A ) ) ) ).
% psubsetE
thf(fact_1044_not__psubset__empty,axiom,
! [A: set_complex] :
~ ( ord_less_set_complex @ A @ bot_bot_set_complex ) ).
% not_psubset_empty
thf(fact_1045_psubset__imp__ex__mem,axiom,
! [A: set_real,B2: set_real] :
( ( ord_less_set_real @ A @ B2 )
=> ? [B6: real] : ( member_real @ B6 @ ( minus_minus_set_real @ B2 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1046_psubset__imp__ex__mem,axiom,
! [A: set_complex,B2: set_complex] :
( ( ord_less_set_complex @ A @ B2 )
=> ? [B6: complex] : ( member_complex @ B6 @ ( minus_811609699411566653omplex @ B2 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1047_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_1048_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1049_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1050_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_1051_one__neq__neg__one,axiom,
( one_one_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% one_neq_neg_one
thf(fact_1052_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_1053_span__mul,axiom,
! [X2: complex,S2: set_complex,C: real] :
( ( member_complex @ X2 @ ( real_V8921647422947696435omplex @ S2 ) )
=> ( member_complex @ ( real_V2046097035970521341omplex @ C @ X2 ) @ ( real_V8921647422947696435omplex @ S2 ) ) ) ).
% span_mul
thf(fact_1054_span__mul,axiom,
! [X2: real,S2: set_real,C: real] :
( ( member_real @ X2 @ ( real_V5325414057265605809n_real @ S2 ) )
=> ( member_real @ ( real_V1485227260804924795R_real @ C @ X2 ) @ ( real_V5325414057265605809n_real @ S2 ) ) ) ).
% span_mul
thf(fact_1055_Ints__1,axiom,
member_real @ one_one_real @ ring_1_Ints_real ).
% Ints_1
thf(fact_1056_Ints__1,axiom,
member_int @ one_one_int @ ring_1_Ints_int ).
% Ints_1
thf(fact_1057_Ints__1,axiom,
member_complex @ one_one_complex @ ring_1_Ints_complex ).
% Ints_1
thf(fact_1058_scaleR__left__le__one__le,axiom,
! [X2: complex,A2: real] :
( ( ord_less_eq_complex @ zero_zero_complex @ X2 )
=> ( ( ord_less_eq_real @ A2 @ one_one_real )
=> ( ord_less_eq_complex @ ( real_V2046097035970521341omplex @ A2 @ X2 ) @ X2 ) ) ) ).
% scaleR_left_le_one_le
thf(fact_1059_scaleR__left__le__one__le,axiom,
! [X2: real,A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ A2 @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A2 @ X2 ) @ X2 ) ) ) ).
% scaleR_left_le_one_le
thf(fact_1060_pth__4_I1_J,axiom,
! [X2: complex] :
( ( real_V2046097035970521341omplex @ zero_zero_real @ X2 )
= zero_zero_complex ) ).
% pth_4(1)
thf(fact_1061_pth__4_I1_J,axiom,
! [X2: real] :
( ( real_V1485227260804924795R_real @ zero_zero_real @ X2 )
= zero_zero_real ) ).
% pth_4(1)
thf(fact_1062_less__iff__diff__less__0,axiom,
( ord_less_complex
= ( ^ [A4: complex,B5: complex] : ( ord_less_complex @ ( minus_minus_complex @ A4 @ B5 ) @ zero_zero_complex ) ) ) ).
% less_iff_diff_less_0
thf(fact_1063_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A4: real,B5: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B5 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_1064_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B5: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B5 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_1065_scaleR__left__diff__distrib,axiom,
! [A2: real,B: real,X2: complex] :
( ( real_V2046097035970521341omplex @ ( minus_minus_real @ A2 @ B ) @ X2 )
= ( minus_minus_complex @ ( real_V2046097035970521341omplex @ A2 @ X2 ) @ ( real_V2046097035970521341omplex @ B @ X2 ) ) ) ).
