%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP040^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n015.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 3.24s 1.81s
% Output : Refutation 3.24s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP040^2 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n015.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:24:16 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.19/0.46 thf(spec,logic,$$dhol).
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % File : ITP040^2 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.46 % Domain : Interactive Theorem Proving
% 0.19/0.46 % Problem : Sledgehammer Coincidence problem prob_718__7224732_1
% 0.19/0.46 % Version : Especial.
% 0.19/0.46 % English :
% 0.19/0.46
% 0.19/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.46 % Source : [Des21]
% 0.19/0.46 % Names : Coincidence/prob_718__7224732_1 [Des21]
% 0.19/0.46
% 0.19/0.46 % Status : ContradictoryAxioms
% 0.19/0.46 % Rating : 0.33 v8.2.0, 0.00 v8.1.0, 0.25 v7.5.0
% 0.19/0.46 % Syntax : Number of formulae : 416 ( 108 unt; 83 typ; 0 def)
% 0.19/0.46 % Number of atoms : 987 ( 291 equ; 0 cnn)
% 0.19/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.19/0.46 % Number of connectives : 7700 ( 76 ~; 11 |; 113 &;7111 @)
% 0.19/0.46 % ( 0 <=>; 389 =>; 0 <=; 0 <~>)
% 0.19/0.46 % Maximal formula depth : 28 ( 8 avg)
% 0.19/0.46 % Number of types : 7 ( 6 usr)
% 0.19/0.47 % Number of type conns : 246 ( 246 >; 0 *; 0 +; 0 <<)
% 0.19/0.47 % Number of symbols : 80 ( 77 usr; 14 con; 0-7 aty)
% 0.19/0.47 % Number of variables : 1358 ( 155 ^;1095 !; 10 ?;1358 :)
% 0.19/0.47 % ( 98 !>; 0 ?*; 0 @-; 0 @+)
% 0.19/0.47 % SPC : TH1_CAX_EQU_NAR
% 0.19/0.47
% 0.19/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:27:45.025
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 % Could-be-implicit typings (13)
% 0.19/0.47 thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext,type,
% 0.19/0.47 denota1663640101rp_ext: $tType > $tType > $tType > $tType > $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_t_Finite__Cartesian__Product_Ovec,type,
% 0.19/0.47 finite_Cartesian_vec: $tType > $tType > $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_t_Product__Type_Ounit,type,
% 0.19/0.47 product_unit: $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_t_Product__Type_Oprod,type,
% 0.19/0.47 product_prod: $tType > $tType > $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_t_Syntax_Oformula,type,
% 0.19/0.47 formula: $tType > $tType > $tType > $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_t_Sum__Type_Osum,type,
% 0.19/0.47 sum_sum: $tType > $tType > $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_t_Syntax_OODE,type,
% 0.19/0.47 ode: $tType > $tType > $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_t_Real_Oreal,type,
% 0.19/0.47 real: $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_t_Set_Oset,type,
% 0.19/0.47 set: $tType > $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_tf_d,type,
% 0.19/0.47 d: $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_tf_c,type,
% 0.19/0.47 c: $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_tf_b,type,
% 0.19/0.47 b: $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_tf_a,type,
% 0.19/0.47 a: $tType ).
% 0.19/0.47
% 0.19/0.47 % Explicit typings (70)
% 0.19/0.47 thf(sy_cl_Cardinality_OCARD__1,type,
% 0.19/0.47 cARD_1:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
% 0.19/0.47 ordere890947078_space:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_HOL_Otype,type,
% 0.19/0.47 type:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Groups_Ozero,type,
% 0.19/0.47 zero:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Groups_Ouminus,type,
% 0.19/0.47 uminus:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.19/0.47 ord:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.19/0.47 order:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Groups_Ogroup__add,type,
% 0.19/0.47 group_add:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.19/0.47 finite_finite:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.19/0.47 linorder:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.19/0.47 preorder:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 0.19/0.47 boolean_algebra:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.19/0.47 ordered_ab_group_add:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.19/0.47 linord219039673up_add:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.19/0.47 canoni770627133id_add:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Lattice__Algebras_Olattice__ab__group__add,type,
% 0.19/0.47 lattic1601792062up_add:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd,type,
% 0.19/0.47 euclid1678468529ng_gcd:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring,type,
% 0.19/0.47 euclid1155270486miring:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__fml,type,
% 0.19/0.47 coinci1993344360de_fml:
% 0.19/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Denotational__Semantics_OODE__sem,type,
% 0.19/0.47 denotational_ODE_sem:
% 0.19/0.47 !>[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 ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Denotational__Semantics_OODE__vars,type,
% 0.19/0.47 denota811733865E_vars:
% 0.19/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( set @ C ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Denotational__Semantics_OVSagree,type,
% 0.19/0.47 denotational_VSagree:
% 0.19/0.47 !>[C: $tType] : ( ( finite_Cartesian_vec @ real @ C ) > ( finite_Cartesian_vec @ real @ C ) > ( set @ C ) > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Denotational__Semantics_OVagree,type,
% 0.19/0.47 denotational_Vagree:
% 0.19/0.47 !>[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 ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Denotational__Semantics_Oconcrete__v,type,
% 0.19/0.47 denota1738237502rete_v:
% 0.19/0.47 !>[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 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Denotational__Semantics_Oconcrete__v__rel,type,
% 0.19/0.47 denota2028625979_v_rel:
% 0.19/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) > ( product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Denotational__Semantics_Ofml__sem,type,
% 0.19/0.47 denotational_fml_sem:
% 0.19/0.47 !>[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 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Denotational__Semantics_Omk__v,type,
% 0.19/0.47 denotational_mk_v:
% 0.19/0.47 !>[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 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Denotational__Semantics_Omk__xode,type,
% 0.19/0.47 denotational_mk_xode:
% 0.19/0.47 !>[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 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Denotational__Semantics_Orepd,type,
% 0.19/0.47 denotational_repd:
% 0.19/0.47 !>[C: $tType] : ( ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > C > real > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Denotational__Semantics_Orepv,type,
% 0.19/0.47 denotational_repv:
% 0.19/0.47 !>[C: $tType] : ( ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > C > real > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Denotational__Semantics_OsemBV,type,
% 0.19/0.47 denotational_semBV:
% 0.19/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Finite__Cartesian__Product_Omap__matrix,type,
% 0.19/0.47 finite1614961371matrix:
% 0.19/0.47 !>[A: $tType,B: $tType,I: $tType,J: $tType] : ( ( A > B ) > ( finite_Cartesian_vec @ ( finite_Cartesian_vec @ A @ I ) @ J ) > ( finite_Cartesian_vec @ ( finite_Cartesian_vec @ B @ I ) @ J ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__lambda,type,
% 0.19/0.47 finite1990238425lambda:
% 0.19/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( finite_Cartesian_vec @ A @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__nth,type,
% 0.19/0.47 finite1433825200ec_nth:
% 0.19/0.47 !>[A: $tType,B: $tType] : ( ( finite_Cartesian_vec @ A @ B ) > B > A ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_FuncSet_OPi,type,
% 0.19/0.47 pi:
% 0.19/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( A > B ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.19/0.47 uminus_uminus:
% 0.19/0.47 !>[A: $tType] : ( A > A ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.19/0.47 zero_zero:
% 0.19/0.47 !>[A: $tType] : A ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_If,type,
% 0.19/0.47 if:
% 0.19/0.47 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Initial__Value__Problem_Osolves__ode,type,
% 0.19/0.47 initia1685620758es_ode:
% 0.19/0.47 !>[A: $tType] : ( ( real > A ) > ( real > A > A ) > ( set @ real ) > ( set @ A ) > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.19/0.47 ord_less_eq:
% 0.19/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Product__Type_OPair,type,
% 0.19/0.47 product_Pair:
% 0.19/0.47 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.19/0.47 product_rec_prod:
% 0.19/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.19/0.47 product_fst:
% 0.19/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.19/0.47 product_snd:
% 0.19/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Set_OCollect,type,
% 0.19/0.47 collect:
% 0.19/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Set_Oimage,type,
% 0.19/0.47 image:
% 0.19/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
% 0.19/0.47 set_or331188842AtMost:
% 0.19/0.47 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Static__Semantics_OBVO,type,
% 0.19/0.47 static_BVO:
% 0.19/0.47 !>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Static__Semantics_OFVF,type,
% 0.19/0.47 static_FVF:
% 0.19/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Static__Semantics_OFVO,type,
% 0.19/0.47 static_FVO:
% 0.19/0.47 !>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > ( set @ C ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Sum__Type_OInl,type,
% 0.19/0.47 sum_Inl:
% 0.19/0.47 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Sum__Type_OInr,type,
% 0.19/0.47 sum_Inr:
% 0.19/0.47 !>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Syntax_OEquiv,type,
% 0.19/0.47 equiv:
% 0.19/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Syntax_Ofsafe,type,
% 0.19/0.47 fsafe:
% 0.19/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Syntax_Oosafe,type,
% 0.19/0.47 osafe:
% 0.19/0.47 !>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Wellfounded_Oaccp,type,
% 0.19/0.47 accp:
% 0.19/0.47 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_member,type,
% 0.19/0.47 member:
% 0.19/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_v_I____,type,
% 0.19/0.47 i: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
% 0.19/0.47
% 0.19/0.47 thf(sy_v_ODE____,type,
% 0.19/0.47 ode2: ode @ a @ c ).
% 0.19/0.47
% 0.19/0.47 thf(sy_v_ODEa____,type,
% 0.19/0.47 oDEa: ode @ a @ c ).
% 0.19/0.47
% 0.19/0.47 thf(sy_v_P____,type,
% 0.19/0.47 p: formula @ a @ b @ c ).
% 0.19/0.47
% 0.19/0.47 thf(sy_v__092_060phi_062_H____,type,
% 0.19/0.47 phi: formula @ a @ b @ c ).
% 0.19/0.47
% 0.19/0.47 thf(sy_v_aa____,type,
% 0.19/0.47 aa: finite_Cartesian_vec @ real @ c ).
% 0.19/0.47
% 0.19/0.47 thf(sy_v_ab____,type,
% 0.19/0.47 ab: finite_Cartesian_vec @ real @ c ).
% 0.19/0.47
% 0.19/0.47 thf(sy_v_ba____,type,
% 0.19/0.47 ba: finite_Cartesian_vec @ real @ c ).
% 0.19/0.47
% 0.19/0.47 thf(sy_v_bb____,type,
% 0.19/0.47 bb: finite_Cartesian_vec @ real @ c ).
% 0.19/0.47
% 0.19/0.47 thf(sy_v_i____,type,
% 0.19/0.47 i2: c ).
% 0.19/0.47
% 0.19/0.47 thf(sy_v_sol____,type,
% 0.19/0.47 sol: real > ( finite_Cartesian_vec @ real @ c ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_v_t____,type,
% 0.19/0.47 t: real ).
% 0.19/0.47
% 0.19/0.47 thf(sy_v_x____,type,
% 0.19/0.47 x: real ).
% 0.19/0.47
% 0.19/0.47 % Relevant facts (255)
% 0.19/0.47 thf(fact_0_ode__to__fvo,axiom,
% 0.19/0.47 ! [C: $tType,A: $tType,B: $tType,X: A,I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A] :
% 0.19/0.47 ( ( member @ A @ X @ ( denota811733865E_vars @ B @ C @ A @ I2 @ ODE ) )
% 0.19/0.47 => ( member @ A @ X @ ( static_FVO @ B @ A @ ODE ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ode_to_fvo
% 0.19/0.47 thf(fact_1_osafe,axiom,
% 0.19/0.47 osafe @ a @ c @ oDEa ).
% 0.19/0.47
% 0.19/0.47 % osafe
% 0.19/0.47 thf(fact_2_eqP,axiom,
% 0.19/0.47 p = phi ).
% 0.19/0.47
% 0.19/0.47 % eqP
% 0.19/0.47 thf(fact_3_x,axiom,
% 0.19/0.47 ord_less_eq @ real @ ( zero_zero @ real ) @ x ).
% 0.19/0.47
% 0.19/0.47 % x
% 0.19/0.47 thf(fact_4_fsafe,axiom,
% 0.19/0.47 fsafe @ a @ b @ c @ phi ).
% 0.19/0.47
% 0.19/0.47 % fsafe
% 0.19/0.47 thf(fact_5__092_060open_062_IInl_Ai_A_092_060in_062_AsemBV_AI_AODE_J_A_061_A_IInr_Ai_A_092_060in_062_AsemBV_AI_AODE_J_092_060close_062,axiom,
% 0.19/0.47 ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) )
% 0.19/0.47 = ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % \<open>(Inl i \<in> semBV I ODE) = (Inr i \<in> semBV I ODE)\<close>
% 0.19/0.47 thf(fact_6_VA,axiom,
% 0.19/0.47 ! [X2: real] :
% 0.19/0.47 ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ X2 ) )
% 0.19/0.47 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb )
% 0.19/0.47 @ ( finite1990238425lambda @ c @ real
% 0.19/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ X2 ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
% 0.19/0.47 @ ( static_FVF @ a @ b @ c @ phi ) ) ).
% 0.19/0.47
% 0.19/0.47 % VA
% 0.19/0.47 thf(fact_7_prod_Ocollapse,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.19/0.47 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.19/0.47 = Prod ) ).
% 0.19/0.47
% 0.19/0.47 % prod.collapse
% 0.19/0.47 thf(fact_8_bvo__to__fvo,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,X: A,ODE: ode @ B @ A] :
% 0.19/0.47 ( ( member @ ( sum_sum @ A @ A ) @ ( sum_Inl @ A @ A @ X ) @ ( static_BVO @ B @ A @ ODE ) )
% 0.19/0.47 => ( member @ A @ X @ ( static_FVO @ B @ A @ ODE ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bvo_to_fvo
% 0.19/0.47 thf(fact_9_vec__lambda__eta,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( finite_finite @ B )
% 0.19/0.47 => ! [G: finite_Cartesian_vec @ A @ B] :
% 0.19/0.47 ( ( finite1990238425lambda @ B @ A @ ( finite1433825200ec_nth @ A @ B @ G ) )
% 0.19/0.47 = G ) ) ).
% 0.19/0.47
% 0.19/0.47 % vec_lambda_eta
% 0.19/0.47 thf(fact_10_Inl__Inr__False,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,X: A,Y: B] :
% 0.19/0.47 ( ( sum_Inl @ A @ B @ X )
% 0.19/0.47 != ( sum_Inr @ B @ A @ Y ) ) ).
% 0.19/0.47
% 0.19/0.47 % Inl_Inr_False
% 0.19/0.47 thf(fact_11_Inr__Inl__False,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,X: B,Y: A] :
% 0.19/0.47 ( ( sum_Inr @ B @ A @ X )
% 0.19/0.47 != ( sum_Inl @ A @ B @ Y ) ) ).
% 0.19/0.47
% 0.19/0.47 % Inr_Inl_False
% 0.19/0.47 thf(fact_12_vec__lambda__beta,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( finite_finite @ B )
% 0.19/0.47 => ! [G: B > A,I4: B] :
% 0.19/0.47 ( ( finite1433825200ec_nth @ A @ B @ ( finite1990238425lambda @ B @ A @ G ) @ I4 )
% 0.19/0.47 = ( G @ I4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % vec_lambda_beta
% 0.19/0.47 thf(fact_13_alt__sem__lemma,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,C: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ C )
% 0.19/0.47 & ( finite_finite @ A )
% 0.19/0.47 & ( finite_finite @ B ) )
% 0.19/0.47 => ! [ODE: ode @ A @ C,I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,Sol: real > ( finite_Cartesian_vec @ real @ C ),T2: real,Ab: finite_Cartesian_vec @ real @ C] :
% 0.19/0.47 ( ( osafe @ A @ C @ ODE )
% 0.19/0.47 => ( ( denotational_ODE_sem @ A @ B @ C @ I2 @ ODE @ ( Sol @ T2 ) )
% 0.19/0.47 = ( denotational_ODE_sem @ A @ B @ C @ I2 @ ODE
% 0.19/0.47 @ ( finite1990238425lambda @ C @ real
% 0.19/0.47 @ ^ [I3: C] : ( if @ real @ ( member @ C @ I3 @ ( static_FVO @ A @ C @ ODE ) ) @ ( finite1433825200ec_nth @ real @ C @ ( Sol @ T2 ) @ I3 ) @ ( finite1433825200ec_nth @ real @ C @ Ab @ I3 ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % alt_sem_lemma
% 0.19/0.47 thf(fact_14_snd__zero,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ( zero @ A )
% 0.19/0.47 & ( zero @ B ) )
% 0.19/0.47 => ( ( product_snd @ B @ A @ ( zero_zero @ ( product_prod @ B @ A ) ) )
% 0.19/0.47 = ( zero_zero @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % snd_zero
% 0.19/0.47 thf(fact_15_fst__zero,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ( zero @ A )
% 0.19/0.47 & ( zero @ B ) )
% 0.19/0.47 => ( ( product_fst @ A @ B @ ( zero_zero @ ( product_prod @ A @ B ) ) )
% 0.19/0.47 = ( zero_zero @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % fst_zero
% 0.19/0.47 thf(fact_16_zero__index,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ( zero @ A )
% 0.19/0.47 & ( finite_finite @ B ) )
% 0.19/0.47 => ! [I4: B] :
% 0.19/0.47 ( ( finite1433825200ec_nth @ A @ B @ ( zero_zero @ ( finite_Cartesian_vec @ A @ B ) ) @ I4 )
% 0.19/0.47 = ( zero_zero @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % zero_index
% 0.19/0.47 thf(fact_17_hpsafe__Evolve_OIH,axiom,
% 0.19/0.47 coinci1993344360de_fml @ a @ b @ c @ p ).
% 0.19/0.47
% 0.19/0.47 % hpsafe_Evolve.IH
% 0.19/0.47 thf(fact_18_hpsafe__Evolve_Ohyps_I1_J,axiom,
% 0.19/0.47 osafe @ a @ c @ ode2 ).
% 0.19/0.47
% 0.19/0.47 % hpsafe_Evolve.hyps(1)
% 0.19/0.47 thf(fact_19_hpsafe__Evolve_Ohyps_I2_J,axiom,
% 0.19/0.47 fsafe @ a @ b @ c @ p ).
% 0.19/0.47
% 0.19/0.47 % hpsafe_Evolve.hyps(2)
% 0.19/0.47 thf(fact_20_old_Oprod_Oinject,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.19/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.19/0.47 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.19/0.47 = ( ( A2 = A3 )
% 0.19/0.47 & ( B2 = B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % old.prod.inject
% 0.19/0.47 thf(fact_21_prod_Oinject,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 0.19/0.47 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.19/0.47 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.19/0.47 = ( ( X1 = Y1 )
% 0.19/0.47 & ( X22 = Y2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod.inject
% 0.19/0.47 thf(fact_22_t,axiom,
% 0.19/0.47 ord_less_eq @ real @ x @ t ).
% 0.19/0.47
% 0.19/0.47 % t
% 0.19/0.47 thf(fact_23_all,axiom,
% 0.19/0.47 ! [I5: c] :
% 0.19/0.47 ( ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ I5 ) @ ( static_BVO @ a @ c @ oDEa ) )
% 0.19/0.47 => ( ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I5 )
% 0.19/0.47 = ( finite1433825200ec_nth @ real @ c @ ab @ I5 ) ) )
% 0.19/0.47 & ( ( member @ ( sum_sum @ c @ d ) @ ( sum_Inl @ c @ d @ I5 ) @ ( image @ c @ ( sum_sum @ c @ d ) @ ( sum_Inl @ c @ d ) @ ( static_FVO @ a @ c @ oDEa ) ) )
% 0.19/0.47 => ( ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I5 )
% 0.19/0.47 = ( finite1433825200ec_nth @ real @ c @ ab @ I5 ) ) )
% 0.19/0.47 & ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ I5 ) @ ( static_FVF @ a @ b @ c @ phi ) )
% 0.19/0.47 => ( ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I5 )
% 0.19/0.47 = ( finite1433825200ec_nth @ real @ c @ ab @ I5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % all
% 0.19/0.47 thf(fact_24_less__eq__vec__def,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ( ord @ A )
% 0.19/0.47 & ( finite_finite @ B ) )
% 0.19/0.47 => ( ( ord_less_eq @ ( finite_Cartesian_vec @ A @ B ) )
% 0.19/0.47 = ( ^ [X3: finite_Cartesian_vec @ A @ B,Y3: finite_Cartesian_vec @ A @ B] :
% 0.19/0.47 ! [I3: B] : ( ord_less_eq @ A @ ( finite1433825200ec_nth @ A @ B @ X3 @ I3 ) @ ( finite1433825200ec_nth @ A @ B @ Y3 @ I3 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_eq_vec_def
% 0.19/0.47 thf(fact_25_old_Oprod_Oinducts,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.19/0.47 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.19/0.47 => ( P @ Prod ) ) ).
% 0.19/0.47
% 0.19/0.47 % old.prod.inducts
% 0.19/0.47 thf(fact_26_old_Oprod_Oexhaust,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.19/0.47 ~ ! [A4: A,B4: B] :
% 0.19/0.47 ( Y
% 0.19/0.47 != ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% 0.19/0.47
% 0.19/0.47 % old.prod.exhaust
% 0.19/0.47 thf(fact_27_prod__induct7,axiom,
% 0.19/0.47 ! [G2: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) )] :
% 0.19/0.47 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G2 ) @ E2 @ ( product_Pair @ F @ G2 @ F2 @ G3 ) ) ) ) ) ) )
% 0.19/0.47 => ( P @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_induct7
% 0.19/0.47 thf(fact_28_prod__induct6,axiom,
% 0.19/0.47 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.19/0.47 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( 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 ) ) ) ) ) )
% 0.19/0.47 => ( P @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_induct6
% 0.19/0.47 thf(fact_29_prod__induct5,axiom,
% 0.19/0.47 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.19/0.47 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.19/0.47 => ( P @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_induct5
% 0.19/0.47 thf(fact_30_prod__induct4,axiom,
% 0.19/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.19/0.47 ( ! [A4: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 0.19/0.47 => ( P @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_induct4
% 0.19/0.47 thf(fact_31_prod__induct3,axiom,
% 0.19/0.47 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.19/0.47 ( ! [A4: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 0.19/0.47 => ( P @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_induct3
% 0.19/0.47 thf(fact_32_prod__cases7,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) )] :
% 0.19/0.47 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G3: G2] :
% 0.19/0.47 ( Y
% 0.19/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G2 ) @ E2 @ ( product_Pair @ F @ G2 @ F2 @ G3 ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases7
% 0.19/0.47 thf(fact_33_prod__cases6,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.19/0.47 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
% 0.19/0.47 ( Y
% 0.19/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( 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 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases6
% 0.19/0.47 thf(fact_34_prod__cases5,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.19/0.47 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
% 0.19/0.47 ( Y
% 0.19/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases5
% 0.19/0.47 thf(fact_35_prod__cases4,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.19/0.47 ~ ! [A4: A,B4: B,C2: C,D2: D] :
% 0.19/0.47 ( Y
% 0.19/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases4
% 0.19/0.47 thf(fact_36_prod__cases3,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.19/0.47 ~ ! [A4: A,B4: B,C2: C] :
% 0.19/0.47 ( Y
% 0.19/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases3
% 0.19/0.47 thf(fact_37_Pair__inject,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.19/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.19/0.47 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.19/0.47 => ~ ( ( A2 = A3 )
% 0.19/0.47 => ( B2 != B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Pair_inject
% 0.19/0.47 thf(fact_38_prod__cases,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.19/0.47 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.19/0.47 => ( P @ P2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases
% 0.19/0.47 thf(fact_39_surj__pair,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.19/0.47 ? [X4: A,Y4: B] :
% 0.19/0.47 ( P2
% 0.19/0.47 = ( product_Pair @ A @ B @ X4 @ Y4 ) ) ).
% 0.19/0.47
% 0.19/0.47 % surj_pair
% 0.19/0.47 thf(fact_40_vec__nth__inject,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( finite_finite @ B )
% 0.19/0.47 => ! [X: finite_Cartesian_vec @ A @ B,Y: finite_Cartesian_vec @ A @ B] :
% 0.19/0.47 ( ( ( finite1433825200ec_nth @ A @ B @ X )
% 0.19/0.47 = ( finite1433825200ec_nth @ A @ B @ Y ) )
% 0.19/0.47 = ( X = Y ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % vec_nth_inject
% 0.19/0.47 thf(fact_41_cond__component,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( finite_finite @ B )
% 0.19/0.47 => ! [B2: $o,X: finite_Cartesian_vec @ A @ B,Y: finite_Cartesian_vec @ A @ B,I4: B] :
% 0.19/0.47 ( ( B2
% 0.19/0.47 => ( ( finite1433825200ec_nth @ A @ B @ ( if @ ( finite_Cartesian_vec @ A @ B ) @ B2 @ X @ Y ) @ I4 )
% 0.19/0.47 = ( finite1433825200ec_nth @ A @ B @ X @ I4 ) ) )
% 0.19/0.47 & ( ~ B2
% 0.19/0.47 => ( ( finite1433825200ec_nth @ A @ B @ ( if @ ( finite_Cartesian_vec @ A @ B ) @ B2 @ X @ Y ) @ I4 )
% 0.19/0.47 = ( finite1433825200ec_nth @ A @ B @ Y @ I4 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % cond_component
% 0.19/0.47 thf(fact_42_vec__eq__iff,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( finite_finite @ B )
% 0.19/0.47 => ( ( ^ [Y5: finite_Cartesian_vec @ A @ B,Z: finite_Cartesian_vec @ A @ B] : ( Y5 = Z ) )
% 0.19/0.47 = ( ^ [X3: finite_Cartesian_vec @ A @ B,Y3: finite_Cartesian_vec @ A @ B] :
% 0.19/0.47 ! [I3: B] :
% 0.19/0.47 ( ( finite1433825200ec_nth @ A @ B @ X3 @ I3 )
% 0.19/0.47 = ( finite1433825200ec_nth @ A @ B @ Y3 @ I3 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % vec_eq_iff
% 0.19/0.47 thf(fact_43_fun__cong__unused__0,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType] :
% 0.19/0.47 ( ( zero @ B )
% 0.19/0.47 => ! [F3: ( A > B ) > C,G: C] :
% 0.19/0.47 ( ( F3
% 0.19/0.47 = ( ^ [X3: A > B] : G ) )
% 0.19/0.47 => ( ( F3
% 0.19/0.47 @ ^ [X3: A] : ( zero_zero @ B ) )
% 0.19/0.47 = G ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % fun_cong_unused_0
% 0.19/0.47 thf(fact_44_zero__prod__def,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( zero @ B )
% 0.19/0.47 & ( zero @ A ) )
% 0.19/0.47 => ( ( zero_zero @ ( product_prod @ A @ B ) )
% 0.19/0.47 = ( product_Pair @ A @ B @ ( zero_zero @ A ) @ ( zero_zero @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % zero_prod_def
% 0.19/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.19/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.19/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.19/0.47 = ( P @ A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % mem_Collect_eq
% 0.19/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] :
% 0.19/0.47 ( ( collect @ A
% 0.19/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ A5 ) )
% 0.19/0.47 = A5 ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_mem_eq
% 0.19/0.47 thf(fact_47_Collect__cong,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.47 ( ! [X4: A] :
% 0.19/0.47 ( ( P @ X4 )
% 0.19/0.47 = ( Q @ X4 ) )
% 0.19/0.47 => ( ( collect @ A @ P )
% 0.19/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_cong
% 0.19/0.47 thf(fact_48_ext,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F3: A > B,G: A > B] :
% 0.19/0.47 ( ! [X4: A] :
% 0.19/0.47 ( ( F3 @ X4 )
% 0.19/0.47 = ( G @ X4 ) )
% 0.19/0.47 => ( F3 = G ) ) ).
% 0.19/0.47
% 0.19/0.47 % ext
% 0.19/0.47 thf(fact_49_fst__conv,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 0.19/0.47 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.19/0.47 = X1 ) ).
% 0.19/0.47
% 0.19/0.47 % fst_conv
% 0.19/0.47 thf(fact_50_fst__eqD,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
% 0.19/0.47 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 0.19/0.47 = A2 )
% 0.19/0.47 => ( X = A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % fst_eqD
% 0.19/0.47 thf(fact_51_snd__conv,axiom,
% 0.19/0.47 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 0.19/0.47 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 0.19/0.47 = X22 ) ).
% 0.19/0.47
% 0.19/0.47 % snd_conv
% 0.19/0.47 thf(fact_52_snd__eqD,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
% 0.19/0.47 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.19/0.47 = A2 )
% 0.19/0.47 => ( Y = A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % snd_eqD
% 0.19/0.47 thf(fact_53_prod__eq__iff,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ^ [Y5: product_prod @ A @ B,Z: product_prod @ A @ B] : ( Y5 = Z ) )
% 0.19/0.47 = ( ^ [S: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 0.19/0.47 ( ( ( product_fst @ A @ B @ S )
% 0.19/0.47 = ( product_fst @ A @ B @ T3 ) )
% 0.19/0.47 & ( ( product_snd @ A @ B @ S )
% 0.19/0.47 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_eq_iff
% 0.19/0.47 thf(fact_54_prod_Oexpand,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 0.19/0.47 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.19/0.47 = ( product_fst @ A @ B @ Prod2 ) )
% 0.19/0.47 & ( ( product_snd @ A @ B @ Prod )
% 0.19/0.47 = ( product_snd @ A @ B @ Prod2 ) ) )
% 0.19/0.47 => ( Prod = Prod2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod.expand
% 0.19/0.47 thf(fact_55_prod__eqI,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 0.19/0.47 ( ( ( product_fst @ A @ B @ P2 )
% 0.19/0.47 = ( product_fst @ A @ B @ Q2 ) )
% 0.19/0.47 => ( ( ( product_snd @ A @ B @ P2 )
% 0.19/0.47 = ( product_snd @ A @ B @ Q2 ) )
% 0.19/0.47 => ( P2 = Q2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_eqI
% 0.19/0.47 thf(fact_56_vec__nth__inverse,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( finite_finite @ B )
% 0.19/0.47 => ! [X: finite_Cartesian_vec @ A @ B] :
% 0.19/0.47 ( ( finite1990238425lambda @ B @ A @ ( finite1433825200ec_nth @ A @ B @ X ) )
% 0.19/0.47 = X ) ) ).
