%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP041^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n018.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 08:55:04 AM UTC 2026
% Result : Theorem 3.09s 1.74s
% Output : Refutation 3.09s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP041^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n018.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:24:01 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.25/0.47 ---- Original DTF file ---
% 0.25/0.47 thf(spec,logic,$$dhol).
% 0.25/0.47 %------------------------------------------------------------------------------
% 0.25/0.47 % File : ITP041^2 : TPTP v9.2.1. Released v7.5.0.
% 0.25/0.47 % Domain : Interactive Theorem Proving
% 0.25/0.47 % Problem : Sledgehammer Coincidence problem prob_905__7236378_1
% 0.25/0.47 % Version : Especial.
% 0.25/0.47 % English :
% 0.25/0.47
% 0.25/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.25/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.25/0.47 % Source : [Des21]
% 0.25/0.47 % Names : Coincidence/prob_905__7236378_1 [Des21]
% 0.25/0.47
% 0.25/0.47 % Status : Theorem
% 0.25/0.47 % Rating : 0.00 v9.1.0, 0.33 v8.1.0, 0.25 v7.5.0
% 0.25/0.47 % Syntax : Number of formulae : 446 ( 118 unt; 105 typ; 0 def)
% 0.25/0.47 % Number of atoms : 978 ( 201 equ; 0 cnn)
% 0.25/0.47 % Maximal formula atoms : 13 ( 2 avg)
% 0.25/0.47 % Number of connectives : 5664 ( 53 ~; 10 |; 134 &;5084 @)
% 0.25/0.47 % ( 0 <=>; 383 =>; 0 <=; 0 <~>)
% 0.25/0.47 % Maximal formula depth : 35 ( 9 avg)
% 0.25/0.47 % Number of types : 7 ( 6 usr)
% 0.25/0.47 % Number of type conns : 298 ( 298 >; 0 *; 0 +; 0 <<)
% 0.25/0.47 % Number of symbols : 100 ( 99 usr; 13 con; 0-13 aty)
% 0.25/0.47 % Number of variables : 1428 ( 92 ^;1202 !; 7 ?;1428 :)
% 0.25/0.47 % ( 127 !>; 0 ?*; 0 @-; 0 @+)
% 0.25/0.47 % SPC : TH1_THM_EQU_NAR
% 0.25/0.47
% 0.25/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:30:21.538
% 0.25/0.47 %------------------------------------------------------------------------------
% 0.25/0.47 % Could-be-implicit typings (17)
% 0.25/0.47 thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext,type,
% 0.25/0.47 denota1663640101rp_ext: $tType > $tType > $tType > $tType > $tType ).
% 0.25/0.47
% 0.25/0.47 thf(ty_t_Finite__Cartesian__Product_Ovec,type,
% 0.25/0.47 finite_Cartesian_vec: $tType > $tType > $tType ).
% 0.25/0.47
% 0.25/0.47 thf(ty_t_Product__Type_Ounit,type,
% 0.25/0.47 product_unit: $tType ).
% 0.25/0.47
% 0.25/0.47 thf(ty_t_Product__Type_Oprod,type,
% 0.25/0.47 product_prod: $tType > $tType > $tType ).
% 0.25/0.47
% 0.25/0.47 thf(ty_t_Syntax_Oformula,type,
% 0.25/0.47 formula: $tType > $tType > $tType > $tType ).
% 0.25/0.47
% 0.25/0.47 thf(ty_t_Sum__Type_Osum,type,
% 0.25/0.47 sum_sum: $tType > $tType > $tType ).
% 0.25/0.47
% 0.25/0.47 thf(ty_t_Filter_Ofilter,type,
% 0.25/0.47 filter: $tType > $tType ).
% 0.25/0.47
% 0.25/0.47 thf(ty_t_Syntax_Otrm,type,
% 0.25/0.47 trm: $tType > $tType > $tType ).
% 0.25/0.47
% 0.25/0.47 thf(ty_t_Syntax_OODE,type,
% 0.25/0.47 ode: $tType > $tType > $tType ).
% 0.25/0.47
% 0.25/0.47 thf(ty_t_Syntax_Ohp,type,
% 0.25/0.47 hp: $tType > $tType > $tType > $tType ).
% 0.25/0.47
% 0.25/0.47 thf(ty_t_Real_Oreal,type,
% 0.25/0.47 real: $tType ).
% 0.25/0.47
% 0.25/0.47 thf(ty_t_List_Olist,type,
% 0.25/0.47 list: $tType > $tType ).
% 0.25/0.47
% 0.25/0.47 thf(ty_t_Set_Oset,type,
% 0.25/0.47 set: $tType > $tType ).
% 0.25/0.47
% 0.25/0.47 thf(ty_tf_d,type,
% 0.25/0.47 d: $tType ).
% 0.25/0.47
% 0.25/0.47 thf(ty_tf_c,type,
% 0.25/0.47 c: $tType ).
% 0.25/0.47
% 0.25/0.47 thf(ty_tf_b,type,
% 0.25/0.47 b: $tType ).
% 0.25/0.47
% 0.25/0.47 thf(ty_tf_a,type,
% 0.25/0.47 a: $tType ).
% 0.25/0.47
% 0.25/0.47 % Explicit typings (88)
% 0.25/0.47 thf(sy_cl_Cardinality_OCARD__1,type,
% 0.25/0.47 cARD_1:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
% 0.25/0.47 ordere890947078_space:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_HOL_Otype,type,
% 0.25/0.47 type:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Groups_Ozero,type,
% 0.25/0.47 zero:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Groups_Ouminus,type,
% 0.25/0.47 uminus:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.25/0.47 ord:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.25/0.47 order:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Groups_Ogroup__add,type,
% 0.25/0.47 group_add:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Orderings_Ono__bot,type,
% 0.25/0.47 no_bot:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Orderings_Ono__top,type,
% 0.25/0.47 no_top:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.25/0.47 finite_finite:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.25/0.47 linorder:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.25/0.47 preorder:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 0.25/0.47 boolean_algebra:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.25/0.47 semilattice_sup:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.25/0.47 ordered_ab_group_add:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.25/0.47 linord219039673up_add:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.25/0.47 linord1659791738miring:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Topological__Spaces_Otopological__space,type,
% 0.25/0.47 topolo503727757_space:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.25/0.47 canoni770627133id_add:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Lattice__Algebras_Olattice__ab__group__add,type,
% 0.25/0.47 lattic1601792062up_add:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
% 0.25/0.47 real_V55928688vector:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd,type,
% 0.25/0.47 euclid1678468529ng_gcd:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
% 0.25/0.47 unique1598680935umeral:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring,type,
% 0.25/0.47 euclid1155270486miring:
% 0.25/0.47 !>[A: $tType] : $o ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.25/0.47 bNF_Ca1785829860lChain:
% 0.25/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__fml,type,
% 0.25/0.47 coinci1993344360de_fml:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__hp,type,
% 0.25/0.47 coinci32832645ide_hp:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__hp_H,type,
% 0.25/0.47 coinci554967470ide_hp:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Oode__sem__equiv,type,
% 0.25/0.47 coinci1495174342_equiv:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Denotational__Semantics_OIagree,type,
% 0.25/0.47 denotational_Iagree:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Denotational__Semantics_OODE__sem,type,
% 0.25/0.47 denotational_ODE_sem:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( finite_Cartesian_vec @ real @ C ) > ( finite_Cartesian_vec @ real @ C ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Denotational__Semantics_OODE__vars,type,
% 0.25/0.47 denota811733865E_vars:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( set @ C ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Denotational__Semantics_OVSagree,type,
% 0.25/0.47 denotational_VSagree:
% 0.25/0.47 !>[C: $tType] : ( ( finite_Cartesian_vec @ real @ C ) > ( finite_Cartesian_vec @ real @ C ) > ( set @ C ) > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Denotational__Semantics_OVagree,type,
% 0.25/0.47 denotational_Vagree:
% 0.25/0.47 !>[C: $tType] : ( ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( set @ ( sum_sum @ C @ C ) ) > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Denotational__Semantics_Ofml__sem,type,
% 0.25/0.47 denotational_fml_sem:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( formula @ A @ B @ C ) > ( set @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Denotational__Semantics_Omk__v,type,
% 0.25/0.47 denotational_mk_v:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( finite_Cartesian_vec @ real @ C ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Denotational__Semantics_Omk__xode,type,
% 0.25/0.47 denotational_mk_xode:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( finite_Cartesian_vec @ real @ C ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Denotational__Semantics_OsemBV,type,
% 0.25/0.47 denotational_semBV:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Deriv_Ohas__vector__derivative,type,
% 0.25/0.47 has_ve2132708402vative:
% 0.25/0.47 !>[B: $tType] : ( ( real > B ) > B > ( filter @ real ) > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
% 0.25/0.47 unique455577585es_aux:
% 0.25/0.47 !>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.25/0.47 uminus_uminus:
% 0.25/0.47 !>[A: $tType] : ( A > A ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.25/0.47 zero_zero:
% 0.25/0.47 !>[A: $tType] : A ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_HOL_OThe,type,
% 0.25/0.47 the:
% 0.25/0.47 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Ids_Oids,type,
% 0.25/0.47 ids:
% 0.25/0.47 !>[Sz: $tType,Sf: $tType,Sc: $tType] : ( Sz > Sz > Sz > Sf > Sf > Sf > Sc > Sc > Sc > Sc > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Initial__Value__Problem_Osolves__ode,type,
% 0.25/0.47 initia1685620758es_ode:
% 0.25/0.47 !>[A: $tType] : ( ( real > A ) > ( real > A > A ) > ( set @ real ) > ( set @ A ) > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.25/0.47 sup_sup:
% 0.25/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.25/0.47 ord_less_eq:
% 0.25/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Product__Type_OPair,type,
% 0.25/0.47 product_Pair:
% 0.25/0.47 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.25/0.47 produc2004651681e_prod:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.25/0.47 product_rec_prod:
% 0.25/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
% 0.25/0.47 product_rec_set_prod:
% 0.25/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
% 0.25/0.47 product_rec_set_unit:
% 0.25/0.47 !>[T: $tType] : ( T > product_unit > T > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
% 0.25/0.47 product_rec_unit:
% 0.25/0.47 !>[T: $tType] : ( T > product_unit > T ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Set_OCollect,type,
% 0.25/0.47 collect:
% 0.25/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Set_Oimage,type,
% 0.25/0.47 image:
% 0.25/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
% 0.25/0.47 set_ord_atLeast:
% 0.25/0.47 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
% 0.25/0.47 set_or331188842AtMost:
% 0.25/0.47 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Set__Interval_Oord__class_OatMost,type,
% 0.25/0.47 set_ord_atMost:
% 0.25/0.47 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Static__Semantics_OBVO,type,
% 0.25/0.47 static_BVO:
% 0.25/0.47 !>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Static__Semantics_OFVF,type,
% 0.25/0.47 static_FVF:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Static__Semantics_OFVO,type,
% 0.25/0.47 static_FVO:
% 0.25/0.47 !>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > ( set @ C ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Static__Semantics_OSIGF,type,
% 0.25/0.47 static_SIGF:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Sum__Type_OInl,type,
% 0.25/0.47 sum_Inl:
% 0.25/0.47 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Sum__Type_OInr,type,
% 0.25/0.47 sum_Inr:
% 0.25/0.47 !>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Syntax_OEquiv,type,
% 0.25/0.47 equiv:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Syntax_OImplies,type,
% 0.25/0.47 implies:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Syntax_OODE__dom,type,
% 0.25/0.47 oDE_dom:
% 0.25/0.47 !>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > ( set @ C ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Syntax_Ofsafe,type,
% 0.25/0.47 fsafe:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Syntax_Ohp_OEvolveODE,type,
% 0.25/0.47 evolveODE:
% 0.25/0.47 !>[A: $tType,C: $tType,B: $tType] : ( ( ode @ A @ C ) > ( formula @ A @ B @ C ) > ( hp @ A @ B @ C ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Syntax_Ohpsafe,type,
% 0.25/0.47 hpsafe:
% 0.25/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Syntax_Oosafe,type,
% 0.25/0.47 osafe:
% 0.25/0.47 !>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
% 0.25/0.47 topolo507301023within:
% 0.25/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( filter @ A ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_c_member,type,
% 0.25/0.47 member:
% 0.25/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_v_I____,type,
% 0.25/0.47 i: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
% 0.25/0.47
% 0.25/0.47 thf(sy_v_J____,type,
% 0.25/0.47 j: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
% 0.25/0.47
% 0.25/0.47 thf(sy_v_ODE____,type,
% 0.25/0.47 ode2: ode @ a @ c ).
% 0.25/0.47
% 0.25/0.47 thf(sy_v_P____,type,
% 0.25/0.47 p: formula @ a @ b @ c ).
% 0.25/0.47
% 0.25/0.47 thf(sy_v_V____,type,
% 0.25/0.47 v: set @ ( sum_sum @ c @ c ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_v_a____,type,
% 0.25/0.47 a2: finite_Cartesian_vec @ real @ c ).
% 0.25/0.47
% 0.25/0.47 thf(sy_v_aa____,type,
% 0.25/0.47 aa: finite_Cartesian_vec @ real @ c ).
% 0.25/0.47
% 0.25/0.47 thf(sy_v_ab____,type,
% 0.25/0.47 ab: finite_Cartesian_vec @ real @ c ).
% 0.25/0.47
% 0.25/0.47 thf(sy_v_b____,type,
% 0.25/0.47 b2: finite_Cartesian_vec @ real @ c ).
% 0.25/0.47
% 0.25/0.47 thf(sy_v_ba____,type,
% 0.25/0.47 ba: finite_Cartesian_vec @ real @ c ).
% 0.25/0.47
% 0.25/0.47 thf(sy_v_bb____,type,
% 0.25/0.47 bb: finite_Cartesian_vec @ real @ c ).
% 0.25/0.47
% 0.25/0.47 thf(sy_v_s____,type,
% 0.25/0.47 s: real ).
% 0.25/0.47
% 0.25/0.47 thf(sy_v_sol____,type,
% 0.25/0.47 sol: real > ( finite_Cartesian_vec @ real @ c ) ).
% 0.25/0.47
% 0.25/0.47 thf(sy_v_t____,type,
% 0.25/0.47 t: real ).
% 0.25/0.47
% 0.25/0.47 % Relevant facts (255)
% 0.25/0.47 thf(fact_0_osafe,axiom,
% 0.25/0.47 osafe @ a @ c @ ode2 ).
% 0.25/0.47
% 0.25/0.47 % osafe
% 0.25/0.47 thf(fact_1__092_060open_062Vagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Iaa_M_Aba_J_A_I_N_AsemBV_AJ_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Imk__xode_AJ_AODE_A_Isol_As_J_J_A_IsemBV_AJ_AODE_J_092_060close_062,axiom,
% 0.25/0.47 ( ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ j @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( sol @ s ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( uminus_uminus @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ j @ ode2 ) ) )
% 0.25/0.47 & ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ j @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( sol @ s ) ) @ ( denotational_mk_xode @ a @ b @ c @ j @ ode2 @ ( sol @ s ) ) @ ( denotational_semBV @ a @ b @ c @ j @ ode2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % \<open>Vagree (mk_v J ODE (aa, ba) (sol s)) (aa, ba) (- semBV J ODE) \<and> Vagree (mk_v J ODE (aa, ba) (sol s)) (mk_xode J ODE (sol s)) (semBV J ODE)\<close>
% 0.25/0.47 thf(fact_2__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_Asa_J_J_A_Iaa_M_Aba_J_A_I_N_AsemBV_AJ_AODE_J_092_060close_062,axiom,
% 0.25/0.47 ! [S: real] : ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ j @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( sol @ S ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( uminus_uminus @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ j @ ode2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % \<open>\<And>sa. Vagree (mk_v J ODE (aa, ba) (sol sa)) (aa, ba) (- semBV J ODE)\<close>
% 0.25/0.47 thf(fact_3_mk__v__agree,axiom,
% 0.25/0.47 ! [C: $tType,A: $tType,B: $tType] :
% 0.25/0.47 ( ( ( finite_finite @ B )
% 0.25/0.47 & ( finite_finite @ A )
% 0.25/0.47 & ( finite_finite @ C ) )
% 0.25/0.47 => ! [I: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A,Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Sol: finite_Cartesian_vec @ real @ A] :
% 0.25/0.47 ( ( denotational_Vagree @ A @ ( denotational_mk_v @ B @ C @ A @ I @ ODE @ Nu @ Sol ) @ Nu @ ( uminus_uminus @ ( set @ ( sum_sum @ A @ A ) ) @ ( denotational_semBV @ B @ C @ A @ I @ ODE ) ) )
% 0.25/0.47 & ( denotational_Vagree @ A @ ( denotational_mk_v @ B @ C @ A @ I @ ODE @ Nu @ Sol ) @ ( denotational_mk_xode @ B @ C @ A @ I @ ODE @ Sol ) @ ( denotational_semBV @ B @ C @ A @ I @ ODE ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % mk_v_agree
% 0.25/0.47 thf(fact_4_prod_Oinject,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 0.25/0.47 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 0.25/0.47 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.25/0.47 = ( ( X1 = Y1 )
% 0.25/0.47 & ( X2 = Y2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.inject
% 0.25/0.47 thf(fact_5_old_Oprod_Oinject,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.25/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.25/0.47 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.25/0.47 = ( ( A2 = A3 )
% 0.25/0.47 & ( B2 = B3 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % old.prod.inject
% 0.25/0.47 thf(fact_6_mk__v__exists,axiom,
% 0.25/0.47 ! [C: $tType,A: $tType,B: $tType] :
% 0.25/0.47 ( ( ( finite_finite @ B )
% 0.25/0.47 & ( finite_finite @ A )
% 0.25/0.47 & ( finite_finite @ C ) )
% 0.25/0.47 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A,Sol: finite_Cartesian_vec @ real @ A] :
% 0.25/0.47 ? [Omega: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
% 0.25/0.47 ( ( denotational_Vagree @ A @ Omega @ Nu @ ( uminus_uminus @ ( set @ ( sum_sum @ A @ A ) ) @ ( denotational_semBV @ B @ C @ A @ I @ ODE ) ) )
% 0.25/0.47 & ( denotational_Vagree @ A @ Omega @ ( denotational_mk_xode @ B @ C @ A @ I @ ODE @ Sol ) @ ( denotational_semBV @ B @ C @ A @ I @ ODE ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % mk_v_exists
% 0.25/0.47 thf(fact_7_VA,axiom,
% 0.25/0.47 denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ v ).
% 0.25/0.47
% 0.25/0.47 % VA
% 0.25/0.47 thf(fact_8__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AI_AODE_A_Ia_M_Ab_J_A_Isol_Asa_J_J_A_Imk__xode_AI_AODE_A_Isol_Asa_J_J_A_IsemBV_AI_AODE_J_092_060close_062,axiom,
% 0.25/0.47 ! [S: real] : ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ i @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( sol @ S ) ) @ ( denotational_mk_xode @ a @ b @ c @ i @ ode2 @ ( sol @ S ) ) @ ( denotational_semBV @ a @ b @ c @ i @ ode2 ) ) ).
% 0.25/0.47
% 0.25/0.47 % \<open>\<And>sa. Vagree (mk_v I ODE (a, b) (sol sa)) (mk_xode I ODE (sol sa)) (semBV I ODE)\<close>
% 0.25/0.47 thf(fact_9_VAOV,axiom,
% 0.25/0.47 denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( static_BVO @ a @ c @ ode2 ) ).
% 0.25/0.47
% 0.25/0.47 % VAOV
% 0.25/0.47 thf(fact_10_agree__comm,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( finite_finite @ A )
% 0.25/0.47 => ! [A4: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),B4: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),V: set @ ( sum_sum @ A @ A )] :
% 0.25/0.47 ( ( denotational_Vagree @ A @ A4 @ B4 @ V )
% 0.25/0.47 => ( denotational_Vagree @ A @ B4 @ A4 @ V ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % agree_comm
% 0.25/0.47 thf(fact_11_agree__refl,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( finite_finite @ A )
% 0.25/0.47 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A4: set @ ( sum_sum @ A @ A )] : ( denotational_Vagree @ A @ Nu @ Nu @ A4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % agree_refl
% 0.25/0.47 thf(fact_12__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AI_AODE_A_Ia_M_Ab_J_A_Isol_Asa_J_J_A_Ia_M_Ab_J_A_I_N_AsemBV_AI_AODE_J_092_060close_062,axiom,
% 0.25/0.47 ! [S: real] : ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ i @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( sol @ S ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( uminus_uminus @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ i @ ode2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % \<open>\<And>sa. Vagree (mk_v I ODE (a, b) (sol sa)) (a, b) (- semBV I ODE)\<close>
% 0.25/0.47 thf(fact_13_surj__pair,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 0.25/0.47 ? [X: A,Y: B] :
% 0.25/0.47 ( P
% 0.25/0.47 = ( product_Pair @ A @ B @ X @ Y ) ) ).
% 0.25/0.47
% 0.25/0.47 % surj_pair
% 0.25/0.47 thf(fact_14_prod__cases,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 0.25/0.47 ( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.25/0.47 => ( P2 @ P ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_cases
% 0.25/0.47 thf(fact_15_OVsub,axiom,
% 0.25/0.47 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_BVO @ a @ c @ ode2 ) @ v ).
% 0.25/0.47
% 0.25/0.47 % OVsub
% 0.25/0.47 thf(fact_16_veq,axiom,
% 0.25/0.47 ( ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb )
% 0.25/0.47 = ( denotational_mk_v @ a @ b @ c @ i @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( sol @ t ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % veq
% 0.25/0.47 thf(fact_17_old_Oprod_Oinducts,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.25/0.47 ( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.25/0.47 => ( P2 @ Prod ) ) ).
% 0.25/0.47
% 0.25/0.47 % old.prod.inducts
% 0.25/0.47 thf(fact_18_old_Oprod_Oexhaust,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 0.25/0.47 ~ ! [A5: A,B5: B] :
% 0.25/0.47 ( Y3
% 0.25/0.47 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% 0.25/0.47
% 0.25/0.47 % old.prod.exhaust
% 0.25/0.47 thf(fact_19_prod__induct7,axiom,
% 0.25/0.47 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.25/0.47 ( ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 0.25/0.47 => ( P2 @ X3 ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_induct7
% 0.25/0.47 thf(fact_20_prod__induct6,axiom,
% 0.25/0.47 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.25/0.47 ( ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 0.25/0.47 => ( P2 @ X3 ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_induct6
% 0.25/0.47 thf(fact_21_prod__induct5,axiom,
% 0.25/0.47 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.25/0.47 ( ! [A5: A,B5: B,C2: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.25/0.47 => ( P2 @ X3 ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_induct5
% 0.25/0.47 thf(fact_22_prod__induct4,axiom,
% 0.25/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.25/0.47 ( ! [A5: A,B5: B,C2: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 0.25/0.47 => ( P2 @ X3 ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_induct4
% 0.25/0.47 thf(fact_23_prod__induct3,axiom,
% 0.25/0.47 ! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.25/0.47 ( ! [A5: A,B5: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) )
% 0.25/0.47 => ( P2 @ X3 ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_induct3
% 0.25/0.47 thf(fact_24_prod__cases7,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.25/0.47 ~ ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
% 0.25/0.47 ( Y3
% 0.25/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_cases7
% 0.25/0.47 thf(fact_25_prod__cases6,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.25/0.47 ~ ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F] :
% 0.25/0.47 ( Y3
% 0.25/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_cases6
% 0.25/0.47 thf(fact_26_prod__cases5,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.25/0.47 ~ ! [A5: A,B5: B,C2: C,D2: D,E2: E] :
% 0.25/0.47 ( Y3
% 0.25/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_cases5
% 0.25/0.47 thf(fact_27_prod__cases4,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.25/0.47 ~ ! [A5: A,B5: B,C2: C,D2: D] :
% 0.25/0.47 ( Y3
% 0.25/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_cases4
% 0.25/0.47 thf(fact_28_prod__cases3,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.25/0.47 ~ ! [A5: A,B5: B,C2: C] :
% 0.25/0.47 ( Y3
% 0.25/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_cases3
% 0.25/0.47 thf(fact_29_Pair__inject,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.25/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.25/0.47 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.25/0.47 => ~ ( ( A2 = A3 )
% 0.25/0.47 => ( B2 != B3 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Pair_inject
% 0.25/0.47 thf(fact_30_OVsub_H_H,axiom,
% 0.25/0.47 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ i @ ode2 ) @ v ).
% 0.25/0.47
% 0.25/0.47 % OVsub''
% 0.25/0.47 thf(fact_31_semBVsub,axiom,
% 0.25/0.47 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ i @ ode2 ) @ ( static_BVO @ a @ c @ ode2 ) ).
% 0.25/0.47
% 0.25/0.47 % semBVsub
% 0.25/0.47 thf(fact_32_uminus__Pair,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( ( uminus @ B )
% 0.25/0.47 & ( uminus @ A ) )
% 0.25/0.47 => ! [A2: A,B2: B] :
% 0.25/0.47 ( ( uminus_uminus @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.25/0.47 = ( product_Pair @ A @ B @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ B @ B2 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % uminus_Pair
% 0.25/0.47 thf(fact_33_old_Oprod_Orec,axiom,
% 0.25/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.25/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.25/0.47 = ( F1 @ A2 @ B2 ) ) ).
% 0.25/0.47
% 0.25/0.47 % old.prod.rec
% 0.25/0.47 thf(fact_34_ComplI,axiom,
% 0.25/0.47 ! [A: $tType,C3: A,A4: set @ A] :
% 0.25/0.47 ( ~ ( member @ A @ C3 @ A4 )
% 0.25/0.47 => ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % ComplI
% 0.25/0.47 thf(fact_35_Compl__iff,axiom,
% 0.25/0.47 ! [A: $tType,C3: A,A4: set @ A] :
% 0.25/0.47 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
% 0.25/0.47 = ( ~ ( member @ A @ C3 @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Compl_iff
% 0.25/0.47 thf(fact_36_Compl__eq__Compl__iff,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.25/0.47 ( ( ( uminus_uminus @ ( set @ A ) @ A4 )
% 0.25/0.47 = ( uminus_uminus @ ( set @ A ) @ B4 ) )
% 0.25/0.47 = ( A4 = B4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % Compl_eq_Compl_iff
% 0.25/0.47 thf(fact_37_verit__minus__simplify_I4_J,axiom,
% 0.25/0.47 ! [B: $tType] :
% 0.25/0.47 ( ( group_add @ B )
% 0.25/0.47 => ! [B2: B] :
% 0.25/0.47 ( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
% 0.25/0.47 = B2 ) ) ).
% 0.25/0.47
% 0.25/0.47 % verit_minus_simplify(4)
% 0.25/0.47 thf(fact_38_uminus__apply,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( uminus @ B )
% 0.25/0.47 => ( ( uminus_uminus @ ( A > B ) )
% 0.25/0.47 = ( ^ [A6: A > B,X4: A] : ( uminus_uminus @ B @ ( A6 @ X4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % uminus_apply
% 0.25/0.47 thf(fact_39_add_Oinverse__inverse,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( group_add @ A )
% 0.25/0.47 => ! [A2: A] :
% 0.25/0.47 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 0.25/0.47 = A2 ) ) ).
% 0.25/0.47
% 0.25/0.47 % add.inverse_inverse
% 0.25/0.47 thf(fact_40_subset__antisym,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.25/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.25/0.47 => ( A4 = B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % subset_antisym
% 0.25/0.47 thf(fact_41_subsetI,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.25/0.47 ( ! [X: A] :
% 0.25/0.47 ( ( member @ A @ X @ A4 )
% 0.25/0.47 => ( member @ A @ X @ B4 ) )
% 0.25/0.47 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % subsetI
% 0.25/0.47 thf(fact_42_compl__eq__compl__iff,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( boolean_algebra @ A )
% 0.25/0.47 => ! [X3: A,Y3: A] :
% 0.25/0.47 ( ( ( uminus_uminus @ A @ X3 )
% 0.25/0.47 = ( uminus_uminus @ A @ Y3 ) )
% 0.25/0.47 = ( X3 = Y3 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % compl_eq_compl_iff
% 0.25/0.47 thf(fact_43_double__compl,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( boolean_algebra @ A )
% 0.25/0.47 => ! [X3: A] :
% 0.25/0.47 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X3 ) )
% 0.25/0.47 = X3 ) ) ).
% 0.25/0.47
% 0.25/0.47 % double_compl
% 0.25/0.47 thf(fact_44_neg__equal__iff__equal,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( group_add @ A )
% 0.25/0.47 => ! [A2: A,B2: A] :
% 0.25/0.47 ( ( ( uminus_uminus @ A @ A2 )
% 0.25/0.47 = ( uminus_uminus @ A @ B2 ) )
% 0.25/0.47 = ( A2 = B2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % neg_equal_iff_equal
% 0.25/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.25/0.47 ! [A: $tType,A2: A,P2: A > $o] :
% 0.25/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 0.25/0.47 = ( P2 @ A2 ) ) ).
% 0.25/0.47
% 0.25/0.47 % mem_Collect_eq
% 0.25/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A] :
% 0.25/0.47 ( ( collect @ A
% 0.25/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
% 0.25/0.47 = A4 ) ).
% 0.25/0.47
% 0.25/0.47 % Collect_mem_eq
% 0.25/0.47 thf(fact_47_Collect__cong,axiom,
% 0.25/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.25/0.47 ( ! [X: A] :
% 0.25/0.47 ( ( P2 @ X )
% 0.25/0.47 = ( Q @ X ) )
% 0.25/0.47 => ( ( collect @ A @ P2 )
% 0.25/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Collect_cong
% 0.25/0.47 thf(fact_48_ext,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
% 0.25/0.47 ( ! [X: A] :
% 0.25/0.47 ( ( F3 @ X )
% 0.25/0.47 = ( G3 @ X ) )
% 0.25/0.47 => ( F3 = G3 ) ) ).
% 0.25/0.47
% 0.25/0.47 % ext
% 0.25/0.47 thf(fact_49_compl__le__compl__iff,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( boolean_algebra @ A )
% 0.25/0.47 => ! [X3: A,Y3: A] :
% 0.25/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y3 ) )
% 0.25/0.47 = ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % compl_le_compl_iff
% 0.25/0.47 thf(fact_50_neg__le__iff__le,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( ordered_ab_group_add @ A )
% 0.25/0.47 => ! [B2: A,A2: A] :
% 0.25/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 0.25/0.47 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % neg_le_iff_le
% 0.25/0.47 thf(fact_51_Compl__subset__Compl__iff,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
% 0.25/0.47 = ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % Compl_subset_Compl_iff
% 0.25/0.47 thf(fact_52_Compl__anti__mono,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.25/0.47 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Compl_anti_mono
% 0.25/0.47 thf(fact_53_verit__la__disequality,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( linorder @ A )
% 0.25/0.47 => ! [A2: A,B2: A] :
% 0.25/0.47 ( ( A2 = B2 )
% 0.25/0.47 | ~ ( ord_less_eq @ A @ A2 @ B2 )
% 0.25/0.47 | ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % verit_la_disequality
% 0.25/0.47 thf(fact_54_Collect__mono__iff,axiom,
% 0.25/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
% 0.25/0.47 = ( ! [X4: A] :
% 0.25/0.47 ( ( P2 @ X4 )
% 0.25/0.47 => ( Q @ X4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Collect_mono_iff
% 0.25/0.47 thf(fact_55_set__eq__subset,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( ^ [Y4: set @ A,Z: set @ A] : ( Y4 = Z ) )
% 0.25/0.47 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.25/0.47 & ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % set_eq_subset
% 0.25/0.47 thf(fact_56_subset__trans,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A,B4: set @ A,C4: set @ A] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.25/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
% 0.25/0.47 => ( ord_less_eq @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % subset_trans
% 0.25/0.47 thf(fact_57_Collect__mono,axiom,
% 0.25/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.25/0.47 ( ! [X: A] :
% 0.25/0.47 ( ( P2 @ X )
% 0.25/0.47 => ( Q @ X ) )
% 0.25/0.47 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Collect_mono
% 0.25/0.47 thf(fact_58_subset__refl,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% 0.25/0.47
% 0.25/0.47 % subset_refl
% 0.25/0.47 thf(fact_59_subset__iff,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.25/0.47 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.25/0.47 ! [T2: A] :
% 0.25/0.47 ( ( member @ A @ T2 @ A6 )
% 0.25/0.47 => ( member @ A @ T2 @ B6 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % subset_iff
% 0.25/0.47 thf(fact_60_equalityD2,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.25/0.47 ( ( A4 = B4 )
% 0.25/0.47 => ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % equalityD2
% 0.25/0.47 thf(fact_61_equalityD1,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.25/0.47 ( ( A4 = B4 )
% 0.25/0.47 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % equalityD1
% 0.25/0.47 thf(fact_62_subset__eq,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.25/0.47 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.25/0.47 ! [X4: A] :
% 0.25/0.47 ( ( member @ A @ X4 @ A6 )
% 0.25/0.47 => ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % subset_eq
% 0.25/0.47 thf(fact_63_equalityE,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.25/0.47 ( ( A4 = B4 )
% 0.25/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.25/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % equalityE
% 0.25/0.47 thf(fact_64_subsetD,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A,B4: set @ A,C3: A] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.25/0.47 => ( ( member @ A @ C3 @ A4 )
% 0.25/0.47 => ( member @ A @ C3 @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % subsetD
% 0.25/0.47 thf(fact_65_in__mono,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A,B4: set @ A,X3: A] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.25/0.47 => ( ( member @ A @ X3 @ A4 )
% 0.25/0.47 => ( member @ A @ X3 @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % in_mono
% 0.25/0.47 thf(fact_66_le__imp__neg__le,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( ordered_ab_group_add @ A )
% 0.25/0.47 => ! [A2: A,B2: A] :
% 0.25/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.25/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % le_imp_neg_le
% 0.25/0.47 thf(fact_67_minus__le__iff,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( ordered_ab_group_add @ A )
% 0.25/0.47 => ! [A2: A,B2: A] :
% 0.25/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 0.25/0.47 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % minus_le_iff
% 0.25/0.47 thf(fact_68_le__minus__iff,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( ordered_ab_group_add @ A )
% 0.25/0.47 => ! [A2: A,B2: A] :
% 0.25/0.47 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 0.25/0.47 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % le_minus_iff
% 0.25/0.47 thf(fact_69_compl__le__swap2,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( boolean_algebra @ A )
% 0.25/0.47 => ! [Y3: A,X3: A] :
% 0.25/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X3 )
% 0.25/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y3 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % compl_le_swap2
% 0.25/0.47 thf(fact_70_compl__le__swap1,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( boolean_algebra @ A )
% 0.25/0.47 => ! [Y3: A,X3: A] :
% 0.25/0.47 ( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X3 ) )
% 0.25/0.47 => ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % compl_le_swap1
% 0.25/0.47 thf(fact_71_compl__mono,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( boolean_algebra @ A )
% 0.25/0.47 => ! [X3: A,Y3: A] :
% 0.25/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.25/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % compl_mono
% 0.25/0.47 thf(fact_72_agree__sub,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( finite_finite @ A )
% 0.25/0.47 => ! [A4: set @ ( sum_sum @ A @ A ),B4: set @ ( sum_sum @ A @ A ),Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Omega2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ A4 @ B4 )
% 0.25/0.47 => ( ( denotational_Vagree @ A @ Nu @ Omega2 @ B4 )
% 0.25/0.47 => ( denotational_Vagree @ A @ Nu @ Omega2 @ A4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % agree_sub
% 0.25/0.47 thf(fact_73_agree__supset,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( finite_finite @ A )
% 0.25/0.47 => ! [B4: set @ ( sum_sum @ A @ A ),A4: set @ ( sum_sum @ A @ A ),Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ B4 @ A4 )
% 0.25/0.48 => ( ( denotational_Vagree @ A @ Nu @ Nu2 @ A4 )
% 0.25/0.48 => ( denotational_Vagree @ A @ Nu @ Nu2 @ B4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % agree_supset
% 0.25/0.48 thf(fact_74_minus__equation__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( group_add @ A )
% 0.25/0.48 => ! [A2: A,B2: A] :
% 0.25/0.48 ( ( ( uminus_uminus @ A @ A2 )
% 0.25/0.48 = B2 )
% 0.25/0.48 = ( ( uminus_uminus @ A @ B2 )
% 0.25/0.48 = A2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % minus_equation_iff
% 0.25/0.48 thf(fact_75_equation__minus__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( group_add @ A )
% 0.25/0.48 => ! [A2: A,B2: A] :
% 0.25/0.48 ( ( A2
% 0.25/0.48 = ( uminus_uminus @ A @ B2 ) )
% 0.25/0.48 = ( B2
% 0.25/0.48 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % equation_minus_iff
% 0.25/0.48 thf(fact_76_fun__Compl__def,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( uminus @ B )
% 0.25/0.48 => ( ( uminus_uminus @ ( A > B ) )
% 0.25/0.48 = ( ^ [A6: A > B,X4: A] : ( uminus_uminus @ B @ ( A6 @ X4 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % fun_Compl_def
% 0.25/0.48 thf(fact_77_verit__negate__coefficient_I3_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ordered_ab_group_add @ A )
% 0.25/0.48 => ! [A2: A,B2: A] :
% 0.25/0.48 ( ( A2 = B2 )
% 0.25/0.48 => ( ( uminus_uminus @ A @ A2 )
% 0.25/0.48 = ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % verit_negate_coefficient(3)
% 0.25/0.48 thf(fact_78_double__complement,axiom,
% 0.25/0.48 ! [A: $tType,A4: set @ A] :
% 0.25/0.48 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
% 0.25/0.48 = A4 ) ).
% 0.25/0.48
% 0.25/0.48 % double_complement
% 0.25/0.48 thf(fact_79_ComplD,axiom,
% 0.25/0.48 ! [A: $tType,C3: A,A4: set @ A] :
% 0.25/0.48 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
% 0.25/0.48 => ~ ( member @ A @ C3 @ A4 ) ) ).
% 0.25/0.48
% 0.25/0.48 % ComplD
% 0.25/0.48 thf(fact_80_euclid__ext__aux_Ocases,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( euclid1678468529ng_gcd @ A )
% 0.25/0.48 => ! [X3: product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) )] :
% 0.25/0.48 ~ ! [S2: A,S3: A,T3: A,T4: A,R: A,R2: A] :
% 0.25/0.48 ( X3
% 0.25/0.48 != ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) ) @ S2 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) @ S3 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) @ T3 @ ( product_Pair @ A @ ( product_prod @ A @ A ) @ T4 @ ( product_Pair @ A @ A @ R @ R2 ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % euclid_ext_aux.cases
% 0.25/0.48 thf(fact_81_Pair__le,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType] :
% 0.25/0.48 ( ( ( ord @ B )
% 0.25/0.48 & ( ord @ A ) )
% 0.25/0.48 => ! [A2: A,B2: B,C3: A,D3: B] :
% 0.25/0.48 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C3 @ D3 ) )
% 0.25/0.48 = ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.25/0.48 & ( ord_less_eq @ B @ B2 @ D3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Pair_le
% 0.25/0.48 thf(fact_82_order__refl,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( preorder @ A )
% 0.25/0.48 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 0.25/0.48
% 0.25/0.48 % order_refl
% 0.25/0.48 thf(fact_83_Pair__mono,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType] :
% 0.25/0.48 ( ( ( ord @ B )
% 0.25/0.48 & ( ord @ A ) )
% 0.25/0.48 => ! [X3: A,X5: A,Y3: B,Y5: B] :
% 0.25/0.48 ( ( ord_less_eq @ A @ X3 @ X5 )
% 0.25/0.48 => ( ( ord_less_eq @ B @ Y3 @ Y5 )
% 0.25/0.48 => ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Pair_mono
% 0.25/0.48 thf(fact_84_internal__case__prod__conv,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 0.25/0.48 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.25/0.48 = ( C3 @ A2 @ B2 ) ) ).
% 0.25/0.48
% 0.25/0.48 % internal_case_prod_conv
% 0.25/0.48 thf(fact_85_Fsub,axiom,
% 0.25/0.48 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVF @ a @ b @ c @ p ) @ v ).
% 0.25/0.48
% 0.25/0.48 % Fsub
% 0.25/0.48 thf(fact_86_solSem,axiom,
% 0.25/0.48 ! [X6: real] :
% 0.25/0.48 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X6 )
% 0.25/0.48 => ( ( ord_less_eq @ real @ X6 @ t )
% 0.25/0.48 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( denotational_mk_v @ a @ b @ c @ i @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( sol @ X6 ) ) @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % solSem
% 0.25/0.48 thf(fact_87_relChain__def,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( ord @ B )
% 0.25/0.48 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.25/0.48 = ( ^ [R3: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.25/0.48 ! [I2: A,J: A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R3 )
% 0.25/0.48 => ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % relChain_def
% 0.25/0.48 thf(fact_88_hpsafe__Evolve_OIH,axiom,
% 0.25/0.48 coinci1993344360de_fml @ a @ b @ c @ p ).
% 0.25/0.48
% 0.25/0.48 % hpsafe_Evolve.IH
% 0.25/0.48 thf(fact_89_t,axiom,
% 0.25/0.48 ord_less_eq @ real @ ( zero_zero @ real ) @ t ).
% 0.25/0.48
% 0.25/0.48 % t
% 0.25/0.48 thf(fact_90_le__zero__eq,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( canoni770627133id_add @ A )
% 0.25/0.48 => ! [N: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.25/0.48 = ( N
% 0.25/0.48 = ( zero_zero @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % le_zero_eq
% 0.25/0.48 thf(fact_91_add_Oinverse__neutral,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( group_add @ A )
% 0.25/0.48 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 0.25/0.48 = ( zero_zero @ A ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % add.inverse_neutral
% 0.25/0.48 thf(fact_92_neg__0__equal__iff__equal,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( group_add @ A )
% 0.25/0.48 => ! [A2: A] :
% 0.25/0.48 ( ( ( zero_zero @ A )
% 0.25/0.48 = ( uminus_uminus @ A @ A2 ) )
% 0.25/0.48 = ( ( zero_zero @ A )
% 0.25/0.48 = A2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % neg_0_equal_iff_equal
% 0.25/0.48 thf(fact_93_neg__equal__0__iff__equal,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( group_add @ A )
% 0.25/0.48 => ! [A2: A] :
% 0.25/0.48 ( ( ( uminus_uminus @ A @ A2 )
% 0.25/0.48 = ( zero_zero @ A ) )
% 0.25/0.48 = ( A2
% 0.25/0.48 = ( zero_zero @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % neg_equal_0_iff_equal
% 0.25/0.48 thf(fact_94_equal__neg__zero,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linord219039673up_add @ A )
% 0.25/0.48 => ! [A2: A] :
% 0.25/0.48 ( ( A2
% 0.25/0.48 = ( uminus_uminus @ A @ A2 ) )
% 0.25/0.48 = ( A2
% 0.25/0.48 = ( zero_zero @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % equal_neg_zero
% 0.25/0.48 thf(fact_95_neg__equal__zero,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linord219039673up_add @ A )
% 0.25/0.48 => ! [A2: A] :
% 0.25/0.48 ( ( ( uminus_uminus @ A @ A2 )
% 0.25/0.48 = A2 )
% 0.25/0.48 = ( A2
% 0.25/0.48 = ( zero_zero @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % neg_equal_zero
% 0.25/0.48 thf(fact_96_IH_H,axiom,
% 0.25/0.48 ! [A7: finite_Cartesian_vec @ real @ c,B7: finite_Cartesian_vec @ real @ c,Aa: finite_Cartesian_vec @ real @ c,Ba: finite_Cartesian_vec @ real @ c] :
% 0.25/0.48 ( ( denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ A7 @ B7 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ Aa @ Ba ) @ ( static_FVF @ a @ b @ c @ p ) )
% 0.25/0.48 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ A7 @ B7 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) )
% 0.25/0.48 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ Aa @ Ba ) @ ( denotational_fml_sem @ a @ b @ c @ j @ p ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % IH'
% 0.25/0.48 thf(fact_97_neg__less__eq__nonneg,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linord219039673up_add @ A )
% 0.25/0.48 => ! [A2: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 0.25/0.48 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % neg_less_eq_nonneg
% 0.25/0.48 thf(fact_98_less__eq__neg__nonpos,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linord219039673up_add @ A )
% 0.25/0.48 => ! [A2: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 0.25/0.48 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % less_eq_neg_nonpos
% 0.25/0.48 thf(fact_99_neg__le__0__iff__le,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ordered_ab_group_add @ A )
% 0.25/0.48 => ! [A2: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.25/0.48 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % neg_le_0_iff_le
% 0.25/0.48 thf(fact_100_neg__0__le__iff__le,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ordered_ab_group_add @ A )
% 0.25/0.48 => ! [A2: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 0.25/0.48 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % neg_0_le_iff_le
% 0.25/0.48 thf(fact_101_fsafe,axiom,
% 0.25/0.48 fsafe @ a @ b @ c @ p ).
% 0.25/0.48
% 0.25/0.48 % fsafe
% 0.25/0.48 thf(fact_102_zero__prod__def,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType] :
% 0.25/0.48 ( ( ( zero @ B )
% 0.25/0.48 & ( zero @ A ) )
% 0.25/0.48 => ( ( zero_zero @ ( product_prod @ A @ B ) )
% 0.25/0.48 = ( product_Pair @ A @ B @ ( zero_zero @ A ) @ ( zero_zero @ B ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % zero_prod_def
% 0.25/0.48 thf(fact_103_zero__reorient,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( zero @ A )
% 0.25/0.48 => ! [X3: A] :
% 0.25/0.48 ( ( ( zero_zero @ A )
% 0.25/0.48 = X3 )
% 0.25/0.48 = ( X3
% 0.25/0.48 = ( zero_zero @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % zero_reorient
% 0.25/0.48 thf(fact_104_iff__to__impl,axiom,
% 0.25/0.48 ! [A: $tType,C: $tType,B: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ B )
% 0.25/0.48 & ( finite_finite @ C )
% 0.25/0.48 & ( finite_finite @ A ) )
% 0.25/0.48 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I: denota1663640101rp_ext @ B @ C @ A @ product_unit,A4: formula @ B @ C @ A,B4: formula @ B @ C @ A] :
% 0.25/0.48 ( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ A4 ) )
% 0.25/0.48 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ B4 ) ) )
% 0.25/0.48 = ( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ A4 ) )
% 0.25/0.48 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ B4 ) ) )
% 0.25/0.48 & ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ B4 ) )
% 0.25/0.48 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ A4 ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % iff_to_impl
% 0.25/0.48 thf(fact_105_zero__le,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( canoni770627133id_add @ A )
% 0.25/0.48 => ! [X3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ).