% scaleR_left_diff_distrib
thf(fact_1066_scaleR__left__diff__distrib,axiom,
! [A2: real,B: real,X2: real] :
( ( real_V1485227260804924795R_real @ ( minus_minus_real @ A2 @ B ) @ X2 )
= ( minus_minus_real @ ( real_V1485227260804924795R_real @ A2 @ X2 ) @ ( real_V1485227260804924795R_real @ B @ X2 ) ) ) ).
% scaleR_left_diff_distrib
thf(fact_1067_scaleR__left_Odiff,axiom,
! [X2: real,Y: real,Xa: complex] :
( ( real_V2046097035970521341omplex @ ( minus_minus_real @ X2 @ Y ) @ Xa )
= ( minus_minus_complex @ ( real_V2046097035970521341omplex @ X2 @ Xa ) @ ( real_V2046097035970521341omplex @ Y @ Xa ) ) ) ).
% scaleR_left.diff
thf(fact_1068_scaleR__left_Odiff,axiom,
! [X2: real,Y: real,Xa: real] :
( ( real_V1485227260804924795R_real @ ( minus_minus_real @ X2 @ Y ) @ Xa )
= ( minus_minus_real @ ( real_V1485227260804924795R_real @ X2 @ Xa ) @ ( real_V1485227260804924795R_real @ Y @ Xa ) ) ) ).
% scaleR_left.diff
thf(fact_1069_le__minus__one__simps_I2_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% le_minus_one_simps(2)
thf(fact_1070_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_1071_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(4)
thf(fact_1072_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_1073_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_1074_zero__neq__neg__one,axiom,
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% zero_neq_neg_one
thf(fact_1075_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_1076_scaleR__right__mono__neg,axiom,
! [B: real,A2: real,C: complex] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_complex @ C @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( real_V2046097035970521341omplex @ A2 @ C ) @ ( real_V2046097035970521341omplex @ B @ C ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_1077_scaleR__right__mono__neg,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A2 @ C ) @ ( real_V1485227260804924795R_real @ B @ C ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_1078_scaleR__right__mono,axiom,
! [A2: real,B: real,X2: complex] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ X2 )
=> ( ord_less_eq_complex @ ( real_V2046097035970521341omplex @ A2 @ X2 ) @ ( real_V2046097035970521341omplex @ B @ X2 ) ) ) ) ).
% scaleR_right_mono
thf(fact_1079_scaleR__right__mono,axiom,
! [A2: real,B: real,X2: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A2 @ X2 ) @ ( real_V1485227260804924795R_real @ B @ X2 ) ) ) ) ).
% scaleR_right_mono
thf(fact_1080_scaleR__left__mono,axiom,
! [X2: real,Y: real,A2: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A2 @ X2 ) @ ( real_V1485227260804924795R_real @ A2 @ Y ) ) ) ) ).
% scaleR_left_mono
thf(fact_1081_scaleR__left__mono__neg,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A2 ) @ ( real_V1485227260804924795R_real @ C @ B ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_1082_span__breakdown__eq,axiom,
! [X2: complex,A2: complex,S2: set_complex] :
( ( member_complex @ X2 @ ( real_V8921647422947696435omplex @ ( insert_complex @ A2 @ S2 ) ) )
= ( ? [K3: real] : ( member_complex @ ( minus_minus_complex @ X2 @ ( real_V2046097035970521341omplex @ K3 @ A2 ) ) @ ( real_V8921647422947696435omplex @ S2 ) ) ) ) ).
% span_breakdown_eq
thf(fact_1083_span__breakdown__eq,axiom,
! [X2: real,A2: real,S2: set_real] :
( ( member_real @ X2 @ ( real_V5325414057265605809n_real @ ( insert_real @ A2 @ S2 ) ) )
= ( ? [K3: real] : ( member_real @ ( minus_minus_real @ X2 @ ( real_V1485227260804924795R_real @ K3 @ A2 ) ) @ ( real_V5325414057265605809n_real @ S2 ) ) ) ) ).
