TPTP Problem File: ITP117^2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP117^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Minkowskis_Theorem problem prob_290__6248976_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Minkowskis_Theorem/prob_290__6248976_1 [Des21]
% Status : Theorem
% Rating : 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 447 ( 177 unt; 65 typ; 0 def)
% Number of atoms : 829 ( 277 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 4544 ( 28 ~; 1 |; 68 &;4161 @)
% ( 0 <=>; 286 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 166 ( 166 >; 0 *; 0 +; 0 <<)
% Number of symbols : 62 ( 60 usr; 2 con; 0-4 aty)
% Number of variables : 1135 ( 87 ^; 971 !; 22 ?;1135 :)
% ( 55 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:31:28.931
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_t_Extended__Nonnegative__Real_Oennreal,type,
extend1814228343nnreal: $tType ).
thf(ty_t_Finite__Cartesian__Product_Ovec,type,
finite_Cartesian_vec: $tType > $tType > $tType ).
thf(ty_t_Sigma__Algebra_Omeasure,type,
sigma_measure: $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_tf_n,type,
n: $tType ).
% Explicit typings (57)
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
ordere890947078_space:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Countable_Ocountable,type,
countable:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_OSup,type,
complete_Sup:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple1035589618norder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Measure_Ocompletion,type,
complete_completion:
!>[A: $tType] : ( ( sigma_measure @ A ) > ( sigma_measure @ A ) ) ).
thf(sy_c_Countable__Set_Ofrom__nat__into,type,
counta609264050t_into:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lebesgue__Measure_Olborel,type,
lebesgue_lborel:
!>[A: $tType] : ( sigma_measure @ A ) ).
thf(sy_c_Measure__Space_Onull__sets,type,
measure_null_sets:
!>[A: $tType] : ( ( sigma_measure @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Minkowskis__Theorem__Mirabelle__jpyvdnlcjd_Oof__int__vec,type,
minkow1937162798nt_vec:
!>[B: $tType,A: $tType] : ( ( finite_Cartesian_vec @ int @ B ) > ( finite_Cartesian_vec @ A @ B ) ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Series_Osums,type,
sums:
!>[A: $tType] : ( ( nat > A ) > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
thf(sy_c_Sigma__Algebra_Oemeasure,type,
sigma_emeasure:
!>[A: $tType] : ( ( sigma_measure @ A ) > ( set @ A ) > extend1814228343nnreal ) ).
thf(sy_c_Sigma__Algebra_Osets,type,
sigma_sets:
!>[A: $tType] : ( ( sigma_measure @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Sigma__Algebra_Ospace,type,
sigma_space:
!>[A: $tType] : ( ( sigma_measure @ A ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_R____,type,
r: ( finite_Cartesian_vec @ int @ n ) > ( set @ ( finite_Cartesian_vec @ real @ n ) ) ).
thf(sy_v_S,type,
s: set @ ( finite_Cartesian_vec @ real @ n ) ).
thf(sy_v_T_H____,type,
t: ( finite_Cartesian_vec @ int @ n ) > ( set @ ( finite_Cartesian_vec @ real @ n ) ) ).
thf(sy_v_T____,type,
t2: ( finite_Cartesian_vec @ int @ n ) > ( set @ ( finite_Cartesian_vec @ real @ n ) ) ).
thf(sy_v_f____,type,
f: nat > ( finite_Cartesian_vec @ int @ n ) ).
% Relevant facts (254)
thf(fact_0_emeasure__T_H,axiom,
! [A2: finite_Cartesian_vec @ int @ n] :
( ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t @ A2 ) )
= ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t2 @ A2 ) ) ) ).
% emeasure_T'
thf(fact_1__092_060open_062_092_060And_062a_O_AT_H_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
! [A2: finite_Cartesian_vec @ int @ n] : ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( t @ A2 ) @ ( sigma_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) ) ).
% \<open>\<And>a. T' a \<in> sets lebesgue\<close>
thf(fact_2__092_060open_062_092_060And_062a_O_AT_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
! [A2: finite_Cartesian_vec @ int @ n] : ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( t2 @ A2 ) @ ( sigma_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) ) ).
% \<open>\<And>a. T a \<in> sets lebesgue\<close>
thf(fact_3_calculation,axiom,
( sums @ extend1814228343nnreal
@ ^ [N: nat] : ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t2 @ ( f @ N ) ) )
@ ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ s ) ) ).
% calculation
thf(fact_4_assms_I1_J,axiom,
member @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ s @ ( sigma_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) ).
% assms(1)
thf(fact_5_emeasure__T__Int,axiom,
! [A2: finite_Cartesian_vec @ int @ n,B2: finite_Cartesian_vec @ int @ n] :
( ( A2 != B2 )
=> ( ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( inf_inf @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( t2 @ A2 ) @ ( t2 @ B2 ) ) )
= ( zero_zero @ extend1814228343nnreal ) ) ) ).
% emeasure_T_Int
thf(fact_6_T__Int,axiom,
! [A2: finite_Cartesian_vec @ int @ n,B2: finite_Cartesian_vec @ int @ n] :
( ( A2 != B2 )
=> ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( inf_inf @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( t2 @ A2 ) @ ( t2 @ B2 ) ) @ ( measure_null_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) ) ) ).
% T_Int
thf(fact_7_assms_I2_J,axiom,
ord_less @ extend1814228343nnreal @ ( one_one @ extend1814228343nnreal ) @ ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ s ) ).
% assms(2)
thf(fact_8__092_060open_062_092_060And_062a_O_AR_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
! [A2: finite_Cartesian_vec @ int @ n] : ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( r @ A2 ) @ ( sigma_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) ) ).
% \<open>\<And>a. R a \<in> sets lebesgue\<close>
thf(fact_9__092_060open_062_I_092_060lambda_062n_O_Aemeasure_Alebesgue_A_IT_A_If_An_J_J_J_Asums_Aemeasure_Alebesgue_A_I_092_060Union_062n_O_AT_A_If_An_J_J_092_060close_062,axiom,
( sums @ extend1814228343nnreal
@ ^ [N: nat] : ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t2 @ ( f @ N ) ) )
@ ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) )
@ ( complete_Sup_Sup @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
@ ( image @ nat @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
@ ^ [N: nat] : ( t2 @ ( f @ N ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% \<open>(\<lambda>n. emeasure lebesgue (T (f n))) sums emeasure lebesgue (\<Union>n. T (f n))\<close>
thf(fact_10_T_H__def,axiom,
( t
= ( ^ [A3: finite_Cartesian_vec @ int @ n] :
( image @ ( finite_Cartesian_vec @ real @ n ) @ ( finite_Cartesian_vec @ real @ n )
@ ^ [X: finite_Cartesian_vec @ real @ n] : ( minus_minus @ ( finite_Cartesian_vec @ real @ n ) @ X @ ( minkow1937162798nt_vec @ n @ real @ A3 ) )
@ ( t2 @ A3 ) ) ) ) ).
% T'_def
thf(fact_11_f__def,axiom,
( f
= ( counta609264050t_into @ ( finite_Cartesian_vec @ int @ n ) @ ( top_top @ ( set @ ( finite_Cartesian_vec @ int @ n ) ) ) ) ) ).
% f_def
thf(fact_12_T_H__altdef,axiom,
! [A2: finite_Cartesian_vec @ int @ n] :
( ( t @ A2 )
= ( vimage @ ( finite_Cartesian_vec @ real @ n ) @ ( finite_Cartesian_vec @ real @ n )
@ ^ [X: finite_Cartesian_vec @ real @ n] : ( plus_plus @ ( finite_Cartesian_vec @ real @ n ) @ X @ ( minkow1937162798nt_vec @ n @ real @ A2 ) )
@ ( t2 @ A2 ) ) ) ).
% T'_altdef
thf(fact_13_of__int__vec__eq__iff,axiom,
! [A: $tType,N2: $tType] :
( ( ( finite_finite @ N2 )
& ( ring_char_0 @ A ) )
=> ! [A2: finite_Cartesian_vec @ int @ N2,B2: finite_Cartesian_vec @ int @ N2] :
( ( ( minkow1937162798nt_vec @ N2 @ A @ A2 )
= ( minkow1937162798nt_vec @ N2 @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% of_int_vec_eq_iff
thf(fact_14_T__def,axiom,
( t2
= ( ^ [A3: finite_Cartesian_vec @ int @ n] : ( inf_inf @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ s @ ( r @ A3 ) ) ) ) ).