% 0.25/0.48
% 0.25/0.48 % zero_le
% 0.25/0.48 thf(fact_106_le__funD,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( ord @ B )
% 0.25/0.48 => ! [F3: A > B,G3: A > B,X3: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
% 0.25/0.48 => ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % le_funD
% 0.25/0.48 thf(fact_107_le__funE,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( ord @ B )
% 0.25/0.48 => ! [F3: A > B,G3: A > B,X3: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
% 0.25/0.48 => ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % le_funE
% 0.25/0.48 thf(fact_108_le__funI,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( ord @ B )
% 0.25/0.48 => ! [F3: A > B,G3: A > B] :
% 0.25/0.48 ( ! [X: A] : ( ord_less_eq @ B @ ( F3 @ X ) @ ( G3 @ X ) )
% 0.25/0.48 => ( ord_less_eq @ ( A > B ) @ F3 @ G3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % le_funI
% 0.25/0.48 thf(fact_109_le__fun__def,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( ord @ B )
% 0.25/0.48 => ( ( ord_less_eq @ ( A > B ) )
% 0.25/0.48 = ( ^ [F4: A > B,G4: A > B] :
% 0.25/0.48 ! [X4: A] : ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % le_fun_def
% 0.25/0.48 thf(fact_110_order__subst1,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType] :
% 0.25/0.48 ( ( ( order @ B )
% 0.25/0.48 & ( order @ A ) )
% 0.25/0.48 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 0.25/0.48 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
% 0.25/0.48 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 0.25/0.48 => ( ! [X: B,Y: B] :
% 0.25/0.48 ( ( ord_less_eq @ B @ X @ Y )
% 0.25/0.48 => ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y ) ) )
% 0.25/0.48 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % order_subst1
% 0.25/0.48 thf(fact_111_order__subst2,axiom,
% 0.25/0.48 ! [A: $tType,C: $tType] :
% 0.25/0.48 ( ( ( order @ C )
% 0.25/0.48 & ( order @ A ) )
% 0.25/0.48 => ! [A2: A,B2: A,F3: A > C,C3: C] :
% 0.25/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.25/0.48 => ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
% 0.25/0.48 => ( ! [X: A,Y: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ X @ Y )
% 0.25/0.48 => ( ord_less_eq @ C @ ( F3 @ X ) @ ( F3 @ Y ) ) )
% 0.25/0.48 => ( ord_less_eq @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % order_subst2
% 0.25/0.48 thf(fact_112_ord__eq__le__subst,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType] :
% 0.25/0.48 ( ( ( ord @ B )
% 0.25/0.48 & ( ord @ A ) )
% 0.25/0.48 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 0.25/0.48 ( ( A2
% 0.25/0.48 = ( F3 @ B2 ) )
% 0.25/0.48 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 0.25/0.48 => ( ! [X: B,Y: B] :
% 0.25/0.48 ( ( ord_less_eq @ B @ X @ Y )
% 0.25/0.48 => ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y ) ) )
% 0.25/0.48 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ord_eq_le_subst
% 0.25/0.48 thf(fact_113_ord__le__eq__subst,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType] :
% 0.25/0.48 ( ( ( ord @ B )
% 0.25/0.48 & ( ord @ A ) )
% 0.25/0.48 => ! [A2: A,B2: A,F3: A > B,C3: B] :
% 0.25/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.25/0.48 => ( ( ( F3 @ B2 )
% 0.25/0.48 = C3 )
% 0.25/0.48 => ( ! [X: A,Y: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ X @ Y )
% 0.25/0.48 => ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) )
% 0.25/0.48 => ( ord_less_eq @ B @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ord_le_eq_subst
% 0.25/0.48 thf(fact_114_eq__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.25/0.48 = ( ^ [X4: A,Y6: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ X4 @ Y6 )
% 0.25/0.48 & ( ord_less_eq @ A @ Y6 @ X4 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % eq_iff
% 0.25/0.48 thf(fact_115_antisym,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ! [X3: A,Y3: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.25/0.48 => ( X3 = Y3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % antisym
% 0.25/0.48 thf(fact_116_linear,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [X3: A,Y3: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.25/0.48 | ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % linear
% 0.25/0.48 thf(fact_117_eq__refl,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( preorder @ A )
% 0.25/0.48 => ! [X3: A,Y3: A] :
% 0.25/0.48 ( ( X3 = Y3 )
% 0.25/0.48 => ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % eq_refl
% 0.25/0.48 thf(fact_118_le__cases,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [X3: A,Y3: A] :
% 0.25/0.48 ( ~ ( ord_less_eq @ A @ X3 @ Y3 )
% 0.25/0.48 => ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % le_cases
% 0.25/0.48 thf(fact_119_order_Otrans,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ! [A2: A,B2: A,C3: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 0.25/0.48 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % order.trans
% 0.25/0.48 thf(fact_120_le__cases3,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [X3: A,Y3: A,Z2: A] :
% 0.25/0.48 ( ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.25/0.48 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 0.25/0.48 => ( ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.25/0.48 => ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
% 0.25/0.48 => ( ( ( ord_less_eq @ A @ X3 @ Z2 )
% 0.25/0.48 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 0.25/0.48 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 0.25/0.48 => ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
% 0.25/0.48 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.25/0.48 => ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
% 0.25/0.48 => ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
% 0.25/0.48 => ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % le_cases3
% 0.25/0.48 thf(fact_121_antisym__conv,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ! [Y3: A,X3: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.25/0.48 = ( X3 = Y3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % antisym_conv
% 0.25/0.48 thf(fact_122_order__class_Oorder_Oeq__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.25/0.48 = ( ^ [A8: A,B8: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ A8 @ B8 )
% 0.25/0.48 & ( ord_less_eq @ A @ B8 @ A8 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % order_class.order.eq_iff
% 0.25/0.48 thf(fact_123_ord__eq__le__trans,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ord @ A )
% 0.25/0.48 => ! [A2: A,B2: A,C3: A] :
% 0.25/0.48 ( ( A2 = B2 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 0.25/0.48 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ord_eq_le_trans
% 0.25/0.48 thf(fact_124_ord__le__eq__trans,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ord @ A )
% 0.25/0.48 => ! [A2: A,B2: A,C3: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.25/0.48 => ( ( B2 = C3 )
% 0.25/0.48 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ord_le_eq_trans
% 0.25/0.48 thf(fact_125_order__class_Oorder_Oantisym,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ! [A2: A,B2: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.25/0.48 => ( A2 = B2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % order_class.order.antisym
% 0.25/0.48 thf(fact_126_order__trans,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( preorder @ A )
% 0.25/0.48 => ! [X3: A,Y3: A,Z2: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.25/0.48 => ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % order_trans
% 0.25/0.48 thf(fact_127_dual__order_Orefl,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.25/0.48
% 0.25/0.48 % dual_order.refl
% 0.25/0.48 thf(fact_128_linorder__wlog,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [P2: A > A > $o,A2: A,B2: A] :
% 0.25/0.48 ( ! [A5: A,B5: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.25/0.48 => ( P2 @ A5 @ B5 ) )
% 0.25/0.48 => ( ! [A5: A,B5: A] :
% 0.25/0.48 ( ( P2 @ B5 @ A5 )
% 0.25/0.48 => ( P2 @ A5 @ B5 ) )
% 0.25/0.48 => ( P2 @ A2 @ B2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % linorder_wlog
% 0.25/0.48 thf(fact_129_dual__order_Otrans,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ! [B2: A,A2: A,C3: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ C3 @ B2 )
% 0.25/0.48 => ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % dual_order.trans
% 0.25/0.48 thf(fact_130_dual__order_Oeq__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.25/0.48 = ( ^ [A8: A,B8: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ B8 @ A8 )
% 0.25/0.48 & ( ord_less_eq @ A @ A8 @ B8 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % dual_order.eq_iff
% 0.25/0.48 thf(fact_131_dual__order_Oantisym,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ! [B2: A,A2: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.25/0.48 => ( A2 = B2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % dual_order.antisym
% 0.25/0.48 thf(fact_132_gcd_Ocases,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( euclid1155270486miring @ A )
% 0.25/0.48 => ! [X3: product_prod @ A @ A] :
% 0.25/0.48 ~ ! [A5: A,B5: A] :
% 0.25/0.48 ( X3
% 0.25/0.48 != ( product_Pair @ A @ A @ A5 @ B5 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % gcd.cases
% 0.25/0.48 thf(fact_133_iff__sem,axiom,
% 0.25/0.48 ! [A: $tType,C: $tType,B: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ B )
% 0.25/0.48 & ( finite_finite @ C )
% 0.25/0.48 & ( finite_finite @ A ) )
% 0.25/0.48 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I: denota1663640101rp_ext @ B @ C @ A @ product_unit,A4: formula @ B @ C @ A,B4: formula @ B @ C @ A] :
% 0.25/0.48 ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ ( equiv @ B @ C @ A @ A4 @ B4 ) ) )
% 0.25/0.48 = ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ A4 ) )
% 0.25/0.48 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ B4 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % iff_sem
% 0.25/0.48 thf(fact_134_IHF,axiom,
% 0.25/0.48 ! [A7: finite_Cartesian_vec @ real @ c,B7: finite_Cartesian_vec @ real @ c,Aa: finite_Cartesian_vec @ real @ c,Ba: finite_Cartesian_vec @ real @ c] :
% 0.25/0.48 ( ( denotational_Iagree @ a @ b @ c @ i @ j @ ( static_SIGF @ a @ b @ c @ p ) )
% 0.25/0.48 => ( ( denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ A7 @ B7 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ Aa @ Ba ) @ ( static_FVF @ a @ b @ c @ p ) )
% 0.25/0.48 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ A7 @ B7 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) )
% 0.25/0.48 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ Aa @ Ba ) @ ( denotational_fml_sem @ a @ b @ c @ j @ p ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % IHF
% 0.25/0.48 thf(fact_135_minus__le__self__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( lattic1601792062up_add @ A )
% 0.25/0.48 => ! [A2: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 0.25/0.48 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % minus_le_self_iff
% 0.25/0.48 thf(fact_136_le__minus__self__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( lattic1601792062up_add @ A )
% 0.25/0.48 => ! [A2: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 0.25/0.48 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % le_minus_self_iff
% 0.25/0.48 thf(fact_137_real__eq__0__iff__le__ge__0,axiom,
% 0.25/0.48 ! [X3: real] :
% 0.25/0.48 ( ( X3
% 0.25/0.48 = ( zero_zero @ real ) )
% 0.25/0.48 = ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
% 0.25/0.48 & ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( uminus_uminus @ real @ X3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % real_eq_0_iff_le_ge_0
% 0.25/0.48 thf(fact_138_le__numeral__extra_I3_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linord1659791738miring @ A )
% 0.25/0.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % le_numeral_extra(3)
% 0.25/0.48 thf(fact_139_coincide__fml__def,axiom,
% 0.25/0.48 ! [C: $tType,B: $tType,A: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ A )
% 0.25/0.48 & ( finite_finite @ B )
% 0.25/0.48 & ( finite_finite @ C ) )
% 0.25/0.48 => ( ( coinci1993344360de_fml @ A @ B @ C )
% 0.25/0.48 = ( ^ [Phi: formula @ A @ B @ C] :
% 0.25/0.48 ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.25/0.48 ( ( denotational_Iagree @ A @ B @ C @ I3 @ J2 @ ( static_SIGF @ A @ B @ C @ Phi ) )
% 0.25/0.48 => ( ( denotational_Vagree @ C @ Nu3 @ Nu4 @ ( static_FVF @ A @ B @ C @ Phi ) )
% 0.25/0.48 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu3 @ ( denotational_fml_sem @ A @ B @ C @ I3 @ Phi ) )
% 0.25/0.48 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu4 @ ( denotational_fml_sem @ A @ B @ C @ J2 @ Phi ) ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % coincide_fml_def
% 0.25/0.48 thf(fact_140_IAP,axiom,
% 0.25/0.48 denotational_Iagree @ a @ b @ c @ i @ j @ ( static_SIGF @ a @ b @ c @ p ) ).
% 0.25/0.48
% 0.25/0.48 % IAP
% 0.25/0.48 thf(fact_141_Iagree__sub,axiom,
% 0.25/0.48 ! [C: $tType,B: $tType,A: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ A )
% 0.25/0.48 & ( finite_finite @ B )
% 0.25/0.48 & ( finite_finite @ C ) )
% 0.25/0.48 => ! [A4: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ),B4: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ),I: denota1663640101rp_ext @ A @ B @ C @ product_unit,J3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) @ A4 @ B4 )
% 0.25/0.48 => ( ( denotational_Iagree @ A @ B @ C @ I @ J3 @ B4 )
% 0.25/0.48 => ( denotational_Iagree @ A @ B @ C @ I @ J3 @ A4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Iagree_sub
% 0.25/0.48 thf(fact_142_Iagree__comm,axiom,
% 0.25/0.48 ! [C: $tType,B: $tType,A: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ A )
% 0.25/0.48 & ( finite_finite @ B )
% 0.25/0.48 & ( finite_finite @ C ) )
% 0.25/0.48 => ! [A4: denota1663640101rp_ext @ A @ B @ C @ product_unit,B4: denota1663640101rp_ext @ A @ B @ C @ product_unit,V: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] :
% 0.25/0.48 ( ( denotational_Iagree @ A @ B @ C @ A4 @ B4 @ V )
% 0.25/0.48 => ( denotational_Iagree @ A @ B @ C @ B4 @ A4 @ V ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Iagree_comm
% 0.25/0.48 thf(fact_143_Iagree__refl,axiom,
% 0.25/0.48 ! [C: $tType,B: $tType,A: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ A )
% 0.25/0.48 & ( finite_finite @ B )
% 0.25/0.48 & ( finite_finite @ C ) )
% 0.25/0.48 => ! [I: denota1663640101rp_ext @ A @ B @ C @ product_unit,A4: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] : ( denotational_Iagree @ A @ B @ C @ I @ I @ A4 ) ) ).
% 0.25/0.48
% 0.25/0.48 % Iagree_refl
% 0.25/0.48 thf(fact_144_ids_Ocoincide__fml__def,axiom,
% 0.25/0.48 ! [Sc: $tType,Sf: $tType,Sz: $tType,B: $tType,A: $tType,C: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ C )
% 0.25/0.48 & ( finite_finite @ A )
% 0.25/0.48 & ( finite_finite @ B )
% 0.25/0.48 & ( finite_finite @ Sz )
% 0.25/0.48 & ( linorder @ Sz )
% 0.25/0.48 & ( finite_finite @ Sf )
% 0.25/0.48 & ( finite_finite @ Sc ) )
% 0.25/0.48 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Phi2: formula @ A @ B @ C] :
% 0.25/0.48 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.25/0.48 => ( ( coinci1993344360de_fml @ A @ B @ C @ Phi2 )
% 0.25/0.48 = ( ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.25/0.48 ( ( denotational_Iagree @ A @ B @ C @ I3 @ J2 @ ( static_SIGF @ A @ B @ C @ Phi2 ) )
% 0.25/0.48 => ( ( denotational_Vagree @ C @ Nu3 @ Nu4 @ ( static_FVF @ A @ B @ C @ Phi2 ) )
% 0.25/0.48 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu3 @ ( denotational_fml_sem @ A @ B @ C @ I3 @ Phi2 ) )
% 0.25/0.48 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu4 @ ( denotational_fml_sem @ A @ B @ C @ J2 @ Phi2 ) ) ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ids.coincide_fml_def
% 0.25/0.48 thf(fact_145__092_060open_062ode__sem__equiv_A_IEvolveODE_AODE_AP_J_AJ_092_060close_062,axiom,
% 0.25/0.48 coinci1495174342_equiv @ a @ b @ c @ ( evolveODE @ a @ c @ b @ ode2 @ p ) @ j ).
% 0.25/0.48
% 0.25/0.48 % \<open>ode_sem_equiv (EvolveODE ODE P) J\<close>
% 0.25/0.48 thf(fact_146__092_060open_062ode__sem__equiv_A_IEvolveODE_AODE_AP_J_AI_092_060close_062,axiom,
% 0.25/0.48 coinci1495174342_equiv @ a @ b @ c @ ( evolveODE @ a @ c @ b @ ode2 @ p ) @ i ).
% 0.25/0.48
% 0.25/0.48 % \<open>ode_sem_equiv (EvolveODE ODE P) I\<close>
% 0.25/0.48 thf(fact_147_impl__sem,axiom,
% 0.25/0.48 ! [A: $tType,C: $tType,B: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ B )
% 0.25/0.48 & ( finite_finite @ C )
% 0.25/0.48 & ( finite_finite @ A ) )
% 0.25/0.48 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I: denota1663640101rp_ext @ B @ C @ A @ product_unit,A4: formula @ B @ C @ A,B4: formula @ B @ C @ A] :
% 0.25/0.48 ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ ( implies @ B @ C @ A @ A4 @ B4 ) ) )
% 0.25/0.48 = ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ A4 ) )
% 0.25/0.48 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ B4 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % impl_sem
% 0.25/0.48 thf(fact_148_equiv,axiom,
% 0.25/0.48 ! [I4: denota1663640101rp_ext @ a @ b @ c @ product_unit] : ( coinci1495174342_equiv @ a @ b @ c @ ( evolveODE @ a @ c @ b @ ode2 @ p ) @ I4 ) ).
% 0.25/0.48
% 0.25/0.48 % equiv
% 0.25/0.48 thf(fact_149_ids_Oseq__sem_Ocases,axiom,
% 0.25/0.48 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ Sf )
% 0.25/0.48 & ( finite_finite @ Sc )
% 0.25/0.48 & ( finite_finite @ Sz )
% 0.25/0.48 & ( linorder @ Sz ) )
% 0.25/0.48 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X3: product_prod @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) )] :
% 0.25/0.48 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.25/0.48 => ~ ! [I5: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,S4: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
% 0.25/0.48 ( X3
% 0.25/0.48 != ( product_Pair @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) @ I5 @ S4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ids.seq_sem.cases
% 0.25/0.48 thf(fact_150_ids_Oseq__sem_Oinduct,axiom,
% 0.25/0.48 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ Sf )
% 0.25/0.48 & ( finite_finite @ Sc )
% 0.25/0.48 & ( finite_finite @ Sz )
% 0.25/0.48 & ( linorder @ Sz ) )
% 0.25/0.48 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P2: ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) > ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) > $o,A0: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,A1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
% 0.25/0.48 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.25/0.48 => ( ! [I5: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,X_1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] : ( P2 @ I5 @ X_1 )
% 0.25/0.48 => ( P2 @ A0 @ A1 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ids.seq_sem.induct
% 0.25/0.48 thf(fact_151_coincide__hp_H__def,axiom,
% 0.25/0.48 ! [C: $tType,B: $tType,A: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ A )
% 0.25/0.48 & ( finite_finite @ B )
% 0.25/0.48 & ( finite_finite @ C ) )
% 0.25/0.48 => ( ( coinci554967470ide_hp @ A @ B @ C )
% 0.25/0.48 = ( ^ [Alpha: hp @ A @ B @ C] :
% 0.25/0.48 ! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.25/0.48 ( ( coinci32832645ide_hp @ A @ B @ C @ Alpha @ I3 @ J2 )
% 0.25/0.48 & ( coinci1495174342_equiv @ A @ B @ C @ Alpha @ I3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % coincide_hp'_def
% 0.25/0.48 thf(fact_152_hp_Oinject_I5_J,axiom,
% 0.25/0.48 ! [C: $tType,B: $tType,A: $tType,X51: ode @ A @ C,X52: formula @ A @ B @ C,Y51: ode @ A @ C,Y52: formula @ A @ B @ C] :
% 0.25/0.48 ( ( ( evolveODE @ A @ C @ B @ X51 @ X52 )
% 0.25/0.48 = ( evolveODE @ A @ C @ B @ Y51 @ Y52 ) )
% 0.25/0.48 = ( ( X51 = Y51 )
% 0.25/0.48 & ( X52 = Y52 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % hp.inject(5)
% 0.25/0.48 thf(fact_153_sol,axiom,
% 0.25/0.48 ( initia1685620758es_ode @ ( finite_Cartesian_vec @ real @ c ) @ sol
% 0.25/0.48 @ ^ [A8: real] : ( denotational_ODE_sem @ a @ b @ c @ i @ ode2 )
% 0.25/0.48 @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t )
% 0.25/0.48 @ ( collect @ ( finite_Cartesian_vec @ real @ c )
% 0.25/0.48 @ ^ [X4: finite_Cartesian_vec @ real @ c] : ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( denotational_mk_v @ a @ b @ c @ i @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ X4 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % sol
% 0.25/0.48 thf(fact_154_less__eq__set__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) )
% 0.25/0.48 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.25/0.48 ( ord_less_eq @ ( A > $o )
% 0.25/0.48 @ ^ [X4: A] : ( member @ A @ X4 @ A6 )
% 0.25/0.48 @ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % less_eq_set_def
% 0.25/0.48 thf(fact_155_Collect__subset,axiom,
% 0.25/0.48 ! [A: $tType,A4: set @ A,P2: A > $o] :
% 0.25/0.48 ( ord_less_eq @ ( set @ A )
% 0.25/0.48 @ ( collect @ A
% 0.25/0.48 @ ^ [X4: A] :
% 0.25/0.48 ( ( member @ A @ X4 @ A4 )
% 0.25/0.48 & ( P2 @ X4 ) ) )
% 0.25/0.48 @ A4 ) ).
% 0.25/0.48
% 0.25/0.48 % Collect_subset
% 0.25/0.48 thf(fact_156_Compl__eq,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( uminus_uminus @ ( set @ A ) )
% 0.25/0.48 = ( ^ [A6: set @ A] :
% 0.25/0.48 ( collect @ A
% 0.25/0.48 @ ^ [X4: A] :
% 0.25/0.48 ~ ( member @ A @ X4 @ A6 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Compl_eq
% 0.25/0.48 thf(fact_157_Collect__neg__eq,axiom,
% 0.25/0.48 ! [A: $tType,P2: A > $o] :
% 0.25/0.48 ( ( collect @ A
% 0.25/0.48 @ ^ [X4: A] :
% 0.25/0.48 ~ ( P2 @ X4 ) )
% 0.25/0.48 = ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Collect_neg_eq
% 0.25/0.48 thf(fact_158_uminus__set__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( uminus_uminus @ ( set @ A ) )
% 0.25/0.48 = ( ^ [A6: set @ A] :
% 0.25/0.48 ( collect @ A
% 0.25/0.48 @ ( uminus_uminus @ ( A > $o )
% 0.25/0.48 @ ^ [X4: A] : ( member @ A @ X4 @ A6 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % uminus_set_def
% 0.25/0.48 thf(fact_159_mk__xode_Oelims,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,C: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ C )
% 0.25/0.48 & ( finite_finite @ B )
% 0.25/0.48 & ( finite_finite @ A ) )
% 0.25/0.48 => ! [X3: denota1663640101rp_ext @ A @ B @ C @ product_unit,Xa: ode @ A @ C,Xb: finite_Cartesian_vec @ real @ C,Y3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 0.25/0.48 ( ( ( denotational_mk_xode @ A @ B @ C @ X3 @ Xa @ Xb )
% 0.25/0.48 = Y3 )
% 0.25/0.48 => ( Y3
% 0.25/0.48 = ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb @ ( denotational_ODE_sem @ A @ B @ C @ X3 @ Xa @ Xb ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % mk_xode.elims
% 0.25/0.48 thf(fact_160_mk__xode_Osimps,axiom,
% 0.25/0.48 ! [C: $tType,B: $tType,A: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ A )
% 0.25/0.48 & ( finite_finite @ B )
% 0.25/0.48 & ( finite_finite @ C ) )
% 0.25/0.48 => ( ( denotational_mk_xode @ A @ B @ C )
% 0.25/0.48 = ( ^ [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,ODE2: ode @ A @ C,Sol2: finite_Cartesian_vec @ real @ C] : ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Sol2 @ ( denotational_ODE_sem @ A @ B @ C @ I3 @ ODE2 @ Sol2 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % mk_xode.simps
% 0.25/0.48 thf(fact_161_ids_Osingleton_Oinduct,axiom,
% 0.25/0.48 ! [Sf: $tType,Sc: $tType,A: $tType,Sz: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ Sz )
% 0.25/0.48 & ( linorder @ Sz )
% 0.25/0.48 & ( finite_finite @ Sc )
% 0.25/0.48 & ( finite_finite @ Sf ) )
% 0.25/0.48 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P2: ( trm @ A @ Sz ) > Sz > $o,A0: trm @ A @ Sz,A1: Sz] :
% 0.25/0.48 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.25/0.48 => ( ! [T4: trm @ A @ Sz,X_1: Sz] : ( P2 @ T4 @ X_1 )
% 0.25/0.48 => ( P2 @ A0 @ A1 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ids.singleton.induct
% 0.25/0.48 thf(fact_162_ids_Osingleton_Ocases,axiom,
% 0.25/0.48 ! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ Sz )
% 0.25/0.48 & ( linorder @ Sz )
% 0.25/0.48 & ( finite_finite @ Sf )
% 0.25/0.48 & ( finite_finite @ Sc ) )
% 0.25/0.48 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X3: product_prod @ ( trm @ A @ Sz ) @ Sz] :
% 0.25/0.48 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.25/0.48 => ~ ! [T4: trm @ A @ Sz,I6: Sz] :
% 0.25/0.48 ( X3
% 0.25/0.48 != ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ T4 @ I6 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ids.singleton.cases
% 0.25/0.48 thf(fact_163_ids_Ocoincide__hp_H__def,axiom,
% 0.25/0.48 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ A )
% 0.25/0.48 & ( finite_finite @ B )
% 0.25/0.48 & ( finite_finite @ C )
% 0.25/0.48 & ( finite_finite @ Sz )
% 0.25/0.48 & ( linorder @ Sz )
% 0.25/0.48 & ( finite_finite @ Sf )
% 0.25/0.48 & ( finite_finite @ Sc ) )
% 0.25/0.48 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Alpha2: hp @ A @ B @ C] :
% 0.25/0.48 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.25/0.48 => ( ( coinci554967470ide_hp @ A @ B @ C @ Alpha2 )
% 0.25/0.48 = ( ! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.25/0.48 ( ( coinci32832645ide_hp @ A @ B @ C @ Alpha2 @ I3 @ J2 )
% 0.25/0.48 & ( coinci1495174342_equiv @ A @ B @ C @ Alpha2 @ I3 ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ids.coincide_hp'_def
% 0.25/0.48 thf(fact_164_atLeastatMost__subset__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( preorder @ A )
% 0.25/0.48 => ! [A2: A,B2: A,C3: A,D3: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A2 @ B2 ) @ ( set_or331188842AtMost @ A @ C3 @ D3 ) )
% 0.25/0.48 = ( ~ ( ord_less_eq @ A @ A2 @ B2 )
% 0.25/0.48 | ( ( ord_less_eq @ A @ C3 @ A2 )
% 0.25/0.48 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atLeastatMost_subset_iff
% 0.25/0.48 thf(fact_165_atLeastAtMost__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ord @ A )
% 0.25/0.48 => ! [I7: A,L: A,U: A] :
% 0.25/0.48 ( ( member @ A @ I7 @ ( set_or331188842AtMost @ A @ L @ U ) )
% 0.25/0.48 = ( ( ord_less_eq @ A @ L @ I7 )
% 0.25/0.48 & ( ord_less_eq @ A @ I7 @ U ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atLeastAtMost_iff
% 0.25/0.48 thf(fact_166_Icc__eq__Icc,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ! [L: A,H: A,L2: A,H2: A] :
% 0.25/0.48 ( ( ( set_or331188842AtMost @ A @ L @ H )
% 0.25/0.48 = ( set_or331188842AtMost @ A @ L2 @ H2 ) )
% 0.25/0.48 = ( ( ( L = L2 )
% 0.25/0.48 & ( H = H2 ) )
% 0.25/0.48 | ( ~ ( ord_less_eq @ A @ L @ H )
% 0.25/0.48 & ~ ( ord_less_eq @ A @ L2 @ H2 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Icc_eq_Icc
% 0.25/0.48 thf(fact_167_predicate1I,axiom,
% 0.25/0.48 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.25/0.48 ( ! [X: A] :
% 0.25/0.48 ( ( P2 @ X )
% 0.25/0.48 => ( Q @ X ) )
% 0.25/0.48 => ( ord_less_eq @ ( A > $o ) @ P2 @ Q ) ) ).
% 0.25/0.48
% 0.25/0.48 % predicate1I
% 0.25/0.48 thf(fact_168_predicate1D,axiom,
% 0.25/0.48 ! [A: $tType,P2: A > $o,Q: A > $o,X3: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
% 0.25/0.48 => ( ( P2 @ X3 )
% 0.25/0.48 => ( Q @ X3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % predicate1D
% 0.25/0.48 thf(fact_169_rev__predicate1D,axiom,
% 0.25/0.48 ! [A: $tType,P2: A > $o,X3: A,Q: A > $o] :
% 0.25/0.48 ( ( P2 @ X3 )
% 0.25/0.48 => ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
% 0.25/0.48 => ( Q @ X3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % rev_predicate1D
% 0.25/0.48 thf(fact_170_pred__subset__eq,axiom,
% 0.25/0.48 ! [A: $tType,R4: set @ A,S5: set @ A] :
% 0.25/0.48 ( ( ord_less_eq @ ( A > $o )
% 0.25/0.48 @ ^ [X4: A] : ( member @ A @ X4 @ R4 )
% 0.25/0.48 @ ^ [X4: A] : ( member @ A @ X4 @ S5 ) )
% 0.25/0.48 = ( ord_less_eq @ ( set @ A ) @ R4 @ S5 ) ) ).
% 0.25/0.48
% 0.25/0.48 % pred_subset_eq
% 0.25/0.48 thf(fact_171_solDeriv,axiom,
% 0.25/0.48 ! [X6: real] :
% 0.25/0.48 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X6 )
% 0.25/0.48 => ( ( ord_less_eq @ real @ X6 @ t )
% 0.25/0.48 => ( has_ve2132708402vative @ ( finite_Cartesian_vec @ real @ c ) @ sol @ ( denotational_ODE_sem @ a @ b @ c @ i @ ode2 @ ( sol @ X6 ) ) @ ( topolo507301023within @ real @ X6 @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % solDeriv
% 0.25/0.48 thf(fact_172_mk__v__def,axiom,
% 0.25/0.48 ! [C: $tType,B: $tType,A: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ A )
% 0.25/0.48 & ( finite_finite @ B )
% 0.25/0.48 & ( finite_finite @ C ) )
% 0.25/0.48 => ( ( denotational_mk_v @ A @ B @ C )
% 0.25/0.48 = ( ^ [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,ODE2: ode @ A @ C,Nu3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Sol2: finite_Cartesian_vec @ real @ C] :
% 0.25/0.48 ( the @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) )
% 0.25/0.48 @ ^ [Omega3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 0.25/0.48 ( ( denotational_Vagree @ C @ Omega3 @ Nu3 @ ( uminus_uminus @ ( set @ ( sum_sum @ C @ C ) ) @ ( denotational_semBV @ A @ B @ C @ I3 @ ODE2 ) ) )
% 0.25/0.48 & ( denotational_Vagree @ C @ Omega3 @ ( denotational_mk_xode @ A @ B @ C @ I3 @ ODE2 @ Sol2 ) @ ( denotational_semBV @ A @ B @ C @ I3 @ ODE2 ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % mk_v_def
% 0.25/0.48 thf(fact_173_subrelI,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S6: set @ ( product_prod @ A @ B )] :
% 0.25/0.48 ( ! [X: A,Y: B] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R5 )
% 0.25/0.48 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ S6 ) )
% 0.25/0.48 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S6 ) ) ).
% 0.25/0.48
% 0.25/0.48 % subrelI
% 0.25/0.48 thf(fact_174_pred__subset__eq2,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
% 0.25/0.48 ( ( ord_less_eq @ ( A > B > $o )
% 0.25/0.48 @ ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ R4 )
% 0.25/0.48 @ ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ S5 ) )
% 0.25/0.48 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S5 ) ) ).
% 0.25/0.48
% 0.25/0.48 % pred_subset_eq2
% 0.25/0.48 thf(fact_175_pred__equals__eq2,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
% 0.25/0.48 ( ( ( ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ R4 ) )
% 0.25/0.48 = ( ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ S5 ) ) )
% 0.25/0.48 = ( R4 = S5 ) ) ).
% 0.25/0.48
% 0.25/0.48 % pred_equals_eq2
% 0.25/0.48 thf(fact_176_old_Orec__prod__def,axiom,
% 0.25/0.48 ! [T: $tType,B: $tType,A: $tType] :
% 0.25/0.48 ( ( product_rec_prod @ A @ B @ T )
% 0.25/0.48 = ( ^ [F12: A > B > T,X4: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % old.rec_prod_def
% 0.25/0.48 thf(fact_177_has__vector__derivative__Pair,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( ( real_V55928688vector @ A )
% 0.25/0.48 & ( real_V55928688vector @ B ) )
% 0.25/0.48 => ! [F3: real > A,F5: A,X3: real,S6: set @ real,G3: real > B,G5: B] :
% 0.25/0.48 ( ( has_ve2132708402vative @ A @ F3 @ F5 @ ( topolo507301023within @ real @ X3 @ S6 ) )
% 0.25/0.48 => ( ( has_ve2132708402vative @ B @ G3 @ G5 @ ( topolo507301023within @ real @ X3 @ S6 ) )
% 0.25/0.48 => ( has_ve2132708402vative @ ( product_prod @ A @ B )
% 0.25/0.48 @ ^ [X4: real] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
% 0.25/0.48 @ ( product_Pair @ A @ B @ F5 @ G5 )
% 0.25/0.48 @ ( topolo507301023within @ real @ X3 @ S6 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % has_vector_derivative_Pair
% 0.25/0.48 thf(fact_178_has__vector__derivative__minus,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( real_V55928688vector @ A )
% 0.25/0.48 => ! [F3: real > A,F5: A,Net: filter @ real] :
% 0.25/0.48 ( ( has_ve2132708402vative @ A @ F3 @ F5 @ Net )
% 0.25/0.48 => ( has_ve2132708402vative @ A
% 0.25/0.48 @ ^ [X4: real] : ( uminus_uminus @ A @ ( F3 @ X4 ) )
% 0.25/0.48 @ ( uminus_uminus @ A @ F5 )
% 0.25/0.48 @ Net ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % has_vector_derivative_minus
% 0.25/0.48 thf(fact_179_predicate2I,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,P2: A > B > $o,Q: A > B > $o] :
% 0.25/0.48 ( ! [X: A,Y: B] :
% 0.25/0.48 ( ( P2 @ X @ Y )
% 0.25/0.48 => ( Q @ X @ Y ) )
% 0.25/0.48 => ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q ) ) ).
% 0.25/0.48
% 0.25/0.48 % predicate2I
% 0.25/0.48 thf(fact_180_rev__predicate2D,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,P2: A > B > $o,X3: A,Y3: B,Q: A > B > $o] :
% 0.25/0.48 ( ( P2 @ X3 @ Y3 )
% 0.25/0.48 => ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
% 0.25/0.48 => ( Q @ X3 @ Y3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % rev_predicate2D
% 0.25/0.48 thf(fact_181_predicate2D,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,P2: A > B > $o,Q: A > B > $o,X3: A,Y3: B] :
% 0.25/0.48 ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
% 0.25/0.48 => ( ( P2 @ X3 @ Y3 )
% 0.25/0.48 => ( Q @ X3 @ Y3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % predicate2D
% 0.25/0.48 thf(fact_182_has__vector__derivative__const,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( real_V55928688vector @ A )
% 0.25/0.48 => ! [C3: A,Net: filter @ real] :
% 0.25/0.48 ( has_ve2132708402vative @ A
% 0.25/0.48 @ ^ [X4: real] : C3
% 0.25/0.48 @ ( zero_zero @ A )
% 0.25/0.48 @ Net ) ) ).
% 0.25/0.48
% 0.25/0.48 % has_vector_derivative_const
% 0.25/0.48 thf(fact_183_has__vector__derivative__within__subset,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( real_V55928688vector @ A )
% 0.25/0.48 => ! [F3: real > A,F5: A,X3: real,S5: set @ real,T5: set @ real] :
% 0.25/0.48 ( ( has_ve2132708402vative @ A @ F3 @ F5 @ ( topolo507301023within @ real @ X3 @ S5 ) )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ real ) @ T5 @ S5 )
% 0.25/0.48 => ( has_ve2132708402vative @ A @ F3 @ F5 @ ( topolo507301023within @ real @ X3 @ T5 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % has_vector_derivative_within_subset
% 0.25/0.48 thf(fact_184_has__vector__derivative__weaken,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( real_V55928688vector @ A )
% 0.25/0.48 => ! [F3: real > A,D4: A,X3: real,T5: set @ real,S5: set @ real,G3: real > A] :
% 0.25/0.48 ( ( has_ve2132708402vative @ A @ F3 @ D4 @ ( topolo507301023within @ real @ X3 @ T5 ) )
% 0.25/0.48 => ( ( member @ real @ X3 @ S5 )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ real ) @ S5 @ T5 )
% 0.25/0.48 => ( ! [X: real] :
% 0.25/0.48 ( ( member @ real @ X @ S5 )
% 0.25/0.48 => ( ( F3 @ X )
% 0.25/0.48 = ( G3 @ X ) ) )
% 0.25/0.48 => ( has_ve2132708402vative @ A @ G3 @ D4 @ ( topolo507301023within @ real @ X3 @ S5 ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % has_vector_derivative_weaken
% 0.25/0.48 thf(fact_185_solves__ode__supset__range,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( real_V55928688vector @ A )
% 0.25/0.48 => ! [X3: real > A,F3: real > A > A,T5: set @ real,X7: set @ A,Y7: set @ A] :
% 0.25/0.48 ( ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ X7 )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ A ) @ X7 @ Y7 )
% 0.25/0.48 => ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ Y7 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % solves_ode_supset_range
% 0.25/0.48 thf(fact_186_solves__ode__subset,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( real_V55928688vector @ A )
% 0.25/0.48 => ! [X3: real > A,F3: real > A > A,T5: set @ real,X7: set @ A,S5: set @ real] :
% 0.25/0.48 ( ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ X7 )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ real ) @ S5 @ T5 )
% 0.25/0.48 => ( initia1685620758es_ode @ A @ X3 @ F3 @ S5 @ X7 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % solves_ode_subset
% 0.25/0.48 thf(fact_187_solves__ode__on__subset,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( real_V55928688vector @ A )
% 0.25/0.48 => ! [X3: real > A,F3: real > A > A,S5: set @ real,Y7: set @ A,T5: set @ real,X7: set @ A] :
% 0.25/0.48 ( ( initia1685620758es_ode @ A @ X3 @ F3 @ S5 @ Y7 )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ real ) @ T5 @ S5 )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ A ) @ Y7 @ X7 )
% 0.25/0.48 => ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ X7 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % solves_ode_on_subset
% 0.25/0.48 thf(fact_188_at__le,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( topolo503727757_space @ A )
% 0.25/0.48 => ! [S6: set @ A,T6: set @ A,X3: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ S6 @ T6 )
% 0.25/0.48 => ( ord_less_eq @ ( filter @ A ) @ ( topolo507301023within @ A @ X3 @ S6 ) @ ( topolo507301023within @ A @ X3 @ T6 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % at_le
% 0.25/0.48 thf(fact_189_intervalE,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ord @ A )
% 0.25/0.48 => ! [A2: A,X3: A,B2: A] :
% 0.25/0.48 ( ( ( ord_less_eq @ A @ A2 @ X3 )
% 0.25/0.48 & ( ord_less_eq @ A @ X3 @ B2 ) )
% 0.25/0.48 => ( member @ A @ X3 @ ( set_or331188842AtMost @ A @ A2 @ B2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % intervalE
% 0.25/0.48 thf(fact_190_eq__subset,axiom,
% 0.25/0.48 ! [A: $tType,P2: A > A > $o] :
% 0.25/0.48 ( ord_less_eq @ ( A > A > $o )
% 0.25/0.48 @ ^ [Y4: A,Z: A] : ( Y4 = Z )
% 0.25/0.48 @ ^ [A8: A,B8: A] :
% 0.25/0.48 ( ( P2 @ A8 @ B8 )
% 0.25/0.48 | ( A8 = B8 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % eq_subset
% 0.25/0.48 thf(fact_191_conj__subset__def,axiom,
% 0.25/0.48 ! [A: $tType,A4: set @ A,P2: A > $o,Q: A > $o] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ A4
% 0.25/0.48 @ ( collect @ A
% 0.25/0.48 @ ^ [X4: A] :
% 0.25/0.48 ( ( P2 @ X4 )
% 0.25/0.48 & ( Q @ X4 ) ) ) )
% 0.25/0.48 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ P2 ) )
% 0.25/0.48 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ Q ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % conj_subset_def
% 0.25/0.48 thf(fact_192_predicate2D__conj,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,P2: A > B > $o,Q: A > B > $o,R4: $o,X3: A,Y3: B] :
% 0.25/0.48 ( ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
% 0.25/0.48 & R4 )
% 0.25/0.48 => ( R4
% 0.25/0.48 & ( ( P2 @ X3 @ Y3 )
% 0.25/0.48 => ( Q @ X3 @ Y3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % predicate2D_conj
% 0.25/0.48 thf(fact_193_fun__cong__unused__0,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,C: $tType] :
% 0.25/0.48 ( ( zero @ B )
% 0.25/0.48 => ! [F3: ( A > B ) > C,G3: C] :
% 0.25/0.48 ( ( F3
% 0.25/0.48 = ( ^ [X4: A > B] : G3 ) )
% 0.25/0.48 => ( ( F3
% 0.25/0.48 @ ^ [X4: A] : ( zero_zero @ B ) )
% 0.25/0.48 = G3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % fun_cong_unused_0
% 0.25/0.48 thf(fact_194_subset__Collect__iff,axiom,
% 0.25/0.48 ! [A: $tType,B4: set @ A,A4: set @ A,P2: A > $o] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B4
% 0.25/0.48 @ ( collect @ A
% 0.25/0.48 @ ^ [X4: A] :
% 0.25/0.48 ( ( member @ A @ X4 @ A4 )
% 0.25/0.48 & ( P2 @ X4 ) ) ) )
% 0.25/0.48 = ( ! [X4: A] :
% 0.25/0.48 ( ( member @ A @ X4 @ B4 )
% 0.25/0.48 => ( P2 @ X4 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % subset_Collect_iff
% 0.25/0.48 thf(fact_195_subset__CollectI,axiom,
% 0.25/0.48 ! [A: $tType,B4: set @ A,A4: set @ A,Q: A > $o,P2: A > $o] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.25/0.48 => ( ! [X: A] :
% 0.25/0.48 ( ( member @ A @ X @ B4 )
% 0.25/0.48 => ( ( Q @ X )
% 0.25/0.48 => ( P2 @ X ) ) )
% 0.25/0.48 => ( ord_less_eq @ ( set @ A )
% 0.25/0.48 @ ( collect @ A
% 0.25/0.48 @ ^ [X4: A] :
% 0.25/0.48 ( ( member @ A @ X4 @ B4 )
% 0.25/0.48 & ( Q @ X4 ) ) )
% 0.25/0.48 @ ( collect @ A
% 0.25/0.48 @ ^ [X4: A] :
% 0.25/0.48 ( ( member @ A @ X4 @ A4 )
% 0.25/0.48 & ( P2 @ X4 ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % subset_CollectI
% 0.25/0.48 thf(fact_196_prop__restrict,axiom,
% 0.25/0.48 ! [A: $tType,X3: A,Z3: set @ A,X7: set @ A,P2: A > $o] :
% 0.25/0.48 ( ( member @ A @ X3 @ Z3 )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ A ) @ Z3
% 0.25/0.48 @ ( collect @ A
% 0.25/0.48 @ ^ [X4: A] :
% 0.25/0.48 ( ( member @ A @ X4 @ X7 )
% 0.25/0.48 & ( P2 @ X4 ) ) ) )
% 0.25/0.48 => ( P2 @ X3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % prop_restrict
% 0.25/0.48 thf(fact_197_Collect__restrict,axiom,
% 0.25/0.48 ! [A: $tType,X7: set @ A,P2: A > $o] :
% 0.25/0.48 ( ord_less_eq @ ( set @ A )
% 0.25/0.48 @ ( collect @ A
% 0.25/0.48 @ ^ [X4: A] :
% 0.25/0.48 ( ( member @ A @ X4 @ X7 )
% 0.25/0.48 & ( P2 @ X4 ) ) )
% 0.25/0.48 @ X7 ) ).
% 0.25/0.48
% 0.25/0.48 % Collect_restrict
% 0.25/0.48 thf(fact_198_ssubst__Pair__rhs,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,R5: A,S6: B,R4: set @ ( product_prod @ A @ B ),S7: B] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R5 @ S6 ) @ R4 )
% 0.25/0.48 => ( ( S7 = S6 )
% 0.25/0.48 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R5 @ S7 ) @ R4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ssubst_Pair_rhs
% 0.25/0.48 thf(fact_199_hpsafe__ODE__simps,axiom,
% 0.25/0.48 ! [C: $tType,B: $tType,A: $tType,ODE: ode @ A @ C,P: formula @ A @ B @ C] :
% 0.25/0.48 ( ( hpsafe @ A @ B @ C @ ( evolveODE @ A @ C @ B @ ODE @ P ) )
% 0.25/0.48 = ( ( osafe @ A @ C @ ODE )
% 0.25/0.48 & ( fsafe @ A @ B @ C @ P ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % hpsafe_ODE_simps
% 0.25/0.48 thf(fact_200_old_Orec__unit__def,axiom,
% 0.25/0.48 ! [T: $tType] :
% 0.25/0.48 ( ( product_rec_unit @ T )
% 0.25/0.48 = ( ^ [F12: T,X4: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % old.rec_unit_def
% 0.25/0.48 thf(fact_201_hpsafe__fsafe_Ohpsafe__Evolve,axiom,
% 0.25/0.48 ! [C: $tType,B: $tType,A: $tType,ODE: ode @ A @ C,P2: formula @ A @ B @ C] :
% 0.25/0.48 ( ( osafe @ A @ C @ ODE )
% 0.25/0.48 => ( ( fsafe @ A @ B @ C @ P2 )
% 0.25/0.48 => ( hpsafe @ A @ B @ C @ ( evolveODE @ A @ C @ B @ ODE @ P2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % hpsafe_fsafe.hpsafe_Evolve
% 0.25/0.48 thf(fact_202_divides__aux__eq,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( unique1598680935umeral @ A )
% 0.25/0.48 => ! [Q2: A,R5: A] :
% 0.25/0.48 ( ( unique455577585es_aux @ A @ ( product_Pair @ A @ A @ Q2 @ R5 ) )
% 0.25/0.48 = ( R5
% 0.25/0.48 = ( zero_zero @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % divides_aux_eq
% 0.25/0.48 thf(fact_203_refl__ge__eq,axiom,
% 0.25/0.48 ! [A: $tType,R4: A > A > $o] :
% 0.25/0.48 ( ! [X: A] : ( R4 @ X @ X )
% 0.25/0.48 => ( ord_less_eq @ ( A > A > $o )
% 0.25/0.48 @ ^ [Y4: A,Z: A] : ( Y4 = Z )
% 0.25/0.48 @ R4 ) ) ).
