TPTP Problem File: ITP116^2.p
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%------------------------------------------------------------------------------
% File : ITP116^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Minkowskis_Theorem problem prob_203__6248138_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_203__6248138_1 [Des21]
% Status : Theorem
% Rating : 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 467 ( 148 unt; 69 typ; 0 def)
% Number of atoms : 953 ( 268 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 4989 ( 16 ~; 2 |; 95 &;4528 @)
% ( 0 <=>; 348 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 273 ( 273 >; 0 *; 0 +; 0 <<)
% Number of symbols : 68 ( 65 usr; 4 con; 0-5 aty)
% Number of variables : 1225 ( 122 ^;1037 !; 3 ?;1225 :)
% ( 63 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:30:09.913
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
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 (62)
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V2090557954_space:
!>[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_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[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_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_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[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_Topological__Spaces_Ot2__space,type,
topological_t2_space:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__monoid__add,type,
topolo1314133330id_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_Real__Vector__Spaces_Oreal__vector,type,
real_V1076094709vector:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Space_Oeuclidean__space,type,
euclid925273238_space:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Olinorder__topology,type,
topolo2117631714pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo503727757_space:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V55928688vector:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Elementary__Topology_Osecond__countable__topology,type,
elemen1026692323pology:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : $o ).
thf(sy_c_Borel__Space_Ois__borel,type,
borel_is_borel:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( sigma_measure @ A ) > $o ) ).
thf(sy_c_Borel__Space_Otopological__space__class_Oborel,type,
borel_1404511681_borel:
!>[A: $tType] : ( sigma_measure @ A ) ).
thf(sy_c_Complete__Measure_Ocompletion,type,
complete_completion:
!>[A: $tType] : ( ( sigma_measure @ A ) > ( sigma_measure @ A ) ) ).
thf(sy_c_Complete__Measure_Omain__part,type,
complete_main_part:
!>[A: $tType] : ( ( sigma_measure @ A ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > 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_If,type,
if:
!>[A: $tType] : ( $o > A > A > 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_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_OLeast,type,
ord_Least:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( 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_Omeasurable,type,
sigma_measurable:
!>[A: $tType,B: $tType] : ( ( sigma_measure @ A ) > ( sigma_measure @ B ) > ( set @ ( A > B ) ) ) ).
thf(sy_c_Sigma__Algebra_Orestrict__space,type,
sigma_restrict_space:
!>[A: $tType] : ( ( sigma_measure @ A ) > ( set @ A ) > ( sigma_measure @ A ) ) ).
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_Sigma__Algebra_Ovimage__algebra,type,
sigma_vimage_algebra:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( sigma_measure @ B ) > ( sigma_measure @ 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_a____,type,
a: finite_Cartesian_vec @ int @ n ).
% Relevant facts (255)
thf(fact_0_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_1_of__int__vec__eq__iff,axiom,
! [A: $tType,N: $tType] :
( ( ( finite_finite @ N )
& ( ring_char_0 @ A ) )
=> ! [A2: finite_Cartesian_vec @ int @ N,B2: finite_Cartesian_vec @ int @ N] :
( ( ( minkow1937162798nt_vec @ N @ A @ A2 )
= ( minkow1937162798nt_vec @ N @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% of_int_vec_eq_iff
thf(fact_2__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,
! [A3: finite_Cartesian_vec @ int @ n] :
( ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( t2 @ A3 ) @ ( 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 @ A3 ) )
@ ( t2 @ A3 ) )
@ ( 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_3__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_4_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_5__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_6_sets__completionI__sets,axiom,
! [A: $tType,A4: set @ A,M: sigma_measure @ A] :
( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) )
=> ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ ( complete_completion @ A @ M ) ) ) ) ).
% sets_completionI_sets
thf(fact_7__092_060open_062_092_060And_062a_O_A_I_092_060lambda_062x_O_Ax_A_L_Aof__int__vec_Aa_J_A_092_060in_062_Alebesgue_A_092_060rightarrow_062_092_060_094sub_062M_Alebesgue_092_060close_062,axiom,
! [A2: finite_Cartesian_vec @ int @ n] :
( member @ ( ( 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 ) )
@ ( sigma_measurable @ ( finite_Cartesian_vec @ real @ n ) @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) ) ).
% \<open>\<And>a. (\<lambda>x. x + of_int_vec a) \<in> lebesgue \<rightarrow>\<^sub>M lebesgue\<close>
thf(fact_8_vimage__ident,axiom,
! [A: $tType,Y: set @ A] :
( ( vimage @ A @ A
@ ^ [X: A] : X
@ Y )
= Y ) ).
% vimage_ident
thf(fact_9_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_10_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_11_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_12_set__plus__intro,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [A2: A,C: set @ A,B2: A,D: set @ A] :
( ( member @ A @ A2 @ C )
=> ( ( member @ A @ B2 @ D )
=> ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ ( set @ A ) @ C @ D ) ) ) ) ) ).
% set_plus_intro
thf(fact_13_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add_left_cancel
thf(fact_14_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add_right_cancel
thf(fact_15_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_16_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_17_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_18_IntI,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% IntI
thf(fact_19_Int__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
= ( ( member @ A @ C2 @ A4 )
& ( member @ A @ C2 @ B3 ) ) ) ).
% Int_iff
thf(fact_20_space__completion,axiom,
! [A: $tType,M: sigma_measure @ A] :
( ( sigma_space @ A @ ( complete_completion @ A @ M ) )
= ( sigma_space @ A @ M ) ) ).
% space_completion
thf(fact_21_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_22_T__def,axiom,
( t2
= ( ^ [A5: finite_Cartesian_vec @ int @ n] : ( inf_inf @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ s @ ( r @ A5 ) ) ) ) ).
% T_def
thf(fact_23_IntE,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
=> ~ ( ( member @ A @ C2 @ A4 )
=> ~ ( member @ A @ C2 @ B3 ) ) ) ).
% IntE
thf(fact_24_IntD1,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
=> ( member @ A @ C2 @ A4 ) ) ).
% IntD1
thf(fact_25_IntD2,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
=> ( member @ A @ C2 @ B3 ) ) ).
% IntD2
thf(fact_26_Int__assoc,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ C )
= ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B3 @ C ) ) ) ).
% Int_assoc
thf(fact_27_Int__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Int_absorb
thf(fact_28_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_29_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_30_Int__left__commute,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B3 @ C ) )
= ( inf_inf @ ( set @ A ) @ B3 @ ( inf_inf @ ( set @ A ) @ A4 @ C ) ) ) ).
% Int_left_commute
thf(fact_31_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_32_Int__Collect,axiom,
! [A: $tType,X2: A,A4: set @ A,P: A > $o] :
( ( member @ A @ X2 @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X2 @ A4 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_33_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_34_measurable__completion,axiom,
! [A: $tType,B: $tType,F: A > B,M: sigma_measure @ A,N2: sigma_measure @ B] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ M @ N2 ) )
=> ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ ( complete_completion @ A @ M ) @ N2 ) ) ) ).
% measurable_completion
thf(fact_35_vimage__inter__cong,axiom,
! [B: $tType,A: $tType,S: set @ A,F: A > B,G: A > B,Y3: set @ B] :
( ! [W: A] :
( ( member @ A @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ Y3 ) @ S )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G @ Y3 ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_36_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_37_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_38_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.left_commute
thf(fact_39_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B5: A] : ( plus_plus @ A @ B5 @ A5 ) ) ) ) ).
% add.commute
thf(fact_40_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add.right_cancel
thf(fact_41_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add.left_cancel
thf(fact_42_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_43_set__plus__elim,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [X2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ X2 @ ( plus_plus @ ( set @ A ) @ A4 @ B3 ) )
=> ~ ! [A7: A,B6: A] :
( ( X2
= ( plus_plus @ A @ A7 @ B6 ) )
=> ( ( member @ A @ A7 @ A4 )
=> ~ ( member @ A @ B6 @ B3 ) ) ) ) ) ).
% set_plus_elim
thf(fact_44_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B3: A,K: A,B2: A,A2: A] :
( ( B3
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
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] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A,K: A,A2: A,B2: A] :
( ( A4
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A4 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_50_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_51_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_52_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: B > $o,F: A > B,Q: A > $o] :
( ! [X3: A] :
( ( P @ ( F @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
= ( collect @ A @ Q ) ) ) ).
% vimage_Collect
thf(fact_53_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_54_sets_OInt__space__eq2,axiom,
! [A: $tType,X2: set @ A,M: sigma_measure @ A] :
( ( member @ ( set @ A ) @ X2 @ ( sigma_sets @ A @ M ) )
=> ( ( inf_inf @ ( set @ A ) @ X2 @ ( sigma_space @ A @ M ) )
= X2 ) ) ).