% T_def
thf(fact_15_R__Int,axiom,
! [A2: finite_Cartesian_vec @ int @ n,B2: finite_Cartesian_vec @ int @ n] :
( ( A2 != B2 )
=> ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( inf_inf @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( r @ A2 ) @ ( r @ B2 ) ) @ ( measure_null_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) ) ) ).
% R_Int
thf(fact_16_sums__emeasure_H,axiom,
! [A: $tType,B3: nat > ( set @ A ),M: sigma_measure @ A] :
( ! [X2: nat] : ( member @ ( set @ A ) @ ( B3 @ X2 ) @ ( sigma_sets @ A @ M ) )
=> ( ! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ( sigma_emeasure @ A @ M @ ( inf_inf @ ( set @ A ) @ ( B3 @ X2 ) @ ( B3 @ Y ) ) )
= ( zero_zero @ extend1814228343nnreal ) ) )
=> ( sums @ extend1814228343nnreal
@ ^ [X: nat] : ( sigma_emeasure @ A @ M @ ( B3 @ X ) )
@ ( sigma_emeasure @ A @ M @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% sums_emeasure'
thf(fact_17_null__sets__UN,axiom,
! [I: $tType,A: $tType] :
( ( countable @ I )
=> ! [N3: I > ( set @ A ),M: sigma_measure @ A] :
( ! [I2: I] : ( member @ ( set @ A ) @ ( N3 @ I2 ) @ ( measure_null_sets @ A @ M ) )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ I @ ( set @ A ) @ N3 @ ( top_top @ ( set @ I ) ) ) ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% null_sets_UN
thf(fact_18_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple1035589618norder @ A )
=> ! [F: B > A,A4: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A4 ) )
= ( top_top @ A ) )
= ( ! [X: A] :
( ( ord_less @ A @ X @ ( top_top @ A ) )
=> ? [Y2: B] :
( ( member @ B @ Y2 @ A4 )
& ( ord_less @ A @ X @ ( F @ Y2 ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_19_null__setsI,axiom,
! [A: $tType,M: sigma_measure @ A,A4: set @ A] :
( ( ( sigma_emeasure @ A @ M @ A4 )
= ( zero_zero @ extend1814228343nnreal ) )
=> ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) )
=> ( member @ ( set @ A ) @ A4 @ ( measure_null_sets @ A @ M ) ) ) ) ).
% null_setsI
thf(fact_20_image__vimage__eq,axiom,
! [A: $tType,B: $tType,F: B > A,A4: set @ A] :
( ( image @ B @ A @ F @ ( vimage @ B @ A @ F @ A4 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% image_vimage_eq
thf(fact_21__092_060open_062_092_060And_062a_O_AT_Aa_A_092_060in_062_Asets_Alebesgue_A_092_060Longrightarrow_062_A_I_092_060lambda_062x_O_Ax_A_L_Aof__int__vec_Aa_J_A_N_096_AT_Aa_A_092_060inter_062_Aspace_Alebesgue_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
! [A5: finite_Cartesian_vec @ int @ n] :
( ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( t2 @ A5 ) @ ( sigma_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) )
=> ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
@ ( inf_inf @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
@ ( vimage @ ( finite_Cartesian_vec @ real @ n ) @ ( finite_Cartesian_vec @ real @ n )
@ ^ [X: finite_Cartesian_vec @ real @ n] : ( plus_plus @ ( finite_Cartesian_vec @ real @ n ) @ X @ ( minkow1937162798nt_vec @ n @ real @ A5 ) )
@ ( t2 @ A5 ) )
@ ( sigma_space @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) )
@ ( sigma_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) ) ) ).
% \<open>\<And>a. T a \<in> sets lebesgue \<Longrightarrow> (\<lambda>x. x + of_int_vec a) -` T a \<inter> space lebesgue \<in> sets lebesgue\<close>
thf(fact_22_surj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_diff_right
thf(fact_23__092_060open_062_092_060Union_062_A_Irange_AT_J_A_061_A_I_092_060Union_062n_O_AT_A_If_An_J_J_092_060close_062,axiom,
( ( complete_Sup_Sup @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( image @ ( finite_Cartesian_vec @ int @ n ) @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ t2 @ ( top_top @ ( set @ ( finite_Cartesian_vec @ int @ n ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
@ ( image @ nat @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
@ ^ [N: nat] : ( t2 @ ( f @ N ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ).
% \<open>\<Union> (range T) = (\<Union>n. T (f n))\<close>
thf(fact_24_Sup__UNIV,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Sup_UNIV
thf(fact_25_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ! [A4: set @ A] :
( ( ( complete_Sup_Sup @ A @ A4 )
= ( top_top @ A ) )
= ( ! [X: A] :
( ( ord_less @ A @ X @ ( top_top @ A ) )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ A4 )
& ( ord_less @ A @ X @ Y2 ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_26_surj__plus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_plus
thf(fact_27_diff__numeral__special_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% diff_numeral_special(9)
thf(fact_28_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X3: B,A4: set @ B] :
( ( B2
= ( F @ X3 ) )
=> ( ( member @ B @ X3 @ A4 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_29_UNIV__I,axiom,
! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_30_Int__iff,axiom,
! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
= ( ( member @ A @ C @ A4 )
& ( member @ A @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_31_IntI,axiom,
! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C @ A4 )
=> ( ( member @ A @ C @ B3 )
=> ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% IntI
thf(fact_32_Union__iff,axiom,
! [A: $tType,A4: A,C2: set @ ( set @ A )] :
( ( member @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) )
= ( ? [X: set @ A] :
( ( member @ ( set @ A ) @ X @ C2 )
& ( member @ A @ A4 @ X ) ) ) ) ).
% Union_iff
thf(fact_33_UnionI,axiom,
! [A: $tType,X4: set @ A,C2: set @ ( set @ A ),A4: A] :
( ( member @ ( set @ A ) @ X4 @ C2 )
=> ( ( member @ A @ A4 @ X4 )
=> ( member @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) ) ) ) ).
% UnionI
thf(fact_34_UN__ball__bex__simps_I1_J,axiom,
! [A: $tType,A4: set @ ( set @ A ),P: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) )
=> ( P @ X ) ) )
= ( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A4 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ X )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_35_UN__ball__bex__simps_I3_J,axiom,
! [D: $tType,A4: set @ ( set @ D ),P: D > $o] :
( ( ? [X: D] :
( ( member @ D @ X @ ( complete_Sup_Sup @ ( set @ D ) @ A4 ) )
& ( P @ X ) ) )
= ( ? [X: set @ D] :
( ( member @ ( set @ D ) @ X @ A4 )
& ? [Y2: D] :
( ( member @ D @ Y2 @ X )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_36_null__sets_ODiff,axiom,
! [A: $tType,A2: set @ A,M: sigma_measure @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A2 @ ( measure_null_sets @ A @ M ) )
=> ( ( member @ ( set @ A ) @ B2 @ ( measure_null_sets @ A @ M ) )
=> ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% null_sets.Diff
thf(fact_37_vimage__eq,axiom,
! [A: $tType,B: $tType,A2: A,F: A > B,B3: set @ B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ B3 ) )
= ( member @ B @ ( F @ A2 ) @ B3 ) ) ).
% vimage_eq
thf(fact_38_vimageI,axiom,
! [B: $tType,A: $tType,F: B > A,A2: B,B2: A,B3: set @ A] :
( ( ( F @ A2 )
= B2 )
=> ( ( member @ A @ B2 @ B3 )
=> ( member @ B @ A2 @ ( vimage @ B @ A @ F @ B3 ) ) ) ) ).
% vimageI
thf(fact_39_image__ident,axiom,
! [A: $tType,Y3: set @ A] :
( ( image @ A @ A
@ ^ [X: A] : X
@ Y3 )
= Y3 ) ).
% image_ident
thf(fact_40_Sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A6: set @ ( A > B ),X: A] :
( complete_Sup_Sup @ B
@ ( image @ ( A > B ) @ B
@ ^ [F2: A > B] : ( F2 @ X )
@ A6 ) ) ) ) ) ).
% Sup_apply
thf(fact_41_vimage__Collect__eq,axiom,
! [B: $tType,A: $tType,F: A > B,P: B > $o] :
( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
= ( collect @ A
@ ^ [Y2: A] : ( P @ ( F @ Y2 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_42_vimage__ident,axiom,
! [A: $tType,Y3: set @ A] :
( ( vimage @ A @ A
@ ^ [X: A] : X
@ Y3 )
= Y3 ) ).