% span_breakdown_eq
thf(fact_1084_le__minus__one__simps_I1_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% le_minus_one_simps(1)
thf(fact_1085_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_1086_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(3)
thf(fact_1087_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_1088_continuous__on__of__real__id,axiom,
! [S2: set_real] : ( topolo8620507378200602458omplex @ S2 @ real_V4546457046886955230omplex ) ).
% continuous_on_of_real_id
thf(fact_1089_continuous__on__of__real__id,axiom,
! [S2: set_real] : ( topolo5044208981011980120l_real @ S2 @ real_V1803761363581548252l_real ) ).
% continuous_on_of_real_id
thf(fact_1090_scaleR__nonpos__nonpos,axiom,
! [A2: real,B: complex] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_complex @ B @ zero_zero_complex )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( real_V2046097035970521341omplex @ A2 @ B ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_1091_scaleR__nonpos__nonpos,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A2 @ B ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_1092_scaleR__nonpos__nonneg,axiom,
! [A2: real,X2: complex] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ X2 )
=> ( ord_less_eq_complex @ ( real_V2046097035970521341omplex @ A2 @ X2 ) @ zero_zero_complex ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_1093_scaleR__nonpos__nonneg,axiom,
! [A2: real,X2: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A2 @ X2 ) @ zero_zero_real ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_1094_scaleR__nonneg__nonpos,axiom,
! [A2: real,X2: complex] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_complex @ X2 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( real_V2046097035970521341omplex @ A2 @ X2 ) @ zero_zero_complex ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_1095_scaleR__nonneg__nonpos,axiom,
! [A2: real,X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A2 @ X2 ) @ zero_zero_real ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_1096_scaleR__nonneg__nonneg,axiom,
! [A2: real,X2: complex] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ X2 )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( real_V2046097035970521341omplex @ A2 @ X2 ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_1097_scaleR__nonneg__nonneg,axiom,
! [A2: real,X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A2 @ X2 ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_1098_split__scaleR__pos__le,axiom,
! [A2: real,B: complex] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_complex @ zero_zero_complex @ B ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_complex @ B @ zero_zero_complex ) ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( real_V2046097035970521341omplex @ A2 @ B ) ) ) ).
% split_scaleR_pos_le
thf(fact_1099_split__scaleR__pos__le,axiom,
! [A2: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A2 @ B ) ) ) ).
% split_scaleR_pos_le
thf(fact_1100_split__scaleR__neg__le,axiom,
! [A2: real,X2: complex] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_complex @ X2 @ zero_zero_complex ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_complex @ zero_zero_complex @ X2 ) ) )
=> ( ord_less_eq_complex @ ( real_V2046097035970521341omplex @ A2 @ X2 ) @ zero_zero_complex ) ) ).
% split_scaleR_neg_le
thf(fact_1101_split__scaleR__neg__le,axiom,
! [A2: real,X2: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ X2 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ X2 ) ) )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A2 @ X2 ) @ zero_zero_real ) ) ).
% split_scaleR_neg_le
thf(fact_1102_scaleR__mono_H,axiom,
! [A2: real,B: real,C: complex,D: complex] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_complex @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ord_less_eq_complex @ ( real_V2046097035970521341omplex @ A2 @ C ) @ ( real_V2046097035970521341omplex @ B @ D ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_1103_scaleR__mono_H,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A2 @ C ) @ ( real_V1485227260804924795R_real @ B @ D ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_1104_scaleR__mono,axiom,
! [A2: real,B: real,X2: complex,Y: complex] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_complex @ X2 @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ X2 )
=> ( ord_less_eq_complex @ ( real_V2046097035970521341omplex @ A2 @ X2 ) @ ( real_V2046097035970521341omplex @ B @ Y ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_1105_scaleR__mono,axiom,
! [A2: real,B: real,X2: real,Y: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A2 @ X2 ) @ ( real_V1485227260804924795R_real @ B @ Y ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_1106_vector__space__over__itself_Ozero__not__in__Basis,axiom,
~ ( member_complex @ zero_zero_complex @ ( insert_complex @ one_one_complex @ bot_bot_set_complex ) ) ).