% 0.25/0.48
% 0.25/0.48 % refl_ge_eq
% 0.25/0.48 thf(fact_204_ge__eq__refl,axiom,
% 0.25/0.48 ! [A: $tType,R4: A > A > $o,X3: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( A > A > $o )
% 0.25/0.48 @ ^ [Y4: A,Z: A] : ( Y4 = Z )
% 0.25/0.48 @ R4 )
% 0.25/0.48 => ( R4 @ X3 @ X3 ) ) ).
% 0.25/0.48
% 0.25/0.48 % ge_eq_refl
% 0.25/0.48 thf(fact_205_Icc__subset__Ici__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( preorder @ A )
% 0.25/0.48 => ! [L: A,H: A,L2: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ L @ H ) @ ( set_ord_atLeast @ A @ L2 ) )
% 0.25/0.48 = ( ~ ( ord_less_eq @ A @ L @ H )
% 0.25/0.48 | ( ord_less_eq @ A @ L2 @ L ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Icc_subset_Ici_iff
% 0.25/0.48 thf(fact_206_Icc__subset__Iic__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( preorder @ A )
% 0.25/0.48 => ! [L: A,H: A,H2: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ L @ H ) @ ( set_ord_atMost @ A @ H2 ) )
% 0.25/0.48 = ( ~ ( ord_less_eq @ A @ L @ H )
% 0.25/0.48 | ( ord_less_eq @ A @ H @ H2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Icc_subset_Iic_iff
% 0.25/0.48 thf(fact_207_atMost__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ord @ A )
% 0.25/0.48 => ! [I7: A,K: A] :
% 0.25/0.48 ( ( member @ A @ I7 @ ( set_ord_atMost @ A @ K ) )
% 0.25/0.48 = ( ord_less_eq @ A @ I7 @ K ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atMost_iff
% 0.25/0.48 thf(fact_208_atLeast__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ord @ A )
% 0.25/0.48 => ! [I7: A,K: A] :
% 0.25/0.48 ( ( member @ A @ I7 @ ( set_ord_atLeast @ A @ K ) )
% 0.25/0.48 = ( ord_less_eq @ A @ K @ I7 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atLeast_iff
% 0.25/0.48 thf(fact_209_atMost__subset__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( preorder @ A )
% 0.25/0.48 => ! [X3: A,Y3: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X3 ) @ ( set_ord_atMost @ A @ Y3 ) )
% 0.25/0.48 = ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atMost_subset_iff
% 0.25/0.48 thf(fact_210_atLeast__subset__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( preorder @ A )
% 0.25/0.48 => ! [X3: A,Y3: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X3 ) @ ( set_ord_atLeast @ A @ Y3 ) )
% 0.25/0.48 = ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atLeast_subset_iff
% 0.25/0.48 thf(fact_211_not__Ici__le__Icc,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( no_top @ A )
% 0.25/0.48 => ! [L: A,L2: A,H2: A] :
% 0.25/0.48 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_or331188842AtMost @ A @ L2 @ H2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % not_Ici_le_Icc
% 0.25/0.48 thf(fact_212_not__Iic__le__Icc,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( no_bot @ A )
% 0.25/0.48 => ! [H: A,L2: A,H2: A] :
% 0.25/0.48 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_or331188842AtMost @ A @ L2 @ H2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % not_Iic_le_Icc
% 0.25/0.48 thf(fact_213_atLeast__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ord @ A )
% 0.25/0.48 => ( ( set_ord_atLeast @ A )
% 0.25/0.48 = ( ^ [L3: A] : ( collect @ A @ ( ord_less_eq @ A @ L3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atLeast_def
% 0.25/0.48 thf(fact_214_atMost__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ord @ A )
% 0.25/0.48 => ( ( set_ord_atMost @ A )
% 0.25/0.48 = ( ^ [U2: A] :
% 0.25/0.48 ( collect @ A
% 0.25/0.48 @ ^ [X4: A] : ( ord_less_eq @ A @ X4 @ U2 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atMost_def
% 0.25/0.48 thf(fact_215_not__Iic__le__Ici,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( no_bot @ A )
% 0.25/0.48 => ! [H: A,L2: A] :
% 0.25/0.48 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_ord_atLeast @ A @ L2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % not_Iic_le_Ici
% 0.25/0.48 thf(fact_216_not__Ici__le__Iic,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( no_top @ A )
% 0.25/0.48 => ! [L: A,H2: A] :
% 0.25/0.48 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_ord_atMost @ A @ H2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % not_Ici_le_Iic
% 0.25/0.48 thf(fact_217_image__uminus__atMost,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ordered_ab_group_add @ A )
% 0.25/0.48 => ! [X3: A] :
% 0.25/0.48 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atMost @ A @ X3 ) )
% 0.25/0.48 = ( set_ord_atLeast @ A @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_uminus_atMost
% 0.25/0.48 thf(fact_218_image__uminus__atLeast,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ordered_ab_group_add @ A )
% 0.25/0.48 => ! [X3: A] :
% 0.25/0.48 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atLeast @ A @ X3 ) )
% 0.25/0.48 = ( set_ord_atMost @ A @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_uminus_atLeast
% 0.25/0.48 thf(fact_219_image__eqI,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,B2: A,F3: B > A,X3: B,A4: set @ B] :
% 0.25/0.48 ( ( B2
% 0.25/0.48 = ( F3 @ X3 ) )
% 0.25/0.48 => ( ( member @ B @ X3 @ A4 )
% 0.25/0.48 => ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_eqI
% 0.25/0.48 thf(fact_220_image__ident,axiom,
% 0.25/0.48 ! [A: $tType,Y7: set @ A] :
% 0.25/0.48 ( ( image @ A @ A
% 0.25/0.48 @ ^ [X4: A] : X4
% 0.25/0.48 @ Y7 )
% 0.25/0.48 = Y7 ) ).
% 0.25/0.48
% 0.25/0.48 % image_ident
% 0.25/0.48 thf(fact_221_image__uminus__atLeastAtMost,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ordered_ab_group_add @ A )
% 0.25/0.48 => ! [X3: A,Y3: A] :
% 0.25/0.48 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or331188842AtMost @ A @ X3 @ Y3 ) )
% 0.25/0.48 = ( set_or331188842AtMost @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_uminus_atLeastAtMost
% 0.25/0.48 thf(fact_222_subset__image__iff,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,B4: set @ A,F3: B > A,A4: set @ B] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image @ B @ A @ F3 @ A4 ) )
% 0.25/0.48 = ( ? [AA: set @ B] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ B ) @ AA @ A4 )
% 0.25/0.48 & ( B4
% 0.25/0.48 = ( image @ B @ A @ F3 @ AA ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % subset_image_iff
% 0.25/0.48 thf(fact_223_image__subset__iff,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,F3: B > A,A4: set @ B,B4: set @ A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ A4 ) @ B4 )
% 0.25/0.48 = ( ! [X4: B] :
% 0.25/0.48 ( ( member @ B @ X4 @ A4 )
% 0.25/0.48 => ( member @ A @ ( F3 @ X4 ) @ B4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_subset_iff
% 0.25/0.48 thf(fact_224_subset__imageE,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,B4: set @ A,F3: B > A,A4: set @ B] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image @ B @ A @ F3 @ A4 ) )
% 0.25/0.48 => ~ ! [C5: set @ B] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A4 )
% 0.25/0.48 => ( B4
% 0.25/0.48 != ( image @ B @ A @ F3 @ C5 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % subset_imageE
% 0.25/0.48 thf(fact_225_image__subsetI,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,A4: set @ A,F3: A > B,B4: set @ B] :
% 0.25/0.48 ( ! [X: A] :
% 0.25/0.48 ( ( member @ A @ X @ A4 )
% 0.25/0.48 => ( member @ B @ ( F3 @ X ) @ B4 ) )
% 0.25/0.48 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A4 ) @ B4 ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_subsetI
% 0.25/0.48 thf(fact_226_image__mono,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,A4: set @ A,B4: set @ A,F3: A > B] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.25/0.48 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A4 ) @ ( image @ A @ B @ F3 @ B4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_mono
% 0.25/0.48 thf(fact_227_solves__ode__subset__range,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( real_V55928688vector @ A )
% 0.25/0.48 => ! [X3: real > A,F3: real > A > A,T5: set @ real,X7: set @ A,Y7: set @ A] :
% 0.25/0.48 ( ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ X7 )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ real @ A @ X3 @ T5 ) @ Y7 )
% 0.25/0.48 => ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ Y7 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % solves_ode_subset_range
% 0.25/0.48 thf(fact_228_imageI,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,X3: A,A4: set @ A,F3: A > B] :
% 0.25/0.48 ( ( member @ A @ X3 @ A4 )
% 0.25/0.48 => ( member @ B @ ( F3 @ X3 ) @ ( image @ A @ B @ F3 @ A4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % imageI
% 0.25/0.48 thf(fact_229_image__iff,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,Z2: A,F3: B > A,A4: set @ B] :
% 0.25/0.48 ( ( member @ A @ Z2 @ ( image @ B @ A @ F3 @ A4 ) )
% 0.25/0.48 = ( ? [X4: B] :
% 0.25/0.48 ( ( member @ B @ X4 @ A4 )
% 0.25/0.48 & ( Z2
% 0.25/0.48 = ( F3 @ X4 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_iff
% 0.25/0.48 thf(fact_230_bex__imageD,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P2: A > $o] :
% 0.25/0.48 ( ? [X8: A] :
% 0.25/0.48 ( ( member @ A @ X8 @ ( image @ B @ A @ F3 @ A4 ) )
% 0.25/0.48 & ( P2 @ X8 ) )
% 0.25/0.48 => ? [X: B] :
% 0.25/0.48 ( ( member @ B @ X @ A4 )
% 0.25/0.48 & ( P2 @ ( F3 @ X ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % bex_imageD
% 0.25/0.48 thf(fact_231_image__cong,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,M: set @ A,N2: set @ A,F3: A > B,G3: A > B] :
% 0.25/0.48 ( ( M = N2 )
% 0.25/0.48 => ( ! [X: A] :
% 0.25/0.48 ( ( member @ A @ X @ N2 )
% 0.25/0.48 => ( ( F3 @ X )
% 0.25/0.48 = ( G3 @ X ) ) )
% 0.25/0.48 => ( ( image @ A @ B @ F3 @ M )
% 0.25/0.48 = ( image @ A @ B @ G3 @ N2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_cong
% 0.25/0.48 thf(fact_232_ball__imageD,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P2: A > $o] :
% 0.25/0.48 ( ! [X: A] :
% 0.25/0.48 ( ( member @ A @ X @ ( image @ B @ A @ F3 @ A4 ) )
% 0.25/0.48 => ( P2 @ X ) )
% 0.25/0.48 => ! [X8: B] :
% 0.25/0.48 ( ( member @ B @ X8 @ A4 )
% 0.25/0.48 => ( P2 @ ( F3 @ X8 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ball_imageD
% 0.25/0.48 thf(fact_233_rev__image__eqI,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,X3: A,A4: set @ A,B2: B,F3: A > B] :
% 0.25/0.48 ( ( member @ A @ X3 @ A4 )
% 0.25/0.48 => ( ( B2
% 0.25/0.48 = ( F3 @ X3 ) )
% 0.25/0.48 => ( member @ B @ B2 @ ( image @ A @ B @ F3 @ A4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % rev_image_eqI
% 0.25/0.48 thf(fact_234_Compr__image__eq,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P2: A > $o] :
% 0.25/0.48 ( ( collect @ A
% 0.25/0.48 @ ^ [X4: A] :
% 0.25/0.48 ( ( member @ A @ X4 @ ( image @ B @ A @ F3 @ A4 ) )
% 0.25/0.48 & ( P2 @ X4 ) ) )
% 0.25/0.48 = ( image @ B @ A @ F3
% 0.25/0.48 @ ( collect @ B
% 0.25/0.48 @ ^ [X4: B] :
% 0.25/0.48 ( ( member @ B @ X4 @ A4 )
% 0.25/0.48 & ( P2 @ ( F3 @ X4 ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Compr_image_eq
% 0.25/0.48 thf(fact_235_image__image,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,C: $tType,F3: B > A,G3: C > B,A4: set @ C] :
% 0.25/0.48 ( ( image @ B @ A @ F3 @ ( image @ C @ B @ G3 @ A4 ) )
% 0.25/0.48 = ( image @ C @ A
% 0.25/0.48 @ ^ [X4: C] : ( F3 @ ( G3 @ X4 ) )
% 0.25/0.48 @ A4 ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_image
% 0.25/0.48 thf(fact_236_imageE,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,B2: A,F3: B > A,A4: set @ B] :
% 0.25/0.48 ( ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A4 ) )
% 0.25/0.48 => ~ ! [X: B] :
% 0.25/0.48 ( ( B2
% 0.25/0.48 = ( F3 @ X ) )
% 0.25/0.48 => ~ ( member @ B @ X @ A4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % imageE
% 0.25/0.48 thf(fact_237_image__Collect__subsetI,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,P2: A > $o,F3: A > B,B4: set @ B] :
% 0.25/0.48 ( ! [X: A] :
% 0.25/0.48 ( ( P2 @ X )
% 0.25/0.48 => ( member @ B @ ( F3 @ X ) @ B4 ) )
% 0.25/0.48 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ ( collect @ A @ P2 ) ) @ B4 ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_Collect_subsetI
% 0.25/0.48 thf(fact_238_Osub,axiom,
% 0.25/0.48 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( static_FVO @ a @ c @ ode2 ) ) @ v ).
% 0.25/0.48
% 0.25/0.48 % Osub
% 0.25/0.48 thf(fact_239_all__subset__image,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P2: ( set @ A ) > $o] :
% 0.25/0.48 ( ( ! [B6: set @ A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F3 @ A4 ) )
% 0.25/0.48 => ( P2 @ B6 ) ) )
% 0.25/0.48 = ( ! [B6: set @ B] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ B ) @ B6 @ A4 )
% 0.25/0.48 => ( P2 @ ( image @ B @ A @ F3 @ B6 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % all_subset_image
% 0.25/0.48 thf(fact_240_bvo__to__fvo,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,X3: A,ODE: ode @ B @ A] :
% 0.25/0.48 ( ( member @ ( sum_sum @ A @ A ) @ ( sum_Inl @ A @ A @ X3 ) @ ( static_BVO @ B @ A @ ODE ) )
% 0.25/0.48 => ( member @ A @ X3 @ ( static_FVO @ B @ A @ ODE ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % bvo_to_fvo
% 0.25/0.48 thf(fact_241_ids_Obvo__to__fvo,axiom,
% 0.25/0.48 ! [Sz: $tType,Sf: $tType,Sc: $tType,A: $tType,B: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ Sc )
% 0.25/0.48 & ( finite_finite @ Sf )
% 0.25/0.48 & ( finite_finite @ Sz )
% 0.25/0.48 & ( linorder @ Sz ) )
% 0.25/0.48 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X3: A,ODE: ode @ B @ A] :
% 0.25/0.48 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.25/0.48 => ( ( member @ ( sum_sum @ A @ A ) @ ( sum_Inl @ A @ A @ X3 ) @ ( static_BVO @ B @ A @ ODE ) )
% 0.25/0.48 => ( member @ A @ X3 @ ( static_FVO @ B @ A @ ODE ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ids.bvo_to_fvo
% 0.25/0.48 thf(fact_242_VSA,axiom,
% 0.25/0.48 ( denotational_VSagree @ c @ ( sol @ ( zero_zero @ real ) ) @ a2
% 0.25/0.48 @ ( collect @ c
% 0.25/0.48 @ ^ [Uu: c] :
% 0.25/0.48 ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_BVO @ a @ c @ ode2 ) )
% 0.25/0.48 | ( member @ ( sum_sum @ c @ d ) @ ( sum_Inl @ c @ d @ Uu ) @ ( image @ c @ ( sum_sum @ c @ d ) @ ( sum_Inl @ c @ d ) @ ( static_FVO @ a @ c @ ode2 ) ) )
% 0.25/0.48 | ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_FVF @ a @ b @ c @ p ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % VSA
% 0.25/0.48 thf(fact_243_ode__to__fvo,axiom,
% 0.25/0.48 ! [C: $tType,A: $tType,B: $tType,X3: A,I: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A] :
% 0.25/0.48 ( ( member @ A @ X3 @ ( denota811733865E_vars @ B @ C @ A @ I @ ODE ) )
% 0.25/0.48 => ( member @ A @ X3 @ ( static_FVO @ B @ A @ ODE ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ode_to_fvo
% 0.25/0.48 thf(fact_244_VSagree__sub,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( finite_finite @ A )
% 0.25/0.48 => ! [A4: set @ A,B4: set @ A,Nu: finite_Cartesian_vec @ real @ A,Omega2: finite_Cartesian_vec @ real @ A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.25/0.48 => ( ( denotational_VSagree @ A @ Nu @ Omega2 @ B4 )
% 0.25/0.48 => ( denotational_VSagree @ A @ Nu @ Omega2 @ A4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % VSagree_sub
% 0.25/0.48 thf(fact_245_VSagree__supset,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( finite_finite @ A )
% 0.25/0.48 => ! [B4: set @ A,A4: set @ A,Nu: finite_Cartesian_vec @ real @ A,Nu2: finite_Cartesian_vec @ real @ A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.25/0.48 => ( ( denotational_VSagree @ A @ Nu @ Nu2 @ A4 )
% 0.25/0.48 => ( denotational_VSagree @ A @ Nu @ Nu2 @ B4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % VSagree_supset
% 0.25/0.48 thf(fact_246_VSagree__refl,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( finite_finite @ A )
% 0.25/0.48 => ! [Nu: finite_Cartesian_vec @ real @ A,A4: set @ A] : ( denotational_VSagree @ A @ Nu @ Nu @ A4 ) ) ).
% 0.25/0.48
% 0.25/0.48 % VSagree_refl
% 0.25/0.48 thf(fact_247_ids_Oode__to__fvo,axiom,
% 0.25/0.48 ! [C: $tType,Sz: $tType,Sf: $tType,Sc: $tType,A: $tType,B: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ Sc )
% 0.25/0.48 & ( finite_finite @ Sf )
% 0.25/0.48 & ( finite_finite @ Sz )
% 0.25/0.48 & ( linorder @ Sz ) )
% 0.25/0.48 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X3: A,I: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A] :
% 0.25/0.48 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.25/0.48 => ( ( member @ A @ X3 @ ( denota811733865E_vars @ B @ C @ A @ I @ ODE ) )
% 0.25/0.48 => ( member @ A @ X3 @ ( static_FVO @ B @ A @ ODE ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ids.ode_to_fvo
% 0.25/0.48 thf(fact_248_OVsub_H,axiom,
% 0.25/0.48 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( oDE_dom @ a @ c @ ode2 ) ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( oDE_dom @ a @ c @ ode2 ) ) ) @ v ).
% 0.25/0.48
% 0.25/0.48 % OVsub'
% 0.25/0.48 thf(fact_249_ids_Ovne23,axiom,
% 0.25/0.48 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ Sz )
% 0.25/0.48 & ( linorder @ Sz )
% 0.25/0.48 & ( finite_finite @ Sf )
% 0.25/0.48 & ( finite_finite @ Sc ) )
% 0.25/0.48 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.25/0.48 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.25/0.48 => ( Vid2 != Vid3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ids.vne23
% 0.25/0.48 thf(fact_250_sup_Oright__idem,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( semilattice_sup @ A )
% 0.25/0.48 => ! [A2: A,B2: A] :
% 0.25/0.48 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
% 0.25/0.48 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % sup.right_idem
% 0.25/0.48 thf(fact_251_sup__left__idem,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( semilattice_sup @ A )
% 0.25/0.48 => ! [X3: A,Y3: A] :
% 0.25/0.48 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y3 ) )
% 0.25/0.48 = ( sup_sup @ A @ X3 @ Y3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % sup_left_idem
% 0.25/0.48 thf(fact_252_sup_Oleft__idem,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( semilattice_sup @ A )
% 0.25/0.48 => ! [A2: A,B2: A] :
% 0.25/0.48 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
% 0.25/0.48 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % sup.left_idem
% 0.25/0.48 thf(fact_253_sup__idem,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( semilattice_sup @ A )
% 0.25/0.48 => ! [X3: A] :
% 0.25/0.48 ( ( sup_sup @ A @ X3 @ X3 )
% 0.25/0.48 = X3 ) ) ).
% 0.25/0.48
% 0.25/0.48 % sup_idem
% 0.25/0.48 thf(fact_254_sup_Oidem,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( semilattice_sup @ A )
% 0.25/0.48 => ! [A2: A] :
% 0.25/0.48 ( ( sup_sup @ A @ A2 @ A2 )
% 0.25/0.48 = A2 ) ) ).
% 0.25/0.48
% 0.25/0.48 % sup.idem
% 0.25/0.48
% 0.25/0.48 % Subclasses (1)
% 0.25/0.48 thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( finite_finite @ A )
% 0.25/0.48 => ( type @ A ) ) ).
% 0.25/0.48
% 0.25/0.48 % Type constructors (81)
% 0.25/0.48 thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( ordere890947078_space @ A9 )
% 0.25/0.48 & ( finite_finite @ A10 ) )
% 0.25/0.48 => ( ordere890947078_space @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space_1,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( ordere890947078_space @ A9 )
% 0.25/0.48 & ( ordere890947078_space @ A10 ) )
% 0.25/0.48 => ( ordere890947078_space @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_2,axiom,
% 0.25/0.48 ordere890947078_space @ real ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( topolo503727757_space @ A10 )
% 0.25/0.48 => ( topolo503727757_space @ ( A9 > A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( semilattice_sup @ A10 )
% 0.25/0.48 => ( semilattice_sup @ ( A9 > A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( boolean_algebra @ A10 )
% 0.25/0.48 => ( boolean_algebra @ ( A9 > A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( preorder @ A10 )
% 0.25/0.48 => ( preorder @ ( A9 > A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ A9 )
% 0.25/0.48 & ( finite_finite @ A10 ) )
% 0.25/0.48 => ( finite_finite @ ( A9 > A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( order @ A10 )
% 0.25/0.48 => ( order @ ( A9 > A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_fun___Orderings_Oord,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ord @ A10 )
% 0.25/0.48 => ( ord @ ( A9 > A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_fun___Groups_Ouminus,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( uminus @ A10 )
% 0.25/0.48 => ( uminus @ ( A9 > A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_3,axiom,
% 0.25/0.48 ! [A9: $tType] : ( semilattice_sup @ ( set @ A9 ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_4,axiom,
% 0.25/0.48 ! [A9: $tType] : ( boolean_algebra @ ( set @ A9 ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
% 0.25/0.48 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Set_Oset___Finite__Set_Ofinite_6,axiom,
% 0.25/0.48 ! [A9: $tType] :
% 0.25/0.48 ( ( finite_finite @ A9 )
% 0.25/0.48 => ( finite_finite @ ( set @ A9 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Set_Oset___Orderings_Oorder_7,axiom,
% 0.25/0.48 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 0.25/0.48 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Set_Oset___Groups_Ouminus_9,axiom,
% 0.25/0.48 ! [A9: $tType] : ( uminus @ ( set @ A9 ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Set_Oset___Groups_Ozero,axiom,
% 0.25/0.48 ! [A9: $tType] :
% 0.25/0.48 ( ( zero @ A9 )
% 0.25/0.48 => ( zero @ ( set @ A9 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_10,axiom,
% 0.25/0.48 topolo503727757_space @ $o ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_11,axiom,
% 0.25/0.48 semilattice_sup @ $o ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_12,axiom,
% 0.25/0.48 boolean_algebra @ $o ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_HOL_Obool___Orderings_Opreorder_13,axiom,
% 0.25/0.48 preorder @ $o ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.25/0.48 linorder @ $o ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_HOL_Obool___Finite__Set_Ofinite_14,axiom,
% 0.25/0.48 finite_finite @ $o ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_HOL_Obool___Orderings_Oorder_15,axiom,
% 0.25/0.48 order @ $o ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_HOL_Obool___Orderings_Oord_16,axiom,
% 0.25/0.48 ord @ $o ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_HOL_Obool___Groups_Ouminus_17,axiom,
% 0.25/0.48 uminus @ $o ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
% 0.25/0.48 real_V55928688vector @ real ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Real_Oreal___Lattice__Algebras_Olattice__ab__group__add,axiom,
% 0.25/0.48 lattic1601792062up_add @ real ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_18,axiom,
% 0.25/0.48 topolo503727757_space @ real ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring,axiom,
% 0.25/0.48 linord1659791738miring @ real ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
% 0.25/0.48 linord219039673up_add @ real ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
% 0.25/0.48 ordered_ab_group_add @ real ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_19,axiom,
% 0.25/0.48 semilattice_sup @ real ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Real_Oreal___Orderings_Opreorder_20,axiom,
% 0.25/0.48 preorder @ real ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Real_Oreal___Orderings_Olinorder_21,axiom,
% 0.25/0.48 linorder @ real ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Real_Oreal___Orderings_Ono__top,axiom,
% 0.25/0.48 no_top @ real ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Real_Oreal___Orderings_Ono__bot,axiom,
% 0.25/0.48 no_bot @ real ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 0.25/0.48 group_add @ real ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Real_Oreal___Orderings_Oorder_22,axiom,
% 0.25/0.48 order @ real ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Real_Oreal___Orderings_Oord_23,axiom,
% 0.25/0.48 ord @ real ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Real_Oreal___Groups_Ouminus_24,axiom,
% 0.25/0.48 uminus @ real ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Real_Oreal___Groups_Ozero_25,axiom,
% 0.25/0.48 zero @ real ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_26,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ A9 )
% 0.25/0.48 & ( finite_finite @ A10 ) )
% 0.25/0.48 => ( finite_finite @ ( sum_sum @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_27,axiom,
% 0.25/0.48 ! [A9: $tType] : ( semilattice_sup @ ( filter @ A9 ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Filter_Ofilter___Orderings_Opreorder_28,axiom,
% 0.25/0.48 ! [A9: $tType] : ( preorder @ ( filter @ A9 ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Filter_Ofilter___Orderings_Oorder_29,axiom,
% 0.25/0.48 ! [A9: $tType] : ( order @ ( filter @ A9 ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Filter_Ofilter___Orderings_Oord_30,axiom,
% 0.25/0.48 ! [A9: $tType] : ( ord @ ( filter @ A9 ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__normed__vector_31,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( real_V55928688vector @ A9 )
% 0.25/0.48 & ( real_V55928688vector @ A10 ) )
% 0.25/0.48 => ( real_V55928688vector @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_32,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( topolo503727757_space @ A9 )
% 0.25/0.48 & ( topolo503727757_space @ A10 ) )
% 0.25/0.48 => ( topolo503727757_space @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__group__add_33,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( ordere890947078_space @ A9 )
% 0.25/0.48 & ( ordere890947078_space @ A10 ) )
% 0.25/0.48 => ( ordered_ab_group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Oprod___Lattices_Osemilattice__sup_34,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( semilattice_sup @ A9 )
% 0.25/0.48 & ( semilattice_sup @ A10 ) )
% 0.25/0.48 => ( semilattice_sup @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Oprod___Lattices_Oboolean__algebra_35,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( boolean_algebra @ A9 )
% 0.25/0.48 & ( boolean_algebra @ A10 ) )
% 0.25/0.48 => ( boolean_algebra @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_36,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( preorder @ A9 )
% 0.25/0.48 & ( preorder @ A10 ) )
% 0.25/0.48 => ( preorder @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_37,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ A9 )
% 0.25/0.48 & ( finite_finite @ A10 ) )
% 0.25/0.48 => ( finite_finite @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Oprod___Groups_Ogroup__add_38,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( group_add @ A9 )
% 0.25/0.48 & ( group_add @ A10 ) )
% 0.25/0.48 => ( group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Oprod___Orderings_Oorder_39,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( order @ A9 )
% 0.25/0.48 & ( order @ A10 ) )
% 0.25/0.48 => ( order @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Oprod___Orderings_Oord_40,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( ord @ A9 )
% 0.25/0.48 & ( ord @ A10 ) )
% 0.25/0.48 => ( ord @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Oprod___Groups_Ouminus_41,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( uminus @ A9 )
% 0.25/0.48 & ( uminus @ A10 ) )
% 0.25/0.48 => ( uminus @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Oprod___Groups_Ozero_42,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( zero @ A9 )
% 0.25/0.48 & ( zero @ A10 ) )
% 0.25/0.48 => ( zero @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_43,axiom,
% 0.25/0.48 semilattice_sup @ product_unit ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Ounit___Lattices_Oboolean__algebra_44,axiom,
% 0.25/0.48 boolean_algebra @ product_unit ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_45,axiom,
% 0.25/0.48 preorder @ product_unit ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_46,axiom,
% 0.25/0.48 linorder @ product_unit ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_47,axiom,
% 0.25/0.48 finite_finite @ product_unit ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Ounit___Orderings_Oorder_48,axiom,
% 0.25/0.48 order @ product_unit ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Ounit___Orderings_Oord_49,axiom,
% 0.25/0.48 ord @ product_unit ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Product__Type_Ounit___Groups_Ouminus_50,axiom,
% 0.25/0.48 uminus @ product_unit ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Finite__Cartesian__Product_Ovec___Real__Vector__Spaces_Oreal__normed__vector_51,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( real_V55928688vector @ A9 )
% 0.25/0.48 & ( finite_finite @ A10 ) )
% 0.25/0.48 => ( real_V55928688vector @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Finite__Cartesian__Product_Ovec___Topological__Spaces_Otopological__space_52,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( topolo503727757_space @ A9 )
% 0.25/0.48 & ( finite_finite @ A10 ) )
% 0.25/0.48 => ( topolo503727757_space @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__group__add_53,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( ordere890947078_space @ A9 )
% 0.25/0.48 & ( finite_finite @ A10 ) )
% 0.25/0.48 => ( ordered_ab_group_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Osemilattice__sup_54,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( ordere890947078_space @ A9 )
% 0.25/0.48 & ( finite_finite @ A10 ) )
% 0.25/0.48 => ( semilattice_sup @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Opreorder_55,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( order @ A9 )
% 0.25/0.48 & ( finite_finite @ A10 ) )
% 0.25/0.48 => ( preorder @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_56,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( linorder @ A9 )
% 0.25/0.48 & ( cARD_1 @ A10 ) )
% 0.25/0.48 => ( linorder @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_57,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( finite_finite @ A9 )
% 0.25/0.48 & ( finite_finite @ A10 ) )
% 0.25/0.48 => ( finite_finite @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ogroup__add_58,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( group_add @ A9 )
% 0.25/0.48 & ( finite_finite @ A10 ) )
% 0.25/0.48 => ( group_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oorder_59,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( order @ A9 )
% 0.25/0.48 & ( finite_finite @ A10 ) )
% 0.25/0.48 => ( order @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oord_60,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( ord @ A9 )
% 0.25/0.48 & ( finite_finite @ A10 ) )
% 0.25/0.48 => ( ord @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ouminus_61,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( uminus @ A9 )
% 0.25/0.48 & ( finite_finite @ A10 ) )
% 0.25/0.48 => ( uminus @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_62,axiom,
% 0.25/0.48 ! [A9: $tType,A10: $tType] :
% 0.25/0.48 ( ( ( zero @ A9 )
% 0.25/0.48 & ( finite_finite @ A10 ) )
% 0.25/0.48 => ( zero @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Free types (3)
% 0.25/0.48 thf(tfree_0,hypothesis,
% 0.25/0.48 finite_finite @ c ).
% 0.25/0.48
% 0.25/0.48 thf(tfree_1,hypothesis,
% 0.25/0.48 finite_finite @ a ).
% 0.25/0.48
% 0.25/0.48 thf(tfree_2,hypothesis,
% 0.25/0.48 finite_finite @ b ).
% 0.25/0.48
% 0.25/0.48 % Conjectures (1)
% 0.25/0.48 thf(conj_0,conjecture,
% 0.25/0.48 denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ j @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( sol @ s ) ) @ ( denotational_mk_xode @ a @ b @ c @ j @ ode2 @ ( sol @ s ) ) @ ( denotational_semBV @ a @ b @ c @ j @ ode2 ) ).
% 0.25/0.48
% 0.25/0.48 %------------------------------------------------------------------------------
% 0.25/0.48 ------------------------
% 2.84/1.58 ---- Embedded in THF ---
% 2.84/1.58 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.84/1.58 %%% Generated on Tue Feb 24 08:24:02 EST 2026
% 2.84/1.58 %%% using '$$dhol' embedding, version 1.3.0.
% 2.84/1.58 %%% Logic specification used:
% 2.84/1.58 %%% thf(spec, logic, $$dhol).
% 2.84/1.58
% 2.84/1.58 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.c1Zf8XeWV8/DTF2DTF26468.p
% 2.84/1.58 thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext, type, denota1663640101rp_ext: ($tType > ($tType > ($tType > ($tType > $tType))))).
% 2.84/1.58 thf(ty_t_Finite__Cartesian__Product_Ovec, type, finite_Cartesian_vec: ($tType > ($tType > $tType))).
% 2.84/1.58 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 2.84/1.58 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.84/1.58 thf(ty_t_Syntax_Oformula, type, formula: ($tType > ($tType > ($tType > $tType)))).
% 2.84/1.58 thf(ty_t_Sum__Type_Osum, type, sum_sum: ($tType > ($tType > $tType))).
% 2.84/1.58 thf(ty_t_Filter_Ofilter, type, filter: ($tType > $tType)).
% 2.84/1.58 thf(ty_t_Syntax_Otrm, type, trm: ($tType > ($tType > $tType))).
% 2.84/1.58 thf(ty_t_Syntax_OODE, type, ode: ($tType > ($tType > $tType))).
% 2.84/1.58 thf(ty_t_Syntax_Ohp, type, hp: ($tType > ($tType > ($tType > $tType)))).
% 2.84/1.58 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.84/1.58 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.84/1.58 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.84/1.58 thf(ty_tf_d, type, d: $tType).
% 2.84/1.58 thf(ty_tf_c, type, c: $tType).
% 2.84/1.58 thf(ty_tf_b, type, b: $tType).
% 2.84/1.58 thf(ty_tf_a, type, a: $tType).
% 2.84/1.58 thf(sy_cl_Cardinality_OCARD__1, type, cARD_1: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space, type, ordere890947078_space: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Groups_Ouminus, type, uminus: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Lattices_Oboolean__algebra, type, boolean_algebra: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Topological__Spaces_Otopological__space, type, topolo503727757_space: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Lattice__Algebras_Olattice__ab__group__add, type, lattic1601792062up_add: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector, type, real_V55928688vector: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd, type, euclid1678468529ng_gcd: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral, type, unique1598680935umeral: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring, type, euclid1155270486miring: (!> [A:$tType]: ($o))).
% 2.84/1.58 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca1785829860lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 2.84/1.58 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__fml, type, coinci1993344360de_fml: (!> [A:$tType,B:$tType,C:$tType]: (((((formula @ A) @ B) @ C) > $o)))).
% 2.84/1.58 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__hp, type, coinci32832645ide_hp: (!> [A:$tType,B:$tType,C:$tType]: (((((hp @ A) @ B) @ C) > (((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > $o)))))).
% 2.84/1.58 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__hp_H, type, coinci554967470ide_hp: (!> [A:$tType,B:$tType,C:$tType]: (((((hp @ A) @ B) @ C) > $o)))).
% 2.84/1.58 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Oode__sem__equiv, type, coinci1495174342_equiv: (!> [A:$tType,B:$tType,C:$tType]: (((((hp @ A) @ B) @ C) > (((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > $o))))).
% 2.84/1.58 thf(sy_c_Denotational__Semantics_OIagree, type, denotational_Iagree: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > ((set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))) > $o)))))).
% 2.84/1.58 thf(sy_c_Denotational__Semantics_OODE__sem, type, denotational_ODE_sem: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((ode @ A) @ C) > (((finite_Cartesian_vec @ real) @ C) > ((finite_Cartesian_vec @ real) @ C))))))).
% 2.84/1.58 thf(sy_c_Denotational__Semantics_OODE__vars, type, denota811733865E_vars: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((ode @ A) @ C) > (set @ C)))))).
% 2.84/1.58 thf(sy_c_Denotational__Semantics_OVSagree, type, denotational_VSagree: (!> [C:$tType]: ((((finite_Cartesian_vec @ real) @ C) > (((finite_Cartesian_vec @ real) @ C) > ((set @ C) > $o)))))).
% 2.84/1.58 thf(sy_c_Denotational__Semantics_OVagree, type, denotational_Vagree: (!> [C:$tType]: ((((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) > (((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) > ((set @ ((sum_sum @ C) @ C)) > $o)))))).
% 2.84/1.58 thf(sy_c_Denotational__Semantics_Ofml__sem, type, denotational_fml_sem: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > ((((formula @ A) @ B) @ C) > (set @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)))))))).
% 2.84/1.58 thf(sy_c_Denotational__Semantics_Omk__v, type, denotational_mk_v: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((ode @ A) @ C) > (((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) > (((finite_Cartesian_vec @ real) @ C) > ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))))))))).
% 2.84/1.58 thf(sy_c_Denotational__Semantics_Omk__xode, type, denotational_mk_xode: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((ode @ A) @ C) > (((finite_Cartesian_vec @ real) @ C) > ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)))))))).
% 2.84/1.58 thf(sy_c_Denotational__Semantics_OsemBV, type, denotational_semBV: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((ode @ A) @ C) > (set @ ((sum_sum @ C) @ C))))))).
% 2.84/1.58 thf(sy_c_Deriv_Ohas__vector__derivative, type, has_ve2132708402vative: (!> [B:$tType]: (((real > B) > (B > ((filter @ real) > $o)))))).
% 2.84/1.58 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux, type, unique455577585es_aux: (!> [A:$tType]: ((((product_prod @ A) @ A) > $o)))).
% 2.84/1.58 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 2.84/1.58 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.84/1.58 thf(sy_c_HOL_OThe, type, the: (!> [A:$tType]: (((A > $o) > A)))).
% 2.84/1.58 thf(sy_c_Ids_Oids, type, ids: (!> [Sz:$tType,Sf:$tType,Sc:$tType]: ((Sz > (Sz > (Sz > (Sf > (Sf > (Sf > (Sc > (Sc > (Sc > (Sc > $o))))))))))))).
% 2.84/1.58 thf(sy_c_Initial__Value__Problem_Osolves__ode, type, initia1685620758es_ode: (!> [A:$tType]: (((real > A) > ((real > (A > A)) > ((set @ real) > ((set @ A) > $o))))))).
% 2.84/1.58 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 2.84/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.84/1.58 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.84/1.58 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.84/1.58 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.84/1.58 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod, type, product_rec_set_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > (T > $o)))))).
% 2.84/1.58 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit, type, product_rec_set_unit: (!> [T:$tType]: ((T > (product_unit > (T > $o)))))).
% 2.84/1.58 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit, type, product_rec_unit: (!> [T:$tType]: ((T > (product_unit > T))))).
% 2.84/1.58 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.84/1.58 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.84/1.58 thf(sy_c_Set__Interval_Oord__class_OatLeast, type, set_ord_atLeast: (!> [A:$tType]: ((A > (set @ A))))).
% 2.84/1.58 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost, type, set_or331188842AtMost: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 2.84/1.58 thf(sy_c_Set__Interval_Oord__class_OatMost, type, set_ord_atMost: (!> [A:$tType]: ((A > (set @ A))))).
% 2.84/1.58 thf(sy_c_Static__Semantics_OBVO, type, static_BVO: (!> [A:$tType,C:$tType]: ((((ode @ A) @ C) > (set @ ((sum_sum @ C) @ C)))))).
% 2.84/1.58 thf(sy_c_Static__Semantics_OFVF, type, static_FVF: (!> [A:$tType,B:$tType,C:$tType]: (((((formula @ A) @ B) @ C) > (set @ ((sum_sum @ C) @ C)))))).
% 2.84/1.58 thf(sy_c_Static__Semantics_OFVO, type, static_FVO: (!> [A:$tType,C:$tType]: ((((ode @ A) @ C) > (set @ C))))).
% 2.84/1.58 thf(sy_c_Static__Semantics_OSIGF, type, static_SIGF: (!> [A:$tType,B:$tType,C:$tType]: (((((formula @ A) @ B) @ C) > (set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))))))).
% 2.84/1.58 thf(sy_c_Sum__Type_OInl, type, sum_Inl: (!> [A:$tType,B:$tType]: ((A > ((sum_sum @ A) @ B))))).
% 2.84/1.58 thf(sy_c_Sum__Type_OInr, type, sum_Inr: (!> [B:$tType,A:$tType]: ((B > ((sum_sum @ A) @ B))))).
% 2.84/1.58 thf(sy_c_Syntax_OEquiv, type, equiv: (!> [A:$tType,B:$tType,C:$tType]: (((((formula @ A) @ B) @ C) > ((((formula @ A) @ B) @ C) > (((formula @ A) @ B) @ C)))))).
% 2.84/1.58 thf(sy_c_Syntax_OImplies, type, implies: (!> [A:$tType,B:$tType,C:$tType]: (((((formula @ A) @ B) @ C) > ((((formula @ A) @ B) @ C) > (((formula @ A) @ B) @ C)))))).
% 2.84/1.58 thf(sy_c_Syntax_OODE__dom, type, oDE_dom: (!> [A:$tType,C:$tType]: ((((ode @ A) @ C) > (set @ C))))).
% 2.84/1.58 thf(sy_c_Syntax_Ofsafe, type, fsafe: (!> [A:$tType,B:$tType,C:$tType]: (((((formula @ A) @ B) @ C) > $o)))).
% 2.84/1.58 thf(sy_c_Syntax_Ohp_OEvolveODE, type, evolveODE: (!> [A:$tType,C:$tType,B:$tType]: ((((ode @ A) @ C) > ((((formula @ A) @ B) @ C) > (((hp @ A) @ B) @ C)))))).
% 2.84/1.58 thf(sy_c_Syntax_Ohpsafe, type, hpsafe: (!> [A:$tType,B:$tType,C:$tType]: (((((hp @ A) @ B) @ C) > $o)))).
% 2.84/1.58 thf(sy_c_Syntax_Oosafe, type, osafe: (!> [A:$tType,C:$tType]: ((((ode @ A) @ C) > $o)))).
% 2.84/1.58 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within, type, topolo507301023within: (!> [A:$tType]: ((A > ((set @ A) > (filter @ A)))))).
% 2.84/1.58 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.84/1.58 thf(sy_v_I____, type, i: ((((denota1663640101rp_ext @ a) @ b) @ c) @ product_unit)).
% 2.84/1.58 thf(sy_v_J____, type, j: ((((denota1663640101rp_ext @ a) @ b) @ c) @ product_unit)).
% 2.84/1.58 thf(sy_v_ODE____, type, ode2: ((ode @ a) @ c)).
% 2.84/1.58 thf(sy_v_P____, type, p: (((formula @ a) @ b) @ c)).
% 2.84/1.58 thf(sy_v_V____, type, v: (set @ ((sum_sum @ c) @ c))).
% 2.84/1.58 thf(sy_v_a____, type, a2: ((finite_Cartesian_vec @ real) @ c)).
% 2.84/1.58 thf(sy_v_aa____, type, aa: ((finite_Cartesian_vec @ real) @ c)).
% 2.84/1.58 thf(sy_v_ab____, type, ab: ((finite_Cartesian_vec @ real) @ c)).
% 2.84/1.58 thf(sy_v_b____, type, b2: ((finite_Cartesian_vec @ real) @ c)).
% 2.84/1.58 thf(sy_v_ba____, type, ba: ((finite_Cartesian_vec @ real) @ c)).
% 2.84/1.58 thf(sy_v_bb____, type, bb: ((finite_Cartesian_vec @ real) @ c)).
% 2.84/1.58 thf(sy_v_s____, type, s: real).
% 2.84/1.58 thf(sy_v_sol____, type, sol: (real > ((finite_Cartesian_vec @ real) @ c))).
% 2.84/1.58 thf(sy_v_t____, type, t: real).
% 2.84/1.58 thf(fact_0_osafe, axiom, (((osafe @ a) @ c) @ ode2)).
% 2.84/1.58 thf(fact_1__092_060open_062Vagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Iaa_M_Aba_J_A_I_N_AsemBV_AJ_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Imk__xode_AJ_AODE_A_Isol_As_J_J_A_IsemBV_AJ_AODE_J_092_060close_062, axiom, (((((denotational_Vagree @ c) @ (((((((denotational_mk_v @ a) @ b) @ c) @ j) @ ode2) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ aa) @ ba)) @ (sol @ s))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ aa) @ ba)) @ ((uminus_uminus @ (set @ ((sum_sum @ c) @ c))) @ (((((denotational_semBV @ a) @ b) @ c) @ j) @ ode2))) & ((((denotational_Vagree @ c) @ (((((((denotational_mk_v @ a) @ b) @ c) @ j) @ ode2) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ aa) @ ba)) @ (sol @ s))) @ ((((((denotational_mk_xode @ a) @ b) @ c) @ j) @ ode2) @ (sol @ s))) @ (((((denotational_semBV @ a) @ b) @ c) @ j) @ ode2)))).
% 2.84/1.58 thf(fact_2__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_Asa_J_J_A_Iaa_M_Aba_J_A_I_N_AsemBV_AJ_AODE_J_092_060close_062, axiom, (! [S:real]: (((((denotational_Vagree @ c) @ (((((((denotational_mk_v @ a) @ b) @ c) @ j) @ ode2) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ aa) @ ba)) @ (sol @ S))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ aa) @ ba)) @ ((uminus_uminus @ (set @ ((sum_sum @ c) @ c))) @ (((((denotational_semBV @ a) @ b) @ c) @ j) @ ode2)))))).