% vimage_ident
thf(fact_43_Int__UNIV,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( top_top @ ( set @ A ) ) )
= ( ( A4
= ( top_top @ ( set @ A ) ) )
& ( B3
= ( top_top @ ( set @ A ) ) ) ) ) ).
% Int_UNIV
thf(fact_44_ball__UN,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A4: set @ B,P: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A4 ) ) )
=> ( P @ X ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A4 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ ( B3 @ X ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_bex__UN,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A4: set @ B,P: A > $o] :
( ( ? [X: A] :
( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A4 ) ) )
& ( P @ X ) ) )
= ( ? [X: B] :
( ( member @ B @ X @ A4 )
& ? [Y2: A] :
( ( member @ A @ Y2 @ ( B3 @ X ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_50_UN__ball__bex__simps_I2_J,axiom,
! [C3: $tType,B: $tType,B3: B > ( set @ C3 ),A4: set @ B,P: C3 > $o] :
( ( ! [X: C3] :
( ( member @ C3 @ X @ ( complete_Sup_Sup @ ( set @ C3 ) @ ( image @ B @ ( set @ C3 ) @ B3 @ A4 ) ) )
=> ( P @ X ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A4 )
=> ! [Y2: C3] :
( ( member @ C3 @ Y2 @ ( B3 @ X ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_51_UN__ball__bex__simps_I4_J,axiom,
! [F3: $tType,E: $tType,B3: E > ( set @ F3 ),A4: set @ E,P: F3 > $o] :
( ( ? [X: F3] :
( ( member @ F3 @ X @ ( complete_Sup_Sup @ ( set @ F3 ) @ ( image @ E @ ( set @ F3 ) @ B3 @ A4 ) ) )
& ( P @ X ) ) )
= ( ? [X: E] :
( ( member @ E @ X @ A4 )
& ? [Y2: F3] :
( ( member @ F3 @ Y2 @ ( B3 @ X ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_52_vimage__UNIV,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( vimage @ A @ B @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% vimage_UNIV
thf(fact_53_null__sets_OInt,axiom,
! [A: $tType,A2: set @ A,M: sigma_measure @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A2 @ ( measure_null_sets @ A @ M ) )
=> ( ( member @ ( set @ A ) @ B2 @ ( measure_null_sets @ A @ M ) )
=> ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% null_sets.Int
thf(fact_54_vimage__Int,axiom,
! [A: $tType,B: $tType,F: A > B,A4: set @ B,B3: set @ B] :
( ( vimage @ A @ B @ F @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ A4 ) @ ( vimage @ A @ B @ F @ B3 ) ) ) ).
% vimage_Int
thf(fact_55_SUP__identity__eq,axiom,
! [A: $tType] :
( ( complete_Sup @ A )
=> ! [A4: set @ A] :
( ( complete_Sup_Sup @ A
@ ( image @ A @ A
@ ^ [X: A] : X
@ A4 ) )
= ( complete_Sup_Sup @ A @ A4 ) ) ) ).
% SUP_identity_eq
thf(fact_56_SUP__apply,axiom,
! [A: $tType,B: $tType,C3: $tType] :
( ( complete_Sup @ A )
=> ! [F: C3 > B > A,A4: set @ C3,X3: B] :
( ( complete_Sup_Sup @ ( B > A ) @ ( image @ C3 @ ( B > A ) @ F @ A4 ) @ X3 )
= ( complete_Sup_Sup @ A
@ ( image @ C3 @ A
@ ^ [Y2: C3] : ( F @ Y2 @ X3 )
@ A4 ) ) ) ) ).
% SUP_apply
thf(fact_57_UN__iff,axiom,
! [A: $tType,B: $tType,B2: A,B3: B > ( set @ A ),A4: set @ B] :
( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A4 ) ) )
= ( ? [X: B] :
( ( member @ B @ X @ A4 )
& ( member @ A @ B2 @ ( B3 @ X ) ) ) ) ) ).
% UN_iff
thf(fact_58_UN__I,axiom,
! [B: $tType,A: $tType,A2: A,A4: set @ A,B2: B,B3: A > ( set @ B )] :
( ( member @ A @ A2 @ A4 )
=> ( ( member @ B @ B2 @ ( B3 @ A2 ) )
=> ( member @ B @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A4 ) ) ) ) ) ).
% UN_I
thf(fact_59_S__decompose,axiom,
( s
= ( complete_Sup_Sup @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( image @ ( finite_Cartesian_vec @ int @ n ) @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ t2 @ ( top_top @ ( set @ ( finite_Cartesian_vec @ int @ n ) ) ) ) ) ) ).
% S_decompose
thf(fact_60_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S )
= S ) ) ).
% image_add_0
thf(fact_61_null__sets_OInt__space__eq2,axiom,
! [A: $tType,X3: set @ A,M: sigma_measure @ A] :
( ( member @ ( set @ A ) @ X3 @ ( measure_null_sets @ A @ M ) )
=> ( ( inf_inf @ ( set @ A ) @ X3 @ ( sigma_space @ A @ M ) )
= X3 ) ) ).
% null_sets.Int_space_eq2
thf(fact_62_null__sets_OInt__space__eq1,axiom,
! [A: $tType,X3: set @ A,M: sigma_measure @ A] :
( ( member @ ( set @ A ) @ X3 @ ( measure_null_sets @ A @ M ) )
=> ( ( inf_inf @ ( set @ A ) @ ( sigma_space @ A @ M ) @ X3 )
= X3 ) ) ).
% null_sets.Int_space_eq1
thf(fact_63_in__sets__SUP,axiom,
! [B: $tType,A: $tType,I3: A,I4: set @ A,M: A > ( sigma_measure @ B ),Y3: set @ B,X4: set @ B] :
( ( member @ A @ I3 @ I4 )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I4 )
=> ( ( sigma_space @ B @ ( M @ I2 ) )
= Y3 ) )
=> ( ( member @ ( set @ B ) @ X4 @ ( sigma_sets @ B @ ( M @ I3 ) ) )
=> ( member @ ( set @ B ) @ X4 @ ( sigma_sets @ B @ ( complete_Sup_Sup @ ( sigma_measure @ B ) @ ( image @ A @ ( sigma_measure @ B ) @ M @ I4 ) ) ) ) ) ) ) ).
% in_sets_SUP
thf(fact_64_in__sets__Sup,axiom,
! [A: $tType,M: set @ ( sigma_measure @ A ),X4: set @ A,M2: sigma_measure @ A,A4: set @ A] :
( ! [M3: sigma_measure @ A] :
( ( member @ ( sigma_measure @ A ) @ M3 @ M )
=> ( ( sigma_space @ A @ M3 )
= X4 ) )
=> ( ( member @ ( sigma_measure @ A ) @ M2 @ M )
=> ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M2 ) )
=> ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ ( complete_Sup_Sup @ ( sigma_measure @ A ) @ M ) ) ) ) ) ) ).
% in_sets_Sup
thf(fact_65_sets__SUP__cong,axiom,
! [B: $tType,A: $tType,I4: set @ A,M: A > ( sigma_measure @ B ),N3: A > ( sigma_measure @ B )] :
( ! [I2: A] :
( ( member @ A @ I2 @ I4 )
=> ( ( sigma_sets @ B @ ( M @ I2 ) )
= ( sigma_sets @ B @ ( N3 @ I2 ) ) ) )
=> ( ( sigma_sets @ B @ ( complete_Sup_Sup @ ( sigma_measure @ B ) @ ( image @ A @ ( sigma_measure @ B ) @ M @ I4 ) ) )
= ( sigma_sets @ B @ ( complete_Sup_Sup @ ( sigma_measure @ B ) @ ( image @ A @ ( sigma_measure @ B ) @ N3 @ I4 ) ) ) ) ) ).
% sets_SUP_cong
thf(fact_66_Sup__set__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) )
= ( ^ [A6: set @ ( set @ A )] :
( collect @ A
@ ^ [X: A] : ( complete_Sup_Sup @ $o @ ( image @ ( set @ A ) @ $o @ ( member @ A @ X ) @ A6 ) ) ) ) ) ).
% Sup_set_def
thf(fact_67_space__Sup__eq__UN,axiom,
! [A: $tType,M: set @ ( sigma_measure @ A )] :
( ( sigma_space @ A @ ( complete_Sup_Sup @ ( sigma_measure @ A ) @ M ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( sigma_measure @ A ) @ ( set @ A ) @ ( sigma_space @ A ) @ M ) ) ) ).