% vector_space_over_itself.zero_not_in_Basis
thf(fact_1107_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_real @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_1108_Multiseries__Expansion_Ocompare__reals__diff__sgnD_I3_J,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B ) )
=> ( ord_less_real @ B @ A2 ) ) ).
% Multiseries_Expansion.compare_reals_diff_sgnD(3)
thf(fact_1109_Multiseries__Expansion_Ocompare__reals__diff__sgnD_I1_J,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( minus_minus_real @ A2 @ B ) @ zero_zero_real )
=> ( ord_less_real @ A2 @ B ) ) ).
% Multiseries_Expansion.compare_reals_diff_sgnD(1)
thf(fact_1110_Bolzano,axiom,
! [A2: real,B: real,P2: real > real > $o] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [A5: real,B6: real,C4: real] :
( ( P2 @ A5 @ B6 )
=> ( ( P2 @ B6 @ C4 )
=> ( ( ord_less_eq_real @ A5 @ B6 )
=> ( ( ord_less_eq_real @ B6 @ C4 )
=> ( P2 @ A5 @ C4 ) ) ) ) )
=> ( ! [X: real] :
( ( ord_less_eq_real @ A2 @ X )
=> ( ( ord_less_eq_real @ X @ B )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A5: real,B6: real] :
( ( ( ord_less_eq_real @ A5 @ X )
& ( ord_less_eq_real @ X @ B6 )
& ( ord_less_real @ ( minus_minus_real @ B6 @ A5 ) @ D3 ) )
=> ( P2 @ A5 @ B6 ) ) ) ) )
=> ( P2 @ A2 @ B ) ) ) ) ).
% Bolzano
thf(fact_1111_kuhn__labelling__lemma_H,axiom,
! [P2: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X: nat > real] :
( ( P2 @ X )
=> ( P2 @ ( F @ X ) ) )
=> ( ! [X: nat > real] :
( ( P2 @ X )
=> ! [I: nat] :
( ( Q @ I )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X @ I ) )
& ( ord_less_eq_real @ ( X @ I ) @ one_one_real ) ) ) )
=> ? [L: ( nat > real ) > nat > nat] :
( ! [X5: nat > real,I2: nat] : ( ord_less_eq_nat @ ( L @ X5 @ I2 ) @ one_one_nat )
& ! [X5: nat > real,I2: nat] :
( ( ( P2 @ X5 )
& ( Q @ I2 )
& ( ( X5 @ I2 )
= zero_zero_real ) )
=> ( ( L @ X5 @ I2 )
= zero_zero_nat ) )
& ! [X5: nat > real,I2: nat] :
( ( ( P2 @ X5 )
& ( Q @ I2 )
& ( ( X5 @ I2 )
= one_one_real ) )
=> ( ( L @ X5 @ I2 )
= one_one_nat ) )
& ! [X5: nat > real,I2: nat] :
( ( ( P2 @ X5 )
& ( Q @ I2 )
& ( ( L @ X5 @ I2 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X5 @ I2 ) @ ( F @ X5 @ I2 ) ) )
& ! [X5: nat > real,I2: nat] :
( ( ( P2 @ X5 )
& ( Q @ I2 )
& ( ( L @ X5 @ I2 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X5 @ I2 ) @ ( X5 @ I2 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1112_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1113_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1114_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1115_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_1116_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_1117_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1118_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1119_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1120_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1121_diff__le__mono,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% diff_le_mono
thf(fact_1122_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1123_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1124_diff__le__mono2,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1125_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y4: nat] :
( ( P2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X: nat] :
( ( P2 @ X )
& ! [Y6: nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1126_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_1127_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_1128_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_1129_le__trans,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_1130_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1131_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J: nat] :
( ! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1132_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1133_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1134_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1135_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_1136_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1137_diff__less__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1138_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1139_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1140_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1141_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1142_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1143_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ~ ( P2 @ I2 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1144_seq__mono__lemma,axiom,
