TPTP Problem File: SLH0023^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Actuarial_Mathematics/0000_Preliminaries/prob_00164_005139__12839314_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1392 ( 634 unt; 117 typ; 0 def)
% Number of atoms : 3686 (1252 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10654 ( 324 ~; 80 |; 240 &;8620 @)
% ( 0 <=>;1390 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 6 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 412 ( 412 >; 0 *; 0 +; 0 <<)
% Number of symbols : 109 ( 108 usr; 13 con; 0-4 aty)
% Number of variables : 3504 ( 219 ^;3234 !; 51 ?;3504 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:11:57.404
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
sigma_measure_real: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Nat__Onat_J,type,
sigma_measure_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__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__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (108)
thf(sy_c_Elementary__Metric__Spaces_Odiameter_001t__Complex__Ocomplex,type,
elemen7580115971934582161omplex: set_complex > real ).
thf(sy_c_Elementary__Metric__Spaces_Odiameter_001t__Real__Oreal,type,
elemen4332022982980038671r_real: set_real > real ).
thf(sy_c_Elementary__Topology_Ointerior_001t__Nat__Onat,type,
elemen7215728294084146536or_nat: set_nat > set_nat ).
thf(sy_c_Elementary__Topology_Ointerior_001t__Real__Oreal,type,
elemen1149380513509575748r_real: set_real > set_real ).
thf(sy_c_Factorial__Ring_Ofactorial__semiring__class_OGcd__factorial_001t__Nat__Onat,type,
factor8539158941071730396al_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
minus_minus_complex: complex > complex > complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
plus_plus_complex: complex > complex > complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
plus_p7052360327008956141omplex: set_complex > set_complex > set_complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Real__Oreal_J,type,
plus_plus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
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__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_Henstock__Kurzweil__Integration_Ohas__integral_001t__Real__Oreal_001t__Real__Oreal,type,
hensto240673015341029504l_real: ( real > real ) > real > set_real > $o ).
thf(sy_c_Henstock__Kurzweil__Integration_Ointegrable__on_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
hensto5734242992300535224omplex: ( complex > complex ) > set_complex > $o ).
thf(sy_c_Henstock__Kurzweil__Integration_Ointegrable__on_001t__Complex__Ocomplex_001t__Real__Oreal,type,
hensto7712097424450753206x_real: ( complex > real ) > set_complex > $o ).
thf(sy_c_Henstock__Kurzweil__Integration_Ointegrable__on_001t__Real__Oreal_001t__Complex__Ocomplex,type,
hensto7658508923946432566omplex: ( real > complex ) > set_real > $o ).
thf(sy_c_Henstock__Kurzweil__Integration_Ointegrable__on_001t__Real__Oreal_001t__Real__Oreal,type,
hensto5963834015518849588l_real: ( real > real ) > set_real > $o ).
thf(sy_c_Henstock__Kurzweil__Integration_Ointegral_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
hensto5696219843187970890omplex: set_complex > ( complex > complex ) > complex ).
thf(sy_c_Henstock__Kurzweil__Integration_Ointegral_001t__Complex__Ocomplex_001t__Real__Oreal,type,
hensto9151348850405077832x_real: set_complex > ( complex > real ) > real ).
thf(sy_c_Henstock__Kurzweil__Integration_Ointegral_001t__Real__Oreal_001t__Complex__Ocomplex,type,
hensto9097760349900757192omplex: set_real > ( real > complex ) > complex ).
thf(sy_c_Henstock__Kurzweil__Integration_Ointegral_001t__Real__Oreal_001t__Real__Oreal,type,
hensto2714581292692559302l_real: set_real > ( real > real ) > real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Real__Oreal,type,
sup_sup_real: real > real > real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Real__Oreal_J,type,
sup_sup_set_real: set_real > set_real > set_real ).
thf(sy_c_Lebesgue__Measure_Olborel_001t__Real__Oreal,type,
lebesgue_lborel_real: sigma_measure_real ).
thf(sy_c_Measure__Space_Oincreasing_001t__Nat__Onat_001t__Nat__Onat,type,
measur1302623347068717141at_nat: set_set_nat > ( set_nat > nat ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Nat__Onat_001t__Real__Oreal,type,
measur5905188192028735665t_real: set_set_nat > ( set_nat > real ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
measur5248428813077667851et_nat: set_set_nat > ( set_nat > set_nat ) > $o ).
thf(sy_c_Measure__Space_Onull__measure_001t__Real__Oreal,type,
measur3749977956733390128e_real: sigma_measure_real > sigma_measure_real ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
bot_bot_real_o: real > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__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__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Complex__Ocomplex,type,
ord_less_complex: complex > complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $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__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
real_V1022390504157884413omplex: complex > real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
real_V7735802525324610683m_real: real > real ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__empty_001t__Real__Oreal,type,
is_empty_real: set_real > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Real__Oreal,type,
is_singleton_real: set_real > $o ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001t__Real__Oreal,type,
remove_real: real > set_real > set_real ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Real__Oreal,type,
the_elem_real: set_real > real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
set_or66887138388493659n_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Real__Oreal,type,
set_or2392270231875598684t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or7074010630789208630et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
set_or1633881224788618240n_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or8625682525731655386et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Sigma__Algebra_Omeasure_001t__Nat__Onat,type,
sigma_measure_nat2: sigma_measure_nat > set_nat > real ).
thf(sy_c_Sigma__Algebra_Omeasure_001t__Real__Oreal,type,
sigma_measure_real2: sigma_measure_real > set_real > real ).
thf(sy_c_Tagged__Division_Ointerval__lowerbound_001t__Real__Oreal,type,
tagged7185179502841007924d_real: set_real > real ).
thf(sy_c_Tagged__Division_Ointerval__upperbound_001t__Real__Oreal,type,
tagged3620743804205775283d_real: set_real > real ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Complex__Ocomplex_001t__Real__Oreal,type,
topolo8674095878704923098x_real: set_complex > ( complex > real ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Nat__Onat,type,
topolo2287203362918339196al_nat: set_real > ( real > nat ) > $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_Topology__Euclidean__Space_Obox_001t__Real__Oreal,type,
topolo8288580659802485013x_real: real > real > set_real ).
thf(sy_c_Topology__Euclidean__Space_Ocbox_001t__Complex__Ocomplex,type,
topolo4166636743940337274omplex: complex > complex > set_complex ).
thf(sy_c_Topology__Euclidean__Space_Ocbox_001t__Real__Oreal,type,
topolo7804196973972690552x_real: real > real > set_real ).
thf(sy_c_Topology__Euclidean__Space_Oext__cont_001t__Real__Oreal_001t__Real__Oreal,type,
topolo6696879324188940074l_real: ( real > real ) > real > real > real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_a,type,
a: real ).
thf(sy_v_b,type,
b: real ).
thf(sy_v_f,type,
f: real > real ).
% Relevant facts (1274)
thf(fact_0_assms,axiom,
ord_less_eq_real @ b @ a ).
% assms
thf(fact_1_has__integral__is__0,axiom,
! [S: set_real,F: real > real] :
( ! [X: real] :
( ( member_real @ X @ S )
=> ( ( F @ X )
= zero_zero_real ) )
=> ( hensto240673015341029504l_real @ F @ zero_zero_real @ S ) ) ).
% has_integral_is_0
thf(fact_2_has__integral__eq,axiom,
! [S2: set_real,F: real > real,G: real > real,K: real] :
( ! [X: real] :
( ( member_real @ X @ S2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( hensto240673015341029504l_real @ F @ K @ S2 )
=> ( hensto240673015341029504l_real @ G @ K @ S2 ) ) ) ).
% has_integral_eq
thf(fact_3_has__integral__cong,axiom,
! [S2: set_real,F: real > real,G: real > real,I: real] :
( ! [X: real] :
( ( member_real @ X @ S2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( hensto240673015341029504l_real @ F @ I @ S2 )
= ( hensto240673015341029504l_real @ G @ I @ S2 ) ) ) ).
% has_integral_cong
thf(fact_4_has__integral__eq__rhs,axiom,
! [F: real > real,J: real,S: set_real,I: real] :
( ( hensto240673015341029504l_real @ F @ J @ S )
=> ( ( I = J )
=> ( hensto240673015341029504l_real @ F @ I @ S ) ) ) ).
% has_integral_eq_rhs
thf(fact_5_has__integral__unique,axiom,
! [F: real > real,K1: real,I: set_real,K2: real] :
( ( hensto240673015341029504l_real @ F @ K1 @ I )
=> ( ( hensto240673015341029504l_real @ F @ K2 @ I )
=> ( K1 = K2 ) ) ) ).
% has_integral_unique
thf(fact_6_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_7_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_8_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_9_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_10_has__integral__null__real,axiom,
! [A: real,B: real,F: real > real] :
( ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( set_or1222579329274155063t_real @ A @ B ) )
= zero_zero_real )
=> ( hensto240673015341029504l_real @ F @ zero_zero_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% has_integral_null_real
thf(fact_11_has__integral__empty,axiom,
! [F: real > real] : ( hensto240673015341029504l_real @ F @ zero_zero_real @ bot_bot_set_real ) ).
% has_integral_empty
thf(fact_12_has__integral__empty__eq,axiom,
! [F: real > real,I: real] :
( ( hensto240673015341029504l_real @ F @ I @ bot_bot_set_real )
= ( I = zero_zero_real ) ) ).
% has_integral_empty_eq
thf(fact_13_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_14_content__empty,axiom,
( ( sigma_measure_real2 @ lebesgue_lborel_real @ bot_bot_set_real )
= zero_zero_real ) ).
% content_empty
thf(fact_15_content__pos__le,axiom,
! [X3: set_real] : ( ord_less_eq_real @ zero_zero_real @ ( sigma_measure_real2 @ lebesgue_lborel_real @ X3 ) ) ).
% content_pos_le
thf(fact_16_content__real__eq__0,axiom,
! [A: real,B: real] :
( ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( set_or1222579329274155063t_real @ A @ B ) )
= zero_zero_real )
= ( ord_less_eq_real @ B @ A ) ) ).
% content_real_eq_0
thf(fact_17_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_18_has__integral__le,axiom,
! [F: real > real,I: real,S: set_real,G: real > real,J: real] :
( ( hensto240673015341029504l_real @ F @ I @ S )
=> ( ( hensto240673015341029504l_real @ G @ J @ S )
=> ( ! [X: real] :
( ( member_real @ X @ S )
=> ( ord_less_eq_real @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_real @ I @ J ) ) ) ) ).
% has_integral_le
thf(fact_19_has__integral__nonneg,axiom,
! [F: real > real,I: real,S: set_real] :
( ( hensto240673015341029504l_real @ F @ I @ S )
=> ( ! [X: real] :
( ( member_real @ X @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ I ) ) ) ).
% has_integral_nonneg
thf(fact_20_measure__empty,axiom,
! [M: sigma_measure_real] :
( ( sigma_measure_real2 @ M @ bot_bot_set_real )
= zero_zero_real ) ).
% measure_empty
thf(fact_21_measure__empty,axiom,
! [M: sigma_measure_nat] :
( ( sigma_measure_nat2 @ M @ bot_bot_set_nat )
= zero_zero_real ) ).
% measure_empty
thf(fact_22_atLeastatMost__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( set_or4548717258645045905et_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_23_atLeastatMost__empty__iff,axiom,
! [A: real,B: real] :
( ( ( set_or1222579329274155063t_real @ A @ B )
= bot_bot_set_real )
= ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_24_atLeastatMost__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_25_atLeastatMost__empty__iff2,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_set_nat
= ( set_or4548717258645045905et_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_26_atLeastatMost__empty__iff2,axiom,
! [A: real,B: real] :
( ( bot_bot_set_real
= ( set_or1222579329274155063t_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_27_atLeastatMost__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_28_atLeastatMost__empty_H,axiom,
! [A: set_nat,B: set_nat] :
( ~ ( ord_less_eq_set_nat @ A @ B )
=> ( ( set_or4548717258645045905et_nat @ A @ B )
= bot_bot_set_set_nat ) ) ).
% atLeastatMost_empty'
thf(fact_29_atLeastatMost__empty_H,axiom,
! [A: real,B: real] :
( ~ ( ord_less_eq_real @ A @ B )
=> ( ( set_or1222579329274155063t_real @ A @ B )
= bot_bot_set_real ) ) ).
% atLeastatMost_empty'
thf(fact_30_atLeastatMost__empty_H,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty'
thf(fact_31_atLeastatMost__subset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_32_atLeastatMost__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_33_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_34_atLeastAtMost__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_35_atLeastAtMost__iff,axiom,
! [I: real,L: real,U: real] :
( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_eq_real @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_36_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_37_Icc__eq__Icc,axiom,
! [L: set_nat,H: set_nat,L2: set_nat,H2: set_nat] :
( ( ( set_or4548717258645045905et_nat @ L @ H )
= ( set_or4548717258645045905et_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_nat @ L @ H )
& ~ ( ord_less_eq_set_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_38_Icc__eq__Icc,axiom,
! [L: real,H: real,L2: real,H2: real] :
( ( ( set_or1222579329274155063t_real @ L @ H )
= ( set_or1222579329274155063t_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_real @ L @ H )
& ~ ( ord_less_eq_real @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_39_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H )
= ( set_or1269000886237332187st_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat @ L @ H )
& ~ ( ord_less_eq_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_40_measure__nonneg,axiom,
! [M: sigma_measure_real,A2: set_real] : ( ord_less_eq_real @ zero_zero_real @ ( sigma_measure_real2 @ M @ A2 ) ) ).
% measure_nonneg
thf(fact_41_measure__le__0__iff,axiom,
! [M: sigma_measure_real,X3: set_real] :
( ( ord_less_eq_real @ ( sigma_measure_real2 @ M @ X3 ) @ zero_zero_real )
= ( ( sigma_measure_real2 @ M @ X3 )
= zero_zero_real ) ) ).
% measure_le_0_iff
thf(fact_42_empty__iff,axiom,
! [C: real] :
~ ( member_real @ C @ bot_bot_set_real ) ).
% empty_iff
thf(fact_43_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_44_all__not__in__conv,axiom,
! [A2: set_real] :
( ( ! [X4: real] :
~ ( member_real @ X4 @ A2 ) )
= ( A2 = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_45_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X4: nat] :
~ ( member_nat @ X4 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_46_Collect__empty__eq,axiom,
! [P: real > $o] :
( ( ( collect_real @ P )
= bot_bot_set_real )
= ( ! [X4: real] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_47_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_48_subsetI,axiom,
! [A2: set_real,B2: set_real] :
( ! [X: real] :
( ( member_real @ X @ A2 )
=> ( member_real @ X @ B2 ) )
=> ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% subsetI
thf(fact_49_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_50_subset__antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_51_empty__Collect__eq,axiom,
! [P: real > $o] :
( ( bot_bot_set_real
= ( collect_real @ P ) )
= ( ! [X4: real] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_52_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_53_empty__subsetI,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% empty_subsetI
thf(fact_54_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_55_subset__empty,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
= ( A2 = bot_bot_set_real ) ) ).
% subset_empty
thf(fact_56_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_57_in__mono,axiom,
! [A2: set_real,B2: set_real,X2: real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( member_real @ X2 @ A2 )
=> ( member_real @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_58_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_59_subsetD,axiom,
! [A2: set_real,B2: set_real,C: real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B2 ) ) ) ).
% subsetD
thf(fact_60_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_61_equalityE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_62_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_63_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_nat @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_64_equalityD1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_65_equalityD2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_66_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
! [T: real] :
( ( member_real @ T @ A3 )
=> ( member_real @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_67_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_68_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_69_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_71_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_72_subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_73_set__eq__subset,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_74_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_75_has__integral__on__superset,axiom,
! [F: real > real,I: real,S: set_real,T2: set_real] :
( ( hensto240673015341029504l_real @ F @ I @ S )
=> ( ! [X: real] :
( ~ ( member_real @ X @ S )
=> ( ( F @ X )
= zero_zero_real ) )
=> ( ( ord_less_eq_set_real @ S @ T2 )
=> ( hensto240673015341029504l_real @ F @ I @ T2 ) ) ) ) ).
% has_integral_on_superset
thf(fact_76_ex__in__conv,axiom,
! [A2: set_real] :
( ( ? [X4: real] : ( member_real @ X4 @ A2 ) )
= ( A2 != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_77_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X4: nat] : ( member_nat @ X4 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_78_equals0I,axiom,
! [A2: set_real] :
( ! [Y2: real] :
~ ( member_real @ Y2 @ A2 )
=> ( A2 = bot_bot_set_real ) ) ).
% equals0I
thf(fact_79_equals0I,axiom,
! [A2: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_80_equals0D,axiom,
! [A2: set_real,A: real] :
( ( A2 = bot_bot_set_real )
=> ~ ( member_real @ A @ A2 ) ) ).
% equals0D
thf(fact_81_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_82_emptyE,axiom,
! [A: real] :
~ ( member_real @ A @ bot_bot_set_real ) ).
% emptyE
thf(fact_83_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_84_has__integral__subset__le,axiom,
! [S2: set_real,T3: set_real,F: real > real,I: real,J: real] :
( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ( hensto240673015341029504l_real @ F @ I @ S2 )
=> ( ( hensto240673015341029504l_real @ F @ J @ T3 )
=> ( ! [X: real] :
( ( member_real @ X @ T3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) )
=> ( ord_less_eq_real @ I @ J ) ) ) ) ) ).
% has_integral_subset_le
thf(fact_85_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_86_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_87_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_88_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_89_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_90_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_91_increasing__def,axiom,
( measur5905188192028735665t_real
= ( ^ [M2: set_set_nat,Mu: set_nat > real] :
! [X4: set_nat] :
( ( member_set_nat @ X4 @ M2 )
=> ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ M2 )
=> ( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( Mu @ X4 ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_92_increasing__def,axiom,
( measur1302623347068717141at_nat
= ( ^ [M2: set_set_nat,Mu: set_nat > nat] :
! [X4: set_nat] :
( ( member_set_nat @ X4 @ M2 )
=> ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ M2 )
=> ( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( Mu @ X4 ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_93_increasing__def,axiom,
( measur5248428813077667851et_nat
= ( ^ [M2: set_set_nat,Mu: set_nat > set_nat] :
! [X4: set_nat] :
( ( member_set_nat @ X4 @ M2 )
=> ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ M2 )
=> ( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( Mu @ X4 ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_94_increasingD,axiom,
! [M: set_set_nat,F: set_nat > real,X2: set_nat,Y4: set_nat] :
( ( measur5905188192028735665t_real @ M @ F )
=> ( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( member_set_nat @ X2 @ M )
=> ( ( member_set_nat @ Y4 @ M )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_95_increasingD,axiom,
! [M: set_set_nat,F: set_nat > nat,X2: set_nat,Y4: set_nat] :
( ( measur1302623347068717141at_nat @ M @ F )
=> ( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( member_set_nat @ X2 @ M )
=> ( ( member_set_nat @ Y4 @ M )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_96_increasingD,axiom,
! [M: set_set_nat,F: set_nat > set_nat,X2: set_nat,Y4: set_nat] :
( ( measur5248428813077667851et_nat @ M @ F )
=> ( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( member_set_nat @ X2 @ M )
=> ( ( member_set_nat @ Y4 @ M )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_97_content__real__if,axiom,
! [A: real,B: real] :
( ( ( ord_less_eq_real @ A @ B )
=> ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) )
& ( ~ ( ord_less_eq_real @ A @ B )
=> ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( set_or1222579329274155063t_real @ A @ B ) )
= zero_zero_real ) ) ) ).
% content_real_if
thf(fact_98_subset__emptyI,axiom,
! [A2: set_real] :
( ! [X: real] :
~ ( member_real @ X @ A2 )
=> ( ord_less_eq_set_real @ A2 @ bot_bot_set_real ) ) ).
% subset_emptyI
thf(fact_99_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X: nat] :
~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_100_bot_Oextremum__uniqueI,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
=> ( A = bot_bot_set_real ) ) ).
% bot.extremum_uniqueI
thf(fact_101_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_102_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_103_bot_Oextremum__unique,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
= ( A = bot_bot_set_real ) ) ).