% 2.84/1.58 thf(fact_3_mk__v__agree, axiom, (! [C:$tType,A:$tType,B:$tType]: ((((finite_finite @ B) & ((finite_finite @ A) & (finite_finite @ C))) => (! [I:((((denota1663640101rp_ext @ B) @ C) @ A) @ product_unit),ODE:((ode @ B) @ A),Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),Sol:((finite_Cartesian_vec @ real) @ A)]: ((((((denotational_Vagree @ A) @ (((((((denotational_mk_v @ B) @ C) @ A) @ I) @ ODE) @ Nu) @ Sol)) @ Nu) @ ((uminus_uminus @ (set @ ((sum_sum @ A) @ A))) @ (((((denotational_semBV @ B) @ C) @ A) @ I) @ ODE))) & ((((denotational_Vagree @ A) @ (((((((denotational_mk_v @ B) @ C) @ A) @ I) @ ODE) @ Nu) @ Sol)) @ ((((((denotational_mk_xode @ B) @ C) @ A) @ I) @ ODE) @ Sol)) @ (((((denotational_semBV @ B) @ C) @ A) @ I) @ ODE))))))))).
% 2.84/1.58 thf(fact_4_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X2:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X2) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X2 = Y2)))))).
% 2.84/1.58 thf(fact_5_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) = ((A2 = A3) & (B2 = B3)))))).
% 2.84/1.58 thf(fact_6_mk__v__exists, axiom, (! [C:$tType,A:$tType,B:$tType]: ((((finite_finite @ B) & ((finite_finite @ A) & (finite_finite @ C))) => (! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),I:((((denota1663640101rp_ext @ B) @ C) @ A) @ product_unit),ODE:((ode @ B) @ A),Sol:((finite_Cartesian_vec @ real) @ A)]: ((? [Omega:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))]: ((((((denotational_Vagree @ A) @ Omega) @ Nu) @ ((uminus_uminus @ (set @ ((sum_sum @ A) @ A))) @ (((((denotational_semBV @ B) @ C) @ A) @ I) @ ODE))) & ((((denotational_Vagree @ A) @ Omega) @ ((((((denotational_mk_xode @ B) @ C) @ A) @ I) @ ODE) @ Sol)) @ (((((denotational_semBV @ B) @ C) @ A) @ I) @ ODE))))))))))).
% 2.84/1.58 thf(fact_7_VA, axiom, ((((denotational_Vagree @ c) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ a2) @ b2)) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ aa) @ ba)) @ v)).
% 2.84/1.58 thf(fact_8__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AI_AODE_A_Ia_M_Ab_J_A_Isol_Asa_J_J_A_Imk__xode_AI_AODE_A_Isol_Asa_J_J_A_IsemBV_AI_AODE_J_092_060close_062, axiom, (! [S:real]: (((((denotational_Vagree @ c) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ ode2) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ a2) @ b2)) @ (sol @ S))) @ ((((((denotational_mk_xode @ a) @ b) @ c) @ i) @ ode2) @ (sol @ S))) @ (((((denotational_semBV @ a) @ b) @ c) @ i) @ ode2))))).
% 2.84/1.58 thf(fact_9_VAOV, axiom, ((((denotational_Vagree @ c) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ a2) @ b2)) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ aa) @ ba)) @ (((static_BVO @ a) @ c) @ ode2))).
% 2.84/1.58 thf(fact_10_agree__comm, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A4:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),B4:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),V:(set @ ((sum_sum @ A) @ A))]: ((((((denotational_Vagree @ A) @ A4) @ B4) @ V) => ((((denotational_Vagree @ A) @ B4) @ A4) @ V)))))))).
% 2.84/1.58 thf(fact_11_agree__refl, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),A4:(set @ ((sum_sum @ A) @ A))]: (((((denotational_Vagree @ A) @ Nu) @ Nu) @ A4))))))).
% 2.84/1.58 thf(fact_12__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AI_AODE_A_Ia_M_Ab_J_A_Isol_Asa_J_J_A_Ia_M_Ab_J_A_I_N_AsemBV_AI_AODE_J_092_060close_062, axiom, (! [S:real]: (((((denotational_Vagree @ c) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ ode2) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ a2) @ b2)) @ (sol @ S))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ a2) @ b2)) @ ((uminus_uminus @ (set @ ((sum_sum @ c) @ c))) @ (((((denotational_semBV @ a) @ b) @ c) @ i) @ ode2)))))).
% 2.84/1.58 thf(fact_13_surj__pair, axiom, (! [A:$tType,B:$tType,P:((product_prod @ A) @ B)]: ((? [X:A,Y:B]: ((P = ((((product_Pair @ A) @ B) @ X) @ Y))))))).
% 2.84/1.58 thf(fact_14_prod__cases, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),P:((product_prod @ A) @ B)]: (((! [A5:A,B5:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A5) @ B5)))) => (P2 @ P))))).
% 2.84/1.58 thf(fact_15_OVsub, axiom, (((ord_less_eq @ (set @ ((sum_sum @ c) @ c))) @ (((static_BVO @ a) @ c) @ ode2)) @ v)).
% 2.84/1.58 thf(fact_16_veq, axiom, (((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb) = (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ ode2) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ a2) @ b2)) @ (sol @ t)))).
% 2.84/1.58 thf(fact_17_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B5:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A5) @ B5)))) => (P2 @ Prod))))).
% 2.84/1.58 thf(fact_18_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y3:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B5:B]: ((Y3 != ((((product_Pair @ A) @ B) @ A5) @ B5))))))))).
% 2.84/1.58 thf(fact_19_prod__induct7, axiom, (! [G:$tType,F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: (((! [A5:A,B5:B,C2:C,D2:D,E2:E,F2:F,G2:G]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2))))))))) => (P2 @ X3))))).
% 2.84/1.58 thf(fact_20_prod__induct6, axiom, (! [F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: (((! [A5:A,B5:B,C2:C,D2:D,E2:E,F2:F]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2)))))))) => (P2 @ X3))))).
% 2.84/1.58 thf(fact_21_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A5:A,B5:B,C2:C,D2:D,E2:E]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P2 @ X3))))).
% 2.84/1.58 thf(fact_22_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A5:A,B5:B,C2:C,D2:D]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C2) @ D2)))))) => (P2 @ X3))))).
% 2.84/1.58 thf(fact_23_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B5:B,C2:C]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B5) @ C2))))) => (P2 @ X3))))).
% 2.84/1.58 thf(fact_24_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,G:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: ((~ ((! [A5:A,B5:B,C2:C,D2:D,E2:E,F2:F,G2:G]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2)))))))))))))).
% 2.84/1.58 thf(fact_25_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: ((~ ((! [A5:A,B5:B,C2:C,D2:D,E2:E,F2:F]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2))))))))))))).
% 2.84/1.58 thf(fact_26_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A5:A,B5:B,C2:C,D2:D,E2:E]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 2.84/1.58 thf(fact_27_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A5:A,B5:B,C2:C,D2:D]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C2) @ D2))))))))))).
% 2.84/1.58 thf(fact_28_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B5:B,C2:C]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B5) @ C2)))))))))).
% 2.84/1.58 thf(fact_29_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) => (~ (((A2 = A3) => (B2 != B3)))))))).
% 2.84/1.58 thf(fact_30_OVsub_H_H, axiom, (((ord_less_eq @ (set @ ((sum_sum @ c) @ c))) @ (((((denotational_semBV @ a) @ b) @ c) @ i) @ ode2)) @ v)).
% 2.84/1.58 thf(fact_31_semBVsub, axiom, (((ord_less_eq @ (set @ ((sum_sum @ c) @ c))) @ (((((denotational_semBV @ a) @ b) @ c) @ i) @ ode2)) @ (((static_BVO @ a) @ c) @ ode2))).
% 2.84/1.58 thf(fact_32_uminus__Pair, axiom, (! [A:$tType,B:$tType]: ((((uminus @ B) & (uminus @ A)) => (! [A2:A,B2:B]: ((((uminus_uminus @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((((product_Pair @ A) @ B) @ ((uminus_uminus @ A) @ A2)) @ ((uminus_uminus @ B) @ B2))))))))).
% 2.84/1.58 thf(fact_33_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 2.84/1.58 thf(fact_34_ComplI, axiom, (! [A:$tType,C3:A,A4:(set @ A)]: (((~ ((((member @ A) @ C3) @ A4))) => (((member @ A) @ C3) @ ((uminus_uminus @ (set @ A)) @ A4)))))).
% 2.84/1.58 thf(fact_35_Compl__iff, axiom, (! [A:$tType,C3:A,A4:(set @ A)]: (((((member @ A) @ C3) @ ((uminus_uminus @ (set @ A)) @ A4)) = (~ ((((member @ A) @ C3) @ A4))))))).
% 2.84/1.58 thf(fact_36_Compl__eq__Compl__iff, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((uminus_uminus @ (set @ A)) @ A4) = ((uminus_uminus @ (set @ A)) @ B4)) = (A4 = B4))))).
% 2.84/1.58 thf(fact_37_verit__minus__simplify_I4_J, axiom, (! [B:$tType]: (((group_add @ B) => (! [B2:B]: ((((uminus_uminus @ B) @ ((uminus_uminus @ B) @ B2)) = B2))))))).
% 2.84/1.58 thf(fact_38_uminus__apply, axiom, (! [B:$tType,A:$tType]: (((uminus @ B) => ((uminus_uminus @ (A > B)) = (^ [A6:(A > B),X4:A]: (((uminus_uminus @ B) @ (A6 @ X4))))))))).
% 2.84/1.58 thf(fact_39_add_Oinverse__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ A2)) = A2))))))).
% 2.84/1.58 thf(fact_40_subset__antisym, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ A4) => (A4 = B4)))))).
% 2.84/1.58 thf(fact_41_subsetI, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((! [X:A]: (((((member @ A) @ X) @ A4) => (((member @ A) @ X) @ B4)))) => (((ord_less_eq @ (set @ A)) @ A4) @ B4))))).
% 2.84/1.58 thf(fact_42_compl__eq__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X3:A,Y3:A]: (((((uminus_uminus @ A) @ X3) = ((uminus_uminus @ A) @ Y3)) = (X3 = Y3)))))))).
% 2.84/1.58 thf(fact_43_double__compl, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X3:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ X3)) = X3))))))).
% 2.84/1.58 thf(fact_44_neg__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((uminus_uminus @ A) @ A2) = ((uminus_uminus @ A) @ B2)) = (A2 = B2)))))))).
% 2.84/1.58 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P2:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P2)) = (P2 @ A2))))).
% 2.84/1.58 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A4)))) = A4)))).
% 2.84/1.58 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P2 @ X) = (Q @ X)))) => (((collect @ A) @ P2) = ((collect @ A) @ Q)))))).
% 2.84/1.58 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F3:(A > B),G3:(A > B)]: (((! [X:A]: (((F3 @ X) = (G3 @ X)))) => (F3 = G3))))).
% 2.84/1.58 thf(fact_49_compl__le__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X3)) @ ((uminus_uminus @ A) @ Y3)) = (((ord_less_eq @ A) @ Y3) @ X3)))))))).
% 2.84/1.58 thf(fact_50_neg__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.84/1.58 thf(fact_51_Compl__subset__Compl__iff, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A4)) @ ((uminus_uminus @ (set @ A)) @ B4)) = (((ord_less_eq @ (set @ A)) @ B4) @ A4))))).
% 2.84/1.58 thf(fact_52_Compl__anti__mono, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ B4)) @ ((uminus_uminus @ (set @ A)) @ A4)))))).
% 2.84/1.58 thf(fact_53_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((A2 = B2) | ((~ ((((ord_less_eq @ A) @ A2) @ B2))) | (~ ((((ord_less_eq @ A) @ B2) @ A2))))))))))).
% 2.84/1.58 thf(fact_54_Collect__mono__iff, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P2)) @ ((collect @ A) @ Q)) = (! [X4:A]: (((P2 @ X4) => (Q @ X4)))))))).
% 2.84/1.58 thf(fact_55_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y4:(set @ A),Z:(set @ A)]: ((Y4 = Z))) = (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ A6)))))))).
% 2.84/1.58 thf(fact_56_subset__trans, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ C4) => (((ord_less_eq @ (set @ A)) @ A4) @ C4)))))).
% 2.84/1.58 thf(fact_57_Collect__mono, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P2 @ X) => (Q @ X)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P2)) @ ((collect @ A) @ Q)))))).
% 2.84/1.58 thf(fact_58_subset__refl, axiom, (! [A:$tType,A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A4) @ A4)))).
% 2.84/1.58 thf(fact_59_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A6) => (((member @ A) @ T2) @ B6)))))))))).
% 2.84/1.58 thf(fact_60_equalityD2, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (((ord_less_eq @ (set @ A)) @ B4) @ A4))))).
% 2.84/1.58 thf(fact_61_equalityD1, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (((ord_less_eq @ (set @ A)) @ A4) @ B4))))).
% 2.84/1.58 thf(fact_62_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((! [X4:A]: (((((member @ A) @ X4) @ A6) => (((member @ A) @ X4) @ B6)))))))))).
% 2.84/1.58 thf(fact_63_equalityE, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (~ (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (~ ((((ord_less_eq @ (set @ A)) @ B4) @ A4)))))))))).
% 2.84/1.58 thf(fact_64_subsetD, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ C3) @ A4) => (((member @ A) @ C3) @ B4)))))).
% 2.84/1.58 thf(fact_65_in__mono, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ X3) @ A4) => (((member @ A) @ X3) @ B4)))))).
% 2.84/1.58 thf(fact_66_le__imp__neg__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2))))))))).
% 2.84/1.58 thf(fact_67_minus__le__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ A2)))))))).
% 2.84/1.58 thf(fact_68_le__minus__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ B2)) = (((ord_less_eq @ A) @ B2) @ ((uminus_uminus @ A) @ A2))))))))).
% 2.84/1.58 thf(fact_69_compl__le__swap2, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y3:A,X3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y3)) @ X3) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X3)) @ Y3)))))))).
% 2.84/1.58 thf(fact_70_compl__le__swap1, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y3:A,X3:A]: (((((ord_less_eq @ A) @ Y3) @ ((uminus_uminus @ A) @ X3)) => (((ord_less_eq @ A) @ X3) @ ((uminus_uminus @ A) @ Y3))))))))).
% 2.84/1.58 thf(fact_71_compl__mono, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y3)) @ ((uminus_uminus @ A) @ X3))))))))).
% 2.84/1.58 thf(fact_72_agree__sub, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A4:(set @ ((sum_sum @ A) @ A)),B4:(set @ ((sum_sum @ A) @ A)),Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),Omega2:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))]: (((((ord_less_eq @ (set @ ((sum_sum @ A) @ A))) @ A4) @ B4) => (((((denotational_Vagree @ A) @ Nu) @ Omega2) @ B4) => ((((denotational_Vagree @ A) @ Nu) @ Omega2) @ A4))))))))).
% 2.84/1.58 thf(fact_73_agree__supset, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [B4:(set @ ((sum_sum @ A) @ A)),A4:(set @ ((sum_sum @ A) @ A)),Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),Nu2:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))]: (((((ord_less_eq @ (set @ ((sum_sum @ A) @ A))) @ B4) @ A4) => (((((denotational_Vagree @ A) @ Nu) @ Nu2) @ A4) => ((((denotational_Vagree @ A) @ Nu) @ Nu2) @ B4))))))))).
% 2.84/1.58 thf(fact_74_minus__equation__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((uminus_uminus @ A) @ A2) = B2) = (((uminus_uminus @ A) @ B2) = A2)))))))).
% 2.84/1.58 thf(fact_75_equation__minus__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((A2 = ((uminus_uminus @ A) @ B2)) = (B2 = ((uminus_uminus @ A) @ A2))))))))).
% 2.84/1.58 thf(fact_76_fun__Compl__def, axiom, (! [B:$tType,A:$tType]: (((uminus @ B) => ((uminus_uminus @ (A > B)) = (^ [A6:(A > B),X4:A]: (((uminus_uminus @ B) @ (A6 @ X4))))))))).
% 2.84/1.58 thf(fact_77_verit__negate__coefficient_I3_J, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((A2 = B2) => (((uminus_uminus @ A) @ A2) = ((uminus_uminus @ A) @ B2))))))))).
% 2.84/1.58 thf(fact_78_double__complement, axiom, (! [A:$tType,A4:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A4)) = A4)))).
% 2.84/1.58 thf(fact_79_ComplD, axiom, (! [A:$tType,C3:A,A4:(set @ A)]: (((((member @ A) @ C3) @ ((uminus_uminus @ (set @ A)) @ A4)) => (~ ((((member @ A) @ C3) @ A4))))))).
% 2.84/1.58 thf(fact_80_euclid__ext__aux_Ocases, axiom, (! [A:$tType]: (((euclid1678468529ng_gcd @ A) => (! [X3:((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A)))))]: ((~ ((! [S2:A,S3:A,T3:A,T4:A,R:A,R2:A]: ((X3 != ((((product_Pair @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A))))) @ S2) @ ((((product_Pair @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A)))) @ S3) @ ((((product_Pair @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A))) @ T3) @ ((((product_Pair @ A) @ ((product_prod @ A) @ A)) @ T4) @ ((((product_Pair @ A) @ A) @ R) @ R2)))))))))))))))).
% 2.84/1.58 thf(fact_81_Pair__le, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:B,C3:A,D3:B]: (((((ord_less_eq @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Pair @ A) @ B) @ C3) @ D3)) = ((((ord_less_eq @ A) @ A2) @ C3) & (((ord_less_eq @ B) @ B2) @ D3))))))))).
% 2.84/1.58 thf(fact_82_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A]: ((((ord_less_eq @ A) @ X3) @ X3))))))).
% 2.84/1.58 thf(fact_83_Pair__mono, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [X3:A,X5:A,Y3:B,Y5:B]: (((((ord_less_eq @ A) @ X3) @ X5) => ((((ord_less_eq @ B) @ Y3) @ Y5) => (((ord_less_eq @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ ((((product_Pair @ A) @ B) @ X5) @ Y5)))))))))).
% 2.84/1.58 thf(fact_84_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(B > (C > A)),A2:B,B2:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C3) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((C3 @ A2) @ B2))))).
% 2.84/1.58 thf(fact_85_Fsub, axiom, (((ord_less_eq @ (set @ ((sum_sum @ c) @ c))) @ ((((static_FVF @ a) @ b) @ c) @ p)) @ v)).
% 2.84/1.58 thf(fact_86_solSem, axiom, (! [X6:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ X6) => ((((ord_less_eq @ real) @ X6) @ t) => (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ ode2) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ a2) @ b2)) @ (sol @ X6))) @ (((((denotational_fml_sem @ a) @ b) @ c) @ i) @ p))))))).
% 2.84/1.58 thf(fact_87_relChain__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (((bNF_Ca1785829860lChain @ A) @ B) = (^ [R3:(set @ ((product_prod @ A) @ A)),As:(A > B)]: ((! [I2:A,J:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I2) @ J)) @ R3) => (((ord_less_eq @ B) @ (As @ I2)) @ (As @ J)))))))))))).
% 2.84/1.58 thf(fact_88_hpsafe__Evolve_OIH, axiom, ((((coinci1993344360de_fml @ a) @ b) @ c) @ p)).
% 2.84/1.58 thf(fact_89_t, axiom, (((ord_less_eq @ real) @ (zero_zero @ real)) @ t)).
% 2.84/1.58 thf(fact_90_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.84/1.58 thf(fact_91_add_Oinverse__neutral, axiom, (! [A:$tType]: (((group_add @ A) => (((uminus_uminus @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.84/1.58 thf(fact_92_neg__0__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: ((((zero_zero @ A) = ((uminus_uminus @ A) @ A2)) = ((zero_zero @ A) = A2)))))))).
% 2.84/1.58 thf(fact_93_neg__equal__0__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((uminus_uminus @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.84/1.58 thf(fact_94_equal__neg__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((A2 = ((uminus_uminus @ A) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 2.84/1.58 thf(fact_95_neg__equal__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((uminus_uminus @ A) @ A2) = A2) = (A2 = (zero_zero @ A))))))))).
% 2.84/1.58 thf(fact_96_IH_H, axiom, (! [A7:((finite_Cartesian_vec @ real) @ c),B7:((finite_Cartesian_vec @ real) @ c),Aa:((finite_Cartesian_vec @ real) @ c),Ba:((finite_Cartesian_vec @ real) @ c)]: ((((((denotational_Vagree @ c) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ A7) @ B7)) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ Aa) @ Ba)) @ ((((static_FVF @ a) @ b) @ c) @ p)) => ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ A7) @ B7)) @ (((((denotational_fml_sem @ a) @ b) @ c) @ i) @ p)) = (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ Aa) @ Ba)) @ (((((denotational_fml_sem @ a) @ b) @ c) @ j) @ p))))))).
% 2.84/1.58 thf(fact_97_neg__less__eq__nonneg, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.84/1.58 thf(fact_98_less__eq__neg__nonpos, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.84/1.58 thf(fact_99_neg__le__0__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.84/1.58 thf(fact_100_neg__0__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.84/1.58 thf(fact_101_fsafe, axiom, ((((fsafe @ a) @ b) @ c) @ p)).
% 2.84/1.58 thf(fact_102_zero__prod__def, axiom, (! [A:$tType,B:$tType]: ((((zero @ B) & (zero @ A)) => ((zero_zero @ ((product_prod @ A) @ B)) = ((((product_Pair @ A) @ B) @ (zero_zero @ A)) @ (zero_zero @ B))))))).
% 2.84/1.58 thf(fact_103_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X3:A]: ((((zero_zero @ A) = X3) = (X3 = (zero_zero @ A))))))))).
% 2.84/1.58 thf(fact_104_iff__to__impl, axiom, (! [A:$tType,C:$tType,B:$tType]: ((((finite_finite @ B) & ((finite_finite @ C) & (finite_finite @ A))) => (! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),I:((((denota1663640101rp_ext @ B) @ C) @ A) @ product_unit),A4:(((formula @ B) @ C) @ A),B4:(((formula @ B) @ C) @ A)]: ((((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I) @ A4)) = (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I) @ B4))) = (((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I) @ A4)) => (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I) @ B4))) & ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I) @ B4)) => (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I) @ A4))))))))))).
% 2.84/1.58 thf(fact_105_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X3:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3))))))).
% 2.84/1.58 thf(fact_106_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G3:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G3) => (((ord_less_eq @ B) @ (F3 @ X3)) @ (G3 @ X3))))))))).
% 2.84/1.58 thf(fact_107_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G3:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G3) => (((ord_less_eq @ B) @ (F3 @ X3)) @ (G3 @ X3))))))))).
% 2.84/1.58 thf(fact_108_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G3:(A > B)]: (((! [X:A]: ((((ord_less_eq @ B) @ (F3 @ X)) @ (G3 @ X)))) => (((ord_less_eq @ (A > B)) @ F3) @ G3)))))))).
% 2.84/1.58 thf(fact_109_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F4:(A > B),G4:(A > B)]: ((! [X4:A]: ((((ord_less_eq @ B) @ (F4 @ X4)) @ (G4 @ X4))))))))))).
% 2.84/1.58 thf(fact_110_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F3:(B > A),B2:B,C3:B]: (((((ord_less_eq @ A) @ A2) @ (F3 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C3) => ((! [X:B,Y:B]: (((((ord_less_eq @ B) @ X) @ Y) => (((ord_less_eq @ A) @ (F3 @ X)) @ (F3 @ Y))))) => (((ord_less_eq @ A) @ A2) @ (F3 @ C3))))))))))).
% 2.84/1.58 thf(fact_111_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F3:(A > C),C3:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F3 @ B2)) @ C3) => ((! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ C) @ (F3 @ X)) @ (F3 @ Y))))) => (((ord_less_eq @ C) @ (F3 @ A2)) @ C3)))))))))).
% 2.84/1.58 thf(fact_112_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F3:(B > A),B2:B,C3:B]: (((A2 = (F3 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C3) => ((! [X:B,Y:B]: (((((ord_less_eq @ B) @ X) @ Y) => (((ord_less_eq @ A) @ (F3 @ X)) @ (F3 @ Y))))) => (((ord_less_eq @ A) @ A2) @ (F3 @ C3))))))))))).
% 2.84/1.58 thf(fact_113_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F3:(A > B),C3:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F3 @ B2) = C3) => ((! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ B) @ (F3 @ X)) @ (F3 @ Y))))) => (((ord_less_eq @ B) @ (F3 @ A2)) @ C3)))))))))).
% 2.84/1.58 thf(fact_114_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [X4:A,Y6:A]: (((((ord_less_eq @ A) @ X4) @ Y6) & (((ord_less_eq @ A) @ Y6) @ X4))))))))).
% 2.84/1.58 thf(fact_115_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ X3) => (X3 = Y3))))))))).
% 2.84/1.58 thf(fact_116_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) | (((ord_less_eq @ A) @ Y3) @ X3)))))))).
% 2.84/1.58 thf(fact_117_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y3:A]: (((X3 = Y3) => (((ord_less_eq @ A) @ X3) @ Y3)))))))).
% 2.84/1.58 thf(fact_118_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y3))) => (((ord_less_eq @ A) @ Y3) @ X3)))))))).
% 2.84/1.58 thf(fact_119_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 2.84/1.58 thf(fact_120_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A,Z2:A]: ((((((ord_less_eq @ A) @ X3) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ Z2)))) => (((((ord_less_eq @ A) @ Y3) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Z2)))) => (((((ord_less_eq @ A) @ X3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y3)))) => (((((ord_less_eq @ A) @ Z2) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ X3)))) => (((((ord_less_eq @ A) @ Y3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X3)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Y3))))))))))))))))).
% 2.84/1.58 thf(fact_121_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y3:A,X3:A]: (((((ord_less_eq @ A) @ Y3) @ X3) => ((((ord_less_eq @ A) @ X3) @ Y3) = (X3 = Y3))))))))).
% 2.84/1.58 thf(fact_122_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [A8:A,B8:A]: (((((ord_less_eq @ A) @ A8) @ B8) & (((ord_less_eq @ A) @ B8) @ A8))))))))).
% 2.84/1.58 thf(fact_123_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C3:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 2.84/1.58 thf(fact_124_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 2.84/1.58 thf(fact_125_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.84/1.58 thf(fact_126_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y3:A,Z2:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ Z2) => (((ord_less_eq @ A) @ X3) @ Z2))))))))).
% 2.84/1.58 thf(fact_127_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.84/1.58 thf(fact_128_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P2:(A > (A > $o)),A2:A,B2:A]: (((! [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) => ((P2 @ A5) @ B5)))) => ((! [A5:A,B5:A]: ((((P2 @ B5) @ A5) => ((P2 @ A5) @ B5)))) => ((P2 @ A2) @ B2))))))))).
% 2.84/1.58 thf(fact_129_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C3:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C3) @ B2) => (((ord_less_eq @ A) @ C3) @ A2))))))))).
% 2.84/1.58 thf(fact_130_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [A8:A,B8:A]: (((((ord_less_eq @ A) @ B8) @ A8) & (((ord_less_eq @ A) @ A8) @ B8))))))))).
% 2.84/1.58 thf(fact_131_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.84/1.58 thf(fact_132_gcd_Ocases, axiom, (! [A:$tType]: (((euclid1155270486miring @ A) => (! [X3:((product_prod @ A) @ A)]: ((~ ((! [A5:A,B5:A]: ((X3 != ((((product_Pair @ A) @ A) @ A5) @ B5)))))))))))).
% 2.84/1.58 thf(fact_133_iff__sem, axiom, (! [A:$tType,C:$tType,B:$tType]: ((((finite_finite @ B) & ((finite_finite @ C) & (finite_finite @ A))) => (! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),I:((((denota1663640101rp_ext @ B) @ C) @ A) @ product_unit),A4:(((formula @ B) @ C) @ A),B4:(((formula @ B) @ C) @ A)]: (((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I) @ (((((equiv @ B) @ C) @ A) @ A4) @ B4))) = ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I) @ A4)) = (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I) @ B4)))))))))).
% 2.84/1.58 thf(fact_134_IHF, axiom, (! [A7:((finite_Cartesian_vec @ real) @ c),B7:((finite_Cartesian_vec @ real) @ c),Aa:((finite_Cartesian_vec @ real) @ c),Ba:((finite_Cartesian_vec @ real) @ c)]: ((((((((denotational_Iagree @ a) @ b) @ c) @ i) @ j) @ ((((static_SIGF @ a) @ b) @ c) @ p)) => (((((denotational_Vagree @ c) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ A7) @ B7)) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ Aa) @ Ba)) @ ((((static_FVF @ a) @ b) @ c) @ p)) => ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ A7) @ B7)) @ (((((denotational_fml_sem @ a) @ b) @ c) @ i) @ p)) = (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ Aa) @ Ba)) @ (((((denotational_fml_sem @ a) @ b) @ c) @ j) @ p)))))))).
% 2.84/1.58 thf(fact_135_minus__le__self__iff, axiom, (! [A:$tType]: (((lattic1601792062up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.84/1.58 thf(fact_136_le__minus__self__iff, axiom, (! [A:$tType]: (((lattic1601792062up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.84/1.58 thf(fact_137_real__eq__0__iff__le__ge__0, axiom, (! [X3:real]: (((X3 = (zero_zero @ real)) = ((((ord_less_eq @ real) @ (zero_zero @ real)) @ X3) & (((ord_less_eq @ real) @ (zero_zero @ real)) @ ((uminus_uminus @ real) @ X3))))))).
% 2.84/1.58 thf(fact_138_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.84/1.58 thf(fact_139_coincide__fml__def, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => ((((coinci1993344360de_fml @ A) @ B) @ C) = (^ [Phi:(((formula @ A) @ B) @ C)]: ((! [Nu3:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),Nu4:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),I3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),J2:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: ((((((((denotational_Iagree @ A) @ B) @ C) @ I3) @ J2) @ ((((static_SIGF @ A) @ B) @ C) @ Phi)) => (((((denotational_Vagree @ C) @ Nu3) @ Nu4) @ ((((static_FVF @ A) @ B) @ C) @ Phi)) => ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ Nu3) @ (((((denotational_fml_sem @ A) @ B) @ C) @ I3) @ Phi)) = (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ Nu4) @ (((((denotational_fml_sem @ A) @ B) @ C) @ J2) @ Phi)))))))))))))).
% 2.84/1.58 thf(fact_140_IAP, axiom, ((((((denotational_Iagree @ a) @ b) @ c) @ i) @ j) @ ((((static_SIGF @ a) @ b) @ c) @ p))).
% 2.84/1.58 thf(fact_141_Iagree__sub, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [A4:(set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))),B4:(set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))),I:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),J3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: (((((ord_less_eq @ (set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C)))) @ A4) @ B4) => (((((((denotational_Iagree @ A) @ B) @ C) @ I) @ J3) @ B4) => ((((((denotational_Iagree @ A) @ B) @ C) @ I) @ J3) @ A4))))))))).
% 2.84/1.58 thf(fact_142_Iagree__comm, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [A4:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),B4:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),V:(set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C)))]: ((((((((denotational_Iagree @ A) @ B) @ C) @ A4) @ B4) @ V) => ((((((denotational_Iagree @ A) @ B) @ C) @ B4) @ A4) @ V)))))))).
% 2.84/1.58 thf(fact_143_Iagree__refl, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [I:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),A4:(set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C)))]: (((((((denotational_Iagree @ A) @ B) @ C) @ I) @ I) @ A4))))))).
% 2.84/1.58 thf(fact_144_ids_Ocoincide__fml__def, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,B:$tType,A:$tType,C:$tType]: ((((finite_finite @ C) & ((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,Phi2:(((formula @ A) @ B) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((((coinci1993344360de_fml @ A) @ B) @ C) @ Phi2) = (! [Nu3:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),Nu4:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),I3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),J2:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: ((((((((denotational_Iagree @ A) @ B) @ C) @ I3) @ J2) @ ((((static_SIGF @ A) @ B) @ C) @ Phi2)) => (((((denotational_Vagree @ C) @ Nu3) @ Nu4) @ ((((static_FVF @ A) @ B) @ C) @ Phi2)) => ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ Nu3) @ (((((denotational_fml_sem @ A) @ B) @ C) @ I3) @ Phi2)) = (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ Nu4) @ (((((denotational_fml_sem @ A) @ B) @ C) @ J2) @ Phi2))))))))))))))).
% 2.84/1.58 thf(fact_145__092_060open_062ode__sem__equiv_A_IEvolveODE_AODE_AP_J_AJ_092_060close_062, axiom, (((((coinci1495174342_equiv @ a) @ b) @ c) @ (((((evolveODE @ a) @ c) @ b) @ ode2) @ p)) @ j)).
% 2.84/1.58 thf(fact_146__092_060open_062ode__sem__equiv_A_IEvolveODE_AODE_AP_J_AI_092_060close_062, axiom, (((((coinci1495174342_equiv @ a) @ b) @ c) @ (((((evolveODE @ a) @ c) @ b) @ ode2) @ p)) @ i)).
% 2.84/1.58 thf(fact_147_impl__sem, axiom, (! [A:$tType,C:$tType,B:$tType]: ((((finite_finite @ B) & ((finite_finite @ C) & (finite_finite @ A))) => (! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),I:((((denota1663640101rp_ext @ B) @ C) @ A) @ product_unit),A4:(((formula @ B) @ C) @ A),B4:(((formula @ B) @ C) @ A)]: (((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I) @ (((((implies @ B) @ C) @ A) @ A4) @ B4))) = ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I) @ A4)) => (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I) @ B4)))))))))).
% 2.84/1.58 thf(fact_148_equiv, axiom, (! [I4:((((denota1663640101rp_ext @ a) @ b) @ c) @ product_unit)]: ((((((coinci1495174342_equiv @ a) @ b) @ c) @ (((((evolveODE @ a) @ c) @ b) @ ode2) @ p)) @ I4)))).
% 2.84/1.58 thf(fact_149_ids_Oseq__sem_Ocases, axiom, (! [Sz:$tType,Sc:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sc) & ((finite_finite @ Sz) & (linorder @ Sz)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X3:((product_prod @ ((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit)) @ ((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz))))]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (~ ((! [I5:((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit),S4:((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz)))]: ((X3 != ((((product_Pair @ ((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit)) @ ((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz)))) @ I5) @ S4))))))))))))).
% 2.84/1.58 thf(fact_150_ids_Oseq__sem_Oinduct, axiom, (! [Sz:$tType,Sc:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sc) & ((finite_finite @ Sz) & (linorder @ Sz)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,P2:(((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit) > (((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz))) > $o)),A0:((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit),A1:((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz)))]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((! [I5:((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit),X_1:((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz)))]: (((P2 @ I5) @ X_1))) => ((P2 @ A0) @ A1))))))))).
% 2.84/1.58 thf(fact_151_coincide__hp_H__def, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => ((((coinci554967470ide_hp @ A) @ B) @ C) = (^ [Alpha:(((hp @ A) @ B) @ C)]: ((! [I3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),J2:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: ((((((((coinci32832645ide_hp @ A) @ B) @ C) @ Alpha) @ I3) @ J2) & (((((coinci1495174342_equiv @ A) @ B) @ C) @ Alpha) @ I3))))))))))).
% 2.84/1.58 thf(fact_152_hp_Oinject_I5_J, axiom, (! [C:$tType,B:$tType,A:$tType,X51:((ode @ A) @ C),X52:(((formula @ A) @ B) @ C),Y51:((ode @ A) @ C),Y52:(((formula @ A) @ B) @ C)]: ((((((((evolveODE @ A) @ C) @ B) @ X51) @ X52) = (((((evolveODE @ A) @ C) @ B) @ Y51) @ Y52)) = ((X51 = Y51) & (X52 = Y52)))))).
% 2.84/1.58 thf(fact_153_sol, axiom, (((((initia1685620758es_ode @ ((finite_Cartesian_vec @ real) @ c)) @ sol) @ (^ [A8:real]: ((((((denotational_ODE_sem @ a) @ b) @ c) @ i) @ ode2)))) @ (((set_or331188842AtMost @ real) @ (zero_zero @ real)) @ t)) @ ((collect @ ((finite_Cartesian_vec @ real) @ c)) @ (^ [X4:((finite_Cartesian_vec @ real) @ c)]: ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ ode2) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ a2) @ b2)) @ X4)) @ (((((denotational_fml_sem @ a) @ b) @ c) @ i) @ p))))))).
% 2.84/1.58 thf(fact_154_less__eq__set__def, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((((ord_less_eq @ (A > $o)) @ (^ [X4:A]: ((((member @ A) @ X4) @ A6)))) @ (^ [X4:A]: ((((member @ A) @ X4) @ B6)))))))))).
% 2.84/1.58 thf(fact_155_Collect__subset, axiom, (! [A:$tType,A4:(set @ A),P2:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ A4) & (P2 @ X4)))))) @ A4)))).
% 2.84/1.58 thf(fact_156_Compl__eq, axiom, (! [A:$tType]: (((uminus_uminus @ (set @ A)) = (^ [A6:(set @ A)]: (((collect @ A) @ (^ [X4:A]: ((~ ((((member @ A) @ X4) @ A6)))))))))))).
% 2.84/1.58 thf(fact_157_Collect__neg__eq, axiom, (! [A:$tType,P2:(A > $o)]: ((((collect @ A) @ (^ [X4:A]: ((~ ((P2 @ X4)))))) = ((uminus_uminus @ (set @ A)) @ ((collect @ A) @ P2)))))).
% 2.84/1.58 thf(fact_158_uminus__set__def, axiom, (! [A:$tType]: (((uminus_uminus @ (set @ A)) = (^ [A6:(set @ A)]: (((collect @ A) @ ((uminus_uminus @ (A > $o)) @ (^ [X4:A]: ((((member @ A) @ X4) @ A6))))))))))).
% 2.84/1.58 thf(fact_159_mk__xode_Oelims, axiom, (! [A:$tType,B:$tType,C:$tType]: ((((finite_finite @ C) & ((finite_finite @ B) & (finite_finite @ A))) => (! [X3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),Xa:((ode @ A) @ C),Xb:((finite_Cartesian_vec @ real) @ C),Y3:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))]: (((((((((denotational_mk_xode @ A) @ B) @ C) @ X3) @ Xa) @ Xb) = Y3) => (Y3 = ((((product_Pair @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ Xb) @ ((((((denotational_ODE_sem @ A) @ B) @ C) @ X3) @ Xa) @ Xb)))))))))).
% 2.84/1.58 thf(fact_160_mk__xode_Osimps, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => ((((denotational_mk_xode @ A) @ B) @ C) = (^ [I3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),ODE2:((ode @ A) @ C),Sol2:((finite_Cartesian_vec @ real) @ C)]: (((((product_Pair @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ Sol2) @ ((((((denotational_ODE_sem @ A) @ B) @ C) @ I3) @ ODE2) @ Sol2))))))))).
% 2.84/1.58 thf(fact_161_ids_Osingleton_Oinduct, axiom, (! [Sf:$tType,Sc:$tType,A:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sc) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,P2:(((trm @ A) @ Sz) > (Sz > $o)),A0:((trm @ A) @ Sz),A1:Sz]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((! [T4:((trm @ A) @ Sz),X_1:Sz]: (((P2 @ T4) @ X_1))) => ((P2 @ A0) @ A1))))))))).
% 2.84/1.58 thf(fact_162_ids_Osingleton_Ocases, axiom, (! [Sc:$tType,Sf:$tType,A:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X3:((product_prod @ ((trm @ A) @ Sz)) @ Sz)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (~ ((! [T4:((trm @ A) @ Sz),I6:Sz]: ((X3 != ((((product_Pair @ ((trm @ A) @ Sz)) @ Sz) @ T4) @ I6))))))))))))).
% 2.84/1.58 thf(fact_163_ids_Ocoincide__hp_H__def, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ C) & ((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,Alpha2:(((hp @ A) @ B) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((((coinci554967470ide_hp @ A) @ B) @ C) @ Alpha2) = (! [I3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),J2:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: ((((((((coinci32832645ide_hp @ A) @ B) @ C) @ Alpha2) @ I3) @ J2) & (((((coinci1495174342_equiv @ A) @ B) @ C) @ Alpha2) @ I3)))))))))))).
% 2.84/1.58 thf(fact_164_atLeastatMost__subset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B2:A,C3:A,D3:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or331188842AtMost @ A) @ A2) @ B2)) @ (((set_or331188842AtMost @ A) @ C3) @ D3)) = ((~ ((((ord_less_eq @ A) @ A2) @ B2))) | ((((ord_less_eq @ A) @ C3) @ A2) & (((ord_less_eq @ A) @ B2) @ D3)))))))))).
% 2.84/1.58 thf(fact_165_atLeastAtMost__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I7:A,L:A,U:A]: (((((member @ A) @ I7) @ (((set_or331188842AtMost @ A) @ L) @ U)) = ((((ord_less_eq @ A) @ L) @ I7) & (((ord_less_eq @ A) @ I7) @ U))))))))).
% 2.84/1.58 thf(fact_166_Icc__eq__Icc, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,H:A,L2:A,H2:A]: ((((((set_or331188842AtMost @ A) @ L) @ H) = (((set_or331188842AtMost @ A) @ L2) @ H2)) = (((L = L2) & (H = H2)) | ((~ ((((ord_less_eq @ A) @ L) @ H))) & (~ ((((ord_less_eq @ A) @ L2) @ H2)))))))))))).
% 2.84/1.58 thf(fact_167_predicate1I, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P2 @ X) => (Q @ X)))) => (((ord_less_eq @ (A > $o)) @ P2) @ Q))))).
% 2.84/1.58 thf(fact_168_predicate1D, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o),X3:A]: (((((ord_less_eq @ (A > $o)) @ P2) @ Q) => ((P2 @ X3) => (Q @ X3)))))).
% 2.84/1.58 thf(fact_169_rev__predicate1D, axiom, (! [A:$tType,P2:(A > $o),X3:A,Q:(A > $o)]: (((P2 @ X3) => ((((ord_less_eq @ (A > $o)) @ P2) @ Q) => (Q @ X3)))))).
% 2.84/1.58 thf(fact_170_pred__subset__eq, axiom, (! [A:$tType,R4:(set @ A),S5:(set @ A)]: (((((ord_less_eq @ (A > $o)) @ (^ [X4:A]: ((((member @ A) @ X4) @ R4)))) @ (^ [X4:A]: ((((member @ A) @ X4) @ S5)))) = (((ord_less_eq @ (set @ A)) @ R4) @ S5))))).
% 2.84/1.58 thf(fact_171_solDeriv, axiom, (! [X6:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ X6) => ((((ord_less_eq @ real) @ X6) @ t) => ((((has_ve2132708402vative @ ((finite_Cartesian_vec @ real) @ c)) @ sol) @ ((((((denotational_ODE_sem @ a) @ b) @ c) @ i) @ ode2) @ (sol @ X6))) @ (((topolo507301023within @ real) @ X6) @ (((set_or331188842AtMost @ real) @ (zero_zero @ real)) @ t)))))))).
% 2.84/1.58 thf(fact_172_mk__v__def, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => ((((denotational_mk_v @ A) @ B) @ C) = (^ [I3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),ODE2:((ode @ A) @ C),Nu3:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),Sol2:((finite_Cartesian_vec @ real) @ C)]: (((the @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ (^ [Omega3:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))]: ((((((denotational_Vagree @ C) @ Omega3) @ Nu3) @ ((uminus_uminus @ (set @ ((sum_sum @ C) @ C))) @ (((((denotational_semBV @ A) @ B) @ C) @ I3) @ ODE2))) & ((((denotational_Vagree @ C) @ Omega3) @ ((((((denotational_mk_xode @ A) @ B) @ C) @ I3) @ ODE2) @ Sol2)) @ (((((denotational_semBV @ A) @ B) @ C) @ I3) @ ODE2))))))))))))).
% 2.84/1.58 thf(fact_173_subrelI, axiom, (! [B:$tType,A:$tType,R5:(set @ ((product_prod @ A) @ B)),S6:(set @ ((product_prod @ A) @ B))]: (((! [X:A,Y:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ R5) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ S6)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R5) @ S6))))).
% 2.84/1.58 thf(fact_174_pred__subset__eq2, axiom, (! [B:$tType,A:$tType,R4:(set @ ((product_prod @ A) @ B)),S5:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (A > (B > $o))) @ (^ [X4:A,Y6:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y6)) @ R4)))) @ (^ [X4:A,Y6:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y6)) @ S5)))) = (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R4) @ S5))))).
% 2.84/1.58 thf(fact_175_pred__equals__eq2, axiom, (! [B:$tType,A:$tType,R4:(set @ ((product_prod @ A) @ B)),S5:(set @ ((product_prod @ A) @ B))]: ((((^ [X4:A,Y6:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y6)) @ R4))) = (^ [X4:A,Y6:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y6)) @ S5)))) = (R4 = S5))))).
% 2.84/1.58 thf(fact_176_old_Orec__prod__def, axiom, (! [T:$tType,B:$tType,A:$tType]: (((((product_rec_prod @ A) @ B) @ T) = (^ [F12:(A > (B > T)),X4:((product_prod @ A) @ B)]: (((the @ T) @ (((((product_rec_set_prod @ A) @ B) @ T) @ F12) @ X4)))))))).
% 2.84/1.58 thf(fact_177_has__vector__derivative__Pair, axiom, (! [B:$tType,A:$tType]: ((((real_V55928688vector @ A) & (real_V55928688vector @ B)) => (! [F3:(real > A),F5:A,X3:real,S6:(set @ real),G3:(real > B),G5:B]: ((((((has_ve2132708402vative @ A) @ F3) @ F5) @ (((topolo507301023within @ real) @ X3) @ S6)) => (((((has_ve2132708402vative @ B) @ G3) @ G5) @ (((topolo507301023within @ real) @ X3) @ S6)) => ((((has_ve2132708402vative @ ((product_prod @ A) @ B)) @ (^ [X4:real]: (((((product_Pair @ A) @ B) @ (F3 @ X4)) @ (G3 @ X4))))) @ ((((product_Pair @ A) @ B) @ F5) @ G5)) @ (((topolo507301023within @ real) @ X3) @ S6)))))))))).