% sets.Int_space_eq2
thf(fact_55_sets_OInt__space__eq1,axiom,
! [A: $tType,X2: set @ A,M: sigma_measure @ A] :
( ( member @ ( set @ A ) @ X2 @ ( sigma_sets @ A @ M ) )
=> ( ( inf_inf @ ( set @ A ) @ ( sigma_space @ A @ M ) @ X2 )
= X2 ) ) ).
% sets.Int_space_eq1
thf(fact_56_measurable__sets,axiom,
! [B: $tType,A: $tType,F: A > B,M: sigma_measure @ A,A4: sigma_measure @ B,S: set @ B] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ M @ A4 ) )
=> ( ( member @ ( set @ B ) @ S @ ( sigma_sets @ B @ A4 ) )
=> ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ S ) @ ( sigma_space @ A @ M ) ) @ ( sigma_sets @ A @ M ) ) ) ) ).
% measurable_sets
thf(fact_57_measurableI,axiom,
! [A: $tType,B: $tType,M: sigma_measure @ A,F: A > B,N2: sigma_measure @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( sigma_space @ A @ M ) )
=> ( member @ B @ ( F @ X3 ) @ ( sigma_space @ B @ N2 ) ) )
=> ( ! [A8: set @ B] :
( ( member @ ( set @ B ) @ A8 @ ( sigma_sets @ B @ N2 ) )
=> ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ A8 ) @ ( sigma_space @ A @ M ) ) @ ( sigma_sets @ A @ M ) ) )
=> ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ M @ N2 ) ) ) ) ).
% measurableI
thf(fact_58_sets_Otop,axiom,
! [A: $tType,M: sigma_measure @ A] : ( member @ ( set @ A ) @ ( sigma_space @ A @ M ) @ ( sigma_sets @ A @ M ) ) ).
% sets.top
thf(fact_59_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_60_measurable__If__set,axiom,
! [A: $tType,B: $tType,F: A > B,M: sigma_measure @ A,M2: sigma_measure @ B,G: A > B,A4: set @ A] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ M @ M2 ) )
=> ( ( member @ ( A > B ) @ G @ ( sigma_measurable @ A @ B @ M @ M2 ) )
=> ( ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ ( sigma_space @ A @ M ) ) @ ( sigma_sets @ A @ M ) )
=> ( member @ ( A > B )
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ A4 ) @ ( F @ X ) @ ( G @ X ) )
@ ( sigma_measurable @ A @ B @ M @ M2 ) ) ) ) ) ).
% measurable_If_set
thf(fact_61_measurable__sets__Collect,axiom,
! [B: $tType,A: $tType,F: A > B,M: sigma_measure @ A,N2: sigma_measure @ B,P: B > $o] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ M @ N2 ) )
=> ( ( member @ ( set @ B )
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ ( sigma_space @ B @ N2 ) )
& ( P @ X ) ) )
@ ( sigma_sets @ B @ N2 ) )
=> ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( P @ ( F @ X ) ) ) )
@ ( sigma_sets @ A @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_62_measurable__If,axiom,
! [A: $tType,B: $tType,F: A > B,M: sigma_measure @ A,M2: sigma_measure @ B,G: A > B,P: A > $o] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ M @ M2 ) )
=> ( ( member @ ( A > B ) @ G @ ( sigma_measurable @ A @ B @ M @ M2 ) )
=> ( ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( P @ X ) ) )
@ ( sigma_sets @ A @ M ) )
=> ( member @ ( A > B )
@ ^ [X: A] : ( if @ B @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ ( sigma_measurable @ A @ B @ M @ M2 ) ) ) ) ) ).
% measurable_If
thf(fact_63_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X: A] : ( inf_inf @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% inf_apply
thf(fact_64_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y3 ) @ Y3 )
= ( inf_inf @ A @ X2 @ Y3 ) ) ) ).
% inf_right_idem
thf(fact_65_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% inf.right_idem
thf(fact_66_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y3: A] :
( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ X2 @ Y3 ) )
= ( inf_inf @ A @ X2 @ Y3 ) ) ) ).
% inf_left_idem
thf(fact_67_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B2 ) )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% inf.left_idem
thf(fact_68_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A] :
( ( inf_inf @ A @ X2 @ X2 )
= X2 ) ) ).
% inf_idem
thf(fact_69_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A] :
( ( inf_inf @ A @ A2 @ A2 )
= A2 ) ) ).
% inf.idem
thf(fact_70_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B4: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A6 )
@ ^ [X: A] : ( member @ A @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_71_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y3 @ Z ) )
= ( inf_inf @ A @ Y3 @ ( inf_inf @ A @ X2 @ Z ) ) ) ) ).
% inf_left_commute
thf(fact_72_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A2 @ C2 ) )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% inf.left_commute
thf(fact_73_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [X: A,Y2: A] : ( inf_inf @ A @ Y2 @ X ) ) ) ) ).
% inf_commute
thf(fact_74_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [A5: A,B5: A] : ( inf_inf @ A @ B5 @ A5 ) ) ) ) ).
% inf.commute
thf(fact_75_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y3 ) @ Z )
= ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y3 @ Z ) ) ) ) ).
% inf_assoc
thf(fact_76_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% inf.assoc
thf(fact_77_boolean__algebra__cancel_Oinf2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,K: A,B2: A,A2: A] :
( ( B3
= ( inf_inf @ A @ K @ B2 ) )
=> ( ( inf_inf @ A @ A2 @ B3 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_78_boolean__algebra__cancel_Oinf1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,K: A,A2: A,B2: A] :
( ( A4
= ( inf_inf @ A @ K @ A2 ) )
=> ( ( inf_inf @ A @ A4 @ B2 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_79_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X: A] : ( inf_inf @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% inf_fun_def
thf(fact_80_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( inf_inf @ A )
= ( ^ [X: A,Y2: A] : ( inf_inf @ A @ Y2 @ X ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_81_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y3 ) @ Z )
= ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y3 @ Z ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_82_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y3 @ Z ) )
= ( inf_inf @ A @ Y3 @ ( inf_inf @ A @ X2 @ Z ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_83_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y3: A] :
( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ X2 @ Y3 ) )
= ( inf_inf @ A @ X2 @ Y3 ) ) ) ).
% inf_sup_aci(4)
thf(fact_84_measurable__compose__rev,axiom,
! [A: $tType,C3: $tType,B: $tType,F: A > B,L2: sigma_measure @ A,N2: sigma_measure @ B,G: C3 > A,M: sigma_measure @ C3] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ L2 @ N2 ) )
=> ( ( member @ ( C3 > A ) @ G @ ( sigma_measurable @ C3 @ A @ M @ L2 ) )
=> ( member @ ( C3 > B )
@ ^ [X: C3] : ( F @ ( G @ X ) )
@ ( sigma_measurable @ C3 @ B @ M @ N2 ) ) ) ) ).
% measurable_compose_rev
thf(fact_85_measurable__compose,axiom,
! [B: $tType,A: $tType,C3: $tType,F: A > B,M: sigma_measure @ A,N2: sigma_measure @ B,G: B > C3,L2: sigma_measure @ C3] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ M @ N2 ) )
=> ( ( member @ ( B > C3 ) @ G @ ( sigma_measurable @ B @ C3 @ N2 @ L2 ) )
=> ( member @ ( A > C3 )
@ ^ [X: A] : ( G @ ( F @ X ) )
@ ( sigma_measurable @ A @ C3 @ M @ L2 ) ) ) ) ).
% measurable_compose
thf(fact_86_measurable__id,axiom,
! [A: $tType,M: sigma_measure @ A] :
( member @ ( A > A )
@ ^ [X: A] : X
@ ( sigma_measurable @ A @ A @ M @ M ) ) ).
% measurable_id
thf(fact_87_measurable__cong__sets,axiom,
! [A: $tType,B: $tType,M: sigma_measure @ A,M2: sigma_measure @ A,N2: sigma_measure @ B,N3: sigma_measure @ B] :
( ( ( sigma_sets @ A @ M )
= ( sigma_sets @ A @ M2 ) )
=> ( ( ( sigma_sets @ B @ N2 )
= ( sigma_sets @ B @ N3 ) )
=> ( ( sigma_measurable @ A @ B @ M @ N2 )
= ( sigma_measurable @ A @ B @ M2 @ N3 ) ) ) ) ).
% measurable_cong_sets
thf(fact_88_sets__eq__imp__space__eq,axiom,
! [A: $tType,M: sigma_measure @ A,M2: sigma_measure @ A] :
( ( ( sigma_sets @ A @ M )
= ( sigma_sets @ A @ M2 ) )
=> ( ( sigma_space @ A @ M )
= ( sigma_space @ A @ M2 ) ) ) ).