% bot.extremum_unique
thf(fact_104_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_105_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_106_bot_Oextremum,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% bot.extremum
thf(fact_107_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_108_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_109_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_110_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_111_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_112_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_113_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_114_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_115_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_116_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_117_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_118_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_119_content__real,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ) ).
% content_real
thf(fact_120_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_121_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_122_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_123_bot__set__def,axiom,
( bot_bot_set_real
= ( collect_real @ bot_bot_real_o ) ) ).
% bot_set_def
thf(fact_124_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_125_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_126_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_127_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_128_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_129_eq__iff__diff__eq__0,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [A4: real,B4: real] :
( ( minus_minus_real @ A4 @ B4 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_130_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_131_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_132_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_133_le__cases3,axiom,
! [X2: real,Y4: real,Z2: real] :
( ( ( ord_less_eq_real @ X2 @ Y4 )
=> ~ ( ord_less_eq_real @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y4 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y4 )
=> ~ ( ord_less_eq_real @ Y4 @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y4 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_134_le__cases3,axiom,
! [X2: nat,Y4: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_135_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
& ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_136_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_137_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
= ( ^ [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_138_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_139_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_140_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_141_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_142_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_143_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_144_order__antisym,axiom,
! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_145_order__antisym,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_146_order__antisym,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_147_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_148_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_149_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_150_order__trans,axiom,
! [X2: real,Y4: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ Z2 )
=> ( ord_less_eq_real @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_151_order__trans,axiom,
! [X2: nat,Y4: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_152_order__trans,axiom,
! [X2: set_nat,Y4: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_153_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: real,B5: real] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_154_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_155_dual__order_Oeq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_156_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_157_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_158_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_159_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_160_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_161_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_162_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_163_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_164_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_165_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_166_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_167_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_168_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_169_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_170_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_171_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_172_order__subst1,axiom,
! [A: real,F: set_nat > real,B: set_nat,C: set_nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_173_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_174_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_175_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_176_order__subst1,axiom,
! [A: set_nat,F: real > set_nat,B: real,C: real] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_177_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_178_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_179_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_180_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_181_order__subst2,axiom,
! [A: real,B: real,F: real > set_nat,C: set_nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_182_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_183_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_184_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_185_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > real,C: real] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_186_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_187_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_188_order__eq__refl,axiom,
! [X2: real,Y4: real] :
( ( X2 = Y4 )
=> ( ord_less_eq_real @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_189_order__eq__refl,axiom,
! [X2: nat,Y4: nat] :
( ( X2 = Y4 )
=> ( ord_less_eq_nat @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_190_order__eq__refl,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( X2 = Y4 )
=> ( ord_less_eq_set_nat @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_191_linorder__linear,axiom,
! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
| ( ord_less_eq_real @ Y4 @ X2 ) ) ).
% linorder_linear
thf(fact_192_linorder__linear,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X2 ) ) ).
% linorder_linear
thf(fact_193_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_194_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_195_ord__eq__le__subst,axiom,
! [A: set_nat,F: real > set_nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_196_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_197_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_198_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_199_ord__eq__le__subst,axiom,
! [A: real,F: set_nat > real,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_200_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_201_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_202_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_203_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_204_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > set_nat,C: set_nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_205_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_206_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_207_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_208_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > real,C: real] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_209_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_210_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_211_linorder__le__cases,axiom,
! [X2: real,Y4: real] :
( ~ ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ Y4 @ X2 ) ) ).
% linorder_le_cases
thf(fact_212_linorder__le__cases,axiom,
! [X2: nat,Y4: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ).
% linorder_le_cases
thf(fact_213_order__antisym__conv,axiom,
! [Y4: real,X2: real] :
( ( ord_less_eq_real @ Y4 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_214_order__antisym__conv,axiom,
! [Y4: nat,X2: nat] :
( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_215_order__antisym__conv,axiom,
! [Y4: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_216_measure__lborel__Icc,axiom,
! [L: real,U: real] :
( ( ord_less_eq_real @ L @ U )
=> ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( set_or1222579329274155063t_real @ L @ U ) )
= ( minus_minus_real @ U @ L ) ) ) ).
% measure_lborel_Icc
thf(fact_217_diff__shunt__var,axiom,
! [X2: set_real,Y4: set_real] :
( ( ( minus_minus_set_real @ X2 @ Y4 )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ X2 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_218_diff__shunt__var,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( ( minus_minus_set_nat @ X2 @ Y4 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X2 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_219_ge__iff__diff__ge__0,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A4: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A4 @ B4 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_220_has__integral__null__eq,axiom,
! [A: real,B: real,F: real > real,I: real] :
( ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( topolo7804196973972690552x_real @ A @ B ) )
= zero_zero_real )
=> ( ( hensto240673015341029504l_real @ F @ I @ ( topolo7804196973972690552x_real @ A @ B ) )
= ( I = zero_zero_real ) ) ) ).
% has_integral_null_eq
thf(fact_221_has__integral__null,axiom,
! [A: real,B: real,F: real > real] :
( ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( topolo7804196973972690552x_real @ A @ B ) )
= zero_zero_real )
=> ( hensto240673015341029504l_real @ F @ zero_zero_real @ ( topolo7804196973972690552x_real @ A @ B ) ) ) ).
% has_integral_null
thf(fact_222_abs__eq__content,axiom,
! [X2: real,Y4: real] :
( ( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ( abs_abs_real @ ( minus_minus_real @ Y4 @ X2 ) )
= ( sigma_measure_real2 @ lebesgue_lborel_real @ ( set_or1222579329274155063t_real @ X2 @ Y4 ) ) ) )
& ( ~ ( ord_less_eq_real @ X2 @ Y4 )
=> ( ( abs_abs_real @ ( minus_minus_real @ Y4 @ X2 ) )
= ( sigma_measure_real2 @ lebesgue_lborel_real @ ( set_or1222579329274155063t_real @ Y4 @ X2 ) ) ) ) ) ).
% abs_eq_content
thf(fact_223_content__singleton,axiom,
! [A: real] :
( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( insert_real @ A @ bot_bot_set_real ) )
= zero_zero_real ) ).
% content_singleton
thf(fact_224_Set_Ois__empty__def,axiom,
( is_empty_real
= ( ^ [A3: set_real] : ( A3 = bot_bot_set_real ) ) ) ).
% Set.is_empty_def
thf(fact_225_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_226_measure__null__measure,axiom,
! [M: sigma_measure_real,X3: set_real] :
( ( sigma_measure_real2 @ ( measur3749977956733390128e_real @ M ) @ X3 )
= zero_zero_real ) ).
% measure_null_measure
thf(fact_227_Diff__cancel,axiom,
! [A2: set_real] :
( ( minus_minus_set_real @ A2 @ A2 )
= bot_bot_set_real ) ).
% Diff_cancel
thf(fact_228_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_229_empty__Diff,axiom,
! [A2: set_real] :
( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
= bot_bot_set_real ) ).
% empty_Diff
thf(fact_230_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_231_Diff__empty,axiom,
! [A2: set_real] :
( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
= A2 ) ).
% Diff_empty
thf(fact_232_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_233_insert__Diff1,axiom,
! [X2: real,B2: set_real,A2: set_real] :
( ( member_real @ X2 @ B2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_234_Diff__insert0,axiom,
! [X2: real,A2: set_real,B2: set_real] :
( ~ ( member_real @ X2 @ A2 )
=> ( ( minus_minus_set_real @ A2 @ ( insert_real @ X2 @ B2 ) )
= ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_235_insertCI,axiom,
! [A: real,B2: set_real,B: real] :
( ( ~ ( member_real @ A @ B2 )
=> ( A = B ) )
=> ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).
% insertCI
thf(fact_236_insert__iff,axiom,
! [A: real,B: real,A2: set_real] :
( ( member_real @ A @ ( insert_real @ B @ A2 ) )
= ( ( A = B )
| ( member_real @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_237_abs__idempotent,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_idempotent
thf(fact_238_Diff__eq__empty__iff,axiom,
! [A2: set_real,B2: set_real] :
( ( ( minus_minus_set_real @ A2 @ B2 )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_239_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_240_abs__0__eq,axiom,
! [A: real] :
( ( zero_zero_real
= ( abs_abs_real @ A ) )
= ( A = zero_zero_real ) ) ).
% abs_0_eq
thf(fact_241_abs__eq__0,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0
thf(fact_242_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_zero
thf(fact_243_insert__subset,axiom,
! [X2: real,A2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ ( insert_real @ X2 @ A2 ) @ B2 )
= ( ( member_real @ X2 @ B2 )
& ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_244_insert__subset,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A2 ) @ B2 )
= ( ( member_nat @ X2 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_245_insert__Diff__single,axiom,
! [A: real,A2: set_real] :
( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( insert_real @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_246_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_247_singletonI,axiom,
! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).
% singletonI
thf(fact_248_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_249_abs__le__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_le_zero_iff
thf(fact_250_abs__le__self__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% abs_le_self_iff
thf(fact_251_abs__of__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_252_singleton__insert__inj__eq,axiom,
! [B: real,A: real,A2: set_real] :
( ( ( insert_real @ B @ bot_bot_set_real )
= ( insert_real @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_253_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_254_singleton__insert__inj__eq_H,axiom,
! [A: real,A2: set_real,B: real] :
( ( ( insert_real @ A @ A2 )
= ( insert_real @ B @ bot_bot_set_real ) )
= ( ( A = B )
& ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_255_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_256_atLeastAtMost__singleton__iff,axiom,
! [A: real,B: real,C: real] :
( ( ( set_or1222579329274155063t_real @ A @ B )
= ( insert_real @ C @ bot_bot_set_real ) )
= ( ( A = B )
& ( B = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_257_atLeastAtMost__singleton__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= ( insert_nat @ C @ bot_bot_set_nat ) )
= ( ( A = B )
& ( B = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_258_atLeastAtMost__singleton,axiom,
! [A: real] :
( ( set_or1222579329274155063t_real @ A @ A )
= ( insert_real @ A @ bot_bot_set_real ) ) ).
% atLeastAtMost_singleton
thf(fact_259_atLeastAtMost__singleton,axiom,
! [A: nat] :
( ( set_or1269000886237332187st_nat @ A @ A )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% atLeastAtMost_singleton
thf(fact_260_has__integral__refl_I1_J,axiom,
! [F: real > real,A: real] : ( hensto240673015341029504l_real @ F @ zero_zero_real @ ( topolo7804196973972690552x_real @ A @ A ) ) ).
% has_integral_refl(1)
thf(fact_261_has__integral__refl_I2_J,axiom,
! [F: real > real,A: real] : ( hensto240673015341029504l_real @ F @ zero_zero_real @ ( insert_real @ A @ bot_bot_set_real ) ) ).
% has_integral_refl(2)
thf(fact_262_measure__lborel__singleton,axiom,
! [X2: real] :
( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( insert_real @ X2 @ bot_bot_set_real ) )
= zero_zero_real ) ).
% measure_lborel_singleton
thf(fact_263_insert__Diff__if,axiom,
! [X2: real,B2: set_real,A2: set_real] :
( ( ( member_real @ X2 @ B2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_real @ A2 @ B2 ) ) )
& ( ~ ( member_real @ X2 @ B2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X2 @ A2 ) @ B2 )
= ( insert_real @ X2 @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_264_insertE,axiom,
! [A: real,B: real,A2: set_real] :
( ( member_real @ A @ ( insert_real @ B @ A2 ) )
=> ( ( A != B )
=> ( member_real @ A @ A2 ) ) ) ).
% insertE
thf(fact_265_insertI1,axiom,
! [A: real,B2: set_real] : ( member_real @ A @ ( insert_real @ A @ B2 ) ) ).
% insertI1
thf(fact_266_insertI2,axiom,
! [A: real,B2: set_real,B: real] :
( ( member_real @ A @ B2 )
=> ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).
% insertI2
thf(fact_267_Set_Oset__insert,axiom,
! [X2: real,A2: set_real] :
( ( member_real @ X2 @ A2 )
=> ~ ! [B6: set_real] :
( ( A2
= ( insert_real @ X2 @ B6 ) )
=> ( member_real @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_268_insert__ident,axiom,
! [X2: real,A2: set_real,B2: set_real] :
( ~ ( member_real @ X2 @ A2 )
=> ( ~ ( member_real @ X2 @ B2 )
=> ( ( ( insert_real @ X2 @ A2 )
= ( insert_real @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_269_insert__absorb,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ( ( insert_real @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_270_insert__eq__iff,axiom,
! [A: real,A2: set_real,B: real,B2: set_real] :
( ~ ( member_real @ A @ A2 )
=> ( ~ ( member_real @ B @ B2 )
=> ( ( ( insert_real @ A @ A2 )
= ( insert_real @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_real] :
( ( A2
= ( insert_real @ B @ C3 ) )
& ~ ( member_real @ B @ C3 )
& ( B2
= ( insert_real @ A @ C3 ) )
& ~ ( member_real @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_271_mk__disjoint__insert,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ? [B6: set_real] :
( ( A2
= ( insert_real @ A @ B6 ) )
& ~ ( member_real @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_272_subset__Diff__insert,axiom,
! [A2: set_real,B2: set_real,X2: real,C2: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ ( insert_real @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ C2 ) )
& ~ ( member_real @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_273_subset__Diff__insert,axiom,
! [A2: set_nat,B2: set_nat,X2: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C2 ) )
& ~ ( member_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_274_Diff__insert,axiom,
! [A2: set_real,A: real,B2: set_real] :
( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
= ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ B2 ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).
% Diff_insert
thf(fact_275_Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_276_insert__Diff,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_277_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_278_Diff__insert2,axiom,
! [A2: set_real,A: real,B2: set_real] :
( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
= ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_279_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_280_Diff__insert__absorb,axiom,
! [X2: real,A2: set_real] :
( ~ ( member_real @ X2 @ A2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X2 @ A2 ) @ ( insert_real @ X2 @ bot_bot_set_real ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_281_Diff__insert__absorb,axiom,
! [X2: nat,A2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_282_subset__insert__iff,axiom,
! [A2: set_real,X2: real,B2: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X2 @ B2 ) )
= ( ( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B2 ) )
& ( ~ ( member_real @ X2 @ A2 )
=> ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_283_subset__insert__iff,axiom,
! [A2: set_nat,X2: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B2 ) )
= ( ( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_284_Diff__single__insert,axiom,
! [A2: set_real,X2: real,B2: set_real] :
( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B2 )
=> ( ord_less_eq_set_real @ A2 @ ( insert_real @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_285_Diff__single__insert,axiom,
! [A2: set_nat,X2: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_286_abs__ge__self,axiom,
! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% abs_ge_self
thf(fact_287_abs__le__D1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% abs_le_D1
thf(fact_288_abs__minus__commute,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
= ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_289_interval__cbox,axiom,
set_or1222579329274155063t_real = topolo7804196973972690552x_real ).
% interval_cbox
thf(fact_290_insert__mono,axiom,
! [C2: set_nat,D2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C2 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_291_subset__insert,axiom,
! [X2: real,A2: set_real,B2: set_real] :
( ~ ( member_real @ X2 @ A2 )
=> ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X2 @ B2 ) )
= ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_292_subset__insert,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_293_subset__insertI,axiom,
! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).
% subset_insertI
thf(fact_294_subset__insertI2,axiom,
! [A2: set_nat,B2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_295_insert__subsetI,axiom,
! [X2: real,A2: set_real,X3: set_real] :
( ( member_real @ X2 @ A2 )
=> ( ( ord_less_eq_set_real @ X3 @ A2 )
=> ( ord_less_eq_set_real @ ( insert_real @ X2 @ X3 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_296_insert__subsetI,axiom,
! [X2: nat,A2: set_nat,X3: set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ X3 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_297_singleton__inject,axiom,
! [A: real,B: real] :
( ( ( insert_real @ A @ bot_bot_set_real )
= ( insert_real @ B @ bot_bot_set_real ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_298_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_299_insert__not__empty,axiom,
! [A: real,A2: set_real] :
( ( insert_real @ A @ A2 )
!= bot_bot_set_real ) ).
% insert_not_empty
thf(fact_300_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_301_doubleton__eq__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( insert_real @ A @ ( insert_real @ B @ bot_bot_set_real ) )
= ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_302_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_303_singleton__iff,axiom,
! [B: real,A: real] :
( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_304_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_305_singletonD,axiom,
! [B: real,A: real] :
( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_306_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_307_Diff__mono,axiom,
! [A2: set_nat,C2: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_308_Diff__subset,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_309_double__diff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_310_has__integral__unique__cbox,axiom,
! [F: real > real,K1: real,A: real,B: real,K2: real] :
( ( hensto240673015341029504l_real @ F @ K1 @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ( hensto240673015341029504l_real @ F @ K2 @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( K1 = K2 ) ) ) ).
% has_integral_unique_cbox
thf(fact_311_abs__ge__zero,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% abs_ge_zero
thf(fact_312_abs__triangle__ineq2__sym,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_313_abs__triangle__ineq3,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_314_abs__triangle__ineq2,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_315_subset__singletonD,axiom,
! [A2: set_real,X2: real] :
( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X2 @ bot_bot_set_real ) )
=> ( ( A2 = bot_bot_set_real )
| ( A2
= ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ).
% subset_singletonD
thf(fact_316_subset__singletonD,axiom,
! [A2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_317_subset__singleton__iff,axiom,
! [X3: set_real,A: real] :
( ( ord_less_eq_set_real @ X3 @ ( insert_real @ A @ bot_bot_set_real ) )
= ( ( X3 = bot_bot_set_real )
| ( X3
= ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% subset_singleton_iff
thf(fact_318_subset__singleton__iff,axiom,
! [X3: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X3 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X3 = bot_bot_set_nat )
| ( X3
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_319_has__integral__is__0__cbox,axiom,
! [A: real,B: real,F: real > real] :
( ! [X: real] :
( ( member_real @ X @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ( F @ X )
= zero_zero_real ) )
=> ( hensto240673015341029504l_real @ F @ zero_zero_real @ ( topolo7804196973972690552x_real @ A @ B ) ) ) ).
% has_integral_is_0_cbox
thf(fact_320_atLeastAtMost__singleton_H,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( set_or1222579329274155063t_real @ A @ B )
= ( insert_real @ A @ bot_bot_set_real ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_321_atLeastAtMost__singleton_H,axiom,
! [A: nat,B: nat] :
( ( A = B )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_322_content__0__subset,axiom,
! [A: real,B: real,S2: set_real] :
( ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( topolo7804196973972690552x_real @ A @ B ) )
= zero_zero_real )
=> ( ( ord_less_eq_set_real @ S2 @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ( sigma_measure_real2 @ lebesgue_lborel_real @ S2 )
= zero_zero_real ) ) ) ).
% content_0_subset
thf(fact_323_content__subset,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( topolo7804196973972690552x_real @ A @ B ) @ ( topolo7804196973972690552x_real @ C @ D ) )
=> ( ord_less_eq_real @ ( sigma_measure_real2 @ lebesgue_lborel_real @ ( topolo7804196973972690552x_real @ A @ B ) ) @ ( sigma_measure_real2 @ lebesgue_lborel_real @ ( topolo7804196973972690552x_real @ C @ D ) ) ) ) ).
% content_subset
thf(fact_324_property__empty__interval,axiom,
! [P: set_real > $o] :
( ! [A5: real,B5: real] :
( ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( topolo7804196973972690552x_real @ A5 @ B5 ) )
= zero_zero_real )
=> ( P @ ( topolo7804196973972690552x_real @ A5 @ B5 ) ) )
=> ( P @ bot_bot_set_real ) ) ).
% property_empty_interval
thf(fact_325_cbox__idem,axiom,
! [A: real] :
( ( topolo7804196973972690552x_real @ A @ A )
= ( insert_real @ A @ bot_bot_set_real ) ) ).
% cbox_idem
thf(fact_326_abs__0,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_0
thf(fact_327_set__zero,axiom,
( zero_zero_set_real
= ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) ).