% 0.19/0.47
% 0.19/0.47 % vec_nth_inverse
% 0.19/0.47 thf(fact_57_vec__lambda__unique,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( finite_finite @ A )
% 0.19/0.47 => ! [F3: finite_Cartesian_vec @ B @ A,G: A > B] :
% 0.19/0.47 ( ( ! [I3: A] :
% 0.19/0.47 ( ( finite1433825200ec_nth @ B @ A @ F3 @ I3 )
% 0.19/0.47 = ( G @ I3 ) ) )
% 0.19/0.47 = ( ( finite1990238425lambda @ A @ B @ G )
% 0.19/0.47 = F3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % vec_lambda_unique
% 0.19/0.47 thf(fact_58_obj__sumE__f,axiom,
% 0.19/0.47 ! [A: $tType,C: $tType,B: $tType,S2: B,F3: ( sum_sum @ A @ C ) > B,P: $o] :
% 0.19/0.47 ( ! [X4: A] :
% 0.19/0.47 ( ( S2
% 0.19/0.47 = ( F3 @ ( sum_Inl @ A @ C @ X4 ) ) )
% 0.19/0.47 => P )
% 0.19/0.47 => ( ! [X4: C] :
% 0.19/0.47 ( ( S2
% 0.19/0.47 = ( F3 @ ( sum_Inr @ C @ A @ X4 ) ) )
% 0.19/0.47 => P )
% 0.19/0.47 => ! [X5: sum_sum @ A @ C] :
% 0.19/0.47 ( ( S2
% 0.19/0.47 = ( F3 @ X5 ) )
% 0.19/0.47 => P ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % obj_sumE_f
% 0.19/0.47 thf(fact_59_zero__vec__def,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ B )
% 0.19/0.47 & ( zero @ A ) )
% 0.19/0.47 => ( ( zero_zero @ ( finite_Cartesian_vec @ A @ B ) )
% 0.19/0.47 = ( finite1990238425lambda @ B @ A
% 0.19/0.47 @ ^ [I3: B] : ( zero_zero @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % zero_vec_def
% 0.19/0.47 thf(fact_60_surjective__pairing,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.19/0.47 ( T2
% 0.19/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % surjective_pairing
% 0.19/0.47 thf(fact_61_prod_Oexhaust__sel,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.19/0.47 ( Prod
% 0.19/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod.exhaust_sel
% 0.19/0.47 thf(fact_62_mk__v__concrete,axiom,
% 0.19/0.47 ! [A: $tType,C: $tType,B: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ B )
% 0.19/0.47 & ( finite_finite @ C )
% 0.19/0.47 & ( finite_finite @ A ) )
% 0.19/0.47 => ( ( denotational_mk_v @ B @ C @ A )
% 0.19/0.47 = ( ^ [I6: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE2: ode @ B @ A,Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Sol2: finite_Cartesian_vec @ real @ A] :
% 0.19/0.47 ( product_Pair @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )
% 0.19/0.47 @ ( finite1990238425lambda @ A @ real
% 0.19/0.47 @ ^ [I3: A] : ( finite1433825200ec_nth @ real @ A @ ( if @ ( finite_Cartesian_vec @ real @ A ) @ ( member @ ( sum_sum @ A @ A ) @ ( sum_Inl @ A @ A @ I3 ) @ ( denotational_semBV @ B @ C @ A @ I6 @ ODE2 ) ) @ Sol2 @ ( product_fst @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ Nu ) ) @ I3 ) )
% 0.19/0.47 @ ( finite1990238425lambda @ A @ real
% 0.19/0.47 @ ^ [I3: A] : ( finite1433825200ec_nth @ real @ A @ ( if @ ( finite_Cartesian_vec @ real @ A ) @ ( member @ ( sum_sum @ A @ A ) @ ( sum_Inr @ A @ A @ I3 ) @ ( denotational_semBV @ B @ C @ A @ I6 @ ODE2 ) ) @ ( denotational_ODE_sem @ B @ C @ A @ I6 @ ODE2 @ Sol2 ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ Nu ) ) @ I3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % mk_v_concrete
% 0.19/0.47 thf(fact_63__092_060open_062Vagree_A_Imk__v_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_Ax_A_E_Ai_Aelse_Aab_A_E_Ai_J_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_N_AsemBV_AI_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_Ax_A_E_Ai_Aelse_Aab_A_E_Ai_J_J_A_Imk__xode_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_Ax_A_E_Ai_Aelse_Aab_A_E_Ai_J_J_A_IsemBV_AI_AODE_J_092_060close_062,axiom,
% 0.19/0.47 ( ( denotational_Vagree @ c
% 0.19/0.47 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 0.19/0.47 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.19/0.47 @ ( finite1990238425lambda @ c @ real
% 0.19/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 0.19/0.47 @ bb )
% 0.19/0.47 @ ( finite1990238425lambda @ c @ real
% 0.19/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
% 0.19/0.47 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.19/0.47 @ ( finite1990238425lambda @ c @ real
% 0.19/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 0.19/0.47 @ bb )
% 0.19/0.47 @ ( uminus_uminus @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) ) )
% 0.19/0.47 & ( denotational_Vagree @ c
% 0.19/0.47 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 0.19/0.47 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.19/0.47 @ ( finite1990238425lambda @ c @ real
% 0.19/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 0.19/0.47 @ bb )
% 0.19/0.47 @ ( finite1990238425lambda @ c @ real
% 0.19/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
% 0.19/0.47 @ ( denotational_mk_xode @ a @ b @ c @ i @ oDEa
% 0.19/0.47 @ ( finite1990238425lambda @ c @ real
% 0.19/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
% 0.19/0.47 @ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % \<open>Vagree (mk_v I ODE (\<chi>i. if i \<in> FVO ODE then sol 0 $ i else ab $ i, bb) (\<chi>i. if i \<in> FVO ODE then sol x $ i else ab $ i)) (\<chi>i. if i \<in> FVO ODE then sol 0 $ i else ab $ i, bb) (- semBV I ODE) \<and> Vagree (mk_v I ODE (\<chi>i. if i \<in> FVO ODE then sol 0 $ i else ab $ i, bb) (\<chi>i. if i \<in> FVO ODE then sol x $ i else ab $ i)) (mk_xode I ODE (\<chi>i. if i \<in> FVO ODE then sol x $ i else ab $ i)) (semBV I ODE)\<close>
% 0.19/0.47 thf(fact_64_Vagree__def,axiom,
% 0.19/0.47 ! [C: $tType] :
% 0.19/0.47 ( ( finite_finite @ C )
% 0.19/0.47 => ( ( denotational_Vagree @ C )
% 0.19/0.47 = ( ^ [Nu: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),V: set @ ( sum_sum @ C @ C )] :
% 0.19/0.47 ( ! [I3: C] :
% 0.19/0.47 ( ( member @ ( sum_sum @ C @ C ) @ ( sum_Inl @ C @ C @ I3 ) @ V )
% 0.19/0.47 => ( ( finite1433825200ec_nth @ real @ C @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) @ I3 )
% 0.19/0.47 = ( finite1433825200ec_nth @ real @ C @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu2 ) @ I3 ) ) )
% 0.19/0.47 & ! [I3: C] :
% 0.19/0.47 ( ( member @ ( sum_sum @ C @ C ) @ ( sum_Inr @ C @ C @ I3 ) @ V )
% 0.19/0.47 => ( ( finite1433825200ec_nth @ real @ C @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) @ I3 )
% 0.19/0.47 = ( finite1433825200ec_nth @ real @ C @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu2 ) @ I3 ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Vagree_def
% 0.19/0.47 thf(fact_65__092_060open_062Vagree_A_Imk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_Ax_J_J_A_Iab_M_Abb_J_A_I_N_AsemBV_AI_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_Ax_J_J_A_Imk__xode_AI_AODE_A_Isol_Ax_J_J_A_IsemBV_AI_AODE_J_092_060close_062,axiom,
% 0.19/0.47 ( ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( uminus_uminus @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) ) )
% 0.19/0.47 & ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) @ ( denotational_mk_xode @ a @ b @ c @ i @ oDEa @ ( sol @ x ) ) @ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % \<open>Vagree (mk_v I ODE (ab, bb) (sol x)) (ab, bb) (- semBV I ODE) \<and> Vagree (mk_v I ODE (ab, bb) (sol x)) (mk_xode I ODE (sol x)) (semBV I ODE)\<close>
% 0.19/0.47 thf(fact_66_concrete__v_Osimps,axiom,
% 0.19/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ A )
% 0.19/0.47 & ( finite_finite @ B )
% 0.19/0.47 & ( finite_finite @ C ) )
% 0.19/0.47 => ( ( denota1738237502rete_v @ A @ B @ C )
% 0.19/0.47 = ( ^ [I6: denota1663640101rp_ext @ A @ B @ C @ product_unit,ODE2: ode @ A @ C,Nu: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Sol2: finite_Cartesian_vec @ real @ C] :
% 0.19/0.47 ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )
% 0.19/0.47 @ ( finite1990238425lambda @ C @ real
% 0.19/0.47 @ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inl @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ I6 @ ODE2 ) ) @ Sol2 @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) ) @ I3 ) )
% 0.19/0.47 @ ( finite1990238425lambda @ C @ real
% 0.19/0.47 @ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inr @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ I6 @ ODE2 ) ) @ ( denotational_ODE_sem @ A @ B @ C @ I6 @ ODE2 @ Sol2 ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) ) @ I3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % concrete_v.simps
% 0.19/0.47 thf(fact_67_concrete__v_Oelims,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,C: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ C )
% 0.19/0.47 & ( finite_finite @ A )
% 0.19/0.47 & ( finite_finite @ B ) )
% 0.19/0.47 => ! [X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Xa: ode @ A @ C,Xb: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Xc: finite_Cartesian_vec @ real @ C,Y: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 0.19/0.47 ( ( ( denota1738237502rete_v @ A @ B @ C @ X @ Xa @ Xb @ Xc )
% 0.19/0.47 = Y )
% 0.19/0.47 => ( Y
% 0.19/0.47 = ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )
% 0.19/0.47 @ ( finite1990238425lambda @ C @ real
% 0.19/0.47 @ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inl @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ X @ Xa ) ) @ Xc @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb ) ) @ I3 ) )
% 0.19/0.47 @ ( finite1990238425lambda @ C @ real
% 0.19/0.47 @ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inr @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ X @ Xa ) ) @ ( denotational_ODE_sem @ A @ B @ C @ X @ Xa @ Xc ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb ) ) @ I3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % concrete_v.elims
% 0.19/0.47 thf(fact_68_Pair__le,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( ord @ B )
% 0.19/0.47 & ( ord @ A ) )
% 0.19/0.47 => ! [A2: A,B2: B,C3: A,D3: B] :
% 0.19/0.47 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C3 @ D3 ) )
% 0.19/0.47 = ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.19/0.47 & ( ord_less_eq @ B @ B2 @ D3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Pair_le
% 0.19/0.47 thf(fact_69_le__zero__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( canoni770627133id_add @ A )
% 0.19/0.47 => ! [N: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.19/0.47 = ( N
% 0.19/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_zero_eq
% 0.19/0.47 thf(fact_70_sem,axiom,
% 0.19/0.47 ! [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 )] :
% 0.19/0.47 ( ( denotational_Vagree @ c @ Nu3 @ Nu4 @ ( static_FVF @ a @ b @ c @ p ) )
% 0.19/0.47 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu3 @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) )
% 0.19/0.47 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu4 @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % sem
% 0.19/0.47 thf(fact_71_aaba,axiom,
% 0.19/0.47 ( ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba )
% 0.19/0.47 = ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ t ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % aaba
% 0.19/0.47 thf(fact_72_neg__equal__iff__equal,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( group_add @ A )
% 0.19/0.47 => ! [A2: A,B2: A] :
% 0.19/0.47 ( ( ( uminus_uminus @ A @ A2 )
% 0.19/0.47 = ( uminus_uminus @ A @ B2 ) )
% 0.19/0.47 = ( A2 = B2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % neg_equal_iff_equal
% 0.19/0.47 thf(fact_73_add_Oinverse__inverse,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( group_add @ A )
% 0.19/0.47 => ! [A2: A] :
% 0.19/0.47 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 0.19/0.47 = A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % add.inverse_inverse
% 0.19/0.47 thf(fact_74_add_Oinverse__neutral,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( group_add @ A )
% 0.19/0.47 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 0.19/0.47 = ( zero_zero @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add.inverse_neutral
% 0.19/0.47 thf(fact_75_neg__0__equal__iff__equal,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( group_add @ A )
% 0.19/0.47 => ! [A2: A] :
% 0.19/0.47 ( ( ( zero_zero @ A )
% 0.19/0.47 = ( uminus_uminus @ A @ A2 ) )
% 0.19/0.47 = ( ( zero_zero @ A )
% 0.19/0.47 = A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % neg_0_equal_iff_equal
% 0.19/0.47 thf(fact_76_neg__equal__0__iff__equal,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( group_add @ A )
% 0.19/0.47 => ! [A2: A] :
% 0.19/0.47 ( ( ( uminus_uminus @ A @ A2 )
% 0.19/0.47 = ( zero_zero @ A ) )
% 0.19/0.47 = ( A2
% 0.19/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % neg_equal_0_iff_equal
% 0.19/0.47 thf(fact_77_equal__neg__zero,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linord219039673up_add @ A )
% 0.19/0.47 => ! [A2: A] :
% 0.19/0.47 ( ( A2
% 0.19/0.47 = ( uminus_uminus @ A @ A2 ) )
% 0.19/0.47 = ( A2
% 0.19/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equal_neg_zero
% 0.19/0.47 thf(fact_78_neg__equal__zero,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linord219039673up_add @ A )
% 0.19/0.47 => ! [A2: A] :
% 0.19/0.47 ( ( ( uminus_uminus @ A @ A2 )
% 0.19/0.47 = A2 )
% 0.19/0.47 = ( A2
% 0.19/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % neg_equal_zero
% 0.19/0.47 thf(fact_79_neg__le__iff__le,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A )
% 0.19/0.47 => ! [B2: A,A2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 0.19/0.47 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % neg_le_iff_le
% 0.19/0.47 thf(fact_80_uminus__Pair,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( uminus @ B )
% 0.19/0.47 & ( uminus @ A ) )
% 0.19/0.47 => ! [A2: A,B2: B] :
% 0.19/0.47 ( ( uminus_uminus @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.19/0.47 = ( product_Pair @ A @ B @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ B @ B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % uminus_Pair
% 0.19/0.47 thf(fact_81_vector__uminus__component,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ B )
% 0.19/0.47 & ( uminus @ A ) )
% 0.19/0.47 => ! [X: finite_Cartesian_vec @ A @ B,I4: B] :
% 0.19/0.47 ( ( finite1433825200ec_nth @ A @ B @ ( uminus_uminus @ ( finite_Cartesian_vec @ A @ B ) @ X ) @ I4 )
% 0.19/0.47 = ( uminus_uminus @ A @ ( finite1433825200ec_nth @ A @ B @ X @ I4 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % vector_uminus_component
% 0.19/0.47 thf(fact_82_fst__uminus,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ( uminus @ A )
% 0.19/0.47 & ( uminus @ B ) )
% 0.19/0.47 => ! [X: product_prod @ A @ B] :
% 0.19/0.47 ( ( product_fst @ A @ B @ ( uminus_uminus @ ( product_prod @ A @ B ) @ X ) )
% 0.19/0.47 = ( uminus_uminus @ A @ ( product_fst @ A @ B @ X ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % fst_uminus
% 0.19/0.47 thf(fact_83_snd__uminus,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( uminus @ B )
% 0.19/0.47 & ( uminus @ A ) )
% 0.19/0.47 => ! [X: product_prod @ B @ A] :
% 0.19/0.47 ( ( product_snd @ B @ A @ ( uminus_uminus @ ( product_prod @ B @ A ) @ X ) )
% 0.19/0.47 = ( uminus_uminus @ A @ ( product_snd @ B @ A @ X ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % snd_uminus
% 0.19/0.47 thf(fact_84_neg__less__eq__nonneg,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linord219039673up_add @ A )
% 0.19/0.47 => ! [A2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 0.19/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % neg_less_eq_nonneg
% 0.19/0.47 thf(fact_85_less__eq__neg__nonpos,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linord219039673up_add @ A )
% 0.19/0.47 => ! [A2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 0.19/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_eq_neg_nonpos
% 0.19/0.47 thf(fact_86_neg__le__0__iff__le,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A )
% 0.19/0.47 => ! [A2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.19/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % neg_le_0_iff_le
% 0.19/0.47 thf(fact_87_neg__0__le__iff__le,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A )
% 0.19/0.47 => ! [A2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 0.19/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % neg_0_le_iff_le
% 0.19/0.47 thf(fact_88_allT,axiom,
% 0.19/0.47 ! [S3: real] :
% 0.19/0.47 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ S3 )
% 0.19/0.47 => ( ( ord_less_eq @ real @ S3 @ t )
% 0.19/0.47 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ S3 ) ) @ ( denotational_fml_sem @ a @ b @ c @ i @ phi ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % allT
% 0.19/0.47 thf(fact_89_minus__equation__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( group_add @ A )
% 0.19/0.47 => ! [A2: A,B2: A] :
% 0.19/0.47 ( ( ( uminus_uminus @ A @ A2 )
% 0.19/0.47 = B2 )
% 0.19/0.47 = ( ( uminus_uminus @ A @ B2 )
% 0.19/0.47 = A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % minus_equation_iff
% 0.19/0.47 thf(fact_90_equation__minus__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( group_add @ A )
% 0.19/0.47 => ! [A2: A,B2: A] :
% 0.19/0.47 ( ( A2
% 0.19/0.47 = ( uminus_uminus @ A @ B2 ) )
% 0.19/0.47 = ( B2
% 0.19/0.47 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equation_minus_iff
% 0.19/0.47 thf(fact_91_iff__to__impl,axiom,
% 0.19/0.47 ! [A: $tType,C: $tType,B: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ B )
% 0.19/0.47 & ( finite_finite @ C )
% 0.19/0.47 & ( finite_finite @ A ) )
% 0.19/0.47 => ! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,A5: formula @ B @ C @ A,B5: formula @ B @ C @ A] :
% 0.19/0.47 ( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A5 ) )
% 0.19/0.47 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B5 ) ) )
% 0.19/0.47 = ( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A5 ) )
% 0.19/0.47 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B5 ) ) )
% 0.19/0.47 & ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B5 ) )
% 0.19/0.47 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A5 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % iff_to_impl
% 0.19/0.47 thf(fact_92_le__imp__neg__le,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A )
% 0.19/0.47 => ! [A2: A,B2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_imp_neg_le
% 0.19/0.47 thf(fact_93_minus__le__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A )
% 0.19/0.47 => ! [A2: A,B2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 0.19/0.47 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % minus_le_iff
% 0.19/0.47 thf(fact_94_le__minus__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A )
% 0.19/0.47 => ! [A2: A,B2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 0.19/0.47 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_minus_iff
% 0.19/0.47 thf(fact_95_agree__supset,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( finite_finite @ A )
% 0.19/0.47 => ! [B5: set @ ( sum_sum @ A @ A ),A5: set @ ( sum_sum @ A @ A ),Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu6: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ B5 @ A5 )
% 0.19/0.47 => ( ( denotational_Vagree @ A @ Nu5 @ Nu6 @ A5 )
% 0.19/0.47 => ( denotational_Vagree @ A @ Nu5 @ Nu6 @ B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % agree_supset
% 0.19/0.47 thf(fact_96_agree__sub,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( finite_finite @ A )
% 0.19/0.47 => ! [A5: set @ ( sum_sum @ A @ A ),B5: set @ ( sum_sum @ A @ A ),Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Omega: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ A5 @ B5 )
% 0.19/0.47 => ( ( denotational_Vagree @ A @ Nu5 @ Omega @ B5 )
% 0.19/0.47 => ( denotational_Vagree @ A @ Nu5 @ Omega @ A5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % agree_sub
% 0.19/0.47 thf(fact_97_mk__v__exists,axiom,
% 0.19/0.47 ! [C: $tType,A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ B )
% 0.19/0.47 & ( finite_finite @ A )
% 0.19/0.47 & ( finite_finite @ C ) )
% 0.19/0.47 => ! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A,Sol: finite_Cartesian_vec @ real @ A] :
% 0.19/0.47 ? [Omega2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
% 0.19/0.47 ( ( denotational_Vagree @ A @ Omega2 @ Nu5 @ ( uminus_uminus @ ( set @ ( sum_sum @ A @ A ) ) @ ( denotational_semBV @ B @ C @ A @ I2 @ ODE ) ) )
% 0.19/0.47 & ( denotational_Vagree @ A @ Omega2 @ ( denotational_mk_xode @ B @ C @ A @ I2 @ ODE @ Sol ) @ ( denotational_semBV @ B @ C @ A @ I2 @ ODE ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % mk_v_exists
% 0.19/0.47 thf(fact_98_uminus__vec__def,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ( uminus @ A )
% 0.19/0.47 & ( finite_finite @ B ) )
% 0.19/0.47 => ( ( uminus_uminus @ ( finite_Cartesian_vec @ A @ B ) )
% 0.19/0.47 = ( ^ [X3: finite_Cartesian_vec @ A @ B] :
% 0.19/0.47 ( finite1990238425lambda @ B @ A
% 0.19/0.47 @ ^ [I3: B] : ( uminus_uminus @ A @ ( finite1433825200ec_nth @ A @ B @ X3 @ I3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % uminus_vec_def
% 0.19/0.47 thf(fact_99_uminus__prod__def,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ( uminus @ A )
% 0.19/0.47 & ( uminus @ B ) )
% 0.19/0.47 => ( ( uminus_uminus @ ( product_prod @ A @ B ) )
% 0.19/0.47 = ( ^ [X3: product_prod @ A @ B] : ( product_Pair @ A @ B @ ( uminus_uminus @ A @ ( product_fst @ A @ B @ X3 ) ) @ ( uminus_uminus @ B @ ( product_snd @ A @ B @ X3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % uminus_prod_def
% 0.19/0.47 thf(fact_100_mk__v__agree,axiom,
% 0.19/0.47 ! [C: $tType,A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ B )
% 0.19/0.47 & ( finite_finite @ A )
% 0.19/0.47 & ( finite_finite @ C ) )
% 0.19/0.47 => ! [I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A,Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Sol: finite_Cartesian_vec @ real @ A] :
% 0.19/0.47 ( ( denotational_Vagree @ A @ ( denotational_mk_v @ B @ C @ A @ I2 @ ODE @ Nu5 @ Sol ) @ Nu5 @ ( uminus_uminus @ ( set @ ( sum_sum @ A @ A ) ) @ ( denotational_semBV @ B @ C @ A @ I2 @ ODE ) ) )
% 0.19/0.47 & ( denotational_Vagree @ A @ ( denotational_mk_v @ B @ C @ A @ I2 @ ODE @ Nu5 @ Sol ) @ ( denotational_mk_xode @ B @ C @ A @ I2 @ ODE @ Sol ) @ ( denotational_semBV @ B @ C @ A @ I2 @ ODE ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % mk_v_agree
% 0.19/0.47 thf(fact_101_mk__xode_Oelims,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ C )
% 0.19/0.47 & ( finite_finite @ B )
% 0.19/0.47 & ( finite_finite @ A ) )
% 0.19/0.47 => ! [X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Xa: ode @ A @ C,Xb: finite_Cartesian_vec @ real @ C,Y: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 0.19/0.47 ( ( ( denotational_mk_xode @ A @ B @ C @ X @ Xa @ Xb )
% 0.19/0.47 = Y )
% 0.19/0.47 => ( Y
% 0.19/0.47 = ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb @ ( denotational_ODE_sem @ A @ B @ C @ X @ Xa @ Xb ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % mk_xode.elims
% 0.19/0.47 thf(fact_102_mk__xode_Osimps,axiom,
% 0.19/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ A )
% 0.19/0.47 & ( finite_finite @ B )
% 0.19/0.47 & ( finite_finite @ C ) )
% 0.19/0.47 => ( ( denotational_mk_xode @ A @ B @ C )
% 0.19/0.47 = ( ^ [I6: 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 @ I6 @ ODE2 @ Sol2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % mk_xode.simps
% 0.19/0.47 thf(fact_103_zero__reorient,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( zero @ A )
% 0.19/0.47 => ! [X: A] :
% 0.19/0.47 ( ( ( zero_zero @ A )
% 0.19/0.47 = X )
% 0.19/0.47 = ( X
% 0.19/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % zero_reorient
% 0.19/0.47 thf(fact_104_agree__refl,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( finite_finite @ A )
% 0.19/0.47 => ! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A5: set @ ( sum_sum @ A @ A )] : ( denotational_Vagree @ A @ Nu5 @ Nu5 @ A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % agree_refl
% 0.19/0.47 thf(fact_105_agree__comm,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( finite_finite @ A )
% 0.19/0.47 => ! [A5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),B5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),V2: set @ ( sum_sum @ A @ A )] :
% 0.19/0.47 ( ( denotational_Vagree @ A @ A5 @ B5 @ V2 )
% 0.19/0.47 => ( denotational_Vagree @ A @ B5 @ A5 @ V2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % agree_comm
% 0.19/0.47 thf(fact_106_zero__le,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( canoni770627133id_add @ A )
% 0.19/0.47 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % zero_le
% 0.19/0.47 thf(fact_107_Pair__mono,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( ord @ B )
% 0.19/0.47 & ( ord @ A ) )
% 0.19/0.47 => ! [X: A,X6: A,Y: B,Y6: B] :
% 0.19/0.47 ( ( ord_less_eq @ A @ X @ X6 )
% 0.19/0.47 => ( ( ord_less_eq @ B @ Y @ Y6 )
% 0.19/0.47 => ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( product_Pair @ A @ B @ X6 @ Y6 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Pair_mono
% 0.19/0.47 thf(fact_108_fst__mono,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ( ord @ A )