% 2.84/1.58 thf(fact_178_has__vector__derivative__minus, axiom, (! [A:$tType]: (((real_V55928688vector @ A) => (! [F3:(real > A),F5:A,Net:(filter @ real)]: ((((((has_ve2132708402vative @ A) @ F3) @ F5) @ Net) => ((((has_ve2132708402vative @ A) @ (^ [X4:real]: (((uminus_uminus @ A) @ (F3 @ X4))))) @ ((uminus_uminus @ A) @ F5)) @ Net)))))))).
% 2.84/1.58 thf(fact_179_predicate2I, axiom, (! [B:$tType,A:$tType,P2:(A > (B > $o)),Q:(A > (B > $o))]: (((! [X:A,Y:B]: ((((P2 @ X) @ Y) => ((Q @ X) @ Y)))) => (((ord_less_eq @ (A > (B > $o))) @ P2) @ Q))))).
% 2.84/1.58 thf(fact_180_rev__predicate2D, axiom, (! [A:$tType,B:$tType,P2:(A > (B > $o)),X3:A,Y3:B,Q:(A > (B > $o))]: ((((P2 @ X3) @ Y3) => ((((ord_less_eq @ (A > (B > $o))) @ P2) @ Q) => ((Q @ X3) @ Y3)))))).
% 2.84/1.58 thf(fact_181_predicate2D, axiom, (! [A:$tType,B:$tType,P2:(A > (B > $o)),Q:(A > (B > $o)),X3:A,Y3:B]: (((((ord_less_eq @ (A > (B > $o))) @ P2) @ Q) => (((P2 @ X3) @ Y3) => ((Q @ X3) @ Y3)))))).
% 2.84/1.58 thf(fact_182_has__vector__derivative__const, axiom, (! [A:$tType]: (((real_V55928688vector @ A) => (! [C3:A,Net:(filter @ real)]: (((((has_ve2132708402vative @ A) @ (^ [X4:real]: (C3))) @ (zero_zero @ A)) @ Net))))))).
% 2.84/1.58 thf(fact_183_has__vector__derivative__within__subset, axiom, (! [A:$tType]: (((real_V55928688vector @ A) => (! [F3:(real > A),F5:A,X3:real,S5:(set @ real),T5:(set @ real)]: ((((((has_ve2132708402vative @ A) @ F3) @ F5) @ (((topolo507301023within @ real) @ X3) @ S5)) => ((((ord_less_eq @ (set @ real)) @ T5) @ S5) => ((((has_ve2132708402vative @ A) @ F3) @ F5) @ (((topolo507301023within @ real) @ X3) @ T5)))))))))).
% 2.84/1.58 thf(fact_184_has__vector__derivative__weaken, axiom, (! [A:$tType]: (((real_V55928688vector @ A) => (! [F3:(real > A),D4:A,X3:real,T5:(set @ real),S5:(set @ real),G3:(real > A)]: ((((((has_ve2132708402vative @ A) @ F3) @ D4) @ (((topolo507301023within @ real) @ X3) @ T5)) => ((((member @ real) @ X3) @ S5) => ((((ord_less_eq @ (set @ real)) @ S5) @ T5) => ((! [X:real]: (((((member @ real) @ X) @ S5) => ((F3 @ X) = (G3 @ X))))) => ((((has_ve2132708402vative @ A) @ G3) @ D4) @ (((topolo507301023within @ real) @ X3) @ S5)))))))))))).
% 2.84/1.58 thf(fact_185_solves__ode__supset__range, axiom, (! [A:$tType]: (((real_V55928688vector @ A) => (! [X3:(real > A),F3:(real > (A > A)),T5:(set @ real),X7:(set @ A),Y7:(set @ A)]: (((((((initia1685620758es_ode @ A) @ X3) @ F3) @ T5) @ X7) => ((((ord_less_eq @ (set @ A)) @ X7) @ Y7) => (((((initia1685620758es_ode @ A) @ X3) @ F3) @ T5) @ Y7))))))))).
% 2.84/1.58 thf(fact_186_solves__ode__subset, axiom, (! [A:$tType]: (((real_V55928688vector @ A) => (! [X3:(real > A),F3:(real > (A > A)),T5:(set @ real),X7:(set @ A),S5:(set @ real)]: (((((((initia1685620758es_ode @ A) @ X3) @ F3) @ T5) @ X7) => ((((ord_less_eq @ (set @ real)) @ S5) @ T5) => (((((initia1685620758es_ode @ A) @ X3) @ F3) @ S5) @ X7))))))))).
% 2.84/1.58 thf(fact_187_solves__ode__on__subset, axiom, (! [A:$tType]: (((real_V55928688vector @ A) => (! [X3:(real > A),F3:(real > (A > A)),S5:(set @ real),Y7:(set @ A),T5:(set @ real),X7:(set @ A)]: (((((((initia1685620758es_ode @ A) @ X3) @ F3) @ S5) @ Y7) => ((((ord_less_eq @ (set @ real)) @ T5) @ S5) => ((((ord_less_eq @ (set @ A)) @ Y7) @ X7) => (((((initia1685620758es_ode @ A) @ X3) @ F3) @ T5) @ X7)))))))))).
% 2.84/1.58 thf(fact_188_at__le, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [S6:(set @ A),T6:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ S6) @ T6) => (((ord_less_eq @ (filter @ A)) @ (((topolo507301023within @ A) @ X3) @ S6)) @ (((topolo507301023within @ A) @ X3) @ T6))))))))).
% 2.84/1.58 thf(fact_189_intervalE, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,X3:A,B2:A]: ((((((ord_less_eq @ A) @ A2) @ X3) & (((ord_less_eq @ A) @ X3) @ B2)) => (((member @ A) @ X3) @ (((set_or331188842AtMost @ A) @ A2) @ B2))))))))).
% 2.84/1.58 thf(fact_190_eq__subset, axiom, (! [A:$tType,P2:(A > (A > $o))]: ((((ord_less_eq @ (A > (A > $o))) @ (^ [Y4:A,Z:A]: ((Y4 = Z)))) @ (^ [A8:A,B8:A]: ((((P2 @ A8) @ B8) | (A8 = B8)))))))).
% 2.84/1.58 thf(fact_191_conj__subset__def, axiom, (! [A:$tType,A4:(set @ A),P2:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ A4) @ ((collect @ A) @ (^ [X4:A]: (((P2 @ X4) & (Q @ X4)))))) = ((((ord_less_eq @ (set @ A)) @ A4) @ ((collect @ A) @ P2)) & (((ord_less_eq @ (set @ A)) @ A4) @ ((collect @ A) @ Q))))))).
% 2.84/1.58 thf(fact_192_predicate2D__conj, axiom, (! [A:$tType,B:$tType,P2:(A > (B > $o)),Q:(A > (B > $o)),R4:$o,X3:A,Y3:B]: ((((((ord_less_eq @ (A > (B > $o))) @ P2) @ Q) & R4) => (R4 & (((P2 @ X3) @ Y3) => ((Q @ X3) @ Y3))))))).
% 2.84/1.58 thf(fact_193_fun__cong__unused__0, axiom, (! [A:$tType,B:$tType,C:$tType]: (((zero @ B) => (! [F3:((A > B) > C),G3:C]: (((F3 = (^ [X4:(A > B)]: (G3))) => ((F3 @ (^ [X4:A]: ((zero_zero @ B)))) = G3)))))))).
% 2.84/1.58 thf(fact_194_subset__Collect__iff, axiom, (! [A:$tType,B4:(set @ A),A4:(set @ A),P2:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B4) @ A4) => ((((ord_less_eq @ (set @ A)) @ B4) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ A4) & (P2 @ X4)))))) = (! [X4:A]: (((((member @ A) @ X4) @ B4) => (P2 @ X4))))))))).
% 2.84/1.58 thf(fact_195_subset__CollectI, axiom, (! [A:$tType,B4:(set @ A),A4:(set @ A),Q:(A > $o),P2:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B4) @ A4) => ((! [X:A]: (((((member @ A) @ X) @ B4) => ((Q @ X) => (P2 @ X))))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ B4) & (Q @ X4)))))) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ A4) & (P2 @ X4))))))))))).
% 2.84/1.58 thf(fact_196_prop__restrict, axiom, (! [A:$tType,X3:A,Z3:(set @ A),X7:(set @ A),P2:(A > $o)]: (((((member @ A) @ X3) @ Z3) => ((((ord_less_eq @ (set @ A)) @ Z3) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ X7) & (P2 @ X4)))))) => (P2 @ X3)))))).
% 2.84/1.58 thf(fact_197_Collect__restrict, axiom, (! [A:$tType,X7:(set @ A),P2:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ X7) & (P2 @ X4)))))) @ X7)))).
% 2.84/1.58 thf(fact_198_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R5:A,S6:B,R4:(set @ ((product_prod @ A) @ B)),S7:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R5) @ S6)) @ R4) => ((S7 = S6) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R5) @ S7)) @ R4)))))).
% 2.84/1.58 thf(fact_199_hpsafe__ODE__simps, axiom, (! [C:$tType,B:$tType,A:$tType,ODE:((ode @ A) @ C),P:(((formula @ A) @ B) @ C)]: ((((((hpsafe @ A) @ B) @ C) @ (((((evolveODE @ A) @ C) @ B) @ ODE) @ P)) = ((((osafe @ A) @ C) @ ODE) & ((((fsafe @ A) @ B) @ C) @ P)))))).
% 2.84/1.58 thf(fact_200_old_Orec__unit__def, axiom, (! [T:$tType]: (((product_rec_unit @ T) = (^ [F12:T,X4:product_unit]: (((the @ T) @ (((product_rec_set_unit @ T) @ F12) @ X4)))))))).
% 2.84/1.58 thf(fact_201_hpsafe__fsafe_Ohpsafe__Evolve, axiom, (! [C:$tType,B:$tType,A:$tType,ODE:((ode @ A) @ C),P2:(((formula @ A) @ B) @ C)]: (((((osafe @ A) @ C) @ ODE) => (((((fsafe @ A) @ B) @ C) @ P2) => ((((hpsafe @ A) @ B) @ C) @ (((((evolveODE @ A) @ C) @ B) @ ODE) @ P2))))))).
% 2.84/1.58 thf(fact_202_divides__aux__eq, axiom, (! [A:$tType]: (((unique1598680935umeral @ A) => (! [Q2:A,R5:A]: ((((unique455577585es_aux @ A) @ ((((product_Pair @ A) @ A) @ Q2) @ R5)) = (R5 = (zero_zero @ A))))))))).
% 2.84/1.58 thf(fact_203_refl__ge__eq, axiom, (! [A:$tType,R4:(A > (A > $o))]: (((! [X:A]: (((R4 @ X) @ X))) => (((ord_less_eq @ (A > (A > $o))) @ (^ [Y4:A,Z:A]: ((Y4 = Z)))) @ R4))))).
% 2.84/1.58 thf(fact_204_ge__eq__refl, axiom, (! [A:$tType,R4:(A > (A > $o)),X3:A]: (((((ord_less_eq @ (A > (A > $o))) @ (^ [Y4:A,Z:A]: ((Y4 = Z)))) @ R4) => ((R4 @ X3) @ X3))))).
% 2.84/1.58 thf(fact_205_Icc__subset__Ici__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [L:A,H:A,L2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or331188842AtMost @ A) @ L) @ H)) @ ((set_ord_atLeast @ A) @ L2)) = ((~ ((((ord_less_eq @ A) @ L) @ H))) | (((ord_less_eq @ A) @ L2) @ L))))))))).
% 2.84/1.58 thf(fact_206_Icc__subset__Iic__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [L:A,H:A,H2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or331188842AtMost @ A) @ L) @ H)) @ ((set_ord_atMost @ A) @ H2)) = ((~ ((((ord_less_eq @ A) @ L) @ H))) | (((ord_less_eq @ A) @ H) @ H2))))))))).
% 2.84/1.58 thf(fact_207_atMost__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I7:A,K:A]: (((((member @ A) @ I7) @ ((set_ord_atMost @ A) @ K)) = (((ord_less_eq @ A) @ I7) @ K)))))))).
% 2.84/1.58 thf(fact_208_atLeast__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I7:A,K:A]: (((((member @ A) @ I7) @ ((set_ord_atLeast @ A) @ K)) = (((ord_less_eq @ A) @ K) @ I7)))))))).
% 2.84/1.58 thf(fact_209_atMost__subset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y3:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ X3)) @ ((set_ord_atMost @ A) @ Y3)) = (((ord_less_eq @ A) @ X3) @ Y3)))))))).
% 2.84/1.58 thf(fact_210_atLeast__subset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y3:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atLeast @ A) @ X3)) @ ((set_ord_atLeast @ A) @ Y3)) = (((ord_less_eq @ A) @ Y3) @ X3)))))))).
% 2.84/1.58 thf(fact_211_not__Ici__le__Icc, axiom, (! [A:$tType]: (((no_top @ A) => (! [L:A,L2:A,H2:A]: ((~ ((((ord_less_eq @ (set @ A)) @ ((set_ord_atLeast @ A) @ L)) @ (((set_or331188842AtMost @ A) @ L2) @ H2)))))))))).
% 2.84/1.58 thf(fact_212_not__Iic__le__Icc, axiom, (! [A:$tType]: (((no_bot @ A) => (! [H:A,L2:A,H2:A]: ((~ ((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ H)) @ (((set_or331188842AtMost @ A) @ L2) @ H2)))))))))).
% 2.84/1.58 thf(fact_213_atLeast__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_ord_atLeast @ A) = (^ [L3:A]: (((collect @ A) @ ((ord_less_eq @ A) @ L3))))))))).
% 2.84/1.58 thf(fact_214_atMost__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_ord_atMost @ A) = (^ [U2:A]: (((collect @ A) @ (^ [X4:A]: ((((ord_less_eq @ A) @ X4) @ U2))))))))))).
% 2.84/1.58 thf(fact_215_not__Iic__le__Ici, axiom, (! [A:$tType]: (((no_bot @ A) => (! [H:A,L2:A]: ((~ ((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ H)) @ ((set_ord_atLeast @ A) @ L2)))))))))).
% 2.84/1.58 thf(fact_216_not__Ici__le__Iic, axiom, (! [A:$tType]: (((no_top @ A) => (! [L:A,H2:A]: ((~ ((((ord_less_eq @ (set @ A)) @ ((set_ord_atLeast @ A) @ L)) @ ((set_ord_atMost @ A) @ H2)))))))))).
% 2.84/1.58 thf(fact_217_image__uminus__atMost, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X3:A]: ((((((image @ A) @ A) @ (uminus_uminus @ A)) @ ((set_ord_atMost @ A) @ X3)) = ((set_ord_atLeast @ A) @ ((uminus_uminus @ A) @ X3))))))))).
% 2.84/1.58 thf(fact_218_image__uminus__atLeast, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X3:A]: ((((((image @ A) @ A) @ (uminus_uminus @ A)) @ ((set_ord_atLeast @ A) @ X3)) = ((set_ord_atMost @ A) @ ((uminus_uminus @ A) @ X3))))))))).
% 2.84/1.58 thf(fact_219_image__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F3:(B > A),X3:B,A4:(set @ B)]: (((B2 = (F3 @ X3)) => ((((member @ B) @ X3) @ A4) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F3) @ A4))))))).
% 2.84/1.58 thf(fact_220_image__ident, axiom, (! [A:$tType,Y7:(set @ A)]: ((((((image @ A) @ A) @ (^ [X4:A]: (X4))) @ Y7) = Y7)))).
% 2.84/1.58 thf(fact_221_image__uminus__atLeastAtMost, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X3:A,Y3:A]: ((((((image @ A) @ A) @ (uminus_uminus @ A)) @ (((set_or331188842AtMost @ A) @ X3) @ Y3)) = (((set_or331188842AtMost @ A) @ ((uminus_uminus @ A) @ Y3)) @ ((uminus_uminus @ A) @ X3))))))))).
% 2.84/1.58 thf(fact_222_subset__image__iff, axiom, (! [A:$tType,B:$tType,B4:(set @ A),F3:(B > A),A4:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B4) @ ((((image @ B) @ A) @ F3) @ A4)) = (? [AA:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ AA) @ A4) & (B4 = ((((image @ B) @ A) @ F3) @ AA))))))))).
% 2.84/1.58 thf(fact_223_image__subset__iff, axiom, (! [A:$tType,B:$tType,F3:(B > A),A4:(set @ B),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F3) @ A4)) @ B4) = (! [X4:B]: (((((member @ B) @ X4) @ A4) => (((member @ A) @ (F3 @ X4)) @ B4)))))))).
% 2.84/1.58 thf(fact_224_subset__imageE, axiom, (! [A:$tType,B:$tType,B4:(set @ A),F3:(B > A),A4:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B4) @ ((((image @ B) @ A) @ F3) @ A4)) => (~ ((! [C5:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C5) @ A4) => (B4 != ((((image @ B) @ A) @ F3) @ C5))))))))))).
% 2.84/1.58 thf(fact_225_image__subsetI, axiom, (! [A:$tType,B:$tType,A4:(set @ A),F3:(A > B),B4:(set @ B)]: (((! [X:A]: (((((member @ A) @ X) @ A4) => (((member @ B) @ (F3 @ X)) @ B4)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F3) @ A4)) @ B4))))).
% 2.84/1.58 thf(fact_226_image__mono, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B4:(set @ A),F3:(A > B)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F3) @ A4)) @ ((((image @ A) @ B) @ F3) @ B4)))))).
% 2.84/1.58 thf(fact_227_solves__ode__subset__range, axiom, (! [A:$tType]: (((real_V55928688vector @ A) => (! [X3:(real > A),F3:(real > (A > A)),T5:(set @ real),X7:(set @ A),Y7:(set @ A)]: (((((((initia1685620758es_ode @ A) @ X3) @ F3) @ T5) @ X7) => ((((ord_less_eq @ (set @ A)) @ ((((image @ real) @ A) @ X3) @ T5)) @ Y7) => (((((initia1685620758es_ode @ A) @ X3) @ F3) @ T5) @ Y7))))))))).
% 2.84/1.58 thf(fact_228_imageI, axiom, (! [B:$tType,A:$tType,X3:A,A4:(set @ A),F3:(A > B)]: (((((member @ A) @ X3) @ A4) => (((member @ B) @ (F3 @ X3)) @ ((((image @ A) @ B) @ F3) @ A4)))))).
% 2.84/1.58 thf(fact_229_image__iff, axiom, (! [A:$tType,B:$tType,Z2:A,F3:(B > A),A4:(set @ B)]: (((((member @ A) @ Z2) @ ((((image @ B) @ A) @ F3) @ A4)) = (? [X4:B]: (((((member @ B) @ X4) @ A4) & (Z2 = (F3 @ X4))))))))).
% 2.84/1.58 thf(fact_230_bex__imageD, axiom, (! [A:$tType,B:$tType,F3:(B > A),A4:(set @ B),P2:(A > $o)]: (((? [X8:A]: (((((member @ A) @ X8) @ ((((image @ B) @ A) @ F3) @ A4)) & (P2 @ X8)))) => (? [X:B]: (((((member @ B) @ X) @ A4) & (P2 @ (F3 @ X))))))))).
% 2.84/1.58 thf(fact_231_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N2:(set @ A),F3:(A > B),G3:(A > B)]: (((M = N2) => ((! [X:A]: (((((member @ A) @ X) @ N2) => ((F3 @ X) = (G3 @ X))))) => (((((image @ A) @ B) @ F3) @ M) = ((((image @ A) @ B) @ G3) @ N2))))))).
% 2.84/1.58 thf(fact_232_ball__imageD, axiom, (! [A:$tType,B:$tType,F3:(B > A),A4:(set @ B),P2:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ ((((image @ B) @ A) @ F3) @ A4)) => (P2 @ X)))) => (! [X8:B]: (((((member @ B) @ X8) @ A4) => (P2 @ (F3 @ X8))))))))).
% 2.84/1.58 thf(fact_233_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X3:A,A4:(set @ A),B2:B,F3:(A > B)]: (((((member @ A) @ X3) @ A4) => ((B2 = (F3 @ X3)) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ F3) @ A4))))))).
% 2.84/1.58 thf(fact_234_Compr__image__eq, axiom, (! [A:$tType,B:$tType,F3:(B > A),A4:(set @ B),P2:(A > $o)]: ((((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ ((((image @ B) @ A) @ F3) @ A4)) & (P2 @ X4))))) = ((((image @ B) @ A) @ F3) @ ((collect @ B) @ (^ [X4:B]: (((((member @ B) @ X4) @ A4) & (P2 @ (F3 @ X4))))))))))).
% 2.84/1.58 thf(fact_235_image__image, axiom, (! [A:$tType,B:$tType,C:$tType,F3:(B > A),G3:(C > B),A4:(set @ C)]: ((((((image @ B) @ A) @ F3) @ ((((image @ C) @ B) @ G3) @ A4)) = ((((image @ C) @ A) @ (^ [X4:C]: ((F3 @ (G3 @ X4))))) @ A4))))).
% 2.84/1.58 thf(fact_236_imageE, axiom, (! [A:$tType,B:$tType,B2:A,F3:(B > A),A4:(set @ B)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ F3) @ A4)) => (~ ((! [X:B]: (((B2 = (F3 @ X)) => (~ ((((member @ B) @ X) @ A4)))))))))))).
% 2.84/1.58 thf(fact_237_image__Collect__subsetI, axiom, (! [A:$tType,B:$tType,P2:(A > $o),F3:(A > B),B4:(set @ B)]: (((! [X:A]: (((P2 @ X) => (((member @ B) @ (F3 @ X)) @ B4)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F3) @ ((collect @ A) @ P2))) @ B4))))).
% 2.84/1.58 thf(fact_238_Osub, axiom, (((ord_less_eq @ (set @ ((sum_sum @ c) @ c))) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inl @ c) @ c)) @ (((static_FVO @ a) @ c) @ ode2))) @ v)).
% 2.84/1.58 thf(fact_239_all__subset__image, axiom, (! [A:$tType,B:$tType,F3:(B > A),A4:(set @ B),P2:((set @ A) > $o)]: (((! [B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B6) @ ((((image @ B) @ A) @ F3) @ A4)) => (P2 @ B6)))) = (! [B6:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ B6) @ A4) => (P2 @ ((((image @ B) @ A) @ F3) @ B6))))))))).
% 2.84/1.58 thf(fact_240_bvo__to__fvo, axiom, (! [A:$tType,B:$tType,X3:A,ODE:((ode @ B) @ A)]: (((((member @ ((sum_sum @ A) @ A)) @ (((sum_Inl @ A) @ A) @ X3)) @ (((static_BVO @ B) @ A) @ ODE)) => (((member @ A) @ X3) @ (((static_FVO @ B) @ A) @ ODE)))))).
% 2.84/1.58 thf(fact_241_ids_Obvo__to__fvo, axiom, (! [Sz:$tType,Sf:$tType,Sc:$tType,A:$tType,B:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sf) & ((finite_finite @ Sz) & (linorder @ Sz)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X3:A,ODE:((ode @ B) @ A)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((member @ ((sum_sum @ A) @ A)) @ (((sum_Inl @ A) @ A) @ X3)) @ (((static_BVO @ B) @ A) @ ODE)) => (((member @ A) @ X3) @ (((static_FVO @ B) @ A) @ ODE)))))))))).
% 2.84/1.58 thf(fact_242_VSA, axiom, ((((denotational_VSagree @ c) @ (sol @ (zero_zero @ real))) @ a2) @ ((collect @ c) @ (^ [Uu:c]: (((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ Uu)) @ (((static_BVO @ a) @ c) @ ode2)) | ((((member @ ((sum_sum @ c) @ d)) @ (((sum_Inl @ c) @ d) @ Uu)) @ ((((image @ c) @ ((sum_sum @ c) @ d)) @ ((sum_Inl @ c) @ d)) @ (((static_FVO @ a) @ c) @ ode2))) | (((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ Uu)) @ ((((static_FVF @ a) @ b) @ c) @ p))))))))).
% 2.84/1.58 thf(fact_243_ode__to__fvo, axiom, (! [C:$tType,A:$tType,B:$tType,X3:A,I:((((denota1663640101rp_ext @ B) @ C) @ A) @ product_unit),ODE:((ode @ B) @ A)]: (((((member @ A) @ X3) @ (((((denota811733865E_vars @ B) @ C) @ A) @ I) @ ODE)) => (((member @ A) @ X3) @ (((static_FVO @ B) @ A) @ ODE)))))).
% 2.84/1.58 thf(fact_244_VSagree__sub, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A4:(set @ A),B4:(set @ A),Nu:((finite_Cartesian_vec @ real) @ A),Omega2:((finite_Cartesian_vec @ real) @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((((denotational_VSagree @ A) @ Nu) @ Omega2) @ B4) => ((((denotational_VSagree @ A) @ Nu) @ Omega2) @ A4))))))))).
% 2.84/1.58 thf(fact_245_VSagree__supset, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [B4:(set @ A),A4:(set @ A),Nu:((finite_Cartesian_vec @ real) @ A),Nu2:((finite_Cartesian_vec @ real) @ A)]: (((((ord_less_eq @ (set @ A)) @ B4) @ A4) => (((((denotational_VSagree @ A) @ Nu) @ Nu2) @ A4) => ((((denotational_VSagree @ A) @ Nu) @ Nu2) @ B4))))))))).
% 2.84/1.58 thf(fact_246_VSagree__refl, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [Nu:((finite_Cartesian_vec @ real) @ A),A4:(set @ A)]: (((((denotational_VSagree @ A) @ Nu) @ Nu) @ A4))))))).
% 2.84/1.58 thf(fact_247_ids_Oode__to__fvo, axiom, (! [C:$tType,Sz:$tType,Sf:$tType,Sc:$tType,A:$tType,B:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sf) & ((finite_finite @ Sz) & (linorder @ Sz)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X3:A,I:((((denota1663640101rp_ext @ B) @ C) @ A) @ product_unit),ODE:((ode @ B) @ A)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((member @ A) @ X3) @ (((((denota811733865E_vars @ B) @ C) @ A) @ I) @ ODE)) => (((member @ A) @ X3) @ (((static_FVO @ B) @ A) @ ODE)))))))))).
% 2.84/1.58 thf(fact_248_OVsub_H, axiom, (((ord_less_eq @ (set @ ((sum_sum @ c) @ c))) @ (((sup_sup @ (set @ ((sum_sum @ c) @ c))) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inl @ c) @ c)) @ (((oDE_dom @ a) @ c) @ ode2))) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inr @ c) @ c)) @ (((oDE_dom @ a) @ c) @ ode2)))) @ v)).
% 2.84/1.58 thf(fact_249_ids_Ovne23, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid2 != Vid3)))))))).
% 2.84/1.58 thf(fact_250_sup_Oright__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ B2) = (((sup_sup @ A) @ A2) @ B2)))))))).
% 2.84/1.58 thf(fact_251_sup__left__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,Y3:A]: (((((sup_sup @ A) @ X3) @ (((sup_sup @ A) @ X3) @ Y3)) = (((sup_sup @ A) @ X3) @ Y3)))))))).
% 2.84/1.58 thf(fact_252_sup_Oleft__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A]: (((((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ A2) @ B2)) = (((sup_sup @ A) @ A2) @ B2)))))))).
% 2.84/1.58 thf(fact_253_sup__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A]: (((((sup_sup @ A) @ X3) @ X3) = X3))))))).
% 2.84/1.58 thf(fact_254_sup_Oidem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A]: (((((sup_sup @ A) @ A2) @ A2) = A2))))))).
% 2.84/1.58 thf(subcl_Finite__Set_Ofinite___HOL_Otype, axiom, (! [A:$tType]: (((finite_finite @ A) => (type @ A))))).
% 2.84/1.58 thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space, axiom, (! [A9:$tType,A10:$tType]: ((((ordere890947078_space @ A9) & (finite_finite @ A10)) => (ordere890947078_space @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space_1, axiom, (! [A9:$tType,A10:$tType]: ((((ordere890947078_space @ A9) & (ordere890947078_space @ A10)) => (ordere890947078_space @ ((product_prod @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_2, axiom, (ordere890947078_space @ real)).
% 2.84/1.58 thf(tcon_fun___Topological__Spaces_Otopological__space, axiom, (! [A9:$tType,A10:$tType]: (((topolo503727757_space @ A10) => (topolo503727757_space @ (A9 > A10)))))).
% 2.84/1.58 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A9:$tType,A10:$tType]: (((semilattice_sup @ A10) => (semilattice_sup @ (A9 > A10)))))).
% 2.84/1.58 thf(tcon_fun___Lattices_Oboolean__algebra, axiom, (! [A9:$tType,A10:$tType]: (((boolean_algebra @ A10) => (boolean_algebra @ (A9 > A10)))))).
% 2.84/1.58 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A9:$tType,A10:$tType]: (((preorder @ A10) => (preorder @ (A9 > A10)))))).
% 2.84/1.58 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ (A9 > A10)))))).
% 2.84/1.58 thf(tcon_fun___Orderings_Oorder, axiom, (! [A9:$tType,A10:$tType]: (((order @ A10) => (order @ (A9 > A10)))))).
% 2.84/1.58 thf(tcon_fun___Orderings_Oord, axiom, (! [A9:$tType,A10:$tType]: (((ord @ A10) => (ord @ (A9 > A10)))))).
% 2.84/1.58 thf(tcon_fun___Groups_Ouminus, axiom, (! [A9:$tType,A10:$tType]: (((uminus @ A10) => (uminus @ (A9 > A10)))))).
% 2.84/1.58 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_3, axiom, (! [A9:$tType]: ((semilattice_sup @ (set @ A9))))).
% 2.84/1.58 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_4, axiom, (! [A9:$tType]: ((boolean_algebra @ (set @ A9))))).
% 2.84/1.58 thf(tcon_Set_Oset___Orderings_Opreorder_5, axiom, (! [A9:$tType]: ((preorder @ (set @ A9))))).
% 2.84/1.58 thf(tcon_Set_Oset___Finite__Set_Ofinite_6, axiom, (! [A9:$tType]: (((finite_finite @ A9) => (finite_finite @ (set @ A9)))))).
% 2.84/1.58 thf(tcon_Set_Oset___Orderings_Oorder_7, axiom, (! [A9:$tType]: ((order @ (set @ A9))))).
% 2.84/1.58 thf(tcon_Set_Oset___Orderings_Oord_8, axiom, (! [A9:$tType]: ((ord @ (set @ A9))))).
% 2.84/1.58 thf(tcon_Set_Oset___Groups_Ouminus_9, axiom, (! [A9:$tType]: ((uminus @ (set @ A9))))).
% 2.84/1.58 thf(tcon_Set_Oset___Groups_Ozero, axiom, (! [A9:$tType]: (((zero @ A9) => (zero @ (set @ A9)))))).
% 2.84/1.58 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_10, axiom, (topolo503727757_space @ $o)).
% 2.84/1.58 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_11, axiom, (semilattice_sup @ $o)).
% 2.84/1.58 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_12, axiom, (boolean_algebra @ $o)).
% 2.84/1.58 thf(tcon_HOL_Obool___Orderings_Opreorder_13, axiom, (preorder @ $o)).
% 2.84/1.58 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 2.84/1.58 thf(tcon_HOL_Obool___Finite__Set_Ofinite_14, axiom, (finite_finite @ $o)).
% 2.84/1.58 thf(tcon_HOL_Obool___Orderings_Oorder_15, axiom, (order @ $o)).
% 2.84/1.58 thf(tcon_HOL_Obool___Orderings_Oord_16, axiom, (ord @ $o)).
% 2.84/1.58 thf(tcon_HOL_Obool___Groups_Ouminus_17, axiom, (uminus @ $o)).
% 2.84/1.58 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector, axiom, (real_V55928688vector @ real)).
% 2.84/1.58 thf(tcon_Real_Oreal___Lattice__Algebras_Olattice__ab__group__add, axiom, (lattic1601792062up_add @ real)).
% 2.84/1.58 thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_18, axiom, (topolo503727757_space @ real)).
% 2.84/1.58 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ real)).
% 2.84/1.58 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add, axiom, (linord219039673up_add @ real)).
% 2.84/1.58 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add, axiom, (ordered_ab_group_add @ real)).
% 2.84/1.58 thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_19, axiom, (semilattice_sup @ real)).
% 2.84/1.58 thf(tcon_Real_Oreal___Orderings_Opreorder_20, axiom, (preorder @ real)).
% 2.84/1.58 thf(tcon_Real_Oreal___Orderings_Olinorder_21, axiom, (linorder @ real)).
% 2.84/1.58 thf(tcon_Real_Oreal___Orderings_Ono__top, axiom, (no_top @ real)).
% 2.84/1.58 thf(tcon_Real_Oreal___Orderings_Ono__bot, axiom, (no_bot @ real)).
% 2.84/1.58 thf(tcon_Real_Oreal___Groups_Ogroup__add, axiom, (group_add @ real)).
% 2.84/1.58 thf(tcon_Real_Oreal___Orderings_Oorder_22, axiom, (order @ real)).
% 2.84/1.58 thf(tcon_Real_Oreal___Orderings_Oord_23, axiom, (ord @ real)).
% 2.84/1.58 thf(tcon_Real_Oreal___Groups_Ouminus_24, axiom, (uminus @ real)).
% 2.84/1.58 thf(tcon_Real_Oreal___Groups_Ozero_25, axiom, (zero @ real)).
% 2.84/1.58 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_26, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ ((sum_sum @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_27, axiom, (! [A9:$tType]: ((semilattice_sup @ (filter @ A9))))).
% 2.84/1.58 thf(tcon_Filter_Ofilter___Orderings_Opreorder_28, axiom, (! [A9:$tType]: ((preorder @ (filter @ A9))))).
% 2.84/1.58 thf(tcon_Filter_Ofilter___Orderings_Oorder_29, axiom, (! [A9:$tType]: ((order @ (filter @ A9))))).
% 2.84/1.58 thf(tcon_Filter_Ofilter___Orderings_Oord_30, axiom, (! [A9:$tType]: ((ord @ (filter @ A9))))).
% 2.84/1.58 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__normed__vector_31, axiom, (! [A9:$tType,A10:$tType]: ((((real_V55928688vector @ A9) & (real_V55928688vector @ A10)) => (real_V55928688vector @ ((product_prod @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_32, axiom, (! [A9:$tType,A10:$tType]: ((((topolo503727757_space @ A9) & (topolo503727757_space @ A10)) => (topolo503727757_space @ ((product_prod @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__group__add_33, axiom, (! [A9:$tType,A10:$tType]: ((((ordere890947078_space @ A9) & (ordere890947078_space @ A10)) => (ordered_ab_group_add @ ((product_prod @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Product__Type_Oprod___Lattices_Osemilattice__sup_34, axiom, (! [A9:$tType,A10:$tType]: ((((semilattice_sup @ A9) & (semilattice_sup @ A10)) => (semilattice_sup @ ((product_prod @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Product__Type_Oprod___Lattices_Oboolean__algebra_35, axiom, (! [A9:$tType,A10:$tType]: ((((boolean_algebra @ A9) & (boolean_algebra @ A10)) => (boolean_algebra @ ((product_prod @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_36, axiom, (! [A9:$tType,A10:$tType]: ((((preorder @ A9) & (preorder @ A10)) => (preorder @ ((product_prod @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_37, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ ((product_prod @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Product__Type_Oprod___Groups_Ogroup__add_38, axiom, (! [A9:$tType,A10:$tType]: ((((group_add @ A9) & (group_add @ A10)) => (group_add @ ((product_prod @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Product__Type_Oprod___Orderings_Oorder_39, axiom, (! [A9:$tType,A10:$tType]: ((((order @ A9) & (order @ A10)) => (order @ ((product_prod @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Product__Type_Oprod___Orderings_Oord_40, axiom, (! [A9:$tType,A10:$tType]: ((((ord @ A9) & (ord @ A10)) => (ord @ ((product_prod @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Product__Type_Oprod___Groups_Ouminus_41, axiom, (! [A9:$tType,A10:$tType]: ((((uminus @ A9) & (uminus @ A10)) => (uminus @ ((product_prod @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Product__Type_Oprod___Groups_Ozero_42, axiom, (! [A9:$tType,A10:$tType]: ((((zero @ A9) & (zero @ A10)) => (zero @ ((product_prod @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_43, axiom, (semilattice_sup @ product_unit)).
% 2.84/1.58 thf(tcon_Product__Type_Ounit___Lattices_Oboolean__algebra_44, axiom, (boolean_algebra @ product_unit)).
% 2.84/1.58 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_45, axiom, (preorder @ product_unit)).
% 2.84/1.58 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_46, axiom, (linorder @ product_unit)).
% 2.84/1.58 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_47, axiom, (finite_finite @ product_unit)).
% 2.84/1.58 thf(tcon_Product__Type_Ounit___Orderings_Oorder_48, axiom, (order @ product_unit)).
% 2.84/1.58 thf(tcon_Product__Type_Ounit___Orderings_Oord_49, axiom, (ord @ product_unit)).
% 2.84/1.58 thf(tcon_Product__Type_Ounit___Groups_Ouminus_50, axiom, (uminus @ product_unit)).
% 2.84/1.58 thf(tcon_Finite__Cartesian__Product_Ovec___Real__Vector__Spaces_Oreal__normed__vector_51, axiom, (! [A9:$tType,A10:$tType]: ((((real_V55928688vector @ A9) & (finite_finite @ A10)) => (real_V55928688vector @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Finite__Cartesian__Product_Ovec___Topological__Spaces_Otopological__space_52, axiom, (! [A9:$tType,A10:$tType]: ((((topolo503727757_space @ A9) & (finite_finite @ A10)) => (topolo503727757_space @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__group__add_53, axiom, (! [A9:$tType,A10:$tType]: ((((ordere890947078_space @ A9) & (finite_finite @ A10)) => (ordered_ab_group_add @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Osemilattice__sup_54, axiom, (! [A9:$tType,A10:$tType]: ((((ordere890947078_space @ A9) & (finite_finite @ A10)) => (semilattice_sup @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Opreorder_55, axiom, (! [A9:$tType,A10:$tType]: ((((order @ A9) & (finite_finite @ A10)) => (preorder @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_56, axiom, (! [A9:$tType,A10:$tType]: ((((linorder @ A9) & (cARD_1 @ A10)) => (linorder @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_57, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ogroup__add_58, axiom, (! [A9:$tType,A10:$tType]: ((((group_add @ A9) & (finite_finite @ A10)) => (group_add @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oorder_59, axiom, (! [A9:$tType,A10:$tType]: ((((order @ A9) & (finite_finite @ A10)) => (order @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oord_60, axiom, (! [A9:$tType,A10:$tType]: ((((ord @ A9) & (finite_finite @ A10)) => (ord @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ouminus_61, axiom, (! [A9:$tType,A10:$tType]: ((((uminus @ A9) & (finite_finite @ A10)) => (uminus @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 2.84/1.58 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_62, axiom, (! [A9:$tType,A10:$tType]: ((((zero @ A9) & (finite_finite @ A10)) => (zero @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 2.84/1.58 thf(tfree_0, hypothesis, (finite_finite @ c)).
% 2.84/1.58 thf(tfree_1, hypothesis, (finite_finite @ a)).
% 2.84/1.58 thf(tfree_2, hypothesis, (finite_finite @ b)).
% 2.84/1.58 thf(conj_0, conjecture, ((((denotational_Vagree @ c) @ (((((((denotational_mk_v @ a) @ b) @ c) @ j) @ ode2) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ aa) @ ba)) @ (sol @ s))) @ ((((((denotational_mk_xode @ a) @ b) @ c) @ j) @ ode2) @ (sol @ s))) @ (((((denotational_semBV @ a) @ b) @ c) @ j) @ ode2))).
% 2.84/1.58 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.c1Zf8XeWV8/DTF2DTF26468.p
% 2.84/1.58 ------------------------
% 3.03/1.64 ---- Cleaned THF ---
% 3.03/1.64 thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext,type,
% 3.03/1.64 denota1663640101rp_ext: $tType > $tType > $tType > $tType > $tType ).
% 3.03/1.64
% 3.03/1.64 thf(ty_t_Finite__Cartesian__Product_Ovec,type,
% 3.03/1.64 finite_Cartesian_vec: $tType > $tType > $tType ).
% 3.03/1.64
% 3.03/1.64 thf(ty_t_Product__Type_Ounit,type,
% 3.03/1.64 product_unit: $tType ).
% 3.03/1.64
% 3.03/1.64 thf(ty_t_Product__Type_Oprod,type,
% 3.03/1.64 product_prod: $tType > $tType > $tType ).
% 3.03/1.64
% 3.03/1.64 thf(ty_t_Syntax_Oformula,type,
% 3.03/1.64 formula: $tType > $tType > $tType > $tType ).
% 3.03/1.64
% 3.03/1.64 thf(ty_t_Sum__Type_Osum,type,
% 3.03/1.64 sum_sum: $tType > $tType > $tType ).
% 3.03/1.64
% 3.03/1.64 thf(ty_t_Filter_Ofilter,type,
% 3.03/1.64 filter: $tType > $tType ).
% 3.03/1.64
% 3.03/1.64 thf(ty_t_Syntax_Otrm,type,
% 3.03/1.64 trm: $tType > $tType > $tType ).
% 3.03/1.64
% 3.03/1.64 thf(ty_t_Syntax_OODE,type,
% 3.03/1.64 ode: $tType > $tType > $tType ).
% 3.03/1.64
% 3.03/1.64 thf(ty_t_Syntax_Ohp,type,
% 3.03/1.64 hp: $tType > $tType > $tType > $tType ).
% 3.03/1.64
% 3.03/1.64 thf(ty_t_Real_Oreal,type,
% 3.03/1.64 real: $tType ).
% 3.03/1.64
% 3.03/1.64 thf(ty_t_List_Olist,type,
% 3.03/1.64 list: $tType > $tType ).
% 3.03/1.64
% 3.03/1.64 thf(ty_t_Set_Oset,type,
% 3.03/1.64 set: $tType > $tType ).
% 3.03/1.64
% 3.03/1.64 thf(ty_tf_d,type,
% 3.03/1.64 d: $tType ).
% 3.03/1.64
% 3.03/1.64 thf(ty_tf_c,type,
% 3.03/1.64 c: $tType ).
% 3.03/1.64
% 3.03/1.64 thf(ty_tf_b,type,
% 3.03/1.64 b: $tType ).
% 3.03/1.64
% 3.03/1.64 thf(ty_tf_a,type,
% 3.03/1.64 a: $tType ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Cardinality_OCARD__1,type,
% 3.03/1.64 cARD_1:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
% 3.03/1.64 ordere890947078_space:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_HOL_Otype,type,
% 3.03/1.64 type:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Groups_Ozero,type,
% 3.03/1.64 zero:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Groups_Ouminus,type,
% 3.03/1.64 uminus:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Orderings_Oord,type,
% 3.03/1.64 ord:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Orderings_Oorder,type,
% 3.03/1.64 order:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Groups_Ogroup__add,type,
% 3.03/1.64 group_add:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Orderings_Ono__bot,type,
% 3.03/1.64 no_bot:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Orderings_Ono__top,type,
% 3.03/1.64 no_top:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Finite__Set_Ofinite,type,
% 3.03/1.64 finite_finite:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Orderings_Olinorder,type,
% 3.03/1.64 linorder:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Orderings_Opreorder,type,
% 3.03/1.64 preorder:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 3.03/1.64 boolean_algebra:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 3.03/1.64 semilattice_sup:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 3.03/1.64 ordered_ab_group_add:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 3.03/1.64 linord219039673up_add:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 3.03/1.64 linord1659791738miring:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Topological__Spaces_Otopological__space,type,
% 3.03/1.64 topolo503727757_space:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 3.03/1.64 canoni770627133id_add:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Lattice__Algebras_Olattice__ab__group__add,type,
% 3.03/1.64 lattic1601792062up_add:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
% 3.03/1.64 real_V55928688vector:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd,type,
% 3.03/1.64 euclid1678468529ng_gcd:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
% 3.03/1.64 unique1598680935umeral:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring,type,
% 3.03/1.64 euclid1155270486miring:
% 3.03/1.64 !>[A: $tType] : $o ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 3.03/1.64 bNF_Ca1785829860lChain:
% 3.03/1.64 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__fml,type,
% 3.03/1.64 coinci1993344360de_fml:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__hp,type,
% 3.03/1.64 coinci32832645ide_hp:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__hp_H,type,
% 3.03/1.64 coinci554967470ide_hp:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Oode__sem__equiv,type,
% 3.03/1.64 coinci1495174342_equiv:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Denotational__Semantics_OIagree,type,
% 3.03/1.64 denotational_Iagree:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Denotational__Semantics_OODE__sem,type,
% 3.03/1.64 denotational_ODE_sem:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( finite_Cartesian_vec @ real @ C ) > ( finite_Cartesian_vec @ real @ C ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Denotational__Semantics_OODE__vars,type,
% 3.03/1.64 denota811733865E_vars:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( set @ C ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Denotational__Semantics_OVSagree,type,
% 3.03/1.64 denotational_VSagree:
% 3.03/1.64 !>[C: $tType] : ( ( finite_Cartesian_vec @ real @ C ) > ( finite_Cartesian_vec @ real @ C ) > ( set @ C ) > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Denotational__Semantics_OVagree,type,
% 3.03/1.64 denotational_Vagree:
% 3.03/1.64 !>[C: $tType] : ( ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( set @ ( sum_sum @ C @ C ) ) > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Denotational__Semantics_Ofml__sem,type,
% 3.03/1.64 denotational_fml_sem:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( formula @ A @ B @ C ) > ( set @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Denotational__Semantics_Omk__v,type,
% 3.03/1.64 denotational_mk_v:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( finite_Cartesian_vec @ real @ C ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Denotational__Semantics_Omk__xode,type,
% 3.03/1.64 denotational_mk_xode:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( finite_Cartesian_vec @ real @ C ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Denotational__Semantics_OsemBV,type,
% 3.03/1.64 denotational_semBV:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Deriv_Ohas__vector__derivative,type,
% 3.03/1.64 has_ve2132708402vative:
% 3.03/1.64 !>[B: $tType] : ( ( real > B ) > B > ( filter @ real ) > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
% 3.03/1.64 unique455577585es_aux:
% 3.03/1.64 !>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 3.03/1.64 uminus_uminus:
% 3.03/1.64 !>[A: $tType] : ( A > A ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 3.03/1.64 zero_zero:
% 3.03/1.64 !>[A: $tType] : A ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_HOL_OThe,type,
% 3.03/1.64 the:
% 3.03/1.64 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Ids_Oids,type,
% 3.03/1.64 ids:
% 3.03/1.64 !>[Sz: $tType,Sf: $tType,Sc: $tType] : ( Sz > Sz > Sz > Sf > Sf > Sf > Sc > Sc > Sc > Sc > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Initial__Value__Problem_Osolves__ode,type,
% 3.03/1.64 initia1685620758es_ode:
% 3.03/1.64 !>[A: $tType] : ( ( real > A ) > ( real > A > A ) > ( set @ real ) > ( set @ A ) > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Lattices_Osup__class_Osup,type,
% 3.03/1.64 sup_sup:
% 3.03/1.64 !>[A: $tType] : ( A > A > A ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.03/1.64 ord_less_eq:
% 3.03/1.64 !>[A: $tType] : ( A > A > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Product__Type_OPair,type,
% 3.03/1.64 product_Pair:
% 3.03/1.64 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 3.03/1.64 produc2004651681e_prod:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 3.03/1.64 product_rec_prod:
% 3.03/1.64 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
% 3.03/1.64 product_rec_set_prod:
% 3.03/1.64 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
% 3.03/1.64 product_rec_set_unit:
% 3.03/1.64 !>[T: $tType] : ( T > product_unit > T > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
% 3.03/1.64 product_rec_unit:
% 3.03/1.64 !>[T: $tType] : ( T > product_unit > T ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Set_OCollect,type,
% 3.03/1.64 collect:
% 3.03/1.64 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Set_Oimage,type,
% 3.03/1.64 image:
% 3.03/1.64 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
% 3.03/1.64 set_ord_atLeast:
% 3.03/1.64 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
% 3.03/1.64 set_or331188842AtMost:
% 3.03/1.64 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Set__Interval_Oord__class_OatMost,type,
% 3.03/1.64 set_ord_atMost:
% 3.03/1.64 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Static__Semantics_OBVO,type,
% 3.03/1.64 static_BVO:
% 3.03/1.64 !>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Static__Semantics_OFVF,type,
% 3.03/1.64 static_FVF:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Static__Semantics_OFVO,type,
% 3.03/1.64 static_FVO:
% 3.03/1.64 !>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > ( set @ C ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Static__Semantics_OSIGF,type,
% 3.03/1.64 static_SIGF:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Sum__Type_OInl,type,
% 3.03/1.64 sum_Inl:
% 3.03/1.64 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Sum__Type_OInr,type,
% 3.03/1.64 sum_Inr:
% 3.03/1.64 !>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Syntax_OEquiv,type,
% 3.03/1.64 equiv:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Syntax_OImplies,type,
% 3.03/1.64 implies:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Syntax_OODE__dom,type,
% 3.03/1.64 oDE_dom:
% 3.03/1.64 !>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > ( set @ C ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Syntax_Ofsafe,type,
% 3.03/1.64 fsafe:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Syntax_Ohp_OEvolveODE,type,
% 3.03/1.64 evolveODE:
% 3.03/1.64 !>[A: $tType,C: $tType,B: $tType] : ( ( ode @ A @ C ) > ( formula @ A @ B @ C ) > ( hp @ A @ B @ C ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Syntax_Ohpsafe,type,
% 3.03/1.64 hpsafe:
% 3.03/1.64 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Syntax_Oosafe,type,
% 3.03/1.64 osafe:
% 3.03/1.64 !>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
% 3.03/1.64 topolo507301023within:
% 3.03/1.64 !>[A: $tType] : ( A > ( set @ A ) > ( filter @ A ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_c_member,type,
% 3.03/1.64 member:
% 3.03/1.64 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_v_I____,type,
% 3.03/1.64 i: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
% 3.03/1.64
% 3.03/1.64 thf(sy_v_J____,type,
% 3.03/1.64 j: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
% 3.03/1.64
% 3.03/1.64 thf(sy_v_ODE____,type,
% 3.03/1.64 ode2: ode @ a @ c ).