% set_zero
thf(fact_328_set__zero,axiom,
( zero_zero_set_nat
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% set_zero
thf(fact_329_box__real_I2_J,axiom,
topolo7804196973972690552x_real = set_or1222579329274155063t_real ).
% box_real(2)
thf(fact_330_abs__abs,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_abs
thf(fact_331_the__elem__eq,axiom,
! [X2: real] :
( ( the_elem_real @ ( insert_real @ X2 @ bot_bot_set_real ) )
= X2 ) ).
% the_elem_eq
thf(fact_332_the__elem__eq,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_333_interval__bounds__nz__content_I1_J,axiom,
! [A: real,B: real] :
( ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( topolo7804196973972690552x_real @ A @ B ) )
!= zero_zero_real )
=> ( ( tagged3620743804205775283d_real @ ( topolo7804196973972690552x_real @ A @ B ) )
= B ) ) ).
% interval_bounds_nz_content(1)
thf(fact_334_interval__bounds__nz__content_I2_J,axiom,
! [A: real,B: real] :
( ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( topolo7804196973972690552x_real @ A @ B ) )
!= zero_zero_real )
=> ( ( tagged7185179502841007924d_real @ ( topolo7804196973972690552x_real @ A @ B ) )
= A ) ) ).
% interval_bounds_nz_content(2)
thf(fact_335_Diff__iff,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
= ( ( member_real @ C @ A2 )
& ~ ( member_real @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_336_DiffI,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ A2 )
=> ( ~ ( member_real @ C @ B2 )
=> ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_337_mem__box__real_I2_J,axiom,
! [X2: real,A: real,B: real] :
( ( member_real @ X2 @ ( topolo7804196973972690552x_real @ A @ B ) )
= ( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_eq_real @ X2 @ B ) ) ) ).
% mem_box_real(2)
thf(fact_338_DiffD2,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ~ ( member_real @ C @ B2 ) ) ).
% DiffD2
thf(fact_339_DiffD1,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ( member_real @ C @ A2 ) ) ).
% DiffD1
thf(fact_340_DiffE,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ~ ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B2 ) ) ) ).
% DiffE
thf(fact_341_abs__eq__0__iff,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0_iff
thf(fact_342_eq__cbox,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( topolo7804196973972690552x_real @ A @ B )
= ( topolo7804196973972690552x_real @ C @ D ) )
= ( ( ( ( topolo7804196973972690552x_real @ A @ B )
= bot_bot_set_real )
& ( ( topolo7804196973972690552x_real @ C @ D )
= bot_bot_set_real ) )
| ( ( A = C )
& ( B = D ) ) ) ) ).
% eq_cbox
thf(fact_343_interval__bounds_H_I2_J,axiom,
! [A: real,B: real] :
( ( ( topolo7804196973972690552x_real @ A @ B )
!= bot_bot_set_real )
=> ( ( tagged7185179502841007924d_real @ ( topolo7804196973972690552x_real @ A @ B ) )
= A ) ) ).
% interval_bounds'(2)
thf(fact_344_interval__bounds_H_I1_J,axiom,
! [A: real,B: real] :
( ( ( topolo7804196973972690552x_real @ A @ B )
!= bot_bot_set_real )
=> ( ( tagged3620743804205775283d_real @ ( topolo7804196973972690552x_real @ A @ B ) )
= B ) ) ).
% interval_bounds'(1)
thf(fact_345_is__singleton__the__elem,axiom,
( is_singleton_real
= ( ^ [A3: set_real] :
( A3
= ( insert_real @ ( the_elem_real @ A3 ) @ bot_bot_set_real ) ) ) ) ).
% is_singleton_the_elem
thf(fact_346_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_347_bot__empty__eq,axiom,
( bot_bot_real_o
= ( ^ [X4: real] : ( member_real @ X4 @ bot_bot_set_real ) ) ) ).
% bot_empty_eq
thf(fact_348_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X4: nat] : ( member_nat @ X4 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_349_Collect__empty__eq__bot,axiom,
! [P: real > $o] :
( ( ( collect_real @ P )
= bot_bot_set_real )
= ( P = bot_bot_real_o ) ) ).
% Collect_empty_eq_bot
thf(fact_350_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_351_is__singletonI,axiom,
! [X2: real] : ( is_singleton_real @ ( insert_real @ X2 @ bot_bot_set_real ) ) ).
% is_singletonI
thf(fact_352_is__singletonI,axiom,
! [X2: nat] : ( is_singleton_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_353_remove__def,axiom,
( remove_real
= ( ^ [X4: real,A3: set_real] : ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ).
% remove_def
thf(fact_354_remove__def,axiom,
( remove_nat
= ( ^ [X4: nat,A3: set_nat] : ( minus_minus_set_nat @ A3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_355_member__remove,axiom,
! [X2: real,Y4: real,A2: set_real] :
( ( member_real @ X2 @ ( remove_real @ Y4 @ A2 ) )
= ( ( member_real @ X2 @ A2 )
& ( X2 != Y4 ) ) ) ).
% member_remove
thf(fact_356_is__singletonI_H,axiom,
! [A2: set_real] :
( ( A2 != bot_bot_set_real )
=> ( ! [X: real,Y2: real] :
( ( member_real @ X @ A2 )
=> ( ( member_real @ Y2 @ A2 )
=> ( X = Y2 ) ) )
=> ( is_singleton_real @ A2 ) ) ) ).
% is_singletonI'
thf(fact_357_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X: nat,Y2: nat] :
( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y2 @ A2 )
=> ( X = Y2 ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_358_is__singletonE,axiom,
! [A2: set_real] :
( ( is_singleton_real @ A2 )
=> ~ ! [X: real] :
( A2
!= ( insert_real @ X @ bot_bot_set_real ) ) ) ).
% is_singletonE
thf(fact_359_is__singletonE,axiom,
! [A2: set_nat] :
( ( is_singleton_nat @ A2 )
=> ~ ! [X: nat] :
( A2
!= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_360_is__singleton__def,axiom,
( is_singleton_real
= ( ^ [A3: set_real] :
? [X4: real] :
( A3
= ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ).
% is_singleton_def
thf(fact_361_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X4: nat] :
( A3
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_362_integral__null,axiom,
! [A: real,B: real,F: real > real] :
( ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( topolo7804196973972690552x_real @ A @ B ) )
= zero_zero_real )
=> ( ( hensto2714581292692559302l_real @ ( topolo7804196973972690552x_real @ A @ B ) @ F )
= zero_zero_real ) ) ).
% integral_null
thf(fact_363_measure__lborel__Ioc,axiom,
! [L: real,U: real] :
( ( ord_less_eq_real @ L @ U )
=> ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( set_or2392270231875598684t_real @ L @ U ) )
= ( minus_minus_real @ U @ L ) ) ) ).
% measure_lborel_Ioc
thf(fact_364_measure__lborel__Ioo,axiom,
! [L: real,U: real] :
( ( ord_less_eq_real @ L @ U )
=> ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( set_or1633881224788618240n_real @ L @ U ) )
= ( minus_minus_real @ U @ L ) ) ) ).
% measure_lborel_Ioo
thf(fact_365_content__eq__0__interior,axiom,
! [A: real,B: real] :
( ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( topolo7804196973972690552x_real @ A @ B ) )
= zero_zero_real )
= ( ( elemen1149380513509575748r_real @ ( topolo7804196973972690552x_real @ A @ B ) )
= bot_bot_set_real ) ) ).
% content_eq_0_interior
thf(fact_366_measure__lborel__Ico,axiom,
! [L: real,U: real] :
( ( ord_less_eq_real @ L @ U )
=> ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( set_or66887138388493659n_real @ L @ U ) )
= ( minus_minus_real @ U @ L ) ) ) ).
% measure_lborel_Ico
thf(fact_367_sum__content_Obox__empty__imp,axiom,
! [A: real,B: real] :
( ( ( topolo8288580659802485013x_real @ A @ B )
= bot_bot_set_real )
=> ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( topolo7804196973972690552x_real @ A @ B ) )
= zero_zero_real ) ) ).
% sum_content.box_empty_imp
thf(fact_368_Gcd__factorial__eq__0__iff,axiom,
! [A2: set_nat] :
( ( ( factor8539158941071730396al_nat @ A2 )
= zero_zero_nat )
= ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% Gcd_factorial_eq_0_iff
thf(fact_369_box__real_I1_J,axiom,
topolo8288580659802485013x_real = set_or1633881224788618240n_real ).
% box_real(1)
thf(fact_370_box__idem,axiom,
! [A: real] :
( ( topolo8288580659802485013x_real @ A @ A )
= bot_bot_set_real ) ).
% box_idem
thf(fact_371_integral__unique,axiom,
! [F: real > real,Y4: real,K: set_real] :
( ( hensto240673015341029504l_real @ F @ Y4 @ K )
=> ( ( hensto2714581292692559302l_real @ K @ F )
= Y4 ) ) ).
% integral_unique
thf(fact_372_ivl__subset,axiom,
! [I: real,J: real,M3: real,N: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ I @ J ) @ ( set_or66887138388493659n_real @ M3 @ N ) )
= ( ( ord_less_eq_real @ J @ I )
| ( ( ord_less_eq_real @ M3 @ I )
& ( ord_less_eq_real @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_373_ivl__subset,axiom,
! [I: nat,J: nat,M3: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M3 @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M3 @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_374_atLeastLessThan__empty,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( set_or3540276404033026485et_nat @ A @ B )
= bot_bot_set_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_375_atLeastLessThan__empty,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( set_or66887138388493659n_real @ A @ B )
= bot_bot_set_real ) ) ).
% atLeastLessThan_empty
thf(fact_376_atLeastLessThan__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( set_or4665077453230672383an_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_377_Henstock__Kurzweil__Integration_Ointegral__empty,axiom,
! [F: real > real] :
( ( hensto2714581292692559302l_real @ bot_bot_set_real @ F )
= zero_zero_real ) ).
% Henstock_Kurzweil_Integration.integral_empty
thf(fact_378_greaterThanLessThan__empty,axiom,
! [L: set_nat,K: set_nat] :
( ( ord_less_eq_set_nat @ L @ K )
=> ( ( set_or8625682525731655386et_nat @ K @ L )
= bot_bot_set_set_nat ) ) ).
% greaterThanLessThan_empty
thf(fact_379_greaterThanLessThan__empty,axiom,
! [L: real,K: real] :
( ( ord_less_eq_real @ L @ K )
=> ( ( set_or1633881224788618240n_real @ K @ L )
= bot_bot_set_real ) ) ).
% greaterThanLessThan_empty
thf(fact_380_greaterThanLessThan__empty,axiom,
! [L: nat,K: nat] :
( ( ord_less_eq_nat @ L @ K )
=> ( ( set_or5834768355832116004an_nat @ K @ L )
= bot_bot_set_nat ) ) ).
% greaterThanLessThan_empty
thf(fact_381_greaterThanLessThan__empty__iff,axiom,
! [A: real,B: real] :
( ( ( set_or1633881224788618240n_real @ A @ B )
= bot_bot_set_real )
= ( ord_less_eq_real @ B @ A ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_382_greaterThanLessThan__empty__iff2,axiom,
! [A: real,B: real] :
( ( bot_bot_set_real
= ( set_or1633881224788618240n_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_383_ivl__diff,axiom,
! [I: real,N: real,M3: real] :
( ( ord_less_eq_real @ I @ N )
=> ( ( minus_minus_set_real @ ( set_or66887138388493659n_real @ I @ M3 ) @ ( set_or66887138388493659n_real @ I @ N ) )
= ( set_or66887138388493659n_real @ N @ M3 ) ) ) ).
% ivl_diff
thf(fact_384_ivl__diff,axiom,
! [I: nat,N: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M3 ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M3 ) ) ) ).
% ivl_diff
thf(fact_385_integral__refl,axiom,
! [A: real,F: real > real] :
( ( hensto2714581292692559302l_real @ ( topolo7804196973972690552x_real @ A @ A ) @ F )
= zero_zero_real ) ).
% integral_refl
thf(fact_386_greaterThanAtMost__empty,axiom,
! [L: set_nat,K: set_nat] :
( ( ord_less_eq_set_nat @ L @ K )
=> ( ( set_or7074010630789208630et_nat @ K @ L )
= bot_bot_set_set_nat ) ) ).
% greaterThanAtMost_empty
thf(fact_387_greaterThanAtMost__empty,axiom,
! [L: real,K: real] :
( ( ord_less_eq_real @ L @ K )
=> ( ( set_or2392270231875598684t_real @ K @ L )
= bot_bot_set_real ) ) ).
% greaterThanAtMost_empty
thf(fact_388_greaterThanAtMost__empty,axiom,
! [L: nat,K: nat] :
( ( ord_less_eq_nat @ L @ K )
=> ( ( set_or6659071591806873216st_nat @ K @ L )
= bot_bot_set_nat ) ) ).
% greaterThanAtMost_empty
thf(fact_389_eq__box__cbox,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( topolo8288580659802485013x_real @ A @ B )
= ( topolo7804196973972690552x_real @ C @ D ) )
= ( ( ( topolo8288580659802485013x_real @ A @ B )
= bot_bot_set_real )
& ( ( topolo7804196973972690552x_real @ C @ D )
= bot_bot_set_real ) ) ) ).
% eq_box_cbox
thf(fact_390_eq__cbox__box,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( topolo7804196973972690552x_real @ A @ B )
= ( topolo8288580659802485013x_real @ C @ D ) )
= ( ( ( topolo7804196973972690552x_real @ A @ B )
= bot_bot_set_real )
& ( ( topolo8288580659802485013x_real @ C @ D )
= bot_bot_set_real ) ) ) ).
% eq_cbox_box
thf(fact_391_interior__cbox,axiom,
! [A: real,B: real] :
( ( elemen1149380513509575748r_real @ ( topolo7804196973972690552x_real @ A @ B ) )
= ( topolo8288580659802485013x_real @ A @ B ) ) ).
% interior_cbox
thf(fact_392_integral__singleton,axiom,
! [A: real,F: real > real] :
( ( hensto2714581292692559302l_real @ ( insert_real @ A @ bot_bot_set_real ) @ F )
= zero_zero_real ) ).
% integral_singleton
thf(fact_393_Henstock__Kurzweil__Integration_Ointegral__cong,axiom,
! [S2: set_real,F: real > real,G: real > real] :
( ! [X: real] :
( ( member_real @ X @ S2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( hensto2714581292692559302l_real @ S2 @ F )
= ( hensto2714581292692559302l_real @ S2 @ G ) ) ) ).
% Henstock_Kurzweil_Integration.integral_cong
thf(fact_394_integral__open__interval,axiom,
! [A: real,B: real,F: real > real] :
( ( hensto2714581292692559302l_real @ ( topolo8288580659802485013x_real @ A @ B ) @ F )
= ( hensto2714581292692559302l_real @ ( topolo7804196973972690552x_real @ A @ B ) @ F ) ) ).
% integral_open_interval
thf(fact_395_eq__box,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( topolo8288580659802485013x_real @ A @ B )
= ( topolo8288580659802485013x_real @ C @ D ) )
= ( ( ( ( topolo8288580659802485013x_real @ A @ B )
= bot_bot_set_real )
& ( ( topolo8288580659802485013x_real @ C @ D )
= bot_bot_set_real ) )
| ( ( A = C )
& ( B = D ) ) ) ) ).
% eq_box
thf(fact_396_Ioc__inj,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( set_or2392270231875598684t_real @ A @ B )
= ( set_or2392270231875598684t_real @ C @ D ) )
= ( ( ( ord_less_eq_real @ B @ A )
& ( ord_less_eq_real @ D @ C ) )
| ( ( A = C )
& ( B = D ) ) ) ) ).
% Ioc_inj
thf(fact_397_Ioc__inj,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or6659071591806873216st_nat @ A @ B )
= ( set_or6659071591806873216st_nat @ C @ D ) )
= ( ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ D @ C ) )
| ( ( A = C )
& ( B = D ) ) ) ) ).
% Ioc_inj
thf(fact_398_atLeastLessThan__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A @ B ) @ ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_eq_real @ B @ A )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_399_atLeastLessThan__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_400_box__subset__cbox,axiom,
! [A: real,B: real] : ( ord_less_eq_set_real @ ( topolo8288580659802485013x_real @ A @ B ) @ ( topolo7804196973972690552x_real @ A @ B ) ) ).
% box_subset_cbox
thf(fact_401_has__integral__open__interval,axiom,
! [F: real > real,Y4: real,A: real,B: real] :
( ( hensto240673015341029504l_real @ F @ Y4 @ ( topolo8288580659802485013x_real @ A @ B ) )
= ( hensto240673015341029504l_real @ F @ Y4 @ ( topolo7804196973972690552x_real @ A @ B ) ) ) ).
% has_integral_open_interval
thf(fact_402_has__integral__spike__interior,axiom,
! [F: real > real,Y4: real,A: real,B: real,G: real > real] :
( ( hensto240673015341029504l_real @ F @ Y4 @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ! [X: real] :
( ( member_real @ X @ ( topolo8288580659802485013x_real @ A @ B ) )
=> ( ( G @ X )
= ( F @ X ) ) )
=> ( hensto240673015341029504l_real @ G @ Y4 @ ( topolo7804196973972690552x_real @ A @ B ) ) ) ) ).
% has_integral_spike_interior
thf(fact_403_has__integral__spike__interior__eq,axiom,
! [A: real,B: real,G: real > real,F: real > real,Y4: real] :
( ! [X: real] :
( ( member_real @ X @ ( topolo8288580659802485013x_real @ A @ B ) )
=> ( ( G @ X )
= ( F @ X ) ) )
=> ( ( hensto240673015341029504l_real @ F @ Y4 @ ( topolo7804196973972690552x_real @ A @ B ) )
= ( hensto240673015341029504l_real @ G @ Y4 @ ( topolo7804196973972690552x_real @ A @ B ) ) ) ) ).
% has_integral_spike_interior_eq
thf(fact_404_Ioc__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or2392270231875598684t_real @ A @ B ) @ ( set_or2392270231875598684t_real @ C @ D ) )
= ( ( ord_less_eq_real @ B @ A )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% Ioc_subset_iff
thf(fact_405_Ioc__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or6659071591806873216st_nat @ A @ B ) @ ( set_or6659071591806873216st_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% Ioc_subset_iff
thf(fact_406_has__integral__Icc__iff__Ioo,axiom,
! [F: real > real,I2: real,A: real,B: real] :
( ( hensto240673015341029504l_real @ F @ I2 @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( hensto240673015341029504l_real @ F @ I2 @ ( set_or1633881224788618240n_real @ A @ B ) ) ) ).
% has_integral_Icc_iff_Ioo
thf(fact_407_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
( set_or66887138388493659n_real
= ( ^ [A4: real,B4: real] : ( minus_minus_set_real @ ( set_or1222579329274155063t_real @ A4 @ B4 ) @ ( insert_real @ B4 @ bot_bot_set_real ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_408_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
( set_or4665077453230672383an_nat
= ( ^ [A4: nat,B4: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A4 @ B4 ) @ ( insert_nat @ B4 @ bot_bot_set_nat ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_409_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
( set_or2392270231875598684t_real
= ( ^ [A4: real,B4: real] : ( minus_minus_set_real @ ( set_or1222579329274155063t_real @ A4 @ B4 ) @ ( insert_real @ A4 @ bot_bot_set_real ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_410_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
( set_or6659071591806873216st_nat
= ( ^ [A4: nat,B4: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A4 @ B4 ) @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_411_atLeastAtMost__diff__ends,axiom,
! [A: real,B: real] :
( ( minus_minus_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( insert_real @ A @ ( insert_real @ B @ bot_bot_set_real ) ) )
= ( set_or1633881224788618240n_real @ A @ B ) ) ).
% atLeastAtMost_diff_ends
thf(fact_412_atLeastAtMost__diff__ends,axiom,
! [A: nat,B: nat] :
( ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( set_or5834768355832116004an_nat @ A @ B ) ) ).
% atLeastAtMost_diff_ends
thf(fact_413_interior__singleton,axiom,
! [A: real] :
( ( elemen1149380513509575748r_real @ ( insert_real @ A @ bot_bot_set_real ) )
= bot_bot_set_real ) ).