% sets_eq_imp_space_eq
thf(fact_89_measurable__cong__simp,axiom,
! [A: $tType,B: $tType,M: sigma_measure @ A,N2: sigma_measure @ A,M2: sigma_measure @ B,N3: sigma_measure @ B,F: A > B,G: A > B] :
( ( M = N2 )
=> ( ( M2 = N3 )
=> ( ! [W: A] :
( ( member @ A @ W @ ( sigma_space @ A @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ M @ M2 ) )
= ( member @ ( A > B ) @ G @ ( sigma_measurable @ A @ B @ N2 @ N3 ) ) ) ) ) ) ).
% measurable_cong_simp
thf(fact_90_measurable__space,axiom,
! [A: $tType,B: $tType,F: A > B,M: sigma_measure @ A,A4: sigma_measure @ B,X2: A] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ M @ A4 ) )
=> ( ( member @ A @ X2 @ ( sigma_space @ A @ M ) )
=> ( member @ B @ ( F @ X2 ) @ ( sigma_space @ B @ A4 ) ) ) ) ).
% measurable_space
thf(fact_91_measurable__cong,axiom,
! [A: $tType,B: $tType,M: sigma_measure @ A,F: A > B,G: A > B,M2: sigma_measure @ B] :
( ! [W: A] :
( ( member @ A @ W @ ( sigma_space @ A @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ M @ M2 ) )
= ( member @ ( A > B ) @ G @ ( sigma_measurable @ A @ B @ M @ M2 ) ) ) ) ).
% measurable_cong
thf(fact_92_measurable__ident__sets,axiom,
! [A: $tType,M: sigma_measure @ A,M2: sigma_measure @ A] :
( ( ( sigma_sets @ A @ M )
= ( sigma_sets @ A @ M2 ) )
=> ( member @ ( A > A )
@ ^ [X: A] : X
@ ( sigma_measurable @ A @ A @ M @ M2 ) ) ) ).
% measurable_ident_sets
thf(fact_93_sets_Osets__Collect_I5_J,axiom,
! [A: $tType,M: sigma_measure @ A,Pb: $o] :
( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& Pb ) )
@ ( sigma_sets @ A @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_94_sets_Osets__Collect__imp,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 ) ) )
@ ( sigma_sets @ A @ M ) )
=> ( ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( Q @ X ) ) )
@ ( sigma_sets @ A @ M ) )
=> ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( ( Q @ X )
=> ( P @ X ) ) ) )
@ ( sigma_sets @ A @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_95_sets_Osets__Collect__neg,axiom,
! [A: $tType,M: sigma_measure @ A,P: A > $o] :
( ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( P @ X ) ) )
@ ( sigma_sets @ A @ M ) )
=> ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ~ ( P @ X ) ) )
@ ( sigma_sets @ A @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_96_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 ) ) )
@ ( sigma_sets @ A @ M ) )
=> ( ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( Q @ X ) ) )
@ ( sigma_sets @ A @ M ) )
=> ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( Q @ X )
& ( P @ X ) ) )
@ ( sigma_sets @ A @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_97_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 ) ) )
@ ( sigma_sets @ A @ M ) )
=> ( ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( Q @ X ) ) )
@ ( sigma_sets @ A @ M ) )
=> ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( ( Q @ X )
| ( P @ X ) ) ) )
@ ( sigma_sets @ A @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_98_sets_Osets__Collect__const,axiom,
! [A: $tType,M: sigma_measure @ A,P: $o] :
( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& P ) )
@ ( sigma_sets @ A @ M ) ) ).
% sets.sets_Collect_const
thf(fact_99_measurable__const,axiom,
! [B: $tType,A: $tType,C2: A,M2: sigma_measure @ A,M: sigma_measure @ B] :
( ( member @ A @ C2 @ ( sigma_space @ A @ M2 ) )
=> ( member @ ( B > A )
@ ^ [X: B] : C2
@ ( sigma_measurable @ B @ A @ M @ M2 ) ) ) ).
% measurable_const
thf(fact_100_T_H__def,axiom,
( t
= ( ^ [A5: 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 @ A5 ) )
@ ( t2 @ A5 ) ) ) ) ).
% T'_def
thf(fact_101_inf__Int__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( inf_inf @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ R )
@ ^ [X: A] : ( member @ A @ X @ S ) )
= ( ^ [X: A] : ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_102_main__part__sets,axiom,
! [A: $tType,S: set @ A,M: sigma_measure @ A] :
( ( member @ ( set @ A ) @ S @ ( sigma_sets @ A @ ( complete_completion @ A @ M ) ) )
=> ( member @ ( set @ A ) @ ( complete_main_part @ A @ M @ S ) @ ( sigma_sets @ A @ M ) ) ) ).
% main_part_sets
thf(fact_103_measurable__restrict__space__iff,axiom,
! [A: $tType,B: $tType,Omega: set @ A,M: sigma_measure @ A,C2: B,N2: sigma_measure @ B,F: A > B] :
( ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ Omega @ ( sigma_space @ A @ M ) ) @ ( sigma_sets @ A @ M ) )
=> ( ( member @ B @ C2 @ ( sigma_space @ B @ N2 ) )
=> ( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ ( sigma_restrict_space @ A @ M @ Omega ) @ N2 ) )
= ( member @ ( A > B )
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ Omega ) @ ( F @ X ) @ C2 )
@ ( sigma_measurable @ A @ B @ M @ N2 ) ) ) ) ) ).
% measurable_restrict_space_iff
thf(fact_104_sets__Least,axiom,
! [A: $tType,M: sigma_measure @ A,P: nat > A > $o,A4: set @ nat] :
( ! [I2: nat] :
( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( P @ I2 @ X ) ) )
@ ( sigma_sets @ A @ M ) )
=> ( member @ ( set @ A )
@ ( inf_inf @ ( set @ A )
@ ( vimage @ A @ nat
@ ^ [X: A] :
( ord_Least @ nat
@ ^ [J2: nat] : ( P @ J2 @ X ) )
@ A4 )
@ ( sigma_space @ A @ M ) )
@ ( sigma_sets @ A @ M ) ) ) ).
% sets_Least
thf(fact_105_in__vimage__algebra,axiom,
! [B: $tType,A: $tType,A4: set @ A,M: sigma_measure @ A,F: B > A,X4: set @ B] :
( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) )
=> ( member @ ( set @ B ) @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F @ A4 ) @ X4 ) @ ( sigma_sets @ B @ ( sigma_vimage_algebra @ B @ A @ X4 @ F @ M ) ) ) ) ).
% in_vimage_algebra
thf(fact_106_in__borel__measurable__borel,axiom,
! [A: $tType,B: $tType] :
( ( topolo503727757_space @ B )
=> ! [F: A > B,M: sigma_measure @ A] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ M @ ( borel_1404511681_borel @ B ) ) )
= ( ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ ( sigma_sets @ B @ ( borel_1404511681_borel @ B ) ) )
=> ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ X ) @ ( sigma_space @ A @ M ) ) @ ( sigma_sets @ A @ M ) ) ) ) ) ) ).
% in_borel_measurable_borel
thf(fact_107_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X2: B,A4: set @ B] :
( ( B2
= ( F @ X2 ) )
=> ( ( member @ B @ X2 @ A4 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_108_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A6: A > B,B4: A > B,X: A] : ( minus_minus @ B @ ( A6 @ X ) @ ( B4 @ X ) ) ) ) ) ).
% minus_apply
thf(fact_109_inf1I,axiom,
! [A: $tType,A4: A > $o,X2: A,B3: A > $o] :
( ( A4 @ X2 )
=> ( ( B3 @ X2 )
=> ( inf_inf @ ( A > $o ) @ A4 @ B3 @ X2 ) ) ) ).
% inf1I
thf(fact_110_image__ident,axiom,
! [A: $tType,Y: set @ A] :
( ( image @ A @ A
@ ^ [X: A] : X
@ Y )
= Y ) ).
% image_ident
thf(fact_111_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_112_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_113_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_114_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_115_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_116_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_117_space__vimage__algebra,axiom,
! [B: $tType,A: $tType,X4: set @ A,F: A > B,M: sigma_measure @ B] :
( ( sigma_space @ A @ ( sigma_vimage_algebra @ A @ B @ X4 @ F @ M ) )
= X4 ) ).
% space_vimage_algebra
thf(fact_118_main__part,axiom,
! [A: $tType,S: set @ A,M: sigma_measure @ A] :
( ( member @ ( set @ A ) @ S @ ( sigma_sets @ A @ M ) )
=> ( ( complete_main_part @ A @ M @ S )
= S ) ) ).