! [M2: nat,D: nat > real,E2: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_real @ ( D @ N3 ) @ ( E2 @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_real @ ( E2 @ N3 ) @ ( E2 @ M2 ) ) )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_real @ ( D @ N4 ) @ ( E2 @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1145_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P2 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I2: nat] :
( ( ord_less_nat @ K2 @ I2 )
=> ( P2 @ I2 ) )
=> ( P2 @ K2 ) ) )
=> ( P2 @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1146_ln__Gamma__series__complex__of__real,axiom,
! [X2: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( gamma_3083380197916210424omplex @ ( real_V4546457046886955230omplex @ X2 ) @ N )
= ( real_V4546457046886955230omplex @ ( gamma_2780986569588390390s_real @ X2 @ N ) ) ) ) ) ).
% ln_Gamma_series_complex_of_real
thf(fact_1147_enat__ord__number_I1_J,axiom,
! [M2: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_1148_i0__lb,axiom,
! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% i0_lb
thf(fact_1149_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% ile0_eq
thf(fact_1150_zle__diff1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1151_int__le__induct,axiom,
! [I3: int,K: int,P2: int > $o] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P2 @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P2 @ I )
=> ( P2 @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% int_le_induct
thf(fact_1152_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1153_imp__le__cong,axiom,
! [X2: int,X7: int,P2: $o,P5: $o] :
( ( X2 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P2 = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P5 ) ) ) ) ).
% imp_le_cong
thf(fact_1154_conj__le__cong,axiom,
! [X2: int,X7: int,P2: $o,P5: $o] :
( ( X2 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P2 = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P5 ) ) ) ) ).
% conj_le_cong
thf(fact_1155_verit__la__generic,axiom,
! [A2: int,X2: int] :
( ( ord_less_eq_int @ A2 @ X2 )
| ( A2 = X2 )
| ( ord_less_eq_int @ X2 @ A2 ) ) ).
% verit_la_generic
thf(fact_1156_le__num__One__iff,axiom,
! [X2: num] :
( ( ord_less_eq_num @ X2 @ one )
= ( X2 = one ) ) ).
% le_num_One_iff
thf(fact_1157_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1158_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1159_bot__enat__def,axiom,
bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% bot_enat_def
thf(fact_1160_int__less__induct,axiom,
! [I3: int,K: int,P2: int > $o] :
( ( ord_less_int @ I3 @ K )
=> ( ( P2 @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ I @ K )
=> ( ( P2 @ I )
=> ( P2 @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% int_less_induct
thf(fact_1161_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1162_minus__int__code_I2_J,axiom,
! [L2: int] :
( ( minus_minus_int @ zero_zero_int @ L2 )
= ( uminus_uminus_int @ L2 ) ) ).
% minus_int_code(2)
thf(fact_1163_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_1164_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_1165_semiring__norm_I71_J,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% semiring_norm(71)
thf(fact_1166_semiring__norm_I69_J,axiom,
! [M2: num] :
~ ( ord_less_eq_num @ ( bit0 @ M2 ) @ one ) ).
% semiring_norm(69)
thf(fact_1167_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_1168_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1169_nat__numeral__diff__1,axiom,
! [V: num] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
= ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% nat_numeral_diff_1
thf(fact_1170_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% nat_numeral
thf(fact_1171_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ Z2 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1172_nat__0__iff,axiom,
! [I3: int] :
( ( ( nat2 @ I3 )
= zero_zero_nat )
= ( ord_less_eq_int @ I3 @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1173_zless__nat__conj,axiom,
! [W2: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W2 @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_1174_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= zero_zero_nat ) ).