% interior_singleton
thf(fact_414_interior__atLeastAtMost__real,axiom,
! [A: real,B: real] :
( ( elemen1149380513509575748r_real @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1633881224788618240n_real @ A @ B ) ) ).
% interior_atLeastAtMost_real
thf(fact_415_interior__empty,axiom,
( ( elemen7215728294084146536or_nat @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% interior_empty
thf(fact_416_interior__empty,axiom,
( ( elemen1149380513509575748r_real @ bot_bot_set_real )
= bot_bot_set_real ) ).
% interior_empty
thf(fact_417_greaterThanLessThan__eq__iff,axiom,
! [R: real,S2: real,T3: real,U: real] :
( ( ( set_or1633881224788618240n_real @ R @ S2 )
= ( set_or1633881224788618240n_real @ T3 @ U ) )
= ( ( ( ord_less_eq_real @ S2 @ R )
& ( ord_less_eq_real @ U @ T3 ) )
| ( ( R = T3 )
& ( S2 = U ) ) ) ) ).
% greaterThanLessThan_eq_iff
thf(fact_418_interior__mono,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ord_less_eq_set_nat @ ( elemen7215728294084146536or_nat @ S ) @ ( elemen7215728294084146536or_nat @ T2 ) ) ) ).
% interior_mono
thf(fact_419_interior__mono,axiom,
! [S: set_real,T2: set_real] :
( ( ord_less_eq_set_real @ S @ T2 )
=> ( ord_less_eq_set_real @ ( elemen1149380513509575748r_real @ S ) @ ( elemen1149380513509575748r_real @ T2 ) ) ) ).
% interior_mono
thf(fact_420_interior__subset,axiom,
! [S: set_nat] : ( ord_less_eq_set_nat @ ( elemen7215728294084146536or_nat @ S ) @ S ) ).
% interior_subset
thf(fact_421_interior__subset,axiom,
! [S: set_real] : ( ord_less_eq_set_real @ ( elemen1149380513509575748r_real @ S ) @ S ) ).
% interior_subset
thf(fact_422_ivl__disj__un__singleton_I5_J,axiom,
! [L: real,U: real] :
( ( ord_less_eq_real @ L @ U )
=> ( ( sup_sup_set_real @ ( insert_real @ L @ bot_bot_set_real ) @ ( set_or2392270231875598684t_real @ L @ U ) )
= ( set_or1222579329274155063t_real @ L @ U ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_423_ivl__disj__un__singleton_I5_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( insert_nat @ L @ bot_bot_set_nat ) @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_424_ivl__disj__un__singleton_I6_J,axiom,
! [L: real,U: real] :
( ( ord_less_eq_real @ L @ U )
=> ( ( sup_sup_set_real @ ( set_or66887138388493659n_real @ L @ U ) @ ( insert_real @ U @ bot_bot_set_real ) )
= ( set_or1222579329274155063t_real @ L @ U ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_425_ivl__disj__un__singleton_I6_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ U ) @ ( insert_nat @ U @ bot_bot_set_nat ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_426_has__integral__0__cbox__imp__0,axiom,
! [A: real,B: real,F: real > real,X2: real] :
( ( topolo5044208981011980120l_real @ ( topolo7804196973972690552x_real @ A @ B ) @ F )
=> ( ! [X: real] :
( ( member_real @ X @ ( topolo8288580659802485013x_real @ A @ B ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) )
=> ( ( hensto240673015341029504l_real @ F @ zero_zero_real @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ( ( topolo8288580659802485013x_real @ A @ B )
!= bot_bot_set_real )
=> ( ( member_real @ X2 @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ( F @ X2 )
= zero_zero_real ) ) ) ) ) ) ).
% has_integral_0_cbox_imp_0
thf(fact_427_content__nonneg,axiom,
! [A: real,B: real] :
~ ( ord_less_real @ ( sigma_measure_real2 @ lebesgue_lborel_real @ ( topolo7804196973972690552x_real @ A @ B ) ) @ zero_zero_real ) ).
% content_nonneg
thf(fact_428_content__lt__nz,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( sigma_measure_real2 @ lebesgue_lborel_real @ ( topolo7804196973972690552x_real @ A @ B ) ) )
= ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( topolo7804196973972690552x_real @ A @ B ) )
!= zero_zero_real ) ) ).
% content_lt_nz
thf(fact_429_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1633881224788618240n_real @ A @ B ) @ ( set_or2392270231875598684t_real @ C @ D ) )
= ( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_430_psubsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_431_sup_Oright__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ B )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_432_sup__left__idem,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y4 ) )
= ( sup_sup_set_nat @ X2 @ Y4 ) ) ).
% sup_left_idem
thf(fact_433_sup_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_434_sup__idem,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_435_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_436_Un__iff,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ ( sup_sup_set_real @ A2 @ B2 ) )
= ( ( member_real @ C @ A2 )
| ( member_real @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_437_Un__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_438_UnCI,axiom,
! [C: real,B2: set_real,A2: set_real] :
( ( ~ ( member_real @ C @ B2 )
=> ( member_real @ C @ A2 ) )
=> ( member_real @ C @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_439_UnCI,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_440_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_441_le__sup__iff,axiom,
! [X2: real,Y4: real,Z2: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ X2 @ Y4 ) @ Z2 )
= ( ( ord_less_eq_real @ X2 @ Z2 )
& ( ord_less_eq_real @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_442_le__sup__iff,axiom,
! [X2: nat,Y4: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y4 ) @ Z2 )
= ( ( ord_less_eq_nat @ X2 @ Z2 )
& ( ord_less_eq_nat @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_443_le__sup__iff,axiom,
! [X2: set_nat,Y4: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ Y4 ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X2 @ Z2 )
& ( ord_less_eq_set_nat @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_444_sup_Obounded__iff,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B @ C ) @ A )
= ( ( ord_less_eq_real @ B @ A )
& ( ord_less_eq_real @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_445_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_446_sup_Obounded__iff,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
= ( ( ord_less_eq_set_nat @ B @ A )
& ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_447_sup__bot__left,axiom,
! [X2: set_real] :
( ( sup_sup_set_real @ bot_bot_set_real @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_448_sup__bot__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_449_sup__bot__right,axiom,
! [X2: set_real] :
( ( sup_sup_set_real @ X2 @ bot_bot_set_real )
= X2 ) ).
% sup_bot_right
thf(fact_450_sup__bot__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% sup_bot_right
thf(fact_451_bot__eq__sup__iff,axiom,
! [X2: set_real,Y4: set_real] :
( ( bot_bot_set_real
= ( sup_sup_set_real @ X2 @ Y4 ) )
= ( ( X2 = bot_bot_set_real )
& ( Y4 = bot_bot_set_real ) ) ) ).
% bot_eq_sup_iff
thf(fact_452_bot__eq__sup__iff,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X2 @ Y4 ) )
= ( ( X2 = bot_bot_set_nat )
& ( Y4 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_453_sup__eq__bot__iff,axiom,
! [X2: set_real,Y4: set_real] :
( ( ( sup_sup_set_real @ X2 @ Y4 )
= bot_bot_set_real )
= ( ( X2 = bot_bot_set_real )
& ( Y4 = bot_bot_set_real ) ) ) ).
% sup_eq_bot_iff
thf(fact_454_sup__eq__bot__iff,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( ( sup_sup_set_nat @ X2 @ Y4 )
= bot_bot_set_nat )
= ( ( X2 = bot_bot_set_nat )
& ( Y4 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_455_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_real,B: set_real] :
( ( ( sup_sup_set_real @ A @ B )
= bot_bot_set_real )
= ( ( A = bot_bot_set_real )
& ( B = bot_bot_set_real ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_456_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_457_sup__bot_Oleft__neutral,axiom,
! [A: set_real] :
( ( sup_sup_set_real @ bot_bot_set_real @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_458_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_459_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_real,B: set_real] :
( ( bot_bot_set_real
= ( sup_sup_set_real @ A @ B ) )
= ( ( A = bot_bot_set_real )
& ( B = bot_bot_set_real ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_460_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_461_sup__bot_Oright__neutral,axiom,
! [A: set_real] :
( ( sup_sup_set_real @ A @ bot_bot_set_real )
= A ) ).
% sup_bot.right_neutral
thf(fact_462_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_463_Un__subset__iff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_nat @ A2 @ C2 )
& ( ord_less_eq_set_nat @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_464_Un__empty,axiom,
! [A2: set_real,B2: set_real] :
( ( ( sup_sup_set_real @ A2 @ B2 )
= bot_bot_set_real )
= ( ( A2 = bot_bot_set_real )
& ( B2 = bot_bot_set_real ) ) ) ).
% Un_empty
thf(fact_465_Un__empty,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_466_Un__insert__right,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_467_Un__insert__left,axiom,
! [A: nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B2 ) @ C2 )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_468_Un__Diff__cancel2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_nat @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_469_Un__Diff__cancel,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_470_greaterThanLessThan__iff,axiom,
! [I: real,L: real,U: real] :
( ( member_real @ I @ ( set_or1633881224788618240n_real @ L @ U ) )
= ( ( ord_less_real @ L @ I )
& ( ord_less_real @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_471_greaterThanLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_472_mem__box__real_I1_J,axiom,
! [X2: real,A: real,B: real] :
( ( member_real @ X2 @ ( topolo8288580659802485013x_real @ A @ B ) )
= ( ( ord_less_real @ A @ X2 )
& ( ord_less_real @ X2 @ B ) ) ) ).
% mem_box_real(1)
thf(fact_473_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_474_zero__less__abs__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
= ( A != zero_zero_real ) ) ).
% zero_less_abs_iff
thf(fact_475_atLeastatMost__empty,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( set_or1222579329274155063t_real @ A @ B )
= bot_bot_set_real ) ) ).
% atLeastatMost_empty
thf(fact_476_atLeastatMost__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_477_atLeastLessThan__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or3540276404033026485et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_set_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_478_atLeastLessThan__iff,axiom,
! [I: real,L: real,U: real] :
( ( member_real @ I @ ( set_or66887138388493659n_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_real @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_479_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_480_atLeastLessThan__empty__iff,axiom,
! [A: real,B: real] :
( ( ( set_or66887138388493659n_real @ A @ B )
= bot_bot_set_real )
= ( ~ ( ord_less_real @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_481_atLeastLessThan__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_482_atLeastLessThan__empty__iff2,axiom,
! [A: real,B: real] :
( ( bot_bot_set_real
= ( set_or66887138388493659n_real @ A @ B ) )
= ( ~ ( ord_less_real @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_483_atLeastLessThan__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or4665077453230672383an_nat @ A @ B ) )
= ( ~ ( ord_less_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_484_greaterThanAtMost__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or7074010630789208630et_nat @ L @ U ) )
= ( ( ord_less_set_nat @ L @ I )
& ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_485_greaterThanAtMost__iff,axiom,
! [I: real,L: real,U: real] :
( ( member_real @ I @ ( set_or2392270231875598684t_real @ L @ U ) )
= ( ( ord_less_real @ L @ I )
& ( ord_less_eq_real @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_486_greaterThanAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_487_greaterThanAtMost__empty__iff,axiom,
! [K: real,L: real] :
( ( ( set_or2392270231875598684t_real @ K @ L )
= bot_bot_set_real )
= ( ~ ( ord_less_real @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_488_greaterThanAtMost__empty__iff,axiom,
! [K: nat,L: nat] :
( ( ( set_or6659071591806873216st_nat @ K @ L )
= bot_bot_set_nat )
= ( ~ ( ord_less_nat @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_489_greaterThanAtMost__empty__iff2,axiom,
! [K: real,L: real] :
( ( bot_bot_set_real
= ( set_or2392270231875598684t_real @ K @ L ) )
= ( ~ ( ord_less_real @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_490_greaterThanAtMost__empty__iff2,axiom,
! [K: nat,L: nat] :
( ( bot_bot_set_nat
= ( set_or6659071591806873216st_nat @ K @ L ) )
= ( ~ ( ord_less_nat @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_491_integrable__integral,axiom,
! [F: real > real,I: set_real] :
( ( hensto5963834015518849588l_real @ F @ I )
=> ( hensto240673015341029504l_real @ F @ ( hensto2714581292692559302l_real @ I @ F ) @ I ) ) ).
% integrable_integral
thf(fact_492_integral__less__real,axiom,
! [A: real,B: real,F: real > real,G: real > real] :
( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ G )
=> ( ( ( set_or1633881224788618240n_real @ A @ B )
!= bot_bot_set_real )
=> ( ! [X: real] :
( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( ord_less_real @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_real @ ( hensto2714581292692559302l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F ) @ ( hensto2714581292692559302l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ G ) ) ) ) ) ) ).
% integral_less_real
thf(fact_493_sup_Ostrict__coboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_494_sup_Ostrict__coboundedI2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ ( sup_sup_real @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_495_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_496_sup_Ostrict__coboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ C @ A )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_497_sup_Ostrict__coboundedI1,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ A )
=> ( ord_less_real @ C @ ( sup_sup_real @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_498_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_499_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( A4
= ( sup_sup_set_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_500_sup_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [B4: real,A4: real] :
( ( A4
= ( sup_sup_real @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_501_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( A4
= ( sup_sup_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_502_sup_Ostrict__boundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_set_nat @ B @ A )
=> ~ ( ord_less_set_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_503_sup_Ostrict__boundedE,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( sup_sup_real @ B @ C ) @ A )
=> ~ ( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_504_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_505_sup__left__commute,axiom,
! [X2: set_nat,Y4: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y4 @ Z2 ) )
= ( sup_sup_set_nat @ Y4 @ ( sup_sup_set_nat @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_506_sup_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C ) )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_507_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( B2
= ( sup_sup_set_nat @ K @ B ) )
=> ( ( sup_sup_set_nat @ A @ B2 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_508_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( A2
= ( sup_sup_set_nat @ K @ A ) )
=> ( ( sup_sup_set_nat @ A2 @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_509_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X4 ) ) ) ).
% sup_commute
thf(fact_510_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_511_sup_Oabsorb4,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_512_sup_Oabsorb4,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( sup_sup_real @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_513_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_514_sup_Oabsorb3,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_515_sup_Oabsorb3,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( sup_sup_real @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_516_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_517_less__supI2,axiom,
! [X2: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ X2 @ B )
=> ( ord_less_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_518_less__supI2,axiom,
! [X2: real,B: real,A: real] :
( ( ord_less_real @ X2 @ B )
=> ( ord_less_real @ X2 @ ( sup_sup_real @ A @ B ) ) ) ).
% less_supI2
thf(fact_519_less__supI2,axiom,
! [X2: nat,B: nat,A: nat] :
( ( ord_less_nat @ X2 @ B )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_520_less__supI1,axiom,
! [X2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ X2 @ A )
=> ( ord_less_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_521_less__supI1,axiom,
! [X2: real,A: real,B: real] :
( ( ord_less_real @ X2 @ A )
=> ( ord_less_real @ X2 @ ( sup_sup_real @ A @ B ) ) ) ).
% less_supI1
thf(fact_522_less__supI1,axiom,
! [X2: nat,A: nat,B: nat] :
( ( ord_less_nat @ X2 @ A )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_523_sup__assoc,axiom,
! [X2: set_nat,Y4: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y4 ) @ Z2 )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y4 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_524_sup_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_525_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_526_inf__sup__aci_I6_J,axiom,
! [X2: set_nat,Y4: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y4 ) @ Z2 )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_527_inf__sup__aci_I7_J,axiom,
! [X2: set_nat,Y4: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y4 @ Z2 ) )
= ( sup_sup_set_nat @ Y4 @ ( sup_sup_set_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_528_inf__sup__aci_I8_J,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y4 ) )
= ( sup_sup_set_nat @ X2 @ Y4 ) ) ).
% inf_sup_aci(8)
thf(fact_529_Un__left__commute,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_530_Un__left__absorb,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_531_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_532_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_533_Un__assoc,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_534_ball__Un,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o] :
( ( ! [X4: nat] :
( ( member_nat @ X4 @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ X4 ) )
& ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_535_bex__Un,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o] :
( ( ? [X4: nat] :
( ( member_nat @ X4 @ ( sup_sup_set_nat @ A2 @ B2 ) )
& ( P @ X4 ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) )
| ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_536_UnI2,axiom,
! [C: real,B2: set_real,A2: set_real] :
( ( member_real @ C @ B2 )
=> ( member_real @ C @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_537_UnI2,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_538_UnI1,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ A2 )
=> ( member_real @ C @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_539_UnI1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_540_UnE,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ ( sup_sup_set_real @ A2 @ B2 ) )
=> ( ~ ( member_real @ C @ A2 )
=> ( member_real @ C @ B2 ) ) ) ).
% UnE
thf(fact_541_UnE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_542_integrable__continuous__interval,axiom,
! [A: real,B: real,F: real > real] :
( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( hensto5963834015518849588l_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% integrable_continuous_interval
thf(fact_543_order__less__imp__not__less,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ~ ( ord_less_real @ Y4 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_544_order__less__imp__not__less,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_545_order__less__imp__not__eq2,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( Y4 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_546_order__less__imp__not__eq2,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( Y4 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_547_order__less__imp__not__eq,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( X2 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_548_order__less__imp__not__eq,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( X2 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_549_linorder__less__linear,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
| ( X2 = Y4 )
| ( ord_less_real @ Y4 @ X2 ) ) ).
% linorder_less_linear
thf(fact_550_linorder__less__linear,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
| ( X2 = Y4 )
| ( ord_less_nat @ Y4 @ X2 ) ) ).
% linorder_less_linear
thf(fact_551_order__less__imp__triv,axiom,
! [X2: real,Y4: real,P: $o] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ( ord_less_real @ Y4 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_552_order__less__imp__triv,axiom,
! [X2: nat,Y4: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_553_order__less__not__sym,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ~ ( ord_less_real @ Y4 @ X2 ) ) ).
% order_less_not_sym
thf(fact_554_order__less__not__sym,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X2 ) ) ).
% order_less_not_sym
thf(fact_555_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_556_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_557_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_558_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_559_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_560_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_561_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_562_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_563_order__less__irrefl,axiom,
! [X2: real] :
~ ( ord_less_real @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_564_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_565_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_566_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_567_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_568_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_569_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_570_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_571_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_572_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_573_order__less__trans,axiom,
! [X2: real,Y4: real,Z2: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ( ord_less_real @ Y4 @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_574_order__less__trans,axiom,
! [X2: nat,Y4: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_575_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_576_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_577_linorder__neq__iff,axiom,
! [X2: real,Y4: real] :
( ( X2 != Y4 )
= ( ( ord_less_real @ X2 @ Y4 )
| ( ord_less_real @ Y4 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_578_linorder__neq__iff,axiom,
! [X2: nat,Y4: nat] :
( ( X2 != Y4 )
= ( ( ord_less_nat @ X2 @ Y4 )
| ( ord_less_nat @ Y4 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_579_order__less__asym,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ~ ( ord_less_real @ Y4 @ X2 ) ) ).
% order_less_asym
thf(fact_580_order__less__asym,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X2 ) ) ).
% order_less_asym
thf(fact_581_linorder__neqE,axiom,
! [X2: real,Y4: real] :
( ( X2 != Y4 )
=> ( ~ ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_real @ Y4 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_582_linorder__neqE,axiom,
! [X2: nat,Y4: nat] :
( ( X2 != Y4 )
=> ( ~ ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_nat @ Y4 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_583_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_584_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_585_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_586_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_587_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_588_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_589_not__less__iff__gr__or__eq,axiom,
! [X2: real,Y4: real] :
( ( ~ ( ord_less_real @ X2 @ Y4 ) )
= ( ( ord_less_real @ Y4 @ X2 )
| ( X2 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_590_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y4 ) )
= ( ( ord_less_nat @ Y4 @ X2 )
| ( X2 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_591_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_592_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_593_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A5: real,B5: real] :
( ( ord_less_real @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: real] : ( P @ A5 @ A5 )
=> ( ! [A5: real,B5: real] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_594_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_595_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ~ ( P3 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_596_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_597_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_598_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_599_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_600_linorder__cases,axiom,
! [X2: real,Y4: real] :
( ~ ( ord_less_real @ X2 @ Y4 )
=> ( ( X2 != Y4 )
=> ( ord_less_real @ Y4 @ X2 ) ) ) ).
