%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP117^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:41 AM UTC 2026
% Result : Theorem 3.01s 1.71s
% Output : Refutation 3.01s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP117^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.34 % Computer : n011.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 08:26:32 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.28/0.46 ---- Original DTF file ---
% 0.28/0.46 thf(spec,logic,$$dhol).
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % File : ITP117^2 : TPTP v9.2.1. Released v7.5.0.
% 0.28/0.46 % Domain : Interactive Theorem Proving
% 0.28/0.46 % Problem : Sledgehammer Minkowskis_Theorem problem prob_290__6248976_1
% 0.28/0.46 % Version : Especial.
% 0.28/0.46 % English :
% 0.28/0.46
% 0.28/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.28/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.28/0.46 % Source : [Des21]
% 0.28/0.46 % Names : Minkowskis_Theorem/prob_290__6248976_1 [Des21]
% 0.28/0.46
% 0.28/0.46 % Status : Theorem
% 0.28/0.46 % Rating : 0.00 v8.1.0, 0.25 v7.5.0
% 0.28/0.46 % Syntax : Number of formulae : 447 ( 177 unt; 65 typ; 0 def)
% 0.28/0.46 % Number of atoms : 829 ( 277 equ; 0 cnn)
% 0.28/0.46 % Maximal formula atoms : 6 ( 2 avg)
% 0.28/0.46 % Number of connectives : 4544 ( 28 ~; 1 |; 68 &;4161 @)
% 0.28/0.46 % ( 0 <=>; 286 =>; 0 <=; 0 <~>)
% 0.28/0.46 % Maximal formula depth : 18 ( 6 avg)
% 0.28/0.46 % Number of types : 6 ( 5 usr)
% 0.28/0.46 % Number of type conns : 166 ( 166 >; 0 *; 0 +; 0 <<)
% 0.28/0.46 % Number of symbols : 62 ( 60 usr; 2 con; 0-4 aty)
% 0.28/0.46 % Number of variables : 1135 ( 87 ^; 971 !; 22 ?;1135 :)
% 0.28/0.46 % ( 55 !>; 0 ?*; 0 @-; 0 @+)
% 0.28/0.46 % SPC : TH1_THM_EQU_NAR
% 0.28/0.46
% 0.28/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:31:28.931
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % Could-be-implicit typings (8)
% 0.28/0.46 thf(ty_t_Extended__Nonnegative__Real_Oennreal,type,
% 0.28/0.46 extend1814228343nnreal: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Finite__Cartesian__Product_Ovec,type,
% 0.28/0.46 finite_Cartesian_vec: $tType > $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Sigma__Algebra_Omeasure,type,
% 0.28/0.46 sigma_measure: $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Real_Oreal,type,
% 0.28/0.46 real: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Set_Oset,type,
% 0.28/0.46 set: $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Nat_Onat,type,
% 0.28/0.46 nat: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Int_Oint,type,
% 0.28/0.46 int: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_tf_n,type,
% 0.28/0.46 n: $tType ).
% 0.28/0.46
% 0.28/0.46 % Explicit typings (57)
% 0.28/0.46 thf(sy_cl_Rings_Oring__1,type,
% 0.28/0.46 ring_1:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
% 0.28/0.46 ordere890947078_space:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_HOL_Otype,type,
% 0.28/0.46 type:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oone,type,
% 0.28/0.46 one:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ozero,type,
% 0.28/0.46 zero:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Nat_Oring__char__0,type,
% 0.28/0.46 ring_char_0:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Num_Oneg__numeral,type,
% 0.28/0.46 neg_numeral:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.28/0.46 group_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.28/0.46 finite_finite:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Omonoid__add,type,
% 0.28/0.46 monoid_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Countable_Ocountable,type,
% 0.28/0.46 countable:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oab__group__add,type,
% 0.28/0.46 ab_group_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.28/0.46 semigroup_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Complete__Lattices_OSup,type,
% 0.28/0.46 complete_Sup:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.28/0.46 comm_monoid_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.28/0.46 ab_semigroup_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 0.28/0.46 comm_monoid_diff:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.28/0.46 cancel_semigroup_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.28/0.46 ordered_ab_group_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.28/0.46 cancel1352612707id_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.28/0.46 cancel146912293up_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.28/0.46 linord219039673up_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.28/0.46 ordere779506340up_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.28/0.46 linord1659791738miring:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 0.28/0.46 comple187826305attice:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
% 0.28/0.46 comple1035589618norder:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.28/0.46 canoni770627133id_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.28/0.46 ordere236663937imp_le:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 0.28/0.46 ordere223160158up_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 0.28/0.46 strict2144017051up_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 0.28/0.46 ordere516151231imp_le:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
% 0.28/0.46 complete_Sup_Sup:
% 0.28/0.46 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Complete__Measure_Ocompletion,type,
% 0.28/0.46 complete_completion:
% 0.28/0.46 !>[A: $tType] : ( ( sigma_measure @ A ) > ( sigma_measure @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Countable__Set_Ofrom__nat__into,type,
% 0.28/0.46 counta609264050t_into:
% 0.28/0.46 !>[A: $tType] : ( ( set @ A ) > nat > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.28/0.46 minus_minus:
% 0.28/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.28/0.46 one_one:
% 0.28/0.46 !>[A: $tType] : A ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.28/0.46 plus_plus:
% 0.28/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.28/0.46 zero_zero:
% 0.28/0.46 !>[A: $tType] : A ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.28/0.46 inf_inf:
% 0.28/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Lebesgue__Measure_Olborel,type,
% 0.28/0.46 lebesgue_lborel:
% 0.28/0.46 !>[A: $tType] : ( sigma_measure @ A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Measure__Space_Onull__sets,type,
% 0.28/0.46 measure_null_sets:
% 0.28/0.46 !>[A: $tType] : ( ( sigma_measure @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Minkowskis__Theorem__Mirabelle__jpyvdnlcjd_Oof__int__vec,type,
% 0.28/0.46 minkow1937162798nt_vec:
% 0.28/0.46 !>[B: $tType,A: $tType] : ( ( finite_Cartesian_vec @ int @ B ) > ( finite_Cartesian_vec @ A @ B ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.28/0.46 ord_less:
% 0.28/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.28/0.46 top_top:
% 0.28/0.46 !>[A: $tType] : A ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Series_Osums,type,
% 0.28/0.46 sums:
% 0.28/0.46 !>[A: $tType] : ( ( nat > A ) > A > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Set_OCollect,type,
% 0.28/0.46 collect:
% 0.28/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Set_Oimage,type,
% 0.28/0.46 image:
% 0.28/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Set_Ovimage,type,
% 0.28/0.46 vimage:
% 0.28/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Sigma__Algebra_Oemeasure,type,
% 0.28/0.46 sigma_emeasure:
% 0.28/0.46 !>[A: $tType] : ( ( sigma_measure @ A ) > ( set @ A ) > extend1814228343nnreal ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Sigma__Algebra_Osets,type,
% 0.28/0.46 sigma_sets:
% 0.28/0.46 !>[A: $tType] : ( ( sigma_measure @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Sigma__Algebra_Ospace,type,
% 0.28/0.46 sigma_space:
% 0.28/0.46 !>[A: $tType] : ( ( sigma_measure @ A ) > ( set @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_member,type,
% 0.28/0.46 member:
% 0.28/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_R____,type,
% 0.28/0.46 r: ( finite_Cartesian_vec @ int @ n ) > ( set @ ( finite_Cartesian_vec @ real @ n ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_S,type,
% 0.28/0.46 s: set @ ( finite_Cartesian_vec @ real @ n ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_T_H____,type,
% 0.28/0.46 t: ( finite_Cartesian_vec @ int @ n ) > ( set @ ( finite_Cartesian_vec @ real @ n ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_T____,type,
% 0.28/0.46 t2: ( finite_Cartesian_vec @ int @ n ) > ( set @ ( finite_Cartesian_vec @ real @ n ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_f____,type,
% 0.28/0.46 f: nat > ( finite_Cartesian_vec @ int @ n ) ).
% 0.28/0.46
% 0.28/0.46 % Relevant facts (254)
% 0.28/0.46 thf(fact_0_emeasure__T_H,axiom,
% 0.28/0.46 ! [A2: finite_Cartesian_vec @ int @ n] :
% 0.28/0.46 ( ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t @ A2 ) )
% 0.28/0.46 = ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t2 @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % emeasure_T'
% 0.28/0.46 thf(fact_1__092_060open_062_092_060And_062a_O_AT_H_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
% 0.28/0.46 ! [A2: finite_Cartesian_vec @ int @ n] : ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( t @ A2 ) @ ( sigma_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % \<open>\<And>a. T' a \<in> sets lebesgue\<close>
% 0.28/0.46 thf(fact_2__092_060open_062_092_060And_062a_O_AT_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
% 0.28/0.46 ! [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 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % \<open>\<And>a. T a \<in> sets lebesgue\<close>
% 0.28/0.46 thf(fact_3_calculation,axiom,
% 0.28/0.46 ( sums @ extend1814228343nnreal
% 0.28/0.46 @ ^ [N: nat] : ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t2 @ ( f @ N ) ) )
% 0.28/0.46 @ ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ s ) ) ).
% 0.28/0.46
% 0.28/0.46 % calculation
% 0.28/0.46 thf(fact_4_assms_I1_J,axiom,
% 0.28/0.46 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 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % assms(1)
% 0.28/0.46 thf(fact_5_emeasure__T__Int,axiom,
% 0.28/0.46 ! [A2: finite_Cartesian_vec @ int @ n,B2: finite_Cartesian_vec @ int @ n] :
% 0.28/0.46 ( ( A2 != B2 )
% 0.28/0.46 => ( ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( inf_inf @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( t2 @ A2 ) @ ( t2 @ B2 ) ) )
% 0.28/0.46 = ( zero_zero @ extend1814228343nnreal ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % emeasure_T_Int
% 0.28/0.46 thf(fact_6_T__Int,axiom,
% 0.28/0.46 ! [A2: finite_Cartesian_vec @ int @ n,B2: finite_Cartesian_vec @ int @ n] :
% 0.28/0.46 ( ( A2 != B2 )
% 0.28/0.46 => ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( inf_inf @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( t2 @ A2 ) @ ( t2 @ B2 ) ) @ ( measure_null_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % T_Int
% 0.28/0.46 thf(fact_7_assms_I2_J,axiom,
% 0.28/0.46 ord_less @ extend1814228343nnreal @ ( one_one @ extend1814228343nnreal ) @ ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ s ) ).
% 0.28/0.46
% 0.28/0.46 % assms(2)
% 0.28/0.46 thf(fact_8__092_060open_062_092_060And_062a_O_AR_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
% 0.28/0.46 ! [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 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % \<open>\<And>a. R a \<in> sets lebesgue\<close>
% 0.28/0.46 thf(fact_9__092_060open_062_I_092_060lambda_062n_O_Aemeasure_Alebesgue_A_IT_A_If_An_J_J_J_Asums_Aemeasure_Alebesgue_A_I_092_060Union_062n_O_AT_A_If_An_J_J_092_060close_062,axiom,
% 0.28/0.46 ( sums @ extend1814228343nnreal
% 0.28/0.46 @ ^ [N: nat] : ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t2 @ ( f @ N ) ) )
% 0.28/0.46 @ ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) )
% 0.28/0.46 @ ( complete_Sup_Sup @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
% 0.28/0.46 @ ( image @ nat @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
% 0.28/0.46 @ ^ [N: nat] : ( t2 @ ( f @ N ) )
% 0.28/0.46 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % \<open>(\<lambda>n. emeasure lebesgue (T (f n))) sums emeasure lebesgue (\<Union>n. T (f n))\<close>
% 0.28/0.46 thf(fact_10_T_H__def,axiom,
% 0.28/0.46 ( t
% 0.28/0.46 = ( ^ [A3: finite_Cartesian_vec @ int @ n] :
% 0.28/0.46 ( image @ ( finite_Cartesian_vec @ real @ n ) @ ( finite_Cartesian_vec @ real @ n )
% 0.28/0.46 @ ^ [X: finite_Cartesian_vec @ real @ n] : ( minus_minus @ ( finite_Cartesian_vec @ real @ n ) @ X @ ( minkow1937162798nt_vec @ n @ real @ A3 ) )
% 0.28/0.46 @ ( t2 @ A3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % T'_def
% 0.28/0.46 thf(fact_11_f__def,axiom,
% 0.28/0.46 ( f
% 0.28/0.46 = ( counta609264050t_into @ ( finite_Cartesian_vec @ int @ n ) @ ( top_top @ ( set @ ( finite_Cartesian_vec @ int @ n ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % f_def
% 0.28/0.46 thf(fact_12_T_H__altdef,axiom,
% 0.28/0.46 ! [A2: finite_Cartesian_vec @ int @ n] :
% 0.28/0.46 ( ( t @ A2 )
% 0.28/0.46 = ( vimage @ ( finite_Cartesian_vec @ real @ n ) @ ( finite_Cartesian_vec @ real @ n )
% 0.28/0.46 @ ^ [X: finite_Cartesian_vec @ real @ n] : ( plus_plus @ ( finite_Cartesian_vec @ real @ n ) @ X @ ( minkow1937162798nt_vec @ n @ real @ A2 ) )
% 0.28/0.46 @ ( t2 @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % T'_altdef
% 0.28/0.46 thf(fact_13_of__int__vec__eq__iff,axiom,
% 0.28/0.46 ! [A: $tType,N2: $tType] :
% 0.28/0.46 ( ( ( finite_finite @ N2 )
% 0.28/0.46 & ( ring_char_0 @ A ) )
% 0.28/0.46 => ! [A2: finite_Cartesian_vec @ int @ N2,B2: finite_Cartesian_vec @ int @ N2] :
% 0.28/0.46 ( ( ( minkow1937162798nt_vec @ N2 @ A @ A2 )
% 0.28/0.46 = ( minkow1937162798nt_vec @ N2 @ A @ B2 ) )
% 0.28/0.46 = ( A2 = B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % of_int_vec_eq_iff
% 0.28/0.46 thf(fact_14_T__def,axiom,
% 0.28/0.46 ( t2
% 0.28/0.46 = ( ^ [A3: finite_Cartesian_vec @ int @ n] : ( inf_inf @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ s @ ( r @ A3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % T_def
% 0.28/0.46 thf(fact_15_R__Int,axiom,
% 0.28/0.46 ! [A2: finite_Cartesian_vec @ int @ n,B2: finite_Cartesian_vec @ int @ n] :
% 0.28/0.46 ( ( A2 != B2 )
% 0.28/0.46 => ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( inf_inf @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( r @ A2 ) @ ( r @ B2 ) ) @ ( measure_null_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % R_Int
% 0.28/0.46 thf(fact_16_sums__emeasure_H,axiom,
% 0.28/0.46 ! [A: $tType,B3: nat > ( set @ A ),M: sigma_measure @ A] :
% 0.28/0.46 ( ! [X2: nat] : ( member @ ( set @ A ) @ ( B3 @ X2 ) @ ( sigma_sets @ A @ M ) )
% 0.28/0.46 => ( ! [X2: nat,Y: nat] :
% 0.28/0.46 ( ( X2 != Y )
% 0.28/0.46 => ( ( sigma_emeasure @ A @ M @ ( inf_inf @ ( set @ A ) @ ( B3 @ X2 ) @ ( B3 @ Y ) ) )
% 0.28/0.46 = ( zero_zero @ extend1814228343nnreal ) ) )
% 0.28/0.46 => ( sums @ extend1814228343nnreal
% 0.28/0.46 @ ^ [X: nat] : ( sigma_emeasure @ A @ M @ ( B3 @ X ) )
% 0.28/0.46 @ ( sigma_emeasure @ A @ M @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sums_emeasure'
% 0.28/0.46 thf(fact_17_null__sets__UN,axiom,
% 0.28/0.46 ! [I: $tType,A: $tType] :
% 0.28/0.46 ( ( countable @ I )
% 0.28/0.46 => ! [N3: I > ( set @ A ),M: sigma_measure @ A] :
% 0.28/0.46 ( ! [I2: I] : ( member @ ( set @ A ) @ ( N3 @ I2 ) @ ( measure_null_sets @ A @ M ) )
% 0.28/0.46 => ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ I @ ( set @ A ) @ N3 @ ( top_top @ ( set @ I ) ) ) ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % null_sets_UN
% 0.28/0.46 thf(fact_18_SUP__eq__top__iff,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( comple1035589618norder @ A )
% 0.28/0.46 => ! [F: B > A,A4: set @ B] :
% 0.28/0.46 ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A4 ) )
% 0.28/0.46 = ( top_top @ A ) )
% 0.28/0.46 = ( ! [X: A] :
% 0.28/0.46 ( ( ord_less @ A @ X @ ( top_top @ A ) )
% 0.28/0.46 => ? [Y2: B] :
% 0.28/0.46 ( ( member @ B @ Y2 @ A4 )
% 0.28/0.46 & ( ord_less @ A @ X @ ( F @ Y2 ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % SUP_eq_top_iff
% 0.28/0.46 thf(fact_19_null__setsI,axiom,
% 0.28/0.46 ! [A: $tType,M: sigma_measure @ A,A4: set @ A] :
% 0.28/0.46 ( ( ( sigma_emeasure @ A @ M @ A4 )
% 0.28/0.46 = ( zero_zero @ extend1814228343nnreal ) )
% 0.28/0.46 => ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) )
% 0.28/0.46 => ( member @ ( set @ A ) @ A4 @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % null_setsI
% 0.28/0.46 thf(fact_20_image__vimage__eq,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,F: B > A,A4: set @ A] :
% 0.28/0.46 ( ( image @ B @ A @ F @ ( vimage @ B @ A @ F @ A4 ) )
% 0.28/0.46 = ( inf_inf @ ( set @ A ) @ A4 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % image_vimage_eq
% 0.28/0.46 thf(fact_21__092_060open_062_092_060And_062a_O_AT_Aa_A_092_060in_062_Asets_Alebesgue_A_092_060Longrightarrow_062_A_I_092_060lambda_062x_O_Ax_A_L_Aof__int__vec_Aa_J_A_N_096_AT_Aa_A_092_060inter_062_Aspace_Alebesgue_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
% 0.28/0.46 ! [A5: finite_Cartesian_vec @ int @ n] :
% 0.28/0.46 ( ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( t2 @ A5 ) @ ( sigma_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) )
% 0.28/0.46 => ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
% 0.28/0.46 @ ( inf_inf @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
% 0.28/0.46 @ ( vimage @ ( finite_Cartesian_vec @ real @ n ) @ ( finite_Cartesian_vec @ real @ n )
% 0.28/0.46 @ ^ [X: finite_Cartesian_vec @ real @ n] : ( plus_plus @ ( finite_Cartesian_vec @ real @ n ) @ X @ ( minkow1937162798nt_vec @ n @ real @ A5 ) )
% 0.28/0.46 @ ( t2 @ A5 ) )
% 0.28/0.46 @ ( sigma_space @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) )
% 0.28/0.46 @ ( sigma_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % \<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>
% 0.28/0.46 thf(fact_22_surj__diff__right,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ab_group_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( image @ A @ A
% 0.28/0.46 @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
% 0.28/0.46 @ ( top_top @ ( set @ A ) ) )
% 0.28/0.46 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % surj_diff_right
% 0.28/0.46 thf(fact_23__092_060open_062_092_060Union_062_A_Irange_AT_J_A_061_A_I_092_060Union_062n_O_AT_A_If_An_J_J_092_060close_062,axiom,
% 0.28/0.46 ( ( complete_Sup_Sup @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( image @ ( finite_Cartesian_vec @ int @ n ) @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ t2 @ ( top_top @ ( set @ ( finite_Cartesian_vec @ int @ n ) ) ) ) )
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
% 0.28/0.46 @ ( image @ nat @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
% 0.28/0.46 @ ^ [N: nat] : ( t2 @ ( f @ N ) )
% 0.28/0.46 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % \<open>\<Union> (range T) = (\<Union>n. T (f n))\<close>
% 0.28/0.46 thf(fact_24_Sup__UNIV,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( comple187826305attice @ A )
% 0.28/0.46 => ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
% 0.28/0.46 = ( top_top @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Sup_UNIV
% 0.28/0.46 thf(fact_25_Sup__eq__top__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( comple1035589618norder @ A )
% 0.28/0.46 => ! [A4: set @ A] :
% 0.28/0.46 ( ( ( complete_Sup_Sup @ A @ A4 )
% 0.28/0.46 = ( top_top @ A ) )
% 0.28/0.46 = ( ! [X: A] :
% 0.28/0.46 ( ( ord_less @ A @ X @ ( top_top @ A ) )
% 0.28/0.46 => ? [Y2: A] :
% 0.28/0.46 ( ( member @ A @ Y2 @ A4 )
% 0.28/0.46 & ( ord_less @ A @ X @ Y2 ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Sup_eq_top_iff
% 0.28/0.46 thf(fact_26_surj__plus,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ab_group_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
% 0.28/0.46 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % surj_plus
% 0.28/0.46 thf(fact_27_diff__numeral__special_I9_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( neg_numeral @ A )
% 0.28/0.46 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 0.28/0.46 = ( zero_zero @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_numeral_special(9)
% 0.28/0.46 thf(fact_28_image__eqI,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,B2: A,F: B > A,X3: B,A4: set @ B] :
% 0.28/0.46 ( ( B2
% 0.28/0.46 = ( F @ X3 ) )
% 0.28/0.46 => ( ( member @ B @ X3 @ A4 )
% 0.28/0.46 => ( member @ A @ B2 @ ( image @ B @ A @ F @ A4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % image_eqI
% 0.28/0.46 thf(fact_29_UNIV__I,axiom,
% 0.28/0.46 ! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UNIV_I
% 0.28/0.46 thf(fact_30_Int__iff,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
% 0.28/0.46 = ( ( member @ A @ C @ A4 )
% 0.28/0.46 & ( member @ A @ C @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_iff
% 0.28/0.46 thf(fact_31_IntI,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ A4 )
% 0.28/0.46 => ( ( member @ A @ C @ B3 )
% 0.28/0.46 => ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % IntI
% 0.28/0.46 thf(fact_32_Union__iff,axiom,
% 0.28/0.46 ! [A: $tType,A4: A,C2: set @ ( set @ A )] :
% 0.28/0.46 ( ( member @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) )
% 0.28/0.46 = ( ? [X: set @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ X @ C2 )
% 0.28/0.46 & ( member @ A @ A4 @ X ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Union_iff
% 0.28/0.46 thf(fact_33_UnionI,axiom,
% 0.28/0.46 ! [A: $tType,X4: set @ A,C2: set @ ( set @ A ),A4: A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ X4 @ C2 )
% 0.28/0.46 => ( ( member @ A @ A4 @ X4 )
% 0.28/0.46 => ( member @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UnionI
% 0.28/0.46 thf(fact_34_UN__ball__bex__simps_I1_J,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ ( set @ A ),P: A > $o] :
% 0.28/0.46 ( ( ! [X: A] :
% 0.28/0.46 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) )
% 0.28/0.46 => ( P @ X ) ) )
% 0.28/0.46 = ( ! [X: set @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ X @ A4 )
% 0.28/0.46 => ! [Y2: A] :
% 0.28/0.46 ( ( member @ A @ Y2 @ X )
% 0.28/0.46 => ( P @ Y2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UN_ball_bex_simps(1)
% 0.28/0.46 thf(fact_35_UN__ball__bex__simps_I3_J,axiom,
% 0.28/0.46 ! [D: $tType,A4: set @ ( set @ D ),P: D > $o] :
% 0.28/0.46 ( ( ? [X: D] :
% 0.28/0.46 ( ( member @ D @ X @ ( complete_Sup_Sup @ ( set @ D ) @ A4 ) )
% 0.28/0.46 & ( P @ X ) ) )
% 0.28/0.46 = ( ? [X: set @ D] :
% 0.28/0.46 ( ( member @ ( set @ D ) @ X @ A4 )
% 0.28/0.46 & ? [Y2: D] :
% 0.28/0.46 ( ( member @ D @ Y2 @ X )
% 0.28/0.46 & ( P @ Y2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UN_ball_bex_simps(3)
% 0.28/0.46 thf(fact_36_null__sets_ODiff,axiom,
% 0.28/0.46 ! [A: $tType,A2: set @ A,M: sigma_measure @ A,B2: set @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ A2 @ ( measure_null_sets @ A @ M ) )
% 0.28/0.46 => ( ( member @ ( set @ A ) @ B2 @ ( measure_null_sets @ A @ M ) )
% 0.28/0.46 => ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % null_sets.Diff
% 0.28/0.46 thf(fact_37_vimage__eq,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,A2: A,F: A > B,B3: set @ B] :
% 0.28/0.46 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ B3 ) )
% 0.28/0.46 = ( member @ B @ ( F @ A2 ) @ B3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % vimage_eq
% 0.28/0.46 thf(fact_38_vimageI,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,F: B > A,A2: B,B2: A,B3: set @ A] :
% 0.28/0.46 ( ( ( F @ A2 )
% 0.28/0.46 = B2 )
% 0.28/0.46 => ( ( member @ A @ B2 @ B3 )
% 0.28/0.46 => ( member @ B @ A2 @ ( vimage @ B @ A @ F @ B3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % vimageI
% 0.28/0.46 thf(fact_39_image__ident,axiom,
% 0.28/0.46 ! [A: $tType,Y3: set @ A] :
% 0.28/0.46 ( ( image @ A @ A
% 0.28/0.46 @ ^ [X: A] : X
% 0.28/0.46 @ Y3 )
% 0.28/0.46 = Y3 ) ).
% 0.28/0.46
% 0.28/0.46 % image_ident
% 0.28/0.46 thf(fact_40_Sup__apply,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( complete_Sup @ B )
% 0.28/0.46 => ( ( complete_Sup_Sup @ ( A > B ) )
% 0.28/0.46 = ( ^ [A6: set @ ( A > B ),X: A] :
% 0.28/0.46 ( complete_Sup_Sup @ B
% 0.28/0.46 @ ( image @ ( A > B ) @ B
% 0.28/0.46 @ ^ [F2: A > B] : ( F2 @ X )
% 0.28/0.46 @ A6 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Sup_apply
% 0.28/0.46 thf(fact_41_vimage__Collect__eq,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,F: A > B,P: B > $o] :
% 0.28/0.46 ( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
% 0.28/0.46 = ( collect @ A
% 0.28/0.46 @ ^ [Y2: A] : ( P @ ( F @ Y2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % vimage_Collect_eq
% 0.28/0.46 thf(fact_42_vimage__ident,axiom,
% 0.28/0.46 ! [A: $tType,Y3: set @ A] :
% 0.28/0.46 ( ( vimage @ A @ A
% 0.28/0.46 @ ^ [X: A] : X
% 0.28/0.46 @ Y3 )
% 0.28/0.46 = Y3 ) ).
% 0.28/0.46
% 0.28/0.46 % vimage_ident
% 0.28/0.46 thf(fact_43_Int__UNIV,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
% 0.28/0.46 = ( top_top @ ( set @ A ) ) )
% 0.28/0.46 = ( ( A4
% 0.28/0.46 = ( top_top @ ( set @ A ) ) )
% 0.28/0.46 & ( B3
% 0.28/0.46 = ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_UNIV
% 0.28/0.46 thf(fact_44_ball__UN,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,B3: B > ( set @ A ),A4: set @ B,P: A > $o] :
% 0.28/0.46 ( ( ! [X: A] :
% 0.28/0.46 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A4 ) ) )
% 0.28/0.46 => ( P @ X ) ) )
% 0.28/0.46 = ( ! [X: B] :
% 0.28/0.46 ( ( member @ B @ X @ A4 )
% 0.28/0.46 => ! [Y2: A] :
% 0.28/0.46 ( ( member @ A @ Y2 @ ( B3 @ X ) )
% 0.28/0.46 => ( P @ Y2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ball_UN
% 0.28/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.28/0.46 ! [A: $tType,A2: A,P: A > $o] :
% 0.28/0.46 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.28/0.46 = ( P @ A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % mem_Collect_eq
% 0.28/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A] :
% 0.28/0.46 ( ( collect @ A
% 0.28/0.46 @ ^ [X: A] : ( member @ A @ X @ A4 ) )
% 0.28/0.46 = A4 ) ).
% 0.28/0.46
% 0.28/0.46 % Collect_mem_eq
% 0.28/0.46 thf(fact_47_Collect__cong,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.46 ( ! [X2: A] :
% 0.28/0.46 ( ( P @ X2 )
% 0.28/0.46 = ( Q @ X2 ) )
% 0.28/0.46 => ( ( collect @ A @ P )
% 0.28/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Collect_cong
% 0.28/0.46 thf(fact_48_ext,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.28/0.46 ( ! [X2: A] :
% 0.28/0.46 ( ( F @ X2 )
% 0.28/0.46 = ( G @ X2 ) )
% 0.28/0.46 => ( F = G ) ) ).
% 0.28/0.46
% 0.28/0.46 % ext
% 0.28/0.46 thf(fact_49_bex__UN,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,B3: B > ( set @ A ),A4: set @ B,P: A > $o] :
% 0.28/0.46 ( ( ? [X: A] :
% 0.28/0.46 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A4 ) ) )
% 0.28/0.46 & ( P @ X ) ) )
% 0.28/0.46 = ( ? [X: B] :
% 0.28/0.46 ( ( member @ B @ X @ A4 )
% 0.28/0.46 & ? [Y2: A] :
% 0.28/0.46 ( ( member @ A @ Y2 @ ( B3 @ X ) )
% 0.28/0.46 & ( P @ Y2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % bex_UN
% 0.28/0.46 thf(fact_50_UN__ball__bex__simps_I2_J,axiom,
% 0.28/0.46 ! [C3: $tType,B: $tType,B3: B > ( set @ C3 ),A4: set @ B,P: C3 > $o] :
% 0.28/0.46 ( ( ! [X: C3] :
% 0.28/0.46 ( ( member @ C3 @ X @ ( complete_Sup_Sup @ ( set @ C3 ) @ ( image @ B @ ( set @ C3 ) @ B3 @ A4 ) ) )
% 0.28/0.46 => ( P @ X ) ) )
% 0.28/0.46 = ( ! [X: B] :
% 0.28/0.46 ( ( member @ B @ X @ A4 )
% 0.28/0.46 => ! [Y2: C3] :
% 0.28/0.46 ( ( member @ C3 @ Y2 @ ( B3 @ X ) )
% 0.28/0.46 => ( P @ Y2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UN_ball_bex_simps(2)
% 0.28/0.46 thf(fact_51_UN__ball__bex__simps_I4_J,axiom,
% 0.28/0.46 ! [F3: $tType,E: $tType,B3: E > ( set @ F3 ),A4: set @ E,P: F3 > $o] :
% 0.28/0.46 ( ( ? [X: F3] :
% 0.28/0.46 ( ( member @ F3 @ X @ ( complete_Sup_Sup @ ( set @ F3 ) @ ( image @ E @ ( set @ F3 ) @ B3 @ A4 ) ) )
% 0.28/0.46 & ( P @ X ) ) )
% 0.28/0.46 = ( ? [X: E] :
% 0.28/0.46 ( ( member @ E @ X @ A4 )
% 0.28/0.46 & ? [Y2: F3] :
% 0.28/0.46 ( ( member @ F3 @ Y2 @ ( B3 @ X ) )
% 0.28/0.46 & ( P @ Y2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UN_ball_bex_simps(4)
% 0.28/0.46 thf(fact_52_vimage__UNIV,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,F: A > B] :
% 0.28/0.46 ( ( vimage @ A @ B @ F @ ( top_top @ ( set @ B ) ) )
% 0.28/0.46 = ( top_top @ ( set @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % vimage_UNIV
% 0.28/0.46 thf(fact_53_null__sets_OInt,axiom,
% 0.28/0.46 ! [A: $tType,A2: set @ A,M: sigma_measure @ A,B2: set @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ A2 @ ( measure_null_sets @ A @ M ) )
% 0.28/0.46 => ( ( member @ ( set @ A ) @ B2 @ ( measure_null_sets @ A @ M ) )
% 0.28/0.46 => ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % null_sets.Int
% 0.28/0.46 thf(fact_54_vimage__Int,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,F: A > B,A4: set @ B,B3: set @ B] :
% 0.28/0.46 ( ( vimage @ A @ B @ F @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) )
% 0.28/0.46 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ A4 ) @ ( vimage @ A @ B @ F @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % vimage_Int
% 0.28/0.46 thf(fact_55_SUP__identity__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( complete_Sup @ A )
% 0.28/0.46 => ! [A4: set @ A] :
% 0.28/0.46 ( ( complete_Sup_Sup @ A
% 0.28/0.46 @ ( image @ A @ A
% 0.28/0.46 @ ^ [X: A] : X
% 0.28/0.46 @ A4 ) )
% 0.28/0.46 = ( complete_Sup_Sup @ A @ A4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % SUP_identity_eq
% 0.28/0.46 thf(fact_56_SUP__apply,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,C3: $tType] :
% 0.28/0.46 ( ( complete_Sup @ A )
% 0.28/0.46 => ! [F: C3 > B > A,A4: set @ C3,X3: B] :
% 0.28/0.46 ( ( complete_Sup_Sup @ ( B > A ) @ ( image @ C3 @ ( B > A ) @ F @ A4 ) @ X3 )
% 0.28/0.46 = ( complete_Sup_Sup @ A
% 0.28/0.46 @ ( image @ C3 @ A
% 0.28/0.46 @ ^ [Y2: C3] : ( F @ Y2 @ X3 )
% 0.28/0.46 @ A4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % SUP_apply
% 0.28/0.46 thf(fact_57_UN__iff,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,B2: A,B3: B > ( set @ A ),A4: set @ B] :
% 0.28/0.46 ( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A4 ) ) )
% 0.28/0.46 = ( ? [X: B] :
% 0.28/0.46 ( ( member @ B @ X @ A4 )
% 0.28/0.46 & ( member @ A @ B2 @ ( B3 @ X ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UN_iff
% 0.28/0.46 thf(fact_58_UN__I,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,A2: A,A4: set @ A,B2: B,B3: A > ( set @ B )] :
% 0.28/0.46 ( ( member @ A @ A2 @ A4 )
% 0.28/0.46 => ( ( member @ B @ B2 @ ( B3 @ A2 ) )
% 0.28/0.46 => ( member @ B @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A4 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UN_I
% 0.28/0.46 thf(fact_59_S__decompose,axiom,
% 0.28/0.46 ( s
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( image @ ( finite_Cartesian_vec @ int @ n ) @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ t2 @ ( top_top @ ( set @ ( finite_Cartesian_vec @ int @ n ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % S_decompose
% 0.28/0.46 thf(fact_60_image__add__0,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( comm_monoid_add @ A )
% 0.28/0.46 => ! [S: set @ A] :
% 0.28/0.46 ( ( image @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S )
% 0.28/0.46 = S ) ) ).