% 3.03/1.64
% 3.03/1.64 thf(sy_v_P____,type,
% 3.03/1.64 p: formula @ a @ b @ c ).
% 3.03/1.64
% 3.03/1.64 thf(sy_v_V____,type,
% 3.03/1.64 v: set @ ( sum_sum @ c @ c ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_v_a____,type,
% 3.03/1.64 a2: finite_Cartesian_vec @ real @ c ).
% 3.03/1.64
% 3.03/1.64 thf(sy_v_aa____,type,
% 3.03/1.64 aa: finite_Cartesian_vec @ real @ c ).
% 3.03/1.64
% 3.03/1.64 thf(sy_v_ab____,type,
% 3.03/1.64 ab: finite_Cartesian_vec @ real @ c ).
% 3.03/1.64
% 3.03/1.64 thf(sy_v_b____,type,
% 3.03/1.64 b2: finite_Cartesian_vec @ real @ c ).
% 3.03/1.64
% 3.03/1.64 thf(sy_v_ba____,type,
% 3.03/1.64 ba: finite_Cartesian_vec @ real @ c ).
% 3.03/1.64
% 3.03/1.64 thf(sy_v_bb____,type,
% 3.03/1.64 bb: finite_Cartesian_vec @ real @ c ).
% 3.03/1.64
% 3.03/1.64 thf(sy_v_s____,type,
% 3.03/1.64 s: real ).
% 3.03/1.64
% 3.03/1.64 thf(sy_v_sol____,type,
% 3.03/1.64 sol: real > ( finite_Cartesian_vec @ real @ c ) ).
% 3.03/1.64
% 3.03/1.64 thf(sy_v_t____,type,
% 3.03/1.64 t: real ).
% 3.03/1.64
% 3.03/1.64 thf(fact_0_osafe,axiom,
% 3.03/1.64 osafe @ a @ c @ ode2 ).
% 3.03/1.64
% 3.03/1.64 thf(fact_1__092_060open_062Vagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Iaa_M_Aba_J_A_I_N_AsemBV_AJ_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Imk__xode_AJ_AODE_A_Isol_As_J_J_A_IsemBV_AJ_AODE_J_092_060close_062,axiom,
% 3.03/1.64 ( ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ j @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( sol @ s ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( uminus_uminus @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ j @ ode2 ) ) )
% 3.03/1.64 & ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ j @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( sol @ s ) ) @ ( denotational_mk_xode @ a @ b @ c @ j @ ode2 @ ( sol @ s ) ) @ ( denotational_semBV @ a @ b @ c @ j @ ode2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_2__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_Asa_J_J_A_Iaa_M_Aba_J_A_I_N_AsemBV_AJ_AODE_J_092_060close_062,axiom,
% 3.03/1.64 ! [S: real] : ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ j @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( sol @ S ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( uminus_uminus @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ j @ ode2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_3_mk__v__agree,axiom,
% 3.03/1.64 ! [C: $tType,A: $tType,B: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ B )
% 3.03/1.64 & ( finite_finite @ A )
% 3.03/1.64 & ( finite_finite @ C ) )
% 3.03/1.64 => ! [I: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A,Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Sol: finite_Cartesian_vec @ real @ A] :
% 3.03/1.64 ( ( denotational_Vagree @ A @ ( denotational_mk_v @ B @ C @ A @ I @ ODE @ Nu @ Sol ) @ Nu @ ( uminus_uminus @ ( set @ ( sum_sum @ A @ A ) ) @ ( denotational_semBV @ B @ C @ A @ I @ ODE ) ) )
% 3.03/1.64 & ( denotational_Vagree @ A @ ( denotational_mk_v @ B @ C @ A @ I @ ODE @ Nu @ Sol ) @ ( denotational_mk_xode @ B @ C @ A @ I @ ODE @ Sol ) @ ( denotational_semBV @ B @ C @ A @ I @ ODE ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_4_prod_Oinject,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 3.03/1.64 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 3.03/1.64 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 3.03/1.64 = ( ( X1 = Y1 )
% 3.03/1.64 & ( X2 = Y2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_5_old_Oprod_Oinject,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 3.03/1.64 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 3.03/1.64 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 3.03/1.64 = ( ( A2 = A3 )
% 3.03/1.64 & ( B2 = B3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_6_mk__v__exists,axiom,
% 3.03/1.64 ! [C: $tType,A: $tType,B: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ B )
% 3.03/1.64 & ( finite_finite @ A )
% 3.03/1.64 & ( finite_finite @ C ) )
% 3.03/1.64 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A,Sol: finite_Cartesian_vec @ real @ A] :
% 3.03/1.64 ? [Omega: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
% 3.03/1.64 ( ( denotational_Vagree @ A @ Omega @ Nu @ ( uminus_uminus @ ( set @ ( sum_sum @ A @ A ) ) @ ( denotational_semBV @ B @ C @ A @ I @ ODE ) ) )
% 3.03/1.64 & ( denotational_Vagree @ A @ Omega @ ( denotational_mk_xode @ B @ C @ A @ I @ ODE @ Sol ) @ ( denotational_semBV @ B @ C @ A @ I @ ODE ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_7_VA,axiom,
% 3.03/1.64 denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ v ).
% 3.03/1.64
% 3.03/1.64 thf(fact_8__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AI_AODE_A_Ia_M_Ab_J_A_Isol_Asa_J_J_A_Imk__xode_AI_AODE_A_Isol_Asa_J_J_A_IsemBV_AI_AODE_J_092_060close_062,axiom,
% 3.03/1.64 ! [S: real] : ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ i @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( sol @ S ) ) @ ( denotational_mk_xode @ a @ b @ c @ i @ ode2 @ ( sol @ S ) ) @ ( denotational_semBV @ a @ b @ c @ i @ ode2 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_9_VAOV,axiom,
% 3.03/1.64 denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( static_BVO @ a @ c @ ode2 ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_10_agree__comm,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( finite_finite @ A )
% 3.03/1.64 => ! [A4: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),B4: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),V: set @ ( sum_sum @ A @ A )] :
% 3.03/1.64 ( ( denotational_Vagree @ A @ A4 @ B4 @ V )
% 3.03/1.64 => ( denotational_Vagree @ A @ B4 @ A4 @ V ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_11_agree__refl,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( finite_finite @ A )
% 3.03/1.64 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A4: set @ ( sum_sum @ A @ A )] : ( denotational_Vagree @ A @ Nu @ Nu @ A4 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_12__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AI_AODE_A_Ia_M_Ab_J_A_Isol_Asa_J_J_A_Ia_M_Ab_J_A_I_N_AsemBV_AI_AODE_J_092_060close_062,axiom,
% 3.03/1.64 ! [S: real] : ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ i @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( sol @ S ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( uminus_uminus @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ i @ ode2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_13_surj__pair,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 3.03/1.64 ? [X: A,Y: B] :
% 3.03/1.64 ( P
% 3.03/1.64 = ( product_Pair @ A @ B @ X @ Y ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_14_prod__cases,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 3.03/1.64 ( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 3.03/1.64 => ( P2 @ P ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_15_OVsub,axiom,
% 3.03/1.64 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_BVO @ a @ c @ ode2 ) @ v ).
% 3.03/1.64
% 3.03/1.64 thf(fact_16_veq,axiom,
% 3.03/1.64 ( ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb )
% 3.03/1.64 = ( denotational_mk_v @ a @ b @ c @ i @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( sol @ t ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_17_old_Oprod_Oinducts,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 3.03/1.64 ( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 3.03/1.64 => ( P2 @ Prod ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_18_old_Oprod_Oexhaust,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 3.03/1.64 ~ ! [A5: A,B5: B] :
% 3.03/1.64 ( Y3
% 3.03/1.64 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_19_prod__induct7,axiom,
% 3.03/1.64 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 3.03/1.64 ( ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 3.03/1.64 => ( P2 @ X3 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_20_prod__induct6,axiom,
% 3.03/1.64 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 3.03/1.64 ( ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 3.03/1.64 => ( P2 @ X3 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_21_prod__induct5,axiom,
% 3.03/1.64 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 3.03/1.64 ( ! [A5: A,B5: B,C2: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 3.03/1.64 => ( P2 @ X3 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_22_prod__induct4,axiom,
% 3.03/1.64 ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 3.03/1.64 ( ! [A5: A,B5: B,C2: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 3.03/1.64 => ( P2 @ X3 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_23_prod__induct3,axiom,
% 3.03/1.64 ! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.03/1.64 ( ! [A5: A,B5: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) )
% 3.03/1.64 => ( P2 @ X3 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_24_prod__cases7,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 3.03/1.64 ~ ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
% 3.03/1.64 ( Y3
% 3.03/1.64 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_25_prod__cases6,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 3.03/1.64 ~ ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F] :
% 3.03/1.64 ( Y3
% 3.03/1.64 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_26_prod__cases5,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 3.03/1.64 ~ ! [A5: A,B5: B,C2: C,D2: D,E2: E] :
% 3.03/1.64 ( Y3
% 3.03/1.64 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_27_prod__cases4,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 3.03/1.64 ~ ! [A5: A,B5: B,C2: C,D2: D] :
% 3.03/1.64 ( Y3
% 3.03/1.64 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_28_prod__cases3,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.03/1.64 ~ ! [A5: A,B5: B,C2: C] :
% 3.03/1.64 ( Y3
% 3.03/1.64 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_29_Pair__inject,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 3.03/1.64 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 3.03/1.64 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 3.03/1.64 => ~ ( ( A2 = A3 )
% 3.03/1.64 => ( B2 != B3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_30_OVsub_H_H,axiom,
% 3.03/1.64 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ i @ ode2 ) @ v ).
% 3.03/1.64
% 3.03/1.64 thf(fact_31_semBVsub,axiom,
% 3.03/1.64 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ i @ ode2 ) @ ( static_BVO @ a @ c @ ode2 ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_32_uminus__Pair,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType] :
% 3.03/1.64 ( ( ( uminus @ B )
% 3.03/1.64 & ( uminus @ A ) )
% 3.03/1.64 => ! [A2: A,B2: B] :
% 3.03/1.64 ( ( uminus_uminus @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 3.03/1.64 = ( product_Pair @ A @ B @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ B @ B2 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_33_old_Oprod_Orec,axiom,
% 3.03/1.64 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 3.03/1.64 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 3.03/1.64 = ( F1 @ A2 @ B2 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_34_ComplI,axiom,
% 3.03/1.64 ! [A: $tType,C3: A,A4: set @ A] :
% 3.03/1.64 ( ~ ( member @ A @ C3 @ A4 )
% 3.03/1.64 => ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_35_Compl__iff,axiom,
% 3.03/1.64 ! [A: $tType,C3: A,A4: set @ A] :
% 3.03/1.64 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
% 3.03/1.64 = ( ~ ( member @ A @ C3 @ A4 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_36_Compl__eq__Compl__iff,axiom,
% 3.03/1.64 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 3.03/1.64 ( ( ( uminus_uminus @ ( set @ A ) @ A4 )
% 3.03/1.64 = ( uminus_uminus @ ( set @ A ) @ B4 ) )
% 3.03/1.64 = ( A4 = B4 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_37_verit__minus__simplify_I4_J,axiom,
% 3.03/1.64 ! [B: $tType] :
% 3.03/1.64 ( ( group_add @ B )
% 3.03/1.64 => ! [B2: B] :
% 3.03/1.64 ( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
% 3.03/1.64 = B2 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_38_uminus__apply,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType] :
% 3.03/1.64 ( ( uminus @ B )
% 3.03/1.64 => ( ( uminus_uminus @ ( A > B ) )
% 3.03/1.64 = ( ^ [A6: A > B,X4: A] : ( uminus_uminus @ B @ ( A6 @ X4 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_39_add_Oinverse__inverse,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( group_add @ A )
% 3.03/1.64 => ! [A2: A] :
% 3.03/1.64 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 3.03/1.64 = A2 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_40_subset__antisym,axiom,
% 3.03/1.64 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 3.03/1.64 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 3.03/1.64 => ( A4 = B4 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_41_subsetI,axiom,
% 3.03/1.64 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 3.03/1.64 ( ! [X: A] :
% 3.03/1.64 ( ( member @ A @ X @ A4 )
% 3.03/1.64 => ( member @ A @ X @ B4 ) )
% 3.03/1.64 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_42_compl__eq__compl__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( boolean_algebra @ A )
% 3.03/1.64 => ! [X3: A,Y3: A] :
% 3.03/1.64 ( ( ( uminus_uminus @ A @ X3 )
% 3.03/1.64 = ( uminus_uminus @ A @ Y3 ) )
% 3.03/1.64 = ( X3 = Y3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_43_double__compl,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( boolean_algebra @ A )
% 3.03/1.64 => ! [X3: A] :
% 3.03/1.64 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X3 ) )
% 3.03/1.64 = X3 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_44_neg__equal__iff__equal,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( group_add @ A )
% 3.03/1.64 => ! [A2: A,B2: A] :
% 3.03/1.64 ( ( ( uminus_uminus @ A @ A2 )
% 3.03/1.64 = ( uminus_uminus @ A @ B2 ) )
% 3.03/1.64 = ( A2 = B2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_45_mem__Collect__eq,axiom,
% 3.03/1.64 ! [A: $tType,A2: A,P2: A > $o] :
% 3.03/1.64 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 3.03/1.64 = ( P2 @ A2 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_46_Collect__mem__eq,axiom,
% 3.03/1.64 ! [A: $tType,A4: set @ A] :
% 3.03/1.64 ( ( collect @ A
% 3.03/1.64 @ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
% 3.03/1.64 = A4 ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_47_Collect__cong,axiom,
% 3.03/1.64 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 3.03/1.64 ( ! [X: A] :
% 3.03/1.64 ( ( P2 @ X )
% 3.03/1.64 = ( Q @ X ) )
% 3.03/1.64 => ( ( collect @ A @ P2 )
% 3.03/1.64 = ( collect @ A @ Q ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_48_ext,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
% 3.03/1.64 ( ! [X: A] :
% 3.03/1.64 ( ( F3 @ X )
% 3.03/1.64 = ( G3 @ X ) )
% 3.03/1.64 => ( F3 = G3 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_49_compl__le__compl__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( boolean_algebra @ A )
% 3.03/1.64 => ! [X3: A,Y3: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y3 ) )
% 3.03/1.64 = ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_50_neg__le__iff__le,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ordered_ab_group_add @ A )
% 3.03/1.64 => ! [B2: A,A2: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 3.03/1.64 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_51_Compl__subset__Compl__iff,axiom,
% 3.03/1.64 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
% 3.03/1.64 = ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_52_Compl__anti__mono,axiom,
% 3.03/1.64 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 3.03/1.64 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_53_verit__la__disequality,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( linorder @ A )
% 3.03/1.64 => ! [A2: A,B2: A] :
% 3.03/1.64 ( ( A2 = B2 )
% 3.03/1.64 | ~ ( ord_less_eq @ A @ A2 @ B2 )
% 3.03/1.64 | ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_54_Collect__mono__iff,axiom,
% 3.03/1.64 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
% 3.03/1.64 = ( ! [X4: A] :
% 3.03/1.64 ( ( P2 @ X4 )
% 3.03/1.64 => ( Q @ X4 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_55_set__eq__subset,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ^ [Y4: set @ A,Z: set @ A] : ( Y4 = Z ) )
% 3.03/1.64 = ( ^ [A6: set @ A,B6: set @ A] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.03/1.64 & ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_56_subset__trans,axiom,
% 3.03/1.64 ! [A: $tType,A4: set @ A,B4: set @ A,C4: set @ A] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 3.03/1.64 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
% 3.03/1.64 => ( ord_less_eq @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_57_Collect__mono,axiom,
% 3.03/1.64 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 3.03/1.64 ( ! [X: A] :
% 3.03/1.64 ( ( P2 @ X )
% 3.03/1.64 => ( Q @ X ) )
% 3.03/1.64 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_58_subset__refl,axiom,
% 3.03/1.64 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_59_subset__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) )
% 3.03/1.64 = ( ^ [A6: set @ A,B6: set @ A] :
% 3.03/1.64 ! [T2: A] :
% 3.03/1.64 ( ( member @ A @ T2 @ A6 )
% 3.03/1.64 => ( member @ A @ T2 @ B6 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_60_equalityD2,axiom,
% 3.03/1.64 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 3.03/1.64 ( ( A4 = B4 )
% 3.03/1.64 => ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_61_equalityD1,axiom,
% 3.03/1.64 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 3.03/1.64 ( ( A4 = B4 )
% 3.03/1.64 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_62_subset__eq,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) )
% 3.03/1.64 = ( ^ [A6: set @ A,B6: set @ A] :
% 3.03/1.64 ! [X4: A] :
% 3.03/1.64 ( ( member @ A @ X4 @ A6 )
% 3.03/1.64 => ( member @ A @ X4 @ B6 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_63_equalityE,axiom,
% 3.03/1.64 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 3.03/1.64 ( ( A4 = B4 )
% 3.03/1.64 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 3.03/1.64 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_64_subsetD,axiom,
% 3.03/1.64 ! [A: $tType,A4: set @ A,B4: set @ A,C3: A] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 3.03/1.64 => ( ( member @ A @ C3 @ A4 )
% 3.03/1.64 => ( member @ A @ C3 @ B4 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_65_in__mono,axiom,
% 3.03/1.64 ! [A: $tType,A4: set @ A,B4: set @ A,X3: A] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 3.03/1.64 => ( ( member @ A @ X3 @ A4 )
% 3.03/1.64 => ( member @ A @ X3 @ B4 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_66_le__imp__neg__le,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ordered_ab_group_add @ A )
% 3.03/1.64 => ! [A2: A,B2: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.03/1.64 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_67_minus__le__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ordered_ab_group_add @ A )
% 3.03/1.64 => ! [A2: A,B2: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 3.03/1.64 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_68_le__minus__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ordered_ab_group_add @ A )
% 3.03/1.64 => ! [A2: A,B2: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 3.03/1.64 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_69_compl__le__swap2,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( boolean_algebra @ A )
% 3.03/1.64 => ! [Y3: A,X3: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X3 )
% 3.03/1.64 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_70_compl__le__swap1,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( boolean_algebra @ A )
% 3.03/1.64 => ! [Y3: A,X3: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X3 ) )
% 3.03/1.64 => ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_71_compl__mono,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( boolean_algebra @ A )
% 3.03/1.64 => ! [X3: A,Y3: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.03/1.64 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_72_agree__sub,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( finite_finite @ A )
% 3.03/1.64 => ! [A4: set @ ( sum_sum @ A @ A ),B4: set @ ( sum_sum @ A @ A ),Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Omega2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ A4 @ B4 )
% 3.03/1.64 => ( ( denotational_Vagree @ A @ Nu @ Omega2 @ B4 )
% 3.03/1.64 => ( denotational_Vagree @ A @ Nu @ Omega2 @ A4 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_73_agree__supset,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( finite_finite @ A )
% 3.03/1.64 => ! [B4: set @ ( sum_sum @ A @ A ),A4: set @ ( sum_sum @ A @ A ),Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ B4 @ A4 )
% 3.03/1.64 => ( ( denotational_Vagree @ A @ Nu @ Nu2 @ A4 )
% 3.03/1.64 => ( denotational_Vagree @ A @ Nu @ Nu2 @ B4 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_74_minus__equation__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( group_add @ A )
% 3.03/1.64 => ! [A2: A,B2: A] :
% 3.03/1.64 ( ( ( uminus_uminus @ A @ A2 )
% 3.03/1.64 = B2 )
% 3.03/1.64 = ( ( uminus_uminus @ A @ B2 )
% 3.03/1.64 = A2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_75_equation__minus__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( group_add @ A )
% 3.03/1.64 => ! [A2: A,B2: A] :
% 3.03/1.64 ( ( A2
% 3.03/1.64 = ( uminus_uminus @ A @ B2 ) )
% 3.03/1.64 = ( B2
% 3.03/1.64 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_76_fun__Compl__def,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType] :
% 3.03/1.64 ( ( uminus @ B )
% 3.03/1.64 => ( ( uminus_uminus @ ( A > B ) )
% 3.03/1.64 = ( ^ [A6: A > B,X4: A] : ( uminus_uminus @ B @ ( A6 @ X4 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_77_verit__negate__coefficient_I3_J,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ordered_ab_group_add @ A )
% 3.03/1.64 => ! [A2: A,B2: A] :
% 3.03/1.64 ( ( A2 = B2 )
% 3.03/1.64 => ( ( uminus_uminus @ A @ A2 )
% 3.03/1.64 = ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_78_double__complement,axiom,
% 3.03/1.64 ! [A: $tType,A4: set @ A] :
% 3.03/1.64 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
% 3.03/1.64 = A4 ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_79_ComplD,axiom,
% 3.03/1.64 ! [A: $tType,C3: A,A4: set @ A] :
% 3.03/1.64 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
% 3.03/1.64 => ~ ( member @ A @ C3 @ A4 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_80_euclid__ext__aux_Ocases,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( euclid1678468529ng_gcd @ A )
% 3.03/1.64 => ! [X3: product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) )] :
% 3.03/1.64 ~ ! [S2: A,S3: A,T3: A,T4: A,R: A,R2: A] :
% 3.03/1.64 ( X3
% 3.03/1.64 != ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) ) @ S2 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) @ S3 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) @ T3 @ ( product_Pair @ A @ ( product_prod @ A @ A ) @ T4 @ ( product_Pair @ A @ A @ R @ R2 ) ) ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_81_Pair__le,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType] :
% 3.03/1.64 ( ( ( ord @ B )
% 3.03/1.64 & ( ord @ A ) )
% 3.03/1.64 => ! [A2: A,B2: B,C3: A,D3: B] :
% 3.03/1.64 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C3 @ D3 ) )
% 3.03/1.64 = ( ( ord_less_eq @ A @ A2 @ C3 )
% 3.03/1.64 & ( ord_less_eq @ B @ B2 @ D3 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_82_order__refl,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( preorder @ A )
% 3.03/1.64 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_83_Pair__mono,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType] :
% 3.03/1.64 ( ( ( ord @ B )
% 3.03/1.64 & ( ord @ A ) )
% 3.03/1.64 => ! [X3: A,X5: A,Y3: B,Y5: B] :
% 3.03/1.64 ( ( ord_less_eq @ A @ X3 @ X5 )
% 3.03/1.64 => ( ( ord_less_eq @ B @ Y3 @ Y5 )
% 3.03/1.64 => ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_84_internal__case__prod__conv,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 3.03/1.64 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 3.03/1.64 = ( C3 @ A2 @ B2 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_85_Fsub,axiom,
% 3.03/1.64 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVF @ a @ b @ c @ p ) @ v ).
% 3.03/1.64
% 3.03/1.64 thf(fact_86_solSem,axiom,
% 3.03/1.64 ! [X6: real] :
% 3.03/1.64 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X6 )
% 3.03/1.64 => ( ( ord_less_eq @ real @ X6 @ t )
% 3.03/1.64 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( denotational_mk_v @ a @ b @ c @ i @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( sol @ X6 ) ) @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_87_relChain__def,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType] :
% 3.03/1.64 ( ( ord @ B )
% 3.03/1.64 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 3.03/1.64 = ( ^ [R3: set @ ( product_prod @ A @ A ),As: A > B] :
% 3.03/1.64 ! [I2: A,J: A] :
% 3.03/1.64 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R3 )
% 3.03/1.64 => ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_88_hpsafe__Evolve_OIH,axiom,
% 3.03/1.64 coinci1993344360de_fml @ a @ b @ c @ p ).
% 3.03/1.64
% 3.03/1.64 thf(fact_89_t,axiom,
% 3.03/1.64 ord_less_eq @ real @ ( zero_zero @ real ) @ t ).
% 3.03/1.64
% 3.03/1.64 thf(fact_90_le__zero__eq,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( canoni770627133id_add @ A )
% 3.03/1.64 => ! [N: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 3.03/1.64 = ( N
% 3.03/1.64 = ( zero_zero @ A ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_91_add_Oinverse__neutral,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( group_add @ A )
% 3.03/1.64 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 3.03/1.64 = ( zero_zero @ A ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_92_neg__0__equal__iff__equal,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( group_add @ A )
% 3.03/1.64 => ! [A2: A] :
% 3.03/1.64 ( ( ( zero_zero @ A )
% 3.03/1.64 = ( uminus_uminus @ A @ A2 ) )
% 3.03/1.64 = ( ( zero_zero @ A )
% 3.03/1.64 = A2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_93_neg__equal__0__iff__equal,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( group_add @ A )
% 3.03/1.64 => ! [A2: A] :
% 3.03/1.64 ( ( ( uminus_uminus @ A @ A2 )
% 3.03/1.64 = ( zero_zero @ A ) )
% 3.03/1.64 = ( A2
% 3.03/1.64 = ( zero_zero @ A ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_94_equal__neg__zero,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( linord219039673up_add @ A )
% 3.03/1.64 => ! [A2: A] :
% 3.03/1.64 ( ( A2
% 3.03/1.64 = ( uminus_uminus @ A @ A2 ) )
% 3.03/1.64 = ( A2
% 3.03/1.64 = ( zero_zero @ A ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_95_neg__equal__zero,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( linord219039673up_add @ A )
% 3.03/1.64 => ! [A2: A] :
% 3.03/1.64 ( ( ( uminus_uminus @ A @ A2 )
% 3.03/1.64 = A2 )
% 3.03/1.64 = ( A2
% 3.03/1.64 = ( zero_zero @ A ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_96_IH_H,axiom,
% 3.03/1.64 ! [A7: finite_Cartesian_vec @ real @ c,B7: finite_Cartesian_vec @ real @ c,Aa: finite_Cartesian_vec @ real @ c,Ba: finite_Cartesian_vec @ real @ c] :
% 3.03/1.64 ( ( denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ A7 @ B7 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ Aa @ Ba ) @ ( static_FVF @ a @ b @ c @ p ) )
% 3.03/1.64 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ A7 @ B7 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) )
% 3.03/1.64 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ Aa @ Ba ) @ ( denotational_fml_sem @ a @ b @ c @ j @ p ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_97_neg__less__eq__nonneg,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( linord219039673up_add @ A )
% 3.03/1.64 => ! [A2: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 3.03/1.64 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_98_less__eq__neg__nonpos,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( linord219039673up_add @ A )
% 3.03/1.64 => ! [A2: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 3.03/1.64 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_99_neg__le__0__iff__le,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ordered_ab_group_add @ A )
% 3.03/1.64 => ! [A2: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 3.03/1.64 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_100_neg__0__le__iff__le,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ordered_ab_group_add @ A )
% 3.03/1.64 => ! [A2: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 3.03/1.64 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_101_fsafe,axiom,
% 3.03/1.64 fsafe @ a @ b @ c @ p ).
% 3.03/1.64
% 3.03/1.64 thf(fact_102_zero__prod__def,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType] :
% 3.03/1.64 ( ( ( zero @ B )
% 3.03/1.64 & ( zero @ A ) )
% 3.03/1.64 => ( ( zero_zero @ ( product_prod @ A @ B ) )
% 3.03/1.64 = ( product_Pair @ A @ B @ ( zero_zero @ A ) @ ( zero_zero @ B ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_103_zero__reorient,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( zero @ A )
% 3.03/1.64 => ! [X3: A] :
% 3.03/1.64 ( ( ( zero_zero @ A )
% 3.03/1.64 = X3 )
% 3.03/1.64 = ( X3
% 3.03/1.64 = ( zero_zero @ A ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_104_iff__to__impl,axiom,
% 3.03/1.64 ! [A: $tType,C: $tType,B: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ B )
% 3.03/1.64 & ( finite_finite @ C )
% 3.03/1.64 & ( finite_finite @ A ) )
% 3.03/1.64 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I: denota1663640101rp_ext @ B @ C @ A @ product_unit,A4: formula @ B @ C @ A,B4: formula @ B @ C @ A] :
% 3.03/1.64 ( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ A4 ) )
% 3.03/1.64 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ B4 ) ) )
% 3.03/1.64 = ( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ A4 ) )
% 3.03/1.64 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ B4 ) ) )
% 3.03/1.64 & ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ B4 ) )
% 3.03/1.64 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ A4 ) ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_105_zero__le,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( canoni770627133id_add @ A )
% 3.03/1.64 => ! [X3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_106_le__funD,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType] :
% 3.03/1.64 ( ( ord @ B )
% 3.03/1.64 => ! [F3: A > B,G3: A > B,X3: A] :
% 3.03/1.64 ( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
% 3.03/1.64 => ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_107_le__funE,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType] :
% 3.03/1.64 ( ( ord @ B )
% 3.03/1.64 => ! [F3: A > B,G3: A > B,X3: A] :
% 3.03/1.64 ( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
% 3.03/1.64 => ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_108_le__funI,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType] :
% 3.03/1.64 ( ( ord @ B )
% 3.03/1.64 => ! [F3: A > B,G3: A > B] :
% 3.03/1.64 ( ! [X: A] : ( ord_less_eq @ B @ ( F3 @ X ) @ ( G3 @ X ) )
% 3.03/1.64 => ( ord_less_eq @ ( A > B ) @ F3 @ G3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_109_le__fun__def,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType] :
% 3.03/1.64 ( ( ord @ B )
% 3.03/1.64 => ( ( ord_less_eq @ ( A > B ) )
% 3.03/1.64 = ( ^ [F4: A > B,G4: A > B] :
% 3.03/1.64 ! [X4: A] : ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_110_order__subst1,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType] :
% 3.03/1.64 ( ( ( order @ B )
% 3.03/1.64 & ( order @ A ) )
% 3.03/1.64 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 3.03/1.64 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
% 3.03/1.64 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 3.03/1.64 => ( ! [X: B,Y: B] :
% 3.03/1.64 ( ( ord_less_eq @ B @ X @ Y )
% 3.03/1.64 => ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y ) ) )
% 3.03/1.64 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_111_order__subst2,axiom,
% 3.03/1.64 ! [A: $tType,C: $tType] :
% 3.03/1.64 ( ( ( order @ C )
% 3.03/1.64 & ( order @ A ) )
% 3.03/1.64 => ! [A2: A,B2: A,F3: A > C,C3: C] :
% 3.03/1.64 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.03/1.64 => ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
% 3.03/1.64 => ( ! [X: A,Y: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ X @ Y )
% 3.03/1.64 => ( ord_less_eq @ C @ ( F3 @ X ) @ ( F3 @ Y ) ) )
% 3.03/1.64 => ( ord_less_eq @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_112_ord__eq__le__subst,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType] :
% 3.03/1.64 ( ( ( ord @ B )
% 3.03/1.64 & ( ord @ A ) )
% 3.03/1.64 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 3.03/1.64 ( ( A2
% 3.03/1.64 = ( F3 @ B2 ) )
% 3.03/1.64 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 3.03/1.64 => ( ! [X: B,Y: B] :
% 3.03/1.64 ( ( ord_less_eq @ B @ X @ Y )
% 3.03/1.64 => ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y ) ) )
% 3.03/1.64 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_113_ord__le__eq__subst,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType] :
% 3.03/1.64 ( ( ( ord @ B )
% 3.03/1.64 & ( ord @ A ) )
% 3.03/1.64 => ! [A2: A,B2: A,F3: A > B,C3: B] :
% 3.03/1.64 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.03/1.64 => ( ( ( F3 @ B2 )
% 3.03/1.64 = C3 )
% 3.03/1.64 => ( ! [X: A,Y: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ X @ Y )
% 3.03/1.64 => ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) )
% 3.03/1.64 => ( ord_less_eq @ B @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_114_eq__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( order @ A )
% 3.03/1.64 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 3.03/1.64 = ( ^ [X4: A,Y6: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ X4 @ Y6 )
% 3.03/1.64 & ( ord_less_eq @ A @ Y6 @ X4 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_115_antisym,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( order @ A )
% 3.03/1.64 => ! [X3: A,Y3: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.03/1.64 => ( ( ord_less_eq @ A @ Y3 @ X3 )
% 3.03/1.64 => ( X3 = Y3 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_116_linear,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( linorder @ A )
% 3.03/1.64 => ! [X3: A,Y3: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.03/1.64 | ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_117_eq__refl,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( preorder @ A )
% 3.03/1.64 => ! [X3: A,Y3: A] :
% 3.03/1.64 ( ( X3 = Y3 )
% 3.03/1.64 => ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_118_le__cases,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( linorder @ A )
% 3.03/1.64 => ! [X3: A,Y3: A] :
% 3.03/1.64 ( ~ ( ord_less_eq @ A @ X3 @ Y3 )
% 3.03/1.64 => ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_119_order_Otrans,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( order @ A )
% 3.03/1.64 => ! [A2: A,B2: A,C3: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.03/1.64 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 3.03/1.64 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_120_le__cases3,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( linorder @ A )
% 3.03/1.64 => ! [X3: A,Y3: A,Z2: A] :
% 3.03/1.64 ( ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.03/1.64 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 3.03/1.64 => ( ( ( ord_less_eq @ A @ Y3 @ X3 )
% 3.03/1.64 => ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
% 3.03/1.64 => ( ( ( ord_less_eq @ A @ X3 @ Z2 )
% 3.03/1.64 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 3.03/1.64 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 3.03/1.64 => ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
% 3.03/1.64 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 3.03/1.64 => ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
% 3.03/1.64 => ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
% 3.03/1.64 => ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_121_antisym__conv,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( order @ A )
% 3.03/1.64 => ! [Y3: A,X3: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ Y3 @ X3 )
% 3.03/1.64 => ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.03/1.64 = ( X3 = Y3 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_122_order__class_Oorder_Oeq__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( order @ A )
% 3.03/1.64 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 3.03/1.64 = ( ^ [A8: A,B8: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ A8 @ B8 )
% 3.03/1.64 & ( ord_less_eq @ A @ B8 @ A8 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_123_ord__eq__le__trans,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ord @ A )
% 3.03/1.64 => ! [A2: A,B2: A,C3: A] :
% 3.03/1.64 ( ( A2 = B2 )
% 3.03/1.64 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 3.03/1.64 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_124_ord__le__eq__trans,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ord @ A )
% 3.03/1.64 => ! [A2: A,B2: A,C3: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.03/1.64 => ( ( B2 = C3 )
% 3.03/1.64 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_125_order__class_Oorder_Oantisym,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( order @ A )
% 3.03/1.64 => ! [A2: A,B2: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.03/1.64 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.03/1.64 => ( A2 = B2 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_126_order__trans,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( preorder @ A )
% 3.03/1.64 => ! [X3: A,Y3: A,Z2: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.03/1.64 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 3.03/1.64 => ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_127_dual__order_Orefl,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( order @ A )
% 3.03/1.64 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_128_linorder__wlog,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( linorder @ A )
% 3.03/1.64 => ! [P2: A > A > $o,A2: A,B2: A] :
% 3.03/1.64 ( ! [A5: A,B5: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ A5 @ B5 )
% 3.03/1.64 => ( P2 @ A5 @ B5 ) )
% 3.03/1.64 => ( ! [A5: A,B5: A] :
% 3.03/1.64 ( ( P2 @ B5 @ A5 )
% 3.03/1.64 => ( P2 @ A5 @ B5 ) )
% 3.03/1.64 => ( P2 @ A2 @ B2 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_129_dual__order_Otrans,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( order @ A )
% 3.03/1.64 => ! [B2: A,A2: A,C3: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.03/1.64 => ( ( ord_less_eq @ A @ C3 @ B2 )
% 3.03/1.64 => ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_130_dual__order_Oeq__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( order @ A )
% 3.03/1.64 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 3.03/1.64 = ( ^ [A8: A,B8: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ B8 @ A8 )
% 3.03/1.64 & ( ord_less_eq @ A @ A8 @ B8 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_131_dual__order_Oantisym,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( order @ A )
% 3.03/1.64 => ! [B2: A,A2: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.03/1.64 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.03/1.64 => ( A2 = B2 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_132_gcd_Ocases,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( euclid1155270486miring @ A )
% 3.03/1.64 => ! [X3: product_prod @ A @ A] :
% 3.03/1.64 ~ ! [A5: A,B5: A] :
% 3.03/1.64 ( X3
% 3.03/1.64 != ( product_Pair @ A @ A @ A5 @ B5 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_133_iff__sem,axiom,
% 3.03/1.64 ! [A: $tType,C: $tType,B: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ B )
% 3.03/1.64 & ( finite_finite @ C )
% 3.03/1.64 & ( finite_finite @ A ) )
% 3.03/1.64 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I: denota1663640101rp_ext @ B @ C @ A @ product_unit,A4: formula @ B @ C @ A,B4: formula @ B @ C @ A] :
% 3.03/1.64 ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ ( equiv @ B @ C @ A @ A4 @ B4 ) ) )
% 3.03/1.64 = ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ A4 ) )
% 3.03/1.64 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ B4 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_134_IHF,axiom,
% 3.03/1.64 ! [A7: finite_Cartesian_vec @ real @ c,B7: finite_Cartesian_vec @ real @ c,Aa: finite_Cartesian_vec @ real @ c,Ba: finite_Cartesian_vec @ real @ c] :
% 3.03/1.64 ( ( denotational_Iagree @ a @ b @ c @ i @ j @ ( static_SIGF @ a @ b @ c @ p ) )
% 3.03/1.64 => ( ( denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ A7 @ B7 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ Aa @ Ba ) @ ( static_FVF @ a @ b @ c @ p ) )
% 3.03/1.64 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ A7 @ B7 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) )
% 3.03/1.64 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ Aa @ Ba ) @ ( denotational_fml_sem @ a @ b @ c @ j @ p ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_135_minus__le__self__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( lattic1601792062up_add @ A )
% 3.03/1.64 => ! [A2: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 3.03/1.64 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_136_le__minus__self__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( lattic1601792062up_add @ A )
% 3.03/1.64 => ! [A2: A] :
% 3.03/1.64 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 3.03/1.64 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_137_real__eq__0__iff__le__ge__0,axiom,
% 3.03/1.64 ! [X3: real] :
% 3.03/1.64 ( ( X3
% 3.03/1.64 = ( zero_zero @ real ) )
% 3.03/1.64 = ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
% 3.03/1.64 & ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( uminus_uminus @ real @ X3 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_138_le__numeral__extra_I3_J,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( linord1659791738miring @ A )
% 3.03/1.64 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_139_coincide__fml__def,axiom,
% 3.03/1.64 ! [C: $tType,B: $tType,A: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ A )
% 3.03/1.64 & ( finite_finite @ B )
% 3.03/1.64 & ( finite_finite @ C ) )
% 3.03/1.64 => ( ( coinci1993344360de_fml @ A @ B @ C )
% 3.03/1.64 = ( ^ [Phi: formula @ A @ B @ C] :
% 3.03/1.64 ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 3.03/1.64 ( ( denotational_Iagree @ A @ B @ C @ I3 @ J2 @ ( static_SIGF @ A @ B @ C @ Phi ) )
% 3.03/1.64 => ( ( denotational_Vagree @ C @ Nu3 @ Nu4 @ ( static_FVF @ A @ B @ C @ Phi ) )
% 3.03/1.64 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu3 @ ( denotational_fml_sem @ A @ B @ C @ I3 @ Phi ) )
% 3.03/1.64 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu4 @ ( denotational_fml_sem @ A @ B @ C @ J2 @ Phi ) ) ) ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_140_IAP,axiom,
% 3.03/1.64 denotational_Iagree @ a @ b @ c @ i @ j @ ( static_SIGF @ a @ b @ c @ p ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_141_Iagree__sub,axiom,
% 3.03/1.64 ! [C: $tType,B: $tType,A: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ A )
% 3.03/1.64 & ( finite_finite @ B )
% 3.03/1.64 & ( finite_finite @ C ) )
% 3.03/1.64 => ! [A4: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ),B4: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ),I: denota1663640101rp_ext @ A @ B @ C @ product_unit,J3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) @ A4 @ B4 )
% 3.03/1.64 => ( ( denotational_Iagree @ A @ B @ C @ I @ J3 @ B4 )
% 3.03/1.64 => ( denotational_Iagree @ A @ B @ C @ I @ J3 @ A4 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_142_Iagree__comm,axiom,
% 3.03/1.64 ! [C: $tType,B: $tType,A: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ A )
% 3.03/1.64 & ( finite_finite @ B )
% 3.03/1.64 & ( finite_finite @ C ) )
% 3.03/1.64 => ! [A4: denota1663640101rp_ext @ A @ B @ C @ product_unit,B4: denota1663640101rp_ext @ A @ B @ C @ product_unit,V: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] :
% 3.03/1.64 ( ( denotational_Iagree @ A @ B @ C @ A4 @ B4 @ V )
% 3.03/1.64 => ( denotational_Iagree @ A @ B @ C @ B4 @ A4 @ V ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_143_Iagree__refl,axiom,
% 3.03/1.64 ! [C: $tType,B: $tType,A: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ A )
% 3.03/1.64 & ( finite_finite @ B )
% 3.03/1.64 & ( finite_finite @ C ) )
% 3.03/1.64 => ! [I: denota1663640101rp_ext @ A @ B @ C @ product_unit,A4: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] : ( denotational_Iagree @ A @ B @ C @ I @ I @ A4 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_144_ids_Ocoincide__fml__def,axiom,
% 3.03/1.64 ! [Sc: $tType,Sf: $tType,Sz: $tType,B: $tType,A: $tType,C: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ C )
% 3.03/1.64 & ( finite_finite @ A )
% 3.03/1.64 & ( finite_finite @ B )
% 3.03/1.64 & ( finite_finite @ Sz )
% 3.03/1.64 & ( linorder @ Sz )
% 3.03/1.64 & ( finite_finite @ Sf )
% 3.03/1.64 & ( finite_finite @ Sc ) )
% 3.03/1.64 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Phi2: formula @ A @ B @ C] :
% 3.03/1.64 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.03/1.64 => ( ( coinci1993344360de_fml @ A @ B @ C @ Phi2 )
% 3.03/1.64 = ( ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 3.03/1.64 ( ( denotational_Iagree @ A @ B @ C @ I3 @ J2 @ ( static_SIGF @ A @ B @ C @ Phi2 ) )
% 3.03/1.64 => ( ( denotational_Vagree @ C @ Nu3 @ Nu4 @ ( static_FVF @ A @ B @ C @ Phi2 ) )
% 3.03/1.64 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu3 @ ( denotational_fml_sem @ A @ B @ C @ I3 @ Phi2 ) )
% 3.03/1.64 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu4 @ ( denotational_fml_sem @ A @ B @ C @ J2 @ Phi2 ) ) ) ) ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_145__092_060open_062ode__sem__equiv_A_IEvolveODE_AODE_AP_J_AJ_092_060close_062,axiom,
% 3.03/1.64 coinci1495174342_equiv @ a @ b @ c @ ( evolveODE @ a @ c @ b @ ode2 @ p ) @ j ).
% 3.03/1.64
% 3.03/1.64 thf(fact_146__092_060open_062ode__sem__equiv_A_IEvolveODE_AODE_AP_J_AI_092_060close_062,axiom,
% 3.03/1.64 coinci1495174342_equiv @ a @ b @ c @ ( evolveODE @ a @ c @ b @ ode2 @ p ) @ i ).