% linorder_cases
thf(fact_601_linorder__cases,axiom,
! [X2: nat,Y4: nat] :
( ~ ( ord_less_nat @ X2 @ Y4 )
=> ( ( X2 != Y4 )
=> ( ord_less_nat @ Y4 @ X2 ) ) ) ).
% linorder_cases
thf(fact_602_antisym__conv3,axiom,
! [Y4: real,X2: real] :
( ~ ( ord_less_real @ Y4 @ X2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y4 ) )
= ( X2 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_603_antisym__conv3,axiom,
! [Y4: nat,X2: nat] :
( ~ ( ord_less_nat @ Y4 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y4 ) )
= ( X2 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_604_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_605_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_606_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_607_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_608_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_609_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_610_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_611_less__imp__neq,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( X2 != Y4 ) ) ).
% less_imp_neq
thf(fact_612_less__imp__neq,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( X2 != Y4 ) ) ).
% less_imp_neq
thf(fact_613_dense,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ? [Z3: real] :
( ( ord_less_real @ X2 @ Z3 )
& ( ord_less_real @ Z3 @ Y4 ) ) ) ).
% dense
thf(fact_614_gt__ex,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% gt_ex
thf(fact_615_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_616_lt__ex,axiom,
! [X2: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X2 ) ).
% lt_ex
thf(fact_617_linorder__neqE__linordered__idom,axiom,
! [X2: real,Y4: real] :
( ( X2 != Y4 )
=> ( ~ ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_real @ Y4 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_618_integrable__continuous,axiom,
! [A: real,B: real,F: real > real] :
( ( topolo5044208981011980120l_real @ ( topolo7804196973972690552x_real @ A @ B ) @ F )
=> ( hensto5963834015518849588l_real @ F @ ( topolo7804196973972690552x_real @ A @ B ) ) ) ).
% integrable_continuous
thf(fact_619_atLeastatMost__psubset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D )
& ( ( ord_less_set_nat @ C @ A )
| ( ord_less_set_nat @ B @ D ) ) ) )
& ( ord_less_eq_set_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_620_atLeastatMost__psubset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D )
& ( ( ord_less_real @ C @ A )
| ( ord_less_real @ B @ D ) ) ) )
& ( ord_less_eq_real @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_621_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D )
& ( ( ord_less_nat @ C @ A )
| ( ord_less_nat @ B @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_622_integral__le,axiom,
! [F: real > real,S: set_real,G: real > real] :
( ( hensto5963834015518849588l_real @ F @ S )
=> ( ( hensto5963834015518849588l_real @ G @ S )
=> ( ! [X: real] :
( ( member_real @ X @ S )
=> ( ord_less_eq_real @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_real @ ( hensto2714581292692559302l_real @ S @ F ) @ ( hensto2714581292692559302l_real @ S @ G ) ) ) ) ) ).
% integral_le
thf(fact_623_inf__sup__ord_I4_J,axiom,
! [Y4: real,X2: real] : ( ord_less_eq_real @ Y4 @ ( sup_sup_real @ X2 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_624_inf__sup__ord_I4_J,axiom,
! [Y4: nat,X2: nat] : ( ord_less_eq_nat @ Y4 @ ( sup_sup_nat @ X2 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_625_inf__sup__ord_I4_J,axiom,
! [Y4: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( sup_sup_set_nat @ X2 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_626_inf__sup__ord_I3_J,axiom,
! [X2: real,Y4: real] : ( ord_less_eq_real @ X2 @ ( sup_sup_real @ X2 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_627_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_628_inf__sup__ord_I3_J,axiom,
! [X2: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_629_le__supE,axiom,
! [A: real,B: real,X2: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ A @ B ) @ X2 )
=> ~ ( ( ord_less_eq_real @ A @ X2 )
=> ~ ( ord_less_eq_real @ B @ X2 ) ) ) ).
% le_supE
thf(fact_630_le__supE,axiom,
! [A: nat,B: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A @ X2 )
=> ~ ( ord_less_eq_nat @ B @ X2 ) ) ) ).
% le_supE
thf(fact_631_le__supE,axiom,
! [A: set_nat,B: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X2 )
=> ~ ( ( ord_less_eq_set_nat @ A @ X2 )
=> ~ ( ord_less_eq_set_nat @ B @ X2 ) ) ) ).
% le_supE
thf(fact_632_le__supI,axiom,
! [A: real,X2: real,B: real] :
( ( ord_less_eq_real @ A @ X2 )
=> ( ( ord_less_eq_real @ B @ X2 )
=> ( ord_less_eq_real @ ( sup_sup_real @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_633_le__supI,axiom,
! [A: nat,X2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X2 )
=> ( ( ord_less_eq_nat @ B @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_634_le__supI,axiom,
! [A: set_nat,X2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ X2 )
=> ( ( ord_less_eq_set_nat @ B @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_635_sup__ge1,axiom,
! [X2: real,Y4: real] : ( ord_less_eq_real @ X2 @ ( sup_sup_real @ X2 @ Y4 ) ) ).
% sup_ge1
thf(fact_636_sup__ge1,axiom,
! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y4 ) ) ).
% sup_ge1
thf(fact_637_sup__ge1,axiom,
! [X2: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y4 ) ) ).
% sup_ge1
thf(fact_638_sup__ge2,axiom,
! [Y4: real,X2: real] : ( ord_less_eq_real @ Y4 @ ( sup_sup_real @ X2 @ Y4 ) ) ).
% sup_ge2
thf(fact_639_sup__ge2,axiom,
! [Y4: nat,X2: nat] : ( ord_less_eq_nat @ Y4 @ ( sup_sup_nat @ X2 @ Y4 ) ) ).
% sup_ge2
thf(fact_640_sup__ge2,axiom,
! [Y4: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( sup_sup_set_nat @ X2 @ Y4 ) ) ).
% sup_ge2
thf(fact_641_le__supI1,axiom,
! [X2: real,A: real,B: real] :
( ( ord_less_eq_real @ X2 @ A )
=> ( ord_less_eq_real @ X2 @ ( sup_sup_real @ A @ B ) ) ) ).
% le_supI1
thf(fact_642_le__supI1,axiom,
! [X2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_643_le__supI1,axiom,
! [X2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_644_le__supI2,axiom,
! [X2: real,B: real,A: real] :
( ( ord_less_eq_real @ X2 @ B )
=> ( ord_less_eq_real @ X2 @ ( sup_sup_real @ A @ B ) ) ) ).
% le_supI2
thf(fact_645_le__supI2,axiom,
! [X2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X2 @ B )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_646_le__supI2,axiom,
! [X2: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ B )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_647_sup_Omono,axiom,
! [C: real,A: real,D: real,B: real] :
( ( ord_less_eq_real @ C @ A )
=> ( ( ord_less_eq_real @ D @ B )
=> ( ord_less_eq_real @ ( sup_sup_real @ C @ D ) @ ( sup_sup_real @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_648_sup_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_649_sup_Omono,axiom,
! [C: set_nat,A: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_650_sup__mono,axiom,
! [A: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ B @ D )
=> ( ord_less_eq_real @ ( sup_sup_real @ A @ B ) @ ( sup_sup_real @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_651_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_652_sup__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_653_sup__least,axiom,
! [Y4: real,X2: real,Z2: real] :
( ( ord_less_eq_real @ Y4 @ X2 )
=> ( ( ord_less_eq_real @ Z2 @ X2 )
=> ( ord_less_eq_real @ ( sup_sup_real @ Y4 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_654_sup__least,axiom,
! [Y4: nat,X2: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y4 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_655_sup__least,axiom,
! [Y4: set_nat,X2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X2 )
=> ( ( ord_less_eq_set_nat @ Z2 @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y4 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_656_le__iff__sup,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y3: real] :
( ( sup_sup_real @ X4 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_657_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y3: nat] :
( ( sup_sup_nat @ X4 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_658_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X4 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_659_sup_OorderE,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( A
= ( sup_sup_real @ A @ B ) ) ) ).
% sup.orderE
thf(fact_660_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_661_sup_OorderE,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( A
= ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_662_sup_OorderI,axiom,
! [A: real,B: real] :
( ( A
= ( sup_sup_real @ A @ B ) )
=> ( ord_less_eq_real @ B @ A ) ) ).
% sup.orderI
thf(fact_663_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_664_sup_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A
= ( sup_sup_set_nat @ A @ B ) )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_665_sup__unique,axiom,
! [F: real > real > real,X2: real,Y4: real] :
( ! [X: real,Y2: real] : ( ord_less_eq_real @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: real,Y2: real] : ( ord_less_eq_real @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ( ( ord_less_eq_real @ Z3 @ X )
=> ( ord_less_eq_real @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_real @ X2 @ Y4 )
= ( F @ X2 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_666_sup__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y4: nat] :
( ! [X: nat,Y2: nat] : ( ord_less_eq_nat @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ Z3 @ X )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_nat @ X2 @ Y4 )
= ( F @ X2 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_667_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X2: set_nat,Y4: set_nat] :
( ! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( ord_less_eq_set_nat @ Z3 @ X )
=> ( ord_less_eq_set_nat @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_set_nat @ X2 @ Y4 )
= ( F @ X2 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_668_sup_Oabsorb1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( sup_sup_real @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_669_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_670_sup_Oabsorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_671_sup_Oabsorb2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( sup_sup_real @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_672_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_673_sup_Oabsorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_674_sup__absorb1,axiom,
! [Y4: real,X2: real] :
( ( ord_less_eq_real @ Y4 @ X2 )
=> ( ( sup_sup_real @ X2 @ Y4 )
= X2 ) ) ).
% sup_absorb1
thf(fact_675_sup__absorb1,axiom,
! [Y4: nat,X2: nat] :
( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y4 )
= X2 ) ) ).
% sup_absorb1
thf(fact_676_sup__absorb1,axiom,
! [Y4: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X2 )
=> ( ( sup_sup_set_nat @ X2 @ Y4 )
= X2 ) ) ).
% sup_absorb1
thf(fact_677_sup__absorb2,axiom,
! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ( sup_sup_real @ X2 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_678_sup__absorb2,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( sup_sup_nat @ X2 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_679_sup__absorb2,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( sup_sup_set_nat @ X2 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_680_sup_OboundedE,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_real @ B @ A )
=> ~ ( ord_less_eq_real @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_681_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_682_sup_OboundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_nat @ B @ A )
=> ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_683_sup_OboundedI,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ A )
=> ( ord_less_eq_real @ ( sup_sup_real @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_684_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_685_sup_OboundedI,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_686_sup_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A4: real] :
( A4
= ( sup_sup_real @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_687_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_688_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( A4
= ( sup_sup_set_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_689_sup_Ocobounded1,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ A @ ( sup_sup_real @ A @ B ) ) ).
% sup.cobounded1
thf(fact_690_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_691_sup_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_692_sup_Ocobounded2,axiom,
! [B: real,A: real] : ( ord_less_eq_real @ B @ ( sup_sup_real @ A @ B ) ) ).
% sup.cobounded2
thf(fact_693_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_694_sup_Ocobounded2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_695_sup_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A4: real] :
( ( sup_sup_real @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_696_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_697_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_698_sup_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B4: real] :
( ( sup_sup_real @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_699_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_700_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_701_sup_OcoboundedI1,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ A )
=> ( ord_less_eq_real @ C @ ( sup_sup_real @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_702_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_703_sup_OcoboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_704_sup_OcoboundedI2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ ( sup_sup_real @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_705_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_706_sup_OcoboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_707_integral__eq__0__iff,axiom,
! [A: real,B: real,F: real > real] :
( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ( ord_less_real @ A @ B )
=> ( ! [X: real] :
( ( member_real @ X @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) )
=> ( ( ( hensto2714581292692559302l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
= zero_zero_real )
= ( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ( F @ X4 )
= zero_zero_real ) ) ) ) ) ) ) ).
% integral_eq_0_iff
thf(fact_708_has__integral__integrable,axiom,
! [F: real > real,I: real,S2: set_real] :
( ( hensto240673015341029504l_real @ F @ I @ S2 )
=> ( hensto5963834015518849588l_real @ F @ S2 ) ) ).
% has_integral_integrable
thf(fact_709_integrable__on__def,axiom,
( hensto5963834015518849588l_real
= ( ^ [F2: real > real,I3: set_real] :
? [Y3: real] : ( hensto240673015341029504l_real @ F2 @ Y3 @ I3 ) ) ) ).
% integrable_on_def
thf(fact_710_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_real] :
( ( sup_sup_set_real @ X2 @ bot_bot_set_real )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_711_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_712_integral__less,axiom,
! [A: real,B: real,F: real > real,G: real > real] :
( ( topolo5044208981011980120l_real @ ( topolo7804196973972690552x_real @ A @ B ) @ F )
=> ( ( topolo5044208981011980120l_real @ ( topolo7804196973972690552x_real @ A @ B ) @ G )
=> ( ( ( topolo8288580659802485013x_real @ A @ B )
!= bot_bot_set_real )
=> ( ! [X: real] :
( ( member_real @ X @ ( topolo8288580659802485013x_real @ A @ B ) )
=> ( ord_less_real @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_real @ ( hensto2714581292692559302l_real @ ( topolo7804196973972690552x_real @ A @ B ) @ F ) @ ( hensto2714581292692559302l_real @ ( topolo7804196973972690552x_real @ A @ B ) @ G ) ) ) ) ) ) ).
% integral_less
thf(fact_713_leD,axiom,
! [Y4: real,X2: real] :
( ( ord_less_eq_real @ Y4 @ X2 )
=> ~ ( ord_less_real @ X2 @ Y4 ) ) ).
% leD
thf(fact_714_leD,axiom,
! [Y4: nat,X2: nat] :
( ( ord_less_eq_nat @ Y4 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y4 ) ) ).
% leD
thf(fact_715_leD,axiom,
! [Y4: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X2 )
=> ~ ( ord_less_set_nat @ X2 @ Y4 ) ) ).
% leD
thf(fact_716_leI,axiom,
! [X2: real,Y4: real] :
( ~ ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ Y4 @ X2 ) ) ).
% leI
thf(fact_717_leI,axiom,
! [X2: nat,Y4: nat] :
( ~ ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ).
% leI
thf(fact_718_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_719_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_720_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_721_antisym__conv1,axiom,
! [X2: real,Y4: real] :
( ~ ( ord_less_real @ X2 @ Y4 )
=> ( ( ord_less_eq_real @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_722_antisym__conv1,axiom,
! [X2: nat,Y4: nat] :
( ~ ( ord_less_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_723_antisym__conv1,axiom,
! [X2: set_nat,Y4: set_nat] :
( ~ ( ord_less_set_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_724_antisym__conv2,axiom,
! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ( ~ ( ord_less_real @ X2 @ Y4 ) )
= ( X2 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_725_antisym__conv2,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y4 ) )
= ( X2 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_726_antisym__conv2,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( ~ ( ord_less_set_nat @ X2 @ Y4 ) )
= ( X2 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_727_dense__ge,axiom,
! [Z2: real,Y4: real] :
( ! [X: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ord_less_eq_real @ Y4 @ X ) )
=> ( ord_less_eq_real @ Y4 @ Z2 ) ) ).
% dense_ge
thf(fact_728_dense__le,axiom,
! [Y4: real,Z2: real] :
( ! [X: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_eq_real @ X @ Z2 ) )
=> ( ord_less_eq_real @ Y4 @ Z2 ) ) ).
% dense_le
thf(fact_729_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_730_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_731_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
& ~ ( ord_less_eq_set_nat @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_732_not__le__imp__less,axiom,
! [Y4: real,X2: real] :
( ~ ( ord_less_eq_real @ Y4 @ X2 )
=> ( ord_less_real @ X2 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_733_not__le__imp__less,axiom,
! [Y4: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y4 @ X2 )
=> ( ord_less_nat @ X2 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_734_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_735_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_736_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_737_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_738_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_739_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_740_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_741_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_742_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_743_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_744_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_745_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_746_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ~ ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_747_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_748_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_749_dense__ge__bounded,axiom,
! [Z2: real,X2: real,Y4: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X2 )
=> ( ord_less_eq_real @ Y4 @ W ) ) )
=> ( ord_less_eq_real @ Y4 @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_750_dense__le__bounded,axiom,
! [X2: real,Y4: real,Z2: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ! [W: real] :
( ( ord_less_real @ X2 @ W )
=> ( ( ord_less_real @ W @ Y4 )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y4 @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_751_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_real @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_752_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_753_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_754_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_755_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_756_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_757_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_758_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_759_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_760_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_761_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_762_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_763_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_764_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_765_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_766_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_767_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_768_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_769_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_770_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_771_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_772_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_773_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_774_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_775_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_776_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_777_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_778_linorder__not__le,axiom,
! [X2: real,Y4: real] :
( ( ~ ( ord_less_eq_real @ X2 @ Y4 ) )
= ( ord_less_real @ Y4 @ X2 ) ) ).
% linorder_not_le
thf(fact_779_linorder__not__le,axiom,
! [X2: nat,Y4: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y4 ) )
= ( ord_less_nat @ Y4 @ X2 ) ) ).
% linorder_not_le
thf(fact_780_linorder__not__less,axiom,
! [X2: real,Y4: real] :
( ( ~ ( ord_less_real @ X2 @ Y4 ) )
= ( ord_less_eq_real @ Y4 @ X2 ) ) ).
% linorder_not_less
thf(fact_781_linorder__not__less,axiom,
! [X2: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y4 ) )
= ( ord_less_eq_nat @ Y4 @ X2 ) ) ).
% linorder_not_less
thf(fact_782_order__less__imp__le,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ X2 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_783_order__less__imp__le,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ X2 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_784_order__less__imp__le,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_nat @ X2 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_785_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_786_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_787_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_788_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_789_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_790_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_791_order__le__less__trans,axiom,
! [X2: real,Y4: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ( ord_less_real @ Y4 @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_792_order__le__less__trans,axiom,
! [X2: nat,Y4: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_793_order__le__less__trans,axiom,
! [X2: set_nat,Y4: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( ord_less_set_nat @ Y4 @ Z2 )
=> ( ord_less_set_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_794_order__less__le__trans,axiom,
! [X2: real,Y4: real,Z2: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_795_order__less__le__trans,axiom,
! [X2: nat,Y4: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_796_order__less__le__trans,axiom,
! [X2: set_nat,Y4: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ Z2 )
=> ( ord_less_set_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_797_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_798_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_799_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_800_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_801_order__le__less__subst1,axiom,
! [A: set_nat,F: real > set_nat,B: real,C: real] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_802_order__le__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_803_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_804_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_805_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > set_nat,C: set_nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_806_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_807_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_808_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_809_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > real,C: real] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_810_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_811_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_812_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_813_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_814_order__less__le__subst1,axiom,
! [A: set_nat,F: real > set_nat,B: real,C: real] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_815_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_816_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_817_order__less__le__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_818_order__less__le__subst1,axiom,
! [A: real,F: set_nat > real,B: set_nat,C: set_nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_819_order__less__le__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_820_order__less__le__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_821_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_822_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_823_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_824_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_825_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > set_nat,C: set_nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_826_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_827_linorder__le__less__linear,axiom,
! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
| ( ord_less_real @ Y4 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_828_linorder__le__less__linear,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
| ( ord_less_nat @ Y4 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_829_order__le__imp__less__or__eq,axiom,
! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ( ord_less_real @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_830_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_nat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_831_order__le__imp__less__or__eq,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( ord_less_set_nat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_832_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_833_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_834_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_835_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_836_gr__implies__not__zero,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_837_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_838_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_839_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_840_subset__UnE,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ~ ! [A6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ A2 )
=> ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ B2 )
=> ( C2
!= ( sup_sup_set_nat @ A6 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_841_Un__absorb2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_842_Un__absorb1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_843_Un__upper2,axiom,
! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_844_Un__upper1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_845_Un__least,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_846_Un__mono,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_847_Un__empty__left,axiom,
! [B2: set_real] :
( ( sup_sup_set_real @ bot_bot_set_real @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_848_Un__empty__left,axiom,
! [B2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_849_Un__empty__right,axiom,
! [A2: set_real] :
( ( sup_sup_set_real @ A2 @ bot_bot_set_real )
= A2 ) ).