% 0.19/0.47 & ( ord @ B ) )
% 0.19/0.47 => ! [X: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 0.19/0.47 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ X @ Y )
% 0.19/0.47 => ( ord_less_eq @ A @ ( product_fst @ A @ B @ X ) @ ( product_fst @ A @ B @ Y ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % fst_mono
% 0.19/0.47 thf(fact_109_snd__mono,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ( ord @ A )
% 0.19/0.47 & ( ord @ B ) )
% 0.19/0.47 => ! [X: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 0.19/0.47 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ X @ Y )
% 0.19/0.47 => ( ord_less_eq @ B @ ( product_snd @ A @ B @ X ) @ ( product_snd @ A @ B @ Y ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % snd_mono
% 0.19/0.47 thf(fact_110_less__eq__prod__def,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ( ord @ A )
% 0.19/0.47 & ( ord @ B ) )
% 0.19/0.47 => ( ( ord_less_eq @ ( product_prod @ A @ B ) )
% 0.19/0.47 = ( ^ [X3: product_prod @ A @ B,Y3: product_prod @ A @ B] :
% 0.19/0.47 ( ( ord_less_eq @ A @ ( product_fst @ A @ B @ X3 ) @ ( product_fst @ A @ B @ Y3 ) )
% 0.19/0.47 & ( ord_less_eq @ B @ ( product_snd @ A @ B @ X3 ) @ ( product_snd @ A @ B @ Y3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_eq_prod_def
% 0.19/0.47 thf(fact_111_ODE__vars__lr,axiom,
% 0.19/0.47 ! [Sc: $tType,Sz: $tType,Sf: $tType,X: Sz,I2: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,ODE: ode @ Sf @ Sz] :
% 0.19/0.47 ( ( member @ ( sum_sum @ Sz @ Sz ) @ ( sum_Inl @ Sz @ Sz @ X ) @ ( denotational_semBV @ Sf @ Sc @ Sz @ I2 @ ODE ) )
% 0.19/0.47 = ( member @ ( sum_sum @ Sz @ Sz ) @ ( sum_Inr @ Sz @ Sz @ X ) @ ( denotational_semBV @ Sf @ Sc @ Sz @ I2 @ ODE ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ODE_vars_lr
% 0.19/0.47 thf(fact_112__092_060open_062_092_060lbrakk_062osafe_AODE_059_Afsafe_A_092_060phi_062_059_AODE_____A_061_AODE_059_AP_A_061_A_092_060phi_062_059_A0_A_092_060le_062_At_059_A_Iaa_M_Aba_J_A_061_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_059_A_Isol_Asolves__ode_A_I_092_060lambda_062a_O_AODE__sem_AI_AODE_J_J_A_1230_O_Ot_125_A_123x_O_Amk__v_AI_AODE_A_Iab_M_Abb_J_Ax_A_092_060in_062_Afml__sem_AI_A_092_060phi_062_125_059_AVSagree_A_Isol_A0_J_Aab_A_123uu___O_AInl_Auu___A_092_060in_062_ABVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AInl_A_096_AFVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AFVF_A_092_060phi_062_125_092_060rbrakk_062_A_092_060Longrightarrow_062_Aab_A_061_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_J_092_060close_062,axiom,
% 0.19/0.47 ( ( osafe @ a @ c @ oDEa )
% 0.19/0.47 => ( ( fsafe @ a @ b @ c @ phi )
% 0.19/0.47 => ( ( ode2 = oDEa )
% 0.19/0.47 => ( ( p = phi )
% 0.19/0.47 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ t )
% 0.19/0.47 => ( ( ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba )
% 0.19/0.47 = ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ t ) ) )
% 0.19/0.47 => ( ( initia1685620758es_ode @ ( finite_Cartesian_vec @ real @ c ) @ sol
% 0.19/0.47 @ ^ [X7: real] : ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa )
% 0.19/0.47 @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t )
% 0.19/0.47 @ ( collect @ ( finite_Cartesian_vec @ real @ c )
% 0.19/0.47 @ ^ [X3: 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 @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ X3 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ phi ) ) ) )
% 0.19/0.47 => ( ( denotational_VSagree @ c @ ( sol @ ( zero_zero @ real ) ) @ ab
% 0.19/0.47 @ ( collect @ c
% 0.19/0.47 @ ^ [Uu: c] :
% 0.19/0.47 ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_BVO @ a @ c @ oDEa ) )
% 0.19/0.47 | ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( static_FVO @ a @ c @ oDEa ) ) )
% 0.19/0.47 | ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_FVF @ a @ b @ c @ phi ) ) ) ) )
% 0.19/0.47 => ( ab
% 0.19/0.47 = ( finite1990238425lambda @ c @ real
% 0.19/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % \<open>\<lbrakk>osafe ODE; fsafe \<phi>; ODE__ = ODE; P = \<phi>; 0 \<le> t; (aa, ba) = mk_v I ODE (ab, bb) (sol t); (sol solves_ode (\<lambda>a. ODE_sem I ODE)) {0..t} {x. mk_v I ODE (ab, bb) x \<in> fml_sem I \<phi>}; VSagree (sol 0) ab {uu_. Inl uu_ \<in> BVO ODE \<or> Inl uu_ \<in> Inl ` FVO ODE \<or> Inl uu_ \<in> FVF \<phi>}\<rbrakk> \<Longrightarrow> ab = (\<chi>i. if i \<in> FVO ODE then sol 0 $ i else ab $ i)\<close>
% 0.19/0.47 thf(fact_113__092_060open_062_092_060lbrakk_062osafe_AODE_059_Afsafe_A_092_060phi_062_059_AODE_____A_061_AODE_059_AP_A_061_A_092_060phi_062_059_A0_A_092_060le_062_At_059_A_Iaa_M_Aba_J_A_061_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_059_A_Isol_Asolves__ode_A_I_092_060lambda_062a_O_AODE__sem_AI_AODE_J_J_A_1230_O_Ot_125_A_123x_O_Amk__v_AI_AODE_A_Iab_M_Abb_J_Ax_A_092_060in_062_Afml__sem_AI_A_092_060phi_062_125_059_AVSagree_A_Isol_A0_J_Aab_A_123uu___O_AInl_Auu___A_092_060in_062_ABVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AInl_A_096_AFVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AFVF_A_092_060phi_062_125_092_060rbrakk_062_A_092_060Longrightarrow_062_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_A_061_Amk__v_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_At_A_E_Ai_Aelse_Aab_A_E_Ai_J_092_060close_062,axiom,
% 0.19/0.47 ( ( osafe @ a @ c @ oDEa )
% 0.19/0.47 => ( ( fsafe @ a @ b @ c @ phi )
% 0.19/0.47 => ( ( ode2 = oDEa )
% 0.19/0.47 => ( ( p = phi )
% 0.19/0.47 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ t )
% 0.19/0.47 => ( ( ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba )
% 0.19/0.47 = ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ t ) ) )
% 0.19/0.47 => ( ( initia1685620758es_ode @ ( finite_Cartesian_vec @ real @ c ) @ sol
% 0.19/0.47 @ ^ [X7: real] : ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa )
% 0.19/0.47 @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t )
% 0.19/0.47 @ ( collect @ ( finite_Cartesian_vec @ real @ c )
% 0.19/0.47 @ ^ [X3: 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 @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ X3 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ phi ) ) ) )
% 0.19/0.47 => ( ( denotational_VSagree @ c @ ( sol @ ( zero_zero @ real ) ) @ ab
% 0.19/0.47 @ ( collect @ c
% 0.19/0.47 @ ^ [Uu: c] :
% 0.19/0.47 ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_BVO @ a @ c @ oDEa ) )
% 0.19/0.47 | ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( static_FVO @ a @ c @ oDEa ) ) )
% 0.19/0.47 | ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_FVF @ a @ b @ c @ phi ) ) ) ) )
% 0.19/0.47 => ( ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ t ) )
% 0.19/0.47 = ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 0.19/0.47 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.19/0.47 @ ( finite1990238425lambda @ c @ real
% 0.19/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 0.19/0.47 @ bb )
% 0.19/0.47 @ ( finite1990238425lambda @ c @ real
% 0.19/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ t ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % \<open>\<lbrakk>osafe ODE; fsafe \<phi>; ODE__ = ODE; P = \<phi>; 0 \<le> t; (aa, ba) = mk_v I ODE (ab, bb) (sol t); (sol solves_ode (\<lambda>a. ODE_sem I ODE)) {0..t} {x. mk_v I ODE (ab, bb) x \<in> fml_sem I \<phi>}; VSagree (sol 0) ab {uu_. Inl uu_ \<in> BVO ODE \<or> Inl uu_ \<in> Inl ` FVO ODE \<or> Inl uu_ \<in> FVF \<phi>}\<rbrakk> \<Longrightarrow> mk_v I ODE (ab, bb) (sol t) = mk_v I ODE (\<chi>i. if i \<in> FVO ODE then sol 0 $ i else ab $ i, bb) (\<chi>i. if i \<in> FVO ODE then sol t $ i else ab $ i)\<close>
% 0.19/0.47 thf(fact_114__092_060open_062_092_060lbrakk_062osafe_AODE_059_Afsafe_A_092_060phi_062_059_AODE_____A_061_AODE_059_AP_A_061_A_092_060phi_062_059_A0_A_092_060le_062_At_059_A_Iaa_M_Aba_J_A_061_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_059_A_Isol_Asolves__ode_A_I_092_060lambda_062a_O_AODE__sem_AI_AODE_J_J_A_1230_O_Ot_125_A_123x_O_Amk__v_AI_AODE_A_Iab_M_Abb_J_Ax_A_092_060in_062_Afml__sem_AI_A_092_060phi_062_125_059_AVSagree_A_Isol_A0_J_Aab_A_123uu___O_AInl_Auu___A_092_060in_062_ABVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AInl_A_096_AFVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AFVF_A_092_060phi_062_125_092_060rbrakk_062_A_092_060Longrightarrow_062_A0_A_092_060le_062_At_092_060close_062,axiom,
% 0.19/0.47 ( ( osafe @ a @ c @ oDEa )
% 0.19/0.47 => ( ( fsafe @ a @ b @ c @ phi )
% 0.19/0.47 => ( ( ode2 = oDEa )
% 0.19/0.47 => ( ( p = phi )
% 0.19/0.47 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ t )
% 0.19/0.47 => ( ( ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba )
% 0.19/0.47 = ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ t ) ) )
% 0.19/0.47 => ( ( initia1685620758es_ode @ ( finite_Cartesian_vec @ real @ c ) @ sol
% 0.19/0.47 @ ^ [X7: real] : ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa )
% 0.19/0.47 @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t )
% 0.19/0.47 @ ( collect @ ( finite_Cartesian_vec @ real @ c )
% 0.19/0.47 @ ^ [X3: 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 @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ X3 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ phi ) ) ) )
% 0.19/0.47 => ( ( denotational_VSagree @ c @ ( sol @ ( zero_zero @ real ) ) @ ab
% 0.19/0.47 @ ( collect @ c
% 0.19/0.47 @ ^ [Uu: c] :
% 0.19/0.47 ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_BVO @ a @ c @ oDEa ) )
% 0.19/0.47 | ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( static_FVO @ a @ c @ oDEa ) ) )
% 0.19/0.47 | ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_FVF @ a @ b @ c @ phi ) ) ) ) )
% 0.19/0.47 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ t ) ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % \<open>\<lbrakk>osafe ODE; fsafe \<phi>; ODE__ = ODE; P = \<phi>; 0 \<le> t; (aa, ba) = mk_v I ODE (ab, bb) (sol t); (sol solves_ode (\<lambda>a. ODE_sem I ODE)) {0..t} {x. mk_v I ODE (ab, bb) x \<in> fml_sem I \<phi>}; VSagree (sol 0) ab {uu_. Inl uu_ \<in> BVO ODE \<or> Inl uu_ \<in> Inl ` FVO ODE \<or> Inl uu_ \<in> FVF \<phi>}\<rbrakk> \<Longrightarrow> 0 \<le> t\<close>
% 0.19/0.47 thf(fact_115_mkV,axiom,
% 0.19/0.47 ( member @ ( real > ( finite_Cartesian_vec @ real @ c ) ) @ sol
% 0.19/0.47 @ ( pi @ real @ ( finite_Cartesian_vec @ real @ c ) @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t )
% 0.19/0.47 @ ^ [Uu: real] :
% 0.19/0.47 ( collect @ ( finite_Cartesian_vec @ real @ c )
% 0.19/0.47 @ ^ [X3: 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 @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ X3 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ phi ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % mkV
% 0.19/0.47 thf(fact_116_euclid__ext__aux_Ocases,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( euclid1678468529ng_gcd @ A )
% 0.19/0.47 => ! [X: product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) )] :
% 0.19/0.47 ~ ! [S4: A,S5: A,T4: A,T5: A,R: A,R2: A] :
% 0.19/0.47 ( X
% 0.19/0.47 != ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) ) @ S4 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) @ S5 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) @ T4 @ ( product_Pair @ A @ ( product_prod @ A @ A ) @ T5 @ ( product_Pair @ A @ A @ R @ R2 ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % euclid_ext_aux.cases
% 0.19/0.47 thf(fact_117_compl__le__compl__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( boolean_algebra @ A )
% 0.19/0.47 => ! [X: A,Y: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
% 0.19/0.47 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % compl_le_compl_iff
% 0.19/0.47 thf(fact_118_iff__sem,axiom,
% 0.19/0.47 ! [A: $tType,C: $tType,B: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ B )
% 0.19/0.47 & ( finite_finite @ C )
% 0.19/0.47 & ( finite_finite @ A ) )
% 0.19/0.47 => ! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,A5: formula @ B @ C @ A,B5: formula @ B @ C @ A] :
% 0.19/0.47 ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ ( equiv @ B @ C @ A @ A5 @ B5 ) ) )
% 0.19/0.47 = ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A5 ) )
% 0.19/0.47 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B5 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % iff_sem
% 0.19/0.47 thf(fact_119_uminus__apply,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( uminus @ B )
% 0.19/0.47 => ( ( uminus_uminus @ ( A > B ) )
% 0.19/0.47 = ( ^ [A6: A > B,X3: A] : ( uminus_uminus @ B @ ( A6 @ X3 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % uminus_apply
% 0.19/0.47 thf(fact_120_double__compl,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( boolean_algebra @ A )
% 0.19/0.47 => ! [X: A] :
% 0.19/0.47 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
% 0.19/0.47 = X ) ) ).
% 0.19/0.47
% 0.19/0.47 % double_compl
% 0.19/0.47 thf(fact_121_compl__eq__compl__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( boolean_algebra @ A )
% 0.19/0.47 => ! [X: A,Y: A] :
% 0.19/0.47 ( ( ( uminus_uminus @ A @ X )
% 0.19/0.47 = ( uminus_uminus @ A @ Y ) )
% 0.19/0.47 = ( X = Y ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % compl_eq_compl_iff
% 0.19/0.47 thf(fact_122_VSagree__sub,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( finite_finite @ A )
% 0.19/0.47 => ! [A5: set @ A,B5: set @ A,Nu5: finite_Cartesian_vec @ real @ A,Omega: finite_Cartesian_vec @ real @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ( denotational_VSagree @ A @ Nu5 @ Omega @ B5 )
% 0.19/0.47 => ( denotational_VSagree @ A @ Nu5 @ Omega @ A5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % VSagree_sub
% 0.19/0.47 thf(fact_123_VSagree__supset,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( finite_finite @ A )
% 0.19/0.47 => ! [B5: set @ A,A5: set @ A,Nu5: finite_Cartesian_vec @ real @ A,Nu6: finite_Cartesian_vec @ real @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
% 0.19/0.47 => ( ( denotational_VSagree @ A @ Nu5 @ Nu6 @ A5 )
% 0.19/0.47 => ( denotational_VSagree @ A @ Nu5 @ Nu6 @ B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % VSagree_supset
% 0.19/0.47 thf(fact_124_VSagree__refl,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( finite_finite @ A )
% 0.19/0.47 => ! [Nu5: finite_Cartesian_vec @ real @ A,A5: set @ A] : ( denotational_VSagree @ A @ Nu5 @ Nu5 @ A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % VSagree_refl
% 0.19/0.47 thf(fact_125_VSagree__def,axiom,
% 0.19/0.47 ! [C: $tType] :
% 0.19/0.47 ( ( finite_finite @ C )
% 0.19/0.47 => ( ( denotational_VSagree @ C )
% 0.19/0.47 = ( ^ [Nu: finite_Cartesian_vec @ real @ C,Nu2: finite_Cartesian_vec @ real @ C,V: set @ C] :
% 0.19/0.47 ! [X3: C] :
% 0.19/0.47 ( ( member @ C @ X3 @ V )
% 0.19/0.47 => ( ( finite1433825200ec_nth @ real @ C @ Nu @ X3 )
% 0.19/0.47 = ( finite1433825200ec_nth @ real @ C @ Nu2 @ X3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % VSagree_def
% 0.19/0.47 thf(fact_126_gcd_Ocases,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( euclid1155270486miring @ A )
% 0.19/0.47 => ! [X: product_prod @ A @ A] :
% 0.19/0.47 ~ ! [A4: A,B4: A] :
% 0.19/0.47 ( X
% 0.19/0.47 != ( product_Pair @ A @ A @ A4 @ B4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % gcd.cases
% 0.19/0.47 thf(fact_127_fun__Compl__def,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( uminus @ B )
% 0.19/0.47 => ( ( uminus_uminus @ ( A > B ) )
% 0.19/0.47 = ( ^ [A6: A > B,X3: A] : ( uminus_uminus @ B @ ( A6 @ X3 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % fun_Compl_def
% 0.19/0.47 thf(fact_128_compl__mono,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( boolean_algebra @ A )
% 0.19/0.47 => ! [X: A,Y: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.19/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % compl_mono
% 0.19/0.47 thf(fact_129_compl__le__swap1,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( boolean_algebra @ A )
% 0.19/0.47 => ! [Y: A,X: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
% 0.19/0.47 => ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % compl_le_swap1
% 0.19/0.47 thf(fact_130_compl__le__swap2,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( boolean_algebra @ A )
% 0.19/0.47 => ! [Y: A,X: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
% 0.19/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % compl_le_swap2
% 0.19/0.47 thf(fact_131_image__uminus__atLeastAtMost,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A )
% 0.19/0.47 => ! [X: A,Y: A] :
% 0.19/0.47 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or331188842AtMost @ A @ X @ Y ) )
% 0.19/0.47 = ( set_or331188842AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_uminus_atLeastAtMost
% 0.19/0.47 thf(fact_132_atLeastatMost__subset__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( preorder @ A )
% 0.19/0.47 => ! [A2: A,B2: A,C3: A,D3: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A2 @ B2 ) @ ( set_or331188842AtMost @ A @ C3 @ D3 ) )
% 0.19/0.47 = ( ~ ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.47 | ( ( ord_less_eq @ A @ C3 @ A2 )
% 0.19/0.47 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % atLeastatMost_subset_iff
% 0.19/0.47 thf(fact_133_Compl__subset__Compl__iff,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) )
% 0.19/0.47 = ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % Compl_subset_Compl_iff
% 0.19/0.47 thf(fact_134_Compl__anti__mono,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Compl_anti_mono
% 0.19/0.47 thf(fact_135_image__eqI,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,B2: A,F3: B > A,X: B,A5: set @ B] :
% 0.19/0.47 ( ( B2
% 0.19/0.47 = ( F3 @ X ) )
% 0.19/0.47 => ( ( member @ B @ X @ A5 )
% 0.19/0.47 => ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_eqI
% 0.19/0.47 thf(fact_136_subset__antisym,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
% 0.19/0.47 => ( A5 = B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_antisym
% 0.19/0.47 thf(fact_137_subsetI,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ! [X4: A] :
% 0.19/0.47 ( ( member @ A @ X4 @ A5 )
% 0.19/0.47 => ( member @ A @ X4 @ B5 ) )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % subsetI
% 0.19/0.47 thf(fact_138_Compl__eq__Compl__iff,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( ( uminus_uminus @ ( set @ A ) @ A5 )
% 0.19/0.47 = ( uminus_uminus @ ( set @ A ) @ B5 ) )
% 0.19/0.47 = ( A5 = B5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % Compl_eq_Compl_iff
% 0.19/0.47 thf(fact_139_Compl__iff,axiom,
% 0.19/0.47 ! [A: $tType,C3: A,A5: set @ A] :
% 0.19/0.47 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
% 0.19/0.47 = ( ~ ( member @ A @ C3 @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Compl_iff
% 0.19/0.47 thf(fact_140_ComplI,axiom,
% 0.19/0.47 ! [A: $tType,C3: A,A5: set @ A] :
% 0.19/0.47 ( ~ ( member @ A @ C3 @ A5 )
% 0.19/0.47 => ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ComplI
% 0.19/0.47 thf(fact_141_image__ident,axiom,
% 0.19/0.47 ! [A: $tType,Y7: set @ A] :
% 0.19/0.47 ( ( image @ A @ A
% 0.19/0.47 @ ^ [X3: A] : X3
% 0.19/0.47 @ Y7 )
% 0.19/0.47 = Y7 ) ).
% 0.19/0.47
% 0.19/0.47 % image_ident
% 0.19/0.47 thf(fact_142_Icc__eq__Icc,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [L: A,H: A,L2: A,H2: A] :
% 0.19/0.47 ( ( ( set_or331188842AtMost @ A @ L @ H )
% 0.19/0.47 = ( set_or331188842AtMost @ A @ L2 @ H2 ) )
% 0.19/0.47 = ( ( ( L = L2 )
% 0.19/0.47 & ( H = H2 ) )
% 0.19/0.47 | ( ~ ( ord_less_eq @ A @ L @ H )
% 0.19/0.47 & ~ ( ord_less_eq @ A @ L2 @ H2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Icc_eq_Icc
% 0.19/0.47 thf(fact_143_atLeastAtMost__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord @ A )
% 0.19/0.47 => ! [I4: A,L: A,U: A] :
% 0.19/0.47 ( ( member @ A @ I4 @ ( set_or331188842AtMost @ A @ L @ U ) )
% 0.19/0.47 = ( ( ord_less_eq @ A @ L @ I4 )
% 0.19/0.47 & ( ord_less_eq @ A @ I4 @ U ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % atLeastAtMost_iff
% 0.19/0.47 thf(fact_144_imageI,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,X: A,A5: set @ A,F3: A > B] :
% 0.19/0.47 ( ( member @ A @ X @ A5 )
% 0.19/0.47 => ( member @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % imageI
% 0.19/0.47 thf(fact_145_image__iff,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,Z2: A,F3: B > A,A5: set @ B] :
% 0.19/0.47 ( ( member @ A @ Z2 @ ( image @ B @ A @ F3 @ A5 ) )
% 0.19/0.47 = ( ? [X3: B] :
% 0.19/0.47 ( ( member @ B @ X3 @ A5 )
% 0.19/0.47 & ( Z2
% 0.19/0.47 = ( F3 @ X3 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_iff
% 0.19/0.47 thf(fact_146_bex__imageD,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P: A > $o] :
% 0.19/0.47 ( ? [X5: A] :
% 0.19/0.47 ( ( member @ A @ X5 @ ( image @ B @ A @ F3 @ A5 ) )
% 0.19/0.47 & ( P @ X5 ) )
% 0.19/0.47 => ? [X4: B] :
% 0.19/0.47 ( ( member @ B @ X4 @ A5 )
% 0.19/0.47 & ( P @ ( F3 @ X4 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bex_imageD
% 0.19/0.47 thf(fact_147_image__cong,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,M: set @ A,N2: set @ A,F3: A > B,G: A > B] :
% 0.19/0.47 ( ( M = N2 )
% 0.19/0.47 => ( ! [X4: A] :
% 0.19/0.47 ( ( member @ A @ X4 @ N2 )
% 0.19/0.47 => ( ( F3 @ X4 )
% 0.19/0.47 = ( G @ X4 ) ) )
% 0.19/0.47 => ( ( image @ A @ B @ F3 @ M )
% 0.19/0.47 = ( image @ A @ B @ G @ N2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_cong
% 0.19/0.47 thf(fact_148_ball__imageD,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P: A > $o] :
% 0.19/0.47 ( ! [X4: A] :
% 0.19/0.47 ( ( member @ A @ X4 @ ( image @ B @ A @ F3 @ A5 ) )
% 0.19/0.47 => ( P @ X4 ) )
% 0.19/0.47 => ! [X5: B] :
% 0.19/0.47 ( ( member @ B @ X5 @ A5 )
% 0.19/0.47 => ( P @ ( F3 @ X5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ball_imageD
% 0.19/0.47 thf(fact_149_rev__image__eqI,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,X: A,A5: set @ A,B2: B,F3: A > B] :
% 0.19/0.47 ( ( member @ A @ X @ A5 )
% 0.19/0.47 => ( ( B2
% 0.19/0.47 = ( F3 @ X ) )
% 0.19/0.47 => ( member @ B @ B2 @ ( image @ A @ B @ F3 @ A5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % rev_image_eqI
% 0.19/0.47 thf(fact_150_Collect__mono__iff,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.19/0.47 = ( ! [X3: A] :
% 0.19/0.47 ( ( P @ X3 )
% 0.19/0.47 => ( Q @ X3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_mono_iff
% 0.19/0.47 thf(fact_151_set__eq__subset,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ^ [Y5: set @ A,Z: set @ A] : ( Y5 = Z ) )
% 0.19/0.47 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.19/0.47 & ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % set_eq_subset
% 0.19/0.47 thf(fact_152_subset__trans,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_trans
% 0.19/0.47 thf(fact_153_Collect__mono,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.47 ( ! [X4: A] :
% 0.19/0.47 ( ( P @ X4 )
% 0.19/0.47 => ( Q @ X4 ) )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_mono
% 0.19/0.47 thf(fact_154_subset__refl,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% 0.19/0.47
% 0.19/0.47 % subset_refl
% 0.19/0.47 thf(fact_155_subset__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.19/0.47 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.19/0.47 ! [T3: A] :
% 0.19/0.47 ( ( member @ A @ T3 @ A6 )
% 0.19/0.47 => ( member @ A @ T3 @ B6 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_iff
% 0.19/0.47 thf(fact_156_equalityD2,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( A5 = B5 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % equalityD2
% 0.19/0.47 thf(fact_157_equalityD1,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( A5 = B5 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % equalityD1
% 0.19/0.47 thf(fact_158_subset__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.19/0.47 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.19/0.47 ! [X3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ A6 )
% 0.19/0.47 => ( member @ A @ X3 @ B6 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_eq
% 0.19/0.47 thf(fact_159_equalityE,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( A5 = B5 )
% 0.19/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equalityE
% 0.19/0.47 thf(fact_160_subsetD,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A,C3: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ( member @ A @ C3 @ A5 )
% 0.19/0.47 => ( member @ A @ C3 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subsetD
% 0.19/0.47 thf(fact_161_in__mono,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A,X: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ( member @ A @ X @ A5 )
% 0.19/0.47 => ( member @ A @ X @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % in_mono
% 0.19/0.47 thf(fact_162_double__complement,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] :
% 0.19/0.47 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
% 0.19/0.47 = A5 ) ).