% space_Sup_eq_UN
thf(fact_68_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_69_Diff__Int__distrib2,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ C2 )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_70_Diff__Int__distrib,axiom,
! [A: $tType,C2: set @ A,A4: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C2 @ A4 ) @ ( inf_inf @ ( set @ A ) @ C2 @ B3 ) ) ) ).
% Diff_Int_distrib
thf(fact_71_Diff__Diff__Int,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ).
% Diff_Diff_Int
thf(fact_72_Diff__Int2,axiom,
! [A: $tType,A4: set @ A,C2: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) @ B3 ) ) ).
% Diff_Int2
thf(fact_73_Int__Diff,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ C2 )
= ( inf_inf @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B3 @ C2 ) ) ) ).
% Int_Diff
thf(fact_74_vimage__Diff,axiom,
! [A: $tType,B: $tType,F: A > B,A4: set @ B,B3: set @ B] :
( ( vimage @ A @ B @ F @ ( minus_minus @ ( set @ B ) @ A4 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( vimage @ A @ B @ F @ A4 ) @ ( vimage @ A @ B @ F @ B3 ) ) ) ).
% vimage_Diff
thf(fact_75_Sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A6: set @ ( A > B ),X: A] :
( complete_Sup_Sup @ B
@ ( image @ ( A > B ) @ B
@ ^ [F2: A > B] : ( F2 @ X )
@ A6 ) ) ) ) ) ).
% Sup_fun_def
thf(fact_76_null__sets_Osets__Collect__disj,axiom,
! [A: $tType,M: sigma_measure @ A,P: A > $o,Q: A > $o] :
( ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( P @ X ) ) )
@ ( measure_null_sets @ A @ M ) )
=> ( ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( Q @ X ) ) )
@ ( measure_null_sets @ A @ M ) )
=> ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( ( Q @ X )
| ( P @ X ) ) ) )
@ ( measure_null_sets @ A @ M ) ) ) ) ).
% null_sets.sets_Collect_disj
thf(fact_77_null__sets_Osets__Collect__conj,axiom,
! [A: $tType,M: sigma_measure @ A,P: A > $o,Q: A > $o] :
( ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( P @ X ) ) )
@ ( measure_null_sets @ A @ M ) )
=> ( ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( Q @ X ) ) )
@ ( measure_null_sets @ A @ M ) )
=> ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( Q @ X )
& ( P @ X ) ) )
@ ( measure_null_sets @ A @ M ) ) ) ) ).
% null_sets.sets_Collect_conj
thf(fact_78_translation__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S2: set @ A,T: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( minus_minus @ ( set @ A ) @ S2 @ T ) )
= ( minus_minus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S2 ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T ) ) ) ) ).
% translation_diff
thf(fact_79_null__set__Diff,axiom,
! [A: $tType,B3: set @ A,M: sigma_measure @ A,A4: set @ A] :
( ( member @ ( set @ A ) @ B3 @ ( measure_null_sets @ A @ M ) )
=> ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) )
=> ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% null_set_Diff
thf(fact_80_Int__Union2,axiom,
! [A: $tType,B3: set @ ( set @ A ),A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) @ A4 )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ ( set @ A ) @ ( set @ A )
@ ^ [C4: set @ A] : ( inf_inf @ ( set @ A ) @ C4 @ A4 )
@ B3 ) ) ) ).
% Int_Union2
thf(fact_81_Int__Union,axiom,
! [A: $tType,A4: set @ A,B3: set @ ( set @ A )] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 ) @ B3 ) ) ) ).
% Int_Union
thf(fact_82_vimage__Union,axiom,
! [A: $tType,B: $tType,F: A > B,A4: set @ ( set @ B )] :
( ( vimage @ A @ B @ F @ ( complete_Sup_Sup @ ( set @ B ) @ A4 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( vimage @ A @ B @ F ) @ A4 ) ) ) ).
% vimage_Union
thf(fact_83_UN__extend__simps_I6_J,axiom,
! [L: $tType,K: $tType,A4: K > ( set @ L ),C2: set @ K,B3: set @ L] :
( ( minus_minus @ ( set @ L ) @ ( complete_Sup_Sup @ ( set @ L ) @ ( image @ K @ ( set @ L ) @ A4 @ C2 ) ) @ B3 )
= ( complete_Sup_Sup @ ( set @ L )
@ ( image @ K @ ( set @ L )
@ ^ [X: K] : ( minus_minus @ ( set @ L ) @ ( A4 @ X ) @ B3 )
@ C2 ) ) ) ).
% UN_extend_simps(6)
thf(fact_84_emeasure__Diff__null__set,axiom,
! [A: $tType,B3: set @ A,M: sigma_measure @ A,A4: set @ A] :
( ( member @ ( set @ A ) @ B3 @ ( measure_null_sets @ A @ M ) )
=> ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) )
=> ( ( sigma_emeasure @ A @ M @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
= ( sigma_emeasure @ A @ M @ A4 ) ) ) ) ).
% emeasure_Diff_null_set
thf(fact_85_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A4: set @ B,B3: set @ B,C2: B > A,D2: B > A,Sup: ( set @ A ) > A] :
( ( A4 = B3 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C2 @ A4 ) )
= ( Sup @ ( image @ B @ A @ D2 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_86_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A4: set @ B,B3: set @ B,C2: B > A,D2: B > A,Inf: ( set @ A ) > A] :
( ( A4 = B3 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C2 @ A4 ) )
= ( Inf @ ( image @ B @ A @ D2 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_87_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X3: A,A4: set @ A,B2: B,F: A > B] :
( ( member @ A @ X3 @ A4 )
=> ( ( B2
= ( F @ X3 ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_88_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( image @ B @ A @ F @ A4 ) )
=> ( P @ X2 ) )
=> ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_89_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N3: set @ A,F: A > B,G: A > B] :
( ( M = N3 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ N3 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_90_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F @ A4 ) )
& ( P @ X5 ) )
=> ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_91_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F: B > A,A4: set @ B] :
( ( member @ A @ Z @ ( image @ B @ A @ F @ A4 ) )
= ( ? [X: B] :
( ( member @ B @ X @ A4 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_92_imageI,axiom,
! [B: $tType,A: $tType,X3: A,A4: set @ A,F: A > B] :
( ( member @ A @ X3 @ A4 )
=> ( member @ B @ ( F @ X3 ) @ ( image @ A @ B @ F @ A4 ) ) ) ).
% imageI
thf(fact_93_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% is_num_normalize(1)
thf(fact_94_UNIV__witness,axiom,
! [A: $tType] :
? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_95_UNIV__eq__I,axiom,
! [A: $tType,A4: set @ A] :
( ! [X2: A] : ( member @ A @ X2 @ A4 )
=> ( ( top_top @ ( set @ A ) )
= A4 ) ) ).
% UNIV_eq_I
thf(fact_96_Int__left__commute,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) )
= ( inf_inf @ ( set @ A ) @ B3 @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_97_Int__left__absorb,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ).
% Int_left_absorb
thf(fact_98_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B4: set @ A] : ( inf_inf @ ( set @ A ) @ B4 @ A6 ) ) ) ).
% Int_commute
thf(fact_99_Int__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Int_absorb
thf(fact_100_Int__assoc,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ C2 )
= ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_101_IntD2,axiom,
! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
=> ( member @ A @ C @ B3 ) ) ).
% IntD2
thf(fact_102_IntD1,axiom,
! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
=> ( member @ A @ C @ A4 ) ) ).
% IntD1
thf(fact_103_IntE,axiom,
! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
=> ~ ( ( member @ A @ C @ A4 )
=> ~ ( member @ A @ C @ B3 ) ) ) ).
% IntE
thf(fact_104_UnionE,axiom,
! [A: $tType,A4: A,C2: set @ ( set @ A )] :
( ( member @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) )
=> ~ ! [X6: set @ A] :
( ( member @ A @ A4 @ X6 )
=> ~ ( member @ ( set @ A ) @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_105_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: B > $o,F: A > B,Q: A > $o] :
( ! [X2: A] :
( ( P @ ( F @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
= ( collect @ A @ Q ) ) ) ).
% vimage_Collect
thf(fact_106_vimageI2,axiom,
! [B: $tType,A: $tType,F: B > A,A2: B,A4: set @ A] :
( ( member @ A @ ( F @ A2 ) @ A4 )
=> ( member @ B @ A2 @ ( vimage @ B @ A @ F @ A4 ) ) ) ).
% vimageI2
thf(fact_107_vimageE,axiom,
! [A: $tType,B: $tType,A2: A,F: A > B,B3: set @ B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ B3 ) )
=> ( member @ B @ ( F @ A2 ) @ B3 ) ) ).