% 0.28/0.46
% 0.28/0.46 % image_add_0
% 0.28/0.46 thf(fact_61_null__sets_OInt__space__eq2,axiom,
% 0.28/0.46 ! [A: $tType,X3: set @ A,M: sigma_measure @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ X3 @ ( measure_null_sets @ A @ M ) )
% 0.28/0.46 => ( ( inf_inf @ ( set @ A ) @ X3 @ ( sigma_space @ A @ M ) )
% 0.28/0.46 = X3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % null_sets.Int_space_eq2
% 0.28/0.46 thf(fact_62_null__sets_OInt__space__eq1,axiom,
% 0.28/0.46 ! [A: $tType,X3: set @ A,M: sigma_measure @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ X3 @ ( measure_null_sets @ A @ M ) )
% 0.28/0.46 => ( ( inf_inf @ ( set @ A ) @ ( sigma_space @ A @ M ) @ X3 )
% 0.28/0.46 = X3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % null_sets.Int_space_eq1
% 0.28/0.46 thf(fact_63_in__sets__SUP,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,I3: A,I4: set @ A,M: A > ( sigma_measure @ B ),Y3: set @ B,X4: set @ B] :
% 0.28/0.46 ( ( member @ A @ I3 @ I4 )
% 0.28/0.46 => ( ! [I2: A] :
% 0.28/0.46 ( ( member @ A @ I2 @ I4 )
% 0.28/0.46 => ( ( sigma_space @ B @ ( M @ I2 ) )
% 0.28/0.46 = Y3 ) )
% 0.28/0.46 => ( ( member @ ( set @ B ) @ X4 @ ( sigma_sets @ B @ ( M @ I3 ) ) )
% 0.28/0.46 => ( member @ ( set @ B ) @ X4 @ ( sigma_sets @ B @ ( complete_Sup_Sup @ ( sigma_measure @ B ) @ ( image @ A @ ( sigma_measure @ B ) @ M @ I4 ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % in_sets_SUP
% 0.28/0.46 thf(fact_64_in__sets__Sup,axiom,
% 0.28/0.46 ! [A: $tType,M: set @ ( sigma_measure @ A ),X4: set @ A,M2: sigma_measure @ A,A4: set @ A] :
% 0.28/0.46 ( ! [M3: sigma_measure @ A] :
% 0.28/0.46 ( ( member @ ( sigma_measure @ A ) @ M3 @ M )
% 0.28/0.46 => ( ( sigma_space @ A @ M3 )
% 0.28/0.46 = X4 ) )
% 0.28/0.46 => ( ( member @ ( sigma_measure @ A ) @ M2 @ M )
% 0.28/0.46 => ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M2 ) )
% 0.28/0.46 => ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ ( complete_Sup_Sup @ ( sigma_measure @ A ) @ M ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % in_sets_Sup
% 0.28/0.46 thf(fact_65_sets__SUP__cong,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,I4: set @ A,M: A > ( sigma_measure @ B ),N3: A > ( sigma_measure @ B )] :
% 0.28/0.46 ( ! [I2: A] :
% 0.28/0.46 ( ( member @ A @ I2 @ I4 )
% 0.28/0.46 => ( ( sigma_sets @ B @ ( M @ I2 ) )
% 0.28/0.46 = ( sigma_sets @ B @ ( N3 @ I2 ) ) ) )
% 0.28/0.46 => ( ( sigma_sets @ B @ ( complete_Sup_Sup @ ( sigma_measure @ B ) @ ( image @ A @ ( sigma_measure @ B ) @ M @ I4 ) ) )
% 0.28/0.46 = ( sigma_sets @ B @ ( complete_Sup_Sup @ ( sigma_measure @ B ) @ ( image @ A @ ( sigma_measure @ B ) @ N3 @ I4 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sets_SUP_cong
% 0.28/0.46 thf(fact_66_Sup__set__def,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( complete_Sup_Sup @ ( set @ A ) )
% 0.28/0.46 = ( ^ [A6: set @ ( set @ A )] :
% 0.28/0.46 ( collect @ A
% 0.28/0.46 @ ^ [X: A] : ( complete_Sup_Sup @ $o @ ( image @ ( set @ A ) @ $o @ ( member @ A @ X ) @ A6 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Sup_set_def
% 0.28/0.46 thf(fact_67_space__Sup__eq__UN,axiom,
% 0.28/0.46 ! [A: $tType,M: set @ ( sigma_measure @ A )] :
% 0.28/0.46 ( ( sigma_space @ A @ ( complete_Sup_Sup @ ( sigma_measure @ A ) @ M ) )
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( sigma_measure @ A ) @ ( set @ A ) @ ( sigma_space @ A ) @ M ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % space_Sup_eq_UN
% 0.28/0.46 thf(fact_68_top__set__def,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( top_top @ ( set @ A ) )
% 0.28/0.46 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % top_set_def
% 0.28/0.46 thf(fact_69_Diff__Int__distrib2,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A,B3: set @ A,C2: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ C2 )
% 0.28/0.46 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Diff_Int_distrib2
% 0.28/0.46 thf(fact_70_Diff__Int__distrib,axiom,
% 0.28/0.46 ! [A: $tType,C2: set @ A,A4: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 0.28/0.46 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C2 @ A4 ) @ ( inf_inf @ ( set @ A ) @ C2 @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Diff_Int_distrib
% 0.28/0.46 thf(fact_71_Diff__Diff__Int,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( minus_minus @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 0.28/0.46 = ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % Diff_Diff_Int
% 0.28/0.46 thf(fact_72_Diff__Int2,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A,C2: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) )
% 0.28/0.46 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) @ B3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % Diff_Int2
% 0.28/0.46 thf(fact_73_Int__Diff,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A,B3: set @ A,C2: set @ A] :
% 0.28/0.46 ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ C2 )
% 0.28/0.46 = ( inf_inf @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B3 @ C2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_Diff
% 0.28/0.46 thf(fact_74_vimage__Diff,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,F: A > B,A4: set @ B,B3: set @ B] :
% 0.28/0.46 ( ( vimage @ A @ B @ F @ ( minus_minus @ ( set @ B ) @ A4 @ B3 ) )
% 0.28/0.46 = ( minus_minus @ ( set @ A ) @ ( vimage @ A @ B @ F @ A4 ) @ ( vimage @ A @ B @ F @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % vimage_Diff
% 0.28/0.46 thf(fact_75_Sup__fun__def,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( complete_Sup @ B )
% 0.28/0.46 => ( ( complete_Sup_Sup @ ( A > B ) )
% 0.28/0.46 = ( ^ [A6: set @ ( A > B ),X: A] :
% 0.28/0.46 ( complete_Sup_Sup @ B
% 0.28/0.46 @ ( image @ ( A > B ) @ B
% 0.28/0.46 @ ^ [F2: A > B] : ( F2 @ X )
% 0.28/0.46 @ A6 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Sup_fun_def
% 0.28/0.46 thf(fact_76_null__sets_Osets__Collect__disj,axiom,
% 0.28/0.46 ! [A: $tType,M: sigma_measure @ A,P: A > $o,Q: A > $o] :
% 0.28/0.46 ( ( member @ ( set @ A )
% 0.28/0.46 @ ( collect @ A
% 0.28/0.46 @ ^ [X: A] :
% 0.28/0.46 ( ( member @ A @ X @ ( sigma_space @ A @ M ) )
% 0.28/0.46 & ( P @ X ) ) )
% 0.28/0.46 @ ( measure_null_sets @ A @ M ) )
% 0.28/0.46 => ( ( member @ ( set @ A )
% 0.28/0.46 @ ( collect @ A
% 0.28/0.46 @ ^ [X: A] :
% 0.28/0.46 ( ( member @ A @ X @ ( sigma_space @ A @ M ) )
% 0.28/0.46 & ( Q @ X ) ) )
% 0.28/0.46 @ ( measure_null_sets @ A @ M ) )
% 0.28/0.46 => ( member @ ( set @ A )
% 0.28/0.46 @ ( collect @ A
% 0.28/0.46 @ ^ [X: A] :
% 0.28/0.46 ( ( member @ A @ X @ ( sigma_space @ A @ M ) )
% 0.28/0.46 & ( ( Q @ X )
% 0.28/0.46 | ( P @ X ) ) ) )
% 0.28/0.46 @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % null_sets.sets_Collect_disj
% 0.28/0.46 thf(fact_77_null__sets_Osets__Collect__conj,axiom,
% 0.28/0.46 ! [A: $tType,M: sigma_measure @ A,P: A > $o,Q: A > $o] :
% 0.28/0.46 ( ( member @ ( set @ A )
% 0.28/0.46 @ ( collect @ A
% 0.28/0.46 @ ^ [X: A] :
% 0.28/0.46 ( ( member @ A @ X @ ( sigma_space @ A @ M ) )
% 0.28/0.46 & ( P @ X ) ) )
% 0.28/0.46 @ ( measure_null_sets @ A @ M ) )
% 0.28/0.46 => ( ( member @ ( set @ A )
% 0.28/0.46 @ ( collect @ A
% 0.28/0.46 @ ^ [X: A] :
% 0.28/0.46 ( ( member @ A @ X @ ( sigma_space @ A @ M ) )
% 0.28/0.46 & ( Q @ X ) ) )
% 0.28/0.46 @ ( measure_null_sets @ A @ M ) )
% 0.28/0.46 => ( member @ ( set @ A )
% 0.28/0.46 @ ( collect @ A
% 0.28/0.46 @ ^ [X: A] :
% 0.28/0.46 ( ( member @ A @ X @ ( sigma_space @ A @ M ) )
% 0.28/0.46 & ( Q @ X )
% 0.28/0.46 & ( P @ X ) ) )
% 0.28/0.46 @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % null_sets.sets_Collect_conj
% 0.28/0.46 thf(fact_78_translation__diff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ab_group_add @ A )
% 0.28/0.46 => ! [A2: A,S2: set @ A,T: set @ A] :
% 0.28/0.46 ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( minus_minus @ ( set @ A ) @ S2 @ T ) )
% 0.28/0.46 = ( minus_minus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S2 ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % translation_diff
% 0.28/0.46 thf(fact_79_null__set__Diff,axiom,
% 0.28/0.46 ! [A: $tType,B3: set @ A,M: sigma_measure @ A,A4: set @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ B3 @ ( measure_null_sets @ A @ M ) )
% 0.28/0.46 => ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) )
% 0.28/0.46 => ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % null_set_Diff
% 0.28/0.46 thf(fact_80_Int__Union2,axiom,
% 0.28/0.46 ! [A: $tType,B3: set @ ( set @ A ),A4: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) @ A4 )
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ A )
% 0.28/0.46 @ ( image @ ( set @ A ) @ ( set @ A )
% 0.28/0.46 @ ^ [C4: set @ A] : ( inf_inf @ ( set @ A ) @ C4 @ A4 )
% 0.28/0.46 @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_Union2
% 0.28/0.46 thf(fact_81_Int__Union,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A,B3: set @ ( set @ A )] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) )
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 ) @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_Union
% 0.28/0.46 thf(fact_82_vimage__Union,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,F: A > B,A4: set @ ( set @ B )] :
% 0.28/0.46 ( ( vimage @ A @ B @ F @ ( complete_Sup_Sup @ ( set @ B ) @ A4 ) )
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( vimage @ A @ B @ F ) @ A4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % vimage_Union
% 0.28/0.46 thf(fact_83_UN__extend__simps_I6_J,axiom,
% 0.28/0.46 ! [L: $tType,K: $tType,A4: K > ( set @ L ),C2: set @ K,B3: set @ L] :
% 0.28/0.46 ( ( minus_minus @ ( set @ L ) @ ( complete_Sup_Sup @ ( set @ L ) @ ( image @ K @ ( set @ L ) @ A4 @ C2 ) ) @ B3 )
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ L )
% 0.28/0.46 @ ( image @ K @ ( set @ L )
% 0.28/0.46 @ ^ [X: K] : ( minus_minus @ ( set @ L ) @ ( A4 @ X ) @ B3 )
% 0.28/0.46 @ C2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UN_extend_simps(6)
% 0.28/0.46 thf(fact_84_emeasure__Diff__null__set,axiom,
% 0.28/0.46 ! [A: $tType,B3: set @ A,M: sigma_measure @ A,A4: set @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ B3 @ ( measure_null_sets @ A @ M ) )
% 0.28/0.46 => ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) )
% 0.28/0.46 => ( ( sigma_emeasure @ A @ M @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 0.28/0.46 = ( sigma_emeasure @ A @ M @ A4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % emeasure_Diff_null_set
% 0.28/0.46 thf(fact_85_Sup_OSUP__cong,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,A4: set @ B,B3: set @ B,C2: B > A,D2: B > A,Sup: ( set @ A ) > A] :
% 0.28/0.46 ( ( A4 = B3 )
% 0.28/0.46 => ( ! [X2: B] :
% 0.28/0.46 ( ( member @ B @ X2 @ B3 )
% 0.28/0.46 => ( ( C2 @ X2 )
% 0.28/0.46 = ( D2 @ X2 ) ) )
% 0.28/0.46 => ( ( Sup @ ( image @ B @ A @ C2 @ A4 ) )
% 0.28/0.46 = ( Sup @ ( image @ B @ A @ D2 @ B3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Sup.SUP_cong
% 0.28/0.46 thf(fact_86_Inf_OINF__cong,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,A4: set @ B,B3: set @ B,C2: B > A,D2: B > A,Inf: ( set @ A ) > A] :
% 0.28/0.46 ( ( A4 = B3 )
% 0.28/0.46 => ( ! [X2: B] :
% 0.28/0.46 ( ( member @ B @ X2 @ B3 )
% 0.28/0.46 => ( ( C2 @ X2 )
% 0.28/0.46 = ( D2 @ X2 ) ) )
% 0.28/0.46 => ( ( Inf @ ( image @ B @ A @ C2 @ A4 ) )
% 0.28/0.46 = ( Inf @ ( image @ B @ A @ D2 @ B3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Inf.INF_cong
% 0.28/0.46 thf(fact_87_rev__image__eqI,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,X3: A,A4: set @ A,B2: B,F: A > B] :
% 0.28/0.46 ( ( member @ A @ X3 @ A4 )
% 0.28/0.46 => ( ( B2
% 0.28/0.46 = ( F @ X3 ) )
% 0.28/0.46 => ( member @ B @ B2 @ ( image @ A @ B @ F @ A4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % rev_image_eqI
% 0.28/0.46 thf(fact_88_ball__imageD,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
% 0.28/0.46 ( ! [X2: A] :
% 0.28/0.46 ( ( member @ A @ X2 @ ( image @ B @ A @ F @ A4 ) )
% 0.28/0.46 => ( P @ X2 ) )
% 0.28/0.46 => ! [X5: B] :
% 0.28/0.46 ( ( member @ B @ X5 @ A4 )
% 0.28/0.46 => ( P @ ( F @ X5 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ball_imageD
% 0.28/0.46 thf(fact_89_image__cong,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,M: set @ A,N3: set @ A,F: A > B,G: A > B] :
% 0.28/0.46 ( ( M = N3 )
% 0.28/0.46 => ( ! [X2: A] :
% 0.28/0.46 ( ( member @ A @ X2 @ N3 )
% 0.28/0.46 => ( ( F @ X2 )
% 0.28/0.46 = ( G @ X2 ) ) )
% 0.28/0.46 => ( ( image @ A @ B @ F @ M )
% 0.28/0.46 = ( image @ A @ B @ G @ N3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % image_cong
% 0.28/0.46 thf(fact_90_bex__imageD,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
% 0.28/0.46 ( ? [X5: A] :
% 0.28/0.46 ( ( member @ A @ X5 @ ( image @ B @ A @ F @ A4 ) )
% 0.28/0.46 & ( P @ X5 ) )
% 0.28/0.46 => ? [X2: B] :
% 0.28/0.46 ( ( member @ B @ X2 @ A4 )
% 0.28/0.46 & ( P @ ( F @ X2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % bex_imageD
% 0.28/0.46 thf(fact_91_image__iff,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,Z: A,F: B > A,A4: set @ B] :
% 0.28/0.46 ( ( member @ A @ Z @ ( image @ B @ A @ F @ A4 ) )
% 0.28/0.46 = ( ? [X: B] :
% 0.28/0.46 ( ( member @ B @ X @ A4 )
% 0.28/0.46 & ( Z
% 0.28/0.46 = ( F @ X ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % image_iff
% 0.28/0.46 thf(fact_92_imageI,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,X3: A,A4: set @ A,F: A > B] :
% 0.28/0.46 ( ( member @ A @ X3 @ A4 )
% 0.28/0.46 => ( member @ B @ ( F @ X3 ) @ ( image @ A @ B @ F @ A4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % imageI
% 0.28/0.46 thf(fact_93_is__num__normalize_I1_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( neg_numeral @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 0.28/0.46 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % is_num_normalize(1)
% 0.28/0.46 thf(fact_94_UNIV__witness,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UNIV_witness
% 0.28/0.46 thf(fact_95_UNIV__eq__I,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A] :
% 0.28/0.46 ( ! [X2: A] : ( member @ A @ X2 @ A4 )
% 0.28/0.46 => ( ( top_top @ ( set @ A ) )
% 0.28/0.46 = A4 ) ) ).
% 0.28/0.46
% 0.28/0.46 % UNIV_eq_I
% 0.28/0.46 thf(fact_96_Int__left__commute,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A,B3: set @ A,C2: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) )
% 0.28/0.46 = ( inf_inf @ ( set @ A ) @ B3 @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_left_commute
% 0.28/0.46 thf(fact_97_Int__left__absorb,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
% 0.28/0.46 = ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_left_absorb
% 0.28/0.46 thf(fact_98_Int__commute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) )
% 0.28/0.46 = ( ^ [A6: set @ A,B4: set @ A] : ( inf_inf @ ( set @ A ) @ B4 @ A6 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_commute
% 0.28/0.46 thf(fact_99_Int__absorb,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ A4 @ A4 )
% 0.28/0.46 = A4 ) ).
% 0.28/0.46
% 0.28/0.46 % Int_absorb
% 0.28/0.46 thf(fact_100_Int__assoc,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A,B3: set @ A,C2: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ C2 )
% 0.28/0.46 = ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_assoc
% 0.28/0.46 thf(fact_101_IntD2,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
% 0.28/0.46 => ( member @ A @ C @ B3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % IntD2
% 0.28/0.46 thf(fact_102_IntD1,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
% 0.28/0.46 => ( member @ A @ C @ A4 ) ) ).
% 0.28/0.46
% 0.28/0.46 % IntD1
% 0.28/0.46 thf(fact_103_IntE,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
% 0.28/0.46 => ~ ( ( member @ A @ C @ A4 )
% 0.28/0.46 => ~ ( member @ A @ C @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % IntE
% 0.28/0.46 thf(fact_104_UnionE,axiom,
% 0.28/0.46 ! [A: $tType,A4: A,C2: set @ ( set @ A )] :
% 0.28/0.46 ( ( member @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) )
% 0.28/0.46 => ~ ! [X6: set @ A] :
% 0.28/0.46 ( ( member @ A @ A4 @ X6 )
% 0.28/0.46 => ~ ( member @ ( set @ A ) @ X6 @ C2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UnionE
% 0.28/0.46 thf(fact_105_vimage__Collect,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,P: B > $o,F: A > B,Q: A > $o] :
% 0.28/0.46 ( ! [X2: A] :
% 0.28/0.46 ( ( P @ ( F @ X2 ) )
% 0.28/0.46 = ( Q @ X2 ) )
% 0.28/0.46 => ( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
% 0.28/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % vimage_Collect
% 0.28/0.46 thf(fact_106_vimageI2,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,F: B > A,A2: B,A4: set @ A] :
% 0.28/0.46 ( ( member @ A @ ( F @ A2 ) @ A4 )
% 0.28/0.46 => ( member @ B @ A2 @ ( vimage @ B @ A @ F @ A4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % vimageI2
% 0.28/0.46 thf(fact_107_vimageE,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,A2: A,F: A > B,B3: set @ B] :
% 0.28/0.46 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ B3 ) )
% 0.28/0.46 => ( member @ B @ ( F @ A2 ) @ B3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % vimageE
% 0.28/0.46 thf(fact_108_vimageD,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,A2: A,F: A > B,A4: set @ B] :
% 0.28/0.46 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ A4 ) )
% 0.28/0.46 => ( member @ B @ ( F @ A2 ) @ A4 ) ) ).
% 0.28/0.46
% 0.28/0.46 % vimageD
% 0.28/0.46 thf(fact_109_Sup_OSUP__identity__eq,axiom,
% 0.28/0.46 ! [A: $tType,Sup: ( set @ A ) > A,A4: set @ A] :
% 0.28/0.46 ( ( Sup
% 0.28/0.46 @ ( image @ A @ A
% 0.28/0.46 @ ^ [X: A] : X
% 0.28/0.46 @ A4 ) )
% 0.28/0.46 = ( Sup @ A4 ) ) ).
% 0.28/0.46
% 0.28/0.46 % Sup.SUP_identity_eq
% 0.28/0.46 thf(fact_110_Inf_OINF__identity__eq,axiom,
% 0.28/0.46 ! [A: $tType,Inf: ( set @ A ) > A,A4: set @ A] :
% 0.28/0.46 ( ( Inf
% 0.28/0.46 @ ( image @ A @ A
% 0.28/0.46 @ ^ [X: A] : X
% 0.28/0.46 @ A4 ) )
% 0.28/0.46 = ( Inf @ A4 ) ) ).
% 0.28/0.46
% 0.28/0.46 % Inf.INF_identity_eq
% 0.28/0.46 thf(fact_111_Compr__image__eq,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
% 0.28/0.46 ( ( collect @ A
% 0.28/0.46 @ ^ [X: A] :
% 0.28/0.46 ( ( member @ A @ X @ ( image @ B @ A @ F @ A4 ) )
% 0.28/0.46 & ( P @ X ) ) )
% 0.28/0.46 = ( image @ B @ A @ F
% 0.28/0.46 @ ( collect @ B
% 0.28/0.46 @ ^ [X: B] :
% 0.28/0.46 ( ( member @ B @ X @ A4 )
% 0.28/0.46 & ( P @ ( F @ X ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Compr_image_eq
% 0.28/0.46 thf(fact_112_image__image,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,C3: $tType,F: B > A,G: C3 > B,A4: set @ C3] :
% 0.28/0.46 ( ( image @ B @ A @ F @ ( image @ C3 @ B @ G @ A4 ) )
% 0.28/0.46 = ( image @ C3 @ A
% 0.28/0.46 @ ^ [X: C3] : ( F @ ( G @ X ) )
% 0.28/0.46 @ A4 ) ) ).
% 0.28/0.46
% 0.28/0.46 % image_image
% 0.28/0.46 thf(fact_113_imageE,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,B2: A,F: B > A,A4: set @ B] :
% 0.28/0.46 ( ( member @ A @ B2 @ ( image @ B @ A @ F @ A4 ) )
% 0.28/0.46 => ~ ! [X2: B] :
% 0.28/0.46 ( ( B2
% 0.28/0.46 = ( F @ X2 ) )
% 0.28/0.46 => ~ ( member @ B @ X2 @ A4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % imageE
% 0.28/0.46 thf(fact_114_UNIV__def,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( top_top @ ( set @ A ) )
% 0.28/0.46 = ( collect @ A
% 0.28/0.46 @ ^ [X: A] : $true ) ) ).
% 0.28/0.46
% 0.28/0.46 % UNIV_def
% 0.28/0.46 thf(fact_115_Collect__conj__eq,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.46 ( ( collect @ A
% 0.28/0.46 @ ^ [X: A] :
% 0.28/0.46 ( ( P @ X )
% 0.28/0.46 & ( Q @ X ) ) )
% 0.28/0.46 = ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Collect_conj_eq
% 0.28/0.46 thf(fact_116_Int__Collect,axiom,
% 0.28/0.46 ! [A: $tType,X3: A,A4: set @ A,P: A > $o] :
% 0.28/0.46 ( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P ) ) )
% 0.28/0.46 = ( ( member @ A @ X3 @ A4 )
% 0.28/0.46 & ( P @ X3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_Collect
% 0.28/0.46 thf(fact_117_Int__def,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) )
% 0.28/0.46 = ( ^ [A6: set @ A,B4: set @ A] :
% 0.28/0.46 ( collect @ A
% 0.28/0.46 @ ^ [X: A] :
% 0.28/0.46 ( ( member @ A @ X @ A6 )
% 0.28/0.46 & ( member @ A @ X @ B4 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_def
% 0.28/0.46 thf(fact_118_vimage__def,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( vimage @ A @ B )
% 0.28/0.46 = ( ^ [F2: A > B,B4: set @ B] :
% 0.28/0.46 ( collect @ A
% 0.28/0.46 @ ^ [X: A] : ( member @ B @ ( F2 @ X ) @ B4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % vimage_def
% 0.28/0.46 thf(fact_119_less__numeral__extra_I3_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linord1659791738miring @ A )
% 0.28/0.46 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_numeral_extra(3)
% 0.28/0.46 thf(fact_120_less__numeral__extra_I4_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linord1659791738miring @ A )
% 0.28/0.46 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_numeral_extra(4)
% 0.28/0.46 thf(fact_121_range__eqI,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,B2: A,F: B > A,X3: B] :
% 0.28/0.46 ( ( B2
% 0.28/0.46 = ( F @ X3 ) )
% 0.28/0.46 => ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % range_eqI
% 0.28/0.46 thf(fact_122_surj__def,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,F: B > A] :
% 0.28/0.46 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.28/0.46 = ( top_top @ ( set @ A ) ) )
% 0.28/0.46 = ( ! [Y2: A] :
% 0.28/0.46 ? [X: B] :
% 0.28/0.46 ( Y2
% 0.28/0.46 = ( F @ X ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % surj_def
% 0.28/0.46 thf(fact_123_rangeI,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,F: B > A,X3: B] : ( member @ A @ ( F @ X3 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % rangeI
% 0.28/0.46 thf(fact_124_surjI,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 0.28/0.46 ( ! [X2: A] :
% 0.28/0.46 ( ( G @ ( F @ X2 ) )
% 0.28/0.46 = X2 )
% 0.28/0.46 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 0.28/0.46 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % surjI
% 0.28/0.46 thf(fact_125_surjE,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,F: B > A,Y4: A] :
% 0.28/0.46 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.28/0.46 = ( top_top @ ( set @ A ) ) )
% 0.28/0.46 => ~ ! [X2: B] :
% 0.28/0.46 ( Y4
% 0.28/0.46 != ( F @ X2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % surjE
% 0.28/0.46 thf(fact_126_surjD,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,F: B > A,Y4: A] :
% 0.28/0.46 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.28/0.46 = ( top_top @ ( set @ A ) ) )
% 0.28/0.46 => ? [X2: B] :
% 0.28/0.46 ( Y4
% 0.28/0.46 = ( F @ X2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % surjD
% 0.28/0.46 thf(fact_127_less__Sup__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( comple1035589618norder @ A )
% 0.28/0.46 => ! [A2: A,S: set @ A] :
% 0.28/0.46 ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ S ) )
% 0.28/0.46 = ( ? [X: A] :
% 0.28/0.46 ( ( member @ A @ X @ S )
% 0.28/0.46 & ( ord_less @ A @ A2 @ X ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_Sup_iff
% 0.28/0.46 thf(fact_128_SUP__cong,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType] :
% 0.28/0.46 ( ( complete_Sup @ A )
% 0.28/0.46 => ! [A4: set @ B,B3: set @ B,C2: B > A,D2: B > A] :
% 0.28/0.46 ( ( A4 = B3 )
% 0.28/0.46 => ( ! [X2: B] :
% 0.28/0.46 ( ( member @ B @ X2 @ B3 )
% 0.28/0.46 => ( ( C2 @ X2 )
% 0.28/0.46 = ( D2 @ X2 ) ) )
% 0.28/0.46 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ C2 @ A4 ) )
% 0.28/0.46 = ( complete_Sup_Sup @ A @ ( image @ B @ A @ D2 @ B3 ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % SUP_cong
% 0.28/0.46 thf(fact_129_Int__UNIV__right,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) )
% 0.28/0.46 = A4 ) ).
% 0.28/0.46
% 0.28/0.46 % Int_UNIV_right
% 0.28/0.46 thf(fact_130_Int__UNIV__left,axiom,
% 0.28/0.46 ! [A: $tType,B3: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 )
% 0.28/0.46 = B3 ) ).
% 0.28/0.46
% 0.28/0.46 % Int_UNIV_left
% 0.28/0.46 thf(fact_131_Union__UNIV,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( complete_Sup_Sup @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
% 0.28/0.46 = ( top_top @ ( set @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Union_UNIV
% 0.28/0.46 thf(fact_132_null__setsD2,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A,M: sigma_measure @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ A4 @ ( measure_null_sets @ A @ M ) )
% 0.28/0.46 => ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % null_setsD2
% 0.28/0.46 thf(fact_133_vimage__inter__cong,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,S: set @ A,F: A > B,G: A > B,Y4: set @ B] :
% 0.28/0.46 ( ! [W: A] :
% 0.28/0.46 ( ( member @ A @ W @ S )
% 0.28/0.46 => ( ( F @ W )
% 0.28/0.46 = ( G @ W ) ) )
% 0.28/0.46 => ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ Y4 ) @ S )
% 0.28/0.46 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G @ Y4 ) @ S ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % vimage_inter_cong
% 0.28/0.46 thf(fact_134_translation__subtract__diff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ab_group_add @ A )
% 0.28/0.46 => ! [A2: A,S2: set @ A,T: set @ A] :
% 0.28/0.46 ( ( image @ A @ A
% 0.28/0.46 @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
% 0.28/0.46 @ ( minus_minus @ ( set @ A ) @ S2 @ T ) )
% 0.28/0.46 = ( minus_minus @ ( set @ A )
% 0.28/0.46 @ ( image @ A @ A
% 0.28/0.46 @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
% 0.28/0.46 @ S2 )
% 0.28/0.46 @ ( image @ A @ A
% 0.28/0.46 @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
% 0.28/0.46 @ T ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % translation_subtract_diff
% 0.28/0.46 thf(fact_135_range__composition,axiom,
% 0.28/0.46 ! [A: $tType,C3: $tType,B: $tType,F: C3 > A,G: B > C3] :
% 0.28/0.46 ( ( image @ B @ A
% 0.28/0.46 @ ^ [X: B] : ( F @ ( G @ X ) )
% 0.28/0.46 @ ( top_top @ ( set @ B ) ) )
% 0.28/0.46 = ( image @ C3 @ A @ F @ ( image @ B @ C3 @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % range_composition
% 0.28/0.46 thf(fact_136_rangeE,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,B2: A,F: B > A] :
% 0.28/0.46 ( ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
% 0.28/0.46 => ~ ! [X2: B] :
% 0.28/0.46 ( B2
% 0.28/0.46 != ( F @ X2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % rangeE
% 0.28/0.46 thf(fact_137_SUP__commute,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,C3: $tType] :
% 0.28/0.46 ( ( comple187826305attice @ A )
% 0.28/0.46 => ! [F: B > C3 > A,B3: set @ C3,A4: set @ B] :
% 0.28/0.46 ( ( complete_Sup_Sup @ A
% 0.28/0.46 @ ( image @ B @ A
% 0.28/0.46 @ ^ [I5: B] : ( complete_Sup_Sup @ A @ ( image @ C3 @ A @ ( F @ I5 ) @ B3 ) )
% 0.28/0.46 @ A4 ) )
% 0.28/0.46 = ( complete_Sup_Sup @ A
% 0.28/0.46 @ ( image @ C3 @ A
% 0.28/0.46 @ ^ [J: C3] :
% 0.28/0.46 ( complete_Sup_Sup @ A
% 0.28/0.46 @ ( image @ B @ A
% 0.28/0.46 @ ^ [I5: B] : ( F @ I5 @ J )
% 0.28/0.46 @ A4 ) )
% 0.28/0.46 @ B3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % SUP_commute
% 0.28/0.46 thf(fact_138_image__Union,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,F: B > A,S: set @ ( set @ B )] :
% 0.28/0.46 ( ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ S ) )
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ S ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % image_Union
% 0.28/0.46 thf(fact_139_UN__UN__flatten,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,C3: $tType,C2: B > ( set @ A ),B3: C3 > ( set @ B ),A4: set @ C3] :
% 0.28/0.46 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C3 @ ( set @ B ) @ B3 @ A4 ) ) ) )
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ A )
% 0.28/0.46 @ ( image @ C3 @ ( set @ A )
% 0.28/0.46 @ ^ [Y2: C3] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C2 @ ( B3 @ Y2 ) ) )
% 0.28/0.46 @ A4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UN_UN_flatten
% 0.28/0.46 thf(fact_140_UN__E,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,B2: A,B3: B > ( set @ A ),A4: set @ B] :
% 0.28/0.46 ( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A4 ) ) )
% 0.28/0.46 => ~ ! [X2: B] :
% 0.28/0.46 ( ( member @ B @ X2 @ A4 )
% 0.28/0.46 => ~ ( member @ A @ B2 @ ( B3 @ X2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UN_E
% 0.28/0.46 thf(fact_141_UN__extend__simps_I8_J,axiom,
% 0.28/0.46 ! [P2: $tType,O: $tType,B3: O > ( set @ P2 ),A4: set @ ( set @ O )] :
% 0.28/0.46 ( ( complete_Sup_Sup @ ( set @ P2 )
% 0.28/0.46 @ ( image @ ( set @ O ) @ ( set @ P2 )
% 0.28/0.46 @ ^ [Y2: set @ O] : ( complete_Sup_Sup @ ( set @ P2 ) @ ( image @ O @ ( set @ P2 ) @ B3 @ Y2 ) )
% 0.28/0.46 @ A4 ) )
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ P2 ) @ ( image @ O @ ( set @ P2 ) @ B3 @ ( complete_Sup_Sup @ ( set @ O ) @ A4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UN_extend_simps(8)
% 0.28/0.46 thf(fact_142_UN__extend__simps_I9_J,axiom,
% 0.28/0.46 ! [S3: $tType,R: $tType,Q2: $tType,C2: R > ( set @ S3 ),B3: Q2 > ( set @ R ),A4: set @ Q2] :
% 0.28/0.46 ( ( complete_Sup_Sup @ ( set @ S3 )
% 0.28/0.46 @ ( image @ Q2 @ ( set @ S3 )
% 0.28/0.46 @ ^ [X: Q2] : ( complete_Sup_Sup @ ( set @ S3 ) @ ( image @ R @ ( set @ S3 ) @ C2 @ ( B3 @ X ) ) )
% 0.28/0.46 @ A4 ) )
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ S3 ) @ ( image @ R @ ( set @ S3 ) @ C2 @ ( complete_Sup_Sup @ ( set @ R ) @ ( image @ Q2 @ ( set @ R ) @ B3 @ A4 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UN_extend_simps(9)
% 0.28/0.46 thf(fact_143_less__numeral__extra_I1_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linord1659791738miring @ A )
% 0.28/0.46 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_numeral_extra(1)
% 0.28/0.46 thf(fact_144_translation__Int,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ab_group_add @ A )
% 0.28/0.46 => ! [A2: A,S2: set @ A,T: set @ A] :
% 0.28/0.46 ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( inf_inf @ ( set @ A ) @ S2 @ T ) )
% 0.28/0.46 = ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S2 ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % translation_Int
% 0.28/0.46 thf(fact_145_surj__image__vimage__eq,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,F: B > A,A4: set @ A] :
% 0.28/0.46 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.28/0.46 = ( top_top @ ( set @ A ) ) )
% 0.28/0.46 => ( ( image @ B @ A @ F @ ( vimage @ B @ A @ F @ A4 ) )
% 0.28/0.46 = A4 ) ) ).
% 0.28/0.46
% 0.28/0.46 % surj_image_vimage_eq
% 0.28/0.46 thf(fact_146_null__set__Int2,axiom,
% 0.28/0.46 ! [A: $tType,B3: set @ A,M: sigma_measure @ A,A4: set @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ B3 @ ( measure_null_sets @ A @ M ) )
% 0.28/0.46 => ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) )
% 0.28/0.46 => ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B3 @ A4 ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % null_set_Int2
% 0.28/0.46 thf(fact_147_null__set__Int1,axiom,
% 0.28/0.46 ! [A: $tType,B3: set @ A,M: sigma_measure @ A,A4: set @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ B3 @ ( measure_null_sets @ A @ M ) )
% 0.28/0.46 => ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) )
% 0.28/0.46 => ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % null_set_Int1
% 0.28/0.46 thf(fact_148_null__setsD1,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A,M: sigma_measure @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ A4 @ ( measure_null_sets @ A @ M ) )
% 0.28/0.46 => ( ( sigma_emeasure @ A @ M @ A4 )
% 0.28/0.46 = ( zero_zero @ extend1814228343nnreal ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % null_setsD1
% 0.28/0.46 thf(fact_149_less__SUP__iff,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType] :
% 0.28/0.46 ( ( comple1035589618norder @ A )
% 0.28/0.46 => ! [A2: A,F: B > A,A4: set @ B] :
% 0.28/0.46 ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A4 ) ) )
% 0.28/0.46 = ( ? [X: B] :
% 0.28/0.46 ( ( member @ B @ X @ A4 )
% 0.28/0.46 & ( ord_less @ A @ A2 @ ( F @ X ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_SUP_iff
% 0.28/0.46 thf(fact_150_SUP__lessD,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( comple187826305attice @ A )
% 0.28/0.46 => ! [F: B > A,A4: set @ B,Y4: A,I3: B] :
% 0.28/0.46 ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A4 ) ) @ Y4 )
% 0.28/0.46 => ( ( member @ B @ I3 @ A4 )
% 0.28/0.46 => ( ord_less @ A @ ( F @ I3 ) @ Y4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % SUP_lessD
% 0.28/0.46 thf(fact_151_translation__subtract__Int,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ab_group_add @ A )
% 0.28/0.46 => ! [A2: A,S2: set @ A,T: set @ A] :
% 0.28/0.46 ( ( image @ A @ A
% 0.28/0.46 @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
% 0.28/0.46 @ ( inf_inf @ ( set @ A ) @ S2 @ T ) )
% 0.28/0.46 = ( inf_inf @ ( set @ A )
% 0.28/0.46 @ ( image @ A @ A
% 0.28/0.46 @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
% 0.28/0.46 @ S2 )
% 0.28/0.46 @ ( image @ A @ A
% 0.28/0.46 @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
% 0.28/0.46 @ T ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % translation_subtract_Int
% 0.28/0.46 thf(fact_152_image__UN,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,C3: $tType,F: B > A,B3: C3 > ( set @ B ),A4: set @ C3] :
% 0.28/0.46 ( ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C3 @ ( set @ B ) @ B3 @ A4 ) ) )
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ A )
% 0.28/0.46 @ ( image @ C3 @ ( set @ A )
% 0.28/0.46 @ ^ [X: C3] : ( image @ B @ A @ F @ ( B3 @ X ) )
% 0.28/0.46 @ A4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % image_UN
% 0.28/0.46 thf(fact_153_UN__extend__simps_I10_J,axiom,
% 0.28/0.46 ! [V: $tType,U: $tType,T2: $tType,B3: U > ( set @ V ),F: T2 > U,A4: set @ T2] :
% 0.28/0.46 ( ( complete_Sup_Sup @ ( set @ V )
% 0.28/0.46 @ ( image @ T2 @ ( set @ V )
% 0.28/0.46 @ ^ [A3: T2] : ( B3 @ ( F @ A3 ) )
% 0.28/0.46 @ A4 ) )
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ V ) @ ( image @ U @ ( set @ V ) @ B3 @ ( image @ T2 @ U @ F @ A4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UN_extend_simps(10)
% 0.28/0.46 thf(fact_154_Int__UN__distrib2,axiom,
% 0.28/0.46 ! [A: $tType,C3: $tType,B: $tType,A4: B > ( set @ A ),I4: set @ B,B3: C3 > ( set @ A ),J2: set @ C3] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A4 @ I4 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ C3 @ ( set @ A ) @ B3 @ J2 ) ) )
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ A )
% 0.28/0.46 @ ( image @ B @ ( set @ A )
% 0.28/0.46 @ ^ [I5: B] :
% 0.28/0.46 ( complete_Sup_Sup @ ( set @ A )
% 0.28/0.46 @ ( image @ C3 @ ( set @ A )
% 0.28/0.46 @ ^ [J: C3] : ( inf_inf @ ( set @ A ) @ ( A4 @ I5 ) @ ( B3 @ J ) )
% 0.28/0.46 @ J2 ) )
% 0.28/0.46 @ I4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_UN_distrib2
% 0.28/0.46 thf(fact_155_Int__UN__distrib,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,B3: set @ A,A4: B > ( set @ A ),I4: set @ B] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A4 @ I4 ) ) )
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ A )
% 0.28/0.46 @ ( image @ B @ ( set @ A )
% 0.28/0.46 @ ^ [I5: B] : ( inf_inf @ ( set @ A ) @ B3 @ ( A4 @ I5 ) )
% 0.28/0.46 @ I4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_UN_distrib
% 0.28/0.46 thf(fact_156_UN__extend__simps_I4_J,axiom,
% 0.28/0.46 ! [H: $tType,G2: $tType,A4: G2 > ( set @ H ),C2: set @ G2,B3: set @ H] :
% 0.28/0.46 ( ( inf_inf @ ( set @ H ) @ ( complete_Sup_Sup @ ( set @ H ) @ ( image @ G2 @ ( set @ H ) @ A4 @ C2 ) ) @ B3 )
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ H )
% 0.28/0.46 @ ( image @ G2 @ ( set @ H )
% 0.28/0.46 @ ^ [X: G2] : ( inf_inf @ ( set @ H ) @ ( A4 @ X ) @ B3 )
% 0.28/0.46 @ C2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UN_extend_simps(4)
% 0.28/0.46 thf(fact_157_UN__extend__simps_I5_J,axiom,
% 0.28/0.46 ! [I: $tType,J3: $tType,A4: set @ I,B3: J3 > ( set @ I ),C2: set @ J3] :
% 0.28/0.46 ( ( inf_inf @ ( set @ I ) @ A4 @ ( complete_Sup_Sup @ ( set @ I ) @ ( image @ J3 @ ( set @ I ) @ B3 @ C2 ) ) )
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ I )
% 0.28/0.46 @ ( image @ J3 @ ( set @ I )
% 0.28/0.46 @ ^ [X: J3] : ( inf_inf @ ( set @ I ) @ A4 @ ( B3 @ X ) )
% 0.28/0.46 @ C2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UN_extend_simps(5)
% 0.28/0.46 thf(fact_158_vimage__UN,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,C3: $tType,F: A > B,B3: C3 > ( set @ B ),A4: set @ C3] :
% 0.28/0.46 ( ( vimage @ A @ B @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C3 @ ( set @ B ) @ B3 @ A4 ) ) )
% 0.28/0.46 = ( complete_Sup_Sup @ ( set @ A )
% 0.28/0.46 @ ( image @ C3 @ ( set @ A )
% 0.28/0.46 @ ^ [X: C3] : ( vimage @ A @ B @ F @ ( B3 @ X ) )
% 0.28/0.46 @ A4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % vimage_UN
% 0.28/0.46 thf(fact_159_SUP__UNION,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,C3: $tType] :
% 0.28/0.46 ( ( comple187826305attice @ A )
% 0.28/0.46 => ! [F: B > A,G: C3 > ( set @ B ),A4: set @ C3] :
% 0.28/0.46 ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C3 @ ( set @ B ) @ G @ A4 ) ) ) )
% 0.28/0.46 = ( complete_Sup_Sup @ A
% 0.28/0.46 @ ( image @ C3 @ A
% 0.28/0.46 @ ^ [Y2: C3] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ ( G @ Y2 ) ) )
% 0.28/0.46 @ A4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % SUP_UNION
% 0.28/0.46 thf(fact_160_null__sets__def,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( measure_null_sets @ A )
% 0.28/0.46 = ( ^ [M4: sigma_measure @ A] :
% 0.28/0.46 ( collect @ ( set @ A )
% 0.28/0.46 @ ^ [N4: set @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ N4 @ ( sigma_sets @ A @ M4 ) )
% 0.28/0.46 & ( ( sigma_emeasure @ A @ M4 @ N4 )
% 0.28/0.46 = ( zero_zero @ extend1814228343nnreal ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % null_sets_def
% 0.28/0.46 thf(fact_161_sets_OInt__space__eq2,axiom,
% 0.28/0.46 ! [A: $tType,X3: set @ A,M: sigma_measure @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ X3 @ ( sigma_sets @ A @ M ) )
% 0.28/0.46 => ( ( inf_inf @ ( set @ A ) @ X3 @ ( sigma_space @ A @ M ) )
% 0.28/0.46 = X3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % sets.Int_space_eq2
% 0.28/0.46 thf(fact_162_sets_OInt__space__eq1,axiom,
% 0.28/0.46 ! [A: $tType,X3: set @ A,M: sigma_measure @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ X3 @ ( sigma_sets @ A @ M ) )
% 0.28/0.46 => ( ( inf_inf @ ( set @ A ) @ ( sigma_space @ A @ M ) @ X3 )
% 0.28/0.46 = X3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % sets.Int_space_eq1
% 0.28/0.46 thf(fact_163_diff__add__zero,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( comm_monoid_diff @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.28/0.46 = ( zero_zero @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_add_zero
% 0.28/0.46 thf(fact_164_diff__gt__0__iff__gt,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordered_ab_group_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
% 0.28/0.46 = ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_gt_0_iff_gt
% 0.28/0.46 thf(fact_165_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linord219039673up_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 0.28/0.46 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % zero_less_double_add_iff_zero_less_single_add
% 0.28/0.46 thf(fact_166_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linord219039673up_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 0.28/0.46 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % double_add_less_zero_iff_single_add_less_zero
% 0.28/0.46 thf(fact_167_less__add__same__cancel2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere516151231imp_le @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
% 0.28/0.46 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_add_same_cancel2
% 0.28/0.46 thf(fact_168_less__add__same__cancel1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere516151231imp_le @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.28/0.46 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_add_same_cancel1
% 0.28/0.46 thf(fact_169_add__less__same__cancel2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere516151231imp_le @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 0.28/0.46 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_less_same_cancel2
% 0.28/0.46 thf(fact_170_add__left__cancel,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel_semigroup_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.28/0.46 = ( plus_plus @ A @ A2 @ C ) )
% 0.28/0.46 = ( B2 = C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_left_cancel
% 0.28/0.46 thf(fact_171_add__right__cancel,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel_semigroup_add @ A )
% 0.28/0.46 => ! [B2: A,A2: A,C: A] :
% 0.28/0.46 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.28/0.46 = ( plus_plus @ A @ C @ A2 ) )
% 0.28/0.46 = ( B2 = C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_right_cancel
% 0.28/0.46 thf(fact_172_DiffI,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ A4 )
% 0.28/0.46 => ( ~ ( member @ A @ C @ B3 )
% 0.28/0.46 => ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % DiffI
% 0.28/0.46 thf(fact_173_Diff__iff,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 0.28/0.46 = ( ( member @ A @ C @ A4 )
% 0.28/0.46 & ~ ( member @ A @ C @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Diff_iff
% 0.28/0.46 thf(fact_174_Diff__idemp,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ B3 )
% 0.28/0.46 = ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % Diff_idemp
% 0.28/0.46 thf(fact_175_not__gr__zero,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( canoni770627133id_add @ A )
% 0.28/0.46 => ! [N5: A] :
% 0.28/0.46 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N5 ) )
% 0.28/0.46 = ( N5
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % not_gr_zero
% 0.28/0.46 thf(fact_176_add_Oleft__neutral,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( monoid_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % add.left_neutral
% 0.28/0.46 thf(fact_177_add_Oright__neutral,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( monoid_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % add.right_neutral
% 0.28/0.46 thf(fact_178_double__zero,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linord219039673up_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( ( plus_plus @ A @ A2 @ A2 )
% 0.28/0.46 = ( zero_zero @ A ) )
% 0.28/0.46 = ( A2
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % double_zero
% 0.28/0.46 thf(fact_179_double__zero__sym,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linord219039673up_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( ( zero_zero @ A )
% 0.28/0.46 = ( plus_plus @ A @ A2 @ A2 ) )
% 0.28/0.46 = ( A2
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % double_zero_sym
% 0.28/0.46 thf(fact_180_add__cancel__left__left,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel1352612707id_add @ A )
% 0.28/0.46 => ! [B2: A,A2: A] :
% 0.28/0.46 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.28/0.46 = A2 )
% 0.28/0.46 = ( B2
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_cancel_left_left
% 0.28/0.46 thf(fact_181_add__cancel__left__right,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel1352612707id_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.28/0.46 = A2 )
% 0.28/0.46 = ( B2
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_cancel_left_right
% 0.28/0.46 thf(fact_182_add__cancel__right__left,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel1352612707id_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( A2
% 0.28/0.46 = ( plus_plus @ A @ B2 @ A2 ) )
% 0.28/0.46 = ( B2
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_cancel_right_left
% 0.28/0.46 thf(fact_183_add__cancel__right__right,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel1352612707id_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( A2
% 0.28/0.46 = ( plus_plus @ A @ A2 @ B2 ) )
% 0.28/0.46 = ( B2
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_cancel_right_right
% 0.28/0.46 thf(fact_184_add__eq__0__iff__both__eq__0,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( canoni770627133id_add @ A )
% 0.28/0.46 => ! [X3: A,Y4: A] :
% 0.28/0.46 ( ( ( plus_plus @ A @ X3 @ Y4 )
% 0.28/0.46 = ( zero_zero @ A ) )
% 0.28/0.46 = ( ( X3
% 0.28/0.46 = ( zero_zero @ A ) )
% 0.28/0.46 & ( Y4
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_eq_0_iff_both_eq_0
% 0.28/0.46 thf(fact_185_zero__eq__add__iff__both__eq__0,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( canoni770627133id_add @ A )
% 0.28/0.46 => ! [X3: A,Y4: A] :
% 0.28/0.46 ( ( ( zero_zero @ A )
% 0.28/0.46 = ( plus_plus @ A @ X3 @ Y4 ) )
% 0.28/0.46 = ( ( X3
% 0.28/0.46 = ( zero_zero @ A ) )
% 0.28/0.46 & ( Y4
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % zero_eq_add_iff_both_eq_0
% 0.28/0.46 thf(fact_186_diff__self,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( minus_minus @ A @ A2 @ A2 )
% 0.28/0.46 = ( zero_zero @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_self
% 0.28/0.46 thf(fact_187_diff__0__right,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_0_right
% 0.28/0.46 thf(fact_188_zero__diff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( comm_monoid_diff @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.46 = ( zero_zero @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % zero_diff
% 0.28/0.46 thf(fact_189_diff__zero,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel1352612707id_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_zero
% 0.28/0.46 thf(fact_190_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel1352612707id_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( minus_minus @ A @ A2 @ A2 )
% 0.28/0.46 = ( zero_zero @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % cancel_comm_monoid_add_class.diff_cancel
% 0.28/0.46 thf(fact_191_add__less__cancel__left,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere236663937imp_le @ A )
% 0.28/0.46 => ! [C: A,A2: A,B2: A] :
% 0.28/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 0.28/0.46 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_less_cancel_left
% 0.28/0.46 thf(fact_192_add__less__cancel__right,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere236663937imp_le @ A )
% 0.28/0.46 => ! [A2: A,C: A,B2: A] :
% 0.28/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 0.28/0.46 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_less_cancel_right
% 0.28/0.46 thf(fact_193_add__diff__cancel,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_diff_cancel
% 0.28/0.46 thf(fact_194_diff__add__cancel,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_add_cancel
% 0.28/0.46 thf(fact_195_add__diff__cancel__left,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel146912293up_add @ A )
% 0.28/0.46 => ! [C: A,A2: A,B2: A] :
% 0.28/0.46 ( ( minus_minus @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 0.28/0.46 = ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_diff_cancel_left
% 0.28/0.46 thf(fact_196_add__diff__cancel__left_H,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel146912293up_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
% 0.28/0.46 = B2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_diff_cancel_left'
% 0.28/0.46 thf(fact_197_add__diff__cancel__right,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel146912293up_add @ A )
% 0.28/0.46 => ! [A2: A,C: A,B2: A] :
% 0.28/0.46 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 0.28/0.46 = ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_diff_cancel_right
% 0.28/0.46 thf(fact_198_add__diff__cancel__right_H,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel146912293up_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_diff_cancel_right'
% 0.28/0.46 thf(fact_199_sets_OInt,axiom,
% 0.28/0.46 ! [A: $tType,A2: set @ A,M: sigma_measure @ A,B2: set @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ A2 @ ( sigma_sets @ A @ M ) )
% 0.28/0.46 => ( ( member @ ( set @ A ) @ B2 @ ( sigma_sets @ A @ M ) )
% 0.28/0.46 => ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ ( sigma_sets @ A @ M ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sets.Int
% 0.28/0.46 thf(fact_200_sets_Otop,axiom,
% 0.28/0.46 ! [A: $tType,M: sigma_measure @ A] : ( member @ ( set @ A ) @ ( sigma_space @ A @ M ) @ ( sigma_sets @ A @ M ) ) ).