% main_part
thf(fact_119_space__restrict__space2,axiom,
! [A: $tType,Omega: set @ A,M: sigma_measure @ A] :
( ( member @ ( set @ A ) @ Omega @ ( sigma_sets @ A @ M ) )
=> ( ( sigma_space @ A @ ( sigma_restrict_space @ A @ M @ Omega ) )
= Omega ) ) ).
% space_restrict_space2
thf(fact_120_inf1E,axiom,
! [A: $tType,A4: A > $o,B3: A > $o,X2: A] :
( ( inf_inf @ ( A > $o ) @ A4 @ B3 @ X2 )
=> ~ ( ( A4 @ X2 )
=> ~ ( B3 @ X2 ) ) ) ).
% inf1E
thf(fact_121_inf1D1,axiom,
! [A: $tType,A4: A > $o,B3: A > $o,X2: A] :
( ( inf_inf @ ( A > $o ) @ A4 @ B3 @ X2 )
=> ( A4 @ X2 ) ) ).
% inf1D1
thf(fact_122_inf1D2,axiom,
! [A: $tType,A4: A > $o,B3: A > $o,X2: A] :
( ( inf_inf @ ( A > $o ) @ A4 @ B3 @ X2 )
=> ( B3 @ X2 ) ) ).
% inf1D2
thf(fact_123_borel__measurable__diff,axiom,
! [A: $tType,B: $tType] :
( ( ( elemen1026692323pology @ B )
& ( real_V55928688vector @ B ) )
=> ! [F: A > B,M: sigma_measure @ A,G: A > B] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ M @ ( borel_1404511681_borel @ B ) ) )
=> ( ( member @ ( A > B ) @ G @ ( sigma_measurable @ A @ B @ M @ ( borel_1404511681_borel @ B ) ) )
=> ( member @ ( A > B )
@ ^ [X: A] : ( minus_minus @ B @ ( F @ X ) @ ( G @ X ) )
@ ( sigma_measurable @ A @ B @ M @ ( borel_1404511681_borel @ B ) ) ) ) ) ) ).
% borel_measurable_diff
thf(fact_124_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A6: A > B,B4: A > B,X: A] : ( minus_minus @ B @ ( A6 @ X ) @ ( B4 @ X ) ) ) ) ) ).
% fun_diff_def
thf(fact_125_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_126_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_127_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,A4: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ F @ A4 ) )
=> ~ ! [X3: B] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member @ B @ X3 @ A4 ) ) ) ).
% imageE
thf(fact_128_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% diff_right_commute
thf(fact_129_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( A2 = B2 )
= ( C2 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_130_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X2: A,A4: set @ A,B2: B,F: A > B] :
( ( member @ A @ X2 @ A4 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_131_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F @ A4 ) )
=> ( P @ X3 ) )
=> ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_132_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N2: set @ A,F: A > B,G: A > B] :
( ( M = N2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_133_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 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_134_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_135_imageI,axiom,
! [B: $tType,A: $tType,X2: A,A4: set @ A,F: A > B] :
( ( member @ A @ X2 @ A4 )
=> ( member @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A4 ) ) ) ).
% imageI
thf(fact_136_borel__measurable__const,axiom,
! [A: $tType,B: $tType] :
( ( topolo503727757_space @ B )
=> ! [C2: B,M: sigma_measure @ A] :
( member @ ( A > B )
@ ^ [X: A] : C2
@ ( sigma_measurable @ A @ B @ M @ ( borel_1404511681_borel @ B ) ) ) ) ).
% borel_measurable_const
thf(fact_137_sets__restrict__space__cong,axiom,
! [A: $tType,M: sigma_measure @ A,N2: sigma_measure @ A,Omega: set @ A] :
( ( ( sigma_sets @ A @ M )
= ( sigma_sets @ A @ N2 ) )
=> ( ( sigma_sets @ A @ ( sigma_restrict_space @ A @ M @ Omega ) )
= ( sigma_sets @ A @ ( sigma_restrict_space @ A @ N2 @ Omega ) ) ) ) ).
% sets_restrict_space_cong
thf(fact_138_restrict__space__sets__cong,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,M: sigma_measure @ A,N2: sigma_measure @ A] :
( ( A4 = B3 )
=> ( ( ( sigma_sets @ A @ M )
= ( sigma_sets @ A @ N2 ) )
=> ( ( sigma_sets @ A @ ( sigma_restrict_space @ A @ M @ A4 ) )
= ( sigma_sets @ A @ ( sigma_restrict_space @ A @ N2 @ B3 ) ) ) ) ) ).
% restrict_space_sets_cong
thf(fact_139_measurable__restrict__space1,axiom,
! [A: $tType,B: $tType,F: A > B,M: sigma_measure @ A,N2: sigma_measure @ B,Omega: set @ A] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ M @ N2 ) )
=> ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ ( sigma_restrict_space @ A @ M @ Omega ) @ N2 ) ) ) ).
% measurable_restrict_space1
thf(fact_140_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ( plus_plus @ A @ C2 @ B2 )
= A2 )
=> ( C2
= ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_141_diff__diff__add,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% diff_diff_add
thf(fact_142_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_143_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_144_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_145_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% add_diff_eq
thf(fact_146_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( A2
= ( minus_minus @ A @ C2 @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= C2 ) ) ) ).
% eq_diff_eq
thf(fact_147_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= C2 )
= ( A2
= ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_148_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,K: A,A2: A,B2: A] :
( ( A4
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( minus_minus @ A @ A4 @ B2 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_149_sets__vimage__algebra__cong,axiom,
! [B: $tType,A: $tType,M: sigma_measure @ A,N2: sigma_measure @ A,X4: set @ B,F: B > A] :
( ( ( sigma_sets @ A @ M )
= ( sigma_sets @ A @ N2 ) )
=> ( ( sigma_sets @ B @ ( sigma_vimage_algebra @ B @ A @ X4 @ F @ M ) )
= ( sigma_sets @ B @ ( sigma_vimage_algebra @ B @ A @ X4 @ F @ N2 ) ) ) ) ).
% sets_vimage_algebra_cong
thf(fact_150_vimage__algebra__cong,axiom,
! [A: $tType,B: $tType,X4: set @ A,Y: set @ A,F: A > B,G: A > B,M: sigma_measure @ B,N2: sigma_measure @ B] :
( ( X4 = Y )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ Y )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( ( sigma_sets @ B @ M )
= ( sigma_sets @ B @ N2 ) )
=> ( ( sigma_vimage_algebra @ A @ B @ X4 @ F @ M )
= ( sigma_vimage_algebra @ A @ B @ Y @ G @ N2 ) ) ) ) ) ).
% vimage_algebra_cong
thf(fact_151_borel__measurable__const__add,axiom,
! [B: $tType,A: $tType] :
( ( real_V55928688vector @ A )
=> ! [F: B > A,M: sigma_measure @ B,A2: A] :
( ( member @ ( B > A ) @ F @ ( sigma_measurable @ B @ A @ M @ ( borel_1404511681_borel @ A ) ) )
=> ( member @ ( B > A )
@ ^ [X: B] : ( plus_plus @ A @ A2 @ ( F @ X ) )
@ ( sigma_measurable @ B @ A @ M @ ( borel_1404511681_borel @ A ) ) ) ) ) ).
% borel_measurable_const_add
thf(fact_152_borel__measurable__add,axiom,
! [A: $tType,B: $tType] :
( ( ( elemen1026692323pology @ B )
& ( topolo1314133330id_add @ B ) )
=> ! [F: A > B,M: sigma_measure @ A,G: A > B] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ M @ ( borel_1404511681_borel @ B ) ) )
=> ( ( member @ ( A > B ) @ G @ ( sigma_measurable @ A @ B @ M @ ( borel_1404511681_borel @ B ) ) )
=> ( member @ ( A > B )
@ ^ [X: A] : ( plus_plus @ B @ ( F @ X ) @ ( G @ X ) )
@ ( sigma_measurable @ A @ B @ M @ ( borel_1404511681_borel @ B ) ) ) ) ) ) ).
% borel_measurable_add
thf(fact_153_borel__measurable__inf,axiom,
! [A: $tType,B: $tType] :
( ( ( elemen1026692323pology @ B )
& ( lattice @ B )
& ( topolo2117631714pology @ B ) )
=> ! [F: A > B,M: sigma_measure @ A,G: A > B] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ M @ ( borel_1404511681_borel @ B ) ) )
=> ( ( member @ ( A > B ) @ G @ ( sigma_measurable @ A @ B @ M @ ( borel_1404511681_borel @ B ) ) )
=> ( member @ ( A > B )
@ ^ [X: A] : ( inf_inf @ B @ ( G @ X ) @ ( F @ X ) )
@ ( sigma_measurable @ A @ B @ M @ ( borel_1404511681_borel @ B ) ) ) ) ) ) ).