% Un_empty_right
thf(fact_850_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_851_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_852_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_853_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_854_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_855_bot_Oextremum__strict,axiom,
! [A: set_real] :
~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% bot.extremum_strict
thf(fact_856_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_857_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_858_bot_Onot__eq__extremum,axiom,
! [A: set_real] :
( ( A != bot_bot_set_real )
= ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% bot.not_eq_extremum
thf(fact_859_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_860_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_861_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_862_subset__psubset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_863_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_864_psubset__subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_865_psubset__imp__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_866_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_867_psubsetE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_868_not__psubset__empty,axiom,
! [A2: set_real] :
~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).
% not_psubset_empty
thf(fact_869_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_870_Un__Diff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ C2 ) @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_871_bgauge__existence__lemma,axiom,
! [S2: set_real,Q2: real > real > $o] :
( ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( Q2 @ D3 @ X4 ) ) ) )
= ( ! [X4: real] :
? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( ( member_real @ X4 @ S2 )
=> ( Q2 @ D3 @ X4 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_872_psubset__imp__ex__mem,axiom,
! [A2: set_real,B2: set_real] :
( ( ord_less_set_real @ A2 @ B2 )
=> ? [B5: real] : ( member_real @ B5 @ ( minus_minus_set_real @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_873_atLeastLessThan__eq__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_874_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_875_Ico__eq__Ico,axiom,
! [L: real,H: real,L2: real,H2: real] :
( ( ( set_or66887138388493659n_real @ L @ H )
= ( set_or66887138388493659n_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_real @ L @ H )
& ~ ( ord_less_real @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_876_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_877_atLeastLessThan__inj_I1_J,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_878_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_879_atLeastLessThan__inj_I2_J,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_880_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_881_ivl__disj__un__two__touch_I2_J,axiom,
! [L: real,M3: real,U: real] :
( ( ord_less_eq_real @ L @ M3 )
=> ( ( ord_less_real @ M3 @ U )
=> ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ M3 ) @ ( set_or66887138388493659n_real @ M3 @ U ) )
= ( set_or66887138388493659n_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_882_ivl__disj__un__two__touch_I2_J,axiom,
! [L: nat,M3: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_nat @ M3 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M3 ) @ ( set_or4665077453230672383an_nat @ M3 @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_883_ivl__disj__un__two__touch_I3_J,axiom,
! [L: real,M3: real,U: real] :
( ( ord_less_real @ L @ M3 )
=> ( ( ord_less_eq_real @ M3 @ U )
=> ( ( sup_sup_set_real @ ( set_or2392270231875598684t_real @ L @ M3 ) @ ( set_or1222579329274155063t_real @ M3 @ U ) )
= ( set_or2392270231875598684t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_884_ivl__disj__un__two__touch_I3_J,axiom,
! [L: nat,M3: nat,U: nat] :
( ( ord_less_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M3 ) @ ( set_or1269000886237332187st_nat @ M3 @ U ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_885_continuous__ge__on__Ioo,axiom,
! [C: real,D: real,G: real > real,A: real,X2: real] :
( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ C @ D ) @ G )
=> ( ! [X: real] :
( ( member_real @ X @ ( set_or1633881224788618240n_real @ C @ D ) )
=> ( ord_less_eq_real @ A @ ( G @ X ) ) )
=> ( ( ord_less_real @ C @ D )
=> ( ( member_real @ X2 @ ( set_or1222579329274155063t_real @ C @ D ) )
=> ( ord_less_eq_real @ A @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_ge_on_Ioo
thf(fact_886_ivl__disj__un__two_I1_J,axiom,
! [L: real,M3: real,U: real] :
( ( ord_less_real @ L @ M3 )
=> ( ( ord_less_eq_real @ M3 @ U )
=> ( ( sup_sup_set_real @ ( set_or1633881224788618240n_real @ L @ M3 ) @ ( set_or66887138388493659n_real @ M3 @ U ) )
= ( set_or1633881224788618240n_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_887_ivl__disj__un__two_I1_J,axiom,
! [L: nat,M3: nat,U: nat] :
( ( ord_less_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ M3 ) @ ( set_or4665077453230672383an_nat @ M3 @ U ) )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_888_ivl__disj__un__two_I2_J,axiom,
! [L: real,M3: real,U: real] :
( ( ord_less_eq_real @ L @ M3 )
=> ( ( ord_less_real @ M3 @ U )
=> ( ( sup_sup_set_real @ ( set_or2392270231875598684t_real @ L @ M3 ) @ ( set_or1633881224788618240n_real @ M3 @ U ) )
= ( set_or1633881224788618240n_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_889_ivl__disj__un__two_I2_J,axiom,
! [L: nat,M3: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_nat @ M3 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M3 ) @ ( set_or5834768355832116004an_nat @ M3 @ U ) )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_890_ivl__disj__un__two__touch_I1_J,axiom,
! [L: real,M3: real,U: real] :
( ( ord_less_real @ L @ M3 )
=> ( ( ord_less_real @ M3 @ U )
=> ( ( sup_sup_set_real @ ( set_or2392270231875598684t_real @ L @ M3 ) @ ( set_or66887138388493659n_real @ M3 @ U ) )
= ( set_or1633881224788618240n_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_891_ivl__disj__un__two__touch_I1_J,axiom,
! [L: nat,M3: nat,U: nat] :
( ( ord_less_nat @ L @ M3 )
=> ( ( ord_less_nat @ M3 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M3 ) @ ( set_or4665077453230672383an_nat @ M3 @ U ) )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_892_integrable__on__superset,axiom,
! [F: real > real,S: set_real,T3: set_real] :
( ( hensto5963834015518849588l_real @ F @ S )
=> ( ! [X: real] :
( ~ ( member_real @ X @ S )
=> ( ( F @ X )
= zero_zero_real ) )
=> ( ( ord_less_eq_set_real @ S @ T3 )
=> ( hensto5963834015518849588l_real @ F @ T3 ) ) ) ) ).
% integrable_on_superset
thf(fact_893_Henstock__Kurzweil__Integration_Ointegral__nonneg,axiom,
! [F: real > real,S: set_real] :
( ( hensto5963834015518849588l_real @ F @ S )
=> ( ! [X: real] :
( ( member_real @ X @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( hensto2714581292692559302l_real @ S @ F ) ) ) ) ).
% Henstock_Kurzweil_Integration.integral_nonneg
thf(fact_894_not__integrable__integral,axiom,
! [F: real > real,I: set_real] :
( ~ ( hensto5963834015518849588l_real @ F @ I )
=> ( ( hensto2714581292692559302l_real @ I @ F )
= zero_zero_real ) ) ).
% not_integrable_integral
thf(fact_895_has__integral__integrable__integral,axiom,
( hensto240673015341029504l_real
= ( ^ [F2: real > real,I3: real,S3: set_real] :
( ( hensto5963834015518849588l_real @ F2 @ S3 )
& ( ( hensto2714581292692559302l_real @ S3 @ F2 )
= I3 ) ) ) ) ).
% has_integral_integrable_integral
thf(fact_896_has__integral__integral,axiom,
( hensto5963834015518849588l_real
= ( ^ [F2: real > real,S3: set_real] : ( hensto240673015341029504l_real @ F2 @ ( hensto2714581292692559302l_real @ S3 @ F2 ) @ S3 ) ) ) ).
% has_integral_integral
thf(fact_897_has__integral__iff,axiom,
( hensto240673015341029504l_real
= ( ^ [F2: real > real,I3: real,S4: set_real] :
( ( hensto5963834015518849588l_real @ F2 @ S4 )
& ( ( hensto2714581292692559302l_real @ S4 @ F2 )
= I3 ) ) ) ) ).
% has_integral_iff
thf(fact_898_ivl__disj__un__two_I4_J,axiom,
! [L: real,M3: real,U: real] :
( ( ord_less_eq_real @ L @ M3 )
=> ( ( ord_less_real @ M3 @ U )
=> ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ M3 ) @ ( set_or1633881224788618240n_real @ M3 @ U ) )
= ( set_or66887138388493659n_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_899_ivl__disj__un__two_I4_J,axiom,
! [L: nat,M3: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_nat @ M3 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M3 ) @ ( set_or5834768355832116004an_nat @ M3 @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_900_ivl__disj__un__singleton_I3_J,axiom,
! [L: real,U: real] :
( ( ord_less_real @ L @ U )
=> ( ( sup_sup_set_real @ ( insert_real @ L @ bot_bot_set_real ) @ ( set_or1633881224788618240n_real @ L @ U ) )
= ( set_or66887138388493659n_real @ L @ U ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_901_ivl__disj__un__singleton_I3_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( insert_nat @ L @ bot_bot_set_nat ) @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_902_ivl__disj__un__two_I5_J,axiom,
! [L: real,M3: real,U: real] :
( ( ord_less_real @ L @ M3 )
=> ( ( ord_less_eq_real @ M3 @ U )
=> ( ( sup_sup_set_real @ ( set_or1633881224788618240n_real @ L @ M3 ) @ ( set_or1222579329274155063t_real @ M3 @ U ) )
= ( set_or2392270231875598684t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_903_ivl__disj__un__two_I5_J,axiom,
! [L: nat,M3: nat,U: nat] :
( ( ord_less_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ M3 ) @ ( set_or1269000886237332187st_nat @ M3 @ U ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_904_ivl__disj__un__singleton_I4_J,axiom,
! [L: real,U: real] :
( ( ord_less_real @ L @ U )
=> ( ( sup_sup_set_real @ ( set_or1633881224788618240n_real @ L @ U ) @ ( insert_real @ U @ bot_bot_set_real ) )
= ( set_or2392270231875598684t_real @ L @ U ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_905_ivl__disj__un__singleton_I4_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ U ) @ ( insert_nat @ U @ bot_bot_set_nat ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_906_ivl__disj__un__two__touch_I4_J,axiom,
! [L: real,M3: real,U: real] :
( ( ord_less_eq_real @ L @ M3 )
=> ( ( ord_less_eq_real @ M3 @ U )
=> ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ M3 ) @ ( set_or1222579329274155063t_real @ M3 @ U ) )
= ( set_or1222579329274155063t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_907_ivl__disj__un__two__touch_I4_J,axiom,
! [L: nat,M3: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M3 ) @ ( set_or1269000886237332187st_nat @ M3 @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_908_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_909_singleton__Un__iff,axiom,
! [X2: real,A2: set_real,B2: set_real] :
( ( ( insert_real @ X2 @ bot_bot_set_real )
= ( sup_sup_set_real @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_real )
& ( B2
= ( insert_real @ X2 @ bot_bot_set_real ) ) )
| ( ( A2
= ( insert_real @ X2 @ bot_bot_set_real ) )
& ( B2 = bot_bot_set_real ) )
| ( ( A2
= ( insert_real @ X2 @ bot_bot_set_real ) )
& ( B2
= ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_910_singleton__Un__iff,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ( ( insert_nat @ X2 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_911_Un__singleton__iff,axiom,
! [A2: set_real,B2: set_real,X2: real] :
( ( ( sup_sup_set_real @ A2 @ B2 )
= ( insert_real @ X2 @ bot_bot_set_real ) )
= ( ( ( A2 = bot_bot_set_real )
& ( B2
= ( insert_real @ X2 @ bot_bot_set_real ) ) )
| ( ( A2
= ( insert_real @ X2 @ bot_bot_set_real ) )
& ( B2 = bot_bot_set_real ) )
| ( ( A2
= ( insert_real @ X2 @ bot_bot_set_real ) )
& ( B2
= ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_912_Un__singleton__iff,axiom,
! [A2: set_nat,B2: set_nat,X2: nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_913_insert__is__Un,axiom,
( insert_real
= ( ^ [A4: real] : ( sup_sup_set_real @ ( insert_real @ A4 @ bot_bot_set_real ) ) ) ) ).
% insert_is_Un
thf(fact_914_insert__is__Un,axiom,
( insert_nat
= ( ^ [A4: nat] : ( sup_sup_set_nat @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_915_Diff__subset__conv,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C2 )
= ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_916_Diff__partition,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_917_abs__not__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% abs_not_less_zero
thf(fact_918_abs__of__pos,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_pos
thf(fact_919_ivl__disj__un__two_I3_J,axiom,
! [L: real,M3: real,U: real] :
( ( ord_less_eq_real @ L @ M3 )
=> ( ( ord_less_eq_real @ M3 @ U )
=> ( ( sup_sup_set_real @ ( set_or66887138388493659n_real @ L @ M3 ) @ ( set_or66887138388493659n_real @ M3 @ U ) )
= ( set_or66887138388493659n_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_920_ivl__disj__un__two_I3_J,axiom,
! [L: nat,M3: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M3 ) @ ( set_or4665077453230672383an_nat @ M3 @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_921_ivl__disj__un__two_I6_J,axiom,
! [L: real,M3: real,U: real] :
( ( ord_less_eq_real @ L @ M3 )
=> ( ( ord_less_eq_real @ M3 @ U )
=> ( ( sup_sup_set_real @ ( set_or2392270231875598684t_real @ L @ M3 ) @ ( set_or2392270231875598684t_real @ M3 @ U ) )
= ( set_or2392270231875598684t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_922_ivl__disj__un__two_I6_J,axiom,
! [L: nat,M3: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M3 ) @ ( set_or6659071591806873216st_nat @ M3 @ U ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_923_measure__nonneg_H,axiom,
! [M: sigma_measure_real,A2: set_real] :
~ ( ord_less_real @ ( sigma_measure_real2 @ M @ A2 ) @ zero_zero_real ) ).
% measure_nonneg'
thf(fact_924_zero__less__measure__iff,axiom,
! [M: sigma_measure_real,A2: set_real] :
( ( ord_less_real @ zero_zero_real @ ( sigma_measure_real2 @ M @ A2 ) )
= ( ( sigma_measure_real2 @ M @ A2 )
!= zero_zero_real ) ) ).
% zero_less_measure_iff
thf(fact_925_integrable__straddle,axiom,
! [S2: set_real,F: real > real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [G2: real > real,H3: real > real,I4: real] :
( ( hensto240673015341029504l_real @ G2 @ I4 @ S2 )
& ? [J2: real] :
( ( hensto240673015341029504l_real @ H3 @ J2 @ S2 )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ I4 @ J2 ) ) @ E )
& ! [X: real] :
( ( member_real @ X @ S2 )
=> ( ( ord_less_eq_real @ ( G2 @ X ) @ ( F @ X ) )
& ( ord_less_eq_real @ ( F @ X ) @ ( H3 @ X ) ) ) ) ) ) )
=> ( hensto5963834015518849588l_real @ F @ S2 ) ) ).
% integrable_straddle
thf(fact_926_integral__subset__le,axiom,
! [S2: set_real,T3: set_real,F: real > real] :
( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ( hensto5963834015518849588l_real @ F @ S2 )
=> ( ( hensto5963834015518849588l_real @ F @ T3 )
=> ( ! [X: real] :
( ( member_real @ X @ T3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) )
=> ( ord_less_eq_real @ ( hensto2714581292692559302l_real @ S2 @ F ) @ ( hensto2714581292692559302l_real @ T3 @ F ) ) ) ) ) ) ).
% integral_subset_le
thf(fact_927_eq__integralD,axiom,
! [K: set_real,F: real > real,Y4: real] :
( ( ( hensto2714581292692559302l_real @ K @ F )
= Y4 )
=> ( ( hensto240673015341029504l_real @ F @ Y4 @ K )
| ( ~ ( hensto5963834015518849588l_real @ F @ K )
& ( Y4 = zero_zero_real ) ) ) ) ).
% eq_integralD
thf(fact_928_continuous__on__ext__cont,axiom,
! [A: real,B: real,F: real > real,S: set_real] :
( ( topolo5044208981011980120l_real @ ( topolo7804196973972690552x_real @ A @ B ) @ F )
=> ( topolo5044208981011980120l_real @ S @ ( topolo6696879324188940074l_real @ F @ A @ B ) ) ) ).
% continuous_on_ext_cont
thf(fact_929_integrable__straddle__interval,axiom,
! [A: real,B: real,F: real > real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [G2: real > real,H3: real > real,I4: real] :
( ( hensto240673015341029504l_real @ G2 @ I4 @ ( topolo7804196973972690552x_real @ A @ B ) )
& ? [J2: real] :
( ( hensto240673015341029504l_real @ H3 @ J2 @ ( topolo7804196973972690552x_real @ A @ B ) )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ I4 @ J2 ) ) @ E )
& ! [X: real] :
( ( member_real @ X @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ( ord_less_eq_real @ ( G2 @ X ) @ ( F @ X ) )
& ( ord_less_eq_real @ ( F @ X ) @ ( H3 @ X ) ) ) ) ) ) )
=> ( hensto5963834015518849588l_real @ F @ ( topolo7804196973972690552x_real @ A @ B ) ) ) ).
% integrable_straddle_interval
thf(fact_930_dense__eq0__I,axiom,
! [X2: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ E ) )
=> ( X2 = zero_zero_real ) ) ).
% dense_eq0_I
thf(fact_931_ivl__disj__un__two_I7_J,axiom,
! [L: real,M3: real,U: real] :
( ( ord_less_eq_real @ L @ M3 )
=> ( ( ord_less_eq_real @ M3 @ U )
=> ( ( sup_sup_set_real @ ( set_or66887138388493659n_real @ L @ M3 ) @ ( set_or1222579329274155063t_real @ M3 @ U ) )
= ( set_or1222579329274155063t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_932_ivl__disj__un__two_I7_J,axiom,
! [L: nat,M3: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M3 ) @ ( set_or1269000886237332187st_nat @ M3 @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_933_le__left__mono,axiom,
! [X2: real,Y4: real,A: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ A )
=> ( ord_less_eq_real @ X2 @ A ) ) ) ).
% le_left_mono
thf(fact_934_le__left__mono,axiom,
! [X2: nat,Y4: nat,A: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ A )
=> ( ord_less_eq_nat @ X2 @ A ) ) ) ).
% le_left_mono
thf(fact_935_le__left__mono,axiom,
! [X2: set_nat,Y4: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ A )
=> ( ord_less_eq_set_nat @ X2 @ A ) ) ) ).
% le_left_mono
thf(fact_936_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1633881224788618240n_real @ A @ B ) @ ( set_or1633881224788618240n_real @ C @ D ) )
= ( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_937_ivl__disj__un__two_I8_J,axiom,
! [L: real,M3: real,U: real] :
( ( ord_less_eq_real @ L @ M3 )
=> ( ( ord_less_eq_real @ M3 @ U )
=> ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ M3 ) @ ( set_or2392270231875598684t_real @ M3 @ U ) )
= ( set_or1222579329274155063t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_938_ivl__disj__un__two_I8_J,axiom,
! [L: nat,M3: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M3 ) @ ( set_or6659071591806873216st_nat @ M3 @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_939_integral__cbox__eq__0__iff,axiom,
! [A: real,B: real,F: real > real] :
( ( topolo5044208981011980120l_real @ ( topolo7804196973972690552x_real @ A @ B ) @ F )
=> ( ( ( topolo8288580659802485013x_real @ A @ B )
!= bot_bot_set_real )
=> ( ! [X: real] :
( ( member_real @ X @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) )
=> ( ( ( hensto2714581292692559302l_real @ ( topolo7804196973972690552x_real @ A @ B ) @ F )
= zero_zero_real )
= ( ! [X4: real] :
( ( member_real @ X4 @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ( F @ X4 )
= zero_zero_real ) ) ) ) ) ) ) ).