% 0.19/0.47
% 0.19/0.47 % double_complement
% 0.19/0.47 thf(fact_163_ComplD,axiom,
% 0.19/0.47 ! [A: $tType,C3: A,A5: set @ A] :
% 0.19/0.47 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
% 0.19/0.47 => ~ ( member @ A @ C3 @ A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % ComplD
% 0.19/0.47 thf(fact_164_imageE,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,B2: A,F3: B > A,A5: set @ B] :
% 0.19/0.47 ( ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A5 ) )
% 0.19/0.47 => ~ ! [X4: B] :
% 0.19/0.47 ( ( B2
% 0.19/0.47 = ( F3 @ X4 ) )
% 0.19/0.47 => ~ ( member @ B @ X4 @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % imageE
% 0.19/0.47 thf(fact_165_image__image,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType,F3: B > A,G: C > B,A5: set @ C] :
% 0.19/0.47 ( ( image @ B @ A @ F3 @ ( image @ C @ B @ G @ A5 ) )
% 0.19/0.47 = ( image @ C @ A
% 0.19/0.47 @ ^ [X3: C] : ( F3 @ ( G @ X3 ) )
% 0.19/0.47 @ A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_image
% 0.19/0.47 thf(fact_166_Compr__image__eq,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P: A > $o] :
% 0.19/0.47 ( ( collect @ A
% 0.19/0.47 @ ^ [X3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ ( image @ B @ A @ F3 @ A5 ) )
% 0.19/0.47 & ( P @ X3 ) ) )
% 0.19/0.47 = ( image @ B @ A @ F3
% 0.19/0.47 @ ( collect @ B
% 0.19/0.47 @ ^ [X3: B] :
% 0.19/0.47 ( ( member @ B @ X3 @ A5 )
% 0.19/0.47 & ( P @ ( F3 @ X3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Compr_image_eq
% 0.19/0.47 thf(fact_167_less__eq__set__def,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.19/0.47 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.19/0.47 ( ord_less_eq @ ( A > $o )
% 0.19/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ A6 )
% 0.19/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ B6 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_eq_set_def
% 0.19/0.47 thf(fact_168_Collect__subset,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,P: A > $o] :
% 0.19/0.47 ( ord_less_eq @ ( set @ A )
% 0.19/0.47 @ ( collect @ A
% 0.19/0.47 @ ^ [X3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ A5 )
% 0.19/0.47 & ( P @ X3 ) ) )
% 0.19/0.47 @ A5 ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_subset
% 0.19/0.47 thf(fact_169_Collect__neg__eq,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o] :
% 0.19/0.47 ( ( collect @ A
% 0.19/0.47 @ ^ [X3: A] :
% 0.19/0.47 ~ ( P @ X3 ) )
% 0.19/0.47 = ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_neg_eq
% 0.19/0.47 thf(fact_170_Compl__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( uminus_uminus @ ( set @ A ) )
% 0.19/0.47 = ( ^ [A6: set @ A] :
% 0.19/0.47 ( collect @ A
% 0.19/0.47 @ ^ [X3: A] :
% 0.19/0.47 ~ ( member @ A @ X3 @ A6 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Compl_eq
% 0.19/0.47 thf(fact_171_uminus__set__def,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( uminus_uminus @ ( set @ A ) )
% 0.19/0.47 = ( ^ [A6: set @ A] :
% 0.19/0.47 ( collect @ A
% 0.19/0.47 @ ( uminus_uminus @ ( A > $o )
% 0.19/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ A6 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % uminus_set_def
% 0.19/0.47 thf(fact_172_image__mono,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,A5: set @ A,B5: set @ A,F3: A > B] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ ( image @ A @ B @ F3 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_mono
% 0.19/0.47 thf(fact_173_image__subsetI,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,A5: set @ A,F3: A > B,B5: set @ B] :
% 0.19/0.47 ( ! [X4: A] :
% 0.19/0.47 ( ( member @ A @ X4 @ A5 )
% 0.19/0.47 => ( member @ B @ ( F3 @ X4 ) @ B5 ) )
% 0.19/0.47 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ B5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_subsetI
% 0.19/0.47 thf(fact_174_subset__imageE,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A5: set @ B] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F3 @ A5 ) )
% 0.19/0.47 => ~ ! [C5: set @ B] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A5 )
% 0.19/0.47 => ( B5
% 0.19/0.47 != ( image @ B @ A @ F3 @ C5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_imageE
% 0.19/0.47 thf(fact_175_image__subset__iff,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,F3: B > A,A5: set @ B,B5: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ A5 ) @ B5 )
% 0.19/0.47 = ( ! [X3: B] :
% 0.19/0.47 ( ( member @ B @ X3 @ A5 )
% 0.19/0.47 => ( member @ A @ ( F3 @ X3 ) @ B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_subset_iff
% 0.19/0.47 thf(fact_176_subset__image__iff,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A5: set @ B] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F3 @ A5 ) )
% 0.19/0.47 = ( ? [AA: set @ B] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ B ) @ AA @ A5 )
% 0.19/0.47 & ( B5
% 0.19/0.47 = ( image @ B @ A @ F3 @ AA ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_image_iff
% 0.19/0.47 thf(fact_177_repd_Oelims,axiom,
% 0.19/0.47 ! [C: $tType] :
% 0.19/0.47 ( ( finite_finite @ C )
% 0.19/0.47 => ! [X: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Xa: C,Xb: real,Y: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 0.19/0.47 ( ( ( denotational_repd @ C @ X @ Xa @ Xb )
% 0.19/0.47 = Y )
% 0.19/0.47 => ( Y
% 0.19/0.47 = ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ X )
% 0.19/0.47 @ ( finite1990238425lambda @ C @ real
% 0.19/0.47 @ ^ [Y3: C] : ( if @ real @ ( Xa = Y3 ) @ Xb @ ( finite1433825200ec_nth @ real @ C @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ X ) @ Y3 ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % repd.elims
% 0.19/0.47 thf(fact_178_repd_Osimps,axiom,
% 0.19/0.47 ! [C: $tType] :
% 0.19/0.47 ( ( finite_finite @ C )
% 0.19/0.47 => ( ( denotational_repd @ C )
% 0.19/0.47 = ( ^ [V3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),X3: C,R3: real] :
% 0.19/0.47 ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V3 )
% 0.19/0.47 @ ( finite1990238425lambda @ C @ real
% 0.19/0.47 @ ^ [Y3: C] : ( if @ real @ ( X3 = Y3 ) @ R3 @ ( finite1433825200ec_nth @ real @ C @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V3 ) @ Y3 ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % repd.simps
% 0.19/0.47 thf(fact_179_nth__map__matrix,axiom,
% 0.19/0.47 ! [A: $tType,C: $tType,D: $tType,B: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ B )
% 0.19/0.47 & ( finite_finite @ C ) )
% 0.19/0.47 => ! [F3: D > A,X: finite_Cartesian_vec @ ( finite_Cartesian_vec @ D @ B ) @ C,I4: C,J2: B] :
% 0.19/0.47 ( ( finite1433825200ec_nth @ A @ B @ ( finite1433825200ec_nth @ ( finite_Cartesian_vec @ A @ B ) @ C @ ( finite1614961371matrix @ D @ A @ B @ C @ F3 @ X ) @ I4 ) @ J2 )
% 0.19/0.47 = ( F3 @ ( finite1433825200ec_nth @ D @ B @ ( finite1433825200ec_nth @ ( finite_Cartesian_vec @ D @ B ) @ C @ X @ I4 ) @ J2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nth_map_matrix
% 0.19/0.47 thf(fact_180_image__subset__iff__funcset,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,F4: B > A,A5: set @ B,B5: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A5 ) @ B5 )
% 0.19/0.47 = ( member @ ( B > A ) @ F4
% 0.19/0.47 @ ( pi @ B @ A @ A5
% 0.19/0.47 @ ^ [Uu: B] : B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_subset_iff_funcset
% 0.19/0.47 thf(fact_181_Pi__I,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,A5: set @ A,F3: A > B,B5: A > ( set @ B )] :
% 0.19/0.47 ( ! [X4: A] :
% 0.19/0.47 ( ( member @ A @ X4 @ A5 )
% 0.19/0.47 => ( member @ B @ ( F3 @ X4 ) @ ( B5 @ X4 ) ) )
% 0.19/0.47 => ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ A5 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Pi_I
% 0.19/0.47 thf(fact_182_PiE,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B5: A > ( set @ B ),X: A] :
% 0.19/0.47 ( ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ A5 @ B5 ) )
% 0.19/0.47 => ( ~ ( member @ B @ ( F3 @ X ) @ ( B5 @ X ) )
% 0.19/0.47 => ~ ( member @ A @ X @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % PiE
% 0.19/0.47 thf(fact_183_Pi__I_H,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,A5: set @ A,F3: A > B,B5: A > ( set @ B )] :
% 0.19/0.47 ( ! [X4: A] :
% 0.19/0.47 ( ( member @ A @ X4 @ A5 )
% 0.19/0.47 => ( member @ B @ ( F3 @ X4 ) @ ( B5 @ X4 ) ) )
% 0.19/0.47 => ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ A5 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Pi_I'
% 0.19/0.47 thf(fact_184_Pi__iff,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F3: A > B,I2: set @ A,X8: A > ( set @ B )] :
% 0.19/0.47 ( ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ I2 @ X8 ) )
% 0.19/0.47 = ( ! [X3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ I2 )
% 0.19/0.47 => ( member @ B @ ( F3 @ X3 ) @ ( X8 @ X3 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Pi_iff
% 0.19/0.47 thf(fact_185_Pi__mem,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B5: A > ( set @ B ),X: A] :
% 0.19/0.47 ( ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ A5 @ B5 ) )
% 0.19/0.47 => ( ( member @ A @ X @ A5 )
% 0.19/0.47 => ( member @ B @ ( F3 @ X ) @ ( B5 @ X ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Pi_mem
% 0.19/0.47 thf(fact_186_Pi__cong,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,A5: set @ A,F3: A > B,G: A > B,B5: A > ( set @ B )] :
% 0.19/0.47 ( ! [W: A] :
% 0.19/0.47 ( ( member @ A @ W @ A5 )
% 0.19/0.47 => ( ( F3 @ W )
% 0.19/0.47 = ( G @ W ) ) )
% 0.19/0.47 => ( ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ A5 @ B5 ) )
% 0.19/0.47 = ( member @ ( A > B ) @ G @ ( pi @ A @ B @ A5 @ B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Pi_cong
% 0.19/0.47 thf(fact_187_funcsetI,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,A5: set @ A,F3: A > B,B5: set @ B] :
% 0.19/0.47 ( ! [X4: A] :
% 0.19/0.47 ( ( member @ A @ X4 @ A5 )
% 0.19/0.47 => ( member @ B @ ( F3 @ X4 ) @ B5 ) )
% 0.19/0.47 => ( member @ ( A > B ) @ F3
% 0.19/0.47 @ ( pi @ A @ B @ A5
% 0.19/0.47 @ ^ [Uu: A] : B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % funcsetI
% 0.19/0.47 thf(fact_188_funcset__id,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] :
% 0.19/0.47 ( member @ ( A > A )
% 0.19/0.47 @ ^ [X3: A] : X3
% 0.19/0.47 @ ( pi @ A @ A @ A5
% 0.19/0.47 @ ^ [Uu: A] : A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % funcset_id
% 0.19/0.47 thf(fact_189_funcset__mem,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B5: set @ B,X: A] :
% 0.19/0.47 ( ( member @ ( A > B ) @ F3
% 0.19/0.47 @ ( pi @ A @ B @ A5
% 0.19/0.47 @ ^ [Uu: A] : B5 ) )
% 0.19/0.47 => ( ( member @ A @ X @ A5 )
% 0.19/0.47 => ( member @ B @ ( F3 @ X ) @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % funcset_mem
% 0.19/0.47 thf(fact_190_Pi__mono,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,A5: set @ A,B5: A > ( set @ B ),C4: A > ( set @ B )] :
% 0.19/0.47 ( ! [X4: A] :
% 0.19/0.47 ( ( member @ A @ X4 @ A5 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ B ) @ ( B5 @ X4 ) @ ( C4 @ X4 ) ) )
% 0.19/0.47 => ( ord_less_eq @ ( set @ ( A > B ) ) @ ( pi @ A @ B @ A5 @ B5 ) @ ( pi @ A @ B @ A5 @ C4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Pi_mono
% 0.19/0.47 thf(fact_191_Pi__anti__mono,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,A7: set @ A,A5: set @ A,B5: A > ( set @ B )] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ A5 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ ( A > B ) ) @ ( pi @ A @ B @ A5 @ B5 ) @ ( pi @ A @ B @ A7 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Pi_anti_mono
% 0.19/0.47 thf(fact_192_map__matrix__def,axiom,
% 0.19/0.47 ! [I: $tType,J: $tType,B: $tType,A: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ J )
% 0.19/0.47 & ( finite_finite @ I ) )
% 0.19/0.47 => ( ( finite1614961371matrix @ A @ B @ I @ J )
% 0.19/0.47 = ( ^ [F5: A > B,X3: finite_Cartesian_vec @ ( finite_Cartesian_vec @ A @ I ) @ J] :
% 0.19/0.47 ( finite1990238425lambda @ J @ ( finite_Cartesian_vec @ B @ I )
% 0.19/0.47 @ ^ [I3: J] :
% 0.19/0.47 ( finite1990238425lambda @ I @ B
% 0.19/0.47 @ ^ [J3: I] : ( F5 @ ( finite1433825200ec_nth @ A @ I @ ( finite1433825200ec_nth @ ( finite_Cartesian_vec @ A @ I ) @ J @ X3 @ I3 ) @ J3 ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % map_matrix_def
% 0.19/0.47 thf(fact_193_funcset__image,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B5: set @ B] :
% 0.19/0.47 ( ( member @ ( A > B ) @ F3
% 0.19/0.47 @ ( pi @ A @ B @ A5
% 0.19/0.47 @ ^ [Uu: A] : B5 ) )
% 0.19/0.47 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ B5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % funcset_image
% 0.19/0.47 thf(fact_194_old_Oprod_Orec,axiom,
% 0.19/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.19/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.19/0.47 = ( F1 @ A2 @ B2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % old.prod.rec
% 0.19/0.47 thf(fact_195_repv_Osimps,axiom,
% 0.19/0.47 ! [C: $tType] :
% 0.19/0.47 ( ( finite_finite @ C )
% 0.19/0.47 => ( ( denotational_repv @ C )
% 0.19/0.47 = ( ^ [V3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),X3: C,R3: real] :
% 0.19/0.47 ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )
% 0.19/0.47 @ ( finite1990238425lambda @ C @ real
% 0.19/0.47 @ ^ [Y3: C] : ( if @ real @ ( X3 = Y3 ) @ R3 @ ( finite1433825200ec_nth @ real @ C @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V3 ) @ Y3 ) ) )
% 0.19/0.47 @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V3 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % repv.simps
% 0.19/0.47 thf(fact_196_repv_Oelims,axiom,
% 0.19/0.47 ! [C: $tType] :
% 0.19/0.47 ( ( finite_finite @ C )
% 0.19/0.47 => ! [X: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Xa: C,Xb: real,Y: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 0.19/0.47 ( ( ( denotational_repv @ C @ X @ Xa @ Xb )
% 0.19/0.47 = Y )
% 0.19/0.47 => ( Y
% 0.19/0.47 = ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )
% 0.19/0.47 @ ( finite1990238425lambda @ C @ real
% 0.19/0.47 @ ^ [Y3: C] : ( if @ real @ ( Xa = Y3 ) @ Xb @ ( finite1433825200ec_nth @ real @ C @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ X ) @ Y3 ) ) )
% 0.19/0.47 @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ X ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % repv.elims
% 0.19/0.47 thf(fact_197_exI__realizer,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,P: A > B > $o,Y: A,X: B] :
% 0.19/0.47 ( ( P @ Y @ X )
% 0.19/0.47 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % exI_realizer
% 0.19/0.47 thf(fact_198_conjI__realizer,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
% 0.19/0.47 ( ( P @ P2 )
% 0.19/0.47 => ( ( Q @ Q2 )
% 0.19/0.47 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
% 0.19/0.47 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % conjI_realizer
% 0.19/0.47 thf(fact_199_exE__realizer_H,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A] :
% 0.19/0.47 ( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
% 0.19/0.47 => ~ ! [X4: B,Y4: A] :
% 0.19/0.47 ~ ( P @ Y4 @ X4 ) ) ).
% 0.19/0.47
% 0.19/0.47 % exE_realizer'
% 0.19/0.47 thf(fact_200_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
% 0.19/0.47 ( ( P @ X @ Y )
% 0.19/0.47 => ( ( A2
% 0.19/0.47 = ( product_Pair @ A @ B @ X @ Y ) )
% 0.19/0.47 => ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % BNF_Greatest_Fixpoint.subst_Pair
% 0.19/0.47 thf(fact_201_concrete__v_Opelims,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,C: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ C )
% 0.19/0.47 & ( finite_finite @ A )
% 0.19/0.47 & ( finite_finite @ B ) )
% 0.19/0.47 => ! [X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Xa: ode @ A @ C,Xb: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Xc: finite_Cartesian_vec @ real @ C,Y: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 0.19/0.47 ( ( ( denota1738237502rete_v @ A @ B @ C @ X @ Xa @ Xb @ Xc )
% 0.19/0.47 = Y )
% 0.19/0.47 => ( ( accp @ ( product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) @ ( denota2028625979_v_rel @ A @ B @ C ) @ ( product_Pair @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ X @ ( product_Pair @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Xa @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb @ Xc ) ) ) )
% 0.30/0.47 => ~ ( ( Y
% 0.30/0.47 = ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )
% 0.30/0.47 @ ( finite1990238425lambda @ C @ real
% 0.30/0.47 @ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inl @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ X @ Xa ) ) @ Xc @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb ) ) @ I3 ) )
% 0.30/0.47 @ ( finite1990238425lambda @ C @ real
% 0.30/0.47 @ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inr @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ X @ Xa ) ) @ ( denotational_ODE_sem @ A @ B @ C @ X @ Xa @ Xc ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb ) ) @ I3 ) ) ) )
% 0.30/0.47 => ~ ( accp @ ( product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) @ ( denota2028625979_v_rel @ A @ B @ C ) @ ( product_Pair @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ X @ ( product_Pair @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Xa @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb @ Xc ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % concrete_v.pelims
% 0.30/0.47 thf(fact_202_not__arg__cong__Inr,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X: A,Y: A] :
% 0.30/0.47 ( ( X != Y )
% 0.30/0.47 => ( ( sum_Inr @ A @ B @ X )
% 0.30/0.47 != ( sum_Inr @ A @ B @ Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_arg_cong_Inr
% 0.30/0.47 thf(fact_203_subset__Collect__iff,axiom,
% 0.30/0.47 ! [A: $tType,B5: set @ A,A5: set @ A,P: A > $o] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B5
% 0.30/0.47 @ ( collect @ A
% 0.30/0.47 @ ^ [X3: A] :
% 0.30/0.47 ( ( member @ A @ X3 @ A5 )
% 0.30/0.47 & ( P @ X3 ) ) ) )
% 0.30/0.47 = ( ! [X3: A] :
% 0.30/0.47 ( ( member @ A @ X3 @ B5 )
% 0.30/0.47 => ( P @ X3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_Collect_iff
% 0.30/0.47 thf(fact_204_subset__CollectI,axiom,
% 0.30/0.47 ! [A: $tType,B5: set @ A,A5: set @ A,Q: A > $o,P: A > $o] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
% 0.30/0.47 => ( ! [X4: A] :
% 0.30/0.47 ( ( member @ A @ X4 @ B5 )
% 0.30/0.47 => ( ( Q @ X4 )
% 0.30/0.47 => ( P @ X4 ) ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A )
% 0.30/0.47 @ ( collect @ A
% 0.30/0.47 @ ^ [X3: A] :
% 0.30/0.47 ( ( member @ A @ X3 @ B5 )
% 0.30/0.47 & ( Q @ X3 ) ) )
% 0.30/0.47 @ ( collect @ A
% 0.30/0.47 @ ^ [X3: A] :
% 0.30/0.47 ( ( member @ A @ X3 @ A5 )
% 0.30/0.47 & ( P @ X3 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_CollectI
% 0.30/0.47 thf(fact_205_obj__sumE,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
% 0.30/0.47 ( ! [X4: A] :
% 0.30/0.47 ( S2
% 0.30/0.47 != ( sum_Inl @ A @ B @ X4 ) )
% 0.30/0.47 => ~ ! [X4: B] :
% 0.30/0.47 ( S2
% 0.30/0.47 != ( sum_Inr @ B @ A @ X4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % obj_sumE
% 0.30/0.47 thf(fact_206_predicate1I,axiom,
% 0.30/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.47 ( ! [X4: A] :
% 0.30/0.47 ( ( P @ X4 )
% 0.30/0.47 => ( Q @ X4 ) )
% 0.30/0.47 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 0.30/0.47
% 0.30/0.47 % predicate1I
% 0.30/0.47 thf(fact_207_minus__le__self__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( lattic1601792062up_add @ A )
% 0.30/0.47 => ! [A2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 0.30/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minus_le_self_iff
% 0.30/0.47 thf(fact_208_order__refl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_refl
% 0.30/0.47 thf(fact_209_order__subst1,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( order @ B )
% 0.30/0.47 & ( order @ A ) )
% 0.30/0.47 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
% 0.30/0.47 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 0.30/0.47 => ( ! [X4: B,Y4: B] :
% 0.30/0.47 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) )
% 0.30/0.47 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_subst1
% 0.30/0.47 thf(fact_210_order__subst2,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType] :
% 0.30/0.47 ( ( ( order @ C )
% 0.30/0.47 & ( order @ A ) )
% 0.30/0.47 => ! [A2: A,B2: A,F3: A > C,C3: C] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
% 0.30/0.47 => ( ! [X4: A,Y4: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.30/0.47 => ( ord_less_eq @ C @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) )
% 0.30/0.47 => ( ord_less_eq @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_subst2
% 0.30/0.47 thf(fact_211_ord__eq__le__subst,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( ord @ B )
% 0.30/0.47 & ( ord @ A ) )
% 0.30/0.47 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 0.30/0.47 ( ( A2
% 0.30/0.47 = ( F3 @ B2 ) )
% 0.30/0.47 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 0.30/0.47 => ( ! [X4: B,Y4: B] :
% 0.30/0.47 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) )
% 0.30/0.47 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_le_subst
% 0.30/0.47 thf(fact_212_ord__le__eq__subst,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( ord @ B )
% 0.30/0.47 & ( ord @ A ) )
% 0.30/0.47 => ! [A2: A,B2: A,F3: A > B,C3: B] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( ( F3 @ B2 )
% 0.30/0.47 = C3 )
% 0.30/0.47 => ( ! [X4: A,Y4: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_le_eq_subst
% 0.30/0.47 thf(fact_213_eq__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 0.30/0.47 = ( ^ [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.47 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_iff
% 0.30/0.47 thf(fact_214_antisym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.47 => ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antisym
% 0.30/0.47 thf(fact_215_linear,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linear
% 0.30/0.47 thf(fact_216_eq__refl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( X = Y )
% 0.30/0.47 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_refl
% 0.30/0.47 thf(fact_217_le__cases,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_cases
% 0.30/0.47 thf(fact_218_order_Otrans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 0.30/0.47 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.trans
% 0.30/0.47 thf(fact_219_le__cases3,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X: A,Y: A,Z2: A] :
% 0.30/0.47 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 0.30/0.47 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 0.30/0.47 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 0.30/0.47 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.30/0.47 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 0.30/0.47 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_cases3
% 0.30/0.47 thf(fact_220_antisym__conv,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [Y: A,X: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.47 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 = ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antisym_conv
% 0.30/0.47 thf(fact_221_order__class_Oorder_Oeq__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 0.30/0.47 = ( ^ [A8: A,B7: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A8 @ B7 )
% 0.30/0.47 & ( ord_less_eq @ A @ B7 @ A8 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_class.order.eq_iff
% 0.30/0.47 thf(fact_222_ord__eq__le__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C3: A] :
% 0.30/0.47 ( ( A2 = B2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 0.30/0.47 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_le_trans
% 0.30/0.47 thf(fact_223_ord__le__eq__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( B2 = C3 )
% 0.30/0.47 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_le_eq_trans
% 0.30/0.47 thf(fact_224_order__class_Oorder_Oantisym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_class.order.antisym
% 0.30/0.47 thf(fact_225_order__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X: A,Y: A,Z2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.30/0.47 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_trans
% 0.30/0.47 thf(fact_226_dual__order_Orefl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.refl
% 0.30/0.47 thf(fact_227_linorder__wlog,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.30/0.47 ( ! [A4: A,B4: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.47 => ( P @ A4 @ B4 ) )
% 0.30/0.47 => ( ! [A4: A,B4: A] :
% 0.30/0.47 ( ( P @ B4 @ A4 )
% 0.30/0.47 => ( P @ A4 @ B4 ) )
% 0.30/0.47 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linorder_wlog
% 0.30/0.47 thf(fact_228_dual__order_Otrans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [B2: A,A2: A,C3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ C3 @ B2 )
% 0.30/0.47 => ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.trans
% 0.30/0.47 thf(fact_229_dual__order_Oeq__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 0.30/0.47 = ( ^ [A8: A,B7: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B7 @ A8 )
% 0.30/0.47 & ( ord_less_eq @ A @ A8 @ B7 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.eq_iff
% 0.30/0.47 thf(fact_230_dual__order_Oantisym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [B2: A,A2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.antisym
% 0.30/0.47 thf(fact_231_predicate1D,axiom,
% 0.30/0.47 ! [A: $tType,P: A > $o,Q: A > $o,X: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.30/0.47 => ( ( P @ X )
% 0.30/0.47 => ( Q @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % predicate1D
% 0.30/0.47 thf(fact_232_rev__predicate1D,axiom,
% 0.30/0.47 ! [A: $tType,P: A > $o,X: A,Q: A > $o] :
% 0.30/0.47 ( ( P @ X )
% 0.30/0.47 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.30/0.47 => ( Q @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % rev_predicate1D
% 0.30/0.47 thf(fact_233_le__funD,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B )
% 0.30/0.47 => ! [F3: A > B,G: A > B,X: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_funD
% 0.30/0.47 thf(fact_234_le__funE,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B )
% 0.30/0.47 => ! [F3: A > B,G: A > B,X: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_funE
% 0.30/0.47 thf(fact_235_le__funI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B )
% 0.30/0.47 => ! [F3: A > B,G: A > B] :
% 0.30/0.47 ( ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
% 0.30/0.47 => ( ord_less_eq @ ( A > B ) @ F3 @ G ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_funI
% 0.30/0.47 thf(fact_236_le__fun__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B )
% 0.30/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.30/0.47 = ( ^ [F5: A > B,G4: A > B] :
% 0.30/0.47 ! [X3: A] : ( ord_less_eq @ B @ ( F5 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_fun_def
% 0.30/0.47 thf(fact_237_le__minus__self__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( lattic1601792062up_add @ A )
% 0.30/0.47 => ! [A2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 0.30/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_minus_self_iff
% 0.30/0.47 thf(fact_238_old_Osum_Oinject_I1_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A2: A,A3: A] :
% 0.30/0.47 ( ( ( sum_Inl @ A @ B @ A2 )
% 0.30/0.47 = ( sum_Inl @ A @ B @ A3 ) )
% 0.30/0.47 = ( A2 = A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.sum.inject(1)
% 0.30/0.47 thf(fact_239_sum_Oinject_I1_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X1: A,Y1: A] :
% 0.30/0.47 ( ( ( sum_Inl @ A @ B @ X1 )
% 0.30/0.47 = ( sum_Inl @ A @ B @ Y1 ) )
% 0.30/0.47 = ( X1 = Y1 ) ) ).
% 0.30/0.47
% 0.30/0.47 % sum.inject(1)
% 0.30/0.47 thf(fact_240_sum_Oinject_I2_J,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X22: B,Y2: B] :
% 0.30/0.47 ( ( ( sum_Inr @ B @ A @ X22 )
% 0.30/0.47 = ( sum_Inr @ B @ A @ Y2 ) )
% 0.30/0.47 = ( X22 = Y2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % sum.inject(2)
% 0.30/0.47 thf(fact_241_old_Osum_Oinject_I2_J,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,B2: B,B3: B] :
% 0.30/0.47 ( ( ( sum_Inr @ B @ A @ B2 )
% 0.30/0.47 = ( sum_Inr @ B @ A @ B3 ) )
% 0.30/0.47 = ( B2 = B3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.sum.inject(2)
% 0.30/0.47 thf(fact_242_Inr__inject,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X: B,Y: B] :
% 0.30/0.47 ( ( ( sum_Inr @ B @ A @ X )
% 0.30/0.47 = ( sum_Inr @ B @ A @ Y ) )
% 0.30/0.47 => ( X = Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % Inr_inject
% 0.30/0.47 thf(fact_243_Inl__inject,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X: A,Y: A] :
% 0.30/0.47 ( ( ( sum_Inl @ A @ B @ X )
% 0.30/0.47 = ( sum_Inl @ A @ B @ Y ) )
% 0.30/0.47 => ( X = Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % Inl_inject
% 0.30/0.47 thf(fact_244_sum_Odistinct_I1_J,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X1: A,X22: B] :
% 0.30/0.47 ( ( sum_Inl @ A @ B @ X1 )
% 0.30/0.47 != ( sum_Inr @ B @ A @ X22 ) ) ).
% 0.30/0.47
% 0.30/0.47 % sum.distinct(1)
% 0.30/0.47 thf(fact_245_old_Osum_Odistinct_I2_J,axiom,
% 0.30/0.47 ! [B8: $tType,A9: $tType,B9: B8,A10: A9] :
% 0.30/0.47 ( ( sum_Inr @ B8 @ A9 @ B9 )
% 0.30/0.47 != ( sum_Inl @ A9 @ B8 @ A10 ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.sum.distinct(2)
% 0.30/0.47 thf(fact_246_old_Osum_Odistinct_I1_J,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A2: A,B3: B] :
% 0.30/0.47 ( ( sum_Inl @ A @ B @ A2 )
% 0.30/0.47 != ( sum_Inr @ B @ A @ B3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.sum.distinct(1)
% 0.30/0.47 thf(fact_247_sumE,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
% 0.30/0.47 ( ! [X4: A] :
% 0.30/0.47 ( S2
% 0.30/0.47 != ( sum_Inl @ A @ B @ X4 ) )
% 0.30/0.47 => ~ ! [Y4: B] :
% 0.30/0.47 ( S2
% 0.30/0.47 != ( sum_Inr @ B @ A @ Y4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sumE
% 0.30/0.47 thf(fact_248_Inr__not__Inl,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,B2: B,A2: A] :
% 0.30/0.47 ( ( sum_Inr @ B @ A @ B2 )
% 0.30/0.47 != ( sum_Inl @ A @ B @ A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Inr_not_Inl
% 0.30/0.47 thf(fact_249_split__sum__ex,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 0.30/0.47 ? [X9: sum_sum @ A @ B] : ( P3 @ X9 ) )
% 0.30/0.47 = ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
% 0.30/0.47 ( ? [X3: A] : ( P4 @ ( sum_Inl @ A @ B @ X3 ) )
% 0.30/0.47 | ? [X3: B] : ( P4 @ ( sum_Inr @ B @ A @ X3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % split_sum_ex
% 0.30/0.47 thf(fact_250_split__sum__all,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 0.30/0.47 ! [X9: sum_sum @ A @ B] : ( P3 @ X9 ) )
% 0.30/0.47 = ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
% 0.30/0.47 ( ! [X3: A] : ( P4 @ ( sum_Inl @ A @ B @ X3 ) )
% 0.30/0.47 & ! [X3: B] : ( P4 @ ( sum_Inr @ B @ A @ X3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % split_sum_all
% 0.30/0.47 thf(fact_251_old_Osum_Oexhaust,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,Y: sum_sum @ A @ B] :
% 0.30/0.47 ( ! [A4: A] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( sum_Inl @ A @ B @ A4 ) )
% 0.30/0.47 => ~ ! [B4: B] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( sum_Inr @ B @ A @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.sum.exhaust
% 0.30/0.47 thf(fact_252_old_Osum_Oinducts,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
% 0.30/0.47 ( ! [A4: A] : ( P @ ( sum_Inl @ A @ B @ A4 ) )
% 0.30/0.47 => ( ! [B4: B] : ( P @ ( sum_Inr @ B @ A @ B4 ) )
% 0.30/0.47 => ( P @ Sum ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.sum.inducts
% 0.30/0.47 thf(fact_253_pred__subset__eq,axiom,
% 0.30/0.47 ! [A: $tType,R4: set @ A,S6: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( A > $o )
% 0.30/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ R4 )
% 0.30/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ S6 ) )
% 0.30/0.47 = ( ord_less_eq @ ( set @ A ) @ R4 @ S6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % pred_subset_eq
% 0.30/0.47 thf(fact_254_verit__minus__simplify_I4_J,axiom,
% 0.30/0.47 ! [B: $tType] :
% 0.30/0.47 ( ( group_add @ B )
% 0.30/0.47 => ! [B2: B] :
% 0.30/0.47 ( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
% 0.30/0.47 = B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % verit_minus_simplify(4)
% 0.30/0.47
% 0.30/0.47 % Subclasses (1)
% 0.30/0.47 thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( finite_finite @ A )
% 0.30/0.47 => ( type @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % Type constructors (59)
% 0.30/0.47 thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( ordere890947078_space @ A9 )
% 0.30/0.47 & ( finite_finite @ A11 ) )
% 0.30/0.47 => ( ordere890947078_space @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space_1,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( ordere890947078_space @ A9 )
% 0.30/0.47 & ( ordere890947078_space @ A11 ) )
% 0.30/0.47 => ( ordere890947078_space @ ( product_prod @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_2,axiom,
% 0.30/0.47 ordere890947078_space @ real ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( boolean_algebra @ A11 )
% 0.30/0.47 => ( boolean_algebra @ ( A9 > A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( preorder @ A11 )
% 0.30/0.47 => ( preorder @ ( A9 > A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( finite_finite @ A9 )
% 0.30/0.47 & ( finite_finite @ A11 ) )
% 0.30/0.47 => ( finite_finite @ ( A9 > A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( order @ A11 )
% 0.30/0.47 => ( order @ ( A9 > A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ord @ A11 )
% 0.30/0.47 => ( ord @ ( A9 > A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Groups_Ouminus,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( uminus @ A11 )
% 0.30/0.47 => ( uminus @ ( A9 > A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_3,axiom,
% 0.30/0.47 ! [A9: $tType] : ( boolean_algebra @ ( set @ A9 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.30/0.47 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
% 0.30/0.47 ! [A9: $tType] :
% 0.30/0.47 ( ( finite_finite @ A9 )
% 0.30/0.47 => ( finite_finite @ ( set @ A9 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 0.30/0.47 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 0.30/0.47 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Groups_Ouminus_8,axiom,
% 0.30/0.47 ! [A9: $tType] : ( uminus @ ( set @ A9 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Groups_Ozero,axiom,
% 0.30/0.47 ! [A9: $tType] :
% 0.30/0.47 ( ( zero @ A9 )
% 0.30/0.47 => ( zero @ ( set @ A9 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_9,axiom,
% 0.30/0.47 boolean_algebra @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
% 0.30/0.47 preorder @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.30/0.47 linorder @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Finite__Set_Ofinite_11,axiom,
% 0.30/0.47 finite_finite @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
% 0.30/0.47 order @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
% 0.30/0.47 ord @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Groups_Ouminus_14,axiom,
% 0.30/0.47 uminus @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Real_Oreal___Lattice__Algebras_Olattice__ab__group__add,axiom,
% 0.30/0.47 lattic1601792062up_add @ real ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
% 0.30/0.47 linord219039673up_add @ real ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
% 0.30/0.47 ordered_ab_group_add @ real ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Real_Oreal___Orderings_Opreorder_15,axiom,
% 0.30/0.47 preorder @ real ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Real_Oreal___Orderings_Olinorder_16,axiom,
% 0.30/0.47 linorder @ real ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 0.30/0.47 group_add @ real ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Real_Oreal___Orderings_Oorder_17,axiom,
% 0.30/0.47 order @ real ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Real_Oreal___Orderings_Oord_18,axiom,
% 0.30/0.47 ord @ real ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Real_Oreal___Groups_Ouminus_19,axiom,
% 0.30/0.47 uminus @ real ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Real_Oreal___Groups_Ozero_20,axiom,
% 0.30/0.47 zero @ real ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_21,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( finite_finite @ A9 )
% 0.30/0.47 & ( finite_finite @ A11 ) )
% 0.30/0.47 => ( finite_finite @ ( sum_sum @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__group__add_22,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( ordere890947078_space @ A9 )
% 0.30/0.47 & ( ordere890947078_space @ A11 ) )
% 0.30/0.47 => ( ordered_ab_group_add @ ( product_prod @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Lattices_Oboolean__algebra_23,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( boolean_algebra @ A9 )
% 0.30/0.47 & ( boolean_algebra @ A11 ) )
% 0.30/0.47 => ( boolean_algebra @ ( product_prod @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_24,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( preorder @ A9 )
% 0.30/0.47 & ( preorder @ A11 ) )
% 0.30/0.47 => ( preorder @ ( product_prod @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_25,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( finite_finite @ A9 )
% 0.30/0.47 & ( finite_finite @ A11 ) )
% 0.30/0.47 => ( finite_finite @ ( product_prod @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Groups_Ogroup__add_26,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( group_add @ A9 )
% 0.30/0.47 & ( group_add @ A11 ) )
% 0.30/0.47 => ( group_add @ ( product_prod @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Orderings_Oorder_27,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( order @ A9 )
% 0.30/0.47 & ( order @ A11 ) )
% 0.30/0.47 => ( order @ ( product_prod @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Orderings_Oord_28,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( ord @ A9 )
% 0.30/0.47 & ( ord @ A11 ) )
% 0.30/0.47 => ( ord @ ( product_prod @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Groups_Ouminus_29,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( uminus @ A9 )
% 0.30/0.47 & ( uminus @ A11 ) )
% 0.30/0.47 => ( uminus @ ( product_prod @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Groups_Ozero_30,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( zero @ A9 )
% 0.30/0.47 & ( zero @ A11 ) )
% 0.30/0.47 => ( zero @ ( product_prod @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Ounit___Lattices_Oboolean__algebra_31,axiom,
% 0.30/0.47 boolean_algebra @ product_unit ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_32,axiom,
% 0.30/0.47 preorder @ product_unit ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_33,axiom,
% 0.30/0.47 linorder @ product_unit ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_34,axiom,
% 0.30/0.47 finite_finite @ product_unit ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Ounit___Orderings_Oorder_35,axiom,
% 0.30/0.47 order @ product_unit ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Ounit___Orderings_Oord_36,axiom,
% 0.30/0.47 ord @ product_unit ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Ounit___Groups_Ouminus_37,axiom,
% 0.30/0.47 uminus @ product_unit ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__group__add_38,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( ordere890947078_space @ A9 )
% 0.30/0.47 & ( finite_finite @ A11 ) )
% 0.30/0.47 => ( ordered_ab_group_add @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Opreorder_39,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( order @ A9 )
% 0.30/0.47 & ( finite_finite @ A11 ) )
% 0.30/0.47 => ( preorder @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_40,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( linorder @ A9 )
% 0.30/0.47 & ( cARD_1 @ A11 ) )
% 0.30/0.47 => ( linorder @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_41,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( finite_finite @ A9 )
% 0.30/0.47 & ( finite_finite @ A11 ) )
% 0.30/0.47 => ( finite_finite @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ogroup__add_42,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( group_add @ A9 )
% 0.30/0.47 & ( finite_finite @ A11 ) )
% 0.30/0.47 => ( group_add @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oorder_43,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( order @ A9 )
% 0.30/0.47 & ( finite_finite @ A11 ) )
% 0.30/0.47 => ( order @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oord_44,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( ord @ A9 )
% 0.30/0.47 & ( finite_finite @ A11 ) )
% 0.30/0.47 => ( ord @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ouminus_45,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( uminus @ A9 )
% 0.30/0.47 & ( finite_finite @ A11 ) )
% 0.30/0.47 => ( uminus @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_46,axiom,
% 0.30/0.47 ! [A9: $tType,A11: $tType] :
% 0.30/0.47 ( ( ( zero @ A9 )
% 0.30/0.47 & ( finite_finite @ A11 ) )
% 0.30/0.47 => ( zero @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Helper facts (3)
% 0.30/0.47 thf(help_If_3_1_T,axiom,
% 0.30/0.47 ! [P: $o] :
% 0.30/0.47 ( ( P = $true )
% 0.30/0.47 | ( P = $false ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(help_If_2_1_T,axiom,
% 0.30/0.47 ! [A: $tType,X: A,Y: A] :
% 0.30/0.47 ( ( if @ A @ $false @ X @ Y )
% 0.30/0.47 = Y ) ).