% borel_measurable_inf
thf(fact_154_sets__restrict__restrict__space,axiom,
! [A: $tType,M: sigma_measure @ A,A4: set @ A,B3: set @ A] :
( ( sigma_sets @ A @ ( sigma_restrict_space @ A @ ( sigma_restrict_space @ A @ M @ A4 ) @ B3 ) )
= ( sigma_sets @ A @ ( sigma_restrict_space @ A @ M @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% sets_restrict_restrict_space
thf(fact_155_space__restrict__space,axiom,
! [A: $tType,M: sigma_measure @ A,Omega: set @ A] :
( ( sigma_space @ A @ ( sigma_restrict_space @ A @ M @ Omega ) )
= ( inf_inf @ ( set @ A ) @ Omega @ ( sigma_space @ A @ M ) ) ) ).
% space_restrict_space
thf(fact_156_measurable__sets__borel,axiom,
! [B: $tType,A: $tType] :
( ( topolo503727757_space @ A )
=> ! [F: A > B,M: sigma_measure @ B,A4: set @ B] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ ( borel_1404511681_borel @ A ) @ M ) )
=> ( ( member @ ( set @ B ) @ A4 @ ( sigma_sets @ B @ M ) )
=> ( member @ ( set @ A ) @ ( vimage @ A @ B @ F @ A4 ) @ ( sigma_sets @ A @ ( borel_1404511681_borel @ A ) ) ) ) ) ) ).
% measurable_sets_borel
thf(fact_157_sets__Collect__restrict__space__iff,axiom,
! [A: $tType,S: set @ A,M: sigma_measure @ A,P: A > $o] :
( ( member @ ( set @ A ) @ S @ ( sigma_sets @ A @ M ) )
=> ( ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ ( sigma_restrict_space @ A @ M @ S ) ) )
& ( P @ X ) ) )
@ ( sigma_sets @ A @ ( sigma_restrict_space @ A @ M @ S ) ) )
= ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( member @ A @ X @ S )
& ( P @ X ) ) )
@ ( sigma_sets @ A @ M ) ) ) ) ).
% sets_Collect_restrict_space_iff
thf(fact_158_borel__measurable__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( elemen1026692323pology @ B )
& ( topolo2117631714pology @ B ) )
=> ! [F: A > B,M: sigma_measure @ A,G: A > B] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ M @ ( borel_1404511681_borel @ B ) ) )
=> ( ( member @ ( A > B ) @ G @ ( sigma_measurable @ A @ B @ M @ ( borel_1404511681_borel @ B ) ) )
=> ( member @ ( set @ A )
@ ( collect @ A
@ ^ [W2: A] :
( ( member @ A @ W2 @ ( sigma_space @ A @ M ) )
& ( ( F @ W2 )
= ( G @ W2 ) ) ) )
@ ( sigma_sets @ A @ M ) ) ) ) ) ).
% borel_measurable_eq
thf(fact_159_borel__measurable__neq,axiom,
! [B: $tType,A: $tType] :
( ( ( elemen1026692323pology @ B )
& ( topolo2117631714pology @ B ) )
=> ! [F: A > B,M: sigma_measure @ A,G: A > B] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ M @ ( borel_1404511681_borel @ B ) ) )
=> ( ( member @ ( A > B ) @ G @ ( sigma_measurable @ A @ B @ M @ ( borel_1404511681_borel @ B ) ) )
=> ( member @ ( set @ A )
@ ( collect @ A
@ ^ [W2: A] :
( ( member @ A @ W2 @ ( sigma_space @ A @ M ) )
& ( ( F @ W2 )
!= ( G @ W2 ) ) ) )
@ ( sigma_sets @ A @ M ) ) ) ) ) ).
% borel_measurable_neq
thf(fact_160_measurable__equality__set,axiom,
! [A: $tType,D3: $tType] :
( ( ( elemen1026692323pology @ A )
& ( topological_t2_space @ A ) )
=> ! [F: D3 > A,M: sigma_measure @ D3,G: D3 > A] :
( ( member @ ( D3 > A ) @ F @ ( sigma_measurable @ D3 @ A @ M @ ( borel_1404511681_borel @ A ) ) )
=> ( ( member @ ( D3 > A ) @ G @ ( sigma_measurable @ D3 @ A @ M @ ( borel_1404511681_borel @ A ) ) )
=> ( member @ ( set @ D3 )
@ ( collect @ D3
@ ^ [X: D3] :
( ( member @ D3 @ X @ ( sigma_space @ D3 @ M ) )
& ( ( F @ X )
= ( G @ X ) ) ) )
@ ( sigma_sets @ D3 @ M ) ) ) ) ) ).
% measurable_equality_set
thf(fact_161_measurable__If__restrict__space__iff,axiom,
! [A: $tType,B: $tType,M: sigma_measure @ A,P: A > $o,F: A > B,G: A > B,N2: sigma_measure @ B] :
( ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ M ) )
& ( P @ X ) ) )
@ ( sigma_sets @ A @ M ) )
=> ( ( member @ ( A > B )
@ ^ [X: A] : ( if @ B @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ ( sigma_measurable @ A @ B @ M @ N2 ) )
= ( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ ( sigma_restrict_space @ A @ M @ ( collect @ A @ P ) ) @ N2 ) )
& ( member @ ( A > B ) @ G
@ ( sigma_measurable @ A @ B
@ ( sigma_restrict_space @ A @ M
@ ( collect @ A
@ ^ [X: A] :
~ ( P @ X ) ) )
@ N2 ) ) ) ) ) ).
% measurable_If_restrict_space_iff
thf(fact_162_space__lebesgue__on,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ! [S: set @ A] :
( ( sigma_space @ A @ ( sigma_restrict_space @ A @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) @ S ) )
= S ) ) ).
% space_lebesgue_on
thf(fact_163_sets__lebesgue__on__refl,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ! [S: set @ A] : ( member @ ( set @ A ) @ S @ ( sigma_sets @ A @ ( sigma_restrict_space @ A @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) @ S ) ) ) ) ).
% sets_lebesgue_on_refl
thf(fact_164_space__lborel,axiom,
! [B: $tType] :
( ( euclid925273238_space @ B )
=> ( ( sigma_space @ B @ ( lebesgue_lborel @ B ) )
= ( sigma_space @ B @ ( borel_1404511681_borel @ B ) ) ) ) ).
% space_lborel
thf(fact_165_measurable__lborel1,axiom,
! [C3: $tType,D3: $tType] :
( ( euclid925273238_space @ D3 )
=> ! [M: sigma_measure @ C3] :
( ( sigma_measurable @ C3 @ D3 @ M @ ( lebesgue_lborel @ D3 ) )
= ( sigma_measurable @ C3 @ D3 @ M @ ( borel_1404511681_borel @ D3 ) ) ) ) ).
% measurable_lborel1
thf(fact_166_measurable__lborel2,axiom,
! [E: $tType,C3: $tType] :
( ( euclid925273238_space @ E )
=> ! [M: sigma_measure @ C3] :
( ( sigma_measurable @ E @ C3 @ ( lebesgue_lborel @ E ) @ M )
= ( sigma_measurable @ E @ C3 @ ( borel_1404511681_borel @ E ) @ M ) ) ) ).
% measurable_lborel2
thf(fact_167_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_168_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_169_sets__lborel,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ( ( sigma_sets @ A @ ( lebesgue_lborel @ A ) )
= ( sigma_sets @ A @ ( borel_1404511681_borel @ A ) ) ) ) ).
% sets_lborel
thf(fact_170_Int__Diff,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ C )
= ( inf_inf @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B3 @ C ) ) ) ).
% Int_Diff
thf(fact_171_Diff__Int2,axiom,
! [A: $tType,A4: set @ A,C: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C ) @ ( inf_inf @ ( set @ A ) @ B3 @ C ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C ) @ B3 ) ) ).
% Diff_Int2
thf(fact_172_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_173_Diff__Int__distrib,axiom,
! [A: $tType,C: set @ A,A4: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ C @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C @ A4 ) @ ( inf_inf @ ( set @ A ) @ C @ B3 ) ) ) ).
% Diff_Int_distrib
thf(fact_174_Diff__Int__distrib2,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ C )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C ) @ ( inf_inf @ ( set @ A ) @ B3 @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_175_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_176_sets__restrict__space,axiom,
! [A: $tType,M: sigma_measure @ A,Omega: set @ A] :
( ( sigma_sets @ A @ ( sigma_restrict_space @ A @ M @ Omega ) )
= ( image @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ Omega ) @ ( sigma_sets @ A @ M ) ) ) ).