% 0.28/0.46
% 0.28/0.46 % sets.top
% 0.28/0.46 thf(fact_201_sets_ODiff,axiom,
% 0.28/0.46 ! [A: $tType,A2: set @ A,M: sigma_measure @ A,B2: set @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ A2 @ ( sigma_sets @ A @ M ) )
% 0.28/0.46 => ( ( member @ ( set @ A ) @ B2 @ ( sigma_sets @ A @ M ) )
% 0.28/0.46 => ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( sigma_sets @ A @ M ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sets.Diff
% 0.28/0.46 thf(fact_202_add__less__same__cancel1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere516151231imp_le @ A )
% 0.28/0.46 => ! [B2: A,A2: A] :
% 0.28/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
% 0.28/0.46 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_less_same_cancel1
% 0.28/0.46 thf(fact_203_sets_Ocompl__sets,axiom,
% 0.28/0.46 ! [A: $tType,A2: set @ A,M: sigma_measure @ A] :
% 0.28/0.46 ( ( member @ ( set @ A ) @ A2 @ ( sigma_sets @ A @ M ) )
% 0.28/0.46 => ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( sigma_space @ A @ M ) @ A2 ) @ ( sigma_sets @ A @ M ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sets.compl_sets
% 0.28/0.46 thf(fact_204_DiffE,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 0.28/0.46 => ~ ( ( member @ A @ C @ A4 )
% 0.28/0.46 => ( member @ A @ C @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % DiffE
% 0.28/0.46 thf(fact_205_DiffD1,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 0.28/0.46 => ( member @ A @ C @ A4 ) ) ).
% 0.28/0.46
% 0.28/0.46 % DiffD1
% 0.28/0.46 thf(fact_206_DiffD2,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 0.28/0.46 => ~ ( member @ A @ C @ B3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % DiffD2
% 0.28/0.46 thf(fact_207_set__diff__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( minus_minus @ ( set @ A ) )
% 0.28/0.46 = ( ^ [A6: set @ A,B4: set @ A] :
% 0.28/0.46 ( collect @ A
% 0.28/0.46 @ ^ [X: A] :
% 0.28/0.46 ( ( member @ A @ X @ A6 )
% 0.28/0.46 & ~ ( member @ A @ X @ B4 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % set_diff_eq
% 0.28/0.46 thf(fact_208_minus__set__def,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( minus_minus @ ( set @ A ) )
% 0.28/0.46 = ( ^ [A6: set @ A,B4: set @ A] :
% 0.28/0.46 ( collect @ A
% 0.28/0.46 @ ( minus_minus @ ( A > $o )
% 0.28/0.46 @ ^ [X: A] : ( member @ A @ X @ A6 )
% 0.28/0.46 @ ^ [X: A] : ( member @ A @ X @ B4 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % minus_set_def
% 0.28/0.46 thf(fact_209_psubset__imp__ex__mem,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 0.28/0.46 => ? [B5: A] : ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % psubset_imp_ex_mem
% 0.28/0.46 thf(fact_210_Sup__bool__def,axiom,
% 0.28/0.46 ( ( complete_Sup_Sup @ $o )
% 0.28/0.46 = ( member @ $o @ $true ) ) ).
% 0.28/0.46
% 0.28/0.46 % Sup_bool_def
% 0.28/0.46 thf(fact_211_zero__reorient,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( zero @ A )
% 0.28/0.46 => ! [X3: A] :
% 0.28/0.46 ( ( ( zero_zero @ A )
% 0.28/0.46 = X3 )
% 0.28/0.46 = ( X3
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % zero_reorient
% 0.28/0.46 thf(fact_212_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ab_semigroup_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 0.28/0.46 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.28/0.46 thf(fact_213_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere779506340up_add @ A )
% 0.28/0.46 => ! [I3: A,J4: A,K2: A,L2: A] :
% 0.28/0.46 ( ( ( I3 = J4 )
% 0.28/0.46 & ( K2 = L2 ) )
% 0.28/0.46 => ( ( plus_plus @ A @ I3 @ K2 )
% 0.28/0.46 = ( plus_plus @ A @ J4 @ L2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_mono_thms_linordered_semiring(4)
% 0.28/0.46 thf(fact_214_group__cancel_Oadd1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( comm_monoid_add @ A )
% 0.28/0.46 => ! [A4: A,K2: A,A2: A,B2: A] :
% 0.28/0.46 ( ( A4
% 0.28/0.46 = ( plus_plus @ A @ K2 @ A2 ) )
% 0.28/0.46 => ( ( plus_plus @ A @ A4 @ B2 )
% 0.28/0.46 = ( plus_plus @ A @ K2 @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % group_cancel.add1
% 0.28/0.46 thf(fact_215_group__cancel_Oadd2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( comm_monoid_add @ A )
% 0.28/0.46 => ! [B3: A,K2: A,B2: A,A2: A] :
% 0.28/0.46 ( ( B3
% 0.28/0.46 = ( plus_plus @ A @ K2 @ B2 ) )
% 0.28/0.46 => ( ( plus_plus @ A @ A2 @ B3 )
% 0.28/0.46 = ( plus_plus @ A @ K2 @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % group_cancel.add2
% 0.28/0.46 thf(fact_216_add_Oassoc,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semigroup_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 0.28/0.46 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add.assoc
% 0.28/0.46 thf(fact_217_add_Oleft__cancel,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.28/0.46 = ( plus_plus @ A @ A2 @ C ) )
% 0.28/0.46 = ( B2 = C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add.left_cancel
% 0.28/0.46 thf(fact_218_add_Oright__cancel,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ! [B2: A,A2: A,C: A] :
% 0.28/0.46 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.28/0.46 = ( plus_plus @ A @ C @ A2 ) )
% 0.28/0.46 = ( B2 = C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add.right_cancel
% 0.28/0.46 thf(fact_219_add_Ocommute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ab_semigroup_add @ A )
% 0.28/0.46 => ( ( plus_plus @ A )
% 0.28/0.46 = ( ^ [A3: A,B6: A] : ( plus_plus @ A @ B6 @ A3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add.commute
% 0.28/0.46 thf(fact_220_add_Oleft__commute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ab_semigroup_add @ A )
% 0.28/0.46 => ! [B2: A,A2: A,C: A] :
% 0.28/0.46 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
% 0.28/0.46 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add.left_commute
% 0.28/0.46 thf(fact_221_add__left__imp__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel_semigroup_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.28/0.46 = ( plus_plus @ A @ A2 @ C ) )
% 0.28/0.46 => ( B2 = C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_left_imp_eq
% 0.28/0.46 thf(fact_222_add__right__imp__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel_semigroup_add @ A )
% 0.28/0.46 => ! [B2: A,A2: A,C: A] :
% 0.28/0.46 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.28/0.46 = ( plus_plus @ A @ C @ A2 ) )
% 0.28/0.46 => ( B2 = C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_right_imp_eq
% 0.28/0.46 thf(fact_223_diff__eq__diff__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A,D3: A] :
% 0.28/0.46 ( ( ( minus_minus @ A @ A2 @ B2 )
% 0.28/0.46 = ( minus_minus @ A @ C @ D3 ) )
% 0.28/0.46 => ( ( A2 = B2 )
% 0.28/0.46 = ( C = D3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_eq_diff_eq
% 0.28/0.46 thf(fact_224_diff__right__commute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel146912293up_add @ A )
% 0.28/0.46 => ! [A2: A,C: A,B2: A] :
% 0.28/0.46 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B2 )
% 0.28/0.46 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_right_commute
% 0.28/0.46 thf(fact_225_one__reorient,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( one @ A )
% 0.28/0.46 => ! [X3: A] :
% 0.28/0.46 ( ( ( one_one @ A )
% 0.28/0.46 = X3 )
% 0.28/0.46 = ( X3
% 0.28/0.46 = ( one_one @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % one_reorient
% 0.28/0.46 thf(fact_226_gr__zeroI,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( canoni770627133id_add @ A )
% 0.28/0.46 => ! [N5: A] :
% 0.28/0.46 ( ( N5
% 0.28/0.46 != ( zero_zero @ A ) )
% 0.28/0.46 => ( ord_less @ A @ ( zero_zero @ A ) @ N5 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % gr_zeroI
% 0.28/0.46 thf(fact_227_not__less__zero,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( canoni770627133id_add @ A )
% 0.28/0.46 => ! [N5: A] :
% 0.28/0.46 ~ ( ord_less @ A @ N5 @ ( zero_zero @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % not_less_zero
% 0.28/0.46 thf(fact_228_gr__implies__not__zero,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( canoni770627133id_add @ A )
% 0.28/0.46 => ! [M2: A,N5: A] :
% 0.28/0.46 ( ( ord_less @ A @ M2 @ N5 )
% 0.28/0.46 => ( N5
% 0.28/0.46 != ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % gr_implies_not_zero
% 0.28/0.46 thf(fact_229_zero__less__iff__neq__zero,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( canoni770627133id_add @ A )
% 0.28/0.46 => ! [N5: A] :
% 0.28/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ N5 )
% 0.28/0.46 = ( N5
% 0.28/0.46 != ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % zero_less_iff_neq_zero
% 0.28/0.46 thf(fact_230_comm__monoid__add__class_Oadd__0,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( comm_monoid_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % comm_monoid_add_class.add_0
% 0.28/0.46 thf(fact_231_add_Ocomm__neutral,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( comm_monoid_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % add.comm_neutral
% 0.28/0.46 thf(fact_232_add_Ogroup__left__neutral,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % add.group_left_neutral
% 0.28/0.46 thf(fact_233_eq__iff__diff__eq__0,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ( ( ^ [Y5: A,Z2: A] : ( Y5 = Z2 ) )
% 0.28/0.46 = ( ^ [A3: A,B6: A] :
% 0.28/0.46 ( ( minus_minus @ A @ A3 @ B6 )
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % eq_iff_diff_eq_0
% 0.28/0.46 thf(fact_234_add__mono__thms__linordered__field_I5_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere223160158up_add @ A )
% 0.28/0.46 => ! [I3: A,J4: A,K2: A,L2: A] :
% 0.28/0.46 ( ( ( ord_less @ A @ I3 @ J4 )
% 0.28/0.46 & ( ord_less @ A @ K2 @ L2 ) )
% 0.28/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I3 @ K2 ) @ ( plus_plus @ A @ J4 @ L2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_mono_thms_linordered_field(5)
% 0.28/0.46 thf(fact_235_add__mono__thms__linordered__field_I2_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere223160158up_add @ A )
% 0.28/0.46 => ! [I3: A,J4: A,K2: A,L2: A] :
% 0.28/0.46 ( ( ( I3 = J4 )
% 0.28/0.46 & ( ord_less @ A @ K2 @ L2 ) )
% 0.28/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I3 @ K2 ) @ ( plus_plus @ A @ J4 @ L2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_mono_thms_linordered_field(2)
% 0.28/0.46 thf(fact_236_add__mono__thms__linordered__field_I1_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere223160158up_add @ A )
% 0.28/0.46 => ! [I3: A,J4: A,K2: A,L2: A] :
% 0.28/0.46 ( ( ( ord_less @ A @ I3 @ J4 )
% 0.28/0.46 & ( K2 = L2 ) )
% 0.28/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I3 @ K2 ) @ ( plus_plus @ A @ J4 @ L2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_mono_thms_linordered_field(1)
% 0.28/0.46 thf(fact_237_add__strict__mono,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( strict2144017051up_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A,D3: A] :
% 0.28/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.46 => ( ( ord_less @ A @ C @ D3 )
% 0.28/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_strict_mono
% 0.28/0.46 thf(fact_238_add__strict__left__mono,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere223160158up_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.46 => ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_strict_left_mono
% 0.28/0.46 thf(fact_239_add__strict__right__mono,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere223160158up_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_strict_right_mono
% 0.28/0.46 thf(fact_240_add__less__imp__less__left,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere236663937imp_le @ A )
% 0.28/0.46 => ! [C: A,A2: A,B2: A] :
% 0.28/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 0.28/0.46 => ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_less_imp_less_left
% 0.28/0.46 thf(fact_241_add__less__imp__less__right,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere236663937imp_le @ A )
% 0.28/0.46 => ! [A2: A,C: A,B2: A] :
% 0.28/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 0.28/0.46 => ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_less_imp_less_right
% 0.28/0.46 thf(fact_242_diff__strict__mono,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordered_ab_group_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,D3: A,C: A] :
% 0.28/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.46 => ( ( ord_less @ A @ D3 @ C )
% 0.28/0.46 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_strict_mono
% 0.28/0.46 thf(fact_243_diff__eq__diff__less,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordered_ab_group_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A,D3: A] :
% 0.28/0.46 ( ( ( minus_minus @ A @ A2 @ B2 )
% 0.28/0.46 = ( minus_minus @ A @ C @ D3 ) )
% 0.28/0.46 => ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.46 = ( ord_less @ A @ C @ D3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_eq_diff_less
% 0.28/0.46 thf(fact_244_diff__strict__left__mono,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordered_ab_group_add @ A )
% 0.28/0.46 => ! [B2: A,A2: A,C: A] :
% 0.28/0.46 ( ( ord_less @ A @ B2 @ A2 )
% 0.28/0.46 => ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_strict_left_mono
% 0.28/0.46 thf(fact_245_diff__strict__right__mono,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordered_ab_group_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.46 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_strict_right_mono
% 0.28/0.46 thf(fact_246_group__cancel_Osub1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ab_group_add @ A )
% 0.28/0.46 => ! [A4: A,K2: A,A2: A,B2: A] :
% 0.28/0.46 ( ( A4
% 0.28/0.46 = ( plus_plus @ A @ K2 @ A2 ) )
% 0.28/0.46 => ( ( minus_minus @ A @ A4 @ B2 )
% 0.28/0.46 = ( plus_plus @ A @ K2 @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % group_cancel.sub1
% 0.28/0.46 thf(fact_247_diff__eq__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( ( minus_minus @ A @ A2 @ B2 )
% 0.28/0.46 = C )
% 0.28/0.46 = ( A2
% 0.28/0.46 = ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_eq_eq
% 0.28/0.46 thf(fact_248_eq__diff__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ! [A2: A,C: A,B2: A] :
% 0.28/0.46 ( ( A2
% 0.28/0.46 = ( minus_minus @ A @ C @ B2 ) )
% 0.28/0.46 = ( ( plus_plus @ A @ A2 @ B2 )
% 0.28/0.46 = C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % eq_diff_eq
% 0.28/0.46 thf(fact_249_add__diff__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C ) )
% 0.28/0.46 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_diff_eq
% 0.28/0.46 thf(fact_250_diff__diff__eq2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C ) )
% 0.28/0.46 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_diff_eq2
% 0.28/0.46 thf(fact_251_diff__add__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ab_group_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C )
% 0.28/0.46 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_add_eq
% 0.28/0.46 thf(fact_252_diff__add__eq__diff__diff__swap,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) )
% 0.28/0.46 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_add_eq_diff_diff_swap
% 0.28/0.46 thf(fact_253_diff__diff__add,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel146912293up_add @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C )
% 0.28/0.46 = ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_diff_add
% 0.28/0.46
% 0.28/0.46 % Subclasses (2)
% 0.28/0.46 thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( finite_finite @ A )
% 0.28/0.46 => ( type @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(subcl_Finite__Set_Ofinite___Countable_Ocountable,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( finite_finite @ A )
% 0.28/0.46 => ( countable @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % Type constructors (124)
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( ordere890947078_space @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( ordere890947078_space @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Rings_Oring__1,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( ring_1 @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( ring_1 @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_1,axiom,
% 0.28/0.46 ordere890947078_space @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Rings_Oring__1_2,axiom,
% 0.28/0.46 ring_1 @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Rings_Oring__1_3,axiom,
% 0.28/0.46 ring_1 @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( comple187826305attice @ A8 )
% 0.28/0.46 => ( comple187826305attice @ ( A7 > A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Complete__Lattices_OSup,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( complete_Sup @ A8 )
% 0.28/0.46 => ( complete_Sup @ ( A7 > A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Countable_Ocountable,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( finite_finite @ A7 )
% 0.28/0.46 & ( countable @ A8 ) )
% 0.28/0.46 => ( countable @ ( A7 > A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( finite_finite @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( finite_finite @ ( A7 > A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 0.28/0.46 ordere516151231imp_le @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 0.28/0.46 strict2144017051up_add @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 0.28/0.46 ordere223160158up_add @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.28/0.46 ordere236663937imp_le @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 0.28/0.46 linord1659791738miring @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
% 0.28/0.46 ordere779506340up_add @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
% 0.28/0.46 linord219039673up_add @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.28/0.46 cancel146912293up_add @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 0.28/0.46 cancel1352612707id_add @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 0.28/0.46 ordered_ab_group_add @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
% 0.28/0.46 cancel_semigroup_add @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
% 0.28/0.46 ab_semigroup_add @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
% 0.28/0.46 comm_monoid_add @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Complete__Lattices_OSup_4,axiom,
% 0.28/0.46 complete_Sup @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
% 0.28/0.46 semigroup_add @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
% 0.28/0.46 ab_group_add @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Countable_Ocountable_5,axiom,
% 0.28/0.46 countable @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
% 0.28/0.46 monoid_add @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 0.28/0.46 group_add @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
% 0.28/0.46 neg_numeral @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
% 0.28/0.46 ring_char_0 @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 0.28/0.46 zero @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Int_Oint___Groups_Oone,axiom,
% 0.28/0.46 one @ int ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_6,axiom,
% 0.28/0.46 ordere516151231imp_le @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_7,axiom,
% 0.28/0.46 strict2144017051up_add @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_8,axiom,
% 0.28/0.46 ordere223160158up_add @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_9,axiom,
% 0.28/0.46 ordere236663937imp_le @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.28/0.46 canoni770627133id_add @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_10,axiom,
% 0.28/0.46 linord1659791738miring @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_11,axiom,
% 0.28/0.46 ordere779506340up_add @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_12,axiom,
% 0.28/0.46 cancel146912293up_add @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_13,axiom,
% 0.28/0.46 cancel1352612707id_add @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_14,axiom,
% 0.28/0.46 cancel_semigroup_add @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 0.28/0.46 comm_monoid_diff @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_15,axiom,
% 0.28/0.46 ab_semigroup_add @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_16,axiom,
% 0.28/0.46 comm_monoid_add @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Complete__Lattices_OSup_17,axiom,
% 0.28/0.46 complete_Sup @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Groups_Osemigroup__add_18,axiom,
% 0.28/0.46 semigroup_add @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Countable_Ocountable_19,axiom,
% 0.28/0.46 countable @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Groups_Omonoid__add_20,axiom,
% 0.28/0.46 monoid_add @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Groups_Ozero_21,axiom,
% 0.28/0.46 zero @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Groups_Oone_22,axiom,
% 0.28/0.46 one @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_23,axiom,
% 0.28/0.46 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Groups_Oab__semigroup__add_24,axiom,
% 0.28/0.46 ! [A7: $tType] :
% 0.28/0.46 ( ( ab_semigroup_add @ A7 )
% 0.28/0.46 => ( ab_semigroup_add @ ( set @ A7 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Groups_Ocomm__monoid__add_25,axiom,
% 0.28/0.46 ! [A7: $tType] :
% 0.28/0.46 ( ( comm_monoid_add @ A7 )
% 0.28/0.46 => ( comm_monoid_add @ ( set @ A7 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Complete__Lattices_OSup_26,axiom,
% 0.28/0.46 ! [A7: $tType] : ( complete_Sup @ ( set @ A7 ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Groups_Osemigroup__add_27,axiom,
% 0.28/0.46 ! [A7: $tType] :
% 0.28/0.46 ( ( semigroup_add @ A7 )
% 0.28/0.46 => ( semigroup_add @ ( set @ A7 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Countable_Ocountable_28,axiom,
% 0.28/0.46 ! [A7: $tType] :
% 0.28/0.46 ( ( finite_finite @ A7 )
% 0.28/0.46 => ( countable @ ( set @ A7 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Groups_Omonoid__add_29,axiom,
% 0.28/0.46 ! [A7: $tType] :
% 0.28/0.46 ( ( monoid_add @ A7 )
% 0.28/0.46 => ( monoid_add @ ( set @ A7 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Finite__Set_Ofinite_30,axiom,
% 0.28/0.46 ! [A7: $tType] :
% 0.28/0.46 ( ( finite_finite @ A7 )
% 0.28/0.46 => ( finite_finite @ ( set @ A7 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Groups_Ozero_31,axiom,
% 0.28/0.46 ! [A7: $tType] :
% 0.28/0.46 ( ( zero @ A7 )
% 0.28/0.46 => ( zero @ ( set @ A7 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Groups_Oone_32,axiom,
% 0.28/0.46 ! [A7: $tType] :
% 0.28/0.46 ( ( one @ A7 )
% 0.28/0.46 => ( one @ ( set @ A7 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_33,axiom,
% 0.28/0.46 comple187826305attice @ $o ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Complete__Lattices_OSup_34,axiom,
% 0.28/0.46 complete_Sup @ $o ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Countable_Ocountable_35,axiom,
% 0.28/0.46 countable @ $o ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Finite__Set_Ofinite_36,axiom,
% 0.28/0.46 finite_finite @ $o ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_37,axiom,
% 0.28/0.46 ordere516151231imp_le @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Ostrict__ordered__ab__semigroup__add_38,axiom,
% 0.28/0.46 strict2144017051up_add @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_39,axiom,
% 0.28/0.46 ordere223160158up_add @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_40,axiom,
% 0.28/0.46 ordere236663937imp_le @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_41,axiom,
% 0.28/0.46 linord1659791738miring @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_42,axiom,
% 0.28/0.46 ordere779506340up_add @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_43,axiom,
% 0.28/0.46 linord219039673up_add @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_44,axiom,
% 0.28/0.46 cancel146912293up_add @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_45,axiom,
% 0.28/0.46 cancel1352612707id_add @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_46,axiom,
% 0.28/0.46 ordered_ab_group_add @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_47,axiom,
% 0.28/0.46 cancel_semigroup_add @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_48,axiom,
% 0.28/0.46 ab_semigroup_add @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_49,axiom,
% 0.28/0.46 comm_monoid_add @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Complete__Lattices_OSup_50,axiom,
% 0.28/0.46 complete_Sup @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Osemigroup__add_51,axiom,
% 0.28/0.46 semigroup_add @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Oab__group__add_52,axiom,
% 0.28/0.46 ab_group_add @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Omonoid__add_53,axiom,
% 0.28/0.46 monoid_add @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Ogroup__add_54,axiom,
% 0.28/0.46 group_add @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Num_Oneg__numeral_55,axiom,
% 0.28/0.46 neg_numeral @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Nat_Oring__char__0_56,axiom,
% 0.28/0.46 ring_char_0 @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Ozero_57,axiom,
% 0.28/0.46 zero @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Real_Oreal___Groups_Oone_58,axiom,
% 0.28/0.46 one @ real ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Sigma__Algebra_Omeasure___Complete__Lattices_Ocomplete__lattice_59,axiom,
% 0.28/0.46 ! [A7: $tType] : ( comple187826305attice @ ( sigma_measure @ A7 ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Sigma__Algebra_Omeasure___Complete__Lattices_OSup_60,axiom,
% 0.28/0.46 ! [A7: $tType] : ( complete_Sup @ ( sigma_measure @ A7 ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__semigroup__monoid__add__imp__le_61,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( ordere890947078_space @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( ordere516151231imp_le @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ostrict__ordered__ab__semigroup__add_62,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( ordere890947078_space @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( strict2144017051up_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__cancel__ab__semigroup__add_63,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( ordere890947078_space @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( ordere223160158up_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__semigroup__add__imp__le_64,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( ordere890947078_space @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( ordere236663937imp_le @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__semigroup__add_65,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( ordere890947078_space @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( ordere779506340up_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocancel__ab__semigroup__add_66,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( cancel146912293up_add @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( cancel146912293up_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocancel__comm__monoid__add_67,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( cancel1352612707id_add @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( cancel1352612707id_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__group__add_68,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( ordere890947078_space @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( ordered_ab_group_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocancel__semigroup__add_69,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( cancel_semigroup_add @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( cancel_semigroup_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oab__semigroup__add_70,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( ab_semigroup_add @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( ab_semigroup_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocomm__monoid__add_71,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( comm_monoid_add @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( comm_monoid_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Complete__Lattices_OSup_72,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( ordere890947078_space @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( complete_Sup @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Osemigroup__add_73,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( semigroup_add @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( semigroup_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oab__group__add_74,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( ab_group_add @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( ab_group_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Countable_Ocountable_75,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( countable @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( countable @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Omonoid__add_76,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( monoid_add @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( monoid_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_77,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( finite_finite @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( finite_finite @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ogroup__add_78,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( group_add @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( group_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Num_Oneg__numeral_79,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( ring_1 @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( neg_numeral @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Nat_Oring__char__0_80,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( ring_char_0 @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( ring_char_0 @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_81,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( zero @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( zero @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oone_82,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ( one @ A7 )
% 0.28/0.46 & ( finite_finite @ A8 ) )
% 0.28/0.46 => ( one @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Ostrict__ordered__ab__semigroup__add_83,axiom,
% 0.28/0.46 strict2144017051up_add @ extend1814228343nnreal ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Ocanonically__ordered__monoid__add_84,axiom,
% 0.28/0.46 canoni770627133id_add @ extend1814228343nnreal ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Extended__Nonnegative__Real_Oennreal___Complete__Lattices_Ocomplete__linorder,axiom,
% 0.28/0.46 comple1035589618norder @ extend1814228343nnreal ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Extended__Nonnegative__Real_Oennreal___Complete__Lattices_Ocomplete__lattice_85,axiom,
% 0.28/0.46 comple187826305attice @ extend1814228343nnreal ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Extended__Nonnegative__Real_Oennreal___Rings_Olinordered__nonzero__semiring_86,axiom,
% 0.28/0.46 linord1659791738miring @ extend1814228343nnreal ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Oordered__ab__semigroup__add_87,axiom,
% 0.28/0.46 ordere779506340up_add @ extend1814228343nnreal ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Oab__semigroup__add_88,axiom,
% 0.28/0.46 ab_semigroup_add @ extend1814228343nnreal ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Ocomm__monoid__add_89,axiom,
% 0.28/0.46 comm_monoid_add @ extend1814228343nnreal ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Extended__Nonnegative__Real_Oennreal___Complete__Lattices_OSup_90,axiom,
% 0.28/0.46 complete_Sup @ extend1814228343nnreal ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Osemigroup__add_91,axiom,
% 0.28/0.46 semigroup_add @ extend1814228343nnreal ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Omonoid__add_92,axiom,
% 0.28/0.46 monoid_add @ extend1814228343nnreal ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Ozero_93,axiom,
% 0.28/0.46 zero @ extend1814228343nnreal ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Oone_94,axiom,
% 0.28/0.46 one @ extend1814228343nnreal ).
% 0.28/0.46
% 0.28/0.46 % Free types (1)
% 0.28/0.46 thf(tfree_0,hypothesis,
% 0.28/0.46 finite_finite @ n ).
% 0.28/0.46
% 0.28/0.46 % Conjectures (1)
% 0.28/0.46 thf(conj_0,conjecture,
% 0.28/0.46 ( ( ^ [N: nat] : ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t2 @ ( f @ N ) ) ) )
% 0.28/0.46 = ( ^ [N: nat] : ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t @ ( f @ N ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 ------------------------
% 2.81/1.56 ---- Embedded in THF ---
% 2.81/1.57 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.81/1.57 %%% Generated on Tue Feb 24 08:26:33 EST 2026
% 2.81/1.57 %%% using '$$dhol' embedding, version 1.3.0.
% 2.81/1.57 %%% Logic specification used:
% 2.81/1.57 %%% thf(spec, logic, $$dhol).
% 2.81/1.57
% 2.81/1.57 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.flmKuaoIzs/DTF2DTF30588.p
% 2.81/1.57 thf(ty_t_Extended__Nonnegative__Real_Oennreal, type, extend1814228343nnreal: $tType).
% 2.81/1.57 thf(ty_t_Finite__Cartesian__Product_Ovec, type, finite_Cartesian_vec: ($tType > ($tType > $tType))).
% 2.81/1.57 thf(ty_t_Sigma__Algebra_Omeasure, type, sigma_measure: ($tType > $tType)).
% 2.81/1.57 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.81/1.57 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.81/1.57 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.81/1.57 thf(ty_t_Int_Oint, type, int: $tType).
% 2.81/1.57 thf(ty_tf_n, type, n: $tType).
% 2.81/1.57 thf(sy_cl_Rings_Oring__1, type, ring_1: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space, type, ordere890947078_space: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Oone, type, one: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Nat_Oring__char__0, type, ring_char_0: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Num_Oneg__numeral, type, neg_numeral: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Omonoid__add, type, monoid_add: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Countable_Ocountable, type, countable: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Oab__group__add, type, ab_group_add: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Complete__Lattices_OSup, type, complete_Sup: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Ocomm__monoid__diff, type, comm_monoid_diff: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple187826305attice: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Complete__Lattices_Ocomplete__linorder, type, comple1035589618norder: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add, type, ordere223160158up_add: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add, type, strict2144017051up_add: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le, type, ordere516151231imp_le: (!> [A:$tType]: ($o))).
% 2.81/1.57 thf(sy_c_Complete__Lattices_OSup__class_OSup, type, complete_Sup_Sup: (!> [A:$tType]: (((set @ A) > A)))).
% 2.81/1.57 thf(sy_c_Complete__Measure_Ocompletion, type, complete_completion: (!> [A:$tType]: (((sigma_measure @ A) > (sigma_measure @ A))))).
% 2.81/1.57 thf(sy_c_Countable__Set_Ofrom__nat__into, type, counta609264050t_into: (!> [A:$tType]: (((set @ A) > (nat > A))))).
% 2.81/1.57 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.81/1.57 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 2.81/1.57 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.81/1.57 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.81/1.57 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))).
% 2.81/1.57 thf(sy_c_Lebesgue__Measure_Olborel, type, lebesgue_lborel: (!> [A:$tType]: ((sigma_measure @ A)))).
% 2.81/1.57 thf(sy_c_Measure__Space_Onull__sets, type, measure_null_sets: (!> [A:$tType]: (((sigma_measure @ A) > (set @ (set @ A)))))).
% 2.81/1.57 thf(sy_c_Minkowskis__Theorem__Mirabelle__jpyvdnlcjd_Oof__int__vec, type, minkow1937162798nt_vec: (!> [B:$tType,A:$tType]: ((((finite_Cartesian_vec @ int) @ B) > ((finite_Cartesian_vec @ A) @ B))))).
% 2.81/1.57 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.81/1.57 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.81/1.57 thf(sy_c_Series_Osums, type, sums: (!> [A:$tType]: (((nat > A) > (A > $o))))).
% 2.81/1.57 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.81/1.57 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.81/1.57 thf(sy_c_Set_Ovimage, type, vimage: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ B) > (set @ A)))))).
% 2.81/1.57 thf(sy_c_Sigma__Algebra_Oemeasure, type, sigma_emeasure: (!> [A:$tType]: (((sigma_measure @ A) > ((set @ A) > extend1814228343nnreal))))).
% 2.81/1.57 thf(sy_c_Sigma__Algebra_Osets, type, sigma_sets: (!> [A:$tType]: (((sigma_measure @ A) > (set @ (set @ A)))))).
% 2.81/1.57 thf(sy_c_Sigma__Algebra_Ospace, type, sigma_space: (!> [A:$tType]: (((sigma_measure @ A) > (set @ A))))).
% 2.81/1.57 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.81/1.57 thf(sy_v_R____, type, r: (((finite_Cartesian_vec @ int) @ n) > (set @ ((finite_Cartesian_vec @ real) @ n)))).
% 2.81/1.57 thf(sy_v_S, type, s: (set @ ((finite_Cartesian_vec @ real) @ n))).
% 2.81/1.57 thf(sy_v_T_H____, type, t: (((finite_Cartesian_vec @ int) @ n) > (set @ ((finite_Cartesian_vec @ real) @ n)))).
% 2.81/1.57 thf(sy_v_T____, type, t2: (((finite_Cartesian_vec @ int) @ n) > (set @ ((finite_Cartesian_vec @ real) @ n)))).
% 2.81/1.57 thf(sy_v_f____, type, f: (nat > ((finite_Cartesian_vec @ int) @ n))).
% 2.81/1.57 thf(fact_0_emeasure__T_H, axiom, (! [A2:((finite_Cartesian_vec @ int) @ n)]: (((((sigma_emeasure @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n)))) @ (t @ A2)) = (((sigma_emeasure @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n)))) @ (t2 @ A2)))))).
% 2.81/1.57 thf(fact_1__092_060open_062_092_060And_062a_O_AT_H_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062, axiom, (! [A2:((finite_Cartesian_vec @ int) @ n)]: ((((member @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ (t @ A2)) @ ((sigma_sets @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n)))))))).
% 2.81/1.57 thf(fact_2__092_060open_062_092_060And_062a_O_AT_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062, axiom, (! [A2:((finite_Cartesian_vec @ int) @ n)]: ((((member @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ (t2 @ A2)) @ ((sigma_sets @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n)))))))).
% 2.81/1.57 thf(fact_3_calculation, axiom, (((sums @ extend1814228343nnreal) @ (^ [N:nat]: ((((sigma_emeasure @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n)))) @ (t2 @ (f @ N)))))) @ (((sigma_emeasure @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n)))) @ s))).
% 2.81/1.57 thf(fact_4_assms_I1_J, axiom, (((member @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ s) @ ((sigma_sets @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n)))))).
% 2.81/1.57 thf(fact_5_emeasure__T__Int, axiom, (! [A2:((finite_Cartesian_vec @ int) @ n),B2:((finite_Cartesian_vec @ int) @ n)]: (((A2 != B2) => ((((sigma_emeasure @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n)))) @ (((inf_inf @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ (t2 @ A2)) @ (t2 @ B2))) = (zero_zero @ extend1814228343nnreal)))))).
% 2.81/1.57 thf(fact_6_T__Int, axiom, (! [A2:((finite_Cartesian_vec @ int) @ n),B2:((finite_Cartesian_vec @ int) @ n)]: (((A2 != B2) => (((member @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ (((inf_inf @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ (t2 @ A2)) @ (t2 @ B2))) @ ((measure_null_sets @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n))))))))).
% 2.81/1.57 thf(fact_7_assms_I2_J, axiom, (((ord_less @ extend1814228343nnreal) @ (one_one @ extend1814228343nnreal)) @ (((sigma_emeasure @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n)))) @ s))).
% 2.81/1.57 thf(fact_8__092_060open_062_092_060And_062a_O_AR_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062, axiom, (! [A2:((finite_Cartesian_vec @ int) @ n)]: ((((member @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ (r @ A2)) @ ((sigma_sets @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n)))))))).
% 2.81/1.57 thf(fact_9__092_060open_062_I_092_060lambda_062n_O_Aemeasure_Alebesgue_A_IT_A_If_An_J_J_J_Asums_Aemeasure_Alebesgue_A_I_092_060Union_062n_O_AT_A_If_An_J_J_092_060close_062, axiom, (((sums @ extend1814228343nnreal) @ (^ [N:nat]: ((((sigma_emeasure @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n)))) @ (t2 @ (f @ N)))))) @ (((sigma_emeasure @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n)))) @ ((complete_Sup_Sup @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ ((((image @ nat) @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ (^ [N:nat]: ((t2 @ (f @ N))))) @ (top_top @ (set @ nat))))))).
% 2.81/1.57 thf(fact_10_T_H__def, axiom, (t = (^ [A3:((finite_Cartesian_vec @ int) @ n)]: (((((image @ ((finite_Cartesian_vec @ real) @ n)) @ ((finite_Cartesian_vec @ real) @ n)) @ (^ [X:((finite_Cartesian_vec @ real) @ n)]: ((((minus_minus @ ((finite_Cartesian_vec @ real) @ n)) @ X) @ (((minkow1937162798nt_vec @ n) @ real) @ A3))))) @ (t2 @ A3)))))).
% 2.81/1.57 thf(fact_11_f__def, axiom, (f = ((counta609264050t_into @ ((finite_Cartesian_vec @ int) @ n)) @ (top_top @ (set @ ((finite_Cartesian_vec @ int) @ n)))))).
% 2.81/1.57 thf(fact_12_T_H__altdef, axiom, (! [A2:((finite_Cartesian_vec @ int) @ n)]: (((t @ A2) = ((((vimage @ ((finite_Cartesian_vec @ real) @ n)) @ ((finite_Cartesian_vec @ real) @ n)) @ (^ [X:((finite_Cartesian_vec @ real) @ n)]: ((((plus_plus @ ((finite_Cartesian_vec @ real) @ n)) @ X) @ (((minkow1937162798nt_vec @ n) @ real) @ A2))))) @ (t2 @ A2)))))).
% 2.81/1.57 thf(fact_13_of__int__vec__eq__iff, axiom, (! [A:$tType,N2:$tType]: ((((finite_finite @ N2) & (ring_char_0 @ A)) => (! [A2:((finite_Cartesian_vec @ int) @ N2),B2:((finite_Cartesian_vec @ int) @ N2)]: ((((((minkow1937162798nt_vec @ N2) @ A) @ A2) = (((minkow1937162798nt_vec @ N2) @ A) @ B2)) = (A2 = B2)))))))).
% 2.81/1.57 thf(fact_14_T__def, axiom, (t2 = (^ [A3:((finite_Cartesian_vec @ int) @ n)]: ((((inf_inf @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ s) @ (r @ A3)))))).
% 2.81/1.57 thf(fact_15_R__Int, axiom, (! [A2:((finite_Cartesian_vec @ int) @ n),B2:((finite_Cartesian_vec @ int) @ n)]: (((A2 != B2) => (((member @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ (((inf_inf @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ (r @ A2)) @ (r @ B2))) @ ((measure_null_sets @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n))))))))).