% vimageE
thf(fact_108_vimageD,axiom,
! [A: $tType,B: $tType,A2: A,F: A > B,A4: set @ B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ A4 ) )
=> ( member @ B @ ( F @ A2 ) @ A4 ) ) ).
% vimageD
thf(fact_109_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A4: set @ A] :
( ( Sup
@ ( image @ A @ A
@ ^ [X: A] : X
@ A4 ) )
= ( Sup @ A4 ) ) ).
% Sup.SUP_identity_eq
thf(fact_110_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A4: set @ A] :
( ( Inf
@ ( image @ A @ A
@ ^ [X: A] : X
@ A4 ) )
= ( Inf @ A4 ) ) ).
% Inf.INF_identity_eq
thf(fact_111_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( image @ B @ A @ F @ A4 ) )
& ( P @ X ) ) )
= ( image @ B @ A @ F
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A4 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_112_image__image,axiom,
! [A: $tType,B: $tType,C3: $tType,F: B > A,G: C3 > B,A4: set @ C3] :
( ( image @ B @ A @ F @ ( image @ C3 @ B @ G @ A4 ) )
= ( image @ C3 @ A
@ ^ [X: C3] : ( F @ ( G @ X ) )
@ A4 ) ) ).
% image_image
thf(fact_113_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,A4: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ F @ A4 ) )
=> ~ ! [X2: B] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member @ B @ X2 @ A4 ) ) ) ).
% imageE
thf(fact_114_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X: A] : $true ) ) ).
% UNIV_def
thf(fact_115_Collect__conj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_116_Int__Collect,axiom,
! [A: $tType,X3: A,A4: set @ A,P: A > $o] :
( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X3 @ A4 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_117_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B4: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A6 )
& ( member @ A @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_118_vimage__def,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F2: A > B,B4: set @ B] :
( collect @ A
@ ^ [X: A] : ( member @ B @ ( F2 @ X ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_119_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_120_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_121_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X3: B] :
( ( B2
= ( F @ X3 ) )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_122_surj__def,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y2: A] :
? [X: B] :
( Y2
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_123_rangeI,axiom,
! [A: $tType,B: $tType,F: B > A,X3: B] : ( member @ A @ ( F @ X3 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_124_surjI,axiom,
! [B: $tType,A: $tType,G: B > A,F: A > B] :
( ! [X2: A] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_125_surjE,axiom,
! [A: $tType,B: $tType,F: B > A,Y4: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X2: B] :
( Y4
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_126_surjD,axiom,
! [A: $tType,B: $tType,F: B > A,Y4: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X2: B] :
( Y4
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_127_less__Sup__iff,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ! [A2: A,S: set @ A] :
( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ S ) )
= ( ? [X: A] :
( ( member @ A @ X @ S )
& ( ord_less @ A @ A2 @ X ) ) ) ) ) ).
% less_Sup_iff
thf(fact_128_SUP__cong,axiom,
! [A: $tType,B: $tType] :
( ( complete_Sup @ A )
=> ! [A4: set @ B,B3: set @ B,C2: B > A,D2: B > A] :
( ( A4 = B3 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ C2 @ A4 ) )
= ( complete_Sup_Sup @ A @ ( image @ B @ A @ D2 @ B3 ) ) ) ) ) ) ).
% SUP_cong
thf(fact_129_Int__UNIV__right,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) )
= A4 ) ).
% Int_UNIV_right
thf(fact_130_Int__UNIV__left,axiom,
! [A: $tType,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 )
= B3 ) ).
% Int_UNIV_left
thf(fact_131_Union__UNIV,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Union_UNIV
thf(fact_132_null__setsD2,axiom,
! [A: $tType,A4: set @ A,M: sigma_measure @ A] :
( ( member @ ( set @ A ) @ A4 @ ( measure_null_sets @ A @ M ) )
=> ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) ) ) ).
% null_setsD2
thf(fact_133_vimage__inter__cong,axiom,
! [B: $tType,A: $tType,S: set @ A,F: A > B,G: A > B,Y4: set @ B] :
( ! [W: A] :
( ( member @ A @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ Y4 ) @ S )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G @ Y4 ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_134_translation__subtract__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S2: set @ A,T: set @ A] :
( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ ( minus_minus @ ( set @ A ) @ S2 @ T ) )
= ( minus_minus @ ( set @ A )
@ ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ S2 )
@ ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ T ) ) ) ) ).
% translation_subtract_diff
thf(fact_135_range__composition,axiom,
! [A: $tType,C3: $tType,B: $tType,F: C3 > A,G: B > C3] :
( ( image @ B @ A
@ ^ [X: B] : ( F @ ( G @ X ) )
@ ( top_top @ ( set @ B ) ) )
= ( image @ C3 @ A @ F @ ( image @ B @ C3 @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_136_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A] :
( ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X2: B] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_137_SUP__commute,axiom,
! [A: $tType,B: $tType,C3: $tType] :
( ( comple187826305attice @ A )
=> ! [F: B > C3 > A,B3: set @ C3,A4: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I5: B] : ( complete_Sup_Sup @ A @ ( image @ C3 @ A @ ( F @ I5 ) @ B3 ) )
@ A4 ) )
= ( complete_Sup_Sup @ A
@ ( image @ C3 @ A
@ ^ [J: C3] :
( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I5: B] : ( F @ I5 @ J )
@ A4 ) )
@ B3 ) ) ) ) ).
% SUP_commute
thf(fact_138_image__Union,axiom,
! [A: $tType,B: $tType,F: B > A,S: set @ ( set @ B )] :
( ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ S ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ S ) ) ) ).
% image_Union
thf(fact_139_UN__UN__flatten,axiom,
! [A: $tType,B: $tType,C3: $tType,C2: B > ( set @ A ),B3: C3 > ( set @ B ),A4: set @ C3] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C3 @ ( set @ B ) @ B3 @ A4 ) ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ C3 @ ( set @ A )
@ ^ [Y2: C3] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C2 @ ( B3 @ Y2 ) ) )
@ A4 ) ) ) ).
% UN_UN_flatten
thf(fact_140_UN__E,axiom,
! [A: $tType,B: $tType,B2: A,B3: B > ( set @ A ),A4: set @ B] :
( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A4 ) ) )
=> ~ ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ~ ( member @ A @ B2 @ ( B3 @ X2 ) ) ) ) ).
% UN_E
thf(fact_141_UN__extend__simps_I8_J,axiom,
! [P2: $tType,O: $tType,B3: O > ( set @ P2 ),A4: set @ ( set @ O )] :
( ( complete_Sup_Sup @ ( set @ P2 )
@ ( image @ ( set @ O ) @ ( set @ P2 )
@ ^ [Y2: set @ O] : ( complete_Sup_Sup @ ( set @ P2 ) @ ( image @ O @ ( set @ P2 ) @ B3 @ Y2 ) )
@ A4 ) )
= ( complete_Sup_Sup @ ( set @ P2 ) @ ( image @ O @ ( set @ P2 ) @ B3 @ ( complete_Sup_Sup @ ( set @ O ) @ A4 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_142_UN__extend__simps_I9_J,axiom,
! [S3: $tType,R: $tType,Q2: $tType,C2: R > ( set @ S3 ),B3: Q2 > ( set @ R ),A4: set @ Q2] :
( ( complete_Sup_Sup @ ( set @ S3 )
@ ( image @ Q2 @ ( set @ S3 )
@ ^ [X: Q2] : ( complete_Sup_Sup @ ( set @ S3 ) @ ( image @ R @ ( set @ S3 ) @ C2 @ ( B3 @ X ) ) )
@ A4 ) )
= ( complete_Sup_Sup @ ( set @ S3 ) @ ( image @ R @ ( set @ S3 ) @ C2 @ ( complete_Sup_Sup @ ( set @ R ) @ ( image @ Q2 @ ( set @ R ) @ B3 @ A4 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_143_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(1)
thf(fact_144_translation__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S2: set @ A,T: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( inf_inf @ ( set @ A ) @ S2 @ T ) )
= ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S2 ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T ) ) ) ) ).