% nat_neg_numeral
thf(fact_1175_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_1176_diff__nat__numeral,axiom,
! [V: num,V2: num] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V2 ) )
= ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V2 ) ) ) ) ).
% diff_nat_numeral
thf(fact_1177_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1178_nat__numeral__as__int,axiom,
( numeral_numeral_nat
= ( ^ [I4: num] : ( nat2 @ ( numeral_numeral_int @ I4 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_1179_nat__mono,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_1180_eq__nat__nat__iff,axiom,
! [Z2: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ( nat2 @ Z2 )
= ( nat2 @ Z5 ) )
= ( Z2 = Z5 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1181_all__nat,axiom,
( ( ^ [P3: nat > $o] :
! [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
! [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P4 @ ( nat2 @ X4 ) ) ) ) ) ).
% all_nat
thf(fact_1182_ex__nat,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
& ( P4 @ ( nat2 @ X4 ) ) ) ) ) ).
% ex_nat
thf(fact_1183_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1184_nat__mono__iff,axiom,
! [Z2: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_1185_nat__le__eq__zle,axiom,
! [W2: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_1186_nat__less__eq__zless,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_1187_nat__diff__distrib_H,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X2 @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_1188_nat__diff__distrib,axiom,
! [Z5: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ord_less_eq_int @ Z5 @ Z2 )
=> ( ( nat2 @ ( minus_minus_int @ Z2 @ Z5 ) )
= ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1189_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_1190_semiring__norm_I73_J,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% semiring_norm(73)
thf(fact_1191_semiring__norm_I72_J,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% semiring_norm(72)
thf(fact_1192_semiring__norm_I70_J,axiom,
! [M2: num] :
~ ( ord_less_eq_num @ ( bit1 @ M2 ) @ one ) ).
% semiring_norm(70)
thf(fact_1193_semiring__norm_I79_J,axiom,
! [M2: num,N: num] :
( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% semiring_norm(79)
thf(fact_1194_semiring__norm_I74_J,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% semiring_norm(74)
thf(fact_1195_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_1196_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_1197_nat__floor__neg,axiom,
! [X2: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
= zero_zero_nat ) ) ).
% nat_floor_neg
thf(fact_1198_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1199_real__add__minus__iff,axiom,
! [X2: real,A2: real] :
( ( ( plus_plus_real @ X2 @ ( uminus_uminus_real @ A2 ) )
= zero_zero_real )
= ( X2 = A2 ) ) ).
% real_add_minus_iff
thf(fact_1200_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1201_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1202_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1203_Re__divide__of__real,axiom,
! [Z2: complex,R2: real] :
( ( re @ ( divide1717551699836669952omplex @ Z2 @ ( real_V4546457046886955230omplex @ R2 ) ) )
= ( divide_divide_real @ ( re @ Z2 ) @ R2 ) ) ).
% Re_divide_of_real
thf(fact_1204_zle__add1__eq__le,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1205_ceiling__minus__divide__eq__div__numeral,axiom,
! [A2: num,B: num] :
( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A2 ) @ ( numeral_numeral_real @ B ) ) ) )
= ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A2 ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% ceiling_minus_divide_eq_div_numeral
thf(fact_1206_floor__minus__divide__eq__div__numeral,axiom,
! [A2: num,B: num] :
( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A2 ) @ ( numeral_numeral_real @ B ) ) ) )
= ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A2 ) ) @ ( numeral_numeral_int @ B ) ) ) ).
% floor_minus_divide_eq_div_numeral
thf(fact_1207_floor__minus__one__divide__eq__div__numeral,axiom,
! [B: num] :
( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
= ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).