% integral_cbox_eq_0_iff
thf(fact_940_psubset__insert__iff,axiom,
! [A2: set_real,X2: real,B2: set_real] :
( ( ord_less_set_real @ A2 @ ( insert_real @ X2 @ B2 ) )
= ( ( ( member_real @ X2 @ B2 )
=> ( ord_less_set_real @ A2 @ B2 ) )
& ( ~ ( member_real @ X2 @ B2 )
=> ( ( ( member_real @ X2 @ A2 )
=> ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B2 ) )
& ( ~ ( member_real @ X2 @ A2 )
=> ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_941_psubset__insert__iff,axiom,
! [A2: set_nat,X2: nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat @ X2 @ B2 ) )
= ( ( ( member_nat @ X2 @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) )
& ( ~ ( member_nat @ X2 @ B2 )
=> ( ( ( member_nat @ X2 @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_942_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or3540276404033026485et_nat @ C @ D ) )
= ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_set_nat @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_943_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or66887138388493659n_real @ C @ D ) )
= ( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_real @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_944_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_nat @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_945_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_946_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1633881224788618240n_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_947_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or2392270231875598684t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_948_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1633881224788618240n_real @ A @ B ) @ ( set_or66887138388493659n_real @ C @ D ) )
= ( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_949_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or2392270231875598684t_real @ A @ B ) @ ( set_or66887138388493659n_real @ C @ D ) )
= ( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_real @ B @ D ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_950_lemma__interval,axiom,
! [A: real,X2: real,B: real] :
( ( ord_less_real @ A @ X2 )
=> ( ( ord_less_real @ X2 @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y5 ) ) @ D4 )
=> ( ( ord_less_eq_real @ A @ Y5 )
& ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).
% lemma_interval
thf(fact_951_lemma__interval__lt,axiom,
! [A: real,X2: real,B: real] :
( ( ord_less_real @ A @ X2 )
=> ( ( ord_less_real @ X2 @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y5 ) ) @ D4 )
=> ( ( ord_less_real @ A @ Y5 )
& ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_952_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A5: real,B5: real,C4: real] :
( ( P @ A5 @ B5 )
=> ( ( P @ B5 @ C4 )
=> ( ( ord_less_eq_real @ A5 @ B5 )
=> ( ( ord_less_eq_real @ B5 @ C4 )
=> ( P @ A5 @ C4 ) ) ) ) )
=> ( ! [X: real] :
( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ X @ B )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [A5: real,B5: real] :
( ( ( ord_less_eq_real @ A5 @ X )
& ( ord_less_eq_real @ X @ B5 )
& ( ord_less_real @ ( minus_minus_real @ B5 @ A5 ) @ D5 ) )
=> ( P @ A5 @ B5 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_953_continuous__on__sing,axiom,
! [X2: real,F: real > real] : ( topolo5044208981011980120l_real @ ( insert_real @ X2 @ bot_bot_set_real ) @ F ) ).
% continuous_on_sing
thf(fact_954_IVT_H,axiom,
! [F: real > nat,A: real,Y4: nat,B: real] :
( ( ord_less_eq_nat @ ( F @ A ) @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2287203362918339196al_nat @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X: real] :
( ( ord_less_eq_real @ A @ X )
& ( ord_less_eq_real @ X @ B )
& ( ( F @ X )
= Y4 ) ) ) ) ) ) ).
% IVT'
thf(fact_955_IVT_H,axiom,
! [F: real > real,A: real,Y4: real,B: real] :
( ( ord_less_eq_real @ ( F @ A ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X: real] :
( ( ord_less_eq_real @ A @ X )
& ( ord_less_eq_real @ X @ B )
& ( ( F @ X )
= Y4 ) ) ) ) ) ) ).
% IVT'
thf(fact_956_IVT2_H,axiom,
! [F: real > nat,B: real,Y4: nat,A: real] :
( ( ord_less_eq_nat @ ( F @ B ) @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ ( F @ A ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2287203362918339196al_nat @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X: real] :
( ( ord_less_eq_real @ A @ X )
& ( ord_less_eq_real @ X @ B )
& ( ( F @ X )
= Y4 ) ) ) ) ) ) ).
% IVT2'
thf(fact_957_IVT2_H,axiom,
! [F: real > real,B: real,Y4: real,A: real] :
( ( ord_less_eq_real @ ( F @ B ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ ( F @ A ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X: real] :
( ( ord_less_eq_real @ A @ X )
& ( ord_less_eq_real @ X @ B )
& ( ( F @ X )
= Y4 ) ) ) ) ) ) ).
% IVT2'
thf(fact_958_psubsetD,axiom,
! [A2: set_real,B2: set_real,C: real] :
( ( ord_less_set_real @ A2 @ B2 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_959_continuous__on__arsinh,axiom,
! [A2: set_real] : ( topolo5044208981011980120l_real @ A2 @ arsinh_real ) ).
% continuous_on_arsinh
thf(fact_960_integrable__continuous__real,axiom,
! [A: real,B: real,F: real > real] :
( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( hensto5963834015518849588l_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% integrable_continuous_real
thf(fact_961_antiderivative__integral__continuous,axiom,
! [A: real,B: real,F: real > real] :
( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ~ ! [G3: real > real] :
~ ! [X6: real] :
( ( member_real @ X6 @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ! [Xa: real] :
( ( member_real @ Xa @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ( ord_less_eq_real @ X6 @ Xa )
=> ( hensto240673015341029504l_real @ F @ ( minus_minus_real @ ( G3 @ Xa ) @ ( G3 @ X6 ) ) @ ( set_or1222579329274155063t_real @ X6 @ Xa ) ) ) ) ) ) ).
% antiderivative_integral_continuous
thf(fact_962_continuous__on__op__minus,axiom,
! [S2: set_real,X2: real] : ( topolo5044208981011980120l_real @ S2 @ ( minus_minus_real @ X2 ) ) ).
% continuous_on_op_minus
thf(fact_963_continuous__on__subset,axiom,
! [S2: set_real,F: real > real,T3: set_real] :
( ( topolo5044208981011980120l_real @ S2 @ F )
=> ( ( ord_less_eq_set_real @ T3 @ S2 )
=> ( topolo5044208981011980120l_real @ T3 @ F ) ) ) ).
% continuous_on_subset
thf(fact_964_continuous__on__empty,axiom,
! [F: real > real] : ( topolo5044208981011980120l_real @ bot_bot_set_real @ F ) ).
% continuous_on_empty
thf(fact_965_indefinite__integral__continuous__left,axiom,
! [F: real > complex,A: real,B: real,C: real,E2: real] :
( ( hensto7658508923946432566omplex @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ( ord_less_real @ A @ C )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ~ ! [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
=> ~ ! [T4: real] :
( ( ( ord_less_real @ ( minus_minus_real @ C @ D4 ) @ T4 )
& ( ord_less_eq_real @ T4 @ C ) )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( hensto9097760349900757192omplex @ ( set_or1222579329274155063t_real @ A @ C ) @ F ) @ ( hensto9097760349900757192omplex @ ( set_or1222579329274155063t_real @ A @ T4 ) @ F ) ) ) @ E2 ) ) ) ) ) ) ) ).
% indefinite_integral_continuous_left
thf(fact_966_indefinite__integral__continuous__left,axiom,
! [F: real > real,A: real,B: real,C: real,E2: real] :
( ( hensto5963834015518849588l_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ( ord_less_real @ A @ C )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ~ ! [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
=> ~ ! [T4: real] :
( ( ( ord_less_real @ ( minus_minus_real @ C @ D4 ) @ T4 )
& ( ord_less_eq_real @ T4 @ C ) )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( hensto2714581292692559302l_real @ ( set_or1222579329274155063t_real @ A @ C ) @ F ) @ ( hensto2714581292692559302l_real @ ( set_or1222579329274155063t_real @ A @ T4 ) @ F ) ) ) @ E2 ) ) ) ) ) ) ) ).
% indefinite_integral_continuous_left
thf(fact_967_diameter__open__interval,axiom,
! [B: real,A: real] :
( ( ( ord_less_real @ B @ A )
=> ( ( elemen4332022982980038671r_real @ ( set_or1633881224788618240n_real @ A @ B ) )
= zero_zero_real ) )
& ( ~ ( ord_less_real @ B @ A )
=> ( ( elemen4332022982980038671r_real @ ( set_or1633881224788618240n_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ) ) ).
% diameter_open_interval
thf(fact_968_diameter__closed__interval,axiom,
! [B: real,A: real] :
( ( ( ord_less_real @ B @ A )
=> ( ( elemen4332022982980038671r_real @ ( set_or1222579329274155063t_real @ A @ B ) )
= zero_zero_real ) )
& ( ~ ( ord_less_real @ B @ A )
=> ( ( elemen4332022982980038671r_real @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ) ) ).
% diameter_closed_interval
thf(fact_969_norm__imp__pos__and__ge,axiom,
! [X2: real,N: real] :
( ( ( real_V7735802525324610683m_real @ X2 )
= N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X2 ) )
& ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X2 ) @ N ) ) ) ).
% norm_imp_pos_and_ge
thf(fact_970_norm__imp__pos__and__ge,axiom,
! [X2: complex,N: real] :
( ( ( real_V1022390504157884413omplex @ X2 )
= N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X2 ) )
& ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ N ) ) ) ).
% norm_imp_pos_and_ge
thf(fact_971_norm__pths_I2_J,axiom,
! [X2: real,Y4: real] :
( ( X2 != Y4 )
= ( ~ ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ zero_zero_real ) ) ) ).
% norm_pths(2)
thf(fact_972_norm__pths_I2_J,axiom,
! [X2: complex,Y4: complex] :
( ( X2 != Y4 )
= ( ~ ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y4 ) ) @ zero_zero_real ) ) ) ).
% norm_pths(2)
thf(fact_973_norm__pths_I1_J,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [X4: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y3 ) ) @ zero_zero_real ) ) ) ).
% norm_pths(1)
thf(fact_974_norm__pths_I1_J,axiom,
( ( ^ [Y: complex,Z: complex] : ( Y = Z ) )
= ( ^ [X4: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y3 ) ) @ zero_zero_real ) ) ) ).
% norm_pths(1)
thf(fact_975_Henstock__Kurzweil__Integration_Ointegral__norm__bound__integral,axiom,
! [F: real > complex,S: set_real,G: real > real] :
( ( hensto7658508923946432566omplex @ F @ S )
=> ( ( hensto5963834015518849588l_real @ G @ S )
=> ( ! [X: real] :
( ( member_real @ X @ S )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X ) ) @ ( G @ X ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( hensto9097760349900757192omplex @ S @ F ) ) @ ( hensto2714581292692559302l_real @ S @ G ) ) ) ) ) ).
% Henstock_Kurzweil_Integration.integral_norm_bound_integral
thf(fact_976_Henstock__Kurzweil__Integration_Ointegral__norm__bound__integral,axiom,
! [F: real > real,S: set_real,G: real > real] :
( ( hensto5963834015518849588l_real @ F @ S )
=> ( ( hensto5963834015518849588l_real @ G @ S )
=> ( ! [X: real] :
( ( member_real @ X @ S )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X ) ) @ ( G @ X ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( hensto2714581292692559302l_real @ S @ F ) ) @ ( hensto2714581292692559302l_real @ S @ G ) ) ) ) ) ).
% Henstock_Kurzweil_Integration.integral_norm_bound_integral
thf(fact_977_integrable__uniform__limit,axiom,
! [A: real,B: real,F: real > real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [G2: real > real] :
( ! [X: real] :
( ( member_real @ X @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( F @ X ) @ ( G2 @ X ) ) ) @ E ) )
& ( hensto5963834015518849588l_real @ G2 @ ( topolo7804196973972690552x_real @ A @ B ) ) ) )
=> ( hensto5963834015518849588l_real @ F @ ( topolo7804196973972690552x_real @ A @ B ) ) ) ).
% integrable_uniform_limit
thf(fact_978_integrable__uniform__limit,axiom,
! [A: real,B: real,F: real > complex] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [G2: real > complex] :
( ! [X: real] :
( ( member_real @ X @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( F @ X ) @ ( G2 @ X ) ) ) @ E ) )
& ( hensto7658508923946432566omplex @ G2 @ ( topolo7804196973972690552x_real @ A @ B ) ) ) )
=> ( hensto7658508923946432566omplex @ F @ ( topolo7804196973972690552x_real @ A @ B ) ) ) ).
% integrable_uniform_limit
thf(fact_979_integrable__uniform__limit__real,axiom,
! [A: real,B: real,F: real > real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [G2: real > real] :
( ! [X: real] :
( ( member_real @ X @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( F @ X ) @ ( G2 @ X ) ) ) @ E ) )
& ( hensto5963834015518849588l_real @ G2 @ ( set_or1222579329274155063t_real @ A @ B ) ) ) )
=> ( hensto5963834015518849588l_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% integrable_uniform_limit_real
thf(fact_980_integrable__uniform__limit__real,axiom,
! [A: real,B: real,F: real > complex] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [G2: real > complex] :
( ! [X: real] :
( ( member_real @ X @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( F @ X ) @ ( G2 @ X ) ) ) @ E ) )
& ( hensto7658508923946432566omplex @ G2 @ ( set_or1222579329274155063t_real @ A @ B ) ) ) )
=> ( hensto7658508923946432566omplex @ F @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% integrable_uniform_limit_real
thf(fact_981_diameter__singleton,axiom,
! [X2: real] :
( ( elemen4332022982980038671r_real @ ( insert_real @ X2 @ bot_bot_set_real ) )
= zero_zero_real ) ).
% diameter_singleton
thf(fact_982_zero__less__norm__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X2 ) )
= ( X2 != zero_zero_real ) ) ).
% zero_less_norm_iff
thf(fact_983_zero__less__norm__iff,axiom,
! [X2: complex] :
( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X2 ) )
= ( X2 != zero_zero_complex ) ) ).
% zero_less_norm_iff
thf(fact_984_norm__le__zero__iff,axiom,
! [X2: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X2 ) @ zero_zero_real )
= ( X2 = zero_zero_real ) ) ).
% norm_le_zero_iff
thf(fact_985_norm__le__zero__iff,axiom,
! [X2: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ zero_zero_real )
= ( X2 = zero_zero_complex ) ) ).
% norm_le_zero_iff
thf(fact_986_diameter__empty,axiom,
( ( elemen4332022982980038671r_real @ bot_bot_set_real )
= zero_zero_real ) ).
% diameter_empty
thf(fact_987_abs__norm__cancel,axiom,
! [A: real] :
( ( abs_abs_real @ ( real_V7735802525324610683m_real @ A ) )
= ( real_V7735802525324610683m_real @ A ) ) ).
% abs_norm_cancel
thf(fact_988_abs__norm__cancel,axiom,
! [A: complex] :
( ( abs_abs_real @ ( real_V1022390504157884413omplex @ A ) )
= ( real_V1022390504157884413omplex @ A ) ) ).
% abs_norm_cancel
thf(fact_989_norm__eq__zero,axiom,
! [X2: real] :
( ( ( real_V7735802525324610683m_real @ X2 )
= zero_zero_real )
= ( X2 = zero_zero_real ) ) ).
% norm_eq_zero
thf(fact_990_norm__eq__zero,axiom,
! [X2: complex] :
( ( ( real_V1022390504157884413omplex @ X2 )
= zero_zero_real )
= ( X2 = zero_zero_complex ) ) ).
% norm_eq_zero
thf(fact_991_norm__zero,axiom,
( ( real_V7735802525324610683m_real @ zero_zero_real )
= zero_zero_real ) ).
% norm_zero
thf(fact_992_norm__zero,axiom,
( ( real_V1022390504157884413omplex @ zero_zero_complex )
= zero_zero_real ) ).
% norm_zero
thf(fact_993_real__norm__def,axiom,
real_V7735802525324610683m_real = abs_abs_real ).
% real_norm_def
thf(fact_994_norm__minus__commute,axiom,
! [A: real,B: real] :
( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) )
= ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ A ) ) ) ).
% norm_minus_commute
thf(fact_995_norm__minus__commute,axiom,
! [A: complex,B: complex] :
( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) )
= ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ A ) ) ) ).
% norm_minus_commute
thf(fact_996_norm__ge__zero,axiom,
! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X2 ) ) ).
% norm_ge_zero
thf(fact_997_norm__ge__zero,axiom,
! [X2: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% norm_ge_zero
thf(fact_998_norm__not__less__zero,axiom,
! [X2: real] :
~ ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ zero_zero_real ) ).
% norm_not_less_zero
thf(fact_999_norm__not__less__zero,axiom,
! [X2: complex] :
~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ zero_zero_real ) ).
% norm_not_less_zero
thf(fact_1000_norm__triangle__ineq2,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% norm_triangle_ineq2
thf(fact_1001_norm__triangle__ineq2,axiom,
! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% norm_triangle_ineq2
thf(fact_1002_norm__triangle__ineq3,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% norm_triangle_ineq3
thf(fact_1003_norm__triangle__ineq3,axiom,
! [A: complex,B: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% norm_triangle_ineq3
thf(fact_1004_diameter__le,axiom,
! [S: set_real,D: real] :
( ( ( S != bot_bot_set_real )
| ( ord_less_eq_real @ zero_zero_real @ D ) )
=> ( ! [X: real,Y2: real] :
( ( member_real @ X @ S )
=> ( ( member_real @ Y2 @ S )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y2 ) ) @ D ) ) )
=> ( ord_less_eq_real @ ( elemen4332022982980038671r_real @ S ) @ D ) ) ) ).
% diameter_le
thf(fact_1005_diameter__le,axiom,
! [S: set_complex,D: real] :
( ( ( S != bot_bot_set_complex )
| ( ord_less_eq_real @ zero_zero_real @ D ) )
=> ( ! [X: complex,Y2: complex] :
( ( member_complex @ X @ S )
=> ( ( member_complex @ Y2 @ S )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y2 ) ) @ D ) ) )
=> ( ord_less_eq_real @ ( elemen7580115971934582161omplex @ S ) @ D ) ) ) ).
% diameter_le
thf(fact_1006_continuous__on__real__range,axiom,
( topolo5044208981011980120l_real
= ( ^ [S3: set_real,F2: real > real] :
! [X4: real] :
( ( member_real @ X4 @ S3 )
=> ! [E3: real] :
( ( ord_less_real @ zero_zero_real @ E3 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [Y3: real] :
( ( member_real @ Y3 @ S3 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ X4 ) ) @ D3 )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( F2 @ Y3 ) @ ( F2 @ X4 ) ) ) @ E3 ) ) ) ) ) ) ) ) ).
% continuous_on_real_range
thf(fact_1007_continuous__on__real__range,axiom,
( topolo8674095878704923098x_real
= ( ^ [S3: set_complex,F2: complex > real] :
! [X4: complex] :
( ( member_complex @ X4 @ S3 )
=> ! [E3: real] :
( ( ord_less_real @ zero_zero_real @ E3 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [Y3: complex] :
( ( member_complex @ Y3 @ S3 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ X4 ) ) @ D3 )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( F2 @ Y3 ) @ ( F2 @ X4 ) ) ) @ E3 ) ) ) ) ) ) ) ) ).
% continuous_on_real_range
thf(fact_1008_indefinite__integral__continuous,axiom,
! [F: real > complex,A: real,B: real,C: real,D: real,Epsilon: real] :
( ( hensto7658508923946432566omplex @ F @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ( member_real @ C @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ( member_real @ D @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ Epsilon )
=> ~ ! [Delta: real] :
( ( ord_less_real @ zero_zero_real @ Delta )
=> ~ ! [C5: real] :
( ( member_real @ C5 @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ! [D6: real] :
( ( member_real @ D6 @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ C5 @ C ) ) @ Delta )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ D6 @ D ) ) @ Delta )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( hensto9097760349900757192omplex @ ( topolo7804196973972690552x_real @ C5 @ D6 ) @ F ) @ ( hensto9097760349900757192omplex @ ( topolo7804196973972690552x_real @ C @ D ) @ F ) ) ) @ Epsilon ) ) ) ) ) ) ) ) ) ) ).