% translation_Int
thf(fact_145_surj__image__vimage__eq,axiom,
! [B: $tType,A: $tType,F: B > A,A4: set @ A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ B @ A @ F @ ( vimage @ B @ A @ F @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_146_null__set__Int2,axiom,
! [A: $tType,B3: set @ A,M: sigma_measure @ A,A4: set @ A] :
( ( member @ ( set @ A ) @ B3 @ ( measure_null_sets @ A @ M ) )
=> ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) )
=> ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B3 @ A4 ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% null_set_Int2
thf(fact_147_null__set__Int1,axiom,
! [A: $tType,B3: set @ A,M: sigma_measure @ A,A4: set @ A] :
( ( member @ ( set @ A ) @ B3 @ ( measure_null_sets @ A @ M ) )
=> ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) )
=> ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% null_set_Int1
thf(fact_148_null__setsD1,axiom,
! [A: $tType,A4: set @ A,M: sigma_measure @ A] :
( ( member @ ( set @ A ) @ A4 @ ( measure_null_sets @ A @ M ) )
=> ( ( sigma_emeasure @ A @ M @ A4 )
= ( zero_zero @ extend1814228343nnreal ) ) ) ).
% null_setsD1
thf(fact_149_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple1035589618norder @ A )
=> ! [A2: A,F: B > A,A4: set @ B] :
( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A4 ) ) )
= ( ? [X: B] :
( ( member @ B @ X @ A4 )
& ( ord_less @ A @ A2 @ ( F @ X ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_150_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple187826305attice @ A )
=> ! [F: B > A,A4: set @ B,Y4: A,I3: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A4 ) ) @ Y4 )
=> ( ( member @ B @ I3 @ A4 )
=> ( ord_less @ A @ ( F @ I3 ) @ Y4 ) ) ) ) ).
% SUP_lessD
thf(fact_151_translation__subtract__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S2: set @ A,T: set @ A] :
( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ ( inf_inf @ ( set @ A ) @ S2 @ T ) )
= ( inf_inf @ ( set @ A )
@ ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ S2 )
@ ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ T ) ) ) ) ).
% translation_subtract_Int
thf(fact_152_image__UN,axiom,
! [A: $tType,B: $tType,C3: $tType,F: B > A,B3: C3 > ( set @ B ),A4: set @ C3] :
( ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C3 @ ( set @ B ) @ B3 @ A4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ C3 @ ( set @ A )
@ ^ [X: C3] : ( image @ B @ A @ F @ ( B3 @ X ) )
@ A4 ) ) ) ).
% image_UN
thf(fact_153_UN__extend__simps_I10_J,axiom,
! [V: $tType,U: $tType,T2: $tType,B3: U > ( set @ V ),F: T2 > U,A4: set @ T2] :
( ( complete_Sup_Sup @ ( set @ V )
@ ( image @ T2 @ ( set @ V )
@ ^ [A3: T2] : ( B3 @ ( F @ A3 ) )
@ A4 ) )
= ( complete_Sup_Sup @ ( set @ V ) @ ( image @ U @ ( set @ V ) @ B3 @ ( image @ T2 @ U @ F @ A4 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_154_Int__UN__distrib2,axiom,
! [A: $tType,C3: $tType,B: $tType,A4: B > ( set @ A ),I4: set @ B,B3: C3 > ( set @ A ),J2: set @ C3] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A4 @ I4 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ C3 @ ( set @ A ) @ B3 @ J2 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [I5: B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image @ C3 @ ( set @ A )
@ ^ [J: C3] : ( inf_inf @ ( set @ A ) @ ( A4 @ I5 ) @ ( B3 @ J ) )
@ J2 ) )
@ I4 ) ) ) ).
% Int_UN_distrib2
thf(fact_155_Int__UN__distrib,axiom,
! [A: $tType,B: $tType,B3: set @ A,A4: B > ( set @ A ),I4: set @ B] :
( ( inf_inf @ ( set @ A ) @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A4 @ I4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [I5: B] : ( inf_inf @ ( set @ A ) @ B3 @ ( A4 @ I5 ) )
@ I4 ) ) ) ).
% Int_UN_distrib
thf(fact_156_UN__extend__simps_I4_J,axiom,
! [H: $tType,G2: $tType,A4: G2 > ( set @ H ),C2: set @ G2,B3: set @ H] :
( ( inf_inf @ ( set @ H ) @ ( complete_Sup_Sup @ ( set @ H ) @ ( image @ G2 @ ( set @ H ) @ A4 @ C2 ) ) @ B3 )
= ( complete_Sup_Sup @ ( set @ H )
@ ( image @ G2 @ ( set @ H )
@ ^ [X: G2] : ( inf_inf @ ( set @ H ) @ ( A4 @ X ) @ B3 )
@ C2 ) ) ) ).
% UN_extend_simps(4)
thf(fact_157_UN__extend__simps_I5_J,axiom,
! [I: $tType,J3: $tType,A4: set @ I,B3: J3 > ( set @ I ),C2: set @ J3] :
( ( inf_inf @ ( set @ I ) @ A4 @ ( complete_Sup_Sup @ ( set @ I ) @ ( image @ J3 @ ( set @ I ) @ B3 @ C2 ) ) )
= ( complete_Sup_Sup @ ( set @ I )
@ ( image @ J3 @ ( set @ I )
@ ^ [X: J3] : ( inf_inf @ ( set @ I ) @ A4 @ ( B3 @ X ) )
@ C2 ) ) ) ).
% UN_extend_simps(5)
thf(fact_158_vimage__UN,axiom,
! [A: $tType,B: $tType,C3: $tType,F: A > B,B3: C3 > ( set @ B ),A4: set @ C3] :
( ( vimage @ A @ B @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C3 @ ( set @ B ) @ B3 @ A4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ C3 @ ( set @ A )
@ ^ [X: C3] : ( vimage @ A @ B @ F @ ( B3 @ X ) )
@ A4 ) ) ) ).
% vimage_UN
thf(fact_159_SUP__UNION,axiom,
! [A: $tType,B: $tType,C3: $tType] :
( ( comple187826305attice @ A )
=> ! [F: B > A,G: C3 > ( set @ B ),A4: set @ C3] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C3 @ ( set @ B ) @ G @ A4 ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ C3 @ A
@ ^ [Y2: C3] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ ( G @ Y2 ) ) )
@ A4 ) ) ) ) ).
% SUP_UNION
thf(fact_160_null__sets__def,axiom,
! [A: $tType] :
( ( measure_null_sets @ A )
= ( ^ [M4: sigma_measure @ A] :
( collect @ ( set @ A )
@ ^ [N4: set @ A] :
( ( member @ ( set @ A ) @ N4 @ ( sigma_sets @ A @ M4 ) )
& ( ( sigma_emeasure @ A @ M4 @ N4 )
= ( zero_zero @ extend1814228343nnreal ) ) ) ) ) ) ).
% null_sets_def
thf(fact_161_sets_OInt__space__eq2,axiom,
! [A: $tType,X3: set @ A,M: sigma_measure @ A] :
( ( member @ ( set @ A ) @ X3 @ ( sigma_sets @ A @ M ) )
=> ( ( inf_inf @ ( set @ A ) @ X3 @ ( sigma_space @ A @ M ) )
= X3 ) ) ).
% sets.Int_space_eq2
thf(fact_162_sets_OInt__space__eq1,axiom,
! [A: $tType,X3: set @ A,M: sigma_measure @ A] :
( ( member @ ( set @ A ) @ X3 @ ( sigma_sets @ A @ M ) )
=> ( ( inf_inf @ ( set @ A ) @ ( sigma_space @ A @ M ) @ X3 )
= X3 ) ) ).
% sets.Int_space_eq1
thf(fact_163_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_164_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_165_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_166_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_167_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_168_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_169_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_170_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% add_left_cancel
thf(fact_171_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% add_right_cancel
thf(fact_172_DiffI,axiom,
! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C @ A4 )
=> ( ~ ( member @ A @ C @ B3 )
=> ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% DiffI
thf(fact_173_Diff__iff,axiom,
! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
= ( ( member @ A @ C @ A4 )
& ~ ( member @ A @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_174_Diff__idemp,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ).
% Diff_idemp
thf(fact_175_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N5: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N5 ) )
= ( N5
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_176_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_177_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_178_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_179_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_180_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [B2: A,A2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_181_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_182_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_183_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_184_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X3: A,Y4: A] :
( ( ( plus_plus @ A @ X3 @ Y4 )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y4
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_185_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X3: A,Y4: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X3 @ Y4 ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y4
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_186_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_187_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_188_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_189_diff__zero,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_190_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_191_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_192_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_193_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel
thf(fact_194_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% diff_add_cancel
thf(fact_195_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [C: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_196_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% add_diff_cancel_left'
thf(fact_197_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,C: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_198_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel_right'
thf(fact_199_sets_OInt,axiom,
! [A: $tType,A2: set @ A,M: sigma_measure @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A2 @ ( sigma_sets @ A @ M ) )
=> ( ( member @ ( set @ A ) @ B2 @ ( sigma_sets @ A @ M ) )
=> ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ ( sigma_sets @ A @ M ) ) ) ) ).