% 3.03/1.64
% 3.03/1.64 thf(fact_147_impl__sem,axiom,
% 3.03/1.64 ! [A: $tType,C: $tType,B: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ B )
% 3.03/1.64 & ( finite_finite @ C )
% 3.03/1.64 & ( finite_finite @ A ) )
% 3.03/1.64 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I: denota1663640101rp_ext @ B @ C @ A @ product_unit,A4: formula @ B @ C @ A,B4: formula @ B @ C @ A] :
% 3.03/1.64 ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ ( implies @ B @ C @ A @ A4 @ B4 ) ) )
% 3.03/1.64 = ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ A4 ) )
% 3.03/1.64 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ B4 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_148_equiv,axiom,
% 3.03/1.64 ! [I4: denota1663640101rp_ext @ a @ b @ c @ product_unit] : ( coinci1495174342_equiv @ a @ b @ c @ ( evolveODE @ a @ c @ b @ ode2 @ p ) @ I4 ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_149_ids_Oseq__sem_Ocases,axiom,
% 3.03/1.64 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ Sf )
% 3.03/1.64 & ( finite_finite @ Sc )
% 3.03/1.64 & ( finite_finite @ Sz )
% 3.03/1.64 & ( linorder @ Sz ) )
% 3.03/1.64 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X3: product_prod @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) )] :
% 3.03/1.64 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.03/1.64 => ~ ! [I5: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,S4: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
% 3.03/1.64 ( X3
% 3.03/1.64 != ( product_Pair @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) @ I5 @ S4 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_150_ids_Oseq__sem_Oinduct,axiom,
% 3.03/1.64 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ Sf )
% 3.03/1.64 & ( finite_finite @ Sc )
% 3.03/1.64 & ( finite_finite @ Sz )
% 3.03/1.64 & ( linorder @ Sz ) )
% 3.03/1.64 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P2: ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) > ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) > $o,A0: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,A1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
% 3.03/1.64 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.03/1.64 => ( ! [I5: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,X_1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] : ( P2 @ I5 @ X_1 )
% 3.03/1.64 => ( P2 @ A0 @ A1 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_151_coincide__hp_H__def,axiom,
% 3.03/1.64 ! [C: $tType,B: $tType,A: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ A )
% 3.03/1.64 & ( finite_finite @ B )
% 3.03/1.64 & ( finite_finite @ C ) )
% 3.03/1.64 => ( ( coinci554967470ide_hp @ A @ B @ C )
% 3.03/1.64 = ( ^ [Alpha: hp @ A @ B @ C] :
% 3.03/1.64 ! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 3.03/1.64 ( ( coinci32832645ide_hp @ A @ B @ C @ Alpha @ I3 @ J2 )
% 3.03/1.64 & ( coinci1495174342_equiv @ A @ B @ C @ Alpha @ I3 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_152_hp_Oinject_I5_J,axiom,
% 3.03/1.64 ! [C: $tType,B: $tType,A: $tType,X51: ode @ A @ C,X52: formula @ A @ B @ C,Y51: ode @ A @ C,Y52: formula @ A @ B @ C] :
% 3.03/1.64 ( ( ( evolveODE @ A @ C @ B @ X51 @ X52 )
% 3.03/1.64 = ( evolveODE @ A @ C @ B @ Y51 @ Y52 ) )
% 3.03/1.64 = ( ( X51 = Y51 )
% 3.03/1.64 & ( X52 = Y52 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_153_sol,axiom,
% 3.03/1.64 ( initia1685620758es_ode @ ( finite_Cartesian_vec @ real @ c ) @ sol
% 3.03/1.64 @ ^ [A8: real] : ( denotational_ODE_sem @ a @ b @ c @ i @ ode2 )
% 3.03/1.64 @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t )
% 3.03/1.64 @ ( collect @ ( finite_Cartesian_vec @ real @ c )
% 3.03/1.64 @ ^ [X4: finite_Cartesian_vec @ real @ c] : ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( denotational_mk_v @ a @ b @ c @ i @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ X4 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_154_less__eq__set__def,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) )
% 3.03/1.64 = ( ^ [A6: set @ A,B6: set @ A] :
% 3.03/1.64 ( ord_less_eq @ ( A > $o )
% 3.03/1.64 @ ^ [X4: A] : ( member @ A @ X4 @ A6 )
% 3.03/1.64 @ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_155_Collect__subset,axiom,
% 3.03/1.64 ! [A: $tType,A4: set @ A,P2: A > $o] :
% 3.03/1.64 ( ord_less_eq @ ( set @ A )
% 3.03/1.64 @ ( collect @ A
% 3.03/1.64 @ ^ [X4: A] :
% 3.03/1.64 ( ( member @ A @ X4 @ A4 )
% 3.03/1.64 & ( P2 @ X4 ) ) )
% 3.03/1.64 @ A4 ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_156_Compl__eq,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( uminus_uminus @ ( set @ A ) )
% 3.03/1.64 = ( ^ [A6: set @ A] :
% 3.03/1.64 ( collect @ A
% 3.03/1.64 @ ^ [X4: A] :
% 3.03/1.64 ~ ( member @ A @ X4 @ A6 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_157_Collect__neg__eq,axiom,
% 3.03/1.64 ! [A: $tType,P2: A > $o] :
% 3.03/1.64 ( ( collect @ A
% 3.03/1.64 @ ^ [X4: A] :
% 3.03/1.64 ~ ( P2 @ X4 ) )
% 3.03/1.64 = ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_158_uminus__set__def,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( uminus_uminus @ ( set @ A ) )
% 3.03/1.64 = ( ^ [A6: set @ A] :
% 3.03/1.64 ( collect @ A
% 3.03/1.64 @ ( uminus_uminus @ ( A > $o )
% 3.03/1.64 @ ^ [X4: A] : ( member @ A @ X4 @ A6 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_159_mk__xode_Oelims,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,C: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ C )
% 3.03/1.64 & ( finite_finite @ B )
% 3.03/1.64 & ( finite_finite @ A ) )
% 3.03/1.64 => ! [X3: denota1663640101rp_ext @ A @ B @ C @ product_unit,Xa: ode @ A @ C,Xb: finite_Cartesian_vec @ real @ C,Y3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 3.03/1.64 ( ( ( denotational_mk_xode @ A @ B @ C @ X3 @ Xa @ Xb )
% 3.03/1.64 = Y3 )
% 3.03/1.64 => ( Y3
% 3.03/1.64 = ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb @ ( denotational_ODE_sem @ A @ B @ C @ X3 @ Xa @ Xb ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_160_mk__xode_Osimps,axiom,
% 3.03/1.64 ! [C: $tType,B: $tType,A: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ A )
% 3.03/1.64 & ( finite_finite @ B )
% 3.03/1.64 & ( finite_finite @ C ) )
% 3.03/1.64 => ( ( denotational_mk_xode @ A @ B @ C )
% 3.03/1.64 = ( ^ [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,ODE2: ode @ A @ C,Sol2: finite_Cartesian_vec @ real @ C] : ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Sol2 @ ( denotational_ODE_sem @ A @ B @ C @ I3 @ ODE2 @ Sol2 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_161_ids_Osingleton_Oinduct,axiom,
% 3.03/1.64 ! [Sf: $tType,Sc: $tType,A: $tType,Sz: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ Sz )
% 3.03/1.64 & ( linorder @ Sz )
% 3.03/1.64 & ( finite_finite @ Sc )
% 3.03/1.64 & ( finite_finite @ Sf ) )
% 3.03/1.64 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P2: ( trm @ A @ Sz ) > Sz > $o,A0: trm @ A @ Sz,A1: Sz] :
% 3.03/1.64 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.03/1.64 => ( ! [T4: trm @ A @ Sz,X_1: Sz] : ( P2 @ T4 @ X_1 )
% 3.03/1.64 => ( P2 @ A0 @ A1 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_162_ids_Osingleton_Ocases,axiom,
% 3.03/1.64 ! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ Sz )
% 3.03/1.64 & ( linorder @ Sz )
% 3.03/1.64 & ( finite_finite @ Sf )
% 3.03/1.64 & ( finite_finite @ Sc ) )
% 3.03/1.64 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X3: product_prod @ ( trm @ A @ Sz ) @ Sz] :
% 3.03/1.64 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.03/1.64 => ~ ! [T4: trm @ A @ Sz,I6: Sz] :
% 3.03/1.64 ( X3
% 3.03/1.64 != ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ T4 @ I6 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_163_ids_Ocoincide__hp_H__def,axiom,
% 3.03/1.64 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ A )
% 3.03/1.64 & ( finite_finite @ B )
% 3.03/1.64 & ( finite_finite @ C )
% 3.03/1.64 & ( finite_finite @ Sz )
% 3.03/1.64 & ( linorder @ Sz )
% 3.03/1.64 & ( finite_finite @ Sf )
% 3.03/1.64 & ( finite_finite @ Sc ) )
% 3.03/1.64 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Alpha2: hp @ A @ B @ C] :
% 3.03/1.64 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.03/1.64 => ( ( coinci554967470ide_hp @ A @ B @ C @ Alpha2 )
% 3.03/1.64 = ( ! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 3.03/1.64 ( ( coinci32832645ide_hp @ A @ B @ C @ Alpha2 @ I3 @ J2 )
% 3.03/1.64 & ( coinci1495174342_equiv @ A @ B @ C @ Alpha2 @ I3 ) ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_164_atLeastatMost__subset__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( preorder @ A )
% 3.03/1.64 => ! [A2: A,B2: A,C3: A,D3: A] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A2 @ B2 ) @ ( set_or331188842AtMost @ A @ C3 @ D3 ) )
% 3.03/1.64 = ( ~ ( ord_less_eq @ A @ A2 @ B2 )
% 3.03/1.64 | ( ( ord_less_eq @ A @ C3 @ A2 )
% 3.03/1.64 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_165_atLeastAtMost__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ord @ A )
% 3.03/1.64 => ! [I7: A,L: A,U: A] :
% 3.03/1.64 ( ( member @ A @ I7 @ ( set_or331188842AtMost @ A @ L @ U ) )
% 3.03/1.64 = ( ( ord_less_eq @ A @ L @ I7 )
% 3.03/1.64 & ( ord_less_eq @ A @ I7 @ U ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_166_Icc__eq__Icc,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( order @ A )
% 3.03/1.64 => ! [L: A,H: A,L2: A,H2: A] :
% 3.03/1.64 ( ( ( set_or331188842AtMost @ A @ L @ H )
% 3.03/1.64 = ( set_or331188842AtMost @ A @ L2 @ H2 ) )
% 3.03/1.64 = ( ( ( L = L2 )
% 3.03/1.64 & ( H = H2 ) )
% 3.03/1.64 | ( ~ ( ord_less_eq @ A @ L @ H )
% 3.03/1.64 & ~ ( ord_less_eq @ A @ L2 @ H2 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_167_predicate1I,axiom,
% 3.03/1.64 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 3.03/1.64 ( ! [X: A] :
% 3.03/1.64 ( ( P2 @ X )
% 3.03/1.64 => ( Q @ X ) )
% 3.03/1.64 => ( ord_less_eq @ ( A > $o ) @ P2 @ Q ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_168_predicate1D,axiom,
% 3.03/1.64 ! [A: $tType,P2: A > $o,Q: A > $o,X3: A] :
% 3.03/1.64 ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
% 3.03/1.64 => ( ( P2 @ X3 )
% 3.03/1.64 => ( Q @ X3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_169_rev__predicate1D,axiom,
% 3.03/1.64 ! [A: $tType,P2: A > $o,X3: A,Q: A > $o] :
% 3.03/1.64 ( ( P2 @ X3 )
% 3.03/1.64 => ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
% 3.03/1.64 => ( Q @ X3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_170_pred__subset__eq,axiom,
% 3.03/1.64 ! [A: $tType,R4: set @ A,S5: set @ A] :
% 3.03/1.64 ( ( ord_less_eq @ ( A > $o )
% 3.03/1.64 @ ^ [X4: A] : ( member @ A @ X4 @ R4 )
% 3.03/1.64 @ ^ [X4: A] : ( member @ A @ X4 @ S5 ) )
% 3.03/1.64 = ( ord_less_eq @ ( set @ A ) @ R4 @ S5 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_171_solDeriv,axiom,
% 3.03/1.64 ! [X6: real] :
% 3.03/1.64 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X6 )
% 3.03/1.64 => ( ( ord_less_eq @ real @ X6 @ t )
% 3.03/1.64 => ( has_ve2132708402vative @ ( finite_Cartesian_vec @ real @ c ) @ sol @ ( denotational_ODE_sem @ a @ b @ c @ i @ ode2 @ ( sol @ X6 ) ) @ ( topolo507301023within @ real @ X6 @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_172_mk__v__def,axiom,
% 3.03/1.64 ! [C: $tType,B: $tType,A: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ A )
% 3.03/1.64 & ( finite_finite @ B )
% 3.03/1.64 & ( finite_finite @ C ) )
% 3.03/1.64 => ( ( denotational_mk_v @ A @ B @ C )
% 3.03/1.64 = ( ^ [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,ODE2: ode @ A @ C,Nu3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Sol2: finite_Cartesian_vec @ real @ C] :
% 3.03/1.64 ( the @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) )
% 3.03/1.64 @ ^ [Omega3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 3.03/1.64 ( ( denotational_Vagree @ C @ Omega3 @ Nu3 @ ( uminus_uminus @ ( set @ ( sum_sum @ C @ C ) ) @ ( denotational_semBV @ A @ B @ C @ I3 @ ODE2 ) ) )
% 3.03/1.64 & ( denotational_Vagree @ C @ Omega3 @ ( denotational_mk_xode @ A @ B @ C @ I3 @ ODE2 @ Sol2 ) @ ( denotational_semBV @ A @ B @ C @ I3 @ ODE2 ) ) ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_173_subrelI,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S6: set @ ( product_prod @ A @ B )] :
% 3.03/1.64 ( ! [X: A,Y: B] :
% 3.03/1.64 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R5 )
% 3.03/1.64 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ S6 ) )
% 3.03/1.64 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S6 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_174_pred__subset__eq2,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
% 3.03/1.64 ( ( ord_less_eq @ ( A > B > $o )
% 3.03/1.64 @ ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ R4 )
% 3.03/1.64 @ ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ S5 ) )
% 3.03/1.64 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S5 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_175_pred__equals__eq2,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
% 3.03/1.64 ( ( ( ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ R4 ) )
% 3.03/1.64 = ( ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ S5 ) ) )
% 3.03/1.64 = ( R4 = S5 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_176_old_Orec__prod__def,axiom,
% 3.03/1.64 ! [T: $tType,B: $tType,A: $tType] :
% 3.03/1.64 ( ( product_rec_prod @ A @ B @ T )
% 3.03/1.64 = ( ^ [F12: A > B > T,X4: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X4 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_177_has__vector__derivative__Pair,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType] :
% 3.03/1.64 ( ( ( real_V55928688vector @ A )
% 3.03/1.64 & ( real_V55928688vector @ B ) )
% 3.03/1.64 => ! [F3: real > A,F5: A,X3: real,S6: set @ real,G3: real > B,G5: B] :
% 3.03/1.64 ( ( has_ve2132708402vative @ A @ F3 @ F5 @ ( topolo507301023within @ real @ X3 @ S6 ) )
% 3.03/1.64 => ( ( has_ve2132708402vative @ B @ G3 @ G5 @ ( topolo507301023within @ real @ X3 @ S6 ) )
% 3.03/1.64 => ( has_ve2132708402vative @ ( product_prod @ A @ B )
% 3.03/1.64 @ ^ [X4: real] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
% 3.03/1.64 @ ( product_Pair @ A @ B @ F5 @ G5 )
% 3.03/1.64 @ ( topolo507301023within @ real @ X3 @ S6 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_178_has__vector__derivative__minus,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( real_V55928688vector @ A )
% 3.03/1.64 => ! [F3: real > A,F5: A,Net: filter @ real] :
% 3.03/1.64 ( ( has_ve2132708402vative @ A @ F3 @ F5 @ Net )
% 3.03/1.64 => ( has_ve2132708402vative @ A
% 3.03/1.64 @ ^ [X4: real] : ( uminus_uminus @ A @ ( F3 @ X4 ) )
% 3.03/1.64 @ ( uminus_uminus @ A @ F5 )
% 3.03/1.64 @ Net ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_179_predicate2I,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType,P2: A > B > $o,Q: A > B > $o] :
% 3.03/1.64 ( ! [X: A,Y: B] :
% 3.03/1.64 ( ( P2 @ X @ Y )
% 3.03/1.64 => ( Q @ X @ Y ) )
% 3.03/1.64 => ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_180_rev__predicate2D,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,P2: A > B > $o,X3: A,Y3: B,Q: A > B > $o] :
% 3.03/1.64 ( ( P2 @ X3 @ Y3 )
% 3.03/1.64 => ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
% 3.03/1.64 => ( Q @ X3 @ Y3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_181_predicate2D,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,P2: A > B > $o,Q: A > B > $o,X3: A,Y3: B] :
% 3.03/1.64 ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
% 3.03/1.64 => ( ( P2 @ X3 @ Y3 )
% 3.03/1.64 => ( Q @ X3 @ Y3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_182_has__vector__derivative__const,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( real_V55928688vector @ A )
% 3.03/1.64 => ! [C3: A,Net: filter @ real] :
% 3.03/1.64 ( has_ve2132708402vative @ A
% 3.03/1.64 @ ^ [X4: real] : C3
% 3.03/1.64 @ ( zero_zero @ A )
% 3.03/1.64 @ Net ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_183_has__vector__derivative__within__subset,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( real_V55928688vector @ A )
% 3.03/1.64 => ! [F3: real > A,F5: A,X3: real,S5: set @ real,T5: set @ real] :
% 3.03/1.64 ( ( has_ve2132708402vative @ A @ F3 @ F5 @ ( topolo507301023within @ real @ X3 @ S5 ) )
% 3.03/1.64 => ( ( ord_less_eq @ ( set @ real ) @ T5 @ S5 )
% 3.03/1.64 => ( has_ve2132708402vative @ A @ F3 @ F5 @ ( topolo507301023within @ real @ X3 @ T5 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_184_has__vector__derivative__weaken,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( real_V55928688vector @ A )
% 3.03/1.64 => ! [F3: real > A,D4: A,X3: real,T5: set @ real,S5: set @ real,G3: real > A] :
% 3.03/1.64 ( ( has_ve2132708402vative @ A @ F3 @ D4 @ ( topolo507301023within @ real @ X3 @ T5 ) )
% 3.03/1.64 => ( ( member @ real @ X3 @ S5 )
% 3.03/1.64 => ( ( ord_less_eq @ ( set @ real ) @ S5 @ T5 )
% 3.03/1.64 => ( ! [X: real] :
% 3.03/1.64 ( ( member @ real @ X @ S5 )
% 3.03/1.64 => ( ( F3 @ X )
% 3.03/1.64 = ( G3 @ X ) ) )
% 3.03/1.64 => ( has_ve2132708402vative @ A @ G3 @ D4 @ ( topolo507301023within @ real @ X3 @ S5 ) ) ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_185_solves__ode__supset__range,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( real_V55928688vector @ A )
% 3.03/1.64 => ! [X3: real > A,F3: real > A > A,T5: set @ real,X7: set @ A,Y7: set @ A] :
% 3.03/1.64 ( ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ X7 )
% 3.03/1.64 => ( ( ord_less_eq @ ( set @ A ) @ X7 @ Y7 )
% 3.03/1.64 => ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ Y7 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_186_solves__ode__subset,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( real_V55928688vector @ A )
% 3.03/1.64 => ! [X3: real > A,F3: real > A > A,T5: set @ real,X7: set @ A,S5: set @ real] :
% 3.03/1.64 ( ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ X7 )
% 3.03/1.64 => ( ( ord_less_eq @ ( set @ real ) @ S5 @ T5 )
% 3.03/1.64 => ( initia1685620758es_ode @ A @ X3 @ F3 @ S5 @ X7 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_187_solves__ode__on__subset,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( real_V55928688vector @ A )
% 3.03/1.64 => ! [X3: real > A,F3: real > A > A,S5: set @ real,Y7: set @ A,T5: set @ real,X7: set @ A] :
% 3.03/1.64 ( ( initia1685620758es_ode @ A @ X3 @ F3 @ S5 @ Y7 )
% 3.03/1.64 => ( ( ord_less_eq @ ( set @ real ) @ T5 @ S5 )
% 3.03/1.64 => ( ( ord_less_eq @ ( set @ A ) @ Y7 @ X7 )
% 3.03/1.64 => ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ X7 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_188_at__le,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( topolo503727757_space @ A )
% 3.03/1.64 => ! [S6: set @ A,T6: set @ A,X3: A] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ S6 @ T6 )
% 3.03/1.64 => ( ord_less_eq @ ( filter @ A ) @ ( topolo507301023within @ A @ X3 @ S6 ) @ ( topolo507301023within @ A @ X3 @ T6 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_189_intervalE,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ord @ A )
% 3.03/1.64 => ! [A2: A,X3: A,B2: A] :
% 3.03/1.64 ( ( ( ord_less_eq @ A @ A2 @ X3 )
% 3.03/1.64 & ( ord_less_eq @ A @ X3 @ B2 ) )
% 3.03/1.64 => ( member @ A @ X3 @ ( set_or331188842AtMost @ A @ A2 @ B2 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_190_eq__subset,axiom,
% 3.03/1.64 ! [A: $tType,P2: A > A > $o] :
% 3.03/1.64 ( ord_less_eq @ ( A > A > $o )
% 3.03/1.64 @ ^ [Y4: A,Z: A] : ( Y4 = Z )
% 3.03/1.64 @ ^ [A8: A,B8: A] :
% 3.03/1.64 ( ( P2 @ A8 @ B8 )
% 3.03/1.64 | ( A8 = B8 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_191_conj__subset__def,axiom,
% 3.03/1.64 ! [A: $tType,A4: set @ A,P2: A > $o,Q: A > $o] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ A4
% 3.03/1.64 @ ( collect @ A
% 3.03/1.64 @ ^ [X4: A] :
% 3.03/1.64 ( ( P2 @ X4 )
% 3.03/1.64 & ( Q @ X4 ) ) ) )
% 3.03/1.64 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ P2 ) )
% 3.03/1.64 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ Q ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_192_predicate2D__conj,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,P2: A > B > $o,Q: A > B > $o,R4: $o,X3: A,Y3: B] :
% 3.03/1.64 ( ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
% 3.03/1.64 & R4 )
% 3.03/1.64 => ( R4
% 3.03/1.64 & ( ( P2 @ X3 @ Y3 )
% 3.03/1.64 => ( Q @ X3 @ Y3 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_193_fun__cong__unused__0,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,C: $tType] :
% 3.03/1.64 ( ( zero @ B )
% 3.03/1.64 => ! [F3: ( A > B ) > C,G3: C] :
% 3.03/1.64 ( ( F3
% 3.03/1.64 = ( ^ [X4: A > B] : G3 ) )
% 3.03/1.64 => ( ( F3
% 3.03/1.64 @ ^ [X4: A] : ( zero_zero @ B ) )
% 3.03/1.64 = G3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_194_subset__Collect__iff,axiom,
% 3.03/1.64 ! [A: $tType,B4: set @ A,A4: set @ A,P2: A > $o] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 3.03/1.64 => ( ( ord_less_eq @ ( set @ A ) @ B4
% 3.03/1.64 @ ( collect @ A
% 3.03/1.64 @ ^ [X4: A] :
% 3.03/1.64 ( ( member @ A @ X4 @ A4 )
% 3.03/1.64 & ( P2 @ X4 ) ) ) )
% 3.03/1.64 = ( ! [X4: A] :
% 3.03/1.64 ( ( member @ A @ X4 @ B4 )
% 3.03/1.64 => ( P2 @ X4 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_195_subset__CollectI,axiom,
% 3.03/1.64 ! [A: $tType,B4: set @ A,A4: set @ A,Q: A > $o,P2: A > $o] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 3.03/1.64 => ( ! [X: A] :
% 3.03/1.64 ( ( member @ A @ X @ B4 )
% 3.03/1.64 => ( ( Q @ X )
% 3.03/1.64 => ( P2 @ X ) ) )
% 3.03/1.64 => ( ord_less_eq @ ( set @ A )
% 3.03/1.64 @ ( collect @ A
% 3.03/1.64 @ ^ [X4: A] :
% 3.03/1.64 ( ( member @ A @ X4 @ B4 )
% 3.03/1.64 & ( Q @ X4 ) ) )
% 3.03/1.64 @ ( collect @ A
% 3.03/1.64 @ ^ [X4: A] :
% 3.03/1.64 ( ( member @ A @ X4 @ A4 )
% 3.03/1.64 & ( P2 @ X4 ) ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_196_prop__restrict,axiom,
% 3.03/1.64 ! [A: $tType,X3: A,Z3: set @ A,X7: set @ A,P2: A > $o] :
% 3.03/1.64 ( ( member @ A @ X3 @ Z3 )
% 3.03/1.64 => ( ( ord_less_eq @ ( set @ A ) @ Z3
% 3.03/1.64 @ ( collect @ A
% 3.03/1.64 @ ^ [X4: A] :
% 3.03/1.64 ( ( member @ A @ X4 @ X7 )
% 3.03/1.64 & ( P2 @ X4 ) ) ) )
% 3.03/1.64 => ( P2 @ X3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_197_Collect__restrict,axiom,
% 3.03/1.64 ! [A: $tType,X7: set @ A,P2: A > $o] :
% 3.03/1.64 ( ord_less_eq @ ( set @ A )
% 3.03/1.64 @ ( collect @ A
% 3.03/1.64 @ ^ [X4: A] :
% 3.03/1.64 ( ( member @ A @ X4 @ X7 )
% 3.03/1.64 & ( P2 @ X4 ) ) )
% 3.03/1.64 @ X7 ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_198_ssubst__Pair__rhs,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType,R5: A,S6: B,R4: set @ ( product_prod @ A @ B ),S7: B] :
% 3.03/1.64 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R5 @ S6 ) @ R4 )
% 3.03/1.64 => ( ( S7 = S6 )
% 3.03/1.64 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R5 @ S7 ) @ R4 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_199_hpsafe__ODE__simps,axiom,
% 3.03/1.64 ! [C: $tType,B: $tType,A: $tType,ODE: ode @ A @ C,P: formula @ A @ B @ C] :
% 3.03/1.64 ( ( hpsafe @ A @ B @ C @ ( evolveODE @ A @ C @ B @ ODE @ P ) )
% 3.03/1.64 = ( ( osafe @ A @ C @ ODE )
% 3.03/1.64 & ( fsafe @ A @ B @ C @ P ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_200_old_Orec__unit__def,axiom,
% 3.03/1.64 ! [T: $tType] :
% 3.03/1.64 ( ( product_rec_unit @ T )
% 3.03/1.64 = ( ^ [F12: T,X4: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X4 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_201_hpsafe__fsafe_Ohpsafe__Evolve,axiom,
% 3.03/1.64 ! [C: $tType,B: $tType,A: $tType,ODE: ode @ A @ C,P2: formula @ A @ B @ C] :
% 3.03/1.64 ( ( osafe @ A @ C @ ODE )
% 3.03/1.64 => ( ( fsafe @ A @ B @ C @ P2 )
% 3.03/1.64 => ( hpsafe @ A @ B @ C @ ( evolveODE @ A @ C @ B @ ODE @ P2 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_202_divides__aux__eq,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( unique1598680935umeral @ A )
% 3.03/1.64 => ! [Q2: A,R5: A] :
% 3.03/1.64 ( ( unique455577585es_aux @ A @ ( product_Pair @ A @ A @ Q2 @ R5 ) )
% 3.03/1.64 = ( R5
% 3.03/1.64 = ( zero_zero @ A ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_203_refl__ge__eq,axiom,
% 3.03/1.64 ! [A: $tType,R4: A > A > $o] :
% 3.03/1.64 ( ! [X: A] : ( R4 @ X @ X )
% 3.03/1.64 => ( ord_less_eq @ ( A > A > $o )
% 3.03/1.64 @ ^ [Y4: A,Z: A] : ( Y4 = Z )
% 3.03/1.64 @ R4 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_204_ge__eq__refl,axiom,
% 3.03/1.64 ! [A: $tType,R4: A > A > $o,X3: A] :
% 3.03/1.64 ( ( ord_less_eq @ ( A > A > $o )
% 3.03/1.64 @ ^ [Y4: A,Z: A] : ( Y4 = Z )
% 3.03/1.64 @ R4 )
% 3.03/1.64 => ( R4 @ X3 @ X3 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_205_Icc__subset__Ici__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( preorder @ A )
% 3.03/1.64 => ! [L: A,H: A,L2: A] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ L @ H ) @ ( set_ord_atLeast @ A @ L2 ) )
% 3.03/1.64 = ( ~ ( ord_less_eq @ A @ L @ H )
% 3.03/1.64 | ( ord_less_eq @ A @ L2 @ L ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_206_Icc__subset__Iic__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( preorder @ A )
% 3.03/1.64 => ! [L: A,H: A,H2: A] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ L @ H ) @ ( set_ord_atMost @ A @ H2 ) )
% 3.03/1.64 = ( ~ ( ord_less_eq @ A @ L @ H )
% 3.03/1.64 | ( ord_less_eq @ A @ H @ H2 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_207_atMost__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ord @ A )
% 3.03/1.64 => ! [I7: A,K: A] :
% 3.03/1.64 ( ( member @ A @ I7 @ ( set_ord_atMost @ A @ K ) )
% 3.03/1.64 = ( ord_less_eq @ A @ I7 @ K ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_208_atLeast__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ord @ A )
% 3.03/1.64 => ! [I7: A,K: A] :
% 3.03/1.64 ( ( member @ A @ I7 @ ( set_ord_atLeast @ A @ K ) )
% 3.03/1.64 = ( ord_less_eq @ A @ K @ I7 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_209_atMost__subset__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( preorder @ A )
% 3.03/1.64 => ! [X3: A,Y3: A] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X3 ) @ ( set_ord_atMost @ A @ Y3 ) )
% 3.03/1.64 = ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_210_atLeast__subset__iff,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( preorder @ A )
% 3.03/1.64 => ! [X3: A,Y3: A] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X3 ) @ ( set_ord_atLeast @ A @ Y3 ) )
% 3.03/1.64 = ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_211_not__Ici__le__Icc,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( no_top @ A )
% 3.03/1.64 => ! [L: A,L2: A,H2: A] :
% 3.03/1.64 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_or331188842AtMost @ A @ L2 @ H2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_212_not__Iic__le__Icc,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( no_bot @ A )
% 3.03/1.64 => ! [H: A,L2: A,H2: A] :
% 3.03/1.64 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_or331188842AtMost @ A @ L2 @ H2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_213_atLeast__def,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ord @ A )
% 3.03/1.64 => ( ( set_ord_atLeast @ A )
% 3.03/1.64 = ( ^ [L3: A] : ( collect @ A @ ( ord_less_eq @ A @ L3 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_214_atMost__def,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ord @ A )
% 3.03/1.64 => ( ( set_ord_atMost @ A )
% 3.03/1.64 = ( ^ [U2: A] :
% 3.03/1.64 ( collect @ A
% 3.03/1.64 @ ^ [X4: A] : ( ord_less_eq @ A @ X4 @ U2 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_215_not__Iic__le__Ici,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( no_bot @ A )
% 3.03/1.64 => ! [H: A,L2: A] :
% 3.03/1.64 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_ord_atLeast @ A @ L2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_216_not__Ici__le__Iic,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( no_top @ A )
% 3.03/1.64 => ! [L: A,H2: A] :
% 3.03/1.64 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_ord_atMost @ A @ H2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_217_image__uminus__atMost,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ordered_ab_group_add @ A )
% 3.03/1.64 => ! [X3: A] :
% 3.03/1.64 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atMost @ A @ X3 ) )
% 3.03/1.64 = ( set_ord_atLeast @ A @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_218_image__uminus__atLeast,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ordered_ab_group_add @ A )
% 3.03/1.64 => ! [X3: A] :
% 3.03/1.64 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atLeast @ A @ X3 ) )
% 3.03/1.64 = ( set_ord_atMost @ A @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_219_image__eqI,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,B2: A,F3: B > A,X3: B,A4: set @ B] :
% 3.03/1.64 ( ( B2
% 3.03/1.64 = ( F3 @ X3 ) )
% 3.03/1.64 => ( ( member @ B @ X3 @ A4 )
% 3.03/1.64 => ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A4 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_220_image__ident,axiom,
% 3.03/1.64 ! [A: $tType,Y7: set @ A] :
% 3.03/1.64 ( ( image @ A @ A
% 3.03/1.64 @ ^ [X4: A] : X4
% 3.03/1.64 @ Y7 )
% 3.03/1.64 = Y7 ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_221_image__uminus__atLeastAtMost,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( ordered_ab_group_add @ A )
% 3.03/1.64 => ! [X3: A,Y3: A] :
% 3.03/1.64 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or331188842AtMost @ A @ X3 @ Y3 ) )
% 3.03/1.64 = ( set_or331188842AtMost @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_222_subset__image__iff,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,B4: set @ A,F3: B > A,A4: set @ B] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image @ B @ A @ F3 @ A4 ) )
% 3.03/1.64 = ( ? [AA: set @ B] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ B ) @ AA @ A4 )
% 3.03/1.64 & ( B4
% 3.03/1.64 = ( image @ B @ A @ F3 @ AA ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_223_image__subset__iff,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,F3: B > A,A4: set @ B,B4: set @ A] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ A4 ) @ B4 )
% 3.03/1.64 = ( ! [X4: B] :
% 3.03/1.64 ( ( member @ B @ X4 @ A4 )
% 3.03/1.64 => ( member @ A @ ( F3 @ X4 ) @ B4 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_224_subset__imageE,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,B4: set @ A,F3: B > A,A4: set @ B] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image @ B @ A @ F3 @ A4 ) )
% 3.03/1.64 => ~ ! [C5: set @ B] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A4 )
% 3.03/1.64 => ( B4
% 3.03/1.64 != ( image @ B @ A @ F3 @ C5 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_225_image__subsetI,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,A4: set @ A,F3: A > B,B4: set @ B] :
% 3.03/1.64 ( ! [X: A] :
% 3.03/1.64 ( ( member @ A @ X @ A4 )
% 3.03/1.64 => ( member @ B @ ( F3 @ X ) @ B4 ) )
% 3.03/1.64 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A4 ) @ B4 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_226_image__mono,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType,A4: set @ A,B4: set @ A,F3: A > B] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 3.03/1.64 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A4 ) @ ( image @ A @ B @ F3 @ B4 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_227_solves__ode__subset__range,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( real_V55928688vector @ A )
% 3.03/1.64 => ! [X3: real > A,F3: real > A > A,T5: set @ real,X7: set @ A,Y7: set @ A] :
% 3.03/1.64 ( ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ X7 )
% 3.03/1.64 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ real @ A @ X3 @ T5 ) @ Y7 )
% 3.03/1.64 => ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ Y7 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_228_imageI,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType,X3: A,A4: set @ A,F3: A > B] :
% 3.03/1.64 ( ( member @ A @ X3 @ A4 )
% 3.03/1.64 => ( member @ B @ ( F3 @ X3 ) @ ( image @ A @ B @ F3 @ A4 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_229_image__iff,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,Z2: A,F3: B > A,A4: set @ B] :
% 3.03/1.64 ( ( member @ A @ Z2 @ ( image @ B @ A @ F3 @ A4 ) )
% 3.03/1.64 = ( ? [X4: B] :
% 3.03/1.64 ( ( member @ B @ X4 @ A4 )
% 3.03/1.64 & ( Z2
% 3.03/1.64 = ( F3 @ X4 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_230_bex__imageD,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P2: A > $o] :
% 3.03/1.64 ( ? [X8: A] :
% 3.03/1.64 ( ( member @ A @ X8 @ ( image @ B @ A @ F3 @ A4 ) )
% 3.03/1.64 & ( P2 @ X8 ) )
% 3.03/1.64 => ? [X: B] :
% 3.03/1.64 ( ( member @ B @ X @ A4 )
% 3.03/1.64 & ( P2 @ ( F3 @ X ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_231_image__cong,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType,M: set @ A,N2: set @ A,F3: A > B,G3: A > B] :
% 3.03/1.64 ( ( M = N2 )
% 3.03/1.64 => ( ! [X: A] :
% 3.03/1.64 ( ( member @ A @ X @ N2 )
% 3.03/1.64 => ( ( F3 @ X )
% 3.03/1.64 = ( G3 @ X ) ) )
% 3.03/1.64 => ( ( image @ A @ B @ F3 @ M )
% 3.03/1.64 = ( image @ A @ B @ G3 @ N2 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_232_ball__imageD,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P2: A > $o] :
% 3.03/1.64 ( ! [X: A] :
% 3.03/1.64 ( ( member @ A @ X @ ( image @ B @ A @ F3 @ A4 ) )
% 3.03/1.64 => ( P2 @ X ) )
% 3.03/1.64 => ! [X8: B] :
% 3.03/1.64 ( ( member @ B @ X8 @ A4 )
% 3.03/1.64 => ( P2 @ ( F3 @ X8 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_233_rev__image__eqI,axiom,
% 3.03/1.64 ! [B: $tType,A: $tType,X3: A,A4: set @ A,B2: B,F3: A > B] :
% 3.03/1.64 ( ( member @ A @ X3 @ A4 )
% 3.03/1.64 => ( ( B2
% 3.03/1.64 = ( F3 @ X3 ) )
% 3.03/1.64 => ( member @ B @ B2 @ ( image @ A @ B @ F3 @ A4 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_234_Compr__image__eq,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P2: A > $o] :
% 3.03/1.64 ( ( collect @ A
% 3.03/1.64 @ ^ [X4: A] :
% 3.03/1.64 ( ( member @ A @ X4 @ ( image @ B @ A @ F3 @ A4 ) )
% 3.03/1.64 & ( P2 @ X4 ) ) )
% 3.03/1.64 = ( image @ B @ A @ F3
% 3.03/1.64 @ ( collect @ B
% 3.03/1.64 @ ^ [X4: B] :
% 3.03/1.64 ( ( member @ B @ X4 @ A4 )
% 3.03/1.64 & ( P2 @ ( F3 @ X4 ) ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_235_image__image,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,C: $tType,F3: B > A,G3: C > B,A4: set @ C] :
% 3.03/1.64 ( ( image @ B @ A @ F3 @ ( image @ C @ B @ G3 @ A4 ) )
% 3.03/1.64 = ( image @ C @ A
% 3.03/1.64 @ ^ [X4: C] : ( F3 @ ( G3 @ X4 ) )
% 3.03/1.64 @ A4 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_236_imageE,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,B2: A,F3: B > A,A4: set @ B] :
% 3.03/1.64 ( ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A4 ) )
% 3.03/1.64 => ~ ! [X: B] :
% 3.03/1.64 ( ( B2
% 3.03/1.64 = ( F3 @ X ) )
% 3.03/1.64 => ~ ( member @ B @ X @ A4 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_237_image__Collect__subsetI,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,P2: A > $o,F3: A > B,B4: set @ B] :
% 3.03/1.64 ( ! [X: A] :
% 3.03/1.64 ( ( P2 @ X )
% 3.03/1.64 => ( member @ B @ ( F3 @ X ) @ B4 ) )
% 3.03/1.64 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ ( collect @ A @ P2 ) ) @ B4 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_238_Osub,axiom,
% 3.03/1.64 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( static_FVO @ a @ c @ ode2 ) ) @ v ).
% 3.03/1.64
% 3.03/1.64 thf(fact_239_all__subset__image,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P2: ( set @ A ) > $o] :
% 3.03/1.64 ( ( ! [B6: set @ A] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F3 @ A4 ) )
% 3.03/1.64 => ( P2 @ B6 ) ) )
% 3.03/1.64 = ( ! [B6: set @ B] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ B ) @ B6 @ A4 )
% 3.03/1.64 => ( P2 @ ( image @ B @ A @ F3 @ B6 ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_240_bvo__to__fvo,axiom,
% 3.03/1.64 ! [A: $tType,B: $tType,X3: A,ODE: ode @ B @ A] :
% 3.03/1.64 ( ( member @ ( sum_sum @ A @ A ) @ ( sum_Inl @ A @ A @ X3 ) @ ( static_BVO @ B @ A @ ODE ) )
% 3.03/1.64 => ( member @ A @ X3 @ ( static_FVO @ B @ A @ ODE ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_241_ids_Obvo__to__fvo,axiom,
% 3.03/1.64 ! [Sz: $tType,Sf: $tType,Sc: $tType,A: $tType,B: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ Sc )
% 3.03/1.64 & ( finite_finite @ Sf )
% 3.03/1.64 & ( finite_finite @ Sz )
% 3.03/1.64 & ( linorder @ Sz ) )
% 3.03/1.64 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X3: A,ODE: ode @ B @ A] :
% 3.03/1.64 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.03/1.64 => ( ( member @ ( sum_sum @ A @ A ) @ ( sum_Inl @ A @ A @ X3 ) @ ( static_BVO @ B @ A @ ODE ) )
% 3.03/1.64 => ( member @ A @ X3 @ ( static_FVO @ B @ A @ ODE ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_242_VSA,axiom,
% 3.03/1.64 ( denotational_VSagree @ c @ ( sol @ ( zero_zero @ real ) ) @ a2
% 3.03/1.64 @ ( collect @ c
% 3.03/1.64 @ ^ [Uu: c] :
% 3.03/1.64 ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_BVO @ a @ c @ ode2 ) )
% 3.03/1.64 | ( member @ ( sum_sum @ c @ d ) @ ( sum_Inl @ c @ d @ Uu ) @ ( image @ c @ ( sum_sum @ c @ d ) @ ( sum_Inl @ c @ d ) @ ( static_FVO @ a @ c @ ode2 ) ) )
% 3.03/1.64 | ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_FVF @ a @ b @ c @ p ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_243_ode__to__fvo,axiom,
% 3.03/1.64 ! [C: $tType,A: $tType,B: $tType,X3: A,I: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A] :
% 3.03/1.64 ( ( member @ A @ X3 @ ( denota811733865E_vars @ B @ C @ A @ I @ ODE ) )
% 3.03/1.64 => ( member @ A @ X3 @ ( static_FVO @ B @ A @ ODE ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_244_VSagree__sub,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( finite_finite @ A )
% 3.03/1.64 => ! [A4: set @ A,B4: set @ A,Nu: finite_Cartesian_vec @ real @ A,Omega2: finite_Cartesian_vec @ real @ A] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 3.03/1.64 => ( ( denotational_VSagree @ A @ Nu @ Omega2 @ B4 )
% 3.03/1.64 => ( denotational_VSagree @ A @ Nu @ Omega2 @ A4 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_245_VSagree__supset,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( finite_finite @ A )
% 3.03/1.64 => ! [B4: set @ A,A4: set @ A,Nu: finite_Cartesian_vec @ real @ A,Nu2: finite_Cartesian_vec @ real @ A] :
% 3.03/1.64 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 3.03/1.64 => ( ( denotational_VSagree @ A @ Nu @ Nu2 @ A4 )
% 3.03/1.64 => ( denotational_VSagree @ A @ Nu @ Nu2 @ B4 ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_246_VSagree__refl,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( finite_finite @ A )
% 3.03/1.64 => ! [Nu: finite_Cartesian_vec @ real @ A,A4: set @ A] : ( denotational_VSagree @ A @ Nu @ Nu @ A4 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_247_ids_Oode__to__fvo,axiom,
% 3.03/1.64 ! [C: $tType,Sz: $tType,Sf: $tType,Sc: $tType,A: $tType,B: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ Sc )
% 3.03/1.64 & ( finite_finite @ Sf )
% 3.03/1.64 & ( finite_finite @ Sz )
% 3.03/1.64 & ( linorder @ Sz ) )
% 3.03/1.64 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X3: A,I: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A] :
% 3.03/1.64 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.03/1.64 => ( ( member @ A @ X3 @ ( denota811733865E_vars @ B @ C @ A @ I @ ODE ) )
% 3.03/1.64 => ( member @ A @ X3 @ ( static_FVO @ B @ A @ ODE ) ) ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_248_OVsub_H,axiom,
% 3.03/1.64 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( oDE_dom @ a @ c @ ode2 ) ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( oDE_dom @ a @ c @ ode2 ) ) ) @ v ).