% floor_minus_one_divide_eq_div_numeral
thf(fact_1208_add__diff__assoc__enat,axiom,
! [Z2: extended_enat,Y: extended_enat,X2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Z2 @ Y )
=> ( ( plus_p3455044024723400733d_enat @ X2 @ ( minus_3235023915231533773d_enat @ Y @ Z2 ) )
= ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X2 @ Y ) @ Z2 ) ) ) ).
% add_diff_assoc_enat
thf(fact_1209_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1210_plus__int__code_I2_J,axiom,
! [L2: int] :
( ( plus_plus_int @ zero_zero_int @ L2 )
= L2 ) ).
% plus_int_code(2)
thf(fact_1211_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1212_trans__le__add2,axiom,
! [I3: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1213_trans__le__add1,axiom,
! [I3: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1214_add__le__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1215_add__le__mono,axiom,
! [I3: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_1216_le__Suc__ex,axiom,
! [K: nat,L2: nat] :
( ( ord_less_eq_nat @ K @ L2 )
=> ? [N3: nat] :
( L2
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1217_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1218_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_1219_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_1220_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_1221_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1222_nat__add__distrib,axiom,
! [Z2: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( nat2 @ ( plus_plus_int @ Z2 @ Z5 ) )
= ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1223_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1224_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ( minus_minus_nat @ J @ I3 )
= K )
= ( J
= ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1225_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1226_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1227_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1228_le__diff__conv,axiom,
! [J: nat,K: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I3 @ K ) ) ) ).
% le_diff_conv
thf(fact_1229_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1230_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X4: real,Y5: real] : ( plus_plus_real @ X4 @ ( uminus_uminus_real @ Y5 ) ) ) ) ).
% minus_real_def
thf(fact_1231_int__ge__induct,axiom,
! [K: int,I3: int,P2: int > $o] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P2 @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P2 @ I )
=> ( P2 @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% int_ge_induct
thf(fact_1232_zless__add1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z2 )
| ( W2 = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1233_int__gr__induct,axiom,
! [K: int,I3: int,P2: int > $o] :
( ( ord_less_int @ K @ I3 )
=> ( ( P2 @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ K @ I )
=> ( ( P2 @ I )
=> ( P2 @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% int_gr_induct
thf(fact_1234_real__0__le__add__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1235_real__add__le__0__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X2 @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X2 ) ) ) ).
% real_add_le_0_iff
thf(fact_1236_real__add__less__0__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X2 @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X2 ) ) ) ).
% real_add_less_0_iff
thf(fact_1237_real__0__less__add__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X2 ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_1238_less__diff__conv2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I3 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1239_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1240_add1__zle__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1241_zless__imp__add1__zle,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1242_int__induct,axiom,
! [P2: int > $o,K: int,I3: int] :
( ( P2 @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P2 @ I )
=> ( P2 @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P2 @ I )
=> ( P2 @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% int_induct
thf(fact_1243_kuhn__lemma,axiom,
! [P: nat,N: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P )
=> ( ! [X: nat > nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_nat @ ( X @ I2 ) @ P ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ( Label @ X @ I )
= zero_zero_nat )
| ( ( Label @ X @ I )
= one_one_nat ) ) ) )
=> ( ! [X: nat > nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_nat @ ( X @ I2 ) @ P ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ( X @ I )
= zero_zero_nat )
=> ( ( Label @ X @ I )
= zero_zero_nat ) ) ) )
=> ( ! [X: nat > nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_nat @ ( X @ I2 ) @ P ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ( X @ I )
= P )
=> ( ( Label @ X @ I )
= one_one_nat ) ) ) )
=> ~ ! [Q2: nat > nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_nat @ ( Q2 @ I2 ) @ P ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ? [R3: nat > nat] :
( ! [J3: nat] :
( ( ord_less_nat @ J3 @ N )