% 2.81/1.57 thf(fact_16_sums__emeasure_H, axiom, (! [A:$tType,B3:(nat > (set @ A)),M:(sigma_measure @ A)]: (((! [X2:nat]: ((((member @ (set @ A)) @ (B3 @ X2)) @ ((sigma_sets @ A) @ M)))) => ((! [X2:nat,Y:nat]: (((X2 != Y) => ((((sigma_emeasure @ A) @ M) @ (((inf_inf @ (set @ A)) @ (B3 @ X2)) @ (B3 @ Y))) = (zero_zero @ extend1814228343nnreal))))) => (((sums @ extend1814228343nnreal) @ (^ [X:nat]: ((((sigma_emeasure @ A) @ M) @ (B3 @ X))))) @ (((sigma_emeasure @ A) @ M) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ B3) @ (top_top @ (set @ nat))))))))))).
% 2.81/1.57 thf(fact_17_null__sets__UN, axiom, (! [I:$tType,A:$tType]: (((countable @ I) => (! [N3:(I > (set @ A)),M:(sigma_measure @ A)]: (((! [I2:I]: ((((member @ (set @ A)) @ (N3 @ I2)) @ ((measure_null_sets @ A) @ M)))) => (((member @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ I) @ (set @ A)) @ N3) @ (top_top @ (set @ I))))) @ ((measure_null_sets @ A) @ M))))))))).
% 2.81/1.57 thf(fact_18_SUP__eq__top__iff, axiom, (! [B:$tType,A:$tType]: (((comple1035589618norder @ A) => (! [F:(B > A),A4:(set @ B)]: (((((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A4)) = (top_top @ A)) = (! [X:A]: (((((ord_less @ A) @ X) @ (top_top @ A)) => (? [Y2:B]: (((((member @ B) @ Y2) @ A4) & (((ord_less @ A) @ X) @ (F @ Y2))))))))))))))).
% 2.81/1.57 thf(fact_19_null__setsI, axiom, (! [A:$tType,M:(sigma_measure @ A),A4:(set @ A)]: ((((((sigma_emeasure @ A) @ M) @ A4) = (zero_zero @ extend1814228343nnreal)) => ((((member @ (set @ A)) @ A4) @ ((sigma_sets @ A) @ M)) => (((member @ (set @ A)) @ A4) @ ((measure_null_sets @ A) @ M))))))).
% 2.81/1.57 thf(fact_20_image__vimage__eq, axiom, (! [A:$tType,B:$tType,F:(B > A),A4:(set @ A)]: ((((((image @ B) @ A) @ F) @ ((((vimage @ B) @ A) @ F) @ A4)) = (((inf_inf @ (set @ A)) @ A4) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))))))).
% 2.81/1.57 thf(fact_21__092_060open_062_092_060And_062a_O_AT_Aa_A_092_060in_062_Asets_Alebesgue_A_092_060Longrightarrow_062_A_I_092_060lambda_062x_O_Ax_A_L_Aof__int__vec_Aa_J_A_N_096_AT_Aa_A_092_060inter_062_Aspace_Alebesgue_A_092_060in_062_Asets_Alebesgue_092_060close_062, axiom, (! [A5:((finite_Cartesian_vec @ int) @ n)]: (((((member @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ (t2 @ A5)) @ ((sigma_sets @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n))))) => (((member @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ (((inf_inf @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ ((((vimage @ ((finite_Cartesian_vec @ real) @ n)) @ ((finite_Cartesian_vec @ real) @ n)) @ (^ [X:((finite_Cartesian_vec @ real) @ n)]: ((((plus_plus @ ((finite_Cartesian_vec @ real) @ n)) @ X) @ (((minkow1937162798nt_vec @ n) @ real) @ A5))))) @ (t2 @ A5))) @ ((sigma_space @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n)))))) @ ((sigma_sets @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n))))))))).
% 2.81/1.57 thf(fact_22_surj__diff__right, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A]: ((((((image @ A) @ A) @ (^ [X:A]: ((((minus_minus @ A) @ X) @ A2)))) @ (top_top @ (set @ A))) = (top_top @ (set @ A))))))))).
% 2.81/1.57 thf(fact_23__092_060open_062_092_060Union_062_A_Irange_AT_J_A_061_A_I_092_060Union_062n_O_AT_A_If_An_J_J_092_060close_062, axiom, (((complete_Sup_Sup @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ ((((image @ ((finite_Cartesian_vec @ int) @ n)) @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ t2) @ (top_top @ (set @ ((finite_Cartesian_vec @ int) @ n))))) = ((complete_Sup_Sup @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ ((((image @ nat) @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ (^ [N:nat]: ((t2 @ (f @ N))))) @ (top_top @ (set @ nat)))))).
% 2.81/1.57 thf(fact_24_Sup__UNIV, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (((complete_Sup_Sup @ A) @ (top_top @ (set @ A))) = (top_top @ A)))))).
% 2.81/1.57 thf(fact_25_Sup__eq__top__iff, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (! [A4:(set @ A)]: (((((complete_Sup_Sup @ A) @ A4) = (top_top @ A)) = (! [X:A]: (((((ord_less @ A) @ X) @ (top_top @ A)) => (? [Y2:A]: (((((member @ A) @ Y2) @ A4) & (((ord_less @ A) @ X) @ Y2)))))))))))))).
% 2.81/1.57 thf(fact_26_surj__plus, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ (top_top @ (set @ A))) = (top_top @ (set @ A))))))))).
% 2.81/1.57 thf(fact_27_diff__numeral__special_I9_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((minus_minus @ A) @ (one_one @ A)) @ (one_one @ A)) = (zero_zero @ A)))))).
% 2.81/1.57 thf(fact_28_image__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A),X3:B,A4:(set @ B)]: (((B2 = (F @ X3)) => ((((member @ B) @ X3) @ A4) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ A4))))))).
% 2.81/1.57 thf(fact_29_UNIV__I, axiom, (! [A:$tType,X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))).
% 2.81/1.57 thf(fact_30_Int__iff, axiom, (! [A:$tType,C:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A4) @ B3)) = ((((member @ A) @ C) @ A4) & (((member @ A) @ C) @ B3)))))).
% 2.81/1.57 thf(fact_31_IntI, axiom, (! [A:$tType,C:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ A4) => ((((member @ A) @ C) @ B3) => (((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A4) @ B3))))))).
% 2.81/1.57 thf(fact_32_Union__iff, axiom, (! [A:$tType,A4:A,C2:(set @ (set @ A))]: (((((member @ A) @ A4) @ ((complete_Sup_Sup @ (set @ A)) @ C2)) = (? [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ C2) & (((member @ A) @ A4) @ X)))))))).
% 2.81/1.57 thf(fact_33_UnionI, axiom, (! [A:$tType,X4:(set @ A),C2:(set @ (set @ A)),A4:A]: (((((member @ (set @ A)) @ X4) @ C2) => ((((member @ A) @ A4) @ X4) => (((member @ A) @ A4) @ ((complete_Sup_Sup @ (set @ A)) @ C2))))))).
% 2.81/1.57 thf(fact_34_UN__ball__bex__simps_I1_J, axiom, (! [A:$tType,A4:(set @ (set @ A)),P:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ ((complete_Sup_Sup @ (set @ A)) @ A4)) => (P @ X)))) = (! [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ A4) => (! [Y2:A]: (((((member @ A) @ Y2) @ X) => (P @ Y2))))))))))).
% 2.81/1.57 thf(fact_35_UN__ball__bex__simps_I3_J, axiom, (! [D:$tType,A4:(set @ (set @ D)),P:(D > $o)]: (((? [X:D]: (((((member @ D) @ X) @ ((complete_Sup_Sup @ (set @ D)) @ A4)) & (P @ X)))) = (? [X:(set @ D)]: (((((member @ (set @ D)) @ X) @ A4) & (? [Y2:D]: (((((member @ D) @ Y2) @ X) & (P @ Y2))))))))))).
% 2.81/1.57 thf(fact_36_null__sets_ODiff, axiom, (! [A:$tType,A2:(set @ A),M:(sigma_measure @ A),B2:(set @ A)]: (((((member @ (set @ A)) @ A2) @ ((measure_null_sets @ A) @ M)) => ((((member @ (set @ A)) @ B2) @ ((measure_null_sets @ A) @ M)) => (((member @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) @ ((measure_null_sets @ A) @ M))))))).
% 2.81/1.57 thf(fact_37_vimage__eq, axiom, (! [A:$tType,B:$tType,A2:A,F:(A > B),B3:(set @ B)]: (((((member @ A) @ A2) @ ((((vimage @ A) @ B) @ F) @ B3)) = (((member @ B) @ (F @ A2)) @ B3))))).
% 2.81/1.57 thf(fact_38_vimageI, axiom, (! [B:$tType,A:$tType,F:(B > A),A2:B,B2:A,B3:(set @ A)]: ((((F @ A2) = B2) => ((((member @ A) @ B2) @ B3) => (((member @ B) @ A2) @ ((((vimage @ B) @ A) @ F) @ B3))))))).
% 2.81/1.57 thf(fact_39_image__ident, axiom, (! [A:$tType,Y3:(set @ A)]: ((((((image @ A) @ A) @ (^ [X:A]: (X))) @ Y3) = Y3)))).
% 2.81/1.57 thf(fact_40_Sup__apply, axiom, (! [B:$tType,A:$tType]: (((complete_Sup @ B) => ((complete_Sup_Sup @ (A > B)) = (^ [A6:(set @ (A > B)),X:A]: (((complete_Sup_Sup @ B) @ ((((image @ (A > B)) @ B) @ (^ [F2:(A > B)]: ((F2 @ X)))) @ A6))))))))).
% 2.81/1.57 thf(fact_41_vimage__Collect__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),P:(B > $o)]: ((((((vimage @ A) @ B) @ F) @ ((collect @ B) @ P)) = ((collect @ A) @ (^ [Y2:A]: ((P @ (F @ Y2))))))))).
% 2.81/1.57 thf(fact_42_vimage__ident, axiom, (! [A:$tType,Y3:(set @ A)]: ((((((vimage @ A) @ A) @ (^ [X:A]: (X))) @ Y3) = Y3)))).
% 2.81/1.57 thf(fact_43_Int__UNIV, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: ((((((inf_inf @ (set @ A)) @ A4) @ B3) = (top_top @ (set @ A))) = ((A4 = (top_top @ (set @ A))) & (B3 = (top_top @ (set @ A)))))))).
% 2.81/1.57 thf(fact_44_ball__UN, axiom, (! [A:$tType,B:$tType,B3:(B > (set @ A)),A4:(set @ B),P:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B3) @ A4))) => (P @ X)))) = (! [X:B]: (((((member @ B) @ X) @ A4) => (! [Y2:A]: (((((member @ A) @ Y2) @ (B3 @ X)) => (P @ Y2))))))))))).
% 2.81/1.57 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.81/1.57 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A4)))) = A4)))).
% 2.81/1.57 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) = (Q @ X2)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.81/1.57 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 2.81/1.57 thf(fact_49_bex__UN, axiom, (! [A:$tType,B:$tType,B3:(B > (set @ A)),A4:(set @ B),P:(A > $o)]: (((? [X:A]: (((((member @ A) @ X) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B3) @ A4))) & (P @ X)))) = (? [X:B]: (((((member @ B) @ X) @ A4) & (? [Y2:A]: (((((member @ A) @ Y2) @ (B3 @ X)) & (P @ Y2))))))))))).
% 2.81/1.57 thf(fact_50_UN__ball__bex__simps_I2_J, axiom, (! [C3:$tType,B:$tType,B3:(B > (set @ C3)),A4:(set @ B),P:(C3 > $o)]: (((! [X:C3]: (((((member @ C3) @ X) @ ((complete_Sup_Sup @ (set @ C3)) @ ((((image @ B) @ (set @ C3)) @ B3) @ A4))) => (P @ X)))) = (! [X:B]: (((((member @ B) @ X) @ A4) => (! [Y2:C3]: (((((member @ C3) @ Y2) @ (B3 @ X)) => (P @ Y2))))))))))).
% 2.81/1.57 thf(fact_51_UN__ball__bex__simps_I4_J, axiom, (! [F3:$tType,E:$tType,B3:(E > (set @ F3)),A4:(set @ E),P:(F3 > $o)]: (((? [X:F3]: (((((member @ F3) @ X) @ ((complete_Sup_Sup @ (set @ F3)) @ ((((image @ E) @ (set @ F3)) @ B3) @ A4))) & (P @ X)))) = (? [X:E]: (((((member @ E) @ X) @ A4) & (? [Y2:F3]: (((((member @ F3) @ Y2) @ (B3 @ X)) & (P @ Y2))))))))))).
% 2.81/1.57 thf(fact_52_vimage__UNIV, axiom, (! [B:$tType,A:$tType,F:(A > B)]: ((((((vimage @ A) @ B) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A)))))).
% 2.81/1.57 thf(fact_53_null__sets_OInt, axiom, (! [A:$tType,A2:(set @ A),M:(sigma_measure @ A),B2:(set @ A)]: (((((member @ (set @ A)) @ A2) @ ((measure_null_sets @ A) @ M)) => ((((member @ (set @ A)) @ B2) @ ((measure_null_sets @ A) @ M)) => (((member @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ B2)) @ ((measure_null_sets @ A) @ M))))))).
% 2.81/1.57 thf(fact_54_vimage__Int, axiom, (! [A:$tType,B:$tType,F:(A > B),A4:(set @ B),B3:(set @ B)]: ((((((vimage @ A) @ B) @ F) @ (((inf_inf @ (set @ B)) @ A4) @ B3)) = (((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ F) @ A4)) @ ((((vimage @ A) @ B) @ F) @ B3)))))).
% 2.81/1.57 thf(fact_55_SUP__identity__eq, axiom, (! [A:$tType]: (((complete_Sup @ A) => (! [A4:(set @ A)]: ((((complete_Sup_Sup @ A) @ ((((image @ A) @ A) @ (^ [X:A]: (X))) @ A4)) = ((complete_Sup_Sup @ A) @ A4)))))))).
% 2.81/1.57 thf(fact_56_SUP__apply, axiom, (! [A:$tType,B:$tType,C3:$tType]: (((complete_Sup @ A) => (! [F:(C3 > (B > A)),A4:(set @ C3),X3:B]: (((((complete_Sup_Sup @ (B > A)) @ ((((image @ C3) @ (B > A)) @ F) @ A4)) @ X3) = ((complete_Sup_Sup @ A) @ ((((image @ C3) @ A) @ (^ [Y2:C3]: (((F @ Y2) @ X3)))) @ A4))))))))).
% 2.81/1.57 thf(fact_57_UN__iff, axiom, (! [A:$tType,B:$tType,B2:A,B3:(B > (set @ A)),A4:(set @ B)]: (((((member @ A) @ B2) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B3) @ A4))) = (? [X:B]: (((((member @ B) @ X) @ A4) & (((member @ A) @ B2) @ (B3 @ X))))))))).
% 2.81/1.57 thf(fact_58_UN__I, axiom, (! [B:$tType,A:$tType,A2:A,A4:(set @ A),B2:B,B3:(A > (set @ B))]: (((((member @ A) @ A2) @ A4) => ((((member @ B) @ B2) @ (B3 @ A2)) => (((member @ B) @ B2) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ B3) @ A4)))))))).
% 2.81/1.57 thf(fact_59_S__decompose, axiom, (s = ((complete_Sup_Sup @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ ((((image @ ((finite_Cartesian_vec @ int) @ n)) @ (set @ ((finite_Cartesian_vec @ real) @ n))) @ t2) @ (top_top @ (set @ ((finite_Cartesian_vec @ int) @ n))))))).
% 2.81/1.57 thf(fact_60_image__add__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [S:(set @ A)]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ (zero_zero @ A))) @ S) = S))))))).
% 2.81/1.57 thf(fact_61_null__sets_OInt__space__eq2, axiom, (! [A:$tType,X3:(set @ A),M:(sigma_measure @ A)]: (((((member @ (set @ A)) @ X3) @ ((measure_null_sets @ A) @ M)) => ((((inf_inf @ (set @ A)) @ X3) @ ((sigma_space @ A) @ M)) = X3))))).
% 2.81/1.57 thf(fact_62_null__sets_OInt__space__eq1, axiom, (! [A:$tType,X3:(set @ A),M:(sigma_measure @ A)]: (((((member @ (set @ A)) @ X3) @ ((measure_null_sets @ A) @ M)) => ((((inf_inf @ (set @ A)) @ ((sigma_space @ A) @ M)) @ X3) = X3))))).
% 2.81/1.57 thf(fact_63_in__sets__SUP, axiom, (! [B:$tType,A:$tType,I3:A,I4:(set @ A),M:(A > (sigma_measure @ B)),Y3:(set @ B),X4:(set @ B)]: (((((member @ A) @ I3) @ I4) => ((! [I2:A]: (((((member @ A) @ I2) @ I4) => (((sigma_space @ B) @ (M @ I2)) = Y3)))) => ((((member @ (set @ B)) @ X4) @ ((sigma_sets @ B) @ (M @ I3))) => (((member @ (set @ B)) @ X4) @ ((sigma_sets @ B) @ ((complete_Sup_Sup @ (sigma_measure @ B)) @ ((((image @ A) @ (sigma_measure @ B)) @ M) @ I4)))))))))).
% 2.81/1.57 thf(fact_64_in__sets__Sup, axiom, (! [A:$tType,M:(set @ (sigma_measure @ A)),X4:(set @ A),M2:(sigma_measure @ A),A4:(set @ A)]: (((! [M3:(sigma_measure @ A)]: (((((member @ (sigma_measure @ A)) @ M3) @ M) => (((sigma_space @ A) @ M3) = X4)))) => ((((member @ (sigma_measure @ A)) @ M2) @ M) => ((((member @ (set @ A)) @ A4) @ ((sigma_sets @ A) @ M2)) => (((member @ (set @ A)) @ A4) @ ((sigma_sets @ A) @ ((complete_Sup_Sup @ (sigma_measure @ A)) @ M))))))))).
% 2.81/1.57 thf(fact_65_sets__SUP__cong, axiom, (! [B:$tType,A:$tType,I4:(set @ A),M:(A > (sigma_measure @ B)),N3:(A > (sigma_measure @ B))]: (((! [I2:A]: (((((member @ A) @ I2) @ I4) => (((sigma_sets @ B) @ (M @ I2)) = ((sigma_sets @ B) @ (N3 @ I2)))))) => (((sigma_sets @ B) @ ((complete_Sup_Sup @ (sigma_measure @ B)) @ ((((image @ A) @ (sigma_measure @ B)) @ M) @ I4))) = ((sigma_sets @ B) @ ((complete_Sup_Sup @ (sigma_measure @ B)) @ ((((image @ A) @ (sigma_measure @ B)) @ N3) @ I4)))))))).
% 2.81/1.57 thf(fact_66_Sup__set__def, axiom, (! [A:$tType]: (((complete_Sup_Sup @ (set @ A)) = (^ [A6:(set @ (set @ A))]: (((collect @ A) @ (^ [X:A]: (((complete_Sup_Sup @ $o) @ ((((image @ (set @ A)) @ $o) @ ((member @ A) @ X)) @ A6))))))))))).
% 2.81/1.57 thf(fact_67_space__Sup__eq__UN, axiom, (! [A:$tType,M:(set @ (sigma_measure @ A))]: ((((sigma_space @ A) @ ((complete_Sup_Sup @ (sigma_measure @ A)) @ M)) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ (sigma_measure @ A)) @ (set @ A)) @ (sigma_space @ A)) @ M)))))).
% 2.81/1.57 thf(fact_68_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.81/1.57 thf(fact_69_Diff__Int__distrib2, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C2:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A4) @ B3)) @ C2) = (((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A4) @ C2)) @ (((inf_inf @ (set @ A)) @ B3) @ C2)))))).
% 2.81/1.57 thf(fact_70_Diff__Int__distrib, axiom, (! [A:$tType,C2:(set @ A),A4:(set @ A),B3:(set @ A)]: (((((inf_inf @ (set @ A)) @ C2) @ (((minus_minus @ (set @ A)) @ A4) @ B3)) = (((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ C2) @ A4)) @ (((inf_inf @ (set @ A)) @ C2) @ B3)))))).
% 2.81/1.57 thf(fact_71_Diff__Diff__Int, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A4) @ (((minus_minus @ (set @ A)) @ A4) @ B3)) = (((inf_inf @ (set @ A)) @ A4) @ B3))))).
% 2.81/1.57 thf(fact_72_Diff__Int2, axiom, (! [A:$tType,A4:(set @ A),C2:(set @ A),B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A4) @ C2)) @ (((inf_inf @ (set @ A)) @ B3) @ C2)) = (((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A4) @ C2)) @ B3))))).
% 2.81/1.57 thf(fact_73_Int__Diff, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C2:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A4) @ B3)) @ C2) = (((inf_inf @ (set @ A)) @ A4) @ (((minus_minus @ (set @ A)) @ B3) @ C2)))))).
% 2.81/1.57 thf(fact_74_vimage__Diff, axiom, (! [A:$tType,B:$tType,F:(A > B),A4:(set @ B),B3:(set @ B)]: ((((((vimage @ A) @ B) @ F) @ (((minus_minus @ (set @ B)) @ A4) @ B3)) = (((minus_minus @ (set @ A)) @ ((((vimage @ A) @ B) @ F) @ A4)) @ ((((vimage @ A) @ B) @ F) @ B3)))))).
% 2.81/1.57 thf(fact_75_Sup__fun__def, axiom, (! [B:$tType,A:$tType]: (((complete_Sup @ B) => ((complete_Sup_Sup @ (A > B)) = (^ [A6:(set @ (A > B)),X:A]: (((complete_Sup_Sup @ B) @ ((((image @ (A > B)) @ B) @ (^ [F2:(A > B)]: ((F2 @ X)))) @ A6))))))))).
% 2.81/1.57 thf(fact_76_null__sets_Osets__Collect__disj, axiom, (! [A:$tType,M:(sigma_measure @ A),P:(A > $o),Q:(A > $o)]: (((((member @ (set @ A)) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ ((sigma_space @ A) @ M)) & (P @ X)))))) @ ((measure_null_sets @ A) @ M)) => ((((member @ (set @ A)) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ ((sigma_space @ A) @ M)) & (Q @ X)))))) @ ((measure_null_sets @ A) @ M)) => (((member @ (set @ A)) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ ((sigma_space @ A) @ M)) & ((Q @ X) | (P @ X))))))) @ ((measure_null_sets @ A) @ M))))))).
% 2.81/1.57 thf(fact_77_null__sets_Osets__Collect__conj, axiom, (! [A:$tType,M:(sigma_measure @ A),P:(A > $o),Q:(A > $o)]: (((((member @ (set @ A)) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ ((sigma_space @ A) @ M)) & (P @ X)))))) @ ((measure_null_sets @ A) @ M)) => ((((member @ (set @ A)) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ ((sigma_space @ A) @ M)) & (Q @ X)))))) @ ((measure_null_sets @ A) @ M)) => (((member @ (set @ A)) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ ((sigma_space @ A) @ M)) & ((Q @ X) & (P @ X))))))) @ ((measure_null_sets @ A) @ M))))))).
% 2.81/1.57 thf(fact_78_translation__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,S2:(set @ A),T:(set @ A)]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ (((minus_minus @ (set @ A)) @ S2) @ T)) = (((minus_minus @ (set @ A)) @ ((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ S2)) @ ((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ T))))))))).
% 2.81/1.57 thf(fact_79_null__set__Diff, axiom, (! [A:$tType,B3:(set @ A),M:(sigma_measure @ A),A4:(set @ A)]: (((((member @ (set @ A)) @ B3) @ ((measure_null_sets @ A) @ M)) => ((((member @ (set @ A)) @ A4) @ ((sigma_sets @ A) @ M)) => (((member @ (set @ A)) @ (((minus_minus @ (set @ A)) @ B3) @ A4)) @ ((measure_null_sets @ A) @ M))))))).
% 2.81/1.57 thf(fact_80_Int__Union2, axiom, (! [A:$tType,B3:(set @ (set @ A)),A4:(set @ A)]: (((((inf_inf @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ B3)) @ A4) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ (set @ A)) @ (set @ A)) @ (^ [C4:(set @ A)]: ((((inf_inf @ (set @ A)) @ C4) @ A4)))) @ B3)))))).
% 2.81/1.57 thf(fact_81_Int__Union, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ (set @ A))]: (((((inf_inf @ (set @ A)) @ A4) @ ((complete_Sup_Sup @ (set @ A)) @ B3)) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ (set @ A)) @ (set @ A)) @ ((inf_inf @ (set @ A)) @ A4)) @ B3)))))).
% 2.81/1.57 thf(fact_82_vimage__Union, axiom, (! [A:$tType,B:$tType,F:(A > B),A4:(set @ (set @ B))]: ((((((vimage @ A) @ B) @ F) @ ((complete_Sup_Sup @ (set @ B)) @ A4)) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ (set @ B)) @ (set @ A)) @ (((vimage @ A) @ B) @ F)) @ A4)))))).
% 2.81/1.57 thf(fact_83_UN__extend__simps_I6_J, axiom, (! [L:$tType,K:$tType,A4:(K > (set @ L)),C2:(set @ K),B3:(set @ L)]: (((((minus_minus @ (set @ L)) @ ((complete_Sup_Sup @ (set @ L)) @ ((((image @ K) @ (set @ L)) @ A4) @ C2))) @ B3) = ((complete_Sup_Sup @ (set @ L)) @ ((((image @ K) @ (set @ L)) @ (^ [X:K]: ((((minus_minus @ (set @ L)) @ (A4 @ X)) @ B3)))) @ C2)))))).
% 2.81/1.57 thf(fact_84_emeasure__Diff__null__set, axiom, (! [A:$tType,B3:(set @ A),M:(sigma_measure @ A),A4:(set @ A)]: (((((member @ (set @ A)) @ B3) @ ((measure_null_sets @ A) @ M)) => ((((member @ (set @ A)) @ A4) @ ((sigma_sets @ A) @ M)) => ((((sigma_emeasure @ A) @ M) @ (((minus_minus @ (set @ A)) @ A4) @ B3)) = (((sigma_emeasure @ A) @ M) @ A4))))))).
% 2.81/1.57 thf(fact_85_Sup_OSUP__cong, axiom, (! [A:$tType,B:$tType,A4:(set @ B),B3:(set @ B),C2:(B > A),D2:(B > A),Sup:((set @ A) > A)]: (((A4 = B3) => ((! [X2:B]: (((((member @ B) @ X2) @ B3) => ((C2 @ X2) = (D2 @ X2))))) => ((Sup @ ((((image @ B) @ A) @ C2) @ A4)) = (Sup @ ((((image @ B) @ A) @ D2) @ B3)))))))).
% 2.81/1.57 thf(fact_86_Inf_OINF__cong, axiom, (! [A:$tType,B:$tType,A4:(set @ B),B3:(set @ B),C2:(B > A),D2:(B > A),Inf:((set @ A) > A)]: (((A4 = B3) => ((! [X2:B]: (((((member @ B) @ X2) @ B3) => ((C2 @ X2) = (D2 @ X2))))) => ((Inf @ ((((image @ B) @ A) @ C2) @ A4)) = (Inf @ ((((image @ B) @ A) @ D2) @ B3)))))))).
% 2.81/1.57 thf(fact_87_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X3:A,A4:(set @ A),B2:B,F:(A > B)]: (((((member @ A) @ X3) @ A4) => ((B2 = (F @ X3)) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ F) @ A4))))))).
% 2.81/1.57 thf(fact_88_ball__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A4:(set @ B),P:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ ((((image @ B) @ A) @ F) @ A4)) => (P @ X2)))) => (! [X5:B]: (((((member @ B) @ X5) @ A4) => (P @ (F @ X5))))))))).
% 2.81/1.57 thf(fact_89_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N3:(set @ A),F:(A > B),G:(A > B)]: (((M = N3) => ((! [X2:A]: (((((member @ A) @ X2) @ N3) => ((F @ X2) = (G @ X2))))) => (((((image @ A) @ B) @ F) @ M) = ((((image @ A) @ B) @ G) @ N3))))))).
% 2.81/1.57 thf(fact_90_bex__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A4:(set @ B),P:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F) @ A4)) & (P @ X5)))) => (? [X2:B]: (((((member @ B) @ X2) @ A4) & (P @ (F @ X2))))))))).
% 2.81/1.57 thf(fact_91_image__iff, axiom, (! [A:$tType,B:$tType,Z:A,F:(B > A),A4:(set @ B)]: (((((member @ A) @ Z) @ ((((image @ B) @ A) @ F) @ A4)) = (? [X:B]: (((((member @ B) @ X) @ A4) & (Z = (F @ X))))))))).
% 2.81/1.57 thf(fact_92_imageI, axiom, (! [B:$tType,A:$tType,X3:A,A4:(set @ A),F:(A > B)]: (((((member @ A) @ X3) @ A4) => (((member @ B) @ (F @ X3)) @ ((((image @ A) @ B) @ F) @ A4)))))).
% 2.81/1.57 thf(fact_93_is__num__normalize_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [A2:A,B2:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.81/1.57 thf(fact_94_UNIV__witness, axiom, (! [A:$tType]: ((? [X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))))).
% 2.81/1.57 thf(fact_95_UNIV__eq__I, axiom, (! [A:$tType,A4:(set @ A)]: (((! [X2:A]: ((((member @ A) @ X2) @ A4))) => ((top_top @ (set @ A)) = A4))))).
% 2.81/1.57 thf(fact_96_Int__left__commute, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C2:(set @ A)]: (((((inf_inf @ (set @ A)) @ A4) @ (((inf_inf @ (set @ A)) @ B3) @ C2)) = (((inf_inf @ (set @ A)) @ B3) @ (((inf_inf @ (set @ A)) @ A4) @ C2)))))).
% 2.81/1.57 thf(fact_97_Int__left__absorb, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A4) @ (((inf_inf @ (set @ A)) @ A4) @ B3)) = (((inf_inf @ (set @ A)) @ A4) @ B3))))).
% 2.81/1.57 thf(fact_98_Int__commute, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A6:(set @ A),B4:(set @ A)]: ((((inf_inf @ (set @ A)) @ B4) @ A6))))))).
% 2.81/1.57 thf(fact_99_Int__absorb, axiom, (! [A:$tType,A4:(set @ A)]: (((((inf_inf @ (set @ A)) @ A4) @ A4) = A4)))).
% 2.81/1.57 thf(fact_100_Int__assoc, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C2:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A4) @ B3)) @ C2) = (((inf_inf @ (set @ A)) @ A4) @ (((inf_inf @ (set @ A)) @ B3) @ C2)))))).
% 2.81/1.57 thf(fact_101_IntD2, axiom, (! [A:$tType,C:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A4) @ B3)) => (((member @ A) @ C) @ B3))))).
% 2.81/1.57 thf(fact_102_IntD1, axiom, (! [A:$tType,C:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A4) @ B3)) => (((member @ A) @ C) @ A4))))).
% 2.81/1.57 thf(fact_103_IntE, axiom, (! [A:$tType,C:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A4) @ B3)) => (~ (((((member @ A) @ C) @ A4) => (~ ((((member @ A) @ C) @ B3)))))))))).
% 2.81/1.57 thf(fact_104_UnionE, axiom, (! [A:$tType,A4:A,C2:(set @ (set @ A))]: (((((member @ A) @ A4) @ ((complete_Sup_Sup @ (set @ A)) @ C2)) => (~ ((! [X6:(set @ A)]: (((((member @ A) @ A4) @ X6) => (~ ((((member @ (set @ A)) @ X6) @ C2)))))))))))).
% 2.81/1.57 thf(fact_105_vimage__Collect, axiom, (! [B:$tType,A:$tType,P:(B > $o),F:(A > B),Q:(A > $o)]: (((! [X2:A]: (((P @ (F @ X2)) = (Q @ X2)))) => (((((vimage @ A) @ B) @ F) @ ((collect @ B) @ P)) = ((collect @ A) @ Q)))))).
% 2.81/1.57 thf(fact_106_vimageI2, axiom, (! [B:$tType,A:$tType,F:(B > A),A2:B,A4:(set @ A)]: (((((member @ A) @ (F @ A2)) @ A4) => (((member @ B) @ A2) @ ((((vimage @ B) @ A) @ F) @ A4)))))).
% 2.81/1.57 thf(fact_107_vimageE, axiom, (! [A:$tType,B:$tType,A2:A,F:(A > B),B3:(set @ B)]: (((((member @ A) @ A2) @ ((((vimage @ A) @ B) @ F) @ B3)) => (((member @ B) @ (F @ A2)) @ B3))))).
% 2.81/1.57 thf(fact_108_vimageD, axiom, (! [A:$tType,B:$tType,A2:A,F:(A > B),A4:(set @ B)]: (((((member @ A) @ A2) @ ((((vimage @ A) @ B) @ F) @ A4)) => (((member @ B) @ (F @ A2)) @ A4))))).
% 2.81/1.57 thf(fact_109_Sup_OSUP__identity__eq, axiom, (! [A:$tType,Sup:((set @ A) > A),A4:(set @ A)]: (((Sup @ ((((image @ A) @ A) @ (^ [X:A]: (X))) @ A4)) = (Sup @ A4))))).
% 2.81/1.57 thf(fact_110_Inf_OINF__identity__eq, axiom, (! [A:$tType,Inf:((set @ A) > A),A4:(set @ A)]: (((Inf @ ((((image @ A) @ A) @ (^ [X:A]: (X))) @ A4)) = (Inf @ A4))))).
% 2.81/1.57 thf(fact_111_Compr__image__eq, axiom, (! [A:$tType,B:$tType,F:(B > A),A4:(set @ B),P:(A > $o)]: ((((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ ((((image @ B) @ A) @ F) @ A4)) & (P @ X))))) = ((((image @ B) @ A) @ F) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ A4) & (P @ (F @ X))))))))))).
% 2.81/1.57 thf(fact_112_image__image, axiom, (! [A:$tType,B:$tType,C3:$tType,F:(B > A),G:(C3 > B),A4:(set @ C3)]: ((((((image @ B) @ A) @ F) @ ((((image @ C3) @ B) @ G) @ A4)) = ((((image @ C3) @ A) @ (^ [X:C3]: ((F @ (G @ X))))) @ A4))))).
% 2.81/1.57 thf(fact_113_imageE, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A),A4:(set @ B)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ A4)) => (~ ((! [X2:B]: (((B2 = (F @ X2)) => (~ ((((member @ B) @ X2) @ A4)))))))))))).
% 2.81/1.57 thf(fact_114_UNIV__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (^ [X:A]: ($true))))))).
% 2.81/1.57 thf(fact_115_Collect__conj__eq, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: ((((collect @ A) @ (^ [X:A]: (((P @ X) & (Q @ X))))) = (((inf_inf @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.81/1.57 thf(fact_116_Int__Collect, axiom, (! [A:$tType,X3:A,A4:(set @ A),P:(A > $o)]: (((((member @ A) @ X3) @ (((inf_inf @ (set @ A)) @ A4) @ ((collect @ A) @ P))) = ((((member @ A) @ X3) @ A4) & (P @ X3)))))).
% 2.81/1.57 thf(fact_117_Int__def, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A6:(set @ A),B4:(set @ A)]: (((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A6) & (((member @ A) @ X) @ B4))))))))))).
% 2.81/1.57 thf(fact_118_vimage__def, axiom, (! [B:$tType,A:$tType]: ((((vimage @ A) @ B) = (^ [F2:(A > B),B4:(set @ B)]: (((collect @ A) @ (^ [X:A]: ((((member @ B) @ (F2 @ X)) @ B4)))))))))).
% 2.81/1.57 thf(fact_119_less__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))).
% 2.81/1.57 thf(fact_120_less__numeral__extra_I4_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (one_one @ A)))))))).
% 2.81/1.57 thf(fact_121_range__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A),X3:B]: (((B2 = (F @ X3)) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))))))).
% 2.81/1.57 thf(fact_122_surj__def, axiom, (! [B:$tType,A:$tType,F:(B > A)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y2:A]: ((? [X:B]: ((Y2 = (F @ X)))))))))).
% 2.81/1.57 thf(fact_123_rangeI, axiom, (! [A:$tType,B:$tType,F:(B > A),X3:B]: ((((member @ A) @ (F @ X3)) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B))))))).
% 2.81/1.57 thf(fact_124_surjI, axiom, (! [B:$tType,A:$tType,G:(B > A),F:(A > B)]: (((! [X2:A]: (((G @ (F @ X2)) = X2))) => (((((image @ B) @ A) @ G) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.81/1.57 thf(fact_125_surjE, axiom, (! [A:$tType,B:$tType,F:(B > A),Y4:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X2:B]: ((Y4 != (F @ X2)))))))))).
% 2.81/1.57 thf(fact_126_surjD, axiom, (! [A:$tType,B:$tType,F:(B > A),Y4:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X2:B]: ((Y4 = (F @ X2)))))))).
% 2.81/1.57 thf(fact_127_less__Sup__iff, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (! [A2:A,S:(set @ A)]: (((((ord_less @ A) @ A2) @ ((complete_Sup_Sup @ A) @ S)) = (? [X:A]: (((((member @ A) @ X) @ S) & (((ord_less @ A) @ A2) @ X))))))))))).
% 2.81/1.57 thf(fact_128_SUP__cong, axiom, (! [A:$tType,B:$tType]: (((complete_Sup @ A) => (! [A4:(set @ B),B3:(set @ B),C2:(B > A),D2:(B > A)]: (((A4 = B3) => ((! [X2:B]: (((((member @ B) @ X2) @ B3) => ((C2 @ X2) = (D2 @ X2))))) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ C2) @ A4)) = ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ D2) @ B3))))))))))).
% 2.81/1.57 thf(fact_129_Int__UNIV__right, axiom, (! [A:$tType,A4:(set @ A)]: (((((inf_inf @ (set @ A)) @ A4) @ (top_top @ (set @ A))) = A4)))).
% 2.81/1.57 thf(fact_130_Int__UNIV__left, axiom, (! [A:$tType,B3:(set @ A)]: (((((inf_inf @ (set @ A)) @ (top_top @ (set @ A))) @ B3) = B3)))).
% 2.81/1.57 thf(fact_131_Union__UNIV, axiom, (! [A:$tType]: ((((complete_Sup_Sup @ (set @ A)) @ (top_top @ (set @ (set @ A)))) = (top_top @ (set @ A)))))).
% 2.81/1.57 thf(fact_132_null__setsD2, axiom, (! [A:$tType,A4:(set @ A),M:(sigma_measure @ A)]: (((((member @ (set @ A)) @ A4) @ ((measure_null_sets @ A) @ M)) => (((member @ (set @ A)) @ A4) @ ((sigma_sets @ A) @ M)))))).
% 2.81/1.57 thf(fact_133_vimage__inter__cong, axiom, (! [B:$tType,A:$tType,S:(set @ A),F:(A > B),G:(A > B),Y4:(set @ B)]: (((! [W:A]: (((((member @ A) @ W) @ S) => ((F @ W) = (G @ W))))) => ((((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ F) @ Y4)) @ S) = (((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ G) @ Y4)) @ S)))))).
% 2.81/1.57 thf(fact_134_translation__subtract__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,S2:(set @ A),T:(set @ A)]: ((((((image @ A) @ A) @ (^ [X:A]: ((((minus_minus @ A) @ X) @ A2)))) @ (((minus_minus @ (set @ A)) @ S2) @ T)) = (((minus_minus @ (set @ A)) @ ((((image @ A) @ A) @ (^ [X:A]: ((((minus_minus @ A) @ X) @ A2)))) @ S2)) @ ((((image @ A) @ A) @ (^ [X:A]: ((((minus_minus @ A) @ X) @ A2)))) @ T))))))))).
% 2.81/1.57 thf(fact_135_range__composition, axiom, (! [A:$tType,C3:$tType,B:$tType,F:(C3 > A),G:(B > C3)]: ((((((image @ B) @ A) @ (^ [X:B]: ((F @ (G @ X))))) @ (top_top @ (set @ B))) = ((((image @ C3) @ A) @ F) @ ((((image @ B) @ C3) @ G) @ (top_top @ (set @ B)))))))).
% 2.81/1.57 thf(fact_136_rangeE, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))) => (~ ((! [X2:B]: ((B2 != (F @ X2)))))))))).
% 2.81/1.57 thf(fact_137_SUP__commute, axiom, (! [A:$tType,B:$tType,C3:$tType]: (((comple187826305attice @ A) => (! [F:(B > (C3 > A)),B3:(set @ C3),A4:(set @ B)]: ((((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [I5:B]: (((complete_Sup_Sup @ A) @ ((((image @ C3) @ A) @ (F @ I5)) @ B3))))) @ A4)) = ((complete_Sup_Sup @ A) @ ((((image @ C3) @ A) @ (^ [J:C3]: (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [I5:B]: (((F @ I5) @ J)))) @ A4))))) @ B3))))))))).
% 2.81/1.57 thf(fact_138_image__Union, axiom, (! [A:$tType,B:$tType,F:(B > A),S:(set @ (set @ B))]: ((((((image @ B) @ A) @ F) @ ((complete_Sup_Sup @ (set @ B)) @ S)) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ (set @ B)) @ (set @ A)) @ (((image @ B) @ A) @ F)) @ S)))))).
% 2.81/1.57 thf(fact_139_UN__UN__flatten, axiom, (! [A:$tType,B:$tType,C3:$tType,C2:(B > (set @ A)),B3:(C3 > (set @ B)),A4:(set @ C3)]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ C2) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ C3) @ (set @ B)) @ B3) @ A4)))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ C3) @ (set @ A)) @ (^ [Y2:C3]: (((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ C2) @ (B3 @ Y2)))))) @ A4)))))).
% 2.81/1.57 thf(fact_140_UN__E, axiom, (! [A:$tType,B:$tType,B2:A,B3:(B > (set @ A)),A4:(set @ B)]: (((((member @ A) @ B2) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B3) @ A4))) => (~ ((! [X2:B]: (((((member @ B) @ X2) @ A4) => (~ ((((member @ A) @ B2) @ (B3 @ X2))))))))))))).