% sets_restrict_space
thf(fact_177_lborelD,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ! [A4: set @ A] :
( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ ( borel_1404511681_borel @ A ) ) )
=> ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ ( lebesgue_lborel @ A ) ) ) ) ) ).
% lborelD
thf(fact_178_measurable__lebesgue__cong,axiom,
! [A: $tType,B: $tType] :
( ( euclid925273238_space @ A )
=> ! [S: set @ A,F: A > B,G: A > B,M: sigma_measure @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ ( sigma_restrict_space @ A @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) @ S ) @ M ) )
= ( member @ ( A > B ) @ G @ ( sigma_measurable @ A @ B @ ( sigma_restrict_space @ A @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) @ S ) @ M ) ) ) ) ) ).
% measurable_lebesgue_cong
thf(fact_179_lborelD__Collect,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ! [P: A > $o] :
( ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ ( borel_1404511681_borel @ A ) ) )
& ( P @ X ) ) )
@ ( sigma_sets @ A @ ( borel_1404511681_borel @ A ) ) )
=> ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( sigma_space @ A @ ( lebesgue_lborel @ A ) ) )
& ( P @ X ) ) )
@ ( sigma_sets @ A @ ( lebesgue_lborel @ A ) ) ) ) ) ).
% lborelD_Collect
thf(fact_180_lebesgue__sets__translation,axiom,
! [B: $tType] :
( ( euclid925273238_space @ B )
=> ! [S: set @ B,A2: B] :
( ( member @ ( set @ B ) @ S @ ( sigma_sets @ B @ ( complete_completion @ B @ ( lebesgue_lborel @ B ) ) ) )
=> ( member @ ( set @ B ) @ ( image @ B @ B @ ( plus_plus @ B @ A2 ) @ S ) @ ( sigma_sets @ B @ ( complete_completion @ B @ ( lebesgue_lborel @ B ) ) ) ) ) ) ).
% lebesgue_sets_translation
thf(fact_181_lebesgue__measurable__vimage__borel,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid925273238_space @ A )
& ( euclid925273238_space @ B ) )
=> ! [F: A > B,T: set @ B] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) @ ( borel_1404511681_borel @ B ) ) )
=> ( ( member @ ( set @ B ) @ T @ ( sigma_sets @ B @ ( borel_1404511681_borel @ B ) ) )
=> ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] : ( member @ B @ ( F @ X ) @ T ) )
@ ( sigma_sets @ A @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) ) ) ) ) ) ).
% lebesgue_measurable_vimage_borel
thf(fact_182_borel__measurable__lebesgue__preimage__borel,axiom,
! [A: $tType,B: $tType] :
( ( ( euclid925273238_space @ B )
& ( euclid925273238_space @ A ) )
=> ! [F: A > B] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) @ ( borel_1404511681_borel @ B ) ) )
= ( ! [T2: set @ B] :
( ( member @ ( set @ B ) @ T2 @ ( sigma_sets @ B @ ( borel_1404511681_borel @ B ) ) )
=> ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] : ( member @ B @ ( F @ X ) @ T2 ) )
@ ( sigma_sets @ A @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) ) ) ) ) ) ) ).
% borel_measurable_lebesgue_preimage_borel
thf(fact_183_borel__measurable__vimage__borel,axiom,
! [A: $tType,B: $tType] :
( ( ( euclid925273238_space @ B )
& ( euclid925273238_space @ A ) )
=> ! [F: A > B,S: set @ A] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ ( sigma_restrict_space @ A @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) @ S ) @ ( borel_1404511681_borel @ B ) ) )
= ( ! [T2: set @ B] :
( ( member @ ( set @ B ) @ T2 @ ( sigma_sets @ B @ ( borel_1404511681_borel @ B ) ) )
=> ( member @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ S )
& ( member @ B @ ( F @ X ) @ T2 ) ) )
@ ( sigma_sets @ A @ ( sigma_restrict_space @ A @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) @ S ) ) ) ) ) ) ) ).
% borel_measurable_vimage_borel
thf(fact_184_restrict__space__eq__vimage__algebra_H,axiom,
! [A: $tType,M: sigma_measure @ A,Omega: set @ A] :
( ( sigma_sets @ A @ ( sigma_restrict_space @ A @ M @ Omega ) )
= ( sigma_sets @ A
@ ( sigma_vimage_algebra @ A @ A @ ( inf_inf @ ( set @ A ) @ Omega @ ( sigma_space @ A @ M ) )
@ ^ [X: A] : X
@ M ) ) ) ).
% restrict_space_eq_vimage_algebra'
thf(fact_185_restrict__restrict__space,axiom,
! [A: $tType,A4: set @ A,M: sigma_measure @ A,B3: set @ A] :
( ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ ( sigma_space @ A @ M ) ) @ ( sigma_sets @ A @ M ) )
=> ( ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B3 @ ( sigma_space @ A @ M ) ) @ ( sigma_sets @ A @ M ) )
=> ( ( sigma_restrict_space @ A @ ( sigma_restrict_space @ A @ M @ A4 ) @ B3 )
= ( sigma_restrict_space @ A @ M @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ) ).
% restrict_restrict_space
thf(fact_186_is__borel__def,axiom,
! [B: $tType,A: $tType] :
( ( topolo503727757_space @ B )
=> ( ( borel_is_borel @ A @ B )
= ( ^ [F2: A > B,M3: sigma_measure @ A] : ( member @ ( A > B ) @ F2 @ ( sigma_measurable @ A @ B @ M3 @ ( borel_1404511681_borel @ B ) ) ) ) ) ) ).
% is_borel_def
thf(fact_187_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_188_Diff__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
= ( ( member @ A @ C2 @ A4 )
& ~ ( member @ A @ C2 @ B3 ) ) ) ).
% Diff_iff
thf(fact_189_DiffI,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ~ ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% DiffI
thf(fact_190_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_191_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_192_DiffD2,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
=> ~ ( member @ A @ C2 @ B3 ) ) ).
% DiffD2
thf(fact_193_DiffD1,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
=> ( member @ A @ C2 @ A4 ) ) ).
% DiffD1
thf(fact_194_DiffE,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
=> ~ ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% DiffE
thf(fact_195_sets__vimage__algebra__space,axiom,
! [A: $tType,B: $tType,X4: set @ A,F: A > B,M: sigma_measure @ B] : ( member @ ( set @ A ) @ X4 @ ( sigma_sets @ A @ ( sigma_vimage_algebra @ A @ B @ X4 @ F @ M ) ) ) ).
% sets_vimage_algebra_space
thf(fact_196_translation__subtract__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S2: set @ A,T3: set @ A] :
( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ ( minus_minus @ ( set @ A ) @ S2 @ T3 ) )
= ( 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 )
@ T3 ) ) ) ) ).
% translation_subtract_diff
thf(fact_197_translation__subtract__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S2: set @ A,T3: set @ A] :
( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ ( inf_inf @ ( set @ A ) @ S2 @ T3 ) )
= ( 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 )
@ T3 ) ) ) ) ).
% translation_subtract_Int
thf(fact_198_translation__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S2: set @ A,T3: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( minus_minus @ ( set @ A ) @ S2 @ T3 ) )
= ( minus_minus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S2 ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T3 ) ) ) ) ).
% translation_diff
thf(fact_199_translation__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S2: set @ A,T3: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( inf_inf @ ( set @ A ) @ S2 @ T3 ) )
= ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S2 ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T3 ) ) ) ) ).
% translation_Int
thf(fact_200_measurable__product__then__coordinatewise,axiom,
! [B: $tType,A: $tType,C3: $tType] :
( ( topolo503727757_space @ C3 )
=> ! [F: A > B > C3,M: sigma_measure @ A,I: B] :
( ( member @ ( A > B > C3 ) @ F @ ( sigma_measurable @ A @ ( B > C3 ) @ M @ ( borel_1404511681_borel @ ( B > C3 ) ) ) )
=> ( member @ ( A > C3 )
@ ^ [X: A] : ( F @ X @ I )
@ ( sigma_measurable @ A @ C3 @ M @ ( borel_1404511681_borel @ C3 ) ) ) ) ) ).
% measurable_product_then_coordinatewise
thf(fact_201_measurable__coordinatewise__then__product,axiom,
! [A: $tType,C3: $tType,B: $tType] :
( ( ( countable @ B )
& ( elemen1026692323pology @ C3 ) )
=> ! [F: A > B > C3,M: sigma_measure @ A] :
( ! [I2: B] :
( member @ ( A > C3 )
@ ^ [X: A] : ( F @ X @ I2 )
@ ( sigma_measurable @ A @ C3 @ M @ ( borel_1404511681_borel @ C3 ) ) )
=> ( member @ ( A > B > C3 ) @ F @ ( sigma_measurable @ A @ ( B > C3 ) @ M @ ( borel_1404511681_borel @ ( B > C3 ) ) ) ) ) ) ).