% 0.30/0.47
% 0.30/0.47 thf(help_If_1_1_T,axiom,
% 0.30/0.47 ! [A: $tType,X: A,Y: A] :
% 0.30/0.47 ( ( if @ A @ $true @ X @ Y )
% 0.30/0.47 = X ) ).
% 0.30/0.47
% 0.30/0.47 % Free types (3)
% 0.30/0.47 thf(tfree_0,hypothesis,
% 0.30/0.47 finite_finite @ c ).
% 0.30/0.47
% 0.30/0.47 thf(tfree_1,hypothesis,
% 0.30/0.47 finite_finite @ a ).
% 0.30/0.47
% 0.30/0.47 thf(tfree_2,hypothesis,
% 0.30/0.47 finite_finite @ b ).
% 0.30/0.47
% 0.30/0.47 % Conjectures (12)
% 0.30/0.47 thf(conj_0,hypothesis,
% 0.30/0.47 member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( static_FVF @ a @ b @ c @ phi ) ).
% 0.30/0.47
% 0.30/0.47 thf(conj_1,hypothesis,
% 0.30/0.47 member @ c @ i2 @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ).
% 0.30/0.47
% 0.30/0.47 thf(conj_2,hypothesis,
% 0.30/0.47 ~ ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(conj_3,hypothesis,
% 0.30/0.47 ( ( ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 0.30/0.47 & ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) ) )
% 0.30/0.47 => ( ( ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
% 0.30/0.47 => ( ( finite1433825200ec_nth @ real @ c
% 0.30/0.47 @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 0.30/0.47 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 0.30/0.47 @ bb )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 0.30/0.47 @ i2 )
% 0.30/0.47 = ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ i2 ) ) )
% 0.30/0.47 & ( ~ ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
% 0.30/0.47 => ( ( finite1433825200ec_nth @ real @ c
% 0.30/0.47 @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 0.30/0.47 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 0.30/0.47 @ bb )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 0.30/0.47 @ i2 )
% 0.30/0.47 = ( finite1433825200ec_nth @ real @ c @ ab @ i2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(conj_4,hypothesis,
% 0.30/0.47 ( ( ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 0.30/0.47 & ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) ) )
% 0.30/0.47 => ( ( finite1433825200ec_nth @ real @ c @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
% 0.30/0.47 = ( finite1433825200ec_nth @ real @ c @ ab @ i2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(conj_5,hypothesis,
% 0.30/0.47 ( ( ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 0.30/0.47 & ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) ) )
% 0.30/0.47 => ( ( finite1433825200ec_nth @ real @ c
% 0.30/0.47 @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 0.30/0.47 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 0.30/0.47 @ bb )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 0.30/0.47 @ i2 )
% 0.30/0.47 = ( finite1433825200ec_nth @ real @ c @ bb @ i2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(conj_6,hypothesis,
% 0.30/0.47 ( ( ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 0.30/0.47 & ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) ) )
% 0.30/0.47 => ( ( finite1433825200ec_nth @ real @ c @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
% 0.30/0.47 = ( finite1433825200ec_nth @ real @ c @ bb @ i2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(conj_7,hypothesis,
% 0.30/0.47 ( ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 0.30/0.47 => ( ( ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
% 0.30/0.47 => ( ( finite1433825200ec_nth @ real @ c
% 0.30/0.47 @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 0.30/0.47 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 0.30/0.47 @ bb )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 0.30/0.47 @ i2 )
% 0.30/0.47 = ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ i2 ) ) )
% 0.30/0.47 & ( ~ ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
% 0.30/0.47 => ( ( finite1433825200ec_nth @ real @ c
% 0.30/0.47 @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 0.30/0.47 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 0.30/0.47 @ bb )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 0.30/0.47 @ i2 )
% 0.30/0.47 = ( finite1433825200ec_nth @ real @ c @ ab @ i2 ) ) ) ) )
% 0.30/0.47 & ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 0.30/0.47 => ( ( ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
% 0.30/0.47 => ( ( finite1433825200ec_nth @ real @ c
% 0.30/0.47 @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 0.30/0.47 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 0.30/0.47 @ bb )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 0.30/0.47 @ i2 )
% 0.30/0.47 = ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ i2 ) ) )
% 0.30/0.47 & ( ~ ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
% 0.30/0.47 => ( ( finite1433825200ec_nth @ real @ c
% 0.30/0.47 @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 0.30/0.47 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 0.30/0.47 @ bb )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 0.30/0.47 @ i2 )
% 0.30/0.47 = ( finite1433825200ec_nth @ real @ c @ ab @ i2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(conj_8,hypothesis,
% 0.30/0.47 ( ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 0.30/0.47 => ( ( finite1433825200ec_nth @ real @ c
% 0.30/0.47 @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 0.30/0.47 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 0.30/0.47 @ bb )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 0.30/0.47 @ i2 )
% 0.30/0.47 = ( finite1433825200ec_nth @ real @ c
% 0.30/0.47 @ ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
% 0.30/0.47 @ i2 ) ) )
% 0.30/0.47 & ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 0.30/0.47 => ( ( finite1433825200ec_nth @ real @ c
% 0.30/0.47 @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 0.30/0.47 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 0.30/0.47 @ bb )
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 0.30/0.47 @ i2 )
% 0.30/0.47 = ( finite1433825200ec_nth @ real @ c
% 0.30/0.47 @ ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa
% 0.30/0.47 @ ( finite1990238425lambda @ c @ real
% 0.30/0.47 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
% 0.30/0.47 @ i2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(conj_9,hypothesis,
% 0.30/0.47 ( ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 0.30/0.47 => ( ( finite1433825200ec_nth @ real @ c @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
% 0.30/0.47 = ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ i2 ) ) )
% 0.30/0.47 & ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 0.30/0.47 => ( ( finite1433825200ec_nth @ real @ c @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
% 0.30/0.47 = ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ i2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(conj_10,hypothesis,
% 0.30/0.47 ( ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 0.30/0.47 => ( ( finite1433825200ec_nth @ real @ c @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
% 0.30/0.47 = ( finite1433825200ec_nth @ real @ c @ ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa @ ( sol @ x ) ) @ i2 ) ) )
% 0.30/0.47 & ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 0.30/0.47 => ( ( finite1433825200ec_nth @ real @ c @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
% 0.30/0.47 = ( finite1433825200ec_nth @ real @ c @ ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa @ ( sol @ x ) ) @ i2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(conj_11,conjecture,
% 0.30/0.47 ( ( finite1433825200ec_nth @ real @ c @ ab @ i2 )
% 0.30/0.47 = ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ i2 ) ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 3.01/1.64 ---- Embedded in THF ---
% 3.01/1.65 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.01/1.65 %%% Generated on Tue Feb 24 08:24:17 EST 2026
% 3.01/1.65 %%% using '$$dhol' embedding, version 1.3.0.
% 3.01/1.65 %%% Logic specification used:
% 3.01/1.65 %%% thf(spec, logic, $$dhol).
% 3.01/1.65
% 3.01/1.65 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.iHbtgXCbd3/DTF2DTF15214.p
% 3.01/1.65 thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext, type, denota1663640101rp_ext: ($tType > ($tType > ($tType > ($tType > $tType))))).
% 3.01/1.65 thf(ty_t_Finite__Cartesian__Product_Ovec, type, finite_Cartesian_vec: ($tType > ($tType > $tType))).
% 3.01/1.65 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 3.01/1.65 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 3.01/1.65 thf(ty_t_Syntax_Oformula, type, formula: ($tType > ($tType > ($tType > $tType)))).
% 3.01/1.65 thf(ty_t_Sum__Type_Osum, type, sum_sum: ($tType > ($tType > $tType))).
% 3.01/1.65 thf(ty_t_Syntax_OODE, type, ode: ($tType > ($tType > $tType))).
% 3.01/1.65 thf(ty_t_Real_Oreal, type, real: $tType).
% 3.01/1.65 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.01/1.65 thf(ty_tf_d, type, d: $tType).
% 3.01/1.65 thf(ty_tf_c, type, c: $tType).
% 3.01/1.65 thf(ty_tf_b, type, b: $tType).
% 3.01/1.65 thf(ty_tf_a, type, a: $tType).
% 3.01/1.65 thf(sy_cl_Cardinality_OCARD__1, type, cARD_1: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space, type, ordere890947078_space: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_cl_Groups_Ouminus, type, uminus: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_cl_Lattices_Oboolean__algebra, type, boolean_algebra: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_cl_Lattice__Algebras_Olattice__ab__group__add, type, lattic1601792062up_add: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd, type, euclid1678468529ng_gcd: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring, type, euclid1155270486miring: (!> [A:$tType]: ($o))).
% 3.01/1.65 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__fml, type, coinci1993344360de_fml: (!> [A:$tType,B:$tType,C:$tType]: (((((formula @ A) @ B) @ C) > $o)))).
% 3.01/1.65 thf(sy_c_Denotational__Semantics_OODE__sem, type, denotational_ODE_sem: (!> [A:$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))))))).
% 3.01/1.65 thf(sy_c_Denotational__Semantics_OODE__vars, type, denota811733865E_vars: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((ode @ A) @ C) > (set @ C)))))).
% 3.01/1.65 thf(sy_c_Denotational__Semantics_OVSagree, type, denotational_VSagree: (!> [C:$tType]: ((((finite_Cartesian_vec @ real) @ C) > (((finite_Cartesian_vec @ real) @ C) > ((set @ C) > $o)))))).
% 3.01/1.65 thf(sy_c_Denotational__Semantics_OVagree, type, denotational_Vagree: (!> [C:$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)) > $o)))))).
% 3.01/1.65 thf(sy_c_Denotational__Semantics_Oconcrete__v, type, denota1738237502rete_v: (!> [A:$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))))))))).
% 3.01/1.65 thf(sy_c_Denotational__Semantics_Oconcrete__v__rel, type, denota2028625979_v_rel: (!> [A:$tType,B:$tType,C:$tType]: ((((product_prod @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ ((product_prod @ ((ode @ A) @ C)) @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((finite_Cartesian_vec @ real) @ C)))) > (((product_prod @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ ((product_prod @ ((ode @ A) @ C)) @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((finite_Cartesian_vec @ real) @ C)))) > $o))))).
% 3.01/1.65 thf(sy_c_Denotational__Semantics_Ofml__sem, type, denotational_fml_sem: (!> [A:$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)))))))).
% 3.01/1.65 thf(sy_c_Denotational__Semantics_Omk__v, type, denotational_mk_v: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((ode @ A) @ C) > (((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) > (((finite_Cartesian_vec @ real) @ C) > ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))))))))).
% 3.01/1.65 thf(sy_c_Denotational__Semantics_Omk__xode, type, denotational_mk_xode: (!> [A:$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)))))))).
% 3.01/1.65 thf(sy_c_Denotational__Semantics_Orepd, type, denotational_repd: (!> [C:$tType]: ((((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) > (C > (real > ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)))))))).
% 3.01/1.65 thf(sy_c_Denotational__Semantics_Orepv, type, denotational_repv: (!> [C:$tType]: ((((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) > (C > (real > ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)))))))).
% 3.01/1.65 thf(sy_c_Denotational__Semantics_OsemBV, type, denotational_semBV: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((ode @ A) @ C) > (set @ ((sum_sum @ C) @ C))))))).
% 3.01/1.65 thf(sy_c_Finite__Cartesian__Product_Omap__matrix, type, finite1614961371matrix: (!> [A:$tType,B:$tType,I:$tType,J:$tType]: (((A > B) > (((finite_Cartesian_vec @ ((finite_Cartesian_vec @ A) @ I)) @ J) > ((finite_Cartesian_vec @ ((finite_Cartesian_vec @ B) @ I)) @ J)))))).
% 3.01/1.65 thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__lambda, type, finite1990238425lambda: (!> [B:$tType,A:$tType]: (((B > A) > ((finite_Cartesian_vec @ A) @ B))))).
% 3.01/1.65 thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__nth, type, finite1433825200ec_nth: (!> [A:$tType,B:$tType]: ((((finite_Cartesian_vec @ A) @ B) > (B > A))))).
% 3.01/1.65 thf(sy_c_FuncSet_OPi, type, pi: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (set @ B)) > (set @ (A > B))))))).
% 3.01/1.65 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 3.01/1.65 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 3.01/1.65 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 3.01/1.65 thf(sy_c_Initial__Value__Problem_Osolves__ode, type, initia1685620758es_ode: (!> [A:$tType]: (((real > A) > ((real > (A > A)) > ((set @ real) > ((set @ A) > $o))))))).
% 3.01/1.65 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 3.01/1.65 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 3.01/1.65 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 3.01/1.65 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 3.01/1.65 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 3.01/1.65 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.01/1.65 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 3.01/1.65 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost, type, set_or331188842AtMost: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 3.01/1.65 thf(sy_c_Static__Semantics_OBVO, type, static_BVO: (!> [A:$tType,C:$tType]: ((((ode @ A) @ C) > (set @ ((sum_sum @ C) @ C)))))).
% 3.01/1.65 thf(sy_c_Static__Semantics_OFVF, type, static_FVF: (!> [A:$tType,B:$tType,C:$tType]: (((((formula @ A) @ B) @ C) > (set @ ((sum_sum @ C) @ C)))))).
% 3.01/1.65 thf(sy_c_Static__Semantics_OFVO, type, static_FVO: (!> [A:$tType,C:$tType]: ((((ode @ A) @ C) > (set @ C))))).
% 3.01/1.65 thf(sy_c_Sum__Type_OInl, type, sum_Inl: (!> [A:$tType,B:$tType]: ((A > ((sum_sum @ A) @ B))))).
% 3.01/1.65 thf(sy_c_Sum__Type_OInr, type, sum_Inr: (!> [B:$tType,A:$tType]: ((B > ((sum_sum @ A) @ B))))).
% 3.01/1.65 thf(sy_c_Syntax_OEquiv, type, equiv: (!> [A:$tType,B:$tType,C:$tType]: (((((formula @ A) @ B) @ C) > ((((formula @ A) @ B) @ C) > (((formula @ A) @ B) @ C)))))).
% 3.01/1.65 thf(sy_c_Syntax_Ofsafe, type, fsafe: (!> [A:$tType,B:$tType,C:$tType]: (((((formula @ A) @ B) @ C) > $o)))).
% 3.01/1.65 thf(sy_c_Syntax_Oosafe, type, osafe: (!> [A:$tType,C:$tType]: ((((ode @ A) @ C) > $o)))).
% 3.01/1.65 thf(sy_c_Wellfounded_Oaccp, type, accp: (!> [A:$tType]: (((A > (A > $o)) > (A > $o))))).
% 3.01/1.65 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.01/1.65 thf(sy_v_I____, type, i: ((((denota1663640101rp_ext @ a) @ b) @ c) @ product_unit)).
% 3.01/1.65 thf(sy_v_ODE____, type, ode2: ((ode @ a) @ c)).
% 3.01/1.65 thf(sy_v_ODEa____, type, oDEa: ((ode @ a) @ c)).
% 3.01/1.65 thf(sy_v_P____, type, p: (((formula @ a) @ b) @ c)).
% 3.01/1.65 thf(sy_v__092_060phi_062_H____, type, phi: (((formula @ a) @ b) @ c)).
% 3.01/1.65 thf(sy_v_aa____, type, aa: ((finite_Cartesian_vec @ real) @ c)).
% 3.01/1.65 thf(sy_v_ab____, type, ab: ((finite_Cartesian_vec @ real) @ c)).
% 3.01/1.65 thf(sy_v_ba____, type, ba: ((finite_Cartesian_vec @ real) @ c)).
% 3.01/1.65 thf(sy_v_bb____, type, bb: ((finite_Cartesian_vec @ real) @ c)).
% 3.01/1.65 thf(sy_v_i____, type, i2: c).
% 3.01/1.65 thf(sy_v_sol____, type, sol: (real > ((finite_Cartesian_vec @ real) @ c))).
% 3.01/1.65 thf(sy_v_t____, type, t: real).
% 3.01/1.65 thf(sy_v_x____, type, x: real).
% 3.01/1.65 thf(fact_0_ode__to__fvo, axiom, (! [C:$tType,A:$tType,B:$tType,X:A,I2:((((denota1663640101rp_ext @ B) @ C) @ A) @ product_unit),ODE:((ode @ B) @ A)]: (((((member @ A) @ X) @ (((((denota811733865E_vars @ B) @ C) @ A) @ I2) @ ODE)) => (((member @ A) @ X) @ (((static_FVO @ B) @ A) @ ODE)))))).
% 3.01/1.65 thf(fact_1_osafe, axiom, (((osafe @ a) @ c) @ oDEa)).
% 3.01/1.65 thf(fact_2_eqP, axiom, (p = phi)).
% 3.01/1.65 thf(fact_3_x, axiom, (((ord_less_eq @ real) @ (zero_zero @ real)) @ x)).
% 3.01/1.65 thf(fact_4_fsafe, axiom, ((((fsafe @ a) @ b) @ c) @ phi)).
% 3.01/1.65 thf(fact_5__092_060open_062_IInl_Ai_A_092_060in_062_AsemBV_AI_AODE_J_A_061_A_IInr_Ai_A_092_060in_062_AsemBV_AI_AODE_J_092_060close_062, axiom, ((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ i2)) @ (((((denotational_semBV @ a) @ b) @ c) @ i) @ oDEa)) = (((member @ ((sum_sum @ c) @ c)) @ (((sum_Inr @ c) @ c) @ i2)) @ (((((denotational_semBV @ a) @ b) @ c) @ i) @ oDEa)))).
% 3.01/1.65 thf(fact_6_VA, axiom, (! [X2:real]: (((((denotational_Vagree @ c) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ (sol @ X2))) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ X2)) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3))))))) @ ((((static_FVF @ a) @ b) @ c) @ phi))))).
% 3.01/1.65 thf(fact_7_prod_Ocollapse, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)) = Prod)))).
% 3.01/1.65 thf(fact_8_bvo__to__fvo, axiom, (! [A:$tType,B:$tType,X:A,ODE:((ode @ B) @ A)]: (((((member @ ((sum_sum @ A) @ A)) @ (((sum_Inl @ A) @ A) @ X)) @ (((static_BVO @ B) @ A) @ ODE)) => (((member @ A) @ X) @ (((static_FVO @ B) @ A) @ ODE)))))).
% 3.01/1.65 thf(fact_9_vec__lambda__eta, axiom, (! [B:$tType,A:$tType]: (((finite_finite @ B) => (! [G:((finite_Cartesian_vec @ A) @ B)]: (((((finite1990238425lambda @ B) @ A) @ (((finite1433825200ec_nth @ A) @ B) @ G)) = G))))))).
% 3.01/1.65 thf(fact_10_Inl__Inr__False, axiom, (! [A:$tType,B:$tType,X:A,Y:B]: (((((sum_Inl @ A) @ B) @ X) != (((sum_Inr @ B) @ A) @ Y))))).
% 3.01/1.65 thf(fact_11_Inr__Inl__False, axiom, (! [B:$tType,A:$tType,X:B,Y:A]: (((((sum_Inr @ B) @ A) @ X) != (((sum_Inl @ A) @ B) @ Y))))).
% 3.01/1.65 thf(fact_12_vec__lambda__beta, axiom, (! [A:$tType,B:$tType]: (((finite_finite @ B) => (! [G:(B > A),I4:B]: ((((((finite1433825200ec_nth @ A) @ B) @ (((finite1990238425lambda @ B) @ A) @ G)) @ I4) = (G @ I4)))))))).
% 3.01/1.65 thf(fact_13_alt__sem__lemma, axiom, (! [B:$tType,A:$tType,C:$tType]: ((((finite_finite @ C) & ((finite_finite @ A) & (finite_finite @ B))) => (! [ODE:((ode @ A) @ C),I2:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),Sol:(real > ((finite_Cartesian_vec @ real) @ C)),T2:real,Ab:((finite_Cartesian_vec @ real) @ C)]: (((((osafe @ A) @ C) @ ODE) => (((((((denotational_ODE_sem @ A) @ B) @ C) @ I2) @ ODE) @ (Sol @ T2)) = ((((((denotational_ODE_sem @ A) @ B) @ C) @ I2) @ ODE) @ (((finite1990238425lambda @ C) @ real) @ (^ [I3:C]: (((((if @ real) @ (((member @ C) @ I3) @ (((static_FVO @ A) @ C) @ ODE))) @ ((((finite1433825200ec_nth @ real) @ C) @ (Sol @ T2)) @ I3)) @ ((((finite1433825200ec_nth @ real) @ C) @ Ab) @ I3)))))))))))))).
% 3.01/1.65 thf(fact_14_snd__zero, axiom, (! [B:$tType,A:$tType]: ((((zero @ A) & (zero @ B)) => ((((product_snd @ B) @ A) @ (zero_zero @ ((product_prod @ B) @ A))) = (zero_zero @ A)))))).
% 3.01/1.65 thf(fact_15_fst__zero, axiom, (! [B:$tType,A:$tType]: ((((zero @ A) & (zero @ B)) => ((((product_fst @ A) @ B) @ (zero_zero @ ((product_prod @ A) @ B))) = (zero_zero @ A)))))).
% 3.01/1.65 thf(fact_16_zero__index, axiom, (! [B:$tType,A:$tType]: ((((zero @ A) & (finite_finite @ B)) => (! [I4:B]: ((((((finite1433825200ec_nth @ A) @ B) @ (zero_zero @ ((finite_Cartesian_vec @ A) @ B))) @ I4) = (zero_zero @ A)))))))).
% 3.01/1.65 thf(fact_17_hpsafe__Evolve_OIH, axiom, ((((coinci1993344360de_fml @ a) @ b) @ c) @ p)).
% 3.01/1.65 thf(fact_18_hpsafe__Evolve_Ohyps_I1_J, axiom, (((osafe @ a) @ c) @ ode2)).
% 3.01/1.65 thf(fact_19_hpsafe__Evolve_Ohyps_I2_J, axiom, ((((fsafe @ a) @ b) @ c) @ p)).
% 3.01/1.65 thf(fact_20_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) = ((A2 = A3) & (B2 = B3)))))).
% 3.01/1.65 thf(fact_21_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X22 = Y2)))))).
% 3.01/1.65 thf(fact_22_t, axiom, (((ord_less_eq @ real) @ x) @ t)).
% 3.01/1.65 thf(fact_23_all, axiom, (! [I5:c]: ((((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ I5)) @ (((static_BVO @ a) @ c) @ oDEa)) => (((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ I5) = ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I5))) & (((((member @ ((sum_sum @ c) @ d)) @ (((sum_Inl @ c) @ d) @ I5)) @ ((((image @ c) @ ((sum_sum @ c) @ d)) @ ((sum_Inl @ c) @ d)) @ (((static_FVO @ a) @ c) @ oDEa))) => (((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ I5) = ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I5))) & ((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ I5)) @ ((((static_FVF @ a) @ b) @ c) @ phi)) => (((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ I5) = ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I5)))))))).
% 3.01/1.65 thf(fact_24_less__eq__vec__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (finite_finite @ B)) => ((ord_less_eq @ ((finite_Cartesian_vec @ A) @ B)) = (^ [X3:((finite_Cartesian_vec @ A) @ B),Y3:((finite_Cartesian_vec @ A) @ B)]: ((! [I3:B]: ((((ord_less_eq @ A) @ ((((finite1433825200ec_nth @ A) @ B) @ X3) @ I3)) @ ((((finite1433825200ec_nth @ A) @ B) @ Y3) @ I3))))))))))).
% 3.01/1.65 thf(fact_25_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A4:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A4) @ B4)))) => (P @ Prod))))).
% 3.01/1.65 thf(fact_26_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A4:A,B4:B]: ((Y != ((((product_Pair @ A) @ B) @ A4) @ B4))))))))).
% 3.01/1.65 thf(fact_27_prod__induct7, axiom, (! [G2:$tType,F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2))))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F,G3:G2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G2)) @ E2) @ ((((product_Pair @ F) @ G2) @ F2) @ G3))))))))) => (P @ X))))).
% 3.01/1.65 thf(fact_28_prod__induct6, axiom, (! [F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2)))))))) => (P @ X))))).
% 3.01/1.65 thf(fact_29_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P @ X))))).
% 3.01/1.65 thf(fact_30_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A4:A,B4:B,C2:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2)))))) => (P @ X))))).
% 3.01/1.65 thf(fact_31_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A4:A,B4:B,C2:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B4) @ C2))))) => (P @ X))))).
% 3.01/1.65 thf(fact_32_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,G2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2))))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F,G3:G2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G2)) @ E2) @ ((((product_Pair @ F) @ G2) @ F2) @ G3)))))))))))))).
% 3.01/1.65 thf(fact_33_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B4) @ ((((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))))))))))))).
% 3.01/1.65 thf(fact_34_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 3.01/1.65 thf(fact_35_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2))))))))))).
% 3.01/1.65 thf(fact_36_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A4:A,B4:B,C2:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B4) @ C2)))))))))).
% 3.01/1.65 thf(fact_37_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)))))))).
% 3.01/1.65 thf(fact_38_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A4:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A4) @ B4)))) => (P @ P2))))).
% 3.01/1.65 thf(fact_39_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X4:A,Y4:B]: ((P2 = ((((product_Pair @ A) @ B) @ X4) @ Y4))))))).