% 2.81/1.57 thf(fact_141_UN__extend__simps_I8_J, axiom, (! [P2:$tType,O:$tType,B3:(O > (set @ P2)),A4:(set @ (set @ O))]: ((((complete_Sup_Sup @ (set @ P2)) @ ((((image @ (set @ O)) @ (set @ P2)) @ (^ [Y2:(set @ O)]: (((complete_Sup_Sup @ (set @ P2)) @ ((((image @ O) @ (set @ P2)) @ B3) @ Y2))))) @ A4)) = ((complete_Sup_Sup @ (set @ P2)) @ ((((image @ O) @ (set @ P2)) @ B3) @ ((complete_Sup_Sup @ (set @ O)) @ A4))))))).
% 2.81/1.57 thf(fact_142_UN__extend__simps_I9_J, axiom, (! [S3:$tType,R:$tType,Q2:$tType,C2:(R > (set @ S3)),B3:(Q2 > (set @ R)),A4:(set @ Q2)]: ((((complete_Sup_Sup @ (set @ S3)) @ ((((image @ Q2) @ (set @ S3)) @ (^ [X:Q2]: (((complete_Sup_Sup @ (set @ S3)) @ ((((image @ R) @ (set @ S3)) @ C2) @ (B3 @ X)))))) @ A4)) = ((complete_Sup_Sup @ (set @ S3)) @ ((((image @ R) @ (set @ S3)) @ C2) @ ((complete_Sup_Sup @ (set @ R)) @ ((((image @ Q2) @ (set @ R)) @ B3) @ A4)))))))).
% 2.81/1.57 thf(fact_143_less__numeral__extra_I1_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 2.81/1.57 thf(fact_144_translation__Int, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,S2:(set @ A),T:(set @ A)]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ (((inf_inf @ (set @ A)) @ S2) @ T)) = (((inf_inf @ (set @ A)) @ ((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ S2)) @ ((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ T))))))))).
% 2.81/1.57 thf(fact_145_surj__image__vimage__eq, axiom, (! [B:$tType,A:$tType,F:(B > A),A4:(set @ A)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((((image @ B) @ A) @ F) @ ((((vimage @ B) @ A) @ F) @ A4)) = A4))))).
% 2.81/1.57 thf(fact_146_null__set__Int2, axiom, (! [A:$tType,B3:(set @ A),M:(sigma_measure @ A),A4:(set @ A)]: (((((member @ (set @ A)) @ B3) @ ((measure_null_sets @ A) @ M)) => ((((member @ (set @ A)) @ A4) @ ((sigma_sets @ A) @ M)) => (((member @ (set @ A)) @ (((inf_inf @ (set @ A)) @ B3) @ A4)) @ ((measure_null_sets @ A) @ M))))))).
% 2.81/1.57 thf(fact_147_null__set__Int1, axiom, (! [A:$tType,B3:(set @ A),M:(sigma_measure @ A),A4:(set @ A)]: (((((member @ (set @ A)) @ B3) @ ((measure_null_sets @ A) @ M)) => ((((member @ (set @ A)) @ A4) @ ((sigma_sets @ A) @ M)) => (((member @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A4) @ B3)) @ ((measure_null_sets @ A) @ M))))))).
% 2.81/1.57 thf(fact_148_null__setsD1, axiom, (! [A:$tType,A4:(set @ A),M:(sigma_measure @ A)]: (((((member @ (set @ A)) @ A4) @ ((measure_null_sets @ A) @ M)) => ((((sigma_emeasure @ A) @ M) @ A4) = (zero_zero @ extend1814228343nnreal)))))).
% 2.81/1.57 thf(fact_149_less__SUP__iff, axiom, (! [A:$tType,B:$tType]: (((comple1035589618norder @ A) => (! [A2:A,F:(B > A),A4:(set @ B)]: (((((ord_less @ A) @ A2) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A4))) = (? [X:B]: (((((member @ B) @ X) @ A4) & (((ord_less @ A) @ A2) @ (F @ X)))))))))))).
% 2.81/1.57 thf(fact_150_SUP__lessD, axiom, (! [B:$tType,A:$tType]: (((comple187826305attice @ A) => (! [F:(B > A),A4:(set @ B),Y4:A,I3:B]: (((((ord_less @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A4))) @ Y4) => ((((member @ B) @ I3) @ A4) => (((ord_less @ A) @ (F @ I3)) @ Y4))))))))).
% 2.81/1.57 thf(fact_151_translation__subtract__Int, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,S2:(set @ A),T:(set @ A)]: ((((((image @ A) @ A) @ (^ [X:A]: ((((minus_minus @ A) @ X) @ A2)))) @ (((inf_inf @ (set @ A)) @ S2) @ T)) = (((inf_inf @ (set @ A)) @ ((((image @ A) @ A) @ (^ [X:A]: ((((minus_minus @ A) @ X) @ A2)))) @ S2)) @ ((((image @ A) @ A) @ (^ [X:A]: ((((minus_minus @ A) @ X) @ A2)))) @ T))))))))).
% 2.81/1.57 thf(fact_152_image__UN, axiom, (! [A:$tType,B:$tType,C3:$tType,F:(B > A),B3:(C3 > (set @ B)),A4:(set @ C3)]: ((((((image @ B) @ A) @ F) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ C3) @ (set @ B)) @ B3) @ A4))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ C3) @ (set @ A)) @ (^ [X:C3]: (((((image @ B) @ A) @ F) @ (B3 @ X))))) @ A4)))))).
% 2.81/1.57 thf(fact_153_UN__extend__simps_I10_J, axiom, (! [V:$tType,U:$tType,T2:$tType,B3:(U > (set @ V)),F:(T2 > U),A4:(set @ T2)]: ((((complete_Sup_Sup @ (set @ V)) @ ((((image @ T2) @ (set @ V)) @ (^ [A3:T2]: ((B3 @ (F @ A3))))) @ A4)) = ((complete_Sup_Sup @ (set @ V)) @ ((((image @ U) @ (set @ V)) @ B3) @ ((((image @ T2) @ U) @ F) @ A4))))))).
% 2.81/1.57 thf(fact_154_Int__UN__distrib2, axiom, (! [A:$tType,C3:$tType,B:$tType,A4:(B > (set @ A)),I4:(set @ B),B3:(C3 > (set @ A)),J2:(set @ C3)]: (((((inf_inf @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ A4) @ I4))) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ C3) @ (set @ A)) @ B3) @ J2))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [I5:B]: (((complete_Sup_Sup @ (set @ A)) @ ((((image @ C3) @ (set @ A)) @ (^ [J:C3]: ((((inf_inf @ (set @ A)) @ (A4 @ I5)) @ (B3 @ J))))) @ J2))))) @ I4)))))).
% 2.81/1.57 thf(fact_155_Int__UN__distrib, axiom, (! [A:$tType,B:$tType,B3:(set @ A),A4:(B > (set @ A)),I4:(set @ B)]: (((((inf_inf @ (set @ A)) @ B3) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ A4) @ I4))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [I5:B]: ((((inf_inf @ (set @ A)) @ B3) @ (A4 @ I5))))) @ I4)))))).
% 2.81/1.57 thf(fact_156_UN__extend__simps_I4_J, axiom, (! [H:$tType,G2:$tType,A4:(G2 > (set @ H)),C2:(set @ G2),B3:(set @ H)]: (((((inf_inf @ (set @ H)) @ ((complete_Sup_Sup @ (set @ H)) @ ((((image @ G2) @ (set @ H)) @ A4) @ C2))) @ B3) = ((complete_Sup_Sup @ (set @ H)) @ ((((image @ G2) @ (set @ H)) @ (^ [X:G2]: ((((inf_inf @ (set @ H)) @ (A4 @ X)) @ B3)))) @ C2)))))).
% 2.81/1.57 thf(fact_157_UN__extend__simps_I5_J, axiom, (! [I:$tType,J3:$tType,A4:(set @ I),B3:(J3 > (set @ I)),C2:(set @ J3)]: (((((inf_inf @ (set @ I)) @ A4) @ ((complete_Sup_Sup @ (set @ I)) @ ((((image @ J3) @ (set @ I)) @ B3) @ C2))) = ((complete_Sup_Sup @ (set @ I)) @ ((((image @ J3) @ (set @ I)) @ (^ [X:J3]: ((((inf_inf @ (set @ I)) @ A4) @ (B3 @ X))))) @ C2)))))).
% 2.81/1.57 thf(fact_158_vimage__UN, axiom, (! [A:$tType,B:$tType,C3:$tType,F:(A > B),B3:(C3 > (set @ B)),A4:(set @ C3)]: ((((((vimage @ A) @ B) @ F) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ C3) @ (set @ B)) @ B3) @ A4))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ C3) @ (set @ A)) @ (^ [X:C3]: (((((vimage @ A) @ B) @ F) @ (B3 @ X))))) @ A4)))))).
% 2.81/1.57 thf(fact_159_SUP__UNION, axiom, (! [A:$tType,B:$tType,C3:$tType]: (((comple187826305attice @ A) => (! [F:(B > A),G:(C3 > (set @ B)),A4:(set @ C3)]: ((((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ C3) @ (set @ B)) @ G) @ A4)))) = ((complete_Sup_Sup @ A) @ ((((image @ C3) @ A) @ (^ [Y2:C3]: (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ (G @ Y2)))))) @ A4))))))))).
% 2.81/1.57 thf(fact_160_null__sets__def, axiom, (! [A:$tType]: (((measure_null_sets @ A) = (^ [M4:(sigma_measure @ A)]: (((collect @ (set @ A)) @ (^ [N4:(set @ A)]: (((((member @ (set @ A)) @ N4) @ ((sigma_sets @ A) @ M4)) & ((((sigma_emeasure @ A) @ M4) @ N4) = (zero_zero @ extend1814228343nnreal)))))))))))).
% 2.81/1.57 thf(fact_161_sets_OInt__space__eq2, axiom, (! [A:$tType,X3:(set @ A),M:(sigma_measure @ A)]: (((((member @ (set @ A)) @ X3) @ ((sigma_sets @ A) @ M)) => ((((inf_inf @ (set @ A)) @ X3) @ ((sigma_space @ A) @ M)) = X3))))).
% 2.81/1.57 thf(fact_162_sets_OInt__space__eq1, axiom, (! [A:$tType,X3:(set @ A),M:(sigma_measure @ A)]: (((((member @ (set @ A)) @ X3) @ ((sigma_sets @ A) @ M)) => ((((inf_inf @ (set @ A)) @ ((sigma_space @ A) @ M)) @ X3) = X3))))).
% 2.81/1.57 thf(fact_163_diff__add__zero, axiom, (! [A:$tType]: (((comm_monoid_diff @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B2)) = (zero_zero @ A)))))))).
% 2.81/1.57 thf(fact_164_diff__gt__0__iff__gt, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A2) @ B2)) = (((ord_less @ A) @ B2) @ A2)))))))).
% 2.81/1.57 thf(fact_165_zero__less__double__add__iff__zero__less__single__add, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.81/1.57 thf(fact_166_double__add__less__zero__iff__single__add__less__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.81/1.57 thf(fact_167_less__add__same__cancel2, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (((plus_plus @ A) @ B2) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))).
% 2.81/1.57 thf(fact_168_less__add__same__cancel1, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B2)) = (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))).
% 2.81/1.57 thf(fact_169_add__less__same__cancel2, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.81/1.57 thf(fact_170_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B2:A,C:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C)) = (B2 = C)))))))).
% 2.81/1.57 thf(fact_171_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A2:A,C:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C) @ A2)) = (B2 = C)))))))).
% 2.81/1.57 thf(fact_172_DiffI, axiom, (! [A:$tType,C:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ A4) => ((~ ((((member @ A) @ C) @ B3))) => (((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A4) @ B3))))))).
% 2.81/1.57 thf(fact_173_Diff__iff, axiom, (! [A:$tType,C:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A4) @ B3)) = ((((member @ A) @ C) @ A4) & (~ ((((member @ A) @ C) @ B3)))))))).
% 2.81/1.57 thf(fact_174_Diff__idemp, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A4) @ B3)) @ B3) = (((minus_minus @ (set @ A)) @ A4) @ B3))))).
% 2.81/1.57 thf(fact_175_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N5:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N5))) = (N5 = (zero_zero @ A))))))))).
% 2.81/1.57 thf(fact_176_add_Oleft__neutral, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))).
% 2.81/1.57 thf(fact_177_add_Oright__neutral, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.81/1.57 thf(fact_178_double__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: ((((((plus_plus @ A) @ A2) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.81/1.57 thf(fact_179_double__zero__sym, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ A2) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 2.81/1.57 thf(fact_180_add__cancel__left__left, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [B2:A,A2:A]: ((((((plus_plus @ A) @ B2) @ A2) = A2) = (B2 = (zero_zero @ A))))))))).
% 2.81/1.57 thf(fact_181_add__cancel__left__right, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A,B2:A]: ((((((plus_plus @ A) @ A2) @ B2) = A2) = (B2 = (zero_zero @ A))))))))).
% 2.81/1.57 thf(fact_182_add__cancel__right__left, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A,B2:A]: (((A2 = (((plus_plus @ A) @ B2) @ A2)) = (B2 = (zero_zero @ A))))))))).
% 2.81/1.57 thf(fact_183_add__cancel__right__right, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A,B2:A]: (((A2 = (((plus_plus @ A) @ A2) @ B2)) = (B2 = (zero_zero @ A))))))))).
% 2.81/1.57 thf(fact_184_add__eq__0__iff__both__eq__0, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X3:A,Y4:A]: ((((((plus_plus @ A) @ X3) @ Y4) = (zero_zero @ A)) = ((X3 = (zero_zero @ A)) & (Y4 = (zero_zero @ A)))))))))).
% 2.81/1.57 thf(fact_185_zero__eq__add__iff__both__eq__0, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X3:A,Y4:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ X3) @ Y4)) = ((X3 = (zero_zero @ A)) & (Y4 = (zero_zero @ A)))))))))).
% 2.81/1.57 thf(fact_186_diff__self, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 2.81/1.57 thf(fact_187_diff__0__right, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.81/1.57 thf(fact_188_zero__diff, axiom, (! [A:$tType]: (((comm_monoid_diff @ A) => (! [A2:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 2.81/1.57 thf(fact_189_diff__zero, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.81/1.57 thf(fact_190_cancel__comm__monoid__add__class_Odiff__cancel, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 2.81/1.57 thf(fact_191_add__less__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2)) = (((ord_less @ A) @ A2) @ B2)))))))).
% 2.81/1.57 thf(fact_192_add__less__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C)) = (((ord_less @ A) @ A2) @ B2)))))))).
% 2.81/1.57 thf(fact_193_add__diff__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = A2))))))).
% 2.81/1.57 thf(fact_194_diff__add__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ B2) = A2))))))).
% 2.81/1.57 thf(fact_195_add__diff__cancel__left, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [C:A,A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2)) = (((minus_minus @ A) @ A2) @ B2)))))))).
% 2.81/1.57 thf(fact_196_add__diff__cancel__left_H, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ A2) = B2))))))).
% 2.81/1.57 thf(fact_197_add__diff__cancel__right, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,C:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C)) = (((minus_minus @ A) @ A2) @ B2)))))))).
% 2.81/1.57 thf(fact_198_add__diff__cancel__right_H, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = A2))))))).
% 2.81/1.57 thf(fact_199_sets_OInt, axiom, (! [A:$tType,A2:(set @ A),M:(sigma_measure @ A),B2:(set @ A)]: (((((member @ (set @ A)) @ A2) @ ((sigma_sets @ A) @ M)) => ((((member @ (set @ A)) @ B2) @ ((sigma_sets @ A) @ M)) => (((member @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ B2)) @ ((sigma_sets @ A) @ M))))))).
% 2.81/1.57 thf(fact_200_sets_Otop, axiom, (! [A:$tType,M:(sigma_measure @ A)]: ((((member @ (set @ A)) @ ((sigma_space @ A) @ M)) @ ((sigma_sets @ A) @ M))))).
% 2.81/1.57 thf(fact_201_sets_ODiff, axiom, (! [A:$tType,A2:(set @ A),M:(sigma_measure @ A),B2:(set @ A)]: (((((member @ (set @ A)) @ A2) @ ((sigma_sets @ A) @ M)) => ((((member @ (set @ A)) @ B2) @ ((sigma_sets @ A) @ M)) => (((member @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) @ ((sigma_sets @ A) @ M))))))).
% 2.81/1.57 thf(fact_202_add__less__same__cancel1, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ B2) @ A2)) @ B2) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.81/1.57 thf(fact_203_sets_Ocompl__sets, axiom, (! [A:$tType,A2:(set @ A),M:(sigma_measure @ A)]: (((((member @ (set @ A)) @ A2) @ ((sigma_sets @ A) @ M)) => (((member @ (set @ A)) @ (((minus_minus @ (set @ A)) @ ((sigma_space @ A) @ M)) @ A2)) @ ((sigma_sets @ A) @ M)))))).
% 2.81/1.57 thf(fact_204_DiffE, axiom, (! [A:$tType,C:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A4) @ B3)) => (~ (((((member @ A) @ C) @ A4) => (((member @ A) @ C) @ B3)))))))).
% 2.81/1.57 thf(fact_205_DiffD1, axiom, (! [A:$tType,C:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A4) @ B3)) => (((member @ A) @ C) @ A4))))).
% 2.81/1.57 thf(fact_206_DiffD2, axiom, (! [A:$tType,C:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A4) @ B3)) => (~ ((((member @ A) @ C) @ B3))))))).
% 2.81/1.57 thf(fact_207_set__diff__eq, axiom, (! [A:$tType]: (((minus_minus @ (set @ A)) = (^ [A6:(set @ A),B4:(set @ A)]: (((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A6) & (~ ((((member @ A) @ X) @ B4))))))))))))).
% 2.81/1.57 thf(fact_208_minus__set__def, axiom, (! [A:$tType]: (((minus_minus @ (set @ A)) = (^ [A6:(set @ A),B4:(set @ A)]: (((collect @ A) @ (((minus_minus @ (A > $o)) @ (^ [X:A]: ((((member @ A) @ X) @ A6)))) @ (^ [X:A]: ((((member @ A) @ X) @ B4))))))))))).
% 2.81/1.57 thf(fact_209_psubset__imp__ex__mem, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B3) => (? [B5:A]: ((((member @ A) @ B5) @ (((minus_minus @ (set @ A)) @ B3) @ A4)))))))).
% 2.81/1.57 thf(fact_210_Sup__bool__def, axiom, ((complete_Sup_Sup @ $o) = ((member @ $o) @ $true))).
% 2.81/1.57 thf(fact_211_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X3:A]: ((((zero_zero @ A) = X3) = (X3 = (zero_zero @ A))))))))).
% 2.81/1.57 thf(fact_212_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A2:A,B2:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.81/1.57 thf(fact_213_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I3:A,J4:A,K2:A,L2:A]: ((((I3 = J4) & (K2 = L2)) => ((((plus_plus @ A) @ I3) @ K2) = (((plus_plus @ A) @ J4) @ L2))))))))).
% 2.81/1.57 thf(fact_214_group__cancel_Oadd1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A4:A,K2:A,A2:A,B2:A]: (((A4 = (((plus_plus @ A) @ K2) @ A2)) => ((((plus_plus @ A) @ A4) @ B2) = (((plus_plus @ A) @ K2) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.81/1.57 thf(fact_215_group__cancel_Oadd2, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [B3:A,K2:A,B2:A,A2:A]: (((B3 = (((plus_plus @ A) @ K2) @ B2)) => ((((plus_plus @ A) @ A2) @ B3) = (((plus_plus @ A) @ K2) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.81/1.57 thf(fact_216_add_Oassoc, axiom, (! [A:$tType]: (((semigroup_add @ A) => (! [A2:A,B2:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.81/1.57 thf(fact_217_add_Oleft__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C)) = (B2 = C)))))))).
% 2.81/1.57 thf(fact_218_add_Oright__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [B2:A,A2:A,C:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C) @ A2)) = (B2 = C)))))))).
% 2.81/1.57 thf(fact_219_add_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((plus_plus @ A) = (^ [A3:A,B6:A]: ((((plus_plus @ A) @ B6) @ A3)))))))).
% 2.81/1.57 thf(fact_220_add_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [B2:A,A2:A,C:A]: (((((plus_plus @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C)) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.81/1.57 thf(fact_221_add__left__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B2:A,C:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C)) => (B2 = C)))))))).
% 2.81/1.57 thf(fact_222_add__right__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A2:A,C:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C) @ A2)) => (B2 = C)))))))).
% 2.81/1.57 thf(fact_223_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C:A,D3:A]: ((((((minus_minus @ A) @ A2) @ B2) = (((minus_minus @ A) @ C) @ D3)) => ((A2 = B2) = (C = D3))))))))).
% 2.81/1.57 thf(fact_224_diff__right__commute, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,C:A,B2:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C)) @ B2) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C)))))))).
% 2.81/1.57 thf(fact_225_one__reorient, axiom, (! [A:$tType]: (((one @ A) => (! [X3:A]: ((((one_one @ A) = X3) = (X3 = (one_one @ A))))))))).
% 2.81/1.57 thf(fact_226_gr__zeroI, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N5:A]: (((N5 != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N5)))))))).
% 2.81/1.57 thf(fact_227_not__less__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N5:A]: ((~ ((((ord_less @ A) @ N5) @ (zero_zero @ A)))))))))).
% 2.81/1.57 thf(fact_228_gr__implies__not__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [M2:A,N5:A]: (((((ord_less @ A) @ M2) @ N5) => (N5 != (zero_zero @ A))))))))).
% 2.81/1.57 thf(fact_229_zero__less__iff__neq__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N5:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N5) = (N5 != (zero_zero @ A))))))))).
% 2.81/1.57 thf(fact_230_comm__monoid__add__class_Oadd__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))).
% 2.81/1.57 thf(fact_231_add_Ocomm__neutral, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.81/1.57 thf(fact_232_add_Ogroup__left__neutral, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))).
% 2.81/1.57 thf(fact_233_eq__iff__diff__eq__0, axiom, (! [A:$tType]: (((group_add @ A) => ((^ [Y5:A,Z2:A]: ((Y5 = Z2))) = (^ [A3:A,B6:A]: (((((minus_minus @ A) @ A3) @ B6) = (zero_zero @ A))))))))).
% 2.81/1.57 thf(fact_234_add__mono__thms__linordered__field_I5_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I3:A,J4:A,K2:A,L2:A]: ((((((ord_less @ A) @ I3) @ J4) & (((ord_less @ A) @ K2) @ L2)) => (((ord_less @ A) @ (((plus_plus @ A) @ I3) @ K2)) @ (((plus_plus @ A) @ J4) @ L2))))))))).
% 2.81/1.57 thf(fact_235_add__mono__thms__linordered__field_I2_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I3:A,J4:A,K2:A,L2:A]: ((((I3 = J4) & (((ord_less @ A) @ K2) @ L2)) => (((ord_less @ A) @ (((plus_plus @ A) @ I3) @ K2)) @ (((plus_plus @ A) @ J4) @ L2))))))))).
% 2.81/1.57 thf(fact_236_add__mono__thms__linordered__field_I1_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I3:A,J4:A,K2:A,L2:A]: ((((((ord_less @ A) @ I3) @ J4) & (K2 = L2)) => (((ord_less @ A) @ (((plus_plus @ A) @ I3) @ K2)) @ (((plus_plus @ A) @ J4) @ L2))))))))).
% 2.81/1.57 thf(fact_237_add__strict__mono, axiom, (! [A:$tType]: (((strict2144017051up_add @ A) => (! [A2:A,B2:A,C:A,D3:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ C) @ D3) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ D3)))))))))).
% 2.81/1.57 thf(fact_238_add__strict__left__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2))))))))).
% 2.81/1.57 thf(fact_239_add__strict__right__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.81/1.57 thf(fact_240_add__less__imp__less__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2)) => (((ord_less @ A) @ A2) @ B2)))))))).
% 2.81/1.57 thf(fact_241_add__less__imp__less__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C)) => (((ord_less @ A) @ A2) @ B2)))))))).
% 2.81/1.57 thf(fact_242_diff__strict__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A,D3:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ D3) @ C) => (((ord_less @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B2) @ D3)))))))))).
% 2.81/1.57 thf(fact_243_diff__eq__diff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A,C:A,D3:A]: ((((((minus_minus @ A) @ A2) @ B2) = (((minus_minus @ A) @ C) @ D3)) => ((((ord_less @ A) @ A2) @ B2) = (((ord_less @ A) @ C) @ D3))))))))).
% 2.81/1.57 thf(fact_244_diff__strict__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A2:A,C:A]: (((((ord_less @ A) @ B2) @ A2) => (((ord_less @ A) @ (((minus_minus @ A) @ C) @ A2)) @ (((minus_minus @ A) @ C) @ B2))))))))).
% 2.81/1.57 thf(fact_245_diff__strict__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B2) @ C))))))))).
% 2.81/1.57 thf(fact_246_group__cancel_Osub1, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A4:A,K2:A,A2:A,B2:A]: (((A4 = (((plus_plus @ A) @ K2) @ A2)) => ((((minus_minus @ A) @ A4) @ B2) = (((plus_plus @ A) @ K2) @ (((minus_minus @ A) @ A2) @ B2)))))))))).
% 2.81/1.57 thf(fact_247_diff__eq__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C:A]: ((((((minus_minus @ A) @ A2) @ B2) = C) = (A2 = (((plus_plus @ A) @ C) @ B2))))))))).
% 2.81/1.57 thf(fact_248_eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,C:A,B2:A]: (((A2 = (((minus_minus @ A) @ C) @ B2)) = ((((plus_plus @ A) @ A2) @ B2) = C)))))))).
% 2.81/1.57 thf(fact_249_add__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C:A]: (((((plus_plus @ A) @ A2) @ (((minus_minus @ A) @ B2) @ C)) = (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C)))))))).
% 2.81/1.57 thf(fact_250_diff__diff__eq2, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C:A]: (((((minus_minus @ A) @ A2) @ (((minus_minus @ A) @ B2) @ C)) = (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ C)) @ B2)))))))).
% 2.81/1.57 thf(fact_251_diff__add__eq, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,B2:A,C:A]: (((((plus_plus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C) = (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ C)) @ B2)))))))).
% 2.81/1.57 thf(fact_252_diff__add__eq__diff__diff__swap, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C:A]: (((((minus_minus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C)) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C)) @ B2)))))))).
% 2.81/1.57 thf(fact_253_diff__diff__add, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,B2:A,C:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C) = (((minus_minus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.81/1.57 thf(subcl_Finite__Set_Ofinite___HOL_Otype, axiom, (! [A:$tType]: (((finite_finite @ A) => (type @ A))))).
% 2.81/1.57 thf(subcl_Finite__Set_Ofinite___Countable_Ocountable, axiom, (! [A:$tType]: (((finite_finite @ A) => (countable @ A))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space, axiom, (! [A7:$tType,A8:$tType]: ((((ordere890947078_space @ A7) & (finite_finite @ A8)) => (ordere890947078_space @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Rings_Oring__1, axiom, (! [A7:$tType,A8:$tType]: ((((ring_1 @ A7) & (finite_finite @ A8)) => (ring_1 @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_1, axiom, (ordere890947078_space @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Rings_Oring__1_2, axiom, (ring_1 @ real)).
% 2.81/1.57 thf(tcon_Int_Oint___Rings_Oring__1_3, axiom, (ring_1 @ int)).
% 2.81/1.57 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A7:$tType,A8:$tType]: (((comple187826305attice @ A8) => (comple187826305attice @ (A7 > A8)))))).
% 2.81/1.57 thf(tcon_fun___Complete__Lattices_OSup, axiom, (! [A7:$tType,A8:$tType]: (((complete_Sup @ A8) => (complete_Sup @ (A7 > A8)))))).
% 2.81/1.57 thf(tcon_fun___Countable_Ocountable, axiom, (! [A7:$tType,A8:$tType]: ((((finite_finite @ A7) & (countable @ A8)) => (countable @ (A7 > A8)))))).
% 2.81/1.57 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A7:$tType,A8:$tType]: ((((finite_finite @ A7) & (finite_finite @ A8)) => (finite_finite @ (A7 > A8)))))).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le, axiom, (ordere516151231imp_le @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add, axiom, (strict2144017051up_add @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add, axiom, (ordere223160158up_add @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le, axiom, (ordere236663937imp_le @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add, axiom, (ordere779506340up_add @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add, axiom, (linord219039673up_add @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add, axiom, (cancel146912293up_add @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add, axiom, (cancel1352612707id_add @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add, axiom, (ordered_ab_group_add @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add, axiom, (cancel_semigroup_add @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Oab__semigroup__add, axiom, (ab_semigroup_add @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add, axiom, (comm_monoid_add @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Complete__Lattices_OSup_4, axiom, (complete_Sup @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Osemigroup__add, axiom, (semigroup_add @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Oab__group__add, axiom, (ab_group_add @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Countable_Ocountable_5, axiom, (countable @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Omonoid__add, axiom, (monoid_add @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Ogroup__add, axiom, (group_add @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Num_Oneg__numeral, axiom, (neg_numeral @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Nat_Oring__char__0, axiom, (ring_char_0 @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Ozero, axiom, (zero @ int)).
% 2.81/1.57 thf(tcon_Int_Oint___Groups_Oone, axiom, (one @ int)).
% 2.81/1.57 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_6, axiom, (ordere516151231imp_le @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_7, axiom, (strict2144017051up_add @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_8, axiom, (ordere223160158up_add @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_9, axiom, (ordere236663937imp_le @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_10, axiom, (linord1659791738miring @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_11, axiom, (ordere779506340up_add @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_12, axiom, (cancel146912293up_add @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_13, axiom, (cancel1352612707id_add @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_14, axiom, (cancel_semigroup_add @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff, axiom, (comm_monoid_diff @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_15, axiom, (ab_semigroup_add @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_16, axiom, (comm_monoid_add @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Complete__Lattices_OSup_17, axiom, (complete_Sup @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Groups_Osemigroup__add_18, axiom, (semigroup_add @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Countable_Ocountable_19, axiom, (countable @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Groups_Omonoid__add_20, axiom, (monoid_add @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Groups_Ozero_21, axiom, (zero @ nat)).
% 2.81/1.57 thf(tcon_Nat_Onat___Groups_Oone_22, axiom, (one @ nat)).
% 2.81/1.57 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_23, axiom, (! [A7:$tType]: ((comple187826305attice @ (set @ A7))))).
% 2.81/1.57 thf(tcon_Set_Oset___Groups_Oab__semigroup__add_24, axiom, (! [A7:$tType]: (((ab_semigroup_add @ A7) => (ab_semigroup_add @ (set @ A7)))))).
% 2.81/1.57 thf(tcon_Set_Oset___Groups_Ocomm__monoid__add_25, axiom, (! [A7:$tType]: (((comm_monoid_add @ A7) => (comm_monoid_add @ (set @ A7)))))).
% 2.81/1.57 thf(tcon_Set_Oset___Complete__Lattices_OSup_26, axiom, (! [A7:$tType]: ((complete_Sup @ (set @ A7))))).
% 2.81/1.57 thf(tcon_Set_Oset___Groups_Osemigroup__add_27, axiom, (! [A7:$tType]: (((semigroup_add @ A7) => (semigroup_add @ (set @ A7)))))).
% 2.81/1.57 thf(tcon_Set_Oset___Countable_Ocountable_28, axiom, (! [A7:$tType]: (((finite_finite @ A7) => (countable @ (set @ A7)))))).
% 2.81/1.57 thf(tcon_Set_Oset___Groups_Omonoid__add_29, axiom, (! [A7:$tType]: (((monoid_add @ A7) => (monoid_add @ (set @ A7)))))).
% 2.81/1.57 thf(tcon_Set_Oset___Finite__Set_Ofinite_30, axiom, (! [A7:$tType]: (((finite_finite @ A7) => (finite_finite @ (set @ A7)))))).
% 2.81/1.57 thf(tcon_Set_Oset___Groups_Ozero_31, axiom, (! [A7:$tType]: (((zero @ A7) => (zero @ (set @ A7)))))).
% 2.81/1.57 thf(tcon_Set_Oset___Groups_Oone_32, axiom, (! [A7:$tType]: (((one @ A7) => (one @ (set @ A7)))))).
% 2.81/1.57 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_33, axiom, (comple187826305attice @ $o)).
% 2.81/1.57 thf(tcon_HOL_Obool___Complete__Lattices_OSup_34, axiom, (complete_Sup @ $o)).
% 2.81/1.57 thf(tcon_HOL_Obool___Countable_Ocountable_35, axiom, (countable @ $o)).
% 2.81/1.57 thf(tcon_HOL_Obool___Finite__Set_Ofinite_36, axiom, (finite_finite @ $o)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_37, axiom, (ordere516151231imp_le @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Ostrict__ordered__ab__semigroup__add_38, axiom, (strict2144017051up_add @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_39, axiom, (ordere223160158up_add @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_40, axiom, (ordere236663937imp_le @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_41, axiom, (linord1659791738miring @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_42, axiom, (ordere779506340up_add @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_43, axiom, (linord219039673up_add @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_44, axiom, (cancel146912293up_add @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_45, axiom, (cancel1352612707id_add @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_46, axiom, (ordered_ab_group_add @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_47, axiom, (cancel_semigroup_add @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_48, axiom, (ab_semigroup_add @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_49, axiom, (comm_monoid_add @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Complete__Lattices_OSup_50, axiom, (complete_Sup @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Osemigroup__add_51, axiom, (semigroup_add @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Oab__group__add_52, axiom, (ab_group_add @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Omonoid__add_53, axiom, (monoid_add @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Ogroup__add_54, axiom, (group_add @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Num_Oneg__numeral_55, axiom, (neg_numeral @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Nat_Oring__char__0_56, axiom, (ring_char_0 @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Ozero_57, axiom, (zero @ real)).
% 2.81/1.57 thf(tcon_Real_Oreal___Groups_Oone_58, axiom, (one @ real)).
% 2.81/1.57 thf(tcon_Sigma__Algebra_Omeasure___Complete__Lattices_Ocomplete__lattice_59, axiom, (! [A7:$tType]: ((comple187826305attice @ (sigma_measure @ A7))))).
% 2.81/1.57 thf(tcon_Sigma__Algebra_Omeasure___Complete__Lattices_OSup_60, axiom, (! [A7:$tType]: ((complete_Sup @ (sigma_measure @ A7))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__semigroup__monoid__add__imp__le_61, axiom, (! [A7:$tType,A8:$tType]: ((((ordere890947078_space @ A7) & (finite_finite @ A8)) => (ordere516151231imp_le @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ostrict__ordered__ab__semigroup__add_62, axiom, (! [A7:$tType,A8:$tType]: ((((ordere890947078_space @ A7) & (finite_finite @ A8)) => (strict2144017051up_add @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__cancel__ab__semigroup__add_63, axiom, (! [A7:$tType,A8:$tType]: ((((ordere890947078_space @ A7) & (finite_finite @ A8)) => (ordere223160158up_add @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__semigroup__add__imp__le_64, axiom, (! [A7:$tType,A8:$tType]: ((((ordere890947078_space @ A7) & (finite_finite @ A8)) => (ordere236663937imp_le @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__semigroup__add_65, axiom, (! [A7:$tType,A8:$tType]: ((((ordere890947078_space @ A7) & (finite_finite @ A8)) => (ordere779506340up_add @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocancel__ab__semigroup__add_66, axiom, (! [A7:$tType,A8:$tType]: ((((cancel146912293up_add @ A7) & (finite_finite @ A8)) => (cancel146912293up_add @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocancel__comm__monoid__add_67, axiom, (! [A7:$tType,A8:$tType]: ((((cancel1352612707id_add @ A7) & (finite_finite @ A8)) => (cancel1352612707id_add @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__group__add_68, axiom, (! [A7:$tType,A8:$tType]: ((((ordere890947078_space @ A7) & (finite_finite @ A8)) => (ordered_ab_group_add @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocancel__semigroup__add_69, axiom, (! [A7:$tType,A8:$tType]: ((((cancel_semigroup_add @ A7) & (finite_finite @ A8)) => (cancel_semigroup_add @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oab__semigroup__add_70, axiom, (! [A7:$tType,A8:$tType]: ((((ab_semigroup_add @ A7) & (finite_finite @ A8)) => (ab_semigroup_add @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocomm__monoid__add_71, axiom, (! [A7:$tType,A8:$tType]: ((((comm_monoid_add @ A7) & (finite_finite @ A8)) => (comm_monoid_add @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Complete__Lattices_OSup_72, axiom, (! [A7:$tType,A8:$tType]: ((((ordere890947078_space @ A7) & (finite_finite @ A8)) => (complete_Sup @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Osemigroup__add_73, axiom, (! [A7:$tType,A8:$tType]: ((((semigroup_add @ A7) & (finite_finite @ A8)) => (semigroup_add @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oab__group__add_74, axiom, (! [A7:$tType,A8:$tType]: ((((ab_group_add @ A7) & (finite_finite @ A8)) => (ab_group_add @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Countable_Ocountable_75, axiom, (! [A7:$tType,A8:$tType]: ((((countable @ A7) & (finite_finite @ A8)) => (countable @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Omonoid__add_76, axiom, (! [A7:$tType,A8:$tType]: ((((monoid_add @ A7) & (finite_finite @ A8)) => (monoid_add @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_77, axiom, (! [A7:$tType,A8:$tType]: ((((finite_finite @ A7) & (finite_finite @ A8)) => (finite_finite @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ogroup__add_78, axiom, (! [A7:$tType,A8:$tType]: ((((group_add @ A7) & (finite_finite @ A8)) => (group_add @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Num_Oneg__numeral_79, axiom, (! [A7:$tType,A8:$tType]: ((((ring_1 @ A7) & (finite_finite @ A8)) => (neg_numeral @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Nat_Oring__char__0_80, axiom, (! [A7:$tType,A8:$tType]: ((((ring_char_0 @ A7) & (finite_finite @ A8)) => (ring_char_0 @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_81, axiom, (! [A7:$tType,A8:$tType]: ((((zero @ A7) & (finite_finite @ A8)) => (zero @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oone_82, axiom, (! [A7:$tType,A8:$tType]: ((((one @ A7) & (finite_finite @ A8)) => (one @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.81/1.57 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Ostrict__ordered__ab__semigroup__add_83, axiom, (strict2144017051up_add @ extend1814228343nnreal)).
% 2.81/1.57 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Ocanonically__ordered__monoid__add_84, axiom, (canoni770627133id_add @ extend1814228343nnreal)).
% 2.81/1.57 thf(tcon_Extended__Nonnegative__Real_Oennreal___Complete__Lattices_Ocomplete__linorder, axiom, (comple1035589618norder @ extend1814228343nnreal)).
% 2.81/1.57 thf(tcon_Extended__Nonnegative__Real_Oennreal___Complete__Lattices_Ocomplete__lattice_85, axiom, (comple187826305attice @ extend1814228343nnreal)).
% 2.81/1.57 thf(tcon_Extended__Nonnegative__Real_Oennreal___Rings_Olinordered__nonzero__semiring_86, axiom, (linord1659791738miring @ extend1814228343nnreal)).
% 2.81/1.57 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Oordered__ab__semigroup__add_87, axiom, (ordere779506340up_add @ extend1814228343nnreal)).
% 2.81/1.57 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Oab__semigroup__add_88, axiom, (ab_semigroup_add @ extend1814228343nnreal)).
% 2.81/1.57 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Ocomm__monoid__add_89, axiom, (comm_monoid_add @ extend1814228343nnreal)).
% 2.81/1.57 thf(tcon_Extended__Nonnegative__Real_Oennreal___Complete__Lattices_OSup_90, axiom, (complete_Sup @ extend1814228343nnreal)).
% 2.81/1.57 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Osemigroup__add_91, axiom, (semigroup_add @ extend1814228343nnreal)).
% 2.81/1.57 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Omonoid__add_92, axiom, (monoid_add @ extend1814228343nnreal)).
% 2.81/1.57 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Ozero_93, axiom, (zero @ extend1814228343nnreal)).
% 2.81/1.57 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Oone_94, axiom, (one @ extend1814228343nnreal)).
% 2.81/1.57 thf(tfree_0, hypothesis, (finite_finite @ n)).
% 2.81/1.57 thf(conj_0, conjecture, ((^ [N:nat]: ((((sigma_emeasure @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n)))) @ (t2 @ (f @ N))))) = (^ [N:nat]: ((((sigma_emeasure @ ((finite_Cartesian_vec @ real) @ n)) @ ((complete_completion @ ((finite_Cartesian_vec @ real) @ n)) @ (lebesgue_lborel @ ((finite_Cartesian_vec @ real) @ n)))) @ (t @ (f @ N))))))).
% 2.81/1.57 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.flmKuaoIzs/DTF2DTF30588.p
% 2.81/1.57 ------------------------
% 2.95/1.61 ---- Cleaned THF ---
% 2.95/1.61 thf(ty_t_Extended__Nonnegative__Real_Oennreal,type,
% 2.95/1.61 extend1814228343nnreal: $tType ).
% 2.95/1.61
% 2.95/1.61 thf(ty_t_Finite__Cartesian__Product_Ovec,type,
% 2.95/1.61 finite_Cartesian_vec: $tType > $tType > $tType ).
% 2.95/1.61
% 2.95/1.61 thf(ty_t_Sigma__Algebra_Omeasure,type,
% 2.95/1.61 sigma_measure: $tType > $tType ).
% 2.95/1.61
% 2.95/1.61 thf(ty_t_Real_Oreal,type,
% 2.95/1.61 real: $tType ).
% 2.95/1.61
% 2.95/1.61 thf(ty_t_Set_Oset,type,
% 2.95/1.61 set: $tType > $tType ).