% measurable_coordinatewise_then_product
thf(fact_202_measurable__product__coordinates,axiom,
! [A: $tType,B: $tType] :
( ( topolo503727757_space @ B )
=> ! [I: A] :
( member @ ( ( A > B ) > B )
@ ^ [X: A > B] : ( X @ I )
@ ( sigma_measurable @ ( A > B ) @ B @ ( borel_1404511681_borel @ ( A > B ) ) @ ( borel_1404511681_borel @ B ) ) ) ) ).
% measurable_product_coordinates
thf(fact_203_translation__invert,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,A4: set @ A,B3: set @ A] :
( ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ A4 )
= ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ B3 ) )
=> ( A4 = B3 ) ) ) ).
% translation_invert
thf(fact_204_translation__assoc,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,A2: A,S: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ B2 ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S ) )
= ( image @ A @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ S ) ) ) ).
% translation_assoc
thf(fact_205_affine__parallel__expl__aux,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [S: set @ A,A2: A,T: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
= ( member @ A @ ( plus_plus @ A @ A2 @ X3 ) @ T ) )
=> ( T
= ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S ) ) ) ) ).
% affine_parallel_expl_aux
thf(fact_206_borel__measurable__if,axiom,
! [A: $tType,B: $tType] :
( ( ( euclid925273238_space @ B )
& ( euclid925273238_space @ A ) )
=> ! [S: set @ A,F: A > B] :
( ( member @ ( set @ A ) @ S @ ( sigma_sets @ A @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) ) )
=> ( ( member @ ( A > B )
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ S ) @ ( F @ X ) @ ( zero_zero @ B ) )
@ ( sigma_measurable @ A @ B @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) @ ( borel_1404511681_borel @ B ) ) )
= ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ ( sigma_restrict_space @ A @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) @ S ) @ ( borel_1404511681_borel @ B ) ) ) ) ) ) ).
% borel_measurable_if
thf(fact_207_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_208_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_209_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_210_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_211_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_212_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_213_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_214_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_215_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X2: A,Y3: A] :
( ( ( plus_plus @ A @ X2 @ Y3 )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_216_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X2: A,Y3: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X2 @ Y3 ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_217_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_218_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_219_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_220_diff__zero,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_221_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_222_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_223_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_224_eq__add__iff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [X2: A,Y3: A] :
( ( X2
= ( plus_plus @ A @ X2 @ Y3 ) )
= ( Y3
= ( zero_zero @ A ) ) ) ) ).
% eq_add_iff
thf(fact_225_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A5: A,B5: A] :
( ( minus_minus @ A @ A5 @ B5 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_226_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_227_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_228_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_229_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X2: A] :
( ( ( zero_zero @ A )
= X2 )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_230_borel__measurable__if__D,axiom,
! [A: $tType,B: $tType] :
( ( ( euclid925273238_space @ B )
& ( euclid925273238_space @ A ) )
=> ! [S: set @ A,F: A > B] :
( ( member @ ( A > B )
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ S ) @ ( F @ X ) @ ( zero_zero @ B ) )
@ ( sigma_measurable @ A @ B @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) @ ( borel_1404511681_borel @ B ) ) )
=> ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ ( sigma_restrict_space @ A @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) @ S ) @ ( borel_1404511681_borel @ B ) ) ) ) ) ).
% borel_measurable_if_D
thf(fact_231_borel__measurable__if__I,axiom,
! [A: $tType,B: $tType] :
( ( ( euclid925273238_space @ B )
& ( euclid925273238_space @ A ) )
=> ! [F: A > B,S: set @ A] :
( ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ ( sigma_restrict_space @ A @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) @ S ) @ ( borel_1404511681_borel @ B ) ) )
=> ( ( member @ ( set @ A ) @ S @ ( sigma_sets @ A @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) ) )
=> ( member @ ( A > B )
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ S ) @ ( F @ X ) @ ( zero_zero @ B ) )
@ ( sigma_measurable @ A @ B @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) @ ( borel_1404511681_borel @ B ) ) ) ) ) ) ).
% borel_measurable_if_I
thf(fact_232_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( zero_zero @ nat ) ) ) ).
% Least_eq_0
thf(fact_233_borel__measurable__if__lebesgue__on,axiom,
! [A: $tType,B: $tType] :
( ( ( euclid925273238_space @ B )
& ( euclid925273238_space @ A ) )
=> ! [S: set @ A,T: set @ A,F: A > B] :
( ( member @ ( set @ A ) @ S @ ( sigma_sets @ A @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) ) )
=> ( ( member @ ( set @ A ) @ T @ ( sigma_sets @ A @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ T )
=> ( ( member @ ( A > B )
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ S ) @ ( F @ X ) @ ( zero_zero @ B ) )
@ ( sigma_measurable @ A @ B @ ( sigma_restrict_space @ A @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) @ T ) @ ( borel_1404511681_borel @ B ) ) )
= ( member @ ( A > B ) @ F @ ( sigma_measurable @ A @ B @ ( sigma_restrict_space @ A @ ( complete_completion @ A @ ( lebesgue_lborel @ A ) ) @ S ) @ ( borel_1404511681_borel @ B ) ) ) ) ) ) ) ) ).
% borel_measurable_if_lebesgue_on
thf(fact_234_subsetI,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ A @ X3 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% subsetI
thf(fact_235_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% subset_antisym
thf(fact_236_diff__0__eq__0,axiom,
! [N4: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N4 )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_237_diff__self__eq__0,axiom,
! [M4: nat] :
( ( minus_minus @ nat @ M4 @ M4 )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_238_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N4: A] :
( ( ord_less_eq @ A @ N4 @ ( zero_zero @ A ) )
= ( N4
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_239_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_240_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_241_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ ( inf_inf @ A @ Y3 @ Z ) )
= ( ( ord_less_eq @ A @ X2 @ Y3 )
& ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% le_inf_iff
thf(fact_242_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% inf.bounded_iff
thf(fact_243_Int__subset__iff,axiom,
! [A: $tType,C: set @ A,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C @ A4 )
& ( ord_less_eq @ ( set @ A ) @ C @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_244_set__plus__mono2,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [C: set @ A,D: set @ A,E2: set @ A,F3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C @ D )
=> ( ( ord_less_eq @ ( set @ A ) @ E2 @ F3 )
=> ( ord_less_eq @ ( set @ A ) @ ( plus_plus @ ( set @ A ) @ C @ E2 ) @ ( plus_plus @ ( set @ A ) @ D @ F3 ) ) ) ) ) ).
% set_plus_mono2
thf(fact_245_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_246_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_247_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_248_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_249_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_250_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_251_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_252_subset__translation__eq,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: A,S2: set @ A,T3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S2 ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T3 ) )
= ( ord_less_eq @ ( set @ A ) @ S2 @ T3 ) ) ) ).
% subset_translation_eq
thf(fact_253_diffs0__imp__equal,axiom,
! [M4: nat,N4: nat] :
( ( ( minus_minus @ nat @ M4 @ N4 )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N4 @ M4 )
= ( zero_zero @ nat ) )
=> ( M4 = N4 ) ) ) ).
% diffs0_imp_equal
thf(fact_254_minus__nat_Odiff__0,axiom,
! [M4: nat] :
( ( minus_minus @ nat @ M4 @ ( zero_zero @ nat ) )
= M4 ) ).