% sets.Int
thf(fact_200_sets_Otop,axiom,
! [A: $tType,M: sigma_measure @ A] : ( member @ ( set @ A ) @ ( sigma_space @ A @ M ) @ ( sigma_sets @ A @ M ) ) ).
% sets.top
thf(fact_201_sets_ODiff,axiom,
! [A: $tType,A2: set @ A,M: sigma_measure @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A2 @ ( sigma_sets @ A @ M ) )
=> ( ( member @ ( set @ A ) @ B2 @ ( sigma_sets @ A @ M ) )
=> ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( sigma_sets @ A @ M ) ) ) ) ).
% sets.Diff
thf(fact_202_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_203_sets_Ocompl__sets,axiom,
! [A: $tType,A2: set @ A,M: sigma_measure @ A] :
( ( member @ ( set @ A ) @ A2 @ ( sigma_sets @ A @ M ) )
=> ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( sigma_space @ A @ M ) @ A2 ) @ ( sigma_sets @ A @ M ) ) ) ).
% sets.compl_sets
thf(fact_204_DiffE,axiom,
! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
=> ~ ( ( member @ A @ C @ A4 )
=> ( member @ A @ C @ B3 ) ) ) ).
% DiffE
thf(fact_205_DiffD1,axiom,
! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
=> ( member @ A @ C @ A4 ) ) ).
% DiffD1
thf(fact_206_DiffD2,axiom,
! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
=> ~ ( member @ A @ C @ B3 ) ) ).
% DiffD2
thf(fact_207_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A6: set @ A,B4: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A6 )
& ~ ( member @ A @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_208_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A6: set @ A,B4: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A6 )
@ ^ [X: A] : ( member @ A @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_209_psubset__imp__ex__mem,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B3 )
=> ? [B5: A] : ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_210_Sup__bool__def,axiom,
( ( complete_Sup_Sup @ $o )
= ( member @ $o @ $true ) ) ).
% Sup_bool_def
thf(fact_211_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X3: A] :
( ( ( zero_zero @ A )
= X3 )
= ( X3
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_212_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_213_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I3: A,J4: A,K2: A,L2: A] :
( ( ( I3 = J4 )
& ( K2 = L2 ) )
=> ( ( plus_plus @ A @ I3 @ K2 )
= ( plus_plus @ A @ J4 @ L2 ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_214_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A,K2: A,A2: A,B2: A] :
( ( A4
= ( plus_plus @ A @ K2 @ A2 ) )
=> ( ( plus_plus @ A @ A4 @ B2 )
= ( plus_plus @ A @ K2 @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_215_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B3: A,K2: A,B2: A,A2: A] :
( ( B3
= ( plus_plus @ A @ K2 @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ K2 @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_216_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add.assoc
thf(fact_217_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% add.left_cancel
thf(fact_218_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% add.right_cancel
thf(fact_219_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,B6: A] : ( plus_plus @ A @ B6 @ A3 ) ) ) ) ).
% add.commute
thf(fact_220_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add.left_commute
thf(fact_221_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
=> ( B2 = C ) ) ) ).
% add_left_imp_eq
thf(fact_222_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
=> ( B2 = C ) ) ) ).
% add_right_imp_eq
thf(fact_223_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C: A,D3: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C @ D3 ) )
=> ( ( A2 = B2 )
= ( C = D3 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_224_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,C: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C ) ) ) ).
% diff_right_commute
thf(fact_225_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X3: A] :
( ( ( one_one @ A )
= X3 )
= ( X3
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_226_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N5: A] :
( ( N5
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N5 ) ) ) ).
% gr_zeroI
thf(fact_227_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N5: A] :
~ ( ord_less @ A @ N5 @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_228_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [M2: A,N5: A] :
( ( ord_less @ A @ M2 @ N5 )
=> ( N5
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_229_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N5 )
= ( N5
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_230_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_231_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_232_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_233_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A3: A,B6: A] :
( ( minus_minus @ A @ A3 @ B6 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_234_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I3: A,J4: A,K2: A,L2: A] :
( ( ( ord_less @ A @ I3 @ J4 )
& ( ord_less @ A @ K2 @ L2 ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I3 @ K2 ) @ ( plus_plus @ A @ J4 @ L2 ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_235_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I3: A,J4: A,K2: A,L2: A] :
( ( ( I3 = J4 )
& ( ord_less @ A @ K2 @ L2 ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I3 @ K2 ) @ ( plus_plus @ A @ J4 @ L2 ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_236_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I3: A,J4: A,K2: A,L2: A] :
( ( ( ord_less @ A @ I3 @ J4 )
& ( K2 = L2 ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I3 @ K2 ) @ ( plus_plus @ A @ J4 @ L2 ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_237_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A )
=> ! [A2: A,B2: A,C: A,D3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_strict_mono
thf(fact_238_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_239_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add_strict_right_mono
thf(fact_240_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_241_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_242_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D3: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ D3 @ C )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_243_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C: A,D3: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C @ D3 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
= ( ord_less @ A @ C @ D3 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_244_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_245_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ C ) ) ) ) ).
% diff_strict_right_mono
thf(fact_246_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,K2: A,A2: A,B2: A] :
( ( A4
= ( plus_plus @ A @ K2 @ A2 ) )
=> ( ( minus_minus @ A @ A4 @ B2 )
= ( plus_plus @ A @ K2 @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_247_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= C )
= ( A2
= ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_248_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,C: A,B2: A] :
( ( A2
= ( minus_minus @ A @ C @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= C ) ) ) ).
% eq_diff_eq
thf(fact_249_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C ) ) ) ).
% add_diff_eq
thf(fact_250_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_251_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_252_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_253_diff__diff__add,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C )
= ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% diff_diff_add
% Subclasses (2)
thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( type @ A ) ) ).
thf(subcl_Finite__Set_Ofinite___Countable_Ocountable,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( countable @ A ) ) ).
% Type constructors (124)
thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
! [A7: $tType,A8: $tType] :
( ( ( ordere890947078_space @ A7 )
& ( finite_finite @ A8 ) )
=> ( ordere890947078_space @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Rings_Oring__1,axiom,
! [A7: $tType,A8: $tType] :
( ( ( ring_1 @ A7 )
& ( finite_finite @ A8 ) )
=> ( ring_1 @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_1,axiom,
ordere890947078_space @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_2,axiom,
ring_1 @ real ).
thf(tcon_Int_Oint___Rings_Oring__1_3,axiom,
ring_1 @ int ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A7: $tType,A8: $tType] :
( ( comple187826305attice @ A8 )
=> ( comple187826305attice @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OSup,axiom,
! [A7: $tType,A8: $tType] :
( ( complete_Sup @ A8 )
=> ( complete_Sup @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Countable_Ocountable,axiom,
! [A7: $tType,A8: $tType] :
( ( ( finite_finite @ A7 )
& ( countable @ A8 ) )
=> ( countable @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A7: $tType,A8: $tType] :
( ( ( finite_finite @ A7 )
& ( finite_finite @ A8 ) )
=> ( finite_finite @ ( A7 > A8 ) ) ) ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Complete__Lattices_OSup_4,axiom,
complete_Sup @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___Countable_Ocountable_5,axiom,
countable @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int ).
thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral @ int ).
thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0 @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_6,axiom,
ordere516151231imp_le @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_7,axiom,
strict2144017051up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_8,axiom,
ordere223160158up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_9,axiom,
ordere236663937imp_le @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_10,axiom,
linord1659791738miring @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_11,axiom,
ordere779506340up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_12,axiom,
cancel146912293up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_13,axiom,
cancel1352612707id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_14,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_15,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_16,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OSup_17,axiom,
complete_Sup @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_18,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Countable_Ocountable_19,axiom,
countable @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_20,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_21,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_22,axiom,
one @ nat ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_23,axiom,
! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Groups_Oab__semigroup__add_24,axiom,
! [A7: $tType] :
( ( ab_semigroup_add @ A7 )
=> ( ab_semigroup_add @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ocomm__monoid__add_25,axiom,
! [A7: $tType] :
( ( comm_monoid_add @ A7 )
=> ( comm_monoid_add @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OSup_26,axiom,
! [A7: $tType] : ( complete_Sup @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Groups_Osemigroup__add_27,axiom,
! [A7: $tType] :
( ( semigroup_add @ A7 )
=> ( semigroup_add @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Countable_Ocountable_28,axiom,
! [A7: $tType] :
( ( finite_finite @ A7 )
=> ( countable @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Groups_Omonoid__add_29,axiom,
! [A7: $tType] :
( ( monoid_add @ A7 )
=> ( monoid_add @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_30,axiom,
! [A7: $tType] :
( ( finite_finite @ A7 )
=> ( finite_finite @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ozero_31,axiom,
! [A7: $tType] :
( ( zero @ A7 )
=> ( zero @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Groups_Oone_32,axiom,
! [A7: $tType] :
( ( one @ A7 )
=> ( one @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_33,axiom,
comple187826305attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OSup_34,axiom,
complete_Sup @ $o ).