=> ( ( ord_less_eq_nat @ ( Q2 @ J3 ) @ ( R3 @ J3 ) )
& ( ord_less_eq_nat @ ( R3 @ J3 ) @ ( plus_plus_nat @ ( Q2 @ J3 ) @ one_one_nat ) ) ) )
& ? [S3: nat > nat] :
( ! [J3: nat] :
( ( ord_less_nat @ J3 @ N )
=> ( ( ord_less_eq_nat @ ( Q2 @ J3 ) @ ( S3 @ J3 ) )
& ( ord_less_eq_nat @ ( S3 @ J3 ) @ ( plus_plus_nat @ ( Q2 @ J3 ) @ one_one_nat ) ) ) )
& ( ( Label @ R3 @ I2 )
!= ( Label @ S3 @ I2 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1244_verit__less__mono__div__int2,axiom,
! [A: int,B2: int,N: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N ) @ ( divide_divide_int @ A @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1245_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1246_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_1247_real__average__minus__second,axiom,
! [B: real,A2: real] :
( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A2 )
= ( divide_divide_real @ ( minus_minus_real @ B @ A2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% real_average_minus_second
thf(fact_1248_div__neg__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L2 @ K )
=> ( ( divide_divide_int @ K @ L2 )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1249_div__pos__pos__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L2 )
=> ( ( divide_divide_int @ K @ L2 )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1250_real__average__minus__first,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A2 @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A2 )
= ( divide_divide_real @ ( minus_minus_real @ B @ A2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% real_average_minus_first
thf(fact_1251_div__le__mono,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1252_div__le__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ).
% div_le_dividend
thf(fact_1253_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ N @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1254_div__le__mono2,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1255_zdiv__mono1,axiom,
! [A2: int,A6: int,B: int] :
( ( ord_less_eq_int @ A2 @ A6 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ ( divide_divide_int @ A6 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1256_zdiv__mono2,axiom,
! [A2: int,B7: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B7 )
=> ( ( ord_less_eq_int @ B7 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ ( divide_divide_int @ A2 @ B7 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1257_zdiv__eq__0__iff,axiom,
! [I3: int,K: int] :
( ( ( divide_divide_int @ I3 @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I3 )
& ( ord_less_int @ I3 @ K ) )
| ( ( ord_less_eq_int @ I3 @ zero_zero_int )
& ( ord_less_int @ K @ I3 ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1258_zdiv__mono1__neg,axiom,
! [A2: int,A6: int,B: int] :
( ( ord_less_eq_int @ A2 @ A6 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B ) @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1259_zdiv__mono2__neg,axiom,
! [A2: int,B7: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B7 )
=> ( ( ord_less_eq_int @ B7 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B7 ) @ ( divide_divide_int @ A2 @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1260_div__int__pos__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) )
= ( ( K = zero_zero_int )
| ( L2 = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L2 ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L2 @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1261_div__nonneg__neg__le0,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1262_div__nonpos__pos__le0,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1263_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I3: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I3 @ K ) )
= ( ord_less_eq_int @ K @ I3 ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1264_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1265_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1266_nonneg1__imp__zdiv__pos__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B ) )
= ( ( ord_less_eq_int @ B @ A2 )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1267_nat__div__distrib_H,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( divide_divide_int @ X2 @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib'
thf(fact_1268_nat__div__distrib,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( nat2 @ ( divide_divide_int @ X2 @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib
thf(fact_1269_div__pos__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
=> ( ( divide_divide_int @ K @ L2 )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_1270_div__pos__geq,axiom,
! [L2: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L2 )
=> ( ( ord_less_eq_int @ L2 @ K )
=> ( ( divide_divide_int @ K @ L2 )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) @ one_one_int ) ) ) ) ).
% div_pos_geq
% Conjectures (1)
thf(conj_0,conjecture,
topolo9015423870875150044omplex @ a @ cotang8298477626502807258omplex ).
%------------------------------------------------------------------------------