% indefinite_integral_continuous
thf(fact_1009_indefinite__integral__continuous,axiom,
! [F: complex > real,A: complex,B: complex,C: complex,D: complex,Epsilon: real] :
( ( hensto7712097424450753206x_real @ F @ ( topolo4166636743940337274omplex @ A @ B ) )
=> ( ( member_complex @ C @ ( topolo4166636743940337274omplex @ A @ B ) )
=> ( ( member_complex @ D @ ( topolo4166636743940337274omplex @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ Epsilon )
=> ~ ! [Delta: real] :
( ( ord_less_real @ zero_zero_real @ Delta )
=> ~ ! [C5: complex] :
( ( member_complex @ C5 @ ( topolo4166636743940337274omplex @ A @ B ) )
=> ! [D6: complex] :
( ( member_complex @ D6 @ ( topolo4166636743940337274omplex @ A @ B ) )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ C5 @ C ) ) @ Delta )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ D6 @ D ) ) @ Delta )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( hensto9151348850405077832x_real @ ( topolo4166636743940337274omplex @ C5 @ D6 ) @ F ) @ ( hensto9151348850405077832x_real @ ( topolo4166636743940337274omplex @ C @ D ) @ F ) ) ) @ Epsilon ) ) ) ) ) ) ) ) ) ) ).
% indefinite_integral_continuous
thf(fact_1010_indefinite__integral__continuous,axiom,
! [F: complex > complex,A: complex,B: complex,C: complex,D: complex,Epsilon: real] :
( ( hensto5734242992300535224omplex @ F @ ( topolo4166636743940337274omplex @ A @ B ) )
=> ( ( member_complex @ C @ ( topolo4166636743940337274omplex @ A @ B ) )
=> ( ( member_complex @ D @ ( topolo4166636743940337274omplex @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ Epsilon )
=> ~ ! [Delta: real] :
( ( ord_less_real @ zero_zero_real @ Delta )
=> ~ ! [C5: complex] :
( ( member_complex @ C5 @ ( topolo4166636743940337274omplex @ A @ B ) )
=> ! [D6: complex] :
( ( member_complex @ D6 @ ( topolo4166636743940337274omplex @ A @ B ) )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ C5 @ C ) ) @ Delta )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ D6 @ D ) ) @ Delta )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( hensto5696219843187970890omplex @ ( topolo4166636743940337274omplex @ C5 @ D6 ) @ F ) @ ( hensto5696219843187970890omplex @ ( topolo4166636743940337274omplex @ C @ D ) @ F ) ) ) @ Epsilon ) ) ) ) ) ) ) ) ) ) ).
% indefinite_integral_continuous
thf(fact_1011_indefinite__integral__continuous,axiom,
! [F: real > real,A: real,B: real,C: real,D: real,Epsilon: real] :
( ( hensto5963834015518849588l_real @ F @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ( member_real @ C @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ( member_real @ D @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ Epsilon )
=> ~ ! [Delta: real] :
( ( ord_less_real @ zero_zero_real @ Delta )
=> ~ ! [C5: real] :
( ( member_real @ C5 @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ! [D6: real] :
( ( member_real @ D6 @ ( topolo7804196973972690552x_real @ A @ B ) )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ C5 @ C ) ) @ Delta )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ D6 @ D ) ) @ Delta )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( hensto2714581292692559302l_real @ ( topolo7804196973972690552x_real @ C5 @ D6 ) @ F ) @ ( hensto2714581292692559302l_real @ ( topolo7804196973972690552x_real @ C @ D ) @ F ) ) ) @ Epsilon ) ) ) ) ) ) ) ) ) ) ).
% indefinite_integral_continuous
thf(fact_1012_vector__choose__size,axiom,
! [C: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ~ ! [X: real] :
( ( real_V7735802525324610683m_real @ X )
!= C ) ) ).
% vector_choose_size
thf(fact_1013_vector__choose__size,axiom,
! [C: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ~ ! [X: complex] :
( ( real_V1022390504157884413omplex @ X )
!= C ) ) ).
% vector_choose_size
thf(fact_1014_indefinite__integral__continuous__right,axiom,
! [F: real > complex,A: real,B: real,C: real,E2: real] :
( ( hensto7658508923946432566omplex @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_real @ C @ B )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ~ ! [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
=> ~ ! [T4: real] :
( ( ( ord_less_eq_real @ C @ T4 )
& ( ord_less_real @ T4 @ ( plus_plus_real @ C @ D4 ) ) )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( hensto9097760349900757192omplex @ ( set_or1222579329274155063t_real @ A @ C ) @ F ) @ ( hensto9097760349900757192omplex @ ( set_or1222579329274155063t_real @ A @ T4 ) @ F ) ) ) @ E2 ) ) ) ) ) ) ) ).
% indefinite_integral_continuous_right
thf(fact_1015_indefinite__integral__continuous__right,axiom,
! [F: real > real,A: real,B: real,C: real,E2: real] :
( ( hensto5963834015518849588l_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_real @ C @ B )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ~ ! [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
=> ~ ! [T4: real] :
( ( ( ord_less_eq_real @ C @ T4 )
& ( ord_less_real @ T4 @ ( plus_plus_real @ C @ D4 ) ) )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( hensto2714581292692559302l_real @ ( set_or1222579329274155063t_real @ A @ C ) @ F ) @ ( hensto2714581292692559302l_real @ ( set_or1222579329274155063t_real @ A @ T4 ) @ F ) ) ) @ E2 ) ) ) ) ) ) ) ).
% indefinite_integral_continuous_right
thf(fact_1016_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1017_add__right__cancel,axiom,
! [B: complex,A: complex,C: complex] :
( ( ( plus_plus_complex @ B @ A )
= ( plus_plus_complex @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1018_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1019_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1020_add__left__cancel,axiom,
! [A: complex,B: complex,C: complex] :
( ( ( plus_plus_complex @ A @ B )
= ( plus_plus_complex @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1021_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1022_set__plus__intro,axiom,
! [A: real,C2: set_real,B: real,D2: set_real] :
( ( member_real @ A @ C2 )
=> ( ( member_real @ B @ D2 )
=> ( member_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_set_real @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_1023_set__plus__intro,axiom,
! [A: complex,C2: set_complex,B: complex,D2: set_complex] :
( ( member_complex @ A @ C2 )
=> ( ( member_complex @ B @ D2 )
=> ( member_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_p7052360327008956141omplex @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_1024_set__plus__intro,axiom,
! [A: nat,C2: set_nat,B: nat,D2: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( member_nat @ B @ D2 )
=> ( member_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_set_nat @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_1025_add__le__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) )
= ( ord_less_eq_complex @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1026_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1027_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1028_add__le__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) )
= ( ord_less_eq_complex @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1029_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1030_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1031_add_Oright__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.right_neutral
thf(fact_1032_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_1033_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1034_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_1035_add__cancel__left__left,axiom,
! [B: complex,A: complex] :
( ( ( plus_plus_complex @ B @ A )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_left
thf(fact_1036_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_1037_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1038_add__cancel__left__right,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_right
thf(fact_1039_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_1040_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1041_add__cancel__right__left,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ B @ A ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_left
thf(fact_1042_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_1043_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1044_add__cancel__right__right,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ A @ B ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_right
thf(fact_1045_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_1046_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1047_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y4: nat] :
( ( ( plus_plus_nat @ X2 @ Y4 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1048_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y4: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y4 ) )
= ( ( X2 = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1049_add__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add_0
thf(fact_1050_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_1051_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1052_add__less__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) )
= ( ord_less_complex @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1053_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1054_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1055_add__less__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) )
= ( ord_less_complex @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1056_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1057_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1058_add__diff__cancel,axiom,
! [A: complex,B: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_1059_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_1060_diff__add__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ ( minus_minus_complex @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_1061_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_1062_add__diff__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) )
= ( minus_minus_complex @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1063_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1064_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1065_add__diff__cancel__left_H,axiom,
! [A: complex,B: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1066_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1067_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1068_add__diff__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) )
= ( minus_minus_complex @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1069_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1070_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1071_add__diff__cancel__right_H,axiom,
! [A: complex,B: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1072_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1073_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1074_abs__add__abs,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
= ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_add_abs
thf(fact_1075_add__le__same__cancel1,axiom,
! [B: complex,A: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ B @ A ) @ B )
= ( ord_less_eq_complex @ A @ zero_zero_complex ) ) ).
% add_le_same_cancel1
thf(fact_1076_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_1077_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1078_add__le__same__cancel2,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A @ B ) @ B )
= ( ord_less_eq_complex @ A @ zero_zero_complex ) ) ).
% add_le_same_cancel2
thf(fact_1079_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_1080_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1081_le__add__same__cancel1,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ A @ ( plus_plus_complex @ A @ B ) )
= ( ord_less_eq_complex @ zero_zero_complex @ B ) ) ).
% le_add_same_cancel1
thf(fact_1082_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_1083_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_1084_le__add__same__cancel2,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ A @ ( plus_plus_complex @ B @ A ) )
= ( ord_less_eq_complex @ zero_zero_complex @ B ) ) ).
% le_add_same_cancel2
thf(fact_1085_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_1086_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_1087_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1088_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1089_add__less__same__cancel1,axiom,
! [B: complex,A: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ B @ A ) @ B )
= ( ord_less_complex @ A @ zero_zero_complex ) ) ).
% add_less_same_cancel1
thf(fact_1090_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_1091_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1092_add__less__same__cancel2,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ A @ B ) @ B )
= ( ord_less_complex @ A @ zero_zero_complex ) ) ).
% add_less_same_cancel2
thf(fact_1093_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_1094_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1095_less__add__same__cancel1,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ A @ ( plus_plus_complex @ A @ B ) )
= ( ord_less_complex @ zero_zero_complex @ B ) ) ).
% less_add_same_cancel1
thf(fact_1096_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_1097_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_1098_less__add__same__cancel2,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ A @ ( plus_plus_complex @ B @ A ) )
= ( ord_less_complex @ zero_zero_complex @ B ) ) ).
% less_add_same_cancel2
thf(fact_1099_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_1100_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_1101_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1102_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1103_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1104_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1105_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1106_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1107_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1108_norm__diff__triangle__ineq,axiom,
! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_1109_norm__diff__triangle__ineq,axiom,
! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_1110_norm__add__leD,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% norm_add_leD
thf(fact_1111_norm__add__leD,axiom,
! [A: complex,B: complex,C: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% norm_add_leD
thf(fact_1112_norm__triangle__le,axiom,
! [X2: real,Y4: real,E2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y4 ) ) @ E2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y4 ) ) @ E2 ) ) ).
% norm_triangle_le
thf(fact_1113_norm__triangle__le,axiom,
! [X2: complex,Y4: complex,E2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y4 ) ) @ E2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y4 ) ) @ E2 ) ) ).
% norm_triangle_le
thf(fact_1114_norm__triangle__ineq,axiom,
! [X2: real,Y4: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y4 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y4 ) ) ) ).
% norm_triangle_ineq
thf(fact_1115_norm__triangle__ineq,axiom,
! [X2: complex,Y4: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y4 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y4 ) ) ) ).
% norm_triangle_ineq
thf(fact_1116_norm__triangle__mono,axiom,
! [A: real,R: real,B: real,S2: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R @ S2 ) ) ) ) ).
% norm_triangle_mono
thf(fact_1117_norm__triangle__mono,axiom,
! [A: complex,R: real,B: complex,S2: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R @ S2 ) ) ) ) ).
% norm_triangle_mono
thf(fact_1118_add__le__imp__le__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) )
=> ( ord_less_eq_complex @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1119_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1120_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1121_add__le__imp__le__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) )
=> ( ord_less_eq_complex @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1122_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1123_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1124_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C6: nat] :
( B4
= ( plus_plus_nat @ A4 @ C6 ) ) ) ) ).
% le_iff_add
thf(fact_1125_add__right__mono,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_eq_complex @ A @ B )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1126_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1127_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1128_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C4: nat] :
( B
!= ( plus_plus_nat @ A @ C4 ) ) ) ).
% less_eqE
thf(fact_1129_add__left__mono,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_eq_complex @ A @ B )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1130_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1131_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1132_add__mono,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ord_less_eq_complex @ A @ B )
=> ( ( ord_less_eq_complex @ C @ D )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1133_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1134_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1135_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_eq_complex @ I @ J )
& ( ord_less_eq_complex @ K @ L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1136_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1137_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1138_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: complex,J: complex,K: complex,L: complex] :
( ( ( I = J )
& ( ord_less_eq_complex @ K @ L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1139_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1140_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1141_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_eq_complex @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1142_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1143_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1144_pth__7_I1_J,axiom,
! [X2: complex] :
( ( plus_plus_complex @ zero_zero_complex @ X2 )
= X2 ) ).
% pth_7(1)
thf(fact_1145_pth__7_I1_J,axiom,
! [X2: real] :
( ( plus_plus_real @ zero_zero_real @ X2 )
= X2 ) ).
% pth_7(1)
thf(fact_1146_pth__d,axiom,
! [X2: complex] :
( ( plus_plus_complex @ X2 @ zero_zero_complex )
= X2 ) ).
% pth_d
thf(fact_1147_pth__d,axiom,
! [X2: real] :
( ( plus_plus_real @ X2 @ zero_zero_real )
= X2 ) ).
% pth_d
thf(fact_1148_comm__monoid__add__class_Oadd__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1149_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1150_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1151_add_Ocomm__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.comm_neutral
thf(fact_1152_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_1153_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1154_add_Ogroup__left__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1155_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1156_add__mono__thms__linordered__field_I5_J,axiom,
! [I: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_complex @ I @ J )
& ( ord_less_complex @ K @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1157_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1158_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1159_add__mono__thms__linordered__field_I2_J,axiom,
! [I: complex,J: complex,K: complex,L: complex] :
( ( ( I = J )
& ( ord_less_complex @ K @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1160_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1161_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1162_add__mono__thms__linordered__field_I1_J,axiom,
! [I: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_complex @ I @ J )
& ( K = L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1163_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1164_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1165_add__strict__mono,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ord_less_complex @ A @ B )
=> ( ( ord_less_complex @ C @ D )
=> ( ord_less_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1166_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1167_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1168_add__strict__left__mono,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_complex @ A @ B )
=> ( ord_less_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1169_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1170_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1171_add__strict__right__mono,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_complex @ A @ B )
=> ( ord_less_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1172_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1173_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1174_add__less__imp__less__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) )
=> ( ord_less_complex @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1175_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1176_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1177_add__less__imp__less__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) )
=> ( ord_less_complex @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1178_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1179_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1180_group__cancel_Osub1,axiom,
! [A2: complex,K: complex,A: complex,B: complex] :
( ( A2
= ( plus_plus_complex @ K @ A ) )
=> ( ( minus_minus_complex @ A2 @ B )
= ( plus_plus_complex @ K @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_1181_group__cancel_Osub1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_1182_sin__bound__lemma,axiom,
! [X2: real,Y4: real,U: real,V: real] :
( ( X2 = Y4 )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X2 @ U ) @ Y4 ) ) @ V ) ) ) ).
% sin_bound_lemma
thf(fact_1183_eq__diff__eq_H,axiom,
! [X2: real,Y4: real,Z2: real] :
( ( X2
= ( minus_minus_real @ Y4 @ Z2 ) )
= ( Y4
= ( plus_plus_real @ X2 @ Z2 ) ) ) ).
% eq_diff_eq'
thf(fact_1184_not__real__square__gt__zero,axiom,
! [X2: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
= ( X2 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1185_square__continuous,axiom,
! [E2: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ Y5 @ X2 ) ) @ D4 )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( times_times_real @ Y5 @ Y5 ) @ ( times_times_real @ X2 @ X2 ) ) ) @ E2 ) ) ) ) ).
% square_continuous
thf(fact_1186_complete__real,axiom,
! [S: set_real] :
( ? [X6: real] : ( member_real @ X6 @ S )
=> ( ? [Z4: real] :
! [X: real] :
( ( member_real @ X @ S )
=> ( ord_less_eq_real @ X @ Z4 ) )
=> ? [Y2: real] :
( ! [X6: real] :
( ( member_real @ X6 @ S )
=> ( ord_less_eq_real @ X6 @ Y2 ) )
& ! [Z4: real] :
( ! [X: real] :
( ( member_real @ X @ S )
=> ( ord_less_eq_real @ X @ Z4 ) )
=> ( ord_less_eq_real @ Y2 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_1187_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% less_eq_real_def
thf(fact_1188_complex__mod__triangle__ineq2,axiom,
! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% complex_mod_triangle_ineq2
thf(fact_1189_seq__mono__lemma,axiom,
! [M3: nat,D: nat > real,E2: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_real @ ( D @ N3 ) @ ( E2 @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_real @ ( E2 @ N3 ) @ ( E2 @ M3 ) ) )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
=> ( ord_less_real @ ( D @ N4 ) @ ( E2 @ M3 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1190_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M4: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1191_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_1192_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1193_diff__is__0__eq_H,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1194_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1195_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1196_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1197_mult__cancel2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M3 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M3 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1198_mult__cancel1,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ( times_times_nat @ K @ M3 )
= ( times_times_nat @ K @ N ) )
= ( ( M3 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1199_mult__0__right,axiom,
! [M3: nat] :
( ( times_times_nat @ M3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1200_mult__is__0,axiom,
! [M3: nat,N: nat] :
( ( ( times_times_nat @ M3 @ N )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1201_nat__0__less__mult__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1202_mult__less__cancel2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M3 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1203_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% finite_greaterThanLessThan
thf(fact_1204_finite__greaterThanAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% finite_greaterThanAtMost
thf(fact_1205_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_1206_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1207_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1208_mult__le__cancel2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M3 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1209_add__gr__0,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1210_add__is__0,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1211_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_1212_zero__less__diff,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M3 ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% zero_less_diff
thf(fact_1213_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1214_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1215_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1216_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_1217_diff__is__0__eq,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% diff_is_0_eq
thf(fact_1218_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1219_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1220_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1221_subset__eq__atLeast0__lessThan__finite,axiom,
! [N6: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_1222_subset__eq__atLeast0__atMost__finite,axiom,
! [N6: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N6 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_1223_Nat_Odiff__cancel,axiom,
! [K: nat,M3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1224_diff__cancel2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% diff_cancel2
thf(fact_1225_diff__add__inverse,axiom,
! [N: nat,M3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M3 ) @ N )
= M3 ) ).
% diff_add_inverse
thf(fact_1226_diff__add__inverse2,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N ) @ N )
= M3 ) ).
% diff_add_inverse2
thf(fact_1227_diff__add__0,axiom,
! [N: nat,M3: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M3 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1228_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1229_add__diff__inverse__nat,axiom,
! [M3: nat,N: nat] :
( ~ ( ord_less_nat @ M3 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M3 @ N ) )
= M3 ) ) ).
% add_diff_inverse_nat
thf(fact_1230_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1231_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1232_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1233_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1234_add__eq__self__zero,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= M3 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1235_atLeastLessThan0,axiom,
! [M3: nat] :
( ( set_or4665077453230672383an_nat @ M3 @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_1236_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1237_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1238_gr__implies__not0,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1239_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1240_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1241_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1242_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1243_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1244_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1245_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1246_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1247_diff__mult__distrib2,axiom,
! [K: nat,M3: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M3 @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1248_diff__mult__distrib,axiom,
! [M3: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M3 @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1249_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M4: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ord_less_nat @ X4 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1250_bounded__nat__set__is__finite,axiom,
! [N6: set_nat,N: nat] :
( ! [X: nat] :
( ( member_nat @ X @ N6 )
=> ( ord_less_nat @ X @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1251_diff__less,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ) ) ).
% diff_less
thf(fact_1252_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1253_diff__less__mono2,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ( ord_less_nat @ M3 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M3 ) ) ) ) ).
% diff_less_mono2
thf(fact_1254_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_1255_diffs0__imp__equal,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M3 )
= zero_zero_nat )
=> ( M3 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1256_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1257_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1258_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1259_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1260_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1261_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1262_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1263_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1264_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1265_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1266_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1267_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1268_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1269_less__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M3 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1270_eq__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M3 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M3 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1271_le__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1272_Nat_Odiff__diff__eq,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M3 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1273_diff__le__mono,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
% Conjectures (1)
thf(conj_0,conjecture,
hensto240673015341029504l_real @ f @ zero_zero_real @ ( set_or1222579329274155063t_real @ a @ b ) ).
%------------------------------------------------------------------------------