% 3.01/1.65 thf(fact_40_vec__nth__inject, axiom, (! [B:$tType,A:$tType]: (((finite_finite @ B) => (! [X:((finite_Cartesian_vec @ A) @ B),Y:((finite_Cartesian_vec @ A) @ B)]: ((((((finite1433825200ec_nth @ A) @ B) @ X) = (((finite1433825200ec_nth @ A) @ B) @ Y)) = (X = Y)))))))).
% 3.01/1.65 thf(fact_41_cond__component, axiom, (! [A:$tType,B:$tType]: (((finite_finite @ B) => (! [B2:$o,X:((finite_Cartesian_vec @ A) @ B),Y:((finite_Cartesian_vec @ A) @ B),I4:B]: (((B2 => (((((finite1433825200ec_nth @ A) @ B) @ ((((if @ ((finite_Cartesian_vec @ A) @ B)) @ B2) @ X) @ Y)) @ I4) = ((((finite1433825200ec_nth @ A) @ B) @ X) @ I4))) & ((~ (B2)) => (((((finite1433825200ec_nth @ A) @ B) @ ((((if @ ((finite_Cartesian_vec @ A) @ B)) @ B2) @ X) @ Y)) @ I4) = ((((finite1433825200ec_nth @ A) @ B) @ Y) @ I4)))))))))).
% 3.01/1.65 thf(fact_42_vec__eq__iff, axiom, (! [B:$tType,A:$tType]: (((finite_finite @ B) => ((^ [Y5:((finite_Cartesian_vec @ A) @ B),Z:((finite_Cartesian_vec @ A) @ B)]: ((Y5 = Z))) = (^ [X3:((finite_Cartesian_vec @ A) @ B),Y3:((finite_Cartesian_vec @ A) @ B)]: ((! [I3:B]: ((((((finite1433825200ec_nth @ A) @ B) @ X3) @ I3) = ((((finite1433825200ec_nth @ A) @ B) @ Y3) @ I3))))))))))).
% 3.01/1.65 thf(fact_43_fun__cong__unused__0, axiom, (! [A:$tType,B:$tType,C:$tType]: (((zero @ B) => (! [F3:((A > B) > C),G:C]: (((F3 = (^ [X3:(A > B)]: (G))) => ((F3 @ (^ [X3:A]: ((zero_zero @ B)))) = G)))))))).
% 3.01/1.65 thf(fact_44_zero__prod__def, axiom, (! [A:$tType,B:$tType]: ((((zero @ B) & (zero @ A)) => ((zero_zero @ ((product_prod @ A) @ B)) = ((((product_Pair @ A) @ B) @ (zero_zero @ A)) @ (zero_zero @ B))))))).
% 3.01/1.65 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 3.01/1.65 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A5:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A5)))) = A5)))).
% 3.01/1.65 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) = (Q @ X4)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 3.01/1.65 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F3:(A > B),G:(A > B)]: (((! [X4:A]: (((F3 @ X4) = (G @ X4)))) => (F3 = G))))).
% 3.01/1.65 thf(fact_49_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X22:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = X1)))).
% 3.01/1.65 thf(fact_50_fst__eqD, axiom, (! [B:$tType,A:$tType,X:A,Y:B,A2:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X) @ Y)) = A2) => (X = A2))))).
% 3.01/1.65 thf(fact_51_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X22:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X22)) = X22)))).
% 3.01/1.65 thf(fact_52_snd__eqD, axiom, (! [B:$tType,A:$tType,X:B,Y:A,A2:A]: ((((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y)) = A2) => (Y = A2))))).
% 3.01/1.65 thf(fact_53_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y5:((product_prod @ A) @ B),Z:((product_prod @ A) @ B)]: ((Y5 = Z))) = (^ [S:((product_prod @ A) @ B),T3:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S) = (((product_fst @ A) @ B) @ T3)) & ((((product_snd @ A) @ B) @ S) = (((product_snd @ A) @ B) @ T3))))))))).
% 3.01/1.65 thf(fact_54_prod_Oexpand, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B),Prod2:((product_prod @ A) @ B)]: (((((((product_fst @ A) @ B) @ Prod) = (((product_fst @ A) @ B) @ Prod2)) & ((((product_snd @ A) @ B) @ Prod) = (((product_snd @ A) @ B) @ Prod2))) => (Prod = Prod2))))).
% 3.01/1.65 thf(fact_55_prod__eqI, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B),Q2:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P2) = (((product_fst @ A) @ B) @ Q2)) => (((((product_snd @ A) @ B) @ P2) = (((product_snd @ A) @ B) @ Q2)) => (P2 = Q2)))))).
% 3.01/1.65 thf(fact_56_vec__nth__inverse, axiom, (! [B:$tType,A:$tType]: (((finite_finite @ B) => (! [X:((finite_Cartesian_vec @ A) @ B)]: (((((finite1990238425lambda @ B) @ A) @ (((finite1433825200ec_nth @ A) @ B) @ X)) = X))))))).
% 3.01/1.65 thf(fact_57_vec__lambda__unique, axiom, (! [A:$tType,B:$tType]: (((finite_finite @ A) => (! [F3:((finite_Cartesian_vec @ B) @ A),G:(A > B)]: (((! [I3:A]: ((((((finite1433825200ec_nth @ B) @ A) @ F3) @ I3) = (G @ I3)))) = ((((finite1990238425lambda @ A) @ B) @ G) = F3)))))))).
% 3.01/1.65 thf(fact_58_obj__sumE__f, axiom, (! [A:$tType,C:$tType,B:$tType,S2:B,F3:(((sum_sum @ A) @ C) > B),P:$o]: (((! [X4:A]: (((S2 = (F3 @ (((sum_Inl @ A) @ C) @ X4))) => P))) => ((! [X4:C]: (((S2 = (F3 @ (((sum_Inr @ C) @ A) @ X4))) => P))) => (! [X5:((sum_sum @ A) @ C)]: (((S2 = (F3 @ X5)) => P)))))))).
% 3.01/1.65 thf(fact_59_zero__vec__def, axiom, (! [A:$tType,B:$tType]: ((((finite_finite @ B) & (zero @ A)) => ((zero_zero @ ((finite_Cartesian_vec @ A) @ B)) = (((finite1990238425lambda @ B) @ A) @ (^ [I3:B]: ((zero_zero @ A))))))))).
% 3.01/1.65 thf(fact_60_surjective__pairing, axiom, (! [B:$tType,A:$tType,T2:((product_prod @ A) @ B)]: ((T2 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T2)) @ (((product_snd @ A) @ B) @ T2)))))).
% 3.01/1.65 thf(fact_61_prod_Oexhaust__sel, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))).
% 3.01/1.65 thf(fact_62_mk__v__concrete, axiom, (! [A:$tType,C:$tType,B:$tType]: ((((finite_finite @ B) & ((finite_finite @ C) & (finite_finite @ A))) => ((((denotational_mk_v @ B) @ C) @ A) = (^ [I6:((((denota1663640101rp_ext @ B) @ C) @ A) @ product_unit),ODE2:((ode @ B) @ A),Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),Sol2:((finite_Cartesian_vec @ real) @ A)]: (((((product_Pair @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)) @ (((finite1990238425lambda @ A) @ real) @ (^ [I3:A]: (((((finite1433825200ec_nth @ real) @ A) @ ((((if @ ((finite_Cartesian_vec @ real) @ A)) @ (((member @ ((sum_sum @ A) @ A)) @ (((sum_Inl @ A) @ A) @ I3)) @ (((((denotational_semBV @ B) @ C) @ A) @ I6) @ ODE2))) @ Sol2) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)) @ Nu))) @ I3))))) @ (((finite1990238425lambda @ A) @ real) @ (^ [I3:A]: (((((finite1433825200ec_nth @ real) @ A) @ ((((if @ ((finite_Cartesian_vec @ real) @ A)) @ (((member @ ((sum_sum @ A) @ A)) @ (((sum_Inr @ A) @ A) @ I3)) @ (((((denotational_semBV @ B) @ C) @ A) @ I6) @ ODE2))) @ ((((((denotational_ODE_sem @ B) @ C) @ A) @ I6) @ ODE2) @ Sol2)) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)) @ Nu))) @ I3)))))))))))).
% 3.01/1.65 thf(fact_63__092_060open_062Vagree_A_Imk__v_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_Ax_A_E_Ai_Aelse_Aab_A_E_Ai_J_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_N_AsemBV_AI_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_Ax_A_E_Ai_Aelse_Aab_A_E_Ai_J_J_A_Imk__xode_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_Ax_A_E_Ai_Aelse_Aab_A_E_Ai_J_J_A_IsemBV_AI_AODE_J_092_060close_062, axiom, (((((denotational_Vagree @ c) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))) @ bb)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3))))))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))) @ bb)) @ ((uminus_uminus @ (set @ ((sum_sum @ c) @ c))) @ (((((denotational_semBV @ a) @ b) @ c) @ i) @ oDEa))) & ((((denotational_Vagree @ c) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))) @ bb)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3))))))) @ ((((((denotational_mk_xode @ a) @ b) @ c) @ i) @ oDEa) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3))))))) @ (((((denotational_semBV @ a) @ b) @ c) @ i) @ oDEa)))).
% 3.01/1.65 thf(fact_64_Vagree__def, axiom, (! [C:$tType]: (((finite_finite @ C) => ((denotational_Vagree @ C) = (^ [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),Nu2:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),V:(set @ ((sum_sum @ C) @ C))]: (((! [I3:C]: (((((member @ ((sum_sum @ C) @ C)) @ (((sum_Inl @ C) @ C) @ I3)) @ V) => (((((finite1433825200ec_nth @ real) @ C) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ Nu)) @ I3) = ((((finite1433825200ec_nth @ real) @ C) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ Nu2)) @ I3))))) & (! [I3:C]: (((((member @ ((sum_sum @ C) @ C)) @ (((sum_Inr @ C) @ C) @ I3)) @ V) => (((((finite1433825200ec_nth @ real) @ C) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ Nu)) @ I3) = ((((finite1433825200ec_nth @ real) @ C) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ Nu2)) @ I3))))))))))))).
% 3.01/1.65 thf(fact_65__092_060open_062Vagree_A_Imk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_Ax_J_J_A_Iab_M_Abb_J_A_I_N_AsemBV_AI_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_Ax_J_J_A_Imk__xode_AI_AODE_A_Isol_Ax_J_J_A_IsemBV_AI_AODE_J_092_060close_062, axiom, (((((denotational_Vagree @ c) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ (sol @ x))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ ((uminus_uminus @ (set @ ((sum_sum @ c) @ c))) @ (((((denotational_semBV @ a) @ b) @ c) @ i) @ oDEa))) & ((((denotational_Vagree @ c) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ (sol @ x))) @ ((((((denotational_mk_xode @ a) @ b) @ c) @ i) @ oDEa) @ (sol @ x))) @ (((((denotational_semBV @ a) @ b) @ c) @ i) @ oDEa)))).
% 3.01/1.65 thf(fact_66_concrete__v_Osimps, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => ((((denota1738237502rete_v @ A) @ B) @ C) = (^ [I6:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),ODE2:((ode @ A) @ C),Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),Sol2:((finite_Cartesian_vec @ real) @ C)]: (((((product_Pair @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ (((finite1990238425lambda @ C) @ real) @ (^ [I3:C]: (((((finite1433825200ec_nth @ real) @ C) @ ((((if @ ((finite_Cartesian_vec @ real) @ C)) @ (((member @ ((sum_sum @ C) @ C)) @ (((sum_Inl @ C) @ C) @ I3)) @ (((((denotational_semBV @ A) @ B) @ C) @ I6) @ ODE2))) @ Sol2) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ Nu))) @ I3))))) @ (((finite1990238425lambda @ C) @ real) @ (^ [I3:C]: (((((finite1433825200ec_nth @ real) @ C) @ ((((if @ ((finite_Cartesian_vec @ real) @ C)) @ (((member @ ((sum_sum @ C) @ C)) @ (((sum_Inr @ C) @ C) @ I3)) @ (((((denotational_semBV @ A) @ B) @ C) @ I6) @ ODE2))) @ ((((((denotational_ODE_sem @ A) @ B) @ C) @ I6) @ ODE2) @ Sol2)) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ Nu))) @ I3)))))))))))).
% 3.01/1.65 thf(fact_67_concrete__v_Oelims, axiom, (! [B:$tType,A:$tType,C:$tType]: ((((finite_finite @ C) & ((finite_finite @ A) & (finite_finite @ B))) => (! [X:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),Xa:((ode @ A) @ C),Xb:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),Xc:((finite_Cartesian_vec @ real) @ C),Y:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))]: ((((((((((denota1738237502rete_v @ A) @ B) @ C) @ X) @ Xa) @ Xb) @ Xc) = Y) => (Y = ((((product_Pair @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ (((finite1990238425lambda @ C) @ real) @ (^ [I3:C]: (((((finite1433825200ec_nth @ real) @ C) @ ((((if @ ((finite_Cartesian_vec @ real) @ C)) @ (((member @ ((sum_sum @ C) @ C)) @ (((sum_Inl @ C) @ C) @ I3)) @ (((((denotational_semBV @ A) @ B) @ C) @ X) @ Xa))) @ Xc) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ Xb))) @ I3))))) @ (((finite1990238425lambda @ C) @ real) @ (^ [I3:C]: (((((finite1433825200ec_nth @ real) @ C) @ ((((if @ ((finite_Cartesian_vec @ real) @ C)) @ (((member @ ((sum_sum @ C) @ C)) @ (((sum_Inr @ C) @ C) @ I3)) @ (((((denotational_semBV @ A) @ B) @ C) @ X) @ Xa))) @ ((((((denotational_ODE_sem @ A) @ B) @ C) @ X) @ Xa) @ Xc)) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ Xb))) @ I3))))))))))))).
% 3.01/1.65 thf(fact_68_Pair__le, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:B,C3:A,D3:B]: (((((ord_less_eq @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Pair @ A) @ B) @ C3) @ D3)) = ((((ord_less_eq @ A) @ A2) @ C3) & (((ord_less_eq @ B) @ B2) @ D3))))))))).
% 3.01/1.65 thf(fact_69_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 3.01/1.65 thf(fact_70_sem, axiom, (! [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))]: ((((((denotational_Vagree @ c) @ Nu3) @ Nu4) @ ((((static_FVF @ a) @ b) @ c) @ p)) => ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ Nu3) @ (((((denotational_fml_sem @ a) @ b) @ c) @ i) @ p)) = (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ Nu4) @ (((((denotational_fml_sem @ a) @ b) @ c) @ i) @ p))))))).
% 3.01/1.65 thf(fact_71_aaba, axiom, (((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ aa) @ ba) = (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ (sol @ t)))).
% 3.01/1.65 thf(fact_72_neg__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((uminus_uminus @ A) @ A2) = ((uminus_uminus @ A) @ B2)) = (A2 = B2)))))))).
% 3.01/1.65 thf(fact_73_add_Oinverse__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ A2)) = A2))))))).
% 3.01/1.65 thf(fact_74_add_Oinverse__neutral, axiom, (! [A:$tType]: (((group_add @ A) => (((uminus_uminus @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 3.01/1.65 thf(fact_75_neg__0__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: ((((zero_zero @ A) = ((uminus_uminus @ A) @ A2)) = ((zero_zero @ A) = A2)))))))).
% 3.01/1.65 thf(fact_76_neg__equal__0__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((uminus_uminus @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 3.01/1.65 thf(fact_77_equal__neg__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((A2 = ((uminus_uminus @ A) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 3.01/1.65 thf(fact_78_neg__equal__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((uminus_uminus @ A) @ A2) = A2) = (A2 = (zero_zero @ A))))))))).
% 3.01/1.65 thf(fact_79_neg__le__iff__le, axiom, (! [A:$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)))))))).
% 3.01/1.65 thf(fact_80_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))))))))).
% 3.01/1.65 thf(fact_81_vector__uminus__component, axiom, (! [A:$tType,B:$tType]: ((((finite_finite @ B) & (uminus @ A)) => (! [X:((finite_Cartesian_vec @ A) @ B),I4:B]: ((((((finite1433825200ec_nth @ A) @ B) @ ((uminus_uminus @ ((finite_Cartesian_vec @ A) @ B)) @ X)) @ I4) = ((uminus_uminus @ A) @ ((((finite1433825200ec_nth @ A) @ B) @ X) @ I4))))))))).
% 3.01/1.65 thf(fact_82_fst__uminus, axiom, (! [B:$tType,A:$tType]: ((((uminus @ A) & (uminus @ B)) => (! [X:((product_prod @ A) @ B)]: (((((product_fst @ A) @ B) @ ((uminus_uminus @ ((product_prod @ A) @ B)) @ X)) = ((uminus_uminus @ A) @ (((product_fst @ A) @ B) @ X))))))))).
% 3.01/1.65 thf(fact_83_snd__uminus, axiom, (! [A:$tType,B:$tType]: ((((uminus @ B) & (uminus @ A)) => (! [X:((product_prod @ B) @ A)]: (((((product_snd @ B) @ A) @ ((uminus_uminus @ ((product_prod @ B) @ A)) @ X)) = ((uminus_uminus @ A) @ (((product_snd @ B) @ A) @ X))))))))).
% 3.01/1.65 thf(fact_84_neg__less__eq__nonneg, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 3.01/1.65 thf(fact_85_less__eq__neg__nonpos, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 3.01/1.65 thf(fact_86_neg__le__0__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 3.01/1.65 thf(fact_87_neg__0__le__iff__le, axiom, (! [A:$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))))))))).
% 3.01/1.65 thf(fact_88_allT, axiom, (! [S3:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ S3) => ((((ord_less_eq @ real) @ S3) @ t) => (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ (sol @ S3))) @ (((((denotational_fml_sem @ a) @ b) @ c) @ i) @ phi))))))).
% 3.01/1.65 thf(fact_89_minus__equation__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((uminus_uminus @ A) @ A2) = B2) = (((uminus_uminus @ A) @ B2) = A2)))))))).
% 3.01/1.65 thf(fact_90_equation__minus__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((A2 = ((uminus_uminus @ A) @ B2)) = (B2 = ((uminus_uminus @ A) @ A2))))))))).
% 3.01/1.65 thf(fact_91_iff__to__impl, axiom, (! [A:$tType,C:$tType,B:$tType]: ((((finite_finite @ B) & ((finite_finite @ C) & (finite_finite @ A))) => (! [Nu5:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),I2:((((denota1663640101rp_ext @ B) @ C) @ A) @ product_unit),A5:(((formula @ B) @ C) @ A),B5:(((formula @ B) @ C) @ A)]: ((((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu5) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ A5)) = (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu5) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ B5))) = (((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu5) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ A5)) => (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu5) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ B5))) & ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu5) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ B5)) => (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu5) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ A5))))))))))).
% 3.01/1.65 thf(fact_92_le__imp__neg__le, axiom, (! [A:$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))))))))).
% 3.01/1.65 thf(fact_93_minus__le__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ A2)))))))).
% 3.01/1.65 thf(fact_94_le__minus__iff, axiom, (! [A:$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))))))))).
% 3.01/1.65 thf(fact_95_agree__supset, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [B5:(set @ ((sum_sum @ A) @ A)),A5:(set @ ((sum_sum @ A) @ A)),Nu5:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),Nu6:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))]: (((((ord_less_eq @ (set @ ((sum_sum @ A) @ A))) @ B5) @ A5) => (((((denotational_Vagree @ A) @ Nu5) @ Nu6) @ A5) => ((((denotational_Vagree @ A) @ Nu5) @ Nu6) @ B5))))))))).
% 3.01/1.65 thf(fact_96_agree__sub, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A5:(set @ ((sum_sum @ A) @ A)),B5:(set @ ((sum_sum @ A) @ A)),Nu5:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),Omega:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))]: (((((ord_less_eq @ (set @ ((sum_sum @ A) @ A))) @ A5) @ B5) => (((((denotational_Vagree @ A) @ Nu5) @ Omega) @ B5) => ((((denotational_Vagree @ A) @ Nu5) @ Omega) @ A5))))))))).
% 3.01/1.65 thf(fact_97_mk__v__exists, axiom, (! [C:$tType,A:$tType,B:$tType]: ((((finite_finite @ B) & ((finite_finite @ A) & (finite_finite @ C))) => (! [Nu5:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),I2:((((denota1663640101rp_ext @ B) @ C) @ A) @ product_unit),ODE:((ode @ B) @ A),Sol:((finite_Cartesian_vec @ real) @ A)]: ((? [Omega2:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))]: ((((((denotational_Vagree @ A) @ Omega2) @ Nu5) @ ((uminus_uminus @ (set @ ((sum_sum @ A) @ A))) @ (((((denotational_semBV @ B) @ C) @ A) @ I2) @ ODE))) & ((((denotational_Vagree @ A) @ Omega2) @ ((((((denotational_mk_xode @ B) @ C) @ A) @ I2) @ ODE) @ Sol)) @ (((((denotational_semBV @ B) @ C) @ A) @ I2) @ ODE))))))))))).
% 3.01/1.65 thf(fact_98_uminus__vec__def, axiom, (! [B:$tType,A:$tType]: ((((uminus @ A) & (finite_finite @ B)) => ((uminus_uminus @ ((finite_Cartesian_vec @ A) @ B)) = (^ [X3:((finite_Cartesian_vec @ A) @ B)]: ((((finite1990238425lambda @ B) @ A) @ (^ [I3:B]: (((uminus_uminus @ A) @ ((((finite1433825200ec_nth @ A) @ B) @ X3) @ I3)))))))))))).
% 3.01/1.65 thf(fact_99_uminus__prod__def, axiom, (! [B:$tType,A:$tType]: ((((uminus @ A) & (uminus @ B)) => ((uminus_uminus @ ((product_prod @ A) @ B)) = (^ [X3:((product_prod @ A) @ B)]: (((((product_Pair @ A) @ B) @ ((uminus_uminus @ A) @ (((product_fst @ A) @ B) @ X3))) @ ((uminus_uminus @ B) @ (((product_snd @ A) @ B) @ X3)))))))))).
% 3.01/1.65 thf(fact_100_mk__v__agree, axiom, (! [C:$tType,A:$tType,B:$tType]: ((((finite_finite @ B) & ((finite_finite @ A) & (finite_finite @ C))) => (! [I2:((((denota1663640101rp_ext @ B) @ C) @ A) @ product_unit),ODE:((ode @ B) @ A),Nu5:((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) @ I2) @ ODE) @ Nu5) @ Sol)) @ Nu5) @ ((uminus_uminus @ (set @ ((sum_sum @ A) @ A))) @ (((((denotational_semBV @ B) @ C) @ A) @ I2) @ ODE))) & ((((denotational_Vagree @ A) @ (((((((denotational_mk_v @ B) @ C) @ A) @ I2) @ ODE) @ Nu5) @ Sol)) @ ((((((denotational_mk_xode @ B) @ C) @ A) @ I2) @ ODE) @ Sol)) @ (((((denotational_semBV @ B) @ C) @ A) @ I2) @ ODE))))))))).
% 3.01/1.65 thf(fact_101_mk__xode_Oelims, axiom, (! [A:$tType,B:$tType,C:$tType]: ((((finite_finite @ C) & ((finite_finite @ B) & (finite_finite @ A))) => (! [X:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),Xa:((ode @ A) @ C),Xb:((finite_Cartesian_vec @ real) @ C),Y:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))]: (((((((((denotational_mk_xode @ A) @ B) @ C) @ X) @ Xa) @ Xb) = Y) => (Y = ((((product_Pair @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ Xb) @ ((((((denotational_ODE_sem @ A) @ B) @ C) @ X) @ Xa) @ Xb)))))))))).
% 3.01/1.65 thf(fact_102_mk__xode_Osimps, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => ((((denotational_mk_xode @ A) @ B) @ C) = (^ [I6:((((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) @ I6) @ ODE2) @ Sol2))))))))).
% 3.01/1.65 thf(fact_103_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 3.01/1.65 thf(fact_104_agree__refl, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [Nu5:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),A5:(set @ ((sum_sum @ A) @ A))]: (((((denotational_Vagree @ A) @ Nu5) @ Nu5) @ A5))))))).
% 3.01/1.65 thf(fact_105_agree__comm, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A5:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),B5:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),V2:(set @ ((sum_sum @ A) @ A))]: ((((((denotational_Vagree @ A) @ A5) @ B5) @ V2) => ((((denotational_Vagree @ A) @ B5) @ A5) @ V2)))))))).
% 3.01/1.65 thf(fact_106_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 3.01/1.65 thf(fact_107_Pair__mono, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [X:A,X6:A,Y:B,Y6:B]: (((((ord_less_eq @ A) @ X) @ X6) => ((((ord_less_eq @ B) @ Y) @ Y6) => (((ord_less_eq @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ ((((product_Pair @ A) @ B) @ X6) @ Y6)))))))))).
% 3.01/1.65 thf(fact_108_fst__mono, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (! [X:((product_prod @ A) @ B),Y:((product_prod @ A) @ B)]: (((((ord_less_eq @ ((product_prod @ A) @ B)) @ X) @ Y) => (((ord_less_eq @ A) @ (((product_fst @ A) @ B) @ X)) @ (((product_fst @ A) @ B) @ Y))))))))).
% 3.01/1.65 thf(fact_109_snd__mono, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (! [X:((product_prod @ A) @ B),Y:((product_prod @ A) @ B)]: (((((ord_less_eq @ ((product_prod @ A) @ B)) @ X) @ Y) => (((ord_less_eq @ B) @ (((product_snd @ A) @ B) @ X)) @ (((product_snd @ A) @ B) @ Y))))))))).
% 3.01/1.65 thf(fact_110_less__eq__prod__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => ((ord_less_eq @ ((product_prod @ A) @ B)) = (^ [X3:((product_prod @ A) @ B),Y3:((product_prod @ A) @ B)]: (((((ord_less_eq @ A) @ (((product_fst @ A) @ B) @ X3)) @ (((product_fst @ A) @ B) @ Y3)) & (((ord_less_eq @ B) @ (((product_snd @ A) @ B) @ X3)) @ (((product_snd @ A) @ B) @ Y3)))))))))).
% 3.01/1.65 thf(fact_111_ODE__vars__lr, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType,X:Sz,I2:((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit),ODE:((ode @ Sf) @ Sz)]: (((((member @ ((sum_sum @ Sz) @ Sz)) @ (((sum_Inl @ Sz) @ Sz) @ X)) @ (((((denotational_semBV @ Sf) @ Sc) @ Sz) @ I2) @ ODE)) = (((member @ ((sum_sum @ Sz) @ Sz)) @ (((sum_Inr @ Sz) @ Sz) @ X)) @ (((((denotational_semBV @ Sf) @ Sc) @ Sz) @ I2) @ ODE)))))).
% 3.01/1.65 thf(fact_112__092_060open_062_092_060lbrakk_062osafe_AODE_059_Afsafe_A_092_060phi_062_059_AODE_____A_061_AODE_059_AP_A_061_A_092_060phi_062_059_A0_A_092_060le_062_At_059_A_Iaa_M_Aba_J_A_061_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_059_A_Isol_Asolves__ode_A_I_092_060lambda_062a_O_AODE__sem_AI_AODE_J_J_A_1230_O_Ot_125_A_123x_O_Amk__v_AI_AODE_A_Iab_M_Abb_J_Ax_A_092_060in_062_Afml__sem_AI_A_092_060phi_062_125_059_AVSagree_A_Isol_A0_J_Aab_A_123uu___O_AInl_Auu___A_092_060in_062_ABVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AInl_A_096_AFVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AFVF_A_092_060phi_062_125_092_060rbrakk_062_A_092_060Longrightarrow_062_Aab_A_061_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_J_092_060close_062, axiom, ((((osafe @ a) @ c) @ oDEa) => (((((fsafe @ a) @ b) @ c) @ phi) => ((ode2 = oDEa) => ((p = phi) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ t) => ((((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ aa) @ ba) = (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ (sol @ t))) => ((((((initia1685620758es_ode @ ((finite_Cartesian_vec @ real) @ c)) @ sol) @ (^ [X7:real]: ((((((denotational_ODE_sem @ a) @ b) @ c) @ i) @ oDEa)))) @ (((set_or331188842AtMost @ real) @ (zero_zero @ real)) @ t)) @ ((collect @ ((finite_Cartesian_vec @ real) @ c)) @ (^ [X3:((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) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ X3)) @ (((((denotational_fml_sem @ a) @ b) @ c) @ i) @ phi)))))) => (((((denotational_VSagree @ c) @ (sol @ (zero_zero @ real))) @ ab) @ ((collect @ c) @ (^ [Uu:c]: (((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ Uu)) @ (((static_BVO @ a) @ c) @ oDEa)) | ((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ Uu)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inl @ c) @ c)) @ (((static_FVO @ a) @ c) @ oDEa))) | (((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ Uu)) @ ((((static_FVF @ a) @ b) @ c) @ phi)))))))) => (ab = (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3))))))))))))))).
% 3.01/1.65 thf(fact_113__092_060open_062_092_060lbrakk_062osafe_AODE_059_Afsafe_A_092_060phi_062_059_AODE_____A_061_AODE_059_AP_A_061_A_092_060phi_062_059_A0_A_092_060le_062_At_059_A_Iaa_M_Aba_J_A_061_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_059_A_Isol_Asolves__ode_A_I_092_060lambda_062a_O_AODE__sem_AI_AODE_J_J_A_1230_O_Ot_125_A_123x_O_Amk__v_AI_AODE_A_Iab_M_Abb_J_Ax_A_092_060in_062_Afml__sem_AI_A_092_060phi_062_125_059_AVSagree_A_Isol_A0_J_Aab_A_123uu___O_AInl_Auu___A_092_060in_062_ABVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AInl_A_096_AFVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AFVF_A_092_060phi_062_125_092_060rbrakk_062_A_092_060Longrightarrow_062_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_A_061_Amk__v_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_At_A_E_Ai_Aelse_Aab_A_E_Ai_J_092_060close_062, axiom, ((((osafe @ a) @ c) @ oDEa) => (((((fsafe @ a) @ b) @ c) @ phi) => ((ode2 = oDEa) => ((p = phi) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ t) => ((((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ aa) @ ba) = (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ (sol @ t))) => ((((((initia1685620758es_ode @ ((finite_Cartesian_vec @ real) @ c)) @ sol) @ (^ [X7:real]: ((((((denotational_ODE_sem @ a) @ b) @ c) @ i) @ oDEa)))) @ (((set_or331188842AtMost @ real) @ (zero_zero @ real)) @ t)) @ ((collect @ ((finite_Cartesian_vec @ real) @ c)) @ (^ [X3:((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) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ X3)) @ (((((denotational_fml_sem @ a) @ b) @ c) @ i) @ phi)))))) => (((((denotational_VSagree @ c) @ (sol @ (zero_zero @ real))) @ ab) @ ((collect @ c) @ (^ [Uu:c]: (((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ Uu)) @ (((static_BVO @ a) @ c) @ oDEa)) | ((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ Uu)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inl @ c) @ c)) @ (((static_FVO @ a) @ c) @ oDEa))) | (((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ Uu)) @ ((((static_FVF @ a) @ b) @ c) @ phi)))))))) => ((((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ (sol @ t)) = (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))) @ bb)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ t)) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))))))))))))).