% 2.95/1.61
% 2.95/1.61 thf(ty_t_Nat_Onat,type,
% 2.95/1.61 nat: $tType ).
% 2.95/1.61
% 2.95/1.61 thf(ty_t_Int_Oint,type,
% 2.95/1.61 int: $tType ).
% 2.95/1.61
% 2.95/1.61 thf(ty_tf_n,type,
% 2.95/1.61 n: $tType ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Rings_Oring__1,type,
% 2.95/1.61 ring_1:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
% 2.95/1.61 ordere890947078_space:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_HOL_Otype,type,
% 2.95/1.61 type:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Oone,type,
% 2.95/1.61 one:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Ozero,type,
% 2.95/1.61 zero:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Nat_Oring__char__0,type,
% 2.95/1.61 ring_char_0:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Num_Oneg__numeral,type,
% 2.95/1.61 neg_numeral:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Ogroup__add,type,
% 2.95/1.61 group_add:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.95/1.61 finite_finite:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Omonoid__add,type,
% 2.95/1.61 monoid_add:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Countable_Ocountable,type,
% 2.95/1.61 countable:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Oab__group__add,type,
% 2.95/1.61 ab_group_add:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Osemigroup__add,type,
% 2.95/1.61 semigroup_add:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Complete__Lattices_OSup,type,
% 2.95/1.61 complete_Sup:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 2.95/1.61 comm_monoid_add:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 2.95/1.61 ab_semigroup_add:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 2.95/1.61 comm_monoid_diff:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 2.95/1.61 cancel_semigroup_add:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.95/1.61 ordered_ab_group_add:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.95/1.61 cancel1352612707id_add:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.95/1.61 cancel146912293up_add:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 2.95/1.61 linord219039673up_add:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 2.95/1.61 ordere779506340up_add:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.95/1.61 linord1659791738miring:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 2.95/1.61 comple187826305attice:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
% 2.95/1.61 comple1035589618norder:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.95/1.61 canoni770627133id_add:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 2.95/1.61 ordere236663937imp_le:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 2.95/1.61 ordere223160158up_add:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 2.95/1.61 strict2144017051up_add:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 2.95/1.61 ordere516151231imp_le:
% 2.95/1.61 !>[A: $tType] : $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
% 2.95/1.61 complete_Sup_Sup:
% 2.95/1.61 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Complete__Measure_Ocompletion,type,
% 2.95/1.61 complete_completion:
% 2.95/1.61 !>[A: $tType] : ( ( sigma_measure @ A ) > ( sigma_measure @ A ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Countable__Set_Ofrom__nat__into,type,
% 2.95/1.61 counta609264050t_into:
% 2.95/1.61 !>[A: $tType] : ( ( set @ A ) > nat > A ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.95/1.61 minus_minus:
% 2.95/1.61 !>[A: $tType] : ( A > A > A ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Groups_Oone__class_Oone,type,
% 2.95/1.61 one_one:
% 2.95/1.61 !>[A: $tType] : A ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.95/1.61 plus_plus:
% 2.95/1.61 !>[A: $tType] : ( A > A > A ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.95/1.61 zero_zero:
% 2.95/1.61 !>[A: $tType] : A ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 2.95/1.61 inf_inf:
% 2.95/1.61 !>[A: $tType] : ( A > A > A ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Lebesgue__Measure_Olborel,type,
% 2.95/1.61 lebesgue_lborel:
% 2.95/1.61 !>[A: $tType] : ( sigma_measure @ A ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Measure__Space_Onull__sets,type,
% 2.95/1.61 measure_null_sets:
% 2.95/1.61 !>[A: $tType] : ( ( sigma_measure @ A ) > ( set @ ( set @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Minkowskis__Theorem__Mirabelle__jpyvdnlcjd_Oof__int__vec,type,
% 2.95/1.61 minkow1937162798nt_vec:
% 2.95/1.61 !>[B: $tType,A: $tType] : ( ( finite_Cartesian_vec @ int @ B ) > ( finite_Cartesian_vec @ A @ B ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.95/1.61 ord_less:
% 2.95/1.61 !>[A: $tType] : ( A > A > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.95/1.61 top_top:
% 2.95/1.61 !>[A: $tType] : A ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Series_Osums,type,
% 2.95/1.61 sums:
% 2.95/1.61 !>[A: $tType] : ( ( nat > A ) > A > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Set_OCollect,type,
% 2.95/1.61 collect:
% 2.95/1.61 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Set_Oimage,type,
% 2.95/1.61 image:
% 2.95/1.61 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Set_Ovimage,type,
% 2.95/1.61 vimage:
% 2.95/1.61 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Sigma__Algebra_Oemeasure,type,
% 2.95/1.61 sigma_emeasure:
% 2.95/1.61 !>[A: $tType] : ( ( sigma_measure @ A ) > ( set @ A ) > extend1814228343nnreal ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Sigma__Algebra_Osets,type,
% 2.95/1.61 sigma_sets:
% 2.95/1.61 !>[A: $tType] : ( ( sigma_measure @ A ) > ( set @ ( set @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Sigma__Algebra_Ospace,type,
% 2.95/1.61 sigma_space:
% 2.95/1.61 !>[A: $tType] : ( ( sigma_measure @ A ) > ( set @ A ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_member,type,
% 2.95/1.61 member:
% 2.95/1.61 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_v_R____,type,
% 2.95/1.61 r: ( finite_Cartesian_vec @ int @ n ) > ( set @ ( finite_Cartesian_vec @ real @ n ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_v_S,type,
% 2.95/1.61 s: set @ ( finite_Cartesian_vec @ real @ n ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_v_T_H____,type,
% 2.95/1.61 t: ( finite_Cartesian_vec @ int @ n ) > ( set @ ( finite_Cartesian_vec @ real @ n ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_v_T____,type,
% 2.95/1.61 t2: ( finite_Cartesian_vec @ int @ n ) > ( set @ ( finite_Cartesian_vec @ real @ n ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_v_f____,type,
% 2.95/1.61 f: nat > ( finite_Cartesian_vec @ int @ n ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_0_emeasure__T_H,axiom,
% 2.95/1.61 ! [A2: finite_Cartesian_vec @ int @ n] :
% 2.95/1.61 ( ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t @ A2 ) )
% 2.95/1.61 = ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t2 @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_1__092_060open_062_092_060And_062a_O_AT_H_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
% 2.95/1.61 ! [A2: finite_Cartesian_vec @ int @ n] : ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( t @ A2 ) @ ( sigma_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_2__092_060open_062_092_060And_062a_O_AT_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
% 2.95/1.61 ! [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 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_3_calculation,axiom,
% 2.95/1.61 ( sums @ extend1814228343nnreal
% 2.95/1.61 @ ^ [N: nat] : ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t2 @ ( f @ N ) ) )
% 2.95/1.61 @ ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ s ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_4_assms_I1_J,axiom,
% 2.95/1.61 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 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_5_emeasure__T__Int,axiom,
% 2.95/1.61 ! [A2: finite_Cartesian_vec @ int @ n,B2: finite_Cartesian_vec @ int @ n] :
% 2.95/1.61 ( ( A2 != B2 )
% 2.95/1.61 => ( ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( inf_inf @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( t2 @ A2 ) @ ( t2 @ B2 ) ) )
% 2.95/1.61 = ( zero_zero @ extend1814228343nnreal ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_6_T__Int,axiom,
% 2.95/1.61 ! [A2: finite_Cartesian_vec @ int @ n,B2: finite_Cartesian_vec @ int @ n] :
% 2.95/1.61 ( ( A2 != B2 )
% 2.95/1.61 => ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( inf_inf @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( t2 @ A2 ) @ ( t2 @ B2 ) ) @ ( measure_null_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_7_assms_I2_J,axiom,
% 2.95/1.61 ord_less @ extend1814228343nnreal @ ( one_one @ extend1814228343nnreal ) @ ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ s ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_8__092_060open_062_092_060And_062a_O_AR_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
% 2.95/1.61 ! [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 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_9__092_060open_062_I_092_060lambda_062n_O_Aemeasure_Alebesgue_A_IT_A_If_An_J_J_J_Asums_Aemeasure_Alebesgue_A_I_092_060Union_062n_O_AT_A_If_An_J_J_092_060close_062,axiom,
% 2.95/1.61 ( sums @ extend1814228343nnreal
% 2.95/1.61 @ ^ [N: nat] : ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t2 @ ( f @ N ) ) )
% 2.95/1.61 @ ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) )
% 2.95/1.61 @ ( complete_Sup_Sup @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
% 2.95/1.61 @ ( image @ nat @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
% 2.95/1.61 @ ^ [N: nat] : ( t2 @ ( f @ N ) )
% 2.95/1.61 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_10_T_H__def,axiom,
% 2.95/1.61 ( t
% 2.95/1.61 = ( ^ [A3: finite_Cartesian_vec @ int @ n] :
% 2.95/1.61 ( image @ ( finite_Cartesian_vec @ real @ n ) @ ( finite_Cartesian_vec @ real @ n )
% 2.95/1.61 @ ^ [X: finite_Cartesian_vec @ real @ n] : ( minus_minus @ ( finite_Cartesian_vec @ real @ n ) @ X @ ( minkow1937162798nt_vec @ n @ real @ A3 ) )
% 2.95/1.61 @ ( t2 @ A3 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_11_f__def,axiom,
% 2.95/1.61 ( f
% 2.95/1.61 = ( counta609264050t_into @ ( finite_Cartesian_vec @ int @ n ) @ ( top_top @ ( set @ ( finite_Cartesian_vec @ int @ n ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_12_T_H__altdef,axiom,
% 2.95/1.61 ! [A2: finite_Cartesian_vec @ int @ n] :
% 2.95/1.61 ( ( t @ A2 )
% 2.95/1.61 = ( vimage @ ( finite_Cartesian_vec @ real @ n ) @ ( finite_Cartesian_vec @ real @ n )
% 2.95/1.61 @ ^ [X: finite_Cartesian_vec @ real @ n] : ( plus_plus @ ( finite_Cartesian_vec @ real @ n ) @ X @ ( minkow1937162798nt_vec @ n @ real @ A2 ) )
% 2.95/1.61 @ ( t2 @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_13_of__int__vec__eq__iff,axiom,
% 2.95/1.61 ! [A: $tType,N2: $tType] :
% 2.95/1.61 ( ( ( finite_finite @ N2 )
% 2.95/1.61 & ( ring_char_0 @ A ) )
% 2.95/1.61 => ! [A2: finite_Cartesian_vec @ int @ N2,B2: finite_Cartesian_vec @ int @ N2] :
% 2.95/1.61 ( ( ( minkow1937162798nt_vec @ N2 @ A @ A2 )
% 2.95/1.61 = ( minkow1937162798nt_vec @ N2 @ A @ B2 ) )
% 2.95/1.61 = ( A2 = B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_14_T__def,axiom,
% 2.95/1.61 ( t2
% 2.95/1.61 = ( ^ [A3: finite_Cartesian_vec @ int @ n] : ( inf_inf @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ s @ ( r @ A3 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_15_R__Int,axiom,
% 2.95/1.61 ! [A2: finite_Cartesian_vec @ int @ n,B2: finite_Cartesian_vec @ int @ n] :
% 2.95/1.61 ( ( A2 != B2 )
% 2.95/1.61 => ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( inf_inf @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( r @ A2 ) @ ( r @ B2 ) ) @ ( measure_null_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_16_sums__emeasure_H,axiom,
% 2.95/1.61 ! [A: $tType,B3: nat > ( set @ A ),M: sigma_measure @ A] :
% 2.95/1.61 ( ! [X2: nat] : ( member @ ( set @ A ) @ ( B3 @ X2 ) @ ( sigma_sets @ A @ M ) )
% 2.95/1.61 => ( ! [X2: nat,Y: nat] :
% 2.95/1.61 ( ( X2 != Y )
% 2.95/1.61 => ( ( sigma_emeasure @ A @ M @ ( inf_inf @ ( set @ A ) @ ( B3 @ X2 ) @ ( B3 @ Y ) ) )
% 2.95/1.61 = ( zero_zero @ extend1814228343nnreal ) ) )
% 2.95/1.61 => ( sums @ extend1814228343nnreal
% 2.95/1.61 @ ^ [X: nat] : ( sigma_emeasure @ A @ M @ ( B3 @ X ) )
% 2.95/1.61 @ ( sigma_emeasure @ A @ M @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_17_null__sets__UN,axiom,
% 2.95/1.61 ! [I: $tType,A: $tType] :
% 2.95/1.61 ( ( countable @ I )
% 2.95/1.61 => ! [N3: I > ( set @ A ),M: sigma_measure @ A] :
% 2.95/1.61 ( ! [I2: I] : ( member @ ( set @ A ) @ ( N3 @ I2 ) @ ( measure_null_sets @ A @ M ) )
% 2.95/1.61 => ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ I @ ( set @ A ) @ N3 @ ( top_top @ ( set @ I ) ) ) ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_18_SUP__eq__top__iff,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( comple1035589618norder @ A )
% 2.95/1.61 => ! [F: B > A,A4: set @ B] :
% 2.95/1.61 ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A4 ) )
% 2.95/1.61 = ( top_top @ A ) )
% 2.95/1.61 = ( ! [X: A] :
% 2.95/1.61 ( ( ord_less @ A @ X @ ( top_top @ A ) )
% 2.95/1.61 => ? [Y2: B] :
% 2.95/1.61 ( ( member @ B @ Y2 @ A4 )
% 2.95/1.61 & ( ord_less @ A @ X @ ( F @ Y2 ) ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_19_null__setsI,axiom,
% 2.95/1.61 ! [A: $tType,M: sigma_measure @ A,A4: set @ A] :
% 2.95/1.61 ( ( ( sigma_emeasure @ A @ M @ A4 )
% 2.95/1.61 = ( zero_zero @ extend1814228343nnreal ) )
% 2.95/1.61 => ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) )
% 2.95/1.61 => ( member @ ( set @ A ) @ A4 @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_20_image__vimage__eq,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,F: B > A,A4: set @ A] :
% 2.95/1.61 ( ( image @ B @ A @ F @ ( vimage @ B @ A @ F @ A4 ) )
% 2.95/1.61 = ( inf_inf @ ( set @ A ) @ A4 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_21__092_060open_062_092_060And_062a_O_AT_Aa_A_092_060in_062_Asets_Alebesgue_A_092_060Longrightarrow_062_A_I_092_060lambda_062x_O_Ax_A_L_Aof__int__vec_Aa_J_A_N_096_AT_Aa_A_092_060inter_062_Aspace_Alebesgue_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
% 2.95/1.61 ! [A5: finite_Cartesian_vec @ int @ n] :
% 2.95/1.61 ( ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( t2 @ A5 ) @ ( sigma_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) )
% 2.95/1.61 => ( member @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
% 2.95/1.61 @ ( inf_inf @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
% 2.95/1.61 @ ( vimage @ ( finite_Cartesian_vec @ real @ n ) @ ( finite_Cartesian_vec @ real @ n )
% 2.95/1.61 @ ^ [X: finite_Cartesian_vec @ real @ n] : ( plus_plus @ ( finite_Cartesian_vec @ real @ n ) @ X @ ( minkow1937162798nt_vec @ n @ real @ A5 ) )
% 2.95/1.61 @ ( t2 @ A5 ) )
% 2.95/1.61 @ ( sigma_space @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) )
% 2.95/1.61 @ ( sigma_sets @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_22_surj__diff__right,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ab_group_add @ A )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( image @ A @ A
% 2.95/1.61 @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
% 2.95/1.61 @ ( top_top @ ( set @ A ) ) )
% 2.95/1.61 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_23__092_060open_062_092_060Union_062_A_Irange_AT_J_A_061_A_I_092_060Union_062n_O_AT_A_If_An_J_J_092_060close_062,axiom,
% 2.95/1.61 ( ( complete_Sup_Sup @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( image @ ( finite_Cartesian_vec @ int @ n ) @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ t2 @ ( top_top @ ( set @ ( finite_Cartesian_vec @ int @ n ) ) ) ) )
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
% 2.95/1.61 @ ( image @ nat @ ( set @ ( finite_Cartesian_vec @ real @ n ) )
% 2.95/1.61 @ ^ [N: nat] : ( t2 @ ( f @ N ) )
% 2.95/1.61 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_24_Sup__UNIV,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( comple187826305attice @ A )
% 2.95/1.61 => ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
% 2.95/1.61 = ( top_top @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_25_Sup__eq__top__iff,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( comple1035589618norder @ A )
% 2.95/1.61 => ! [A4: set @ A] :
% 2.95/1.61 ( ( ( complete_Sup_Sup @ A @ A4 )
% 2.95/1.61 = ( top_top @ A ) )
% 2.95/1.61 = ( ! [X: A] :
% 2.95/1.61 ( ( ord_less @ A @ X @ ( top_top @ A ) )
% 2.95/1.61 => ? [Y2: A] :
% 2.95/1.61 ( ( member @ A @ Y2 @ A4 )
% 2.95/1.61 & ( ord_less @ A @ X @ Y2 ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_26_surj__plus,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ab_group_add @ A )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
% 2.95/1.61 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_27_diff__numeral__special_I9_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( neg_numeral @ A )
% 2.95/1.61 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 2.95/1.61 = ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_28_image__eqI,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,B2: A,F: B > A,X3: B,A4: set @ B] :
% 2.95/1.61 ( ( B2
% 2.95/1.61 = ( F @ X3 ) )
% 2.95/1.61 => ( ( member @ B @ X3 @ A4 )
% 2.95/1.61 => ( member @ A @ B2 @ ( image @ B @ A @ F @ A4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_29_UNIV__I,axiom,
% 2.95/1.61 ! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_30_Int__iff,axiom,
% 2.95/1.61 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 2.95/1.61 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
% 2.95/1.61 = ( ( member @ A @ C @ A4 )
% 2.95/1.61 & ( member @ A @ C @ B3 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_31_IntI,axiom,
% 2.95/1.61 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 2.95/1.61 ( ( member @ A @ C @ A4 )
% 2.95/1.61 => ( ( member @ A @ C @ B3 )
% 2.95/1.61 => ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_32_Union__iff,axiom,
% 2.95/1.61 ! [A: $tType,A4: A,C2: set @ ( set @ A )] :
% 2.95/1.61 ( ( member @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) )
% 2.95/1.61 = ( ? [X: set @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ X @ C2 )
% 2.95/1.61 & ( member @ A @ A4 @ X ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_33_UnionI,axiom,
% 2.95/1.61 ! [A: $tType,X4: set @ A,C2: set @ ( set @ A ),A4: A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ X4 @ C2 )
% 2.95/1.61 => ( ( member @ A @ A4 @ X4 )
% 2.95/1.61 => ( member @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_34_UN__ball__bex__simps_I1_J,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ ( set @ A ),P: A > $o] :
% 2.95/1.61 ( ( ! [X: A] :
% 2.95/1.61 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) )
% 2.95/1.61 => ( P @ X ) ) )
% 2.95/1.61 = ( ! [X: set @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ X @ A4 )
% 2.95/1.61 => ! [Y2: A] :
% 2.95/1.61 ( ( member @ A @ Y2 @ X )
% 2.95/1.61 => ( P @ Y2 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_35_UN__ball__bex__simps_I3_J,axiom,
% 2.95/1.61 ! [D: $tType,A4: set @ ( set @ D ),P: D > $o] :
% 2.95/1.61 ( ( ? [X: D] :
% 2.95/1.61 ( ( member @ D @ X @ ( complete_Sup_Sup @ ( set @ D ) @ A4 ) )
% 2.95/1.61 & ( P @ X ) ) )
% 2.95/1.61 = ( ? [X: set @ D] :
% 2.95/1.61 ( ( member @ ( set @ D ) @ X @ A4 )
% 2.95/1.61 & ? [Y2: D] :
% 2.95/1.61 ( ( member @ D @ Y2 @ X )
% 2.95/1.61 & ( P @ Y2 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_36_null__sets_ODiff,axiom,
% 2.95/1.61 ! [A: $tType,A2: set @ A,M: sigma_measure @ A,B2: set @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ A2 @ ( measure_null_sets @ A @ M ) )
% 2.95/1.61 => ( ( member @ ( set @ A ) @ B2 @ ( measure_null_sets @ A @ M ) )
% 2.95/1.61 => ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_37_vimage__eq,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,A2: A,F: A > B,B3: set @ B] :
% 2.95/1.61 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ B3 ) )
% 2.95/1.61 = ( member @ B @ ( F @ A2 ) @ B3 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_38_vimageI,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType,F: B > A,A2: B,B2: A,B3: set @ A] :
% 2.95/1.61 ( ( ( F @ A2 )
% 2.95/1.61 = B2 )
% 2.95/1.61 => ( ( member @ A @ B2 @ B3 )
% 2.95/1.61 => ( member @ B @ A2 @ ( vimage @ B @ A @ F @ B3 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_39_image__ident,axiom,
% 2.95/1.61 ! [A: $tType,Y3: set @ A] :
% 2.95/1.61 ( ( image @ A @ A
% 2.95/1.61 @ ^ [X: A] : X
% 2.95/1.61 @ Y3 )
% 2.95/1.61 = Y3 ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_40_Sup__apply,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( complete_Sup @ B )
% 2.95/1.61 => ( ( complete_Sup_Sup @ ( A > B ) )
% 2.95/1.61 = ( ^ [A6: set @ ( A > B ),X: A] :
% 2.95/1.61 ( complete_Sup_Sup @ B
% 2.95/1.61 @ ( image @ ( A > B ) @ B
% 2.95/1.61 @ ^ [F2: A > B] : ( F2 @ X )
% 2.95/1.61 @ A6 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_41_vimage__Collect__eq,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType,F: A > B,P: B > $o] :
% 2.95/1.61 ( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
% 2.95/1.61 = ( collect @ A
% 2.95/1.61 @ ^ [Y2: A] : ( P @ ( F @ Y2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_42_vimage__ident,axiom,
% 2.95/1.61 ! [A: $tType,Y3: set @ A] :
% 2.95/1.61 ( ( vimage @ A @ A
% 2.95/1.61 @ ^ [X: A] : X
% 2.95/1.61 @ Y3 )
% 2.95/1.61 = Y3 ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_43_Int__UNIV,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.95/1.61 ( ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
% 2.95/1.61 = ( top_top @ ( set @ A ) ) )
% 2.95/1.61 = ( ( A4
% 2.95/1.61 = ( top_top @ ( set @ A ) ) )
% 2.95/1.61 & ( B3
% 2.95/1.61 = ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_44_ball__UN,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,B3: B > ( set @ A ),A4: set @ B,P: A > $o] :
% 2.95/1.61 ( ( ! [X: A] :
% 2.95/1.61 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A4 ) ) )
% 2.95/1.61 => ( P @ X ) ) )
% 2.95/1.61 = ( ! [X: B] :
% 2.95/1.61 ( ( member @ B @ X @ A4 )
% 2.95/1.61 => ! [Y2: A] :
% 2.95/1.61 ( ( member @ A @ Y2 @ ( B3 @ X ) )
% 2.95/1.61 => ( P @ Y2 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_45_mem__Collect__eq,axiom,
% 2.95/1.61 ! [A: $tType,A2: A,P: A > $o] :
% 2.95/1.61 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.95/1.61 = ( P @ A2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_46_Collect__mem__eq,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A] :
% 2.95/1.61 ( ( collect @ A
% 2.95/1.61 @ ^ [X: A] : ( member @ A @ X @ A4 ) )
% 2.95/1.61 = A4 ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_47_Collect__cong,axiom,
% 2.95/1.61 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.95/1.61 ( ! [X2: A] :
% 2.95/1.61 ( ( P @ X2 )
% 2.95/1.61 = ( Q @ X2 ) )
% 2.95/1.61 => ( ( collect @ A @ P )
% 2.95/1.61 = ( collect @ A @ Q ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_48_ext,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.95/1.61 ( ! [X2: A] :
% 2.95/1.61 ( ( F @ X2 )
% 2.95/1.61 = ( G @ X2 ) )
% 2.95/1.61 => ( F = G ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_49_bex__UN,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,B3: B > ( set @ A ),A4: set @ B,P: A > $o] :
% 2.95/1.61 ( ( ? [X: A] :
% 2.95/1.61 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A4 ) ) )
% 2.95/1.61 & ( P @ X ) ) )
% 2.95/1.61 = ( ? [X: B] :
% 2.95/1.61 ( ( member @ B @ X @ A4 )
% 2.95/1.61 & ? [Y2: A] :
% 2.95/1.61 ( ( member @ A @ Y2 @ ( B3 @ X ) )
% 2.95/1.61 & ( P @ Y2 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_50_UN__ball__bex__simps_I2_J,axiom,
% 2.95/1.61 ! [C3: $tType,B: $tType,B3: B > ( set @ C3 ),A4: set @ B,P: C3 > $o] :
% 2.95/1.61 ( ( ! [X: C3] :
% 2.95/1.61 ( ( member @ C3 @ X @ ( complete_Sup_Sup @ ( set @ C3 ) @ ( image @ B @ ( set @ C3 ) @ B3 @ A4 ) ) )
% 2.95/1.61 => ( P @ X ) ) )
% 2.95/1.61 = ( ! [X: B] :
% 2.95/1.61 ( ( member @ B @ X @ A4 )
% 2.95/1.61 => ! [Y2: C3] :
% 2.95/1.61 ( ( member @ C3 @ Y2 @ ( B3 @ X ) )
% 2.95/1.61 => ( P @ Y2 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_51_UN__ball__bex__simps_I4_J,axiom,
% 2.95/1.61 ! [F3: $tType,E: $tType,B3: E > ( set @ F3 ),A4: set @ E,P: F3 > $o] :
% 2.95/1.61 ( ( ? [X: F3] :
% 2.95/1.61 ( ( member @ F3 @ X @ ( complete_Sup_Sup @ ( set @ F3 ) @ ( image @ E @ ( set @ F3 ) @ B3 @ A4 ) ) )
% 2.95/1.61 & ( P @ X ) ) )
% 2.95/1.61 = ( ? [X: E] :
% 2.95/1.61 ( ( member @ E @ X @ A4 )
% 2.95/1.61 & ? [Y2: F3] :
% 2.95/1.61 ( ( member @ F3 @ Y2 @ ( B3 @ X ) )
% 2.95/1.61 & ( P @ Y2 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_52_vimage__UNIV,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType,F: A > B] :
% 2.95/1.61 ( ( vimage @ A @ B @ F @ ( top_top @ ( set @ B ) ) )
% 2.95/1.61 = ( top_top @ ( set @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_53_null__sets_OInt,axiom,
% 2.95/1.61 ! [A: $tType,A2: set @ A,M: sigma_measure @ A,B2: set @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ A2 @ ( measure_null_sets @ A @ M ) )
% 2.95/1.61 => ( ( member @ ( set @ A ) @ B2 @ ( measure_null_sets @ A @ M ) )
% 2.95/1.61 => ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_54_vimage__Int,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,F: A > B,A4: set @ B,B3: set @ B] :
% 2.95/1.61 ( ( vimage @ A @ B @ F @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) )
% 2.95/1.61 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ A4 ) @ ( vimage @ A @ B @ F @ B3 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_55_SUP__identity__eq,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( complete_Sup @ A )
% 2.95/1.61 => ! [A4: set @ A] :
% 2.95/1.61 ( ( complete_Sup_Sup @ A
% 2.95/1.61 @ ( image @ A @ A
% 2.95/1.61 @ ^ [X: A] : X
% 2.95/1.61 @ A4 ) )
% 2.95/1.61 = ( complete_Sup_Sup @ A @ A4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_56_SUP__apply,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,C3: $tType] :
% 2.95/1.61 ( ( complete_Sup @ A )
% 2.95/1.61 => ! [F: C3 > B > A,A4: set @ C3,X3: B] :
% 2.95/1.61 ( ( complete_Sup_Sup @ ( B > A ) @ ( image @ C3 @ ( B > A ) @ F @ A4 ) @ X3 )
% 2.95/1.61 = ( complete_Sup_Sup @ A
% 2.95/1.61 @ ( image @ C3 @ A
% 2.95/1.61 @ ^ [Y2: C3] : ( F @ Y2 @ X3 )
% 2.95/1.61 @ A4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_57_UN__iff,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,B2: A,B3: B > ( set @ A ),A4: set @ B] :
% 2.95/1.61 ( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A4 ) ) )
% 2.95/1.61 = ( ? [X: B] :
% 2.95/1.61 ( ( member @ B @ X @ A4 )
% 2.95/1.61 & ( member @ A @ B2 @ ( B3 @ X ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_58_UN__I,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType,A2: A,A4: set @ A,B2: B,B3: A > ( set @ B )] :
% 2.95/1.61 ( ( member @ A @ A2 @ A4 )
% 2.95/1.61 => ( ( member @ B @ B2 @ ( B3 @ A2 ) )
% 2.95/1.61 => ( member @ B @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A4 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_59_S__decompose,axiom,
% 2.95/1.61 ( s
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ ( image @ ( finite_Cartesian_vec @ int @ n ) @ ( set @ ( finite_Cartesian_vec @ real @ n ) ) @ t2 @ ( top_top @ ( set @ ( finite_Cartesian_vec @ int @ n ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_60_image__add__0,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( comm_monoid_add @ A )
% 2.95/1.61 => ! [S: set @ A] :
% 2.95/1.61 ( ( image @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S )
% 2.95/1.61 = S ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_61_null__sets_OInt__space__eq2,axiom,
% 2.95/1.61 ! [A: $tType,X3: set @ A,M: sigma_measure @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ X3 @ ( measure_null_sets @ A @ M ) )
% 2.95/1.61 => ( ( inf_inf @ ( set @ A ) @ X3 @ ( sigma_space @ A @ M ) )
% 2.95/1.61 = X3 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_62_null__sets_OInt__space__eq1,axiom,
% 2.95/1.61 ! [A: $tType,X3: set @ A,M: sigma_measure @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ X3 @ ( measure_null_sets @ A @ M ) )
% 2.95/1.61 => ( ( inf_inf @ ( set @ A ) @ ( sigma_space @ A @ M ) @ X3 )
% 2.95/1.61 = X3 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_63_in__sets__SUP,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType,I3: A,I4: set @ A,M: A > ( sigma_measure @ B ),Y3: set @ B,X4: set @ B] :
% 2.95/1.61 ( ( member @ A @ I3 @ I4 )
% 2.95/1.61 => ( ! [I2: A] :
% 2.95/1.61 ( ( member @ A @ I2 @ I4 )
% 2.95/1.61 => ( ( sigma_space @ B @ ( M @ I2 ) )
% 2.95/1.61 = Y3 ) )
% 2.95/1.61 => ( ( member @ ( set @ B ) @ X4 @ ( sigma_sets @ B @ ( M @ I3 ) ) )
% 2.95/1.61 => ( member @ ( set @ B ) @ X4 @ ( sigma_sets @ B @ ( complete_Sup_Sup @ ( sigma_measure @ B ) @ ( image @ A @ ( sigma_measure @ B ) @ M @ I4 ) ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_64_in__sets__Sup,axiom,
% 2.95/1.61 ! [A: $tType,M: set @ ( sigma_measure @ A ),X4: set @ A,M2: sigma_measure @ A,A4: set @ A] :
% 2.95/1.61 ( ! [M3: sigma_measure @ A] :
% 2.95/1.61 ( ( member @ ( sigma_measure @ A ) @ M3 @ M )
% 2.95/1.61 => ( ( sigma_space @ A @ M3 )
% 2.95/1.61 = X4 ) )
% 2.95/1.61 => ( ( member @ ( sigma_measure @ A ) @ M2 @ M )
% 2.95/1.61 => ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M2 ) )
% 2.95/1.61 => ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ ( complete_Sup_Sup @ ( sigma_measure @ A ) @ M ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_65_sets__SUP__cong,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType,I4: set @ A,M: A > ( sigma_measure @ B ),N3: A > ( sigma_measure @ B )] :
% 2.95/1.61 ( ! [I2: A] :
% 2.95/1.61 ( ( member @ A @ I2 @ I4 )
% 2.95/1.61 => ( ( sigma_sets @ B @ ( M @ I2 ) )
% 2.95/1.61 = ( sigma_sets @ B @ ( N3 @ I2 ) ) ) )
% 2.95/1.61 => ( ( sigma_sets @ B @ ( complete_Sup_Sup @ ( sigma_measure @ B ) @ ( image @ A @ ( sigma_measure @ B ) @ M @ I4 ) ) )
% 2.95/1.61 = ( sigma_sets @ B @ ( complete_Sup_Sup @ ( sigma_measure @ B ) @ ( image @ A @ ( sigma_measure @ B ) @ N3 @ I4 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_66_Sup__set__def,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( complete_Sup_Sup @ ( set @ A ) )
% 2.95/1.61 = ( ^ [A6: set @ ( set @ A )] :
% 2.95/1.61 ( collect @ A
% 2.95/1.61 @ ^ [X: A] : ( complete_Sup_Sup @ $o @ ( image @ ( set @ A ) @ $o @ ( member @ A @ X ) @ A6 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_67_space__Sup__eq__UN,axiom,
% 2.95/1.61 ! [A: $tType,M: set @ ( sigma_measure @ A )] :
% 2.95/1.61 ( ( sigma_space @ A @ ( complete_Sup_Sup @ ( sigma_measure @ A ) @ M ) )
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( sigma_measure @ A ) @ ( set @ A ) @ ( sigma_space @ A ) @ M ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_68_top__set__def,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( top_top @ ( set @ A ) )
% 2.95/1.61 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_69_Diff__Int__distrib2,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A,B3: set @ A,C2: set @ A] :
% 2.95/1.61 ( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ C2 )
% 2.95/1.61 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_70_Diff__Int__distrib,axiom,
% 2.95/1.61 ! [A: $tType,C2: set @ A,A4: set @ A,B3: set @ A] :
% 2.95/1.61 ( ( inf_inf @ ( set @ A ) @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 2.95/1.61 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C2 @ A4 ) @ ( inf_inf @ ( set @ A ) @ C2 @ B3 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_71_Diff__Diff__Int,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.95/1.61 ( ( minus_minus @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 2.95/1.61 = ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_72_Diff__Int2,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A,C2: set @ A,B3: set @ A] :
% 2.95/1.61 ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) )
% 2.95/1.61 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) @ B3 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_73_Int__Diff,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A,B3: set @ A,C2: set @ A] :
% 2.95/1.61 ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ C2 )
% 2.95/1.61 = ( inf_inf @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B3 @ C2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_74_vimage__Diff,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,F: A > B,A4: set @ B,B3: set @ B] :
% 2.95/1.61 ( ( vimage @ A @ B @ F @ ( minus_minus @ ( set @ B ) @ A4 @ B3 ) )
% 2.95/1.61 = ( minus_minus @ ( set @ A ) @ ( vimage @ A @ B @ F @ A4 ) @ ( vimage @ A @ B @ F @ B3 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_75_Sup__fun__def,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( complete_Sup @ B )
% 2.95/1.61 => ( ( complete_Sup_Sup @ ( A > B ) )
% 2.95/1.61 = ( ^ [A6: set @ ( A > B ),X: A] :
% 2.95/1.61 ( complete_Sup_Sup @ B
% 2.95/1.61 @ ( image @ ( A > B ) @ B
% 2.95/1.61 @ ^ [F2: A > B] : ( F2 @ X )
% 2.95/1.61 @ A6 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_76_null__sets_Osets__Collect__disj,axiom,
% 2.95/1.61 ! [A: $tType,M: sigma_measure @ A,P: A > $o,Q: A > $o] :
% 2.95/1.61 ( ( member @ ( set @ A )
% 2.95/1.61 @ ( collect @ A
% 2.95/1.61 @ ^ [X: A] :
% 2.95/1.61 ( ( member @ A @ X @ ( sigma_space @ A @ M ) )
% 2.95/1.61 & ( P @ X ) ) )
% 2.95/1.61 @ ( measure_null_sets @ A @ M ) )
% 2.95/1.61 => ( ( member @ ( set @ A )
% 2.95/1.61 @ ( collect @ A
% 2.95/1.61 @ ^ [X: A] :
% 2.95/1.61 ( ( member @ A @ X @ ( sigma_space @ A @ M ) )
% 2.95/1.61 & ( Q @ X ) ) )
% 2.95/1.61 @ ( measure_null_sets @ A @ M ) )
% 2.95/1.61 => ( member @ ( set @ A )
% 2.95/1.61 @ ( collect @ A
% 2.95/1.61 @ ^ [X: A] :
% 2.95/1.61 ( ( member @ A @ X @ ( sigma_space @ A @ M ) )
% 2.95/1.61 & ( ( Q @ X )
% 2.95/1.61 | ( P @ X ) ) ) )
% 2.95/1.61 @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_77_null__sets_Osets__Collect__conj,axiom,
% 2.95/1.61 ! [A: $tType,M: sigma_measure @ A,P: A > $o,Q: A > $o] :
% 2.95/1.61 ( ( member @ ( set @ A )
% 2.95/1.61 @ ( collect @ A
% 2.95/1.61 @ ^ [X: A] :
% 2.95/1.61 ( ( member @ A @ X @ ( sigma_space @ A @ M ) )
% 2.95/1.61 & ( P @ X ) ) )
% 2.95/1.61 @ ( measure_null_sets @ A @ M ) )
% 2.95/1.61 => ( ( member @ ( set @ A )
% 2.95/1.61 @ ( collect @ A
% 2.95/1.61 @ ^ [X: A] :
% 2.95/1.61 ( ( member @ A @ X @ ( sigma_space @ A @ M ) )
% 2.95/1.61 & ( Q @ X ) ) )
% 2.95/1.61 @ ( measure_null_sets @ A @ M ) )
% 2.95/1.61 => ( member @ ( set @ A )
% 2.95/1.61 @ ( collect @ A
% 2.95/1.61 @ ^ [X: A] :
% 2.95/1.61 ( ( member @ A @ X @ ( sigma_space @ A @ M ) )
% 2.95/1.61 & ( Q @ X )
% 2.95/1.61 & ( P @ X ) ) )
% 2.95/1.61 @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_78_translation__diff,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ab_group_add @ A )
% 2.95/1.61 => ! [A2: A,S2: set @ A,T: set @ A] :
% 2.95/1.61 ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( minus_minus @ ( set @ A ) @ S2 @ T ) )
% 2.95/1.61 = ( minus_minus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S2 ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_79_null__set__Diff,axiom,
% 2.95/1.61 ! [A: $tType,B3: set @ A,M: sigma_measure @ A,A4: set @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ B3 @ ( measure_null_sets @ A @ M ) )
% 2.95/1.61 => ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) )
% 2.95/1.61 => ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_80_Int__Union2,axiom,
% 2.95/1.61 ! [A: $tType,B3: set @ ( set @ A ),A4: set @ A] :
% 2.95/1.61 ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) @ A4 )
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ A )
% 2.95/1.61 @ ( image @ ( set @ A ) @ ( set @ A )
% 2.95/1.61 @ ^ [C4: set @ A] : ( inf_inf @ ( set @ A ) @ C4 @ A4 )
% 2.95/1.61 @ B3 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_81_Int__Union,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A,B3: set @ ( set @ A )] :
% 2.95/1.61 ( ( inf_inf @ ( set @ A ) @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) )
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 ) @ B3 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_82_vimage__Union,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,F: A > B,A4: set @ ( set @ B )] :
% 2.95/1.61 ( ( vimage @ A @ B @ F @ ( complete_Sup_Sup @ ( set @ B ) @ A4 ) )
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( vimage @ A @ B @ F ) @ A4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_83_UN__extend__simps_I6_J,axiom,
% 2.95/1.61 ! [L: $tType,K: $tType,A4: K > ( set @ L ),C2: set @ K,B3: set @ L] :
% 2.95/1.61 ( ( minus_minus @ ( set @ L ) @ ( complete_Sup_Sup @ ( set @ L ) @ ( image @ K @ ( set @ L ) @ A4 @ C2 ) ) @ B3 )
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ L )
% 2.95/1.61 @ ( image @ K @ ( set @ L )
% 2.95/1.61 @ ^ [X: K] : ( minus_minus @ ( set @ L ) @ ( A4 @ X ) @ B3 )
% 2.95/1.61 @ C2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_84_emeasure__Diff__null__set,axiom,
% 2.95/1.61 ! [A: $tType,B3: set @ A,M: sigma_measure @ A,A4: set @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ B3 @ ( measure_null_sets @ A @ M ) )
% 2.95/1.61 => ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) )
% 2.95/1.61 => ( ( sigma_emeasure @ A @ M @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 2.95/1.61 = ( sigma_emeasure @ A @ M @ A4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_85_Sup_OSUP__cong,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,A4: set @ B,B3: set @ B,C2: B > A,D2: B > A,Sup: ( set @ A ) > A] :
% 2.95/1.61 ( ( A4 = B3 )
% 2.95/1.61 => ( ! [X2: B] :
% 2.95/1.61 ( ( member @ B @ X2 @ B3 )
% 2.95/1.61 => ( ( C2 @ X2 )
% 2.95/1.61 = ( D2 @ X2 ) ) )
% 2.95/1.61 => ( ( Sup @ ( image @ B @ A @ C2 @ A4 ) )
% 2.95/1.61 = ( Sup @ ( image @ B @ A @ D2 @ B3 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_86_Inf_OINF__cong,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,A4: set @ B,B3: set @ B,C2: B > A,D2: B > A,Inf: ( set @ A ) > A] :
% 2.95/1.61 ( ( A4 = B3 )
% 2.95/1.61 => ( ! [X2: B] :
% 2.95/1.61 ( ( member @ B @ X2 @ B3 )
% 2.95/1.61 => ( ( C2 @ X2 )
% 2.95/1.61 = ( D2 @ X2 ) ) )
% 2.95/1.61 => ( ( Inf @ ( image @ B @ A @ C2 @ A4 ) )
% 2.95/1.61 = ( Inf @ ( image @ B @ A @ D2 @ B3 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_87_rev__image__eqI,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType,X3: A,A4: set @ A,B2: B,F: A > B] :
% 2.95/1.61 ( ( member @ A @ X3 @ A4 )
% 2.95/1.61 => ( ( B2
% 2.95/1.61 = ( F @ X3 ) )
% 2.95/1.61 => ( member @ B @ B2 @ ( image @ A @ B @ F @ A4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_88_ball__imageD,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
% 2.95/1.61 ( ! [X2: A] :
% 2.95/1.61 ( ( member @ A @ X2 @ ( image @ B @ A @ F @ A4 ) )
% 2.95/1.61 => ( P @ X2 ) )
% 2.95/1.61 => ! [X5: B] :
% 2.95/1.61 ( ( member @ B @ X5 @ A4 )
% 2.95/1.61 => ( P @ ( F @ X5 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_89_image__cong,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType,M: set @ A,N3: set @ A,F: A > B,G: A > B] :
% 2.95/1.61 ( ( M = N3 )
% 2.95/1.61 => ( ! [X2: A] :
% 2.95/1.61 ( ( member @ A @ X2 @ N3 )
% 2.95/1.61 => ( ( F @ X2 )
% 2.95/1.61 = ( G @ X2 ) ) )
% 2.95/1.61 => ( ( image @ A @ B @ F @ M )
% 2.95/1.61 = ( image @ A @ B @ G @ N3 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_90_bex__imageD,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
% 2.95/1.61 ( ? [X5: A] :
% 2.95/1.61 ( ( member @ A @ X5 @ ( image @ B @ A @ F @ A4 ) )
% 2.95/1.61 & ( P @ X5 ) )
% 2.95/1.61 => ? [X2: B] :
% 2.95/1.61 ( ( member @ B @ X2 @ A4 )
% 2.95/1.61 & ( P @ ( F @ X2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_91_image__iff,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,Z: A,F: B > A,A4: set @ B] :
% 2.95/1.61 ( ( member @ A @ Z @ ( image @ B @ A @ F @ A4 ) )
% 2.95/1.61 = ( ? [X: B] :
% 2.95/1.61 ( ( member @ B @ X @ A4 )
% 2.95/1.61 & ( Z
% 2.95/1.61 = ( F @ X ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_92_imageI,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType,X3: A,A4: set @ A,F: A > B] :
% 2.95/1.61 ( ( member @ A @ X3 @ A4 )
% 2.95/1.61 => ( member @ B @ ( F @ X3 ) @ ( image @ A @ B @ F @ A4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_93_is__num__normalize_I1_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( neg_numeral @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 2.95/1.61 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_94_UNIV__witness,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_95_UNIV__eq__I,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A] :
% 2.95/1.61 ( ! [X2: A] : ( member @ A @ X2 @ A4 )
% 2.95/1.61 => ( ( top_top @ ( set @ A ) )
% 2.95/1.61 = A4 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_96_Int__left__commute,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A,B3: set @ A,C2: set @ A] :
% 2.95/1.61 ( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) )
% 2.95/1.61 = ( inf_inf @ ( set @ A ) @ B3 @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_97_Int__left__absorb,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.95/1.61 ( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
% 2.95/1.61 = ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_98_Int__commute,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( inf_inf @ ( set @ A ) )
% 2.95/1.61 = ( ^ [A6: set @ A,B4: set @ A] : ( inf_inf @ ( set @ A ) @ B4 @ A6 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_99_Int__absorb,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A] :
% 2.95/1.61 ( ( inf_inf @ ( set @ A ) @ A4 @ A4 )
% 2.95/1.61 = A4 ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_100_Int__assoc,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A,B3: set @ A,C2: set @ A] :
% 2.95/1.61 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ C2 )
% 2.95/1.61 = ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_101_IntD2,axiom,
% 2.95/1.61 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 2.95/1.61 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
% 2.95/1.61 => ( member @ A @ C @ B3 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_102_IntD1,axiom,
% 2.95/1.61 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 2.95/1.61 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
% 2.95/1.61 => ( member @ A @ C @ A4 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_103_IntE,axiom,
% 2.95/1.61 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 2.95/1.61 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
% 2.95/1.61 => ~ ( ( member @ A @ C @ A4 )
% 2.95/1.61 => ~ ( member @ A @ C @ B3 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_104_UnionE,axiom,
% 2.95/1.61 ! [A: $tType,A4: A,C2: set @ ( set @ A )] :
% 2.95/1.61 ( ( member @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) )
% 2.95/1.61 => ~ ! [X6: set @ A] :
% 2.95/1.61 ( ( member @ A @ A4 @ X6 )
% 2.95/1.61 => ~ ( member @ ( set @ A ) @ X6 @ C2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_105_vimage__Collect,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType,P: B > $o,F: A > B,Q: A > $o] :
% 2.95/1.61 ( ! [X2: A] :
% 2.95/1.61 ( ( P @ ( F @ X2 ) )
% 2.95/1.61 = ( Q @ X2 ) )
% 2.95/1.61 => ( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