thf(tcon_HOL_Obool___Countable_Ocountable_35,axiom,
countable @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_36,axiom,
finite_finite @ $o ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_37,axiom,
ordere516151231imp_le @ real ).
thf(tcon_Real_Oreal___Groups_Ostrict__ordered__ab__semigroup__add_38,axiom,
strict2144017051up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_39,axiom,
ordere223160158up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_40,axiom,
ordere236663937imp_le @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_41,axiom,
linord1659791738miring @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_42,axiom,
ordere779506340up_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_43,axiom,
linord219039673up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_44,axiom,
cancel146912293up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_45,axiom,
cancel1352612707id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_46,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_47,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_48,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_49,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Complete__Lattices_OSup_50,axiom,
complete_Sup @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_51,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_52,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_53,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_54,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_55,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_56,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_57,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oone_58,axiom,
one @ real ).
thf(tcon_Sigma__Algebra_Omeasure___Complete__Lattices_Ocomplete__lattice_59,axiom,
! [A7: $tType] : ( comple187826305attice @ ( sigma_measure @ A7 ) ) ).
thf(tcon_Sigma__Algebra_Omeasure___Complete__Lattices_OSup_60,axiom,
! [A7: $tType] : ( complete_Sup @ ( sigma_measure @ A7 ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__semigroup__monoid__add__imp__le_61,axiom,
! [A7: $tType,A8: $tType] :
( ( ( ordere890947078_space @ A7 )
& ( finite_finite @ A8 ) )
=> ( ordere516151231imp_le @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ostrict__ordered__ab__semigroup__add_62,axiom,
! [A7: $tType,A8: $tType] :
( ( ( ordere890947078_space @ A7 )
& ( finite_finite @ A8 ) )
=> ( strict2144017051up_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__cancel__ab__semigroup__add_63,axiom,
! [A7: $tType,A8: $tType] :
( ( ( ordere890947078_space @ A7 )
& ( finite_finite @ A8 ) )
=> ( ordere223160158up_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__semigroup__add__imp__le_64,axiom,
! [A7: $tType,A8: $tType] :
( ( ( ordere890947078_space @ A7 )
& ( finite_finite @ A8 ) )
=> ( ordere236663937imp_le @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__semigroup__add_65,axiom,
! [A7: $tType,A8: $tType] :
( ( ( ordere890947078_space @ A7 )
& ( finite_finite @ A8 ) )
=> ( ordere779506340up_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocancel__ab__semigroup__add_66,axiom,
! [A7: $tType,A8: $tType] :
( ( ( cancel146912293up_add @ A7 )
& ( finite_finite @ A8 ) )
=> ( cancel146912293up_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocancel__comm__monoid__add_67,axiom,
! [A7: $tType,A8: $tType] :
( ( ( cancel1352612707id_add @ A7 )
& ( finite_finite @ A8 ) )
=> ( cancel1352612707id_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__group__add_68,axiom,
! [A7: $tType,A8: $tType] :
( ( ( ordere890947078_space @ A7 )
& ( finite_finite @ A8 ) )
=> ( ordered_ab_group_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocancel__semigroup__add_69,axiom,
! [A7: $tType,A8: $tType] :
( ( ( cancel_semigroup_add @ A7 )
& ( finite_finite @ A8 ) )
=> ( cancel_semigroup_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oab__semigroup__add_70,axiom,
! [A7: $tType,A8: $tType] :
( ( ( ab_semigroup_add @ A7 )
& ( finite_finite @ A8 ) )
=> ( ab_semigroup_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocomm__monoid__add_71,axiom,
! [A7: $tType,A8: $tType] :
( ( ( comm_monoid_add @ A7 )
& ( finite_finite @ A8 ) )
=> ( comm_monoid_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Complete__Lattices_OSup_72,axiom,
! [A7: $tType,A8: $tType] :
( ( ( ordere890947078_space @ A7 )
& ( finite_finite @ A8 ) )
=> ( complete_Sup @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Osemigroup__add_73,axiom,
! [A7: $tType,A8: $tType] :
( ( ( semigroup_add @ A7 )
& ( finite_finite @ A8 ) )
=> ( semigroup_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oab__group__add_74,axiom,
! [A7: $tType,A8: $tType] :
( ( ( ab_group_add @ A7 )
& ( finite_finite @ A8 ) )
=> ( ab_group_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Countable_Ocountable_75,axiom,
! [A7: $tType,A8: $tType] :
( ( ( countable @ A7 )
& ( finite_finite @ A8 ) )
=> ( countable @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Omonoid__add_76,axiom,
! [A7: $tType,A8: $tType] :
( ( ( monoid_add @ A7 )
& ( finite_finite @ A8 ) )
=> ( monoid_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_77,axiom,
! [A7: $tType,A8: $tType] :
( ( ( finite_finite @ A7 )
& ( finite_finite @ A8 ) )
=> ( finite_finite @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ogroup__add_78,axiom,
! [A7: $tType,A8: $tType] :
( ( ( group_add @ A7 )
& ( finite_finite @ A8 ) )
=> ( group_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Num_Oneg__numeral_79,axiom,
! [A7: $tType,A8: $tType] :
( ( ( ring_1 @ A7 )
& ( finite_finite @ A8 ) )
=> ( neg_numeral @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Nat_Oring__char__0_80,axiom,
! [A7: $tType,A8: $tType] :
( ( ( ring_char_0 @ A7 )
& ( finite_finite @ A8 ) )
=> ( ring_char_0 @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_81,axiom,
! [A7: $tType,A8: $tType] :
( ( ( zero @ A7 )
& ( finite_finite @ A8 ) )
=> ( zero @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oone_82,axiom,
! [A7: $tType,A8: $tType] :
( ( ( one @ A7 )
& ( finite_finite @ A8 ) )
=> ( one @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Ostrict__ordered__ab__semigroup__add_83,axiom,
strict2144017051up_add @ extend1814228343nnreal ).
thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Ocanonically__ordered__monoid__add_84,axiom,
canoni770627133id_add @ extend1814228343nnreal ).
thf(tcon_Extended__Nonnegative__Real_Oennreal___Complete__Lattices_Ocomplete__linorder,axiom,
comple1035589618norder @ extend1814228343nnreal ).
thf(tcon_Extended__Nonnegative__Real_Oennreal___Complete__Lattices_Ocomplete__lattice_85,axiom,
comple187826305attice @ extend1814228343nnreal ).
thf(tcon_Extended__Nonnegative__Real_Oennreal___Rings_Olinordered__nonzero__semiring_86,axiom,
linord1659791738miring @ extend1814228343nnreal ).
thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Oordered__ab__semigroup__add_87,axiom,
ordere779506340up_add @ extend1814228343nnreal ).
thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Oab__semigroup__add_88,axiom,
ab_semigroup_add @ extend1814228343nnreal ).
thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Ocomm__monoid__add_89,axiom,
comm_monoid_add @ extend1814228343nnreal ).
thf(tcon_Extended__Nonnegative__Real_Oennreal___Complete__Lattices_OSup_90,axiom,
complete_Sup @ extend1814228343nnreal ).
thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Osemigroup__add_91,axiom,
semigroup_add @ extend1814228343nnreal ).
thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Omonoid__add_92,axiom,
monoid_add @ extend1814228343nnreal ).
thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Ozero_93,axiom,
zero @ extend1814228343nnreal ).
thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Oone_94,axiom,
one @ extend1814228343nnreal ).
% Free types (1)
thf(tfree_0,hypothesis,
finite_finite @ n ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ^ [N: nat] : ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t2 @ ( f @ N ) ) ) )
= ( ^ [N: nat] : ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t @ ( f @ N ) ) ) ) ) ).
%------------------------------------------------------------------------------