% 3.01/1.65 thf(fact_114__092_060open_062_092_060lbrakk_062osafe_AODE_059_Afsafe_A_092_060phi_062_059_AODE_____A_061_AODE_059_AP_A_061_A_092_060phi_062_059_A0_A_092_060le_062_At_059_A_Iaa_M_Aba_J_A_061_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_059_A_Isol_Asolves__ode_A_I_092_060lambda_062a_O_AODE__sem_AI_AODE_J_J_A_1230_O_Ot_125_A_123x_O_Amk__v_AI_AODE_A_Iab_M_Abb_J_Ax_A_092_060in_062_Afml__sem_AI_A_092_060phi_062_125_059_AVSagree_A_Isol_A0_J_Aab_A_123uu___O_AInl_Auu___A_092_060in_062_ABVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AInl_A_096_AFVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AFVF_A_092_060phi_062_125_092_060rbrakk_062_A_092_060Longrightarrow_062_A0_A_092_060le_062_At_092_060close_062, axiom, ((((osafe @ a) @ c) @ oDEa) => (((((fsafe @ a) @ b) @ c) @ phi) => ((ode2 = oDEa) => ((p = phi) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ t) => ((((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ aa) @ ba) = (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ (sol @ t))) => ((((((initia1685620758es_ode @ ((finite_Cartesian_vec @ real) @ c)) @ sol) @ (^ [X7:real]: ((((((denotational_ODE_sem @ a) @ b) @ c) @ i) @ oDEa)))) @ (((set_or331188842AtMost @ real) @ (zero_zero @ real)) @ t)) @ ((collect @ ((finite_Cartesian_vec @ real) @ c)) @ (^ [X3:((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) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ X3)) @ (((((denotational_fml_sem @ a) @ b) @ c) @ i) @ phi)))))) => (((((denotational_VSagree @ c) @ (sol @ (zero_zero @ real))) @ ab) @ ((collect @ c) @ (^ [Uu:c]: (((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ Uu)) @ (((static_BVO @ a) @ c) @ oDEa)) | ((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ Uu)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inl @ c) @ c)) @ (((static_FVO @ a) @ c) @ oDEa))) | (((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ Uu)) @ ((((static_FVF @ a) @ b) @ c) @ phi)))))))) => (((ord_less_eq @ real) @ (zero_zero @ real)) @ t)))))))))).
% 3.01/1.65 thf(fact_115_mkV, axiom, (((member @ (real > ((finite_Cartesian_vec @ real) @ c))) @ sol) @ ((((pi @ real) @ ((finite_Cartesian_vec @ real) @ c)) @ (((set_or331188842AtMost @ real) @ (zero_zero @ real)) @ t)) @ (^ [Uu:real]: (((collect @ ((finite_Cartesian_vec @ real) @ c)) @ (^ [X3:((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) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ X3)) @ (((((denotational_fml_sem @ a) @ b) @ c) @ i) @ phi)))))))))).
% 3.01/1.65 thf(fact_116_euclid__ext__aux_Ocases, axiom, (! [A:$tType]: (((euclid1678468529ng_gcd @ A) => (! [X:((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A)))))]: ((~ ((! [S4:A,S5:A,T4:A,T5:A,R:A,R2:A]: ((X != ((((product_Pair @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A))))) @ S4) @ ((((product_Pair @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A)))) @ S5) @ ((((product_Pair @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A))) @ T4) @ ((((product_Pair @ A) @ ((product_prod @ A) @ A)) @ T5) @ ((((product_Pair @ A) @ A) @ R) @ R2)))))))))))))))).
% 3.01/1.65 thf(fact_117_compl__le__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X)) @ ((uminus_uminus @ A) @ Y)) = (((ord_less_eq @ A) @ Y) @ X)))))))).
% 3.01/1.65 thf(fact_118_iff__sem, axiom, (! [A:$tType,C:$tType,B:$tType]: ((((finite_finite @ B) & ((finite_finite @ C) & (finite_finite @ A))) => (! [Nu5:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),I2:((((denota1663640101rp_ext @ B) @ C) @ A) @ product_unit),A5:(((formula @ B) @ C) @ A),B5:(((formula @ B) @ C) @ A)]: (((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu5) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ (((((equiv @ B) @ C) @ A) @ A5) @ B5))) = ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu5) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ A5)) = (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu5) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ B5)))))))))).
% 3.01/1.65 thf(fact_119_uminus__apply, axiom, (! [B:$tType,A:$tType]: (((uminus @ B) => ((uminus_uminus @ (A > B)) = (^ [A6:(A > B),X3:A]: (((uminus_uminus @ B) @ (A6 @ X3))))))))).
% 3.01/1.65 thf(fact_120_double__compl, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ X)) = X))))))).
% 3.01/1.65 thf(fact_121_compl__eq__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X:A,Y:A]: (((((uminus_uminus @ A) @ X) = ((uminus_uminus @ A) @ Y)) = (X = Y)))))))).
% 3.01/1.65 thf(fact_122_VSagree__sub, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A5:(set @ A),B5:(set @ A),Nu5:((finite_Cartesian_vec @ real) @ A),Omega:((finite_Cartesian_vec @ real) @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => (((((denotational_VSagree @ A) @ Nu5) @ Omega) @ B5) => ((((denotational_VSagree @ A) @ Nu5) @ Omega) @ A5))))))))).
% 3.01/1.65 thf(fact_123_VSagree__supset, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [B5:(set @ A),A5:(set @ A),Nu5:((finite_Cartesian_vec @ real) @ A),Nu6:((finite_Cartesian_vec @ real) @ A)]: (((((ord_less_eq @ (set @ A)) @ B5) @ A5) => (((((denotational_VSagree @ A) @ Nu5) @ Nu6) @ A5) => ((((denotational_VSagree @ A) @ Nu5) @ Nu6) @ B5))))))))).
% 3.01/1.65 thf(fact_124_VSagree__refl, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [Nu5:((finite_Cartesian_vec @ real) @ A),A5:(set @ A)]: (((((denotational_VSagree @ A) @ Nu5) @ Nu5) @ A5))))))).
% 3.01/1.65 thf(fact_125_VSagree__def, axiom, (! [C:$tType]: (((finite_finite @ C) => ((denotational_VSagree @ C) = (^ [Nu:((finite_Cartesian_vec @ real) @ C),Nu2:((finite_Cartesian_vec @ real) @ C),V:(set @ C)]: ((! [X3:C]: (((((member @ C) @ X3) @ V) => (((((finite1433825200ec_nth @ real) @ C) @ Nu) @ X3) = ((((finite1433825200ec_nth @ real) @ C) @ Nu2) @ X3)))))))))))).
% 3.01/1.65 thf(fact_126_gcd_Ocases, axiom, (! [A:$tType]: (((euclid1155270486miring @ A) => (! [X:((product_prod @ A) @ A)]: ((~ ((! [A4:A,B4:A]: ((X != ((((product_Pair @ A) @ A) @ A4) @ B4)))))))))))).
% 3.01/1.65 thf(fact_127_fun__Compl__def, axiom, (! [B:$tType,A:$tType]: (((uminus @ B) => ((uminus_uminus @ (A > B)) = (^ [A6:(A > B),X3:A]: (((uminus_uminus @ B) @ (A6 @ X3))))))))).
% 3.01/1.65 thf(fact_128_compl__mono, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y)) @ ((uminus_uminus @ A) @ X))))))))).
% 3.01/1.65 thf(fact_129_compl__le__swap1, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ ((uminus_uminus @ A) @ X)) => (((ord_less_eq @ A) @ X) @ ((uminus_uminus @ A) @ Y))))))))).
% 3.01/1.65 thf(fact_130_compl__le__swap2, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y)) @ X) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X)) @ Y)))))))).
% 3.01/1.65 thf(fact_131_image__uminus__atLeastAtMost, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A,Y:A]: ((((((image @ A) @ A) @ (uminus_uminus @ A)) @ (((set_or331188842AtMost @ A) @ X) @ Y)) = (((set_or331188842AtMost @ A) @ ((uminus_uminus @ A) @ Y)) @ ((uminus_uminus @ A) @ X))))))))).
% 3.01/1.65 thf(fact_132_atLeastatMost__subset__iff, axiom, (! [A:$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)))))))))).
% 3.01/1.65 thf(fact_133_Compl__subset__Compl__iff, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A5)) @ ((uminus_uminus @ (set @ A)) @ B5)) = (((ord_less_eq @ (set @ A)) @ B5) @ A5))))).
% 3.01/1.65 thf(fact_134_Compl__anti__mono, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ B5)) @ ((uminus_uminus @ (set @ A)) @ A5)))))).
% 3.01/1.65 thf(fact_135_image__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F3:(B > A),X:B,A5:(set @ B)]: (((B2 = (F3 @ X)) => ((((member @ B) @ X) @ A5) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F3) @ A5))))))).
% 3.01/1.65 thf(fact_136_subset__antisym, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ A5) => (A5 = B5)))))).
% 3.01/1.65 thf(fact_137_subsetI, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((! [X4:A]: (((((member @ A) @ X4) @ A5) => (((member @ A) @ X4) @ B5)))) => (((ord_less_eq @ (set @ A)) @ A5) @ B5))))).
% 3.01/1.65 thf(fact_138_Compl__eq__Compl__iff, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((((uminus_uminus @ (set @ A)) @ A5) = ((uminus_uminus @ (set @ A)) @ B5)) = (A5 = B5))))).
% 3.01/1.65 thf(fact_139_Compl__iff, axiom, (! [A:$tType,C3:A,A5:(set @ A)]: (((((member @ A) @ C3) @ ((uminus_uminus @ (set @ A)) @ A5)) = (~ ((((member @ A) @ C3) @ A5))))))).
% 3.01/1.65 thf(fact_140_ComplI, axiom, (! [A:$tType,C3:A,A5:(set @ A)]: (((~ ((((member @ A) @ C3) @ A5))) => (((member @ A) @ C3) @ ((uminus_uminus @ (set @ A)) @ A5)))))).
% 3.01/1.65 thf(fact_141_image__ident, axiom, (! [A:$tType,Y7:(set @ A)]: ((((((image @ A) @ A) @ (^ [X3:A]: (X3))) @ Y7) = Y7)))).
% 3.01/1.65 thf(fact_142_Icc__eq__Icc, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,H:A,L2:A,H2:A]: ((((((set_or331188842AtMost @ A) @ L) @ H) = (((set_or331188842AtMost @ A) @ L2) @ H2)) = (((L = L2) & (H = H2)) | ((~ ((((ord_less_eq @ A) @ L) @ H))) & (~ ((((ord_less_eq @ A) @ L2) @ H2)))))))))))).
% 3.01/1.65 thf(fact_143_atLeastAtMost__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I4:A,L:A,U:A]: (((((member @ A) @ I4) @ (((set_or331188842AtMost @ A) @ L) @ U)) = ((((ord_less_eq @ A) @ L) @ I4) & (((ord_less_eq @ A) @ I4) @ U))))))))).
% 3.01/1.65 thf(fact_144_imageI, axiom, (! [B:$tType,A:$tType,X:A,A5:(set @ A),F3:(A > B)]: (((((member @ A) @ X) @ A5) => (((member @ B) @ (F3 @ X)) @ ((((image @ A) @ B) @ F3) @ A5)))))).
% 3.01/1.65 thf(fact_145_image__iff, axiom, (! [A:$tType,B:$tType,Z2:A,F3:(B > A),A5:(set @ B)]: (((((member @ A) @ Z2) @ ((((image @ B) @ A) @ F3) @ A5)) = (? [X3:B]: (((((member @ B) @ X3) @ A5) & (Z2 = (F3 @ X3))))))))).
% 3.01/1.65 thf(fact_146_bex__imageD, axiom, (! [A:$tType,B:$tType,F3:(B > A),A5:(set @ B),P:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F3) @ A5)) & (P @ X5)))) => (? [X4:B]: (((((member @ B) @ X4) @ A5) & (P @ (F3 @ X4))))))))).
% 3.01/1.65 thf(fact_147_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N2:(set @ A),F3:(A > B),G:(A > B)]: (((M = N2) => ((! [X4:A]: (((((member @ A) @ X4) @ N2) => ((F3 @ X4) = (G @ X4))))) => (((((image @ A) @ B) @ F3) @ M) = ((((image @ A) @ B) @ G) @ N2))))))).
% 3.01/1.65 thf(fact_148_ball__imageD, axiom, (! [A:$tType,B:$tType,F3:(B > A),A5:(set @ B),P:(A > $o)]: (((! [X4:A]: (((((member @ A) @ X4) @ ((((image @ B) @ A) @ F3) @ A5)) => (P @ X4)))) => (! [X5:B]: (((((member @ B) @ X5) @ A5) => (P @ (F3 @ X5))))))))).
% 3.01/1.65 thf(fact_149_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X:A,A5:(set @ A),B2:B,F3:(A > B)]: (((((member @ A) @ X) @ A5) => ((B2 = (F3 @ X)) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ F3) @ A5))))))).
% 3.01/1.65 thf(fact_150_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X3:A]: (((P @ X3) => (Q @ X3)))))))).
% 3.01/1.65 thf(fact_151_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y5:(set @ A),Z:(set @ A)]: ((Y5 = Z))) = (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ A6)))))))).
% 3.01/1.65 thf(fact_152_subset__trans, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ C4) => (((ord_less_eq @ (set @ A)) @ A5) @ C4)))))).
% 3.01/1.65 thf(fact_153_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) => (Q @ X4)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 3.01/1.65 thf(fact_154_subset__refl, axiom, (! [A:$tType,A5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A5) @ A5)))).
% 3.01/1.65 thf(fact_155_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A6) => (((member @ A) @ T3) @ B6)))))))))).
% 3.01/1.65 thf(fact_156_equalityD2, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((A5 = B5) => (((ord_less_eq @ (set @ A)) @ B5) @ A5))))).
% 3.01/1.65 thf(fact_157_equalityD1, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((A5 = B5) => (((ord_less_eq @ (set @ A)) @ A5) @ B5))))).
% 3.01/1.65 thf(fact_158_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((member @ A) @ X3) @ B6)))))))))).
% 3.01/1.65 thf(fact_159_equalityE, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((A5 = B5) => (~ (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => (~ ((((ord_less_eq @ (set @ A)) @ B5) @ A5)))))))))).
% 3.01/1.65 thf(fact_160_subsetD, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((member @ A) @ C3) @ A5) => (((member @ A) @ C3) @ B5)))))).
% 3.01/1.65 thf(fact_161_in__mono, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((member @ A) @ X) @ A5) => (((member @ A) @ X) @ B5)))))).
% 3.01/1.65 thf(fact_162_double__complement, axiom, (! [A:$tType,A5:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A5)) = A5)))).
% 3.01/1.65 thf(fact_163_ComplD, axiom, (! [A:$tType,C3:A,A5:(set @ A)]: (((((member @ A) @ C3) @ ((uminus_uminus @ (set @ A)) @ A5)) => (~ ((((member @ A) @ C3) @ A5))))))).
% 3.01/1.65 thf(fact_164_imageE, axiom, (! [A:$tType,B:$tType,B2:A,F3:(B > A),A5:(set @ B)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ F3) @ A5)) => (~ ((! [X4:B]: (((B2 = (F3 @ X4)) => (~ ((((member @ B) @ X4) @ A5)))))))))))).
% 3.01/1.65 thf(fact_165_image__image, axiom, (! [A:$tType,B:$tType,C:$tType,F3:(B > A),G:(C > B),A5:(set @ C)]: ((((((image @ B) @ A) @ F3) @ ((((image @ C) @ B) @ G) @ A5)) = ((((image @ C) @ A) @ (^ [X3:C]: ((F3 @ (G @ X3))))) @ A5))))).
% 3.01/1.65 thf(fact_166_Compr__image__eq, axiom, (! [A:$tType,B:$tType,F3:(B > A),A5:(set @ B),P:(A > $o)]: ((((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ ((((image @ B) @ A) @ F3) @ A5)) & (P @ X3))))) = ((((image @ B) @ A) @ F3) @ ((collect @ B) @ (^ [X3:B]: (((((member @ B) @ X3) @ A5) & (P @ (F3 @ X3))))))))))).
% 3.01/1.65 thf(fact_167_less__eq__set__def, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((((ord_less_eq @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ A6)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ B6)))))))))).
% 3.01/1.65 thf(fact_168_Collect__subset, axiom, (! [A:$tType,A5:(set @ A),P:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A5) & (P @ X3)))))) @ A5)))).
% 3.01/1.65 thf(fact_169_Collect__neg__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((collect @ A) @ (^ [X3:A]: ((~ ((P @ X3)))))) = ((uminus_uminus @ (set @ A)) @ ((collect @ A) @ P)))))).
% 3.01/1.65 thf(fact_170_Compl__eq, axiom, (! [A:$tType]: (((uminus_uminus @ (set @ A)) = (^ [A6:(set @ A)]: (((collect @ A) @ (^ [X3:A]: ((~ ((((member @ A) @ X3) @ A6)))))))))))).
% 3.01/1.65 thf(fact_171_uminus__set__def, axiom, (! [A:$tType]: (((uminus_uminus @ (set @ A)) = (^ [A6:(set @ A)]: (((collect @ A) @ ((uminus_uminus @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ A6))))))))))).
% 3.01/1.65 thf(fact_172_image__mono, axiom, (! [B:$tType,A:$tType,A5:(set @ A),B5:(set @ A),F3:(A > B)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F3) @ A5)) @ ((((image @ A) @ B) @ F3) @ B5)))))).
% 3.01/1.65 thf(fact_173_image__subsetI, axiom, (! [A:$tType,B:$tType,A5:(set @ A),F3:(A > B),B5:(set @ B)]: (((! [X4:A]: (((((member @ A) @ X4) @ A5) => (((member @ B) @ (F3 @ X4)) @ B5)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F3) @ A5)) @ B5))))).
% 3.01/1.65 thf(fact_174_subset__imageE, axiom, (! [A:$tType,B:$tType,B5:(set @ A),F3:(B > A),A5:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B5) @ ((((image @ B) @ A) @ F3) @ A5)) => (~ ((! [C5:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C5) @ A5) => (B5 != ((((image @ B) @ A) @ F3) @ C5))))))))))).
% 3.01/1.65 thf(fact_175_image__subset__iff, axiom, (! [A:$tType,B:$tType,F3:(B > A),A5:(set @ B),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F3) @ A5)) @ B5) = (! [X3:B]: (((((member @ B) @ X3) @ A5) => (((member @ A) @ (F3 @ X3)) @ B5)))))))).
% 3.01/1.65 thf(fact_176_subset__image__iff, axiom, (! [A:$tType,B:$tType,B5:(set @ A),F3:(B > A),A5:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B5) @ ((((image @ B) @ A) @ F3) @ A5)) = (? [AA:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ AA) @ A5) & (B5 = ((((image @ B) @ A) @ F3) @ AA))))))))).
% 3.01/1.65 thf(fact_177_repd_Oelims, axiom, (! [C:$tType]: (((finite_finite @ C) => (! [X:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),Xa:C,Xb:real,Y:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))]: (((((((denotational_repd @ C) @ X) @ Xa) @ Xb) = Y) => (Y = ((((product_Pair @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ X)) @ (((finite1990238425lambda @ C) @ real) @ (^ [Y3:C]: (((((if @ real) @ (Xa = Y3)) @ Xb) @ ((((finite1433825200ec_nth @ real) @ C) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ X)) @ Y3)))))))))))))).
% 3.01/1.65 thf(fact_178_repd_Osimps, axiom, (! [C:$tType]: (((finite_finite @ C) => ((denotational_repd @ C) = (^ [V3:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),X3:C,R3:real]: (((((product_Pair @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ V3)) @ (((finite1990238425lambda @ C) @ real) @ (^ [Y3:C]: (((((if @ real) @ (X3 = Y3)) @ R3) @ ((((finite1433825200ec_nth @ real) @ C) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ V3)) @ Y3))))))))))))).
% 3.01/1.65 thf(fact_179_nth__map__matrix, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType]: ((((finite_finite @ B) & (finite_finite @ C)) => (! [F3:(D > A),X:((finite_Cartesian_vec @ ((finite_Cartesian_vec @ D) @ B)) @ C),I4:C,J2:B]: ((((((finite1433825200ec_nth @ A) @ B) @ ((((finite1433825200ec_nth @ ((finite_Cartesian_vec @ A) @ B)) @ C) @ ((((((finite1614961371matrix @ D) @ A) @ B) @ C) @ F3) @ X)) @ I4)) @ J2) = (F3 @ ((((finite1433825200ec_nth @ D) @ B) @ ((((finite1433825200ec_nth @ ((finite_Cartesian_vec @ D) @ B)) @ C) @ X) @ I4)) @ J2))))))))).
% 3.01/1.65 thf(fact_180_image__subset__iff__funcset, axiom, (! [A:$tType,B:$tType,F4:(B > A),A5:(set @ B),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F4) @ A5)) @ B5) = (((member @ (B > A)) @ F4) @ ((((pi @ B) @ A) @ A5) @ (^ [Uu:B]: (B5)))))))).
% 3.01/1.65 thf(fact_181_Pi__I, axiom, (! [B:$tType,A:$tType,A5:(set @ A),F3:(A > B),B5:(A > (set @ B))]: (((! [X4:A]: (((((member @ A) @ X4) @ A5) => (((member @ B) @ (F3 @ X4)) @ (B5 @ X4))))) => (((member @ (A > B)) @ F3) @ ((((pi @ A) @ B) @ A5) @ B5)))))).
% 3.01/1.65 thf(fact_182_PiE, axiom, (! [B:$tType,A:$tType,F3:(A > B),A5:(set @ A),B5:(A > (set @ B)),X:A]: (((((member @ (A > B)) @ F3) @ ((((pi @ A) @ B) @ A5) @ B5)) => ((~ ((((member @ B) @ (F3 @ X)) @ (B5 @ X)))) => (~ ((((member @ A) @ X) @ A5)))))))).
% 3.01/1.65 thf(fact_183_Pi__I_H, axiom, (! [B:$tType,A:$tType,A5:(set @ A),F3:(A > B),B5:(A > (set @ B))]: (((! [X4:A]: (((((member @ A) @ X4) @ A5) => (((member @ B) @ (F3 @ X4)) @ (B5 @ X4))))) => (((member @ (A > B)) @ F3) @ ((((pi @ A) @ B) @ A5) @ B5)))))).
% 3.01/1.65 thf(fact_184_Pi__iff, axiom, (! [B:$tType,A:$tType,F3:(A > B),I2:(set @ A),X8:(A > (set @ B))]: (((((member @ (A > B)) @ F3) @ ((((pi @ A) @ B) @ I2) @ X8)) = (! [X3:A]: (((((member @ A) @ X3) @ I2) => (((member @ B) @ (F3 @ X3)) @ (X8 @ X3))))))))).
% 3.01/1.65 thf(fact_185_Pi__mem, axiom, (! [B:$tType,A:$tType,F3:(A > B),A5:(set @ A),B5:(A > (set @ B)),X:A]: (((((member @ (A > B)) @ F3) @ ((((pi @ A) @ B) @ A5) @ B5)) => ((((member @ A) @ X) @ A5) => (((member @ B) @ (F3 @ X)) @ (B5 @ X))))))).
% 3.01/1.65 thf(fact_186_Pi__cong, axiom, (! [B:$tType,A:$tType,A5:(set @ A),F3:(A > B),G:(A > B),B5:(A > (set @ B))]: (((! [W:A]: (((((member @ A) @ W) @ A5) => ((F3 @ W) = (G @ W))))) => ((((member @ (A > B)) @ F3) @ ((((pi @ A) @ B) @ A5) @ B5)) = (((member @ (A > B)) @ G) @ ((((pi @ A) @ B) @ A5) @ B5))))))).
% 3.01/1.65 thf(fact_187_funcsetI, axiom, (! [B:$tType,A:$tType,A5:(set @ A),F3:(A > B),B5:(set @ B)]: (((! [X4:A]: (((((member @ A) @ X4) @ A5) => (((member @ B) @ (F3 @ X4)) @ B5)))) => (((member @ (A > B)) @ F3) @ ((((pi @ A) @ B) @ A5) @ (^ [Uu:A]: (B5)))))))).
% 3.01/1.65 thf(fact_188_funcset__id, axiom, (! [A:$tType,A5:(set @ A)]: ((((member @ (A > A)) @ (^ [X3:A]: (X3))) @ ((((pi @ A) @ A) @ A5) @ (^ [Uu:A]: (A5))))))).
% 3.01/1.65 thf(fact_189_funcset__mem, axiom, (! [A:$tType,B:$tType,F3:(A > B),A5:(set @ A),B5:(set @ B),X:A]: (((((member @ (A > B)) @ F3) @ ((((pi @ A) @ B) @ A5) @ (^ [Uu:A]: (B5)))) => ((((member @ A) @ X) @ A5) => (((member @ B) @ (F3 @ X)) @ B5)))))).
% 3.01/1.65 thf(fact_190_Pi__mono, axiom, (! [B:$tType,A:$tType,A5:(set @ A),B5:(A > (set @ B)),C4:(A > (set @ B))]: (((! [X4:A]: (((((member @ A) @ X4) @ A5) => (((ord_less_eq @ (set @ B)) @ (B5 @ X4)) @ (C4 @ X4))))) => (((ord_less_eq @ (set @ (A > B))) @ ((((pi @ A) @ B) @ A5) @ B5)) @ ((((pi @ A) @ B) @ A5) @ C4)))))).
% 3.01/1.65 thf(fact_191_Pi__anti__mono, axiom, (! [B:$tType,A:$tType,A7:(set @ A),A5:(set @ A),B5:(A > (set @ B))]: (((((ord_less_eq @ (set @ A)) @ A7) @ A5) => (((ord_less_eq @ (set @ (A > B))) @ ((((pi @ A) @ B) @ A5) @ B5)) @ ((((pi @ A) @ B) @ A7) @ B5)))))).
% 3.01/1.65 thf(fact_192_map__matrix__def, axiom, (! [I:$tType,J:$tType,B:$tType,A:$tType]: ((((finite_finite @ J) & (finite_finite @ I)) => (((((finite1614961371matrix @ A) @ B) @ I) @ J) = (^ [F5:(A > B),X3:((finite_Cartesian_vec @ ((finite_Cartesian_vec @ A) @ I)) @ J)]: ((((finite1990238425lambda @ J) @ ((finite_Cartesian_vec @ B) @ I)) @ (^ [I3:J]: ((((finite1990238425lambda @ I) @ B) @ (^ [J3:I]: ((F5 @ ((((finite1433825200ec_nth @ A) @ I) @ ((((finite1433825200ec_nth @ ((finite_Cartesian_vec @ A) @ I)) @ J) @ X3) @ I3)) @ J3))))))))))))))).
% 3.01/1.65 thf(fact_193_funcset__image, axiom, (! [A:$tType,B:$tType,F3:(A > B),A5:(set @ A),B5:(set @ B)]: (((((member @ (A > B)) @ F3) @ ((((pi @ A) @ B) @ A5) @ (^ [Uu:A]: (B5)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F3) @ A5)) @ B5))))).
% 3.01/1.65 thf(fact_194_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 3.01/1.65 thf(fact_195_repv_Osimps, axiom, (! [C:$tType]: (((finite_finite @ C) => ((denotational_repv @ C) = (^ [V3:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),X3:C,R3:real]: (((((product_Pair @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ (((finite1990238425lambda @ C) @ real) @ (^ [Y3:C]: (((((if @ real) @ (X3 = Y3)) @ R3) @ ((((finite1433825200ec_nth @ real) @ C) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ V3)) @ Y3)))))) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ V3))))))))).
% 3.01/1.65 thf(fact_196_repv_Oelims, axiom, (! [C:$tType]: (((finite_finite @ C) => (! [X:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),Xa:C,Xb:real,Y:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))]: (((((((denotational_repv @ C) @ X) @ Xa) @ Xb) = Y) => (Y = ((((product_Pair @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ (((finite1990238425lambda @ C) @ real) @ (^ [Y3:C]: (((((if @ real) @ (Xa = Y3)) @ Xb) @ ((((finite1433825200ec_nth @ real) @ C) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ X)) @ Y3)))))) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ X)))))))))).
% 3.01/1.65 thf(fact_197_exI__realizer, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Y:A,X:B]: ((((P @ Y) @ X) => ((P @ (((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y))) @ (((product_fst @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y))))))).
% 3.01/1.65 thf(fact_198_conjI__realizer, axiom, (! [A:$tType,B:$tType,P:(A > $o),P2:A,Q:(B > $o),Q2:B]: (((P @ P2) => ((Q @ Q2) => ((P @ (((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q2))) & (Q @ (((product_snd @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q2))))))))).