% 2.95/1.61 = ( collect @ A @ Q ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_106_vimageI2,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType,F: B > A,A2: B,A4: set @ A] :
% 2.95/1.61 ( ( member @ A @ ( F @ A2 ) @ A4 )
% 2.95/1.61 => ( member @ B @ A2 @ ( vimage @ B @ A @ F @ A4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_107_vimageE,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,A2: A,F: A > B,B3: set @ B] :
% 2.95/1.61 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ B3 ) )
% 2.95/1.61 => ( member @ B @ ( F @ A2 ) @ B3 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_108_vimageD,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,A2: A,F: A > B,A4: set @ B] :
% 2.95/1.61 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ A4 ) )
% 2.95/1.61 => ( member @ B @ ( F @ A2 ) @ A4 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_109_Sup_OSUP__identity__eq,axiom,
% 2.95/1.61 ! [A: $tType,Sup: ( set @ A ) > A,A4: set @ A] :
% 2.95/1.61 ( ( Sup
% 2.95/1.61 @ ( image @ A @ A
% 2.95/1.61 @ ^ [X: A] : X
% 2.95/1.61 @ A4 ) )
% 2.95/1.61 = ( Sup @ A4 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_110_Inf_OINF__identity__eq,axiom,
% 2.95/1.61 ! [A: $tType,Inf: ( set @ A ) > A,A4: set @ A] :
% 2.95/1.61 ( ( Inf
% 2.95/1.61 @ ( image @ A @ A
% 2.95/1.61 @ ^ [X: A] : X
% 2.95/1.61 @ A4 ) )
% 2.95/1.61 = ( Inf @ A4 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_111_Compr__image__eq,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
% 2.95/1.61 ( ( collect @ A
% 2.95/1.61 @ ^ [X: A] :
% 2.95/1.61 ( ( member @ A @ X @ ( image @ B @ A @ F @ A4 ) )
% 2.95/1.61 & ( P @ X ) ) )
% 2.95/1.61 = ( image @ B @ A @ F
% 2.95/1.61 @ ( collect @ B
% 2.95/1.61 @ ^ [X: B] :
% 2.95/1.61 ( ( member @ B @ X @ A4 )
% 2.95/1.61 & ( P @ ( F @ X ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_112_image__image,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,C3: $tType,F: B > A,G: C3 > B,A4: set @ C3] :
% 2.95/1.61 ( ( image @ B @ A @ F @ ( image @ C3 @ B @ G @ A4 ) )
% 2.95/1.61 = ( image @ C3 @ A
% 2.95/1.61 @ ^ [X: C3] : ( F @ ( G @ X ) )
% 2.95/1.61 @ A4 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_113_imageE,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,B2: A,F: B > A,A4: set @ B] :
% 2.95/1.61 ( ( member @ A @ B2 @ ( image @ B @ A @ F @ A4 ) )
% 2.95/1.61 => ~ ! [X2: B] :
% 2.95/1.61 ( ( B2
% 2.95/1.61 = ( F @ X2 ) )
% 2.95/1.61 => ~ ( member @ B @ X2 @ A4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_114_UNIV__def,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( top_top @ ( set @ A ) )
% 2.95/1.61 = ( collect @ A
% 2.95/1.61 @ ^ [X: A] : $true ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_115_Collect__conj__eq,axiom,
% 2.95/1.61 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.95/1.61 ( ( collect @ A
% 2.95/1.61 @ ^ [X: A] :
% 2.95/1.61 ( ( P @ X )
% 2.95/1.61 & ( Q @ X ) ) )
% 2.95/1.61 = ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_116_Int__Collect,axiom,
% 2.95/1.61 ! [A: $tType,X3: A,A4: set @ A,P: A > $o] :
% 2.95/1.61 ( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P ) ) )
% 2.95/1.61 = ( ( member @ A @ X3 @ A4 )
% 2.95/1.61 & ( P @ X3 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_117_Int__def,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( inf_inf @ ( set @ A ) )
% 2.95/1.61 = ( ^ [A6: set @ A,B4: set @ A] :
% 2.95/1.61 ( collect @ A
% 2.95/1.61 @ ^ [X: A] :
% 2.95/1.61 ( ( member @ A @ X @ A6 )
% 2.95/1.61 & ( member @ A @ X @ B4 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_118_vimage__def,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( vimage @ A @ B )
% 2.95/1.61 = ( ^ [F2: A > B,B4: set @ B] :
% 2.95/1.61 ( collect @ A
% 2.95/1.61 @ ^ [X: A] : ( member @ B @ ( F2 @ X ) @ B4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_119_less__numeral__extra_I3_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linord1659791738miring @ A )
% 2.95/1.61 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_120_less__numeral__extra_I4_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linord1659791738miring @ A )
% 2.95/1.61 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_121_range__eqI,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,B2: A,F: B > A,X3: B] :
% 2.95/1.61 ( ( B2
% 2.95/1.61 = ( F @ X3 ) )
% 2.95/1.61 => ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_122_surj__def,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType,F: B > A] :
% 2.95/1.61 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.95/1.61 = ( top_top @ ( set @ A ) ) )
% 2.95/1.61 = ( ! [Y2: A] :
% 2.95/1.61 ? [X: B] :
% 2.95/1.61 ( Y2
% 2.95/1.61 = ( F @ X ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_123_rangeI,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,F: B > A,X3: B] : ( member @ A @ ( F @ X3 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_124_surjI,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 2.95/1.61 ( ! [X2: A] :
% 2.95/1.61 ( ( G @ ( F @ X2 ) )
% 2.95/1.61 = X2 )
% 2.95/1.61 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 2.95/1.61 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_125_surjE,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,F: B > A,Y4: A] :
% 2.95/1.61 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.95/1.61 = ( top_top @ ( set @ A ) ) )
% 2.95/1.61 => ~ ! [X2: B] :
% 2.95/1.61 ( Y4
% 2.95/1.61 != ( F @ X2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_126_surjD,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,F: B > A,Y4: A] :
% 2.95/1.61 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.95/1.61 = ( top_top @ ( set @ A ) ) )
% 2.95/1.61 => ? [X2: B] :
% 2.95/1.61 ( Y4
% 2.95/1.61 = ( F @ X2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_127_less__Sup__iff,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( comple1035589618norder @ A )
% 2.95/1.61 => ! [A2: A,S: set @ A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ S ) )
% 2.95/1.61 = ( ? [X: A] :
% 2.95/1.61 ( ( member @ A @ X @ S )
% 2.95/1.61 & ( ord_less @ A @ A2 @ X ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_128_SUP__cong,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType] :
% 2.95/1.61 ( ( complete_Sup @ A )
% 2.95/1.61 => ! [A4: set @ B,B3: set @ B,C2: B > A,D2: B > A] :
% 2.95/1.61 ( ( A4 = B3 )
% 2.95/1.61 => ( ! [X2: B] :
% 2.95/1.61 ( ( member @ B @ X2 @ B3 )
% 2.95/1.61 => ( ( C2 @ X2 )
% 2.95/1.61 = ( D2 @ X2 ) ) )
% 2.95/1.61 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ C2 @ A4 ) )
% 2.95/1.61 = ( complete_Sup_Sup @ A @ ( image @ B @ A @ D2 @ B3 ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_129_Int__UNIV__right,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A] :
% 2.95/1.61 ( ( inf_inf @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) )
% 2.95/1.61 = A4 ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_130_Int__UNIV__left,axiom,
% 2.95/1.61 ! [A: $tType,B3: set @ A] :
% 2.95/1.61 ( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 )
% 2.95/1.61 = B3 ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_131_Union__UNIV,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( complete_Sup_Sup @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
% 2.95/1.61 = ( top_top @ ( set @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_132_null__setsD2,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A,M: sigma_measure @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ A4 @ ( measure_null_sets @ A @ M ) )
% 2.95/1.61 => ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_133_vimage__inter__cong,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType,S: set @ A,F: A > B,G: A > B,Y4: set @ B] :
% 2.95/1.61 ( ! [W: A] :
% 2.95/1.61 ( ( member @ A @ W @ S )
% 2.95/1.61 => ( ( F @ W )
% 2.95/1.61 = ( G @ W ) ) )
% 2.95/1.61 => ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ Y4 ) @ S )
% 2.95/1.61 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G @ Y4 ) @ S ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_134_translation__subtract__diff,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ab_group_add @ A )
% 2.95/1.61 => ! [A2: A,S2: set @ A,T: set @ A] :
% 2.95/1.61 ( ( image @ A @ A
% 2.95/1.61 @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
% 2.95/1.61 @ ( minus_minus @ ( set @ A ) @ S2 @ T ) )
% 2.95/1.61 = ( minus_minus @ ( set @ A )
% 2.95/1.61 @ ( image @ A @ A
% 2.95/1.61 @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
% 2.95/1.61 @ S2 )
% 2.95/1.61 @ ( image @ A @ A
% 2.95/1.61 @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
% 2.95/1.61 @ T ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_135_range__composition,axiom,
% 2.95/1.61 ! [A: $tType,C3: $tType,B: $tType,F: C3 > A,G: B > C3] :
% 2.95/1.61 ( ( image @ B @ A
% 2.95/1.61 @ ^ [X: B] : ( F @ ( G @ X ) )
% 2.95/1.61 @ ( top_top @ ( set @ B ) ) )
% 2.95/1.61 = ( image @ C3 @ A @ F @ ( image @ B @ C3 @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_136_rangeE,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,B2: A,F: B > A] :
% 2.95/1.61 ( ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
% 2.95/1.61 => ~ ! [X2: B] :
% 2.95/1.61 ( B2
% 2.95/1.61 != ( F @ X2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_137_SUP__commute,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,C3: $tType] :
% 2.95/1.61 ( ( comple187826305attice @ A )
% 2.95/1.61 => ! [F: B > C3 > A,B3: set @ C3,A4: set @ B] :
% 2.95/1.61 ( ( complete_Sup_Sup @ A
% 2.95/1.61 @ ( image @ B @ A
% 2.95/1.61 @ ^ [I5: B] : ( complete_Sup_Sup @ A @ ( image @ C3 @ A @ ( F @ I5 ) @ B3 ) )
% 2.95/1.61 @ A4 ) )
% 2.95/1.61 = ( complete_Sup_Sup @ A
% 2.95/1.61 @ ( image @ C3 @ A
% 2.95/1.61 @ ^ [J: C3] :
% 2.95/1.61 ( complete_Sup_Sup @ A
% 2.95/1.61 @ ( image @ B @ A
% 2.95/1.61 @ ^ [I5: B] : ( F @ I5 @ J )
% 2.95/1.61 @ A4 ) )
% 2.95/1.61 @ B3 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_138_image__Union,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,F: B > A,S: set @ ( set @ B )] :
% 2.95/1.61 ( ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ S ) )
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ S ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_139_UN__UN__flatten,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,C3: $tType,C2: B > ( set @ A ),B3: C3 > ( set @ B ),A4: set @ C3] :
% 2.95/1.61 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C3 @ ( set @ B ) @ B3 @ A4 ) ) ) )
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ A )
% 2.95/1.61 @ ( image @ C3 @ ( set @ A )
% 2.95/1.61 @ ^ [Y2: C3] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C2 @ ( B3 @ Y2 ) ) )
% 2.95/1.61 @ A4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_140_UN__E,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,B2: A,B3: B > ( set @ A ),A4: set @ B] :
% 2.95/1.61 ( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A4 ) ) )
% 2.95/1.61 => ~ ! [X2: B] :
% 2.95/1.61 ( ( member @ B @ X2 @ A4 )
% 2.95/1.61 => ~ ( member @ A @ B2 @ ( B3 @ X2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_141_UN__extend__simps_I8_J,axiom,
% 2.95/1.61 ! [P2: $tType,O: $tType,B3: O > ( set @ P2 ),A4: set @ ( set @ O )] :
% 2.95/1.61 ( ( complete_Sup_Sup @ ( set @ P2 )
% 2.95/1.61 @ ( image @ ( set @ O ) @ ( set @ P2 )
% 2.95/1.61 @ ^ [Y2: set @ O] : ( complete_Sup_Sup @ ( set @ P2 ) @ ( image @ O @ ( set @ P2 ) @ B3 @ Y2 ) )
% 2.95/1.61 @ A4 ) )
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ P2 ) @ ( image @ O @ ( set @ P2 ) @ B3 @ ( complete_Sup_Sup @ ( set @ O ) @ A4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_142_UN__extend__simps_I9_J,axiom,
% 2.95/1.61 ! [S3: $tType,R: $tType,Q2: $tType,C2: R > ( set @ S3 ),B3: Q2 > ( set @ R ),A4: set @ Q2] :
% 2.95/1.61 ( ( complete_Sup_Sup @ ( set @ S3 )
% 2.95/1.61 @ ( image @ Q2 @ ( set @ S3 )
% 2.95/1.61 @ ^ [X: Q2] : ( complete_Sup_Sup @ ( set @ S3 ) @ ( image @ R @ ( set @ S3 ) @ C2 @ ( B3 @ X ) ) )
% 2.95/1.61 @ A4 ) )
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ S3 ) @ ( image @ R @ ( set @ S3 ) @ C2 @ ( complete_Sup_Sup @ ( set @ R ) @ ( image @ Q2 @ ( set @ R ) @ B3 @ A4 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_143_less__numeral__extra_I1_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linord1659791738miring @ A )
% 2.95/1.61 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_144_translation__Int,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ab_group_add @ A )
% 2.95/1.61 => ! [A2: A,S2: set @ A,T: set @ A] :
% 2.95/1.61 ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( inf_inf @ ( set @ A ) @ S2 @ T ) )
% 2.95/1.61 = ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S2 ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_145_surj__image__vimage__eq,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType,F: B > A,A4: set @ A] :
% 2.95/1.61 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.95/1.61 = ( top_top @ ( set @ A ) ) )
% 2.95/1.61 => ( ( image @ B @ A @ F @ ( vimage @ B @ A @ F @ A4 ) )
% 2.95/1.61 = A4 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_146_null__set__Int2,axiom,
% 2.95/1.61 ! [A: $tType,B3: set @ A,M: sigma_measure @ A,A4: set @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ B3 @ ( measure_null_sets @ A @ M ) )
% 2.95/1.61 => ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) )
% 2.95/1.61 => ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B3 @ A4 ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_147_null__set__Int1,axiom,
% 2.95/1.61 ! [A: $tType,B3: set @ A,M: sigma_measure @ A,A4: set @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ B3 @ ( measure_null_sets @ A @ M ) )
% 2.95/1.61 => ( ( member @ ( set @ A ) @ A4 @ ( sigma_sets @ A @ M ) )
% 2.95/1.61 => ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ ( measure_null_sets @ A @ M ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_148_null__setsD1,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A,M: sigma_measure @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ A4 @ ( measure_null_sets @ A @ M ) )
% 2.95/1.61 => ( ( sigma_emeasure @ A @ M @ A4 )
% 2.95/1.61 = ( zero_zero @ extend1814228343nnreal ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_149_less__SUP__iff,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType] :
% 2.95/1.61 ( ( comple1035589618norder @ A )
% 2.95/1.61 => ! [A2: A,F: B > A,A4: set @ B] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A4 ) ) )
% 2.95/1.61 = ( ? [X: B] :
% 2.95/1.61 ( ( member @ B @ X @ A4 )
% 2.95/1.61 & ( ord_less @ A @ A2 @ ( F @ X ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_150_SUP__lessD,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( comple187826305attice @ A )
% 2.95/1.61 => ! [F: B > A,A4: set @ B,Y4: A,I3: B] :
% 2.95/1.61 ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A4 ) ) @ Y4 )
% 2.95/1.61 => ( ( member @ B @ I3 @ A4 )
% 2.95/1.61 => ( ord_less @ A @ ( F @ I3 ) @ Y4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_151_translation__subtract__Int,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ab_group_add @ A )
% 2.95/1.61 => ! [A2: A,S2: set @ A,T: set @ A] :
% 2.95/1.61 ( ( image @ A @ A
% 2.95/1.61 @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
% 2.95/1.61 @ ( inf_inf @ ( set @ A ) @ S2 @ T ) )
% 2.95/1.61 = ( inf_inf @ ( set @ A )
% 2.95/1.61 @ ( image @ A @ A
% 2.95/1.61 @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
% 2.95/1.61 @ S2 )
% 2.95/1.61 @ ( image @ A @ A
% 2.95/1.61 @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
% 2.95/1.61 @ T ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_152_image__UN,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,C3: $tType,F: B > A,B3: C3 > ( set @ B ),A4: set @ C3] :
% 2.95/1.61 ( ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C3 @ ( set @ B ) @ B3 @ A4 ) ) )
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ A )
% 2.95/1.61 @ ( image @ C3 @ ( set @ A )
% 2.95/1.61 @ ^ [X: C3] : ( image @ B @ A @ F @ ( B3 @ X ) )
% 2.95/1.61 @ A4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_153_UN__extend__simps_I10_J,axiom,
% 2.95/1.61 ! [V: $tType,U: $tType,T2: $tType,B3: U > ( set @ V ),F: T2 > U,A4: set @ T2] :
% 2.95/1.61 ( ( complete_Sup_Sup @ ( set @ V )
% 2.95/1.61 @ ( image @ T2 @ ( set @ V )
% 2.95/1.61 @ ^ [A3: T2] : ( B3 @ ( F @ A3 ) )
% 2.95/1.61 @ A4 ) )
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ V ) @ ( image @ U @ ( set @ V ) @ B3 @ ( image @ T2 @ U @ F @ A4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_154_Int__UN__distrib2,axiom,
% 2.95/1.61 ! [A: $tType,C3: $tType,B: $tType,A4: B > ( set @ A ),I4: set @ B,B3: C3 > ( set @ A ),J2: set @ C3] :
% 2.95/1.61 ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A4 @ I4 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ C3 @ ( set @ A ) @ B3 @ J2 ) ) )
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ A )
% 2.95/1.61 @ ( image @ B @ ( set @ A )
% 2.95/1.61 @ ^ [I5: B] :
% 2.95/1.61 ( complete_Sup_Sup @ ( set @ A )
% 2.95/1.61 @ ( image @ C3 @ ( set @ A )
% 2.95/1.61 @ ^ [J: C3] : ( inf_inf @ ( set @ A ) @ ( A4 @ I5 ) @ ( B3 @ J ) )
% 2.95/1.61 @ J2 ) )
% 2.95/1.61 @ I4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_155_Int__UN__distrib,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,B3: set @ A,A4: B > ( set @ A ),I4: set @ B] :
% 2.95/1.61 ( ( inf_inf @ ( set @ A ) @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A4 @ I4 ) ) )
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ A )
% 2.95/1.61 @ ( image @ B @ ( set @ A )
% 2.95/1.61 @ ^ [I5: B] : ( inf_inf @ ( set @ A ) @ B3 @ ( A4 @ I5 ) )
% 2.95/1.61 @ I4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_156_UN__extend__simps_I4_J,axiom,
% 2.95/1.61 ! [H: $tType,G2: $tType,A4: G2 > ( set @ H ),C2: set @ G2,B3: set @ H] :
% 2.95/1.61 ( ( inf_inf @ ( set @ H ) @ ( complete_Sup_Sup @ ( set @ H ) @ ( image @ G2 @ ( set @ H ) @ A4 @ C2 ) ) @ B3 )
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ H )
% 2.95/1.61 @ ( image @ G2 @ ( set @ H )
% 2.95/1.61 @ ^ [X: G2] : ( inf_inf @ ( set @ H ) @ ( A4 @ X ) @ B3 )
% 2.95/1.61 @ C2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_157_UN__extend__simps_I5_J,axiom,
% 2.95/1.61 ! [I: $tType,J3: $tType,A4: set @ I,B3: J3 > ( set @ I ),C2: set @ J3] :
% 2.95/1.61 ( ( inf_inf @ ( set @ I ) @ A4 @ ( complete_Sup_Sup @ ( set @ I ) @ ( image @ J3 @ ( set @ I ) @ B3 @ C2 ) ) )
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ I )
% 2.95/1.61 @ ( image @ J3 @ ( set @ I )
% 2.95/1.61 @ ^ [X: J3] : ( inf_inf @ ( set @ I ) @ A4 @ ( B3 @ X ) )
% 2.95/1.61 @ C2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_158_vimage__UN,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,C3: $tType,F: A > B,B3: C3 > ( set @ B ),A4: set @ C3] :
% 2.95/1.61 ( ( vimage @ A @ B @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C3 @ ( set @ B ) @ B3 @ A4 ) ) )
% 2.95/1.61 = ( complete_Sup_Sup @ ( set @ A )
% 2.95/1.61 @ ( image @ C3 @ ( set @ A )
% 2.95/1.61 @ ^ [X: C3] : ( vimage @ A @ B @ F @ ( B3 @ X ) )
% 2.95/1.61 @ A4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_159_SUP__UNION,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType,C3: $tType] :
% 2.95/1.61 ( ( comple187826305attice @ A )
% 2.95/1.61 => ! [F: B > A,G: C3 > ( set @ B ),A4: set @ C3] :
% 2.95/1.61 ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C3 @ ( set @ B ) @ G @ A4 ) ) ) )
% 2.95/1.61 = ( complete_Sup_Sup @ A
% 2.95/1.61 @ ( image @ C3 @ A
% 2.95/1.61 @ ^ [Y2: C3] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ ( G @ Y2 ) ) )
% 2.95/1.61 @ A4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_160_null__sets__def,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( measure_null_sets @ A )
% 2.95/1.61 = ( ^ [M4: sigma_measure @ A] :
% 2.95/1.61 ( collect @ ( set @ A )
% 2.95/1.61 @ ^ [N4: set @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ N4 @ ( sigma_sets @ A @ M4 ) )
% 2.95/1.61 & ( ( sigma_emeasure @ A @ M4 @ N4 )
% 2.95/1.61 = ( zero_zero @ extend1814228343nnreal ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_161_sets_OInt__space__eq2,axiom,
% 2.95/1.61 ! [A: $tType,X3: set @ A,M: sigma_measure @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ X3 @ ( sigma_sets @ A @ M ) )
% 2.95/1.61 => ( ( inf_inf @ ( set @ A ) @ X3 @ ( sigma_space @ A @ M ) )
% 2.95/1.61 = X3 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_162_sets_OInt__space__eq1,axiom,
% 2.95/1.61 ! [A: $tType,X3: set @ A,M: sigma_measure @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ X3 @ ( sigma_sets @ A @ M ) )
% 2.95/1.61 => ( ( inf_inf @ ( set @ A ) @ ( sigma_space @ A @ M ) @ X3 )
% 2.95/1.61 = X3 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_163_diff__add__zero,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( comm_monoid_diff @ A )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.95/1.61 = ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_164_diff__gt__0__iff__gt,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordered_ab_group_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
% 2.95/1.61 = ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_165_zero__less__double__add__iff__zero__less__single__add,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linord219039673up_add @ A )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 2.95/1.61 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_166_double__add__less__zero__iff__single__add__less__zero,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linord219039673up_add @ A )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 2.95/1.61 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_167_less__add__same__cancel2,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordere516151231imp_le @ A )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
% 2.95/1.61 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_168_less__add__same__cancel1,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordere516151231imp_le @ A )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.95/1.61 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_169_add__less__same__cancel2,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordere516151231imp_le @ A )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 2.95/1.61 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_170_add__left__cancel,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( cancel_semigroup_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.95/1.61 = ( plus_plus @ A @ A2 @ C ) )
% 2.95/1.61 = ( B2 = C ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_171_add__right__cancel,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( cancel_semigroup_add @ A )
% 2.95/1.61 => ! [B2: A,A2: A,C: A] :
% 2.95/1.61 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.95/1.61 = ( plus_plus @ A @ C @ A2 ) )
% 2.95/1.61 = ( B2 = C ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_172_DiffI,axiom,
% 2.95/1.61 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 2.95/1.61 ( ( member @ A @ C @ A4 )
% 2.95/1.61 => ( ~ ( member @ A @ C @ B3 )
% 2.95/1.61 => ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_173_Diff__iff,axiom,
% 2.95/1.61 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 2.95/1.61 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 2.95/1.61 = ( ( member @ A @ C @ A4 )
% 2.95/1.61 & ~ ( member @ A @ C @ B3 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_174_Diff__idemp,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.95/1.61 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ B3 )
% 2.95/1.61 = ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_175_not__gr__zero,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( canoni770627133id_add @ A )
% 2.95/1.61 => ! [N5: A] :
% 2.95/1.61 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N5 ) )
% 2.95/1.61 = ( N5
% 2.95/1.61 = ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_176_add_Oleft__neutral,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( monoid_add @ A )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 2.95/1.61 = A2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_177_add_Oright__neutral,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( monoid_add @ A )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 2.95/1.61 = A2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_178_double__zero,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linord219039673up_add @ A )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ( plus_plus @ A @ A2 @ A2 )
% 2.95/1.61 = ( zero_zero @ A ) )
% 2.95/1.61 = ( A2
% 2.95/1.61 = ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_179_double__zero__sym,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linord219039673up_add @ A )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ( zero_zero @ A )
% 2.95/1.61 = ( plus_plus @ A @ A2 @ A2 ) )
% 2.95/1.61 = ( A2
% 2.95/1.61 = ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_180_add__cancel__left__left,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( cancel1352612707id_add @ A )
% 2.95/1.61 => ! [B2: A,A2: A] :
% 2.95/1.61 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.95/1.61 = A2 )
% 2.95/1.61 = ( B2
% 2.95/1.61 = ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_181_add__cancel__left__right,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( cancel1352612707id_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.95/1.61 = A2 )
% 2.95/1.61 = ( B2
% 2.95/1.61 = ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_182_add__cancel__right__left,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( cancel1352612707id_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( A2
% 2.95/1.61 = ( plus_plus @ A @ B2 @ A2 ) )
% 2.95/1.61 = ( B2
% 2.95/1.61 = ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_183_add__cancel__right__right,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( cancel1352612707id_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( A2
% 2.95/1.61 = ( plus_plus @ A @ A2 @ B2 ) )
% 2.95/1.61 = ( B2
% 2.95/1.61 = ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_184_add__eq__0__iff__both__eq__0,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( canoni770627133id_add @ A )
% 2.95/1.61 => ! [X3: A,Y4: A] :
% 2.95/1.61 ( ( ( plus_plus @ A @ X3 @ Y4 )
% 2.95/1.61 = ( zero_zero @ A ) )
% 2.95/1.61 = ( ( X3
% 2.95/1.61 = ( zero_zero @ A ) )
% 2.95/1.61 & ( Y4
% 2.95/1.61 = ( zero_zero @ A ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_185_zero__eq__add__iff__both__eq__0,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( canoni770627133id_add @ A )
% 2.95/1.61 => ! [X3: A,Y4: A] :
% 2.95/1.61 ( ( ( zero_zero @ A )
% 2.95/1.61 = ( plus_plus @ A @ X3 @ Y4 ) )
% 2.95/1.61 = ( ( X3
% 2.95/1.61 = ( zero_zero @ A ) )
% 2.95/1.61 & ( Y4
% 2.95/1.61 = ( zero_zero @ A ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_186_diff__self,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( group_add @ A )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( minus_minus @ A @ A2 @ A2 )
% 2.95/1.61 = ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_187_diff__0__right,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( group_add @ A )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 2.95/1.61 = A2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_188_zero__diff,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( comm_monoid_diff @ A )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 2.95/1.61 = ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_189_diff__zero,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( cancel1352612707id_add @ A )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 2.95/1.61 = A2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_190_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( cancel1352612707id_add @ A )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( minus_minus @ A @ A2 @ A2 )
% 2.95/1.61 = ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_191_add__less__cancel__left,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordere236663937imp_le @ A )
% 2.95/1.61 => ! [C: A,A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 2.95/1.61 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_192_add__less__cancel__right,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordere236663937imp_le @ A )
% 2.95/1.61 => ! [A2: A,C: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 2.95/1.61 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_193_add__diff__cancel,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( group_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 2.95/1.61 = A2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_194_diff__add__cancel,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( group_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
% 2.95/1.61 = A2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_195_add__diff__cancel__left,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( cancel146912293up_add @ A )
% 2.95/1.61 => ! [C: A,A2: A,B2: A] :
% 2.95/1.61 ( ( minus_minus @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 2.95/1.61 = ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_196_add__diff__cancel__left_H,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( cancel146912293up_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
% 2.95/1.61 = B2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_197_add__diff__cancel__right,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( cancel146912293up_add @ A )
% 2.95/1.61 => ! [A2: A,C: A,B2: A] :
% 2.95/1.61 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 2.95/1.61 = ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_198_add__diff__cancel__right_H,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( cancel146912293up_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 2.95/1.61 = A2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_199_sets_OInt,axiom,
% 2.95/1.61 ! [A: $tType,A2: set @ A,M: sigma_measure @ A,B2: set @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ A2 @ ( sigma_sets @ A @ M ) )
% 2.95/1.61 => ( ( member @ ( set @ A ) @ B2 @ ( sigma_sets @ A @ M ) )
% 2.95/1.61 => ( member @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ ( sigma_sets @ A @ M ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_200_sets_Otop,axiom,
% 2.95/1.61 ! [A: $tType,M: sigma_measure @ A] : ( member @ ( set @ A ) @ ( sigma_space @ A @ M ) @ ( sigma_sets @ A @ M ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_201_sets_ODiff,axiom,
% 2.95/1.61 ! [A: $tType,A2: set @ A,M: sigma_measure @ A,B2: set @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ A2 @ ( sigma_sets @ A @ M ) )
% 2.95/1.61 => ( ( member @ ( set @ A ) @ B2 @ ( sigma_sets @ A @ M ) )
% 2.95/1.61 => ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( sigma_sets @ A @ M ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_202_add__less__same__cancel1,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordere516151231imp_le @ A )
% 2.95/1.61 => ! [B2: A,A2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
% 2.95/1.61 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_203_sets_Ocompl__sets,axiom,
% 2.95/1.61 ! [A: $tType,A2: set @ A,M: sigma_measure @ A] :
% 2.95/1.61 ( ( member @ ( set @ A ) @ A2 @ ( sigma_sets @ A @ M ) )
% 2.95/1.61 => ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( sigma_space @ A @ M ) @ A2 ) @ ( sigma_sets @ A @ M ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_204_DiffE,axiom,
% 2.95/1.61 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 2.95/1.61 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 2.95/1.61 => ~ ( ( member @ A @ C @ A4 )
% 2.95/1.61 => ( member @ A @ C @ B3 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_205_DiffD1,axiom,
% 2.95/1.61 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 2.95/1.61 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 2.95/1.61 => ( member @ A @ C @ A4 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_206_DiffD2,axiom,
% 2.95/1.61 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 2.95/1.61 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 2.95/1.61 => ~ ( member @ A @ C @ B3 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_207_set__diff__eq,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( minus_minus @ ( set @ A ) )
% 2.95/1.61 = ( ^ [A6: set @ A,B4: set @ A] :
% 2.95/1.61 ( collect @ A
% 2.95/1.61 @ ^ [X: A] :
% 2.95/1.61 ( ( member @ A @ X @ A6 )
% 2.95/1.61 & ~ ( member @ A @ X @ B4 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_208_minus__set__def,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( minus_minus @ ( set @ A ) )
% 2.95/1.61 = ( ^ [A6: set @ A,B4: set @ A] :
% 2.95/1.61 ( collect @ A
% 2.95/1.61 @ ( minus_minus @ ( A > $o )
% 2.95/1.61 @ ^ [X: A] : ( member @ A @ X @ A6 )
% 2.95/1.61 @ ^ [X: A] : ( member @ A @ X @ B4 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_209_psubset__imp__ex__mem,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.95/1.61 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 2.95/1.61 => ? [B5: A] : ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_210_Sup__bool__def,axiom,
% 2.95/1.61 ( ( complete_Sup_Sup @ $o )
% 2.95/1.61 = ( member @ $o @ $true ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_211_zero__reorient,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( zero @ A )
% 2.95/1.61 => ! [X3: A] :
% 2.95/1.61 ( ( ( zero_zero @ A )
% 2.95/1.61 = X3 )
% 2.95/1.61 = ( X3
% 2.95/1.61 = ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_212_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ab_semigroup_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 2.95/1.61 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_213_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordere779506340up_add @ A )
% 2.95/1.61 => ! [I3: A,J4: A,K2: A,L2: A] :
% 2.95/1.61 ( ( ( I3 = J4 )
% 2.95/1.61 & ( K2 = L2 ) )
% 2.95/1.61 => ( ( plus_plus @ A @ I3 @ K2 )
% 2.95/1.61 = ( plus_plus @ A @ J4 @ L2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_214_group__cancel_Oadd1,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( comm_monoid_add @ A )
% 2.95/1.61 => ! [A4: A,K2: A,A2: A,B2: A] :
% 2.95/1.61 ( ( A4
% 2.95/1.61 = ( plus_plus @ A @ K2 @ A2 ) )
% 2.95/1.61 => ( ( plus_plus @ A @ A4 @ B2 )
% 2.95/1.61 = ( plus_plus @ A @ K2 @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_215_group__cancel_Oadd2,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( comm_monoid_add @ A )
% 2.95/1.61 => ! [B3: A,K2: A,B2: A,A2: A] :
% 2.95/1.61 ( ( B3
% 2.95/1.61 = ( plus_plus @ A @ K2 @ B2 ) )
% 2.95/1.61 => ( ( plus_plus @ A @ A2 @ B3 )
% 2.95/1.61 = ( plus_plus @ A @ K2 @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_216_add_Oassoc,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( semigroup_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 2.95/1.61 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_217_add_Oleft__cancel,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( group_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.95/1.61 = ( plus_plus @ A @ A2 @ C ) )
% 2.95/1.61 = ( B2 = C ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_218_add_Oright__cancel,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( group_add @ A )
% 2.95/1.61 => ! [B2: A,A2: A,C: A] :
% 2.95/1.61 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.95/1.61 = ( plus_plus @ A @ C @ A2 ) )
% 2.95/1.61 = ( B2 = C ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_219_add_Ocommute,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ab_semigroup_add @ A )
% 2.95/1.61 => ( ( plus_plus @ A )
% 2.95/1.61 = ( ^ [A3: A,B6: A] : ( plus_plus @ A @ B6 @ A3 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_220_add_Oleft__commute,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ab_semigroup_add @ A )
% 2.95/1.61 => ! [B2: A,A2: A,C: A] :
% 2.95/1.61 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
% 2.95/1.61 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_221_add__left__imp__eq,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( cancel_semigroup_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.95/1.61 = ( plus_plus @ A @ A2 @ C ) )
% 2.95/1.61 => ( B2 = C ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_222_add__right__imp__eq,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( cancel_semigroup_add @ A )
% 2.95/1.61 => ! [B2: A,A2: A,C: A] :
% 2.95/1.61 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.95/1.61 = ( plus_plus @ A @ C @ A2 ) )
% 2.95/1.61 => ( B2 = C ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_223_diff__eq__diff__eq,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( group_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A,D3: A] :
% 2.95/1.61 ( ( ( minus_minus @ A @ A2 @ B2 )
% 2.95/1.61 = ( minus_minus @ A @ C @ D3 ) )
% 2.95/1.61 => ( ( A2 = B2 )
% 2.95/1.61 = ( C = D3 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_224_diff__right__commute,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( cancel146912293up_add @ A )
% 2.95/1.61 => ! [A2: A,C: A,B2: A] :
% 2.95/1.61 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B2 )
% 2.95/1.61 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_225_one__reorient,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( one @ A )
% 2.95/1.61 => ! [X3: A] :
% 2.95/1.61 ( ( ( one_one @ A )
% 2.95/1.61 = X3 )
% 2.95/1.61 = ( X3
% 2.95/1.61 = ( one_one @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_226_gr__zeroI,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( canoni770627133id_add @ A )
% 2.95/1.61 => ! [N5: A] :
% 2.95/1.61 ( ( N5
% 2.95/1.61 != ( zero_zero @ A ) )
% 2.95/1.61 => ( ord_less @ A @ ( zero_zero @ A ) @ N5 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_227_not__less__zero,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( canoni770627133id_add @ A )
% 2.95/1.61 => ! [N5: A] :
% 2.95/1.61 ~ ( ord_less @ A @ N5 @ ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_228_gr__implies__not__zero,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( canoni770627133id_add @ A )
% 2.95/1.61 => ! [M2: A,N5: A] :
% 2.95/1.61 ( ( ord_less @ A @ M2 @ N5 )
% 2.95/1.61 => ( N5
% 2.95/1.61 != ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_229_zero__less__iff__neq__zero,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( canoni770627133id_add @ A )
% 2.95/1.61 => ! [N5: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( zero_zero @ A ) @ N5 )
% 2.95/1.61 = ( N5
% 2.95/1.61 != ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_230_comm__monoid__add__class_Oadd__0,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( comm_monoid_add @ A )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 2.95/1.61 = A2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_231_add_Ocomm__neutral,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( comm_monoid_add @ A )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 2.95/1.61 = A2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_232_add_Ogroup__left__neutral,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( group_add @ A )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 2.95/1.61 = A2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_233_eq__iff__diff__eq__0,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( group_add @ A )
% 2.95/1.61 => ( ( ^ [Y5: A,Z2: A] : ( Y5 = Z2 ) )
% 2.95/1.61 = ( ^ [A3: A,B6: A] :
% 2.95/1.61 ( ( minus_minus @ A @ A3 @ B6 )
% 2.95/1.61 = ( zero_zero @ A ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_234_add__mono__thms__linordered__field_I5_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordere223160158up_add @ A )
% 2.95/1.61 => ! [I3: A,J4: A,K2: A,L2: A] :
% 2.95/1.61 ( ( ( ord_less @ A @ I3 @ J4 )
% 2.95/1.61 & ( ord_less @ A @ K2 @ L2 ) )
% 2.95/1.61 => ( ord_less @ A @ ( plus_plus @ A @ I3 @ K2 ) @ ( plus_plus @ A @ J4 @ L2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_235_add__mono__thms__linordered__field_I2_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordere223160158up_add @ A )
% 2.95/1.61 => ! [I3: A,J4: A,K2: A,L2: A] :
% 2.95/1.61 ( ( ( I3 = J4 )
% 2.95/1.61 & ( ord_less @ A @ K2 @ L2 ) )
% 2.95/1.61 => ( ord_less @ A @ ( plus_plus @ A @ I3 @ K2 ) @ ( plus_plus @ A @ J4 @ L2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_236_add__mono__thms__linordered__field_I1_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordere223160158up_add @ A )
% 2.95/1.61 => ! [I3: A,J4: A,K2: A,L2: A] :
% 2.95/1.61 ( ( ( ord_less @ A @ I3 @ J4 )
% 2.95/1.61 & ( K2 = L2 ) )
% 2.95/1.61 => ( ord_less @ A @ ( plus_plus @ A @ I3 @ K2 ) @ ( plus_plus @ A @ J4 @ L2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_237_add__strict__mono,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( strict2144017051up_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A,D3: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ B2 )
% 2.95/1.61 => ( ( ord_less @ A @ C @ D3 )
% 2.95/1.61 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_238_add__strict__left__mono,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordere223160158up_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ B2 )
% 2.95/1.61 => ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_239_add__strict__right__mono,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordere223160158up_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ B2 )
% 2.95/1.61 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_240_add__less__imp__less__left,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordere236663937imp_le @ A )
% 2.95/1.61 => ! [C: A,A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 2.95/1.61 => ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_241_add__less__imp__less__right,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordere236663937imp_le @ A )
% 2.95/1.61 => ! [A2: A,C: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 2.95/1.61 => ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_242_diff__strict__mono,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordered_ab_group_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,D3: A,C: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ B2 )
% 2.95/1.61 => ( ( ord_less @ A @ D3 @ C )
% 2.95/1.61 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_243_diff__eq__diff__less,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordered_ab_group_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A,D3: A] :
% 2.95/1.61 ( ( ( minus_minus @ A @ A2 @ B2 )
% 2.95/1.61 = ( minus_minus @ A @ C @ D3 ) )
% 2.95/1.61 => ( ( ord_less @ A @ A2 @ B2 )
% 2.95/1.61 = ( ord_less @ A @ C @ D3 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_244_diff__strict__left__mono,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordered_ab_group_add @ A )
% 2.95/1.61 => ! [B2: A,A2: A,C: A] :
% 2.95/1.61 ( ( ord_less @ A @ B2 @ A2 )
% 2.95/1.61 => ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_245_diff__strict__right__mono,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordered_ab_group_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ B2 )
% 2.95/1.61 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_246_group__cancel_Osub1,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ab_group_add @ A )
% 2.95/1.61 => ! [A4: A,K2: A,A2: A,B2: A] :
% 2.95/1.61 ( ( A4
% 2.95/1.61 = ( plus_plus @ A @ K2 @ A2 ) )
% 2.95/1.61 => ( ( minus_minus @ A @ A4 @ B2 )