% 3.03/1.64
% 3.03/1.64 thf(fact_249_ids_Ovne23,axiom,
% 3.03/1.64 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ Sz )
% 3.03/1.64 & ( linorder @ Sz )
% 3.03/1.64 & ( finite_finite @ Sf )
% 3.03/1.64 & ( finite_finite @ Sc ) )
% 3.03/1.64 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.03/1.64 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.03/1.64 => ( Vid2 != Vid3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_250_sup_Oright__idem,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( semilattice_sup @ A )
% 3.03/1.64 => ! [A2: A,B2: A] :
% 3.03/1.64 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
% 3.03/1.64 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_251_sup__left__idem,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( semilattice_sup @ A )
% 3.03/1.64 => ! [X3: A,Y3: A] :
% 3.03/1.64 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y3 ) )
% 3.03/1.64 = ( sup_sup @ A @ X3 @ Y3 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_252_sup_Oleft__idem,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( semilattice_sup @ A )
% 3.03/1.64 => ! [A2: A,B2: A] :
% 3.03/1.64 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
% 3.03/1.64 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_253_sup__idem,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( semilattice_sup @ A )
% 3.03/1.64 => ! [X3: A] :
% 3.03/1.64 ( ( sup_sup @ A @ X3 @ X3 )
% 3.03/1.64 = X3 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(fact_254_sup_Oidem,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( semilattice_sup @ A )
% 3.03/1.64 => ! [A2: A] :
% 3.03/1.64 ( ( sup_sup @ A @ A2 @ A2 )
% 3.03/1.64 = A2 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
% 3.03/1.64 ! [A: $tType] :
% 3.03/1.64 ( ( finite_finite @ A )
% 3.03/1.64 => ( type @ A ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( ordere890947078_space @ A9 )
% 3.03/1.64 & ( finite_finite @ A10 ) )
% 3.03/1.64 => ( ordere890947078_space @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space_1,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( ordere890947078_space @ A9 )
% 3.03/1.64 & ( ordere890947078_space @ A10 ) )
% 3.03/1.64 => ( ordere890947078_space @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_2,axiom,
% 3.03/1.64 ordere890947078_space @ real ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( topolo503727757_space @ A10 )
% 3.03/1.64 => ( topolo503727757_space @ ( A9 > A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( semilattice_sup @ A10 )
% 3.03/1.64 => ( semilattice_sup @ ( A9 > A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( boolean_algebra @ A10 )
% 3.03/1.64 => ( boolean_algebra @ ( A9 > A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( preorder @ A10 )
% 3.03/1.64 => ( preorder @ ( A9 > A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ A9 )
% 3.03/1.64 & ( finite_finite @ A10 ) )
% 3.03/1.64 => ( finite_finite @ ( A9 > A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( order @ A10 )
% 3.03/1.64 => ( order @ ( A9 > A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_fun___Orderings_Oord,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ord @ A10 )
% 3.03/1.64 => ( ord @ ( A9 > A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_fun___Groups_Ouminus,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( uminus @ A10 )
% 3.03/1.64 => ( uminus @ ( A9 > A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_3,axiom,
% 3.03/1.64 ! [A9: $tType] : ( semilattice_sup @ ( set @ A9 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_4,axiom,
% 3.03/1.64 ! [A9: $tType] : ( boolean_algebra @ ( set @ A9 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
% 3.03/1.64 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Set_Oset___Finite__Set_Ofinite_6,axiom,
% 3.03/1.64 ! [A9: $tType] :
% 3.03/1.64 ( ( finite_finite @ A9 )
% 3.03/1.64 => ( finite_finite @ ( set @ A9 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Set_Oset___Orderings_Oorder_7,axiom,
% 3.03/1.64 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 3.03/1.64 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Set_Oset___Groups_Ouminus_9,axiom,
% 3.03/1.64 ! [A9: $tType] : ( uminus @ ( set @ A9 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Set_Oset___Groups_Ozero,axiom,
% 3.03/1.64 ! [A9: $tType] :
% 3.03/1.64 ( ( zero @ A9 )
% 3.03/1.64 => ( zero @ ( set @ A9 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_10,axiom,
% 3.03/1.64 topolo503727757_space @ $o ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_11,axiom,
% 3.03/1.64 semilattice_sup @ $o ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_12,axiom,
% 3.03/1.64 boolean_algebra @ $o ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_HOL_Obool___Orderings_Opreorder_13,axiom,
% 3.03/1.64 preorder @ $o ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 3.03/1.64 linorder @ $o ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_HOL_Obool___Finite__Set_Ofinite_14,axiom,
% 3.03/1.64 finite_finite @ $o ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_HOL_Obool___Orderings_Oorder_15,axiom,
% 3.03/1.64 order @ $o ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_HOL_Obool___Orderings_Oord_16,axiom,
% 3.03/1.64 ord @ $o ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_HOL_Obool___Groups_Ouminus_17,axiom,
% 3.03/1.64 uminus @ $o ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
% 3.03/1.64 real_V55928688vector @ real ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Real_Oreal___Lattice__Algebras_Olattice__ab__group__add,axiom,
% 3.03/1.64 lattic1601792062up_add @ real ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_18,axiom,
% 3.03/1.64 topolo503727757_space @ real ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring,axiom,
% 3.03/1.64 linord1659791738miring @ real ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
% 3.03/1.64 linord219039673up_add @ real ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
% 3.03/1.64 ordered_ab_group_add @ real ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_19,axiom,
% 3.03/1.64 semilattice_sup @ real ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Real_Oreal___Orderings_Opreorder_20,axiom,
% 3.03/1.64 preorder @ real ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Real_Oreal___Orderings_Olinorder_21,axiom,
% 3.03/1.64 linorder @ real ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Real_Oreal___Orderings_Ono__top,axiom,
% 3.03/1.64 no_top @ real ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Real_Oreal___Orderings_Ono__bot,axiom,
% 3.03/1.64 no_bot @ real ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 3.03/1.64 group_add @ real ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Real_Oreal___Orderings_Oorder_22,axiom,
% 3.03/1.64 order @ real ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Real_Oreal___Orderings_Oord_23,axiom,
% 3.03/1.64 ord @ real ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Real_Oreal___Groups_Ouminus_24,axiom,
% 3.03/1.64 uminus @ real ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Real_Oreal___Groups_Ozero_25,axiom,
% 3.03/1.64 zero @ real ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_26,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ A9 )
% 3.03/1.64 & ( finite_finite @ A10 ) )
% 3.03/1.64 => ( finite_finite @ ( sum_sum @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_27,axiom,
% 3.03/1.64 ! [A9: $tType] : ( semilattice_sup @ ( filter @ A9 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Filter_Ofilter___Orderings_Opreorder_28,axiom,
% 3.03/1.64 ! [A9: $tType] : ( preorder @ ( filter @ A9 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Filter_Ofilter___Orderings_Oorder_29,axiom,
% 3.03/1.64 ! [A9: $tType] : ( order @ ( filter @ A9 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Filter_Ofilter___Orderings_Oord_30,axiom,
% 3.03/1.64 ! [A9: $tType] : ( ord @ ( filter @ A9 ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__normed__vector_31,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( real_V55928688vector @ A9 )
% 3.03/1.64 & ( real_V55928688vector @ A10 ) )
% 3.03/1.64 => ( real_V55928688vector @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_32,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( topolo503727757_space @ A9 )
% 3.03/1.64 & ( topolo503727757_space @ A10 ) )
% 3.03/1.64 => ( topolo503727757_space @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__group__add_33,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( ordere890947078_space @ A9 )
% 3.03/1.64 & ( ordere890947078_space @ A10 ) )
% 3.03/1.64 => ( ordered_ab_group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Oprod___Lattices_Osemilattice__sup_34,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( semilattice_sup @ A9 )
% 3.03/1.64 & ( semilattice_sup @ A10 ) )
% 3.03/1.64 => ( semilattice_sup @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Oprod___Lattices_Oboolean__algebra_35,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( boolean_algebra @ A9 )
% 3.03/1.64 & ( boolean_algebra @ A10 ) )
% 3.03/1.64 => ( boolean_algebra @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_36,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( preorder @ A9 )
% 3.03/1.64 & ( preorder @ A10 ) )
% 3.03/1.64 => ( preorder @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_37,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( finite_finite @ A9 )
% 3.03/1.64 & ( finite_finite @ A10 ) )
% 3.03/1.64 => ( finite_finite @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Oprod___Groups_Ogroup__add_38,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( group_add @ A9 )
% 3.03/1.64 & ( group_add @ A10 ) )
% 3.03/1.64 => ( group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Oprod___Orderings_Oorder_39,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( order @ A9 )
% 3.03/1.64 & ( order @ A10 ) )
% 3.03/1.64 => ( order @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Oprod___Orderings_Oord_40,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( ord @ A9 )
% 3.03/1.64 & ( ord @ A10 ) )
% 3.03/1.64 => ( ord @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Oprod___Groups_Ouminus_41,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( uminus @ A9 )
% 3.03/1.64 & ( uminus @ A10 ) )
% 3.03/1.64 => ( uminus @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Oprod___Groups_Ozero_42,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( zero @ A9 )
% 3.03/1.64 & ( zero @ A10 ) )
% 3.03/1.64 => ( zero @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_43,axiom,
% 3.03/1.64 semilattice_sup @ product_unit ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Ounit___Lattices_Oboolean__algebra_44,axiom,
% 3.03/1.64 boolean_algebra @ product_unit ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_45,axiom,
% 3.03/1.64 preorder @ product_unit ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_46,axiom,
% 3.03/1.64 linorder @ product_unit ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_47,axiom,
% 3.03/1.64 finite_finite @ product_unit ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Ounit___Orderings_Oorder_48,axiom,
% 3.03/1.64 order @ product_unit ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Ounit___Orderings_Oord_49,axiom,
% 3.03/1.64 ord @ product_unit ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Product__Type_Ounit___Groups_Ouminus_50,axiom,
% 3.03/1.64 uminus @ product_unit ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Finite__Cartesian__Product_Ovec___Real__Vector__Spaces_Oreal__normed__vector_51,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( real_V55928688vector @ A9 )
% 3.03/1.64 & ( finite_finite @ A10 ) )
% 3.03/1.64 => ( real_V55928688vector @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Finite__Cartesian__Product_Ovec___Topological__Spaces_Otopological__space_52,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( topolo503727757_space @ A9 )
% 3.03/1.64 & ( finite_finite @ A10 ) )
% 3.03/1.64 => ( topolo503727757_space @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__group__add_53,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( ordere890947078_space @ A9 )
% 3.03/1.64 & ( finite_finite @ A10 ) )
% 3.03/1.64 => ( ordered_ab_group_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Osemilattice__sup_54,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.64 ( ( ( ordere890947078_space @ A9 )
% 3.03/1.64 & ( finite_finite @ A10 ) )
% 3.03/1.64 => ( semilattice_sup @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.03/1.64
% 3.03/1.64 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Opreorder_55,axiom,
% 3.03/1.64 ! [A9: $tType,A10: $tType] :
% 3.03/1.65 ( ( ( order @ A9 )
% 3.03/1.65 & ( finite_finite @ A10 ) )
% 3.03/1.65 => ( preorder @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.03/1.65
% 3.03/1.65 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_56,axiom,
% 3.03/1.65 ! [A9: $tType,A10: $tType] :
% 3.03/1.65 ( ( ( linorder @ A9 )
% 3.03/1.65 & ( cARD_1 @ A10 ) )
% 3.03/1.65 => ( linorder @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.03/1.65
% 3.03/1.65 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_57,axiom,
% 3.03/1.65 ! [A9: $tType,A10: $tType] :
% 3.03/1.65 ( ( ( finite_finite @ A9 )
% 3.03/1.65 & ( finite_finite @ A10 ) )
% 3.03/1.65 => ( finite_finite @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.03/1.65
% 3.03/1.65 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ogroup__add_58,axiom,
% 3.03/1.65 ! [A9: $tType,A10: $tType] :
% 3.03/1.65 ( ( ( group_add @ A9 )
% 3.03/1.65 & ( finite_finite @ A10 ) )
% 3.03/1.65 => ( group_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.03/1.65
% 3.03/1.65 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oorder_59,axiom,
% 3.03/1.65 ! [A9: $tType,A10: $tType] :
% 3.03/1.65 ( ( ( order @ A9 )
% 3.03/1.65 & ( finite_finite @ A10 ) )
% 3.03/1.65 => ( order @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.03/1.65
% 3.03/1.65 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oord_60,axiom,
% 3.03/1.65 ! [A9: $tType,A10: $tType] :
% 3.03/1.65 ( ( ( ord @ A9 )
% 3.03/1.65 & ( finite_finite @ A10 ) )
% 3.03/1.65 => ( ord @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.03/1.65
% 3.03/1.65 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ouminus_61,axiom,
% 3.03/1.65 ! [A9: $tType,A10: $tType] :
% 3.03/1.65 ( ( ( uminus @ A9 )
% 3.03/1.65 & ( finite_finite @ A10 ) )
% 3.03/1.65 => ( uminus @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.03/1.65
% 3.03/1.65 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_62,axiom,
% 3.03/1.65 ! [A9: $tType,A10: $tType] :
% 3.03/1.65 ( ( ( zero @ A9 )
% 3.03/1.65 & ( finite_finite @ A10 ) )
% 3.03/1.65 => ( zero @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.03/1.65
% 3.03/1.65 thf(tfree_0,hypothesis,
% 3.03/1.65 finite_finite @ c ).
% 3.03/1.65
% 3.03/1.65 thf(tfree_1,hypothesis,
% 3.03/1.65 finite_finite @ a ).
% 3.03/1.65
% 3.03/1.65 thf(tfree_2,hypothesis,
% 3.03/1.65 finite_finite @ b ).
% 3.03/1.65
% 3.03/1.65 thf(conj_0,conjecture,
% 3.03/1.65 denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ j @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( sol @ s ) ) @ ( denotational_mk_xode @ a @ b @ c @ j @ ode2 @ ( sol @ s ) ) @ ( denotational_semBV @ a @ b @ c @ j @ ode2 ) ).
% 3.03/1.65 ------------------------
% 3.03/1.68 % (26745)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_26468 for (2999ds/4Mi)
% 3.03/1.68 % (26744)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_26468 for (2999ds/183Mi)
% 3.03/1.68 % (26747)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_26468 for (2999ds/2Mi)
% 3.03/1.68 % (26747)Instruction limit reached!
% 3.03/1.68 % (26747)------------------------------
% 3.03/1.68 % (26747)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.68 % (26747)Termination reason: Unknown
% 3.03/1.68 % (26747)Termination phase: shuffling
% 3.03/1.68
% 3.03/1.68 % (26747)Memory used [KB]: 1535
% 3.03/1.68 % (26747)Time elapsed: 0.002 s
% 3.03/1.68 % (26747)Instructions burned: 2 (million)
% 3.03/1.68 % (26747)------------------------------
% 3.03/1.68 % (26747)------------------------------
% 3.03/1.68 % (26745)Instruction limit reached!
% 3.03/1.68 % (26745)------------------------------
% 3.03/1.68 % (26745)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.68 % (26745)Termination reason: Unknown
% 3.03/1.68 % (26745)Termination phase: shuffling
% 3.03/1.68
% 3.03/1.68 % (26745)Memory used [KB]: 1663
% 3.03/1.68 % (26745)Time elapsed: 0.004 s
% 3.03/1.68 % (26745)Instructions burned: 7 (million)
% 3.03/1.68 % (26745)------------------------------
% 3.03/1.68 % (26745)------------------------------
% 3.03/1.69 % (26746)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_26468 for (2999ds/27Mi)
% 3.03/1.69 % (26748)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_26468 for (2999ds/2Mi)
% 3.03/1.69 % (26750)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_26468 for (2999ds/18Mi)
% 3.03/1.69 % (26749)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_26468 for (2999ds/275Mi)
% 3.03/1.69 % (26748)Instruction limit reached!
% 3.03/1.69 % (26748)------------------------------
% 3.03/1.69 % (26748)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.69 % (26748)Termination reason: Unknown
% 3.03/1.69 % (26748)Termination phase: shuffling
% 3.03/1.69
% 3.03/1.69 % (26748)Memory used [KB]: 1535
% 3.03/1.69 % (26748)Time elapsed: 0.004 s
% 3.03/1.69 % (26748)Instructions burned: 3 (million)
% 3.03/1.69 % (26748)------------------------------
% 3.03/1.69 % (26748)------------------------------
% 3.03/1.69 % (26752)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_26468 for (2999ds/37Mi)
% 3.03/1.69 % (26751)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_26468 for (2999ds/3Mi)
% 3.03/1.70 % (26751)Instruction limit reached!
% 3.03/1.70 % (26751)------------------------------
% 3.03/1.70 % (26751)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.70 % (26751)Termination reason: Unknown
% 3.03/1.70 % (26751)Termination phase: shuffling
% 3.03/1.70
% 3.03/1.70 % (26751)Memory used [KB]: 1535
% 3.03/1.70 % (26751)Time elapsed: 0.003 s
% 3.03/1.70 % (26751)Instructions burned: 5 (million)
% 3.03/1.70 % (26751)------------------------------
% 3.03/1.70 % (26751)------------------------------
% 3.09/1.70 % (26750)Instruction limit reached!
% 3.09/1.70 % (26750)------------------------------
% 3.09/1.70 % (26750)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.70 % (26750)Termination reason: Unknown
% 3.09/1.70 % (26750)Termination phase: shuffling
% 3.09/1.70
% 3.09/1.70 % (26750)Memory used [KB]: 1918
% 3.09/1.70 % (26750)Time elapsed: 0.014 s
% 3.09/1.70 % (26750)Instructions burned: 18 (million)
% 3.09/1.70 % (26750)------------------------------
% 3.09/1.70 % (26750)------------------------------
% 3.09/1.70 % (26752)Instruction limit reached!
% 3.09/1.70 % (26752)------------------------------
% 3.09/1.70 % (26752)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.70 % (26752)Termination reason: Unknown
% 3.09/1.70 % (26752)Termination phase: shuffling
% 3.09/1.70
% 3.09/1.70 % (26752)Memory used [KB]: 2302
% 3.09/1.70 % (26752)Time elapsed: 0.012 s
% 3.09/1.70 % (26752)Instructions burned: 38 (million)
% 3.09/1.70 % (26752)------------------------------
% 3.09/1.70 % (26752)------------------------------
% 3.09/1.71 % (26754)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_26468 for (2999ds/3Mi)
% 3.09/1.71 % (26746)Instruction limit reached!
% 3.09/1.71 % (26746)------------------------------
% 3.09/1.71 % (26746)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.71 % (26746)Termination reason: Unknown
% 3.09/1.71 % (26746)Termination phase: shuffling
% 3.09/1.71
% 3.09/1.71 % (26746)Memory used [KB]: 2046
% 3.09/1.71 % (26746)Time elapsed: 0.019 s
% 3.09/1.71 % (26746)Instructions burned: 27 (million)
% 3.09/1.71 % (26746)------------------------------
% 3.09/1.71 % (26746)------------------------------
% 3.09/1.71 % (26754)Instruction limit reached!
% 3.09/1.71 % (26754)------------------------------
% 3.09/1.71 % (26754)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.71 % (26754)Termination reason: Unknown
% 3.09/1.71 % (26754)Termination phase: shuffling
% 3.09/1.71
% 3.09/1.71 % (26754)Memory used [KB]: 1535
% 3.09/1.71 % (26754)Time elapsed: 0.003 s
% 3.09/1.71 % (26754)Instructions burned: 4 (million)
% 3.09/1.71 % (26754)------------------------------
% 3.09/1.71 % (26754)------------------------------
% 3.09/1.71 % (26753)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_26468 for (2999ds/15Mi)
% 3.09/1.71 % (26756)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_26468 for (2999ds/7Mi)
% 3.09/1.72 % (26756)Instruction limit reached!
% 3.09/1.72 % (26756)------------------------------
% 3.09/1.72 % (26756)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.72 % (26756)Termination reason: Unknown
% 3.09/1.72 % (26756)Termination phase: shuffling
% 3.09/1.72
% 3.09/1.72 % (26756)Memory used [KB]: 1663
% 3.09/1.72 % (26756)Time elapsed: 0.004 s
% 3.09/1.72 % (26756)Instructions burned: 9 (million)
% 3.09/1.72 % (26756)------------------------------
% 3.09/1.72 % (26756)------------------------------
% 3.09/1.72 % (26758)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_26468 for (2999ds/3Mi)
% 3.09/1.72 % (26758)Instruction limit reached!
% 3.09/1.72 % (26758)------------------------------
% 3.09/1.72 % (26758)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.72 % (26758)Termination reason: Unknown
% 3.09/1.72 % (26758)Termination phase: shuffling
% 3.09/1.72
% 3.09/1.72 % (26758)Memory used [KB]: 1535
% 3.09/1.72 % (26758)Time elapsed: 0.003 s
% 3.09/1.72 % (26758)Instructions burned: 4 (million)
% 3.09/1.72 % (26758)------------------------------
% 3.09/1.72 % (26758)------------------------------
% 3.09/1.72 % (26753)Instruction limit reached!
% 3.09/1.72 % (26753)------------------------------
% 3.09/1.72 % (26753)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.72 % (26753)Termination reason: Unknown
% 3.09/1.72 % (26753)Termination phase: shuffling
% 3.09/1.72
% 3.09/1.72 % (26753)Memory used [KB]: 1791
% 3.09/1.72 % (26753)Time elapsed: 0.012 s
% 3.09/1.72 % (26753)Instructions burned: 15 (million)
% 3.09/1.72 % (26753)------------------------------
% 3.09/1.72 % (26753)------------------------------
% 3.09/1.72 % (26755)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_26468 for (2999ds/1041Mi)
% 3.09/1.72 % (26757)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_26468 for (2999ds/16Mi)
% 3.09/1.73 % (26759)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_26468 for (2999ds/3Mi)
% 3.09/1.73 % (26759)Instruction limit reached!
% 3.09/1.73 % (26759)------------------------------
% 3.09/1.73 % (26759)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.73 % (26759)Termination reason: Unknown
% 3.09/1.73 % (26759)Termination phase: shuffling
% 3.09/1.73
% 3.09/1.73 % (26759)Memory used [KB]: 1535
% 3.09/1.73 % (26759)Time elapsed: 0.003 s
% 3.09/1.73 % (26759)Instructions burned: 5 (million)
% 3.09/1.73 % (26759)------------------------------
% 3.09/1.73 % (26759)------------------------------
% 3.09/1.73 % (26760)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_26468 for (2999ds/7Mi)
% 3.09/1.73 % (26757)Instruction limit reached!
% 3.09/1.73 % (26757)------------------------------
% 3.09/1.73 % (26757)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.73 % (26757)Termination reason: Unknown
% 3.09/1.73 % (26757)Termination phase: shuffling
% 3.09/1.73
% 3.09/1.73 % (26757)Memory used [KB]: 1791
% 3.09/1.73 % (26757)Time elapsed: 0.008 s
% 3.09/1.73 % (26757)Instructions burned: 16 (million)
% 3.09/1.73 % (26757)------------------------------
% 3.09/1.73 % (26757)------------------------------
% 3.09/1.73 % (26760)Instruction limit reached!
% 3.09/1.73 % (26760)------------------------------
% 3.09/1.73 % (26760)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.73 % (26760)Termination reason: Unknown
% 3.09/1.73 % (26760)Termination phase: shuffling
% 3.09/1.73
% 3.09/1.73 % (26760)Memory used [KB]: 1663
% 3.09/1.73 % (26760)Time elapsed: 0.005 s
% 3.09/1.73 % (26760)Instructions burned: 9 (million)
% 3.09/1.73 % (26760)------------------------------
% 3.09/1.73 % (26760)------------------------------
% 3.09/1.74 % (26761)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_26468 for (2999ds/3Mi)
% 3.09/1.74 % (26762)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_26468 for (2999ds/4Mi)
% 3.09/1.74 % (26761)Instruction limit reached!
% 3.09/1.74 % (26761)------------------------------
% 3.09/1.74 % (26761)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.74 % (26761)Termination reason: Unknown
% 3.09/1.74 % (26761)Termination phase: shuffling
% 3.09/1.74
% 3.09/1.74 % (26761)Memory used [KB]: 1535
% 3.09/1.74 % (26761)Time elapsed: 0.004 s
% 3.09/1.74 % (26761)Instructions burned: 4 (million)
% 3.09/1.74 % (26761)------------------------------
% 3.09/1.74 % (26761)------------------------------
% 3.09/1.74 % (26763)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_26468 for (2999ds/18Mi)
% 3.09/1.74 % (26762)Instruction limit reached!
% 3.09/1.74 % (26762)------------------------------
% 3.09/1.74 % (26762)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.74 % (26762)Termination reason: Unknown
% 3.09/1.74 % (26762)Termination phase: shuffling
% 3.09/1.74
% 3.09/1.74 % (26762)Memory used [KB]: 1535
% 3.09/1.74 % (26762)Time elapsed: 0.003 s
% 3.09/1.74 % (26762)Instructions burned: 6 (million)
% 3.09/1.74 % (26762)------------------------------
% 3.09/1.74 % (26762)------------------------------
% 3.09/1.74 % (26749)First to succeed.
% 3.09/1.74 % (26764)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_26468 for (2999ds/710Mi)
% 3.09/1.74 % (26749)Refutation found. Thanks to Tanya!
% 3.09/1.74 % SZS status Theorem for DTF2THF_26468
% 3.09/1.74 % SZS output start Proof for DTF2THF_26468
% 3.09/1.74 thf(type_def_5, type, product_prod: ($tType * $tType) > $tType).
% 3.09/1.74 thf(type_def_6, type, set: ($tType) > $tType).
% 3.09/1.74 thf(type_def_8, type, formula: ($tType * $tType * $tType) > $tType).
% 3.09/1.74 thf(type_def_9, type, hp: ($tType * $tType * $tType) > $tType).
% 3.09/1.74 thf(type_def_10, type, product_unit: $tType).
% 3.09/1.74 thf(type_def_11, type, denota1663640101rp_ext: ($tType * $tType * $tType * $tType) > $tType).
% 3.09/1.74 thf(type_def_12, type, sum_sum: ($tType * $tType) > $tType).
% 3.09/1.74 thf(type_def_13, type, ode: ($tType * $tType) > $tType).
% 3.09/1.74 thf(type_def_14, type, real: $tType).
% 3.09/1.74 thf(type_def_15, type, finite_Cartesian_vec: ($tType * $tType) > $tType).
% 3.09/1.74 thf(type_def_16, type, filter: ($tType) > $tType).
% 3.09/1.74 thf(type_def_17, type, a: $tType).
% 3.09/1.74 thf(type_def_18, type, b: $tType).
% 3.09/1.74 thf(type_def_19, type, c: $tType).
% 3.09/1.74 thf(type_def_20, type, list: ($tType) > $tType).
% 3.09/1.74 thf(type_def_21, type, trm: ($tType * $tType) > $tType).
% 3.09/1.74 thf(type_def_22, type, d: $tType).
% 3.09/1.74 thf(func_def_0, type, denota1663640101rp_ext: ($tType * $tType * $tType * $tType) > $tType).
% 3.09/1.74 thf(func_def_1, type, finite_Cartesian_vec: ($tType * $tType) > $tType).
% 3.09/1.74 thf(func_def_2, type, product_unit: $tType).
% 3.09/1.74 thf(func_def_3, type, product_prod: ($tType * $tType) > $tType).
% 3.09/1.74 thf(func_def_4, type, formula: ($tType * $tType * $tType) > $tType).
% 3.09/1.74 thf(func_def_5, type, sum_sum: ($tType * $tType) > $tType).
% 3.09/1.74 thf(func_def_6, type, filter: ($tType) > $tType).
% 3.09/1.74 thf(func_def_7, type, trm: ($tType * $tType) > $tType).
% 3.09/1.74 thf(func_def_8, type, ode: ($tType * $tType) > $tType).
% 3.09/1.74 thf(func_def_9, type, hp: ($tType * $tType * $tType) > $tType).
% 3.09/1.74 thf(func_def_10, type, real: $tType).
% 3.09/1.74 thf(func_def_11, type, list: ($tType) > $tType).
% 3.09/1.74 thf(func_def_12, type, set: ($tType) > $tType).
% 3.09/1.74 thf(func_def_13, type, d: $tType).
% 3.09/1.74 thf(func_def_14, type, c: $tType).
% 3.09/1.74 thf(func_def_15, type, b: $tType).
% 3.09/1.74 thf(func_def_16, type, a: $tType).
% 3.09/1.74 thf(func_def_17, type, cARD_1: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_18, type, ordere890947078_space: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_19, type, type: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_20, type, zero: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_21, type, uminus: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_22, type, ord: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_23, type, order: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_24, type, group_add: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_25, type, no_bot: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_26, type, no_top: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_27, type, finite_finite: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_28, type, linorder: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_29, type, preorder: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_30, type, boolean_algebra: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_31, type, semilattice_sup: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_32, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_33, type, linord219039673up_add: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_34, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_35, type, topolo503727757_space: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_36, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_37, type, lattic1601792062up_add: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_38, type, real_V55928688vector: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_39, type, euclid1678468529ng_gcd: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_40, type, unique1598680935umeral: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_41, type, euclid1155270486miring: !>[X0: $tType]:($o)).
% 3.09/1.74 thf(func_def_42, type, bNF_Ca1785829860lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 3.09/1.74 thf(func_def_43, type, coinci1993344360de_fml: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > $o)).
% 3.09/1.74 thf(func_def_44, type, coinci32832645ide_hp: !>[X0: $tType, X1: $tType, X2: $tType]:(hp @ X0 @ X1 @ X2 > denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > $o)).
% 3.09/1.74 thf(func_def_45, type, coinci554967470ide_hp: !>[X0: $tType, X1: $tType, X2: $tType]:(hp @ X0 @ X1 @ X2 > $o)).
% 3.09/1.74 thf(func_def_46, type, coinci1495174342_equiv: !>[X0: $tType, X1: $tType, X2: $tType]:(hp @ X0 @ X1 @ X2 > denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > $o)).
% 3.09/1.74 thf(func_def_47, type, denotational_Iagree: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > set @ sum_sum @ X0 @ sum_sum @ X1 @ X2 > $o)).
% 3.09/1.74 thf(func_def_48, type, denotational_ODE_sem: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > ode @ X0 @ X2 > finite_Cartesian_vec @ real @ X2 > finite_Cartesian_vec @ real @ X2)).
% 3.09/1.74 thf(func_def_49, type, denota811733865E_vars: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > ode @ X0 @ X2 > set @ X2)).
% 3.09/1.74 thf(func_def_50, type, denotational_VSagree: !>[X0: $tType]:(finite_Cartesian_vec @ real @ X0 > finite_Cartesian_vec @ real @ X0 > set @ X0 > $o)).
% 3.09/1.74 thf(func_def_51, type, denotational_Vagree: !>[X0: $tType]:(product_prod @ finite_Cartesian_vec @ real @ X0 @ finite_Cartesian_vec @ real @ X0 > product_prod @ finite_Cartesian_vec @ real @ X0 @ finite_Cartesian_vec @ real @ X0 > set @ sum_sum @ X0 @ X0 > $o)).
% 3.09/1.74 thf(func_def_52, type, denotational_fml_sem: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > formula @ X0 @ X1 @ X2 > set @ product_prod @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2)).
% 3.09/1.74 thf(func_def_53, type, denotational_mk_v: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > ode @ X0 @ X2 > product_prod @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2 > finite_Cartesian_vec @ real @ X2 > product_prod @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2)).
% 3.09/1.74 thf(func_def_54, type, denotational_mk_xode: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > ode @ X0 @ X2 > finite_Cartesian_vec @ real @ X2 > product_prod @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2)).
% 3.09/1.74 thf(func_def_55, type, denotational_semBV: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > ode @ X0 @ X2 > set @ sum_sum @ X2 @ X2)).
% 3.09/1.74 thf(func_def_56, type, has_ve2132708402vative: !>[X0: $tType]:((real > X0) > X0 > filter @ real > $o)).
% 3.09/1.74 thf(func_def_57, type, unique455577585es_aux: !>[X0: $tType]:(product_prod @ X0 @ X0 > $o)).
% 3.09/1.74 thf(func_def_58, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 3.09/1.74 thf(func_def_59, type, zero_zero: !>[X0: $tType]:(X0)).
% 3.09/1.74 thf(func_def_60, type, the: !>[X0: $tType]:((X0 > $o) > X0)).
% 3.09/1.74 thf(func_def_61, type, ids: !>[X0: $tType, X1: $tType, X2: $tType]:(X0 > X0 > X0 > X1 > X1 > X1 > X2 > X2 > X2 > X2 > $o)).
% 3.09/1.74 thf(func_def_62, type, initia1685620758es_ode: !>[X0: $tType]:((real > X0) > (real > X0 > X0) > set @ real > set @ X0 > $o)).
% 3.09/1.74 thf(func_def_63, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.09/1.74 thf(func_def_64, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.09/1.74 thf(func_def_65, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 3.09/1.74 thf(func_def_66, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.09/1.74 thf(func_def_67, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.09/1.74 thf(func_def_68, type, product_rec_set_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2 > $o)).
% 3.09/1.74 thf(func_def_69, type, product_rec_set_unit: !>[X0: $tType]:(X0 > product_unit > X0 > $o)).
% 3.09/1.74 thf(func_def_70, type, product_rec_unit: !>[X0: $tType]:(X0 > product_unit > X0)).
% 3.09/1.74 thf(func_def_71, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.09/1.74 thf(func_def_72, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 3.09/1.74 thf(func_def_73, type, set_ord_atLeast: !>[X0: $tType]:(X0 > set @ X0)).
% 3.09/1.74 thf(func_def_74, type, set_or331188842AtMost: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 3.09/1.74 thf(func_def_75, type, set_ord_atMost: !>[X0: $tType]:(X0 > set @ X0)).
% 3.09/1.74 thf(func_def_76, type, static_BVO: !>[X0: $tType, X1: $tType]:(ode @ X0 @ X1 > set @ sum_sum @ X1 @ X1)).
% 3.09/1.74 thf(func_def_77, type, static_FVF: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > set @ sum_sum @ X2 @ X2)).
% 3.09/1.74 thf(func_def_78, type, static_FVO: !>[X0: $tType, X1: $tType]:(ode @ X0 @ X1 > set @ X1)).
% 3.09/1.74 thf(func_def_79, type, static_SIGF: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > set @ sum_sum @ X0 @ sum_sum @ X1 @ X2)).
% 3.09/1.74 thf(func_def_80, type, sum_Inl: !>[X0: $tType, X1: $tType]:(X0 > sum_sum @ X0 @ X1)).
% 3.09/1.74 thf(func_def_81, type, sum_Inr: !>[X0: $tType, X1: $tType]:(X0 > sum_sum @ X1 @ X0)).
% 3.09/1.74 thf(func_def_82, type, equiv: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > formula @ X0 @ X1 @ X2 > formula @ X0 @ X1 @ X2)).
% 3.09/1.74 thf(func_def_83, type, implies: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > formula @ X0 @ X1 @ X2 > formula @ X0 @ X1 @ X2)).
% 3.09/1.74 thf(func_def_84, type, oDE_dom: !>[X0: $tType, X1: $tType]:(ode @ X0 @ X1 > set @ X1)).
% 3.09/1.74 thf(func_def_85, type, fsafe: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > $o)).
% 3.09/1.74 thf(func_def_86, type, evolveODE: !>[X0: $tType, X1: $tType, X2: $tType]:(ode @ X0 @ X1 > formula @ X0 @ X2 @ X1 > hp @ X0 @ X2 @ X1)).
% 3.09/1.74 thf(func_def_87, type, hpsafe: !>[X0: $tType, X1: $tType, X2: $tType]:(hp @ X0 @ X1 @ X2 > $o)).
% 3.09/1.74 thf(func_def_88, type, osafe: !>[X0: $tType, X1: $tType]:(ode @ X0 @ X1 > $o)).
% 3.09/1.74 thf(func_def_89, type, topolo507301023within: !>[X0: $tType]:(X0 > set @ X0 > filter @ X0)).
% 3.09/1.74 thf(func_def_90, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.09/1.74 thf(func_def_91, type, i: denota1663640101rp_ext @ a @ b @ c @ product_unit).
% 3.09/1.74 thf(func_def_92, type, j: denota1663640101rp_ext @ a @ b @ c @ product_unit).
% 3.09/1.74 thf(func_def_93, type, ode2: ode @ a @ c).
% 3.09/1.74 thf(func_def_94, type, p: formula @ a @ b @ c).
% 3.09/1.74 thf(func_def_95, type, v: set @ sum_sum @ c @ c).
% 3.09/1.74 thf(func_def_96, type, a2: finite_Cartesian_vec @ real @ c).
% 3.09/1.74 thf(func_def_97, type, aa: finite_Cartesian_vec @ real @ c).
% 3.09/1.74 thf(func_def_98, type, ab: finite_Cartesian_vec @ real @ c).
% 3.09/1.74 thf(func_def_99, type, b2: finite_Cartesian_vec @ real @ c).
% 3.09/1.74 thf(func_def_100, type, ba: finite_Cartesian_vec @ real @ c).
% 3.09/1.74 thf(func_def_101, type, bb: finite_Cartesian_vec @ real @ c).
% 3.09/1.74 thf(func_def_102, type, s: real).
% 3.09/1.74 thf(func_def_103, type, sol: real > finite_Cartesian_vec @ real @ c).
% 3.09/1.74 thf(func_def_104, type, t: real).
% 3.09/1.74 thf(func_def_120, type, sK0: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 3.09/1.74 thf(func_def_121, type, sK1: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 3.09/1.74 thf(func_def_122, type, sK2: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 3.09/1.74 thf(func_def_123, type, sK3: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 3.09/1.74 thf(func_def_124, type, sK4: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 3.09/1.74 thf(func_def_125, type, sK5: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 3.09/1.74 thf(func_def_126, type, sK6: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 3.09/1.74 thf(func_def_127, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.09/1.74 thf(func_def_128, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.09/1.74 thf(func_def_129, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.09/1.74 thf(func_def_130, type, sK10: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 3.09/1.74 thf(func_def_131, type, sK11: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 3.09/1.74 thf(func_def_132, type, sK12: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.09/1.74 thf(func_def_133, type, sK13: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.09/1.74 thf(func_def_134, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.09/1.74 thf(func_def_135, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.09/1.74 thf(func_def_136, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.09/1.74 thf(func_def_137, type, sK17: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.09/1.74 thf(func_def_138, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.09/1.74 thf(func_def_139, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.09/1.74 thf(func_def_140, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.09/1.74 thf(func_def_141, type, sK21: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.09/1.74 thf(func_def_142, type, sK22: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.09/1.74 thf(func_def_143, type, sK23: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.09/1.74 thf(func_def_144, type, sK24: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.09/1.74 thf(func_def_145, type, sK25: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 3.09/1.74 thf(func_def_146, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType]:(finite_Cartesian_vec @ real @ X1 > ode @ X2 @ X1 > denota1663640101rp_ext @ X2 @ X0 @ X1 @ product_unit > product_prod @ finite_Cartesian_vec @ real @ X1 @ finite_Cartesian_vec @ real @ X1 > product_prod @ finite_Cartesian_vec @ real @ X1 @ finite_Cartesian_vec @ real @ X1)).
% 3.09/1.74 thf(func_def_147, type, sK27: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 3.09/1.74 thf(func_def_148, type, sK28: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 3.09/1.74 thf(func_def_149, type, sK29: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 3.09/1.74 thf(func_def_150, type, sK30: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0 > $o) > X1)).
% 3.09/1.74 thf(func_def_151, type, sK31: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0 > $o) > X0)).
% 3.09/1.74 thf(func_def_152, type, sK32: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 3.09/1.74 thf(func_def_153, type, sK33: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 3.09/1.74 thf(func_def_154, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 3.09/1.74 thf(func_def_155, type, sK35: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 3.09/1.74 thf(func_def_156, type, sK36: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 3.09/1.74 thf(func_def_157, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 3.09/1.74 thf(func_def_158, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 3.09/1.74 thf(func_def_159, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 3.09/1.74 thf(func_def_160, type, sK40: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 3.09/1.74 thf(func_def_161, type, sK41: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 3.09/1.74 thf(func_def_162, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 3.09/1.74 thf(func_def_163, type, sK43: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.09/1.74 thf(func_def_164, type, sK44: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.09/1.74 thf(func_def_165, type, sK45: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.09/1.74 thf(func_def_166, type, sK46: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 3.09/1.74 thf(func_def_167, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.09/1.74 thf(func_def_168, type, sK48: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.09/1.74 thf(func_def_169, type, sK49: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.09/1.74 thf(func_def_170, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.09/1.74 thf(func_def_171, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.09/1.74 thf(func_def_172, type, sK52: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.09/1.74 thf(func_def_173, type, sK53: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.09/1.74 thf(func_def_174, type, sK54: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.09/1.74 thf(func_def_175, type, sK55: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.09/1.74 thf(func_def_176, type, sK56: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 3.09/1.74 thf(func_def_177, type, sK57: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 3.09/1.74 thf(func_def_178, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 3.09/1.74 thf(func_def_179, type, sK59: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 3.09/1.74 thf(func_def_180, type, sK60: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.09/1.74 thf(func_def_181, type, sK61: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.09/1.74 thf(func_def_182, type, sK62: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.09/1.74 thf(func_def_183, type, sK63: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.09/1.74 thf(func_def_184, type, sK64: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X1)).
% 3.09/1.74 thf(func_def_185, type, sK65: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X0)).
% 3.09/1.74 thf(func_def_187, type, ph67: !>[X0: $tType]:(X0)).
% 3.09/1.74 thf(f1218,plain,(
% 3.09/1.74 $false),
% 3.09/1.74 inference(subsumption_resolution,[status(thm)],[f1171,f1202])).
% 3.09/1.74 thf(f1202,plain,(
% 3.09/1.74 ((denotational_Vagree @ c @ (denotational_mk_v @ a @ b @ c @ j @ ode2 @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ aa @ ba) @ (sol @ s)) @ (denotational_mk_xode @ a @ b @ c @ j @ ode2 @ (sol @ s)) @ (denotational_semBV @ a @ b @ c @ j @ ode2)) = $true)),
% 3.09/1.74 inference(cnf_transformation,[status(thm)],[f673])).
% 3.09/1.74 thf(f673,plain,(
% 3.09/1.74 ((denotational_Vagree @ c @ (denotational_mk_v @ a @ b @ c @ j @ ode2 @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ aa @ ba) @ (sol @ s)) @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ aa @ ba) @ (uminus_uminus @ set @ sum_sum @ c @ c @ (denotational_semBV @ a @ b @ c @ j @ ode2))) = $true) & ((denotational_Vagree @ c @ (denotational_mk_v @ a @ b @ c @ j @ ode2 @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ aa @ ba) @ (sol @ s)) @ (denotational_mk_xode @ a @ b @ c @ j @ ode2 @ (sol @ s)) @ (denotational_semBV @ a @ b @ c @ j @ ode2)) = $true)),
% 3.09/1.74 inference(fool_elimination,[status(thm)],[f672])).
% 3.09/1.74 thf(f672,plain,(
% 3.09/1.74 (denotational_Vagree @ c @ (denotational_mk_v @ a @ b @ c @ j @ ode2 @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ aa @ ba) @ (sol @ s)) @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ aa @ ba) @ (uminus_uminus @ set @ sum_sum @ c @ c @ (denotational_semBV @ a @ b @ c @ j @ ode2))) & (denotational_Vagree @ c @ (denotational_mk_v @ a @ b @ c @ j @ ode2 @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ aa @ ba) @ (sol @ s)) @ (denotational_mk_xode @ a @ b @ c @ j @ ode2 @ (sol @ s)) @ (denotational_semBV @ a @ b @ c @ j @ ode2))),
% 3.09/1.74 inference(rectify,[status(thm)],[f2])).
% 3.09/1.74 thf(f2,axiom,(
% 3.09/1.74 (denotational_Vagree @ c @ (denotational_mk_v @ a @ b @ c @ j @ ode2 @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ aa @ ba) @ (sol @ s)) @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ aa @ ba) @ (uminus_uminus @ set @ sum_sum @ c @ c @ (denotational_semBV @ a @ b @ c @ j @ ode2))) & (denotational_Vagree @ c @ (denotational_mk_v @ a @ b @ c @ j @ ode2 @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ aa @ ba) @ (sol @ s)) @ (denotational_mk_xode @ a @ b @ c @ j @ ode2 @ (sol @ s)) @ (denotational_semBV @ a @ b @ c @ j @ ode2))),
% 3.09/1.74 file('/export/starexec/sandbox2/tmp/tmp.c1Zf8XeWV8/DTF2THF_26468.p',fact_1__092_060open_062Vagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Iaa_M_Aba_J_A_I_N_AsemBV_AJ_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Imk__xode_AJ_AODE_A_Isol_As_J_J_A_IsemBV_AJ_AODE_J_092_060close_062)).
% 3.09/1.74 thf(f1171,plain,(
% 3.09/1.74 ((denotational_Vagree @ c @ (denotational_mk_v @ a @ b @ c @ j @ ode2 @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ aa @ ba) @ (sol @ s)) @ (denotational_mk_xode @ a @ b @ c @ j @ ode2 @ (sol @ s)) @ (denotational_semBV @ a @ b @ c @ j @ ode2)) != $true)),
% 3.09/1.74 inference(cnf_transformation,[status(thm)],[f994])).
% 3.09/1.74 thf(f994,plain,(
% 3.09/1.74 ((denotational_Vagree @ c @ (denotational_mk_v @ a @ b @ c @ j @ ode2 @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ aa @ ba) @ (sol @ s)) @ (denotational_mk_xode @ a @ b @ c @ j @ ode2 @ (sol @ s)) @ (denotational_semBV @ a @ b @ c @ j @ ode2)) != $true)),
% 3.09/1.74 inference(flattening,[status(thm)],[f419])).
% 3.09/1.74 thf(f419,plain,(
% 3.09/1.74 ~((denotational_Vagree @ c @ (denotational_mk_v @ a @ b @ c @ j @ ode2 @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ aa @ ba) @ (sol @ s)) @ (denotational_mk_xode @ a @ b @ c @ j @ ode2 @ (sol @ s)) @ (denotational_semBV @ a @ b @ c @ j @ ode2)) = $true)),
% 3.09/1.74 inference(fool_elimination,[status(thm)],[f418])).
% 3.09/1.74 thf(f418,plain,(
% 3.09/1.74 ~(denotational_Vagree @ c @ (denotational_mk_v @ a @ b @ c @ j @ ode2 @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ aa @ ba) @ (sol @ s)) @ (denotational_mk_xode @ a @ b @ c @ j @ ode2 @ (sol @ s)) @ (denotational_semBV @ a @ b @ c @ j @ ode2))),
% 3.09/1.74 inference(rectify,[status(thm)],[f342])).
% 3.09/1.74 thf(f342,negated_conjecture,(
% 3.09/1.74 ~(denotational_Vagree @ c @ (denotational_mk_v @ a @ b @ c @ j @ ode2 @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ aa @ ba) @ (sol @ s)) @ (denotational_mk_xode @ a @ b @ c @ j @ ode2 @ (sol @ s)) @ (denotational_semBV @ a @ b @ c @ j @ ode2))),
% 3.09/1.74 inference(negated_conjecture,[status(cth)],[f341])).
% 3.09/1.74 thf(f341,conjecture,(
% 3.09/1.74 (denotational_Vagree @ c @ (denotational_mk_v @ a @ b @ c @ j @ ode2 @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ aa @ ba) @ (sol @ s)) @ (denotational_mk_xode @ a @ b @ c @ j @ ode2 @ (sol @ s)) @ (denotational_semBV @ a @ b @ c @ j @ ode2))),
% 3.09/1.74 file('/export/starexec/sandbox2/tmp/tmp.c1Zf8XeWV8/DTF2THF_26468.p',conj_0)).
% 3.09/1.74 % SZS output end Proof for DTF2THF_26468
% 3.09/1.74 % (26749)------------------------------
% 3.09/1.74 % (26749)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.74 % (26749)Termination reason: Refutation
% 3.09/1.74
% 3.09/1.74 % (26749)Memory used [KB]: 7164
% 3.09/1.74 % (26749)Time elapsed: 0.053 s
% 3.09/1.74 % (26749)Instructions burned: 83 (million)
% 3.09/1.74 % (26749)------------------------------
% 3.09/1.74 % (26749)------------------------------
% 3.09/1.74 % (26743)Success in time 0.084 s
%------------------------------------------------------------------------------