% 3.01/1.65 thf(fact_199_exE__realizer_H, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),P2:((product_prod @ B) @ A)]: ((((P @ (((product_snd @ B) @ A) @ P2)) @ (((product_fst @ B) @ A) @ P2)) => (~ ((! [X4:B,Y4:A]: ((~ (((P @ Y4) @ X4))))))))))).
% 3.01/1.65 thf(fact_200_BNF__Greatest__Fixpoint_Osubst__Pair, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),X:A,Y:B,A2:((product_prod @ A) @ B)]: ((((P @ X) @ Y) => ((A2 = ((((product_Pair @ A) @ B) @ X) @ Y)) => ((P @ (((product_fst @ A) @ B) @ A2)) @ (((product_snd @ A) @ B) @ A2))))))).
% 3.01/1.65 thf(fact_201_concrete__v_Opelims, axiom, (! [B:$tType,A:$tType,C:$tType]: ((((finite_finite @ C) & ((finite_finite @ A) & (finite_finite @ B))) => (! [X:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),Xa:((ode @ A) @ C),Xb:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),Xc:((finite_Cartesian_vec @ real) @ C),Y:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))]: ((((((((((denota1738237502rete_v @ A) @ B) @ C) @ X) @ Xa) @ Xb) @ Xc) = Y) => ((((accp @ ((product_prod @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ ((product_prod @ ((ode @ A) @ C)) @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((finite_Cartesian_vec @ real) @ C))))) @ (((denota2028625979_v_rel @ A) @ B) @ C)) @ ((((product_Pair @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ ((product_prod @ ((ode @ A) @ C)) @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((finite_Cartesian_vec @ real) @ C)))) @ X) @ ((((product_Pair @ ((ode @ A) @ C)) @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((finite_Cartesian_vec @ real) @ C))) @ Xa) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((finite_Cartesian_vec @ real) @ C)) @ Xb) @ Xc)))) => (~ (((Y = ((((product_Pair @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ (((finite1990238425lambda @ C) @ real) @ (^ [I3:C]: (((((finite1433825200ec_nth @ real) @ C) @ ((((if @ ((finite_Cartesian_vec @ real) @ C)) @ (((member @ ((sum_sum @ C) @ C)) @ (((sum_Inl @ C) @ C) @ I3)) @ (((((denotational_semBV @ A) @ B) @ C) @ X) @ Xa))) @ Xc) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ Xb))) @ I3))))) @ (((finite1990238425lambda @ C) @ real) @ (^ [I3:C]: (((((finite1433825200ec_nth @ real) @ C) @ ((((if @ ((finite_Cartesian_vec @ real) @ C)) @ (((member @ ((sum_sum @ C) @ C)) @ (((sum_Inr @ C) @ C) @ I3)) @ (((((denotational_semBV @ A) @ B) @ C) @ X) @ Xa))) @ ((((((denotational_ODE_sem @ A) @ B) @ C) @ X) @ Xa) @ Xc)) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ Xb))) @ I3)))))) => (~ ((((accp @ ((product_prod @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ ((product_prod @ ((ode @ A) @ C)) @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((finite_Cartesian_vec @ real) @ C))))) @ (((denota2028625979_v_rel @ A) @ B) @ C)) @ ((((product_Pair @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ ((product_prod @ ((ode @ A) @ C)) @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((finite_Cartesian_vec @ real) @ C)))) @ X) @ ((((product_Pair @ ((ode @ A) @ C)) @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((finite_Cartesian_vec @ real) @ C))) @ Xa) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((finite_Cartesian_vec @ real) @ C)) @ Xb) @ Xc))))))))))))))))).
% 3.01/1.65 thf(fact_202_not__arg__cong__Inr, axiom, (! [B:$tType,A:$tType,X:A,Y:A]: (((X != Y) => ((((sum_Inr @ A) @ B) @ X) != (((sum_Inr @ A) @ B) @ Y)))))).
% 3.01/1.65 thf(fact_203_subset__Collect__iff, axiom, (! [A:$tType,B5:(set @ A),A5:(set @ A),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B5) @ A5) => ((((ord_less_eq @ (set @ A)) @ B5) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A5) & (P @ X3)))))) = (! [X3:A]: (((((member @ A) @ X3) @ B5) => (P @ X3))))))))).
% 3.01/1.65 thf(fact_204_subset__CollectI, axiom, (! [A:$tType,B5:(set @ A),A5:(set @ A),Q:(A > $o),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B5) @ A5) => ((! [X4:A]: (((((member @ A) @ X4) @ B5) => ((Q @ X4) => (P @ X4))))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ B5) & (Q @ X3)))))) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A5) & (P @ X3))))))))))).
% 3.01/1.65 thf(fact_205_obj__sumE, axiom, (! [A:$tType,B:$tType,S2:((sum_sum @ A) @ B)]: (((! [X4:A]: ((S2 != (((sum_Inl @ A) @ B) @ X4)))) => (~ ((! [X4:B]: ((S2 != (((sum_Inr @ B) @ A) @ X4)))))))))).
% 3.01/1.65 thf(fact_206_predicate1I, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) => (Q @ X4)))) => (((ord_less_eq @ (A > $o)) @ P) @ Q))))).
% 3.01/1.65 thf(fact_207_minus__le__self__iff, axiom, (! [A:$tType]: (((lattic1601792062up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 3.01/1.65 thf(fact_208_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 3.01/1.65 thf(fact_209_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F3:(B > A),B2:B,C3:B]: (((((ord_less_eq @ A) @ A2) @ (F3 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C3) => ((! [X4:B,Y4:B]: (((((ord_less_eq @ B) @ X4) @ Y4) => (((ord_less_eq @ A) @ (F3 @ X4)) @ (F3 @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F3 @ C3))))))))))).
% 3.01/1.65 thf(fact_210_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F3:(A > C),C3:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F3 @ B2)) @ C3) => ((! [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) => (((ord_less_eq @ C) @ (F3 @ X4)) @ (F3 @ Y4))))) => (((ord_less_eq @ C) @ (F3 @ A2)) @ C3)))))))))).
% 3.01/1.65 thf(fact_211_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F3:(B > A),B2:B,C3:B]: (((A2 = (F3 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C3) => ((! [X4:B,Y4:B]: (((((ord_less_eq @ B) @ X4) @ Y4) => (((ord_less_eq @ A) @ (F3 @ X4)) @ (F3 @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F3 @ C3))))))))))).
% 3.01/1.65 thf(fact_212_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F3:(A > B),C3:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F3 @ B2) = C3) => ((! [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) => (((ord_less_eq @ B) @ (F3 @ X4)) @ (F3 @ Y4))))) => (((ord_less_eq @ B) @ (F3 @ A2)) @ C3)))))))))).
% 3.01/1.65 thf(fact_213_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z:A]: ((Y5 = Z))) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X3))))))))).
% 3.01/1.65 thf(fact_214_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 3.01/1.65 thf(fact_215_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 3.01/1.65 thf(fact_216_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 3.01/1.65 thf(fact_217_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 3.01/1.65 thf(fact_218_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 3.01/1.65 thf(fact_219_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A,Z2:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z2)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z2)))) => (((((ord_less_eq @ A) @ X) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y)))) => (((((ord_less_eq @ A) @ Z2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 3.01/1.65 thf(fact_220_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 3.01/1.65 thf(fact_221_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z:A]: ((Y5 = Z))) = (^ [A8:A,B7:A]: (((((ord_less_eq @ A) @ A8) @ B7) & (((ord_less_eq @ A) @ B7) @ A8))))))))).
% 3.01/1.65 thf(fact_222_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C3:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 3.01/1.65 thf(fact_223_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 3.01/1.65 thf(fact_224_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 3.01/1.65 thf(fact_225_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z2) => (((ord_less_eq @ A) @ X) @ Z2))))))))).
% 3.01/1.65 thf(fact_226_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 3.01/1.65 thf(fact_227_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((P @ A4) @ B4)))) => ((! [A4:A,B4:A]: ((((P @ B4) @ A4) => ((P @ A4) @ B4)))) => ((P @ A2) @ B2))))))))).
% 3.01/1.65 thf(fact_228_dual__order_Otrans, axiom, (! [A:$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))))))))).
% 3.01/1.65 thf(fact_229_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z:A]: ((Y5 = Z))) = (^ [A8:A,B7:A]: (((((ord_less_eq @ A) @ B7) @ A8) & (((ord_less_eq @ A) @ A8) @ B7))))))))).
% 3.01/1.65 thf(fact_230_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 3.01/1.65 thf(fact_231_predicate1D, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o),X:A]: (((((ord_less_eq @ (A > $o)) @ P) @ Q) => ((P @ X) => (Q @ X)))))).
% 3.01/1.65 thf(fact_232_rev__predicate1D, axiom, (! [A:$tType,P:(A > $o),X:A,Q:(A > $o)]: (((P @ X) => ((((ord_less_eq @ (A > $o)) @ P) @ Q) => (Q @ X)))))).
% 3.01/1.65 thf(fact_233_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G) => (((ord_less_eq @ B) @ (F3 @ X)) @ (G @ X))))))))).
% 3.01/1.65 thf(fact_234_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G) => (((ord_less_eq @ B) @ (F3 @ X)) @ (G @ X))))))))).
% 3.01/1.65 thf(fact_235_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G:(A > B)]: (((! [X4:A]: ((((ord_less_eq @ B) @ (F3 @ X4)) @ (G @ X4)))) => (((ord_less_eq @ (A > B)) @ F3) @ G)))))))).
% 3.01/1.65 thf(fact_236_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F5:(A > B),G4:(A > B)]: ((! [X3:A]: ((((ord_less_eq @ B) @ (F5 @ X3)) @ (G4 @ X3))))))))))).
% 3.01/1.65 thf(fact_237_le__minus__self__iff, axiom, (! [A:$tType]: (((lattic1601792062up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 3.01/1.65 thf(fact_238_old_Osum_Oinject_I1_J, axiom, (! [B:$tType,A:$tType,A2:A,A3:A]: ((((((sum_Inl @ A) @ B) @ A2) = (((sum_Inl @ A) @ B) @ A3)) = (A2 = A3))))).
% 3.01/1.65 thf(fact_239_sum_Oinject_I1_J, axiom, (! [B:$tType,A:$tType,X1:A,Y1:A]: ((((((sum_Inl @ A) @ B) @ X1) = (((sum_Inl @ A) @ B) @ Y1)) = (X1 = Y1))))).
% 3.01/1.65 thf(fact_240_sum_Oinject_I2_J, axiom, (! [A:$tType,B:$tType,X22:B,Y2:B]: ((((((sum_Inr @ B) @ A) @ X22) = (((sum_Inr @ B) @ A) @ Y2)) = (X22 = Y2))))).
% 3.01/1.65 thf(fact_241_old_Osum_Oinject_I2_J, axiom, (! [A:$tType,B:$tType,B2:B,B3:B]: ((((((sum_Inr @ B) @ A) @ B2) = (((sum_Inr @ B) @ A) @ B3)) = (B2 = B3))))).
% 3.01/1.65 thf(fact_242_Inr__inject, axiom, (! [A:$tType,B:$tType,X:B,Y:B]: ((((((sum_Inr @ B) @ A) @ X) = (((sum_Inr @ B) @ A) @ Y)) => (X = Y))))).
% 3.01/1.65 thf(fact_243_Inl__inject, axiom, (! [B:$tType,A:$tType,X:A,Y:A]: ((((((sum_Inl @ A) @ B) @ X) = (((sum_Inl @ A) @ B) @ Y)) => (X = Y))))).
% 3.01/1.65 thf(fact_244_sum_Odistinct_I1_J, axiom, (! [A:$tType,B:$tType,X1:A,X22:B]: (((((sum_Inl @ A) @ B) @ X1) != (((sum_Inr @ B) @ A) @ X22))))).
% 3.01/1.65 thf(fact_245_old_Osum_Odistinct_I2_J, axiom, (! [B8:$tType,A9:$tType,B9:B8,A10:A9]: (((((sum_Inr @ B8) @ A9) @ B9) != (((sum_Inl @ A9) @ B8) @ A10))))).
% 3.01/1.65 thf(fact_246_old_Osum_Odistinct_I1_J, axiom, (! [A:$tType,B:$tType,A2:A,B3:B]: (((((sum_Inl @ A) @ B) @ A2) != (((sum_Inr @ B) @ A) @ B3))))).
% 3.01/1.65 thf(fact_247_sumE, axiom, (! [A:$tType,B:$tType,S2:((sum_sum @ A) @ B)]: (((! [X4:A]: ((S2 != (((sum_Inl @ A) @ B) @ X4)))) => (~ ((! [Y4:B]: ((S2 != (((sum_Inr @ B) @ A) @ Y4)))))))))).
% 3.01/1.65 thf(fact_248_Inr__not__Inl, axiom, (! [B:$tType,A:$tType,B2:B,A2:A]: (((((sum_Inr @ B) @ A) @ B2) != (((sum_Inl @ A) @ B) @ A2))))).
% 3.01/1.65 thf(fact_249_split__sum__ex, axiom, (! [B:$tType,A:$tType]: (((^ [P3:(((sum_sum @ A) @ B) > $o)]: ((? [X9:((sum_sum @ A) @ B)]: ((P3 @ X9))))) = (^ [P4:(((sum_sum @ A) @ B) > $o)]: (((? [X3:A]: ((P4 @ (((sum_Inl @ A) @ B) @ X3)))) | (? [X3:B]: ((P4 @ (((sum_Inr @ B) @ A) @ X3))))))))))).
% 3.01/1.65 thf(fact_250_split__sum__all, axiom, (! [B:$tType,A:$tType]: (((^ [P3:(((sum_sum @ A) @ B) > $o)]: ((! [X9:((sum_sum @ A) @ B)]: ((P3 @ X9))))) = (^ [P4:(((sum_sum @ A) @ B) > $o)]: (((! [X3:A]: ((P4 @ (((sum_Inl @ A) @ B) @ X3)))) & (! [X3:B]: ((P4 @ (((sum_Inr @ B) @ A) @ X3))))))))))).
% 3.01/1.65 thf(fact_251_old_Osum_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((sum_sum @ A) @ B)]: (((! [A4:A]: ((Y != (((sum_Inl @ A) @ B) @ A4)))) => (~ ((! [B4:B]: ((Y != (((sum_Inr @ B) @ A) @ B4)))))))))).
% 3.01/1.65 thf(fact_252_old_Osum_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((sum_sum @ A) @ B) > $o),Sum:((sum_sum @ A) @ B)]: (((! [A4:A]: ((P @ (((sum_Inl @ A) @ B) @ A4)))) => ((! [B4:B]: ((P @ (((sum_Inr @ B) @ A) @ B4)))) => (P @ Sum)))))).
% 3.01/1.65 thf(fact_253_pred__subset__eq, axiom, (! [A:$tType,R4:(set @ A),S6:(set @ A)]: (((((ord_less_eq @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ R4)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ S6)))) = (((ord_less_eq @ (set @ A)) @ R4) @ S6))))).
% 3.01/1.65 thf(fact_254_verit__minus__simplify_I4_J, axiom, (! [B:$tType]: (((group_add @ B) => (! [B2:B]: ((((uminus_uminus @ B) @ ((uminus_uminus @ B) @ B2)) = B2))))))).
% 3.01/1.65 thf(subcl_Finite__Set_Ofinite___HOL_Otype, axiom, (! [A:$tType]: (((finite_finite @ A) => (type @ A))))).
% 3.01/1.65 thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space, axiom, (! [A9:$tType,A11:$tType]: ((((ordere890947078_space @ A9) & (finite_finite @ A11)) => (ordere890947078_space @ ((finite_Cartesian_vec @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space_1, axiom, (! [A9:$tType,A11:$tType]: ((((ordere890947078_space @ A9) & (ordere890947078_space @ A11)) => (ordere890947078_space @ ((product_prod @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_2, axiom, (ordere890947078_space @ real)).
% 3.01/1.65 thf(tcon_fun___Lattices_Oboolean__algebra, axiom, (! [A9:$tType,A11:$tType]: (((boolean_algebra @ A11) => (boolean_algebra @ (A9 > A11)))))).
% 3.01/1.65 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A9:$tType,A11:$tType]: (((preorder @ A11) => (preorder @ (A9 > A11)))))).
% 3.01/1.65 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A9:$tType,A11:$tType]: ((((finite_finite @ A9) & (finite_finite @ A11)) => (finite_finite @ (A9 > A11)))))).
% 3.01/1.65 thf(tcon_fun___Orderings_Oorder, axiom, (! [A9:$tType,A11:$tType]: (((order @ A11) => (order @ (A9 > A11)))))).
% 3.01/1.65 thf(tcon_fun___Orderings_Oord, axiom, (! [A9:$tType,A11:$tType]: (((ord @ A11) => (ord @ (A9 > A11)))))).
% 3.01/1.65 thf(tcon_fun___Groups_Ouminus, axiom, (! [A9:$tType,A11:$tType]: (((uminus @ A11) => (uminus @ (A9 > A11)))))).
% 3.01/1.65 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_3, axiom, (! [A9:$tType]: ((boolean_algebra @ (set @ A9))))).
% 3.01/1.65 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A9:$tType]: ((preorder @ (set @ A9))))).
% 3.01/1.65 thf(tcon_Set_Oset___Finite__Set_Ofinite_5, axiom, (! [A9:$tType]: (((finite_finite @ A9) => (finite_finite @ (set @ A9)))))).
% 3.01/1.65 thf(tcon_Set_Oset___Orderings_Oorder_6, axiom, (! [A9:$tType]: ((order @ (set @ A9))))).
% 3.01/1.65 thf(tcon_Set_Oset___Orderings_Oord_7, axiom, (! [A9:$tType]: ((ord @ (set @ A9))))).
% 3.01/1.65 thf(tcon_Set_Oset___Groups_Ouminus_8, axiom, (! [A9:$tType]: ((uminus @ (set @ A9))))).
% 3.01/1.65 thf(tcon_Set_Oset___Groups_Ozero, axiom, (! [A9:$tType]: (((zero @ A9) => (zero @ (set @ A9)))))).
% 3.01/1.65 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_9, axiom, (boolean_algebra @ $o)).
% 3.01/1.65 thf(tcon_HOL_Obool___Orderings_Opreorder_10, axiom, (preorder @ $o)).
% 3.01/1.65 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 3.01/1.65 thf(tcon_HOL_Obool___Finite__Set_Ofinite_11, axiom, (finite_finite @ $o)).
% 3.01/1.65 thf(tcon_HOL_Obool___Orderings_Oorder_12, axiom, (order @ $o)).
% 3.01/1.65 thf(tcon_HOL_Obool___Orderings_Oord_13, axiom, (ord @ $o)).
% 3.01/1.65 thf(tcon_HOL_Obool___Groups_Ouminus_14, axiom, (uminus @ $o)).
% 3.01/1.65 thf(tcon_Real_Oreal___Lattice__Algebras_Olattice__ab__group__add, axiom, (lattic1601792062up_add @ real)).
% 3.01/1.65 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add, axiom, (linord219039673up_add @ real)).
% 3.01/1.65 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add, axiom, (ordered_ab_group_add @ real)).
% 3.01/1.65 thf(tcon_Real_Oreal___Orderings_Opreorder_15, axiom, (preorder @ real)).
% 3.01/1.65 thf(tcon_Real_Oreal___Orderings_Olinorder_16, axiom, (linorder @ real)).
% 3.01/1.65 thf(tcon_Real_Oreal___Groups_Ogroup__add, axiom, (group_add @ real)).
% 3.01/1.65 thf(tcon_Real_Oreal___Orderings_Oorder_17, axiom, (order @ real)).
% 3.01/1.65 thf(tcon_Real_Oreal___Orderings_Oord_18, axiom, (ord @ real)).
% 3.01/1.65 thf(tcon_Real_Oreal___Groups_Ouminus_19, axiom, (uminus @ real)).
% 3.01/1.65 thf(tcon_Real_Oreal___Groups_Ozero_20, axiom, (zero @ real)).
% 3.01/1.65 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_21, axiom, (! [A9:$tType,A11:$tType]: ((((finite_finite @ A9) & (finite_finite @ A11)) => (finite_finite @ ((sum_sum @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__group__add_22, axiom, (! [A9:$tType,A11:$tType]: ((((ordere890947078_space @ A9) & (ordere890947078_space @ A11)) => (ordered_ab_group_add @ ((product_prod @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Product__Type_Oprod___Lattices_Oboolean__algebra_23, axiom, (! [A9:$tType,A11:$tType]: ((((boolean_algebra @ A9) & (boolean_algebra @ A11)) => (boolean_algebra @ ((product_prod @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_24, axiom, (! [A9:$tType,A11:$tType]: ((((preorder @ A9) & (preorder @ A11)) => (preorder @ ((product_prod @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_25, axiom, (! [A9:$tType,A11:$tType]: ((((finite_finite @ A9) & (finite_finite @ A11)) => (finite_finite @ ((product_prod @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Product__Type_Oprod___Groups_Ogroup__add_26, axiom, (! [A9:$tType,A11:$tType]: ((((group_add @ A9) & (group_add @ A11)) => (group_add @ ((product_prod @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Product__Type_Oprod___Orderings_Oorder_27, axiom, (! [A9:$tType,A11:$tType]: ((((order @ A9) & (order @ A11)) => (order @ ((product_prod @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Product__Type_Oprod___Orderings_Oord_28, axiom, (! [A9:$tType,A11:$tType]: ((((ord @ A9) & (ord @ A11)) => (ord @ ((product_prod @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Product__Type_Oprod___Groups_Ouminus_29, axiom, (! [A9:$tType,A11:$tType]: ((((uminus @ A9) & (uminus @ A11)) => (uminus @ ((product_prod @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Product__Type_Oprod___Groups_Ozero_30, axiom, (! [A9:$tType,A11:$tType]: ((((zero @ A9) & (zero @ A11)) => (zero @ ((product_prod @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Product__Type_Ounit___Lattices_Oboolean__algebra_31, axiom, (boolean_algebra @ product_unit)).
% 3.01/1.65 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_32, axiom, (preorder @ product_unit)).
% 3.01/1.65 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_33, axiom, (linorder @ product_unit)).
% 3.01/1.65 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_34, axiom, (finite_finite @ product_unit)).
% 3.01/1.65 thf(tcon_Product__Type_Ounit___Orderings_Oorder_35, axiom, (order @ product_unit)).
% 3.01/1.65 thf(tcon_Product__Type_Ounit___Orderings_Oord_36, axiom, (ord @ product_unit)).
% 3.01/1.65 thf(tcon_Product__Type_Ounit___Groups_Ouminus_37, axiom, (uminus @ product_unit)).
% 3.01/1.65 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__group__add_38, axiom, (! [A9:$tType,A11:$tType]: ((((ordere890947078_space @ A9) & (finite_finite @ A11)) => (ordered_ab_group_add @ ((finite_Cartesian_vec @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Opreorder_39, axiom, (! [A9:$tType,A11:$tType]: ((((order @ A9) & (finite_finite @ A11)) => (preorder @ ((finite_Cartesian_vec @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_40, axiom, (! [A9:$tType,A11:$tType]: ((((linorder @ A9) & (cARD_1 @ A11)) => (linorder @ ((finite_Cartesian_vec @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_41, axiom, (! [A9:$tType,A11:$tType]: ((((finite_finite @ A9) & (finite_finite @ A11)) => (finite_finite @ ((finite_Cartesian_vec @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ogroup__add_42, axiom, (! [A9:$tType,A11:$tType]: ((((group_add @ A9) & (finite_finite @ A11)) => (group_add @ ((finite_Cartesian_vec @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oorder_43, axiom, (! [A9:$tType,A11:$tType]: ((((order @ A9) & (finite_finite @ A11)) => (order @ ((finite_Cartesian_vec @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oord_44, axiom, (! [A9:$tType,A11:$tType]: ((((ord @ A9) & (finite_finite @ A11)) => (ord @ ((finite_Cartesian_vec @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ouminus_45, axiom, (! [A9:$tType,A11:$tType]: ((((uminus @ A9) & (finite_finite @ A11)) => (uminus @ ((finite_Cartesian_vec @ A9) @ A11)))))).
% 3.01/1.65 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_46, axiom, (! [A9:$tType,A11:$tType]: ((((zero @ A9) & (finite_finite @ A11)) => (zero @ ((finite_Cartesian_vec @ A9) @ A11)))))).
% 3.01/1.65 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 3.01/1.65 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 3.01/1.65 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 3.01/1.65 thf(tfree_0, hypothesis, (finite_finite @ c)).
% 3.01/1.65 thf(tfree_1, hypothesis, (finite_finite @ a)).
% 3.01/1.65 thf(tfree_2, hypothesis, (finite_finite @ b)).
% 3.01/1.65 thf(conj_0, hypothesis, (((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ i2)) @ ((((static_FVF @ a) @ b) @ c) @ phi))).
% 3.01/1.65 thf(conj_1, hypothesis, (((member @ c) @ i2) @ (((((denota811733865E_vars @ a) @ b) @ c) @ i) @ oDEa))).
% 3.01/1.65 thf(conj_2, hypothesis, (~ ((((member @ c) @ i2) @ (((static_FVO @ a) @ c) @ oDEa))))).
% 3.01/1.65 thf(conj_3, hypothesis, (((~ ((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ i2)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inl @ c) @ c)) @ (((((denota811733865E_vars @ a) @ b) @ c) @ i) @ oDEa))))) & (~ ((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ i2)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inr @ c) @ c)) @ (((((denota811733865E_vars @ a) @ b) @ c) @ i) @ oDEa)))))) => (((((member @ c) @ i2) @ (((static_FVO @ a) @ c) @ oDEa)) => (((((finite1433825200ec_nth @ real) @ c) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))) @ bb)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))))) @ i2) = ((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ i2))) & ((~ ((((member @ c) @ i2) @ (((static_FVO @ a) @ c) @ oDEa)))) => (((((finite1433825200ec_nth @ real) @ c) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))) @ bb)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))))) @ i2) = ((((finite1433825200ec_nth @ real) @ c) @ ab) @ i2)))))).
% 3.01/1.65 thf(conj_4, hypothesis, (((~ ((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ i2)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inl @ c) @ c)) @ (((((denota811733865E_vars @ a) @ b) @ c) @ i) @ oDEa))))) & (~ ((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ i2)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inr @ c) @ c)) @ (((((denota811733865E_vars @ a) @ b) @ c) @ i) @ oDEa)))))) => (((((finite1433825200ec_nth @ real) @ c) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ (sol @ x)))) @ i2) = ((((finite1433825200ec_nth @ real) @ c) @ ab) @ i2)))).
% 3.01/1.65 thf(conj_5, hypothesis, (((~ ((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inr @ c) @ c) @ i2)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inl @ c) @ c)) @ (((((denota811733865E_vars @ a) @ b) @ c) @ i) @ oDEa))))) & (~ ((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inr @ c) @ c) @ i2)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inr @ c) @ c)) @ (((((denota811733865E_vars @ a) @ b) @ c) @ i) @ oDEa)))))) => (((((finite1433825200ec_nth @ real) @ c) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))) @ bb)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))))) @ i2) = ((((finite1433825200ec_nth @ real) @ c) @ bb) @ i2)))).
% 3.01/1.65 thf(conj_6, hypothesis, (((~ ((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inr @ c) @ c) @ i2)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inl @ c) @ c)) @ (((((denota811733865E_vars @ a) @ b) @ c) @ i) @ oDEa))))) & (~ ((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inr @ c) @ c) @ i2)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inr @ c) @ c)) @ (((((denota811733865E_vars @ a) @ b) @ c) @ i) @ oDEa)))))) => (((((finite1433825200ec_nth @ real) @ c) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ (sol @ x)))) @ i2) = ((((finite1433825200ec_nth @ real) @ c) @ bb) @ i2)))).
% 3.01/1.65 thf(conj_7, hypothesis, (((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ i2)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inl @ c) @ c)) @ (((((denota811733865E_vars @ a) @ b) @ c) @ i) @ oDEa))) => (((((member @ c) @ i2) @ (((static_FVO @ a) @ c) @ oDEa)) => (((((finite1433825200ec_nth @ real) @ c) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))) @ bb)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))))) @ i2) = ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ i2))) & ((~ ((((member @ c) @ i2) @ (((static_FVO @ a) @ c) @ oDEa)))) => (((((finite1433825200ec_nth @ real) @ c) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))) @ bb)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))))) @ i2) = ((((finite1433825200ec_nth @ real) @ c) @ ab) @ i2))))) & ((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ i2)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inr @ c) @ c)) @ (((((denota811733865E_vars @ a) @ b) @ c) @ i) @ oDEa))) => (((((member @ c) @ i2) @ (((static_FVO @ a) @ c) @ oDEa)) => (((((finite1433825200ec_nth @ real) @ c) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))) @ bb)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))))) @ i2) = ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ i2))) & ((~ ((((member @ c) @ i2) @ (((static_FVO @ a) @ c) @ oDEa)))) => (((((finite1433825200ec_nth @ real) @ c) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))) @ bb)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))))) @ i2) = ((((finite1433825200ec_nth @ real) @ c) @ ab) @ i2))))))).
% 3.01/1.65 thf(conj_8, hypothesis, (((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inr @ c) @ c) @ i2)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inl @ c) @ c)) @ (((((denota811733865E_vars @ a) @ b) @ c) @ i) @ oDEa))) => (((((finite1433825200ec_nth @ real) @ c) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))) @ bb)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))))) @ i2) = ((((finite1433825200ec_nth @ real) @ c) @ ((((((denotational_ODE_sem @ a) @ b) @ c) @ i) @ oDEa) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3))))))) @ i2))) & ((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inr @ c) @ c) @ i2)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inr @ c) @ c)) @ (((((denota811733865E_vars @ a) @ b) @ c) @ i) @ oDEa))) => (((((finite1433825200ec_nth @ real) @ c) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ (zero_zero @ real))) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))) @ bb)) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3)))))))) @ i2) = ((((finite1433825200ec_nth @ real) @ c) @ ((((((denotational_ODE_sem @ a) @ b) @ c) @ i) @ oDEa) @ (((finite1990238425lambda @ c) @ real) @