% 2.95/1.61 = ( plus_plus @ A @ K2 @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_247_diff__eq__eq,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( group_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( ( minus_minus @ A @ A2 @ B2 )
% 2.95/1.61 = C )
% 2.95/1.61 = ( A2
% 2.95/1.61 = ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_248_eq__diff__eq,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( group_add @ A )
% 2.95/1.61 => ! [A2: A,C: A,B2: A] :
% 2.95/1.61 ( ( A2
% 2.95/1.61 = ( minus_minus @ A @ C @ B2 ) )
% 2.95/1.61 = ( ( plus_plus @ A @ A2 @ B2 )
% 2.95/1.61 = C ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_249_add__diff__eq,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( group_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C ) )
% 2.95/1.61 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_250_diff__diff__eq2,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( group_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C ) )
% 2.95/1.61 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_251_diff__add__eq,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ab_group_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C )
% 2.95/1.61 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_252_diff__add__eq__diff__diff__swap,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( group_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) )
% 2.95/1.61 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_253_diff__diff__add,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( cancel146912293up_add @ A )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C )
% 2.95/1.61 = ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( finite_finite @ A )
% 2.95/1.61 => ( type @ A ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(subcl_Finite__Set_Ofinite___Countable_Ocountable,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( finite_finite @ A )
% 2.95/1.61 => ( countable @ A ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( ordere890947078_space @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( ordere890947078_space @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Rings_Oring__1,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( ring_1 @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( ring_1 @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_1,axiom,
% 2.95/1.61 ordere890947078_space @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Rings_Oring__1_2,axiom,
% 2.95/1.61 ring_1 @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Rings_Oring__1_3,axiom,
% 2.95/1.61 ring_1 @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( comple187826305attice @ A8 )
% 2.95/1.61 => ( comple187826305attice @ ( A7 > A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_fun___Complete__Lattices_OSup,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( complete_Sup @ A8 )
% 2.95/1.61 => ( complete_Sup @ ( A7 > A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_fun___Countable_Ocountable,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( finite_finite @ A7 )
% 2.95/1.61 & ( countable @ A8 ) )
% 2.95/1.61 => ( countable @ ( A7 > A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( finite_finite @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( finite_finite @ ( A7 > A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 2.95/1.61 ordere516151231imp_le @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 2.95/1.61 strict2144017051up_add @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 2.95/1.61 ordere223160158up_add @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 2.95/1.61 ordere236663937imp_le @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 2.95/1.61 linord1659791738miring @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
% 2.95/1.61 ordere779506340up_add @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
% 2.95/1.61 linord219039673up_add @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.95/1.61 cancel146912293up_add @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 2.95/1.61 cancel1352612707id_add @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 2.95/1.61 ordered_ab_group_add @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
% 2.95/1.61 cancel_semigroup_add @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
% 2.95/1.61 ab_semigroup_add @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
% 2.95/1.61 comm_monoid_add @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Complete__Lattices_OSup_4,axiom,
% 2.95/1.61 complete_Sup @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
% 2.95/1.61 semigroup_add @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
% 2.95/1.61 ab_group_add @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Countable_Ocountable_5,axiom,
% 2.95/1.61 countable @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
% 2.95/1.61 monoid_add @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 2.95/1.61 group_add @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
% 2.95/1.61 neg_numeral @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
% 2.95/1.61 ring_char_0 @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 2.95/1.61 zero @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Int_Oint___Groups_Oone,axiom,
% 2.95/1.61 one @ int ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_6,axiom,
% 2.95/1.61 ordere516151231imp_le @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_7,axiom,
% 2.95/1.61 strict2144017051up_add @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_8,axiom,
% 2.95/1.61 ordere223160158up_add @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_9,axiom,
% 2.95/1.61 ordere236663937imp_le @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.95/1.61 canoni770627133id_add @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_10,axiom,
% 2.95/1.61 linord1659791738miring @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_11,axiom,
% 2.95/1.61 ordere779506340up_add @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_12,axiom,
% 2.95/1.61 cancel146912293up_add @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_13,axiom,
% 2.95/1.61 cancel1352612707id_add @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_14,axiom,
% 2.95/1.61 cancel_semigroup_add @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 2.95/1.61 comm_monoid_diff @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_15,axiom,
% 2.95/1.61 ab_semigroup_add @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_16,axiom,
% 2.95/1.61 comm_monoid_add @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Complete__Lattices_OSup_17,axiom,
% 2.95/1.61 complete_Sup @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Groups_Osemigroup__add_18,axiom,
% 2.95/1.61 semigroup_add @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Countable_Ocountable_19,axiom,
% 2.95/1.61 countable @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Groups_Omonoid__add_20,axiom,
% 2.95/1.61 monoid_add @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Groups_Ozero_21,axiom,
% 2.95/1.61 zero @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Groups_Oone_22,axiom,
% 2.95/1.61 one @ nat ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_23,axiom,
% 2.95/1.61 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Set_Oset___Groups_Oab__semigroup__add_24,axiom,
% 2.95/1.61 ! [A7: $tType] :
% 2.95/1.61 ( ( ab_semigroup_add @ A7 )
% 2.95/1.61 => ( ab_semigroup_add @ ( set @ A7 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Set_Oset___Groups_Ocomm__monoid__add_25,axiom,
% 2.95/1.61 ! [A7: $tType] :
% 2.95/1.61 ( ( comm_monoid_add @ A7 )
% 2.95/1.61 => ( comm_monoid_add @ ( set @ A7 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Set_Oset___Complete__Lattices_OSup_26,axiom,
% 2.95/1.61 ! [A7: $tType] : ( complete_Sup @ ( set @ A7 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Set_Oset___Groups_Osemigroup__add_27,axiom,
% 2.95/1.61 ! [A7: $tType] :
% 2.95/1.61 ( ( semigroup_add @ A7 )
% 2.95/1.61 => ( semigroup_add @ ( set @ A7 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Set_Oset___Countable_Ocountable_28,axiom,
% 2.95/1.61 ! [A7: $tType] :
% 2.95/1.61 ( ( finite_finite @ A7 )
% 2.95/1.61 => ( countable @ ( set @ A7 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Set_Oset___Groups_Omonoid__add_29,axiom,
% 2.95/1.61 ! [A7: $tType] :
% 2.95/1.61 ( ( monoid_add @ A7 )
% 2.95/1.61 => ( monoid_add @ ( set @ A7 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Set_Oset___Finite__Set_Ofinite_30,axiom,
% 2.95/1.61 ! [A7: $tType] :
% 2.95/1.61 ( ( finite_finite @ A7 )
% 2.95/1.61 => ( finite_finite @ ( set @ A7 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Set_Oset___Groups_Ozero_31,axiom,
% 2.95/1.61 ! [A7: $tType] :
% 2.95/1.61 ( ( zero @ A7 )
% 2.95/1.61 => ( zero @ ( set @ A7 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Set_Oset___Groups_Oone_32,axiom,
% 2.95/1.61 ! [A7: $tType] :
% 2.95/1.61 ( ( one @ A7 )
% 2.95/1.61 => ( one @ ( set @ A7 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_33,axiom,
% 2.95/1.61 comple187826305attice @ $o ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_HOL_Obool___Complete__Lattices_OSup_34,axiom,
% 2.95/1.61 complete_Sup @ $o ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_HOL_Obool___Countable_Ocountable_35,axiom,
% 2.95/1.61 countable @ $o ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_HOL_Obool___Finite__Set_Ofinite_36,axiom,
% 2.95/1.61 finite_finite @ $o ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_37,axiom,
% 2.95/1.61 ordere516151231imp_le @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Ostrict__ordered__ab__semigroup__add_38,axiom,
% 2.95/1.61 strict2144017051up_add @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_39,axiom,
% 2.95/1.61 ordere223160158up_add @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_40,axiom,
% 2.95/1.61 ordere236663937imp_le @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_41,axiom,
% 2.95/1.61 linord1659791738miring @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_42,axiom,
% 2.95/1.61 ordere779506340up_add @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_43,axiom,
% 2.95/1.61 linord219039673up_add @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_44,axiom,
% 2.95/1.61 cancel146912293up_add @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_45,axiom,
% 2.95/1.61 cancel1352612707id_add @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_46,axiom,
% 2.95/1.61 ordered_ab_group_add @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_47,axiom,
% 2.95/1.61 cancel_semigroup_add @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_48,axiom,
% 2.95/1.61 ab_semigroup_add @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_49,axiom,
% 2.95/1.61 comm_monoid_add @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Complete__Lattices_OSup_50,axiom,
% 2.95/1.61 complete_Sup @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Osemigroup__add_51,axiom,
% 2.95/1.61 semigroup_add @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Oab__group__add_52,axiom,
% 2.95/1.61 ab_group_add @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Omonoid__add_53,axiom,
% 2.95/1.61 monoid_add @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Ogroup__add_54,axiom,
% 2.95/1.61 group_add @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Num_Oneg__numeral_55,axiom,
% 2.95/1.61 neg_numeral @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Nat_Oring__char__0_56,axiom,
% 2.95/1.61 ring_char_0 @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Ozero_57,axiom,
% 2.95/1.61 zero @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Real_Oreal___Groups_Oone_58,axiom,
% 2.95/1.61 one @ real ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Sigma__Algebra_Omeasure___Complete__Lattices_Ocomplete__lattice_59,axiom,
% 2.95/1.61 ! [A7: $tType] : ( comple187826305attice @ ( sigma_measure @ A7 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Sigma__Algebra_Omeasure___Complete__Lattices_OSup_60,axiom,
% 2.95/1.61 ! [A7: $tType] : ( complete_Sup @ ( sigma_measure @ A7 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__semigroup__monoid__add__imp__le_61,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( ordere890947078_space @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( ordere516151231imp_le @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ostrict__ordered__ab__semigroup__add_62,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( ordere890947078_space @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( strict2144017051up_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__cancel__ab__semigroup__add_63,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( ordere890947078_space @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( ordere223160158up_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__semigroup__add__imp__le_64,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( ordere890947078_space @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( ordere236663937imp_le @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__semigroup__add_65,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( ordere890947078_space @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( ordere779506340up_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocancel__ab__semigroup__add_66,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( cancel146912293up_add @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( cancel146912293up_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocancel__comm__monoid__add_67,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( cancel1352612707id_add @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( cancel1352612707id_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__group__add_68,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( ordere890947078_space @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( ordered_ab_group_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocancel__semigroup__add_69,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( cancel_semigroup_add @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( cancel_semigroup_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oab__semigroup__add_70,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( ab_semigroup_add @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( ab_semigroup_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocomm__monoid__add_71,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( comm_monoid_add @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( comm_monoid_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Complete__Lattices_OSup_72,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( ordere890947078_space @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( complete_Sup @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Osemigroup__add_73,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( semigroup_add @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( semigroup_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oab__group__add_74,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( ab_group_add @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( ab_group_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Countable_Ocountable_75,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( countable @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( countable @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Omonoid__add_76,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( monoid_add @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( monoid_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_77,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( finite_finite @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( finite_finite @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ogroup__add_78,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( group_add @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( group_add @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Num_Oneg__numeral_79,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( ring_1 @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( neg_numeral @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Nat_Oring__char__0_80,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( ring_char_0 @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( ring_char_0 @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_81,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( zero @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( zero @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oone_82,axiom,
% 2.95/1.61 ! [A7: $tType,A8: $tType] :
% 2.95/1.61 ( ( ( one @ A7 )
% 2.95/1.61 & ( finite_finite @ A8 ) )
% 2.95/1.61 => ( one @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Ostrict__ordered__ab__semigroup__add_83,axiom,
% 2.95/1.61 strict2144017051up_add @ extend1814228343nnreal ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Ocanonically__ordered__monoid__add_84,axiom,
% 2.95/1.61 canoni770627133id_add @ extend1814228343nnreal ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Extended__Nonnegative__Real_Oennreal___Complete__Lattices_Ocomplete__linorder,axiom,
% 2.95/1.61 comple1035589618norder @ extend1814228343nnreal ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Extended__Nonnegative__Real_Oennreal___Complete__Lattices_Ocomplete__lattice_85,axiom,
% 2.95/1.61 comple187826305attice @ extend1814228343nnreal ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Extended__Nonnegative__Real_Oennreal___Rings_Olinordered__nonzero__semiring_86,axiom,
% 2.95/1.61 linord1659791738miring @ extend1814228343nnreal ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Oordered__ab__semigroup__add_87,axiom,
% 2.95/1.61 ordere779506340up_add @ extend1814228343nnreal ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Oab__semigroup__add_88,axiom,
% 2.95/1.61 ab_semigroup_add @ extend1814228343nnreal ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Ocomm__monoid__add_89,axiom,
% 2.95/1.61 comm_monoid_add @ extend1814228343nnreal ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Extended__Nonnegative__Real_Oennreal___Complete__Lattices_OSup_90,axiom,
% 2.95/1.61 complete_Sup @ extend1814228343nnreal ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Osemigroup__add_91,axiom,
% 2.95/1.61 semigroup_add @ extend1814228343nnreal ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Omonoid__add_92,axiom,
% 2.95/1.61 monoid_add @ extend1814228343nnreal ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Ozero_93,axiom,
% 2.95/1.61 zero @ extend1814228343nnreal ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Extended__Nonnegative__Real_Oennreal___Groups_Oone_94,axiom,
% 2.95/1.61 one @ extend1814228343nnreal ).
% 2.95/1.61
% 2.95/1.61 thf(tfree_0,hypothesis,
% 2.95/1.61 finite_finite @ n ).
% 2.95/1.61
% 2.95/1.61 thf(conj_0,conjecture,
% 2.95/1.61 ( ( ^ [N: nat] : ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t2 @ ( f @ N ) ) ) )
% 2.95/1.61 = ( ^ [N: nat] : ( sigma_emeasure @ ( finite_Cartesian_vec @ real @ n ) @ ( complete_completion @ ( finite_Cartesian_vec @ real @ n ) @ ( lebesgue_lborel @ ( finite_Cartesian_vec @ real @ n ) ) ) @ ( t @ ( f @ N ) ) ) ) ) ).
% 2.95/1.61 ------------------------
% 2.95/1.65 % (30874)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/27Mi)
% 2.95/1.65 % (30872)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/183Mi)
% 2.95/1.65 % (30873)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/4Mi)
% 2.95/1.65 % (30873)Instruction limit reached!
% 2.95/1.65 % (30873)------------------------------
% 2.95/1.65 % (30873)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.65 % (30873)Termination reason: Unknown
% 2.95/1.65 % (30873)Termination phase: shuffling
% 2.95/1.65
% 2.95/1.65 % (30873)Memory used [KB]: 1535
% 2.95/1.65 % (30873)Time elapsed: 0.003 s
% 2.95/1.65 % (30873)Instructions burned: 5 (million)
% 2.95/1.65 % (30873)------------------------------
% 2.95/1.65 % (30873)------------------------------
% 2.95/1.65 % (30876)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/2Mi)
% 2.95/1.65 % (30876)Instruction limit reached!
% 2.95/1.65 % (30876)------------------------------
% 2.95/1.65 % (30876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.65 % (30876)Termination reason: Unknown
% 2.95/1.65 % (30876)Termination phase: shuffling
% 2.95/1.65
% 2.95/1.65 % (30876)Memory used [KB]: 1407
% 2.95/1.65 % (30876)Time elapsed: 0.003 s
% 2.95/1.65 % (30876)Instructions burned: 3 (million)
% 2.95/1.65 % (30876)------------------------------
% 2.95/1.65 % (30876)------------------------------
% 2.95/1.65 % (30875)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/2Mi)
% 2.95/1.65 % (30877)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/275Mi)
% 2.95/1.66 % (30878)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/18Mi)
% 2.95/1.66 % (30874)Instruction limit reached!
% 2.95/1.66 % (30874)------------------------------
% 2.95/1.66 % (30874)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.66 % (30874)Termination reason: Unknown
% 2.95/1.66 % (30874)Termination phase: Property scanning
% 2.95/1.66
% 2.95/1.66 % (30874)Memory used [KB]: 1918
% 2.95/1.66 % (30874)Time elapsed: 0.011 s
% 2.95/1.66 % (30874)Instructions burned: 27 (million)
% 2.95/1.66 % (30874)------------------------------
% 2.95/1.66 % (30874)------------------------------
% 2.95/1.66 % (30875)Instruction limit reached!
% 2.95/1.66 % (30875)------------------------------
% 2.95/1.66 % (30875)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.66 % (30875)Termination reason: Unknown
% 2.95/1.66 % (30875)Termination phase: shuffling
% 2.95/1.66
% 2.95/1.66 % (30875)Memory used [KB]: 1407
% 2.95/1.66 % (30875)Time elapsed: 0.004 s
% 2.95/1.66 % (30875)Instructions burned: 2 (million)
% 2.95/1.66 % (30875)------------------------------
% 2.95/1.66 % (30875)------------------------------
% 2.95/1.66 % (30879)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/3Mi)
% 2.95/1.66 % (30879)Instruction limit reached!
% 2.95/1.66 % (30879)------------------------------
% 2.95/1.66 % (30879)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.66 % (30879)Termination reason: Unknown
% 2.95/1.66 % (30879)Termination phase: shuffling
% 2.95/1.66
% 2.95/1.66 % (30879)Memory used [KB]: 1407
% 2.95/1.66 % (30879)Time elapsed: 0.003 s
% 2.95/1.66 % (30879)Instructions burned: 3 (million)
% 2.95/1.66 % (30879)------------------------------
% 2.95/1.66 % (30879)------------------------------
% 2.95/1.66 % (30880)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/37Mi)
% 2.95/1.67 % (30881)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/15Mi)
% 2.95/1.67 % (30878)Instruction limit reached!
% 2.95/1.67 % (30878)------------------------------
% 2.95/1.67 % (30878)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.67 % (30878)Termination reason: Unknown
% 2.95/1.67 % (30878)Termination phase: shuffling
% 2.95/1.67
% 2.95/1.67 % (30878)Memory used [KB]: 1791
% 2.95/1.67 % (30878)Time elapsed: 0.014 s
% 2.95/1.67 % (30878)Instructions burned: 18 (million)
% 2.95/1.67 % (30878)------------------------------
% 2.95/1.67 % (30878)------------------------------
% 2.95/1.67 % (30883)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/1041Mi)
% 2.95/1.67 % (30881)Instruction limit reached!
% 2.95/1.67 % (30881)------------------------------
% 2.95/1.67 % (30881)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.67 % (30881)Termination reason: Unknown
% 2.95/1.67 % (30881)Termination phase: shuffling
% 2.95/1.67
% 2.95/1.67 % (30881)Memory used [KB]: 1791
% 2.95/1.67 % (30881)Time elapsed: 0.007 s
% 2.95/1.67 % (30881)Instructions burned: 17 (million)
% 2.95/1.67 % (30881)------------------------------
% 2.95/1.67 % (30881)------------------------------
% 2.95/1.67 % (30880)Instruction limit reached!
% 2.95/1.67 % (30880)------------------------------
% 2.95/1.67 % (30880)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.67 % (30880)Termination reason: Unknown
% 2.95/1.67 % (30880)Termination phase: shuffling
% 2.95/1.67
% 2.95/1.67 % (30880)Memory used [KB]: 2174
% 2.95/1.67 % (30880)Time elapsed: 0.013 s
% 2.95/1.67 % (30880)Instructions burned: 37 (million)
% 2.95/1.67 % (30880)------------------------------
% 2.95/1.67 % (30880)------------------------------
% 2.95/1.67 % (30882)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/3Mi)
% 2.95/1.68 % (30882)Instruction limit reached!
% 2.95/1.68 % (30882)------------------------------
% 2.95/1.68 % (30882)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.68 % (30882)Termination reason: Unknown
% 2.95/1.68 % (30882)Termination phase: shuffling
% 2.95/1.68
% 2.95/1.68 % (30882)Memory used [KB]: 1407
% 2.95/1.68 % (30882)Time elapsed: 0.004 s
% 2.95/1.68 % (30882)Instructions burned: 3 (million)
% 2.95/1.68 % (30882)------------------------------
% 2.95/1.68 % (30882)------------------------------
% 2.95/1.68 % (30885)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/16Mi)
% 2.95/1.68 % (30886)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/3Mi)
% 2.95/1.68 % (30886)Instruction limit reached!
% 2.95/1.68 % (30886)------------------------------
% 2.95/1.68 % (30886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.68 % (30886)Termination reason: Unknown
% 2.95/1.68 % (30886)Termination phase: shuffling
% 2.95/1.68
% 2.95/1.68 % (30886)Memory used [KB]: 1407
% 2.95/1.68 % (30886)Time elapsed: 0.003 s
% 2.95/1.68 % (30886)Instructions burned: 3 (million)
% 2.95/1.68 % (30886)------------------------------
% 2.95/1.68 % (30886)------------------------------
% 2.95/1.68 % (30884)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/7Mi)
% 2.95/1.69 % (30885)Instruction limit reached!
% 2.95/1.69 % (30885)------------------------------
% 2.95/1.69 % (30885)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.69 % (30885)Termination reason: Unknown
% 2.95/1.69 % (30885)Termination phase: shuffling
% 2.95/1.69
% 2.95/1.69 % (30885)Memory used [KB]: 1791
% 2.95/1.69 % (30885)Time elapsed: 0.007 s
% 2.95/1.69 % (30885)Instructions burned: 17 (million)
% 2.95/1.69 % (30885)------------------------------
% 2.95/1.69 % (30885)------------------------------
% 2.95/1.69 % (30884)Instruction limit reached!
% 2.95/1.69 % (30884)------------------------------
% 2.95/1.69 % (30884)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.69 % (30884)Termination reason: Unknown
% 2.95/1.69 % (30884)Termination phase: shuffling
% 2.95/1.69
% 2.95/1.69 % (30884)Memory used [KB]: 1535
% 2.95/1.69 % (30884)Time elapsed: 0.007 s
% 2.95/1.69 % (30884)Instructions burned: 8 (million)
% 2.95/1.69 % (30884)------------------------------
% 2.95/1.69 % (30884)------------------------------
% 2.95/1.69 % (30887)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/3Mi)
% 2.95/1.69 % (30888)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/7Mi)
% 2.95/1.70 % (30887)Instruction limit reached!
% 2.95/1.70 % (30887)------------------------------
% 2.95/1.70 % (30887)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.70 % (30887)Termination reason: Unknown
% 2.95/1.70 % (30887)Termination phase: shuffling
% 2.95/1.70
% 2.95/1.70 % (30887)Memory used [KB]: 1535
% 2.95/1.70 % (30887)Time elapsed: 0.004 s
% 2.95/1.70 % (30887)Instructions burned: 4 (million)
% 2.95/1.70 % (30887)------------------------------
% 2.95/1.70 % (30887)------------------------------
% 3.01/1.70 % (30889)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/3Mi)
% 3.01/1.70 % (30889)Instruction limit reached!
% 3.01/1.70 % (30889)------------------------------
% 3.01/1.70 % (30889)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.70 % (30889)Termination reason: Unknown
% 3.01/1.70 % (30889)Termination phase: shuffling
% 3.01/1.70
% 3.01/1.70 % (30889)Memory used [KB]: 1407
% 3.01/1.70 % (30889)Time elapsed: 0.002 s
% 3.01/1.70 % (30889)Instructions burned: 3 (million)
% 3.01/1.70 % (30889)------------------------------
% 3.01/1.70 % (30889)------------------------------
% 3.01/1.70 % (30888)Instruction limit reached!
% 3.01/1.70 % (30888)------------------------------
% 3.01/1.70 % (30888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.70 % (30888)Termination reason: Unknown
% 3.01/1.70 % (30888)Termination phase: shuffling
% 3.01/1.70
% 3.01/1.70 % (30888)Memory used [KB]: 1535
% 3.01/1.70 % (30888)Time elapsed: 0.005 s
% 3.01/1.70 % (30888)Instructions burned: 9 (million)
% 3.01/1.70 % (30888)------------------------------
% 3.01/1.70 % (30888)------------------------------
% 3.01/1.70 % (30872)Instruction limit reached!
% 3.01/1.70 % (30872)------------------------------
% 3.01/1.70 % (30872)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.70 % (30872)Termination reason: Unknown
% 3.01/1.70 % (30872)Termination phase: Saturation
% 3.01/1.70
% 3.01/1.70 % (30872)Memory used [KB]: 7291
% 3.01/1.70 % (30872)Time elapsed: 0.058 s
% 3.01/1.70 % (30872)Instructions burned: 183 (million)
% 3.01/1.70 % (30872)------------------------------
% 3.01/1.70 % (30872)------------------------------
% 3.01/1.71 % (30877)First to succeed.
% 3.01/1.71 % (30892)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/710Mi)
% 3.01/1.71 % (30890)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/4Mi)
% 3.01/1.71 % (30893)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/6Mi)
% 3.01/1.71 % (30891)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_30588 for (2999ds/18Mi)
% 3.01/1.71 % (30877)Refutation found. Thanks to Tanya!
% 3.01/1.71 % SZS status Theorem for DTF2THF_30588
% 3.01/1.71 % SZS output start Proof for DTF2THF_30588
% 3.01/1.71 thf(type_def_5, type, set: ($tType) > $tType).
% 3.01/1.71 thf(type_def_7, type, sigma_measure: ($tType) > $tType).
% 3.01/1.71 thf(type_def_8, type, nat: $tType).
% 3.01/1.71 thf(type_def_9, type, int: $tType).
% 3.01/1.71 thf(type_def_10, type, finite_Cartesian_vec: ($tType * $tType) > $tType).
% 3.01/1.71 thf(type_def_11, type, extend1814228343nnreal: $tType).
% 3.01/1.71 thf(type_def_12, type, n: $tType).
% 3.01/1.71 thf(type_def_13, type, real: $tType).
% 3.01/1.71 thf(func_def_0, type, extend1814228343nnreal: $tType).
% 3.01/1.71 thf(func_def_1, type, finite_Cartesian_vec: ($tType * $tType) > $tType).
% 3.01/1.71 thf(func_def_2, type, sigma_measure: ($tType) > $tType).
% 3.01/1.71 thf(func_def_3, type, real: $tType).
% 3.01/1.71 thf(func_def_4, type, set: ($tType) > $tType).
% 3.01/1.71 thf(func_def_5, type, nat: $tType).
% 3.01/1.71 thf(func_def_6, type, int: $tType).
% 3.01/1.71 thf(func_def_7, type, n: $tType).
% 3.01/1.71 thf(func_def_8, type, ring_1: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_9, type, ordere890947078_space: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_10, type, type: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_11, type, one: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_12, type, zero: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_13, type, ring_char_0: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_14, type, neg_numeral: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_15, type, group_add: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_16, type, finite_finite: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_17, type, monoid_add: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_18, type, countable: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_19, type, ab_group_add: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_20, type, semigroup_add: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_21, type, complete_Sup: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_22, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_23, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_24, type, comm_monoid_diff: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_25, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_26, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_27, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_28, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_29, type, linord219039673up_add: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_30, type, ordere779506340up_add: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_31, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_32, type, comple187826305attice: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_33, type, comple1035589618norder: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_34, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_35, type, ordere236663937imp_le: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_36, type, ordere223160158up_add: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_37, type, strict2144017051up_add: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_38, type, ordere516151231imp_le: !>[X0: $tType]:($o)).
% 3.01/1.71 thf(func_def_39, type, complete_Sup_Sup: !>[X0: $tType]:(set @ X0 > X0)).
% 3.01/1.71 thf(func_def_40, type, complete_completion: !>[X0: $tType]:(sigma_measure @ X0 > sigma_measure @ X0)).
% 3.01/1.71 thf(func_def_41, type, counta609264050t_into: !>[X0: $tType]:(set @ X0 > nat > X0)).
% 3.01/1.71 thf(func_def_42, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.01/1.71 thf(func_def_43, type, one_one: !>[X0: $tType]:(X0)).
% 3.01/1.71 thf(func_def_44, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.01/1.71 thf(func_def_45, type, zero_zero: !>[X0: $tType]:(X0)).
% 3.01/1.71 thf(func_def_46, type, inf_inf: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.01/1.71 thf(func_def_47, type, lebesgue_lborel: !>[X0: $tType]:(sigma_measure @ X0)).
% 3.01/1.71 thf(func_def_48, type, measure_null_sets: !>[X0: $tType]:(sigma_measure @ X0 > set @ set @ X0)).
% 3.01/1.71 thf(func_def_49, type, minkow1937162798nt_vec: !>[X0: $tType, X1: $tType]:(finite_Cartesian_vec @ int @ X0 > finite_Cartesian_vec @ X1 @ X0)).
% 3.01/1.71 thf(func_def_50, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.01/1.71 thf(func_def_51, type, top_top: !>[X0: $tType]:(X0)).
% 3.01/1.71 thf(func_def_52, type, sums: !>[X0: $tType]:((nat > X0) > X0 > $o)).
% 3.01/1.71 thf(func_def_53, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.01/1.71 thf(func_def_54, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 3.01/1.71 thf(func_def_55, type, vimage: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X1 > set @ X0)).
% 3.01/1.71 thf(func_def_56, type, sigma_emeasure: !>[X0: $tType]:(sigma_measure @ X0 > set @ X0 > extend1814228343nnreal)).
% 3.01/1.71 thf(func_def_57, type, sigma_sets: !>[X0: $tType]:(sigma_measure @ X0 > set @ set @ X0)).
% 3.01/1.71 thf(func_def_58, type, sigma_space: !>[X0: $tType]:(sigma_measure @ X0 > set @ X0)).
% 3.01/1.71 thf(func_def_59, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.01/1.71 thf(func_def_60, type, r: finite_Cartesian_vec @ int @ n > set @ finite_Cartesian_vec @ real @ n).
% 3.01/1.71 thf(func_def_61, type, s: set @ finite_Cartesian_vec @ real @ n).
% 3.01/1.71 thf(func_def_62, type, t: finite_Cartesian_vec @ int @ n > set @ finite_Cartesian_vec @ real @ n).
% 3.01/1.71 thf(func_def_63, type, t2: finite_Cartesian_vec @ int @ n > set @ finite_Cartesian_vec @ real @ n).
% 3.01/1.71 thf(func_def_64, type, f: nat > finite_Cartesian_vec @ int @ n).
% 3.01/1.71 thf(func_def_76, type, sK0: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X0)).
% 3.01/1.71 thf(func_def_77, type, sK1: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X1)).
% 3.01/1.71 thf(func_def_78, type, sK2: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X0)).
% 3.01/1.71 thf(func_def_79, type, sK3: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > X1)).
% 3.01/1.71 thf(func_def_80, type, sK4: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 3.01/1.71 thf(func_def_81, type, sK5: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > set @ X0) > X0 > X1)).
% 3.01/1.71 thf(func_def_82, type, sK6: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 3.01/1.71 thf(func_def_83, type, sK7: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > X0)).
% 3.01/1.71 thf(func_def_84, type, sK8: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > set @ X0)).
% 3.01/1.71 thf(func_def_85, type, sK9: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > X0)).
% 3.01/1.71 thf(func_def_86, type, sK10: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X1)).
% 3.01/1.71 thf(func_def_87, type, sK11: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X0)).
% 3.01/1.71 thf(func_def_88, type, sK12: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X0)).
% 3.01/1.71 thf(func_def_89, type, sK13: !>[X0: $tType, X1: $tType]:((X0 > X1) > X1)).
% 3.01/1.71 thf(func_def_90, type, sK14: !>[X0: $tType, X1: $tType]:(X1 > (X0 > X1) > X0)).
% 3.01/1.71 thf(func_def_91, type, sK15: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X1)).
% 3.01/1.71 thf(func_def_92, type, sK16: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X0 > X1) > X1)).
% 3.01/1.71 thf(func_def_93, type, sK17: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > X1)).
% 3.01/1.71 thf(func_def_94, type, sK18: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X0)).
% 3.01/1.71 thf(func_def_95, type, sK19: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X1)).
% 3.01/1.71 thf(func_def_96, type, sK20: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X0)).
% 3.01/1.71 thf(func_def_97, type, sK21: !>[X0: $tType]:(X0)).
% 3.01/1.71 thf(func_def_98, type, sK22: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > set @ X0) > X0 > X1)).
% 3.01/1.71 thf(func_def_99, type, sK23: !>[X0: $tType]:(set @ set @ X0 > X0 > set @ X0)).
% 3.01/1.71 thf(func_def_100, type, sK24: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.01/1.71 thf(func_def_101, type, sK25: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X0)).
% 3.01/1.71 thf(func_def_102, type, sK26: !>[X0: $tType, X1: $tType]:((X1 > X0) > X0 > X1)).
% 3.01/1.71 thf(func_def_103, type, sK27: !>[X0: $tType]:(set @ set @ X0 > X0 > set @ X0)).
% 3.01/1.71 thf(func_def_104, type, sK28: !>[X0: $tType, X1: $tType]:((X1 > sigma_measure @ X0) > (X1 > sigma_measure @ X0) > set @ X1 > X1)).
% 3.01/1.71 thf(func_def_105, type, sK29: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > X0 > X1)).
% 3.01/1.71 thf(func_def_106, type, sK30: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 3.01/1.71 thf(func_def_107, type, sK31: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > X0 > X1)).
% 3.01/1.71 thf(func_def_108, type, sK32: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > X0)).
% 3.01/1.71 thf(func_def_109, type, sK33: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > set @ X0)).
% 3.01/1.71 thf(func_def_110, type, sK34: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > X0)).
% 3.01/1.71 thf(func_def_111, type, sK35: !>[X0: $tType]:(set @ X0 > X0)).
% 3.01/1.71 thf(func_def_112, type, sK36: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X1)).
% 3.01/1.71 thf(func_def_113, type, sK37: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X0)).
% 3.01/1.71 thf(func_def_114, type, sK38: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X0)).
% 3.01/1.71 thf(func_def_116, type, ph40: !>[X0: $tType]:(X0)).
% 3.01/1.71 thf(func_def_117, type, sK41: nat).
% 3.01/1.71 thf(f1427,plain,(
% 3.01/1.71 $false),
% 3.01/1.71 inference(subsumption_resolution,[status(thm)],[f1426,f1391])).
% 3.01/1.71 thf(f1391,plain,(
% 3.01/1.71 ( ! [X0 : finite_Cartesian_vec @ int @ n] : (((sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t2 @ X0)) = (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t @ X0)))) )),
% 3.01/1.71 inference(cnf_transformation,[status(thm)],[f1080])).
% 3.01/1.71 thf(f1080,plain,(
% 3.01/1.71 ! [X0 : finite_Cartesian_vec @ int @ n] : ((sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t2 @ X0)) = (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t @ X0)))),
% 3.01/1.71 inference(rectify,[status(thm)],[f1])).
% 3.01/1.71 thf(f1,axiom,(
% 3.01/1.71 ! [X2 : finite_Cartesian_vec @ int @ n] : ((sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t @ X2)) = (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t2 @ X2)))),
% 3.01/1.71 file('/export/starexec/sandbox/tmp/tmp.flmKuaoIzs/DTF2THF_30588.p',fact_0_emeasure__T_H)).
% 3.01/1.71 thf(f1426,plain,(
% 3.01/1.71 ((sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t @ (f @ sK41))) != (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t2 @ (f @ sK41))))),
% 3.01/1.71 inference(beta_eta_normalization,[status(thm)],[f1425])).
% 3.01/1.71 thf(f1425,plain,(
% 3.01/1.71 (((^[Y0 : nat]: (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t @ (f @ Y0)))) @ sK41) != ((^[Y0 : nat]: (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t2 @ (f @ Y0)))) @ sK41))),
% 3.01/1.71 inference(negative_extensionality,[status(thm)],[f1390])).
% 3.01/1.71 thf(f1390,plain,(
% 3.01/1.71 ((^[Y0 : nat]: (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t2 @ (f @ Y0)))) != (^[Y0 : nat]: (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t @ (f @ Y0)))))),
% 3.01/1.71 inference(cnf_transformation,[status(thm)],[f1078])).
% 3.01/1.71 thf(f1078,plain,(
% 3.01/1.71 ((^[Y0 : nat]: (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t2 @ (f @ Y0)))) != (^[Y0 : nat]: (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t @ (f @ Y0)))))),
% 3.01/1.71 inference(flattening,[status(thm)],[f975])).
% 3.01/1.71 thf(f975,plain,(
% 3.01/1.71 ~((^[Y0 : nat]: (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t2 @ (f @ Y0)))) = (^[Y0 : nat]: (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t @ (f @ Y0)))))),
% 3.01/1.71 inference(fool_elimination,[status(thm)],[f974])).
% 3.01/1.71 thf(f974,plain,(
% 3.01/1.71 ~((^[X0 : nat] : (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t @ (f @ X0)))) = (^[X1 : nat] : (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t2 @ (f @ X1)))))),
% 3.01/1.71 inference(rectify,[status(thm)],[f383])).
% 3.01/1.71 thf(f383,negated_conjecture,(
% 3.01/1.71 ~((^[X3 : nat] : (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t @ (f @ X3)))) = (^[X3 : nat] : (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t2 @ (f @ X3)))))),
% 3.01/1.71 inference(negated_conjecture,[status(cth)],[f382])).
% 3.01/1.71 thf(f382,conjecture,(
% 3.01/1.71 ((^[X3 : nat] : (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t @ (f @ X3)))) = (^[X3 : nat] : (sigma_emeasure @ finite_Cartesian_vec @ real @ n @ (complete_completion @ finite_Cartesian_vec @ real @ n @ lebesgue_lborel @ finite_Cartesian_vec @ real @ n) @ (t2 @ (f @ X3)))))),
% 3.01/1.71 file('/export/starexec/sandbox/tmp/tmp.flmKuaoIzs/DTF2THF_30588.p',conj_0)).
% 3.01/1.71 % SZS output end Proof for DTF2THF_30588
% 3.01/1.71 % (30877)------------------------------
% 3.01/1.71 % (30877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.71 % (30877)Termination reason: Refutation
% 3.01/1.71
% 3.01/1.71 % (30877)Memory used [KB]: 6908
% 3.01/1.71 % (30877)Time elapsed: 0.055 s
% 3.01/1.71 % (30877)Instructions burned: 87 (million)
% 3.01/1.71 % (30877)------------------------------
% 3.01/1.71 % (30877)------------------------------
% 3.01/1.71 % (30871)Success in time 0.085 s
%------------------------------------------------------------------------------