% minus_nat.diff_0
% 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 (136)
thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( ordere890947078_space @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Real__Vector__Spaces_Ometric__space,axiom,
! [A9: $tType,A10: $tType] :
( ( ( real_V2090557954_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( real_V2090557954_space @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_1,axiom,
ordere890947078_space @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space_2,axiom,
real_V2090557954_space @ real ).
thf(tcon_fun___Elementary__Topology_Osecond__countable__topology,axiom,
! [A9: $tType,A10: $tType] :
( ( ( countable @ A9 )
& ( elemen1026692323pology @ A10 ) )
=> ( elemen1026692323pology @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
! [A9: $tType,A10: $tType] :
( ( topolo503727757_space @ A10 )
=> ( topolo503727757_space @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A9: $tType,A10: $tType] :
( ( semilattice_inf @ A10 )
=> ( semilattice_inf @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Countable_Ocountable,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( countable @ A10 ) )
=> ( countable @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A9: $tType,A10: $tType] :
( ( lattice @ A10 )
=> ( lattice @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A9: $tType,A10: $tType] :
( ( minus @ A10 )
=> ( minus @ ( A9 > A10 ) ) ) ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Otopological__space_3,axiom,
topolo503727757_space @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
topolo2117631714pology @ 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___Limits_Otopological__monoid__add,axiom,
topolo1314133330id_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
topological_t2_space @ 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___Lattices_Osemilattice__inf_4,axiom,
semilattice_inf @ 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___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___Lattices_Olattice_6,axiom,
lattice @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int ).
thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0 @ int ).
thf(tcon_Int_Oint___Groups_Ominus_7,axiom,
minus @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Groups_Oplus,axiom,
plus @ int ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_8,axiom,
ordere516151231imp_le @ nat ).
thf(tcon_Nat_Onat___Elementary__Topology_Osecond__countable__topology_9,axiom,
elemen1026692323pology @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_10,axiom,
ordere236663937imp_le @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_11,axiom,
topolo503727757_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_12,axiom,
topolo2117631714pology @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_13,axiom,
ordere779506340up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_14,axiom,
cancel146912293up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_15,axiom,
topolo1314133330id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_16,axiom,
cancel1352612707id_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_17,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_18,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_19,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_20,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_21,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_22,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Countable_Ocountable_23,axiom,
countable @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_24,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_25,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_26,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_27,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_28,axiom,
plus @ nat ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_29,axiom,
! [A9: $tType] : ( semilattice_inf @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Groups_Oab__semigroup__add_30,axiom,
! [A9: $tType] :
( ( ab_semigroup_add @ A9 )
=> ( ab_semigroup_add @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ocomm__monoid__add_31,axiom,
! [A9: $tType] :
( ( comm_monoid_add @ A9 )
=> ( comm_monoid_add @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Groups_Osemigroup__add_32,axiom,
! [A9: $tType] :
( ( semigroup_add @ A9 )
=> ( semigroup_add @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Countable_Ocountable_33,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 )
=> ( countable @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Groups_Omonoid__add_34,axiom,
! [A9: $tType] :
( ( monoid_add @ A9 )
=> ( monoid_add @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_35,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 )
=> ( finite_finite @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_36,axiom,
! [A9: $tType] : ( lattice @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_37,axiom,
! [A9: $tType] : ( minus @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Groups_Ozero_38,axiom,
! [A9: $tType] :
( ( zero @ A9 )
=> ( zero @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Groups_Oplus_39,axiom,
! [A9: $tType] :
( ( plus @ A9 )
=> ( plus @ ( set @ A9 ) ) ) ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_40,axiom,
topolo503727757_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_41,axiom,
topolo2117631714pology @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_42,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_43,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Countable_Ocountable_44,axiom,
countable @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_45,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_46,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_47,axiom,
minus @ $o ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_48,axiom,
ordere516151231imp_le @ real ).
thf(tcon_Real_Oreal___Elementary__Topology_Osecond__countable__topology_49,axiom,
elemen1026692323pology @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_50,axiom,
ordere236663937imp_le @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
real_V55928688vector @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_51,axiom,
topolo503727757_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_52,axiom,
topolo2117631714pology @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_53,axiom,
ordere779506340up_add @ real ).
thf(tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space,axiom,
euclid925273238_space @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
real_V1076094709vector @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_54,axiom,
linord219039673up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_55,axiom,
cancel146912293up_add @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__monoid__add_56,axiom,
topolo1314133330id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_57,axiom,
cancel1352612707id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_58,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_59,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_60,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__inf_61,axiom,
semilattice_inf @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_62,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_63,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_64,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_65,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_66,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Lattices_Olattice_67,axiom,
lattice @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_68,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_69,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Ominus_70,axiom,
minus @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_71,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_72,axiom,
plus @ real ).
thf(tcon_Sigma__Algebra_Omeasure___Lattices_Osemilattice__inf_73,axiom,
! [A9: $tType] : ( semilattice_inf @ ( sigma_measure @ A9 ) ) ).
thf(tcon_Sigma__Algebra_Omeasure___Lattices_Olattice_74,axiom,
! [A9: $tType] : ( lattice @ ( sigma_measure @ A9 ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__semigroup__monoid__add__imp__le_75,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( ordere516151231imp_le @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Elementary__Topology_Osecond__countable__topology_76,axiom,
! [A9: $tType,A10: $tType] :
( ( ( euclid925273238_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( elemen1026692323pology @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__semigroup__add__imp__le_77,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( ordere236663937imp_le @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Real__Vector__Spaces_Oreal__normed__vector_78,axiom,
! [A9: $tType,A10: $tType] :
( ( ( real_V55928688vector @ A9 )
& ( finite_finite @ A10 ) )
=> ( real_V55928688vector @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Topological__Spaces_Otopological__space_79,axiom,
! [A9: $tType,A10: $tType] :
( ( ( topolo503727757_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( topolo503727757_space @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__semigroup__add_80,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( ordere779506340up_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Euclidean__Space_Oeuclidean__space_81,axiom,
! [A9: $tType,A10: $tType] :
( ( ( euclid925273238_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( euclid925273238_space @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Real__Vector__Spaces_Oreal__vector_82,axiom,
! [A9: $tType,A10: $tType] :
( ( ( real_V1076094709vector @ A9 )
& ( finite_finite @ A10 ) )
=> ( real_V1076094709vector @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocancel__ab__semigroup__add_83,axiom,
! [A9: $tType,A10: $tType] :
( ( ( cancel146912293up_add @ A9 )
& ( finite_finite @ A10 ) )
=> ( cancel146912293up_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Limits_Otopological__monoid__add_84,axiom,
! [A9: $tType,A10: $tType] :
( ( ( real_V55928688vector @ A9 )
& ( finite_finite @ A10 ) )
=> ( topolo1314133330id_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocancel__comm__monoid__add_85,axiom,
! [A9: $tType,A10: $tType] :
( ( ( cancel1352612707id_add @ A9 )
& ( finite_finite @ A10 ) )
=> ( cancel1352612707id_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Topological__Spaces_Ot2__space_86,axiom,
! [A9: $tType,A10: $tType] :
( ( ( real_V2090557954_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( topological_t2_space @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__group__add_87,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( ordered_ab_group_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocancel__semigroup__add_88,axiom,
! [A9: $tType,A10: $tType] :
( ( ( cancel_semigroup_add @ A9 )
& ( finite_finite @ A10 ) )
=> ( cancel_semigroup_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Osemilattice__inf_89,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( semilattice_inf @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oab__semigroup__add_90,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ab_semigroup_add @ A9 )
& ( finite_finite @ A10 ) )
=> ( ab_semigroup_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocomm__monoid__add_91,axiom,
! [A9: $tType,A10: $tType] :
( ( ( comm_monoid_add @ A9 )
& ( finite_finite @ A10 ) )
=> ( comm_monoid_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Osemigroup__add_92,axiom,
! [A9: $tType,A10: $tType] :
( ( ( semigroup_add @ A9 )
& ( finite_finite @ A10 ) )
=> ( semigroup_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oab__group__add_93,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ab_group_add @ A9 )
& ( finite_finite @ A10 ) )
=> ( ab_group_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Countable_Ocountable_94,axiom,
! [A9: $tType,A10: $tType] :
( ( ( countable @ A9 )
& ( finite_finite @ A10 ) )
=> ( countable @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Omonoid__add_95,axiom,
! [A9: $tType,A10: $tType] :
( ( ( monoid_add @ A9 )
& ( finite_finite @ A10 ) )
=> ( monoid_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_96,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Olattice_97,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( lattice @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ogroup__add_98,axiom,
! [A9: $tType,A10: $tType] :
( ( ( group_add @ A9 )
& ( finite_finite @ A10 ) )
=> ( group_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Nat_Oring__char__0_99,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ring_char_0 @ A9 )
& ( finite_finite @ A10 ) )
=> ( ring_char_0 @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ominus_100,axiom,
! [A9: $tType,A10: $tType] :
( ( ( minus @ A9 )
& ( finite_finite @ A10 ) )
=> ( minus @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_101,axiom,
! [A9: $tType,A10: $tType] :
( ( ( zero @ A9 )
& ( finite_finite @ A10 ) )
=> ( zero @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oplus_102,axiom,
! [A9: $tType,A10: $tType] :
( ( ( plus @ A9 )
& ( finite_finite @ A10 ) )
=> ( plus @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y3: A] :
( ( if @ A @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y3: A] :
( ( if @ A @ $true @ X2 @ Y3 )
= X2 ) ).
% Free types (1)
thf(tfree_0,hypothesis,
finite_finite @ n ).
% Conjectures (1)
thf(conj_0,conjecture,
( member @ ( 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 @ a ) )
@ ( t2 @ a ) )
@ ( sigma_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) ) ).
%------------------------------------------------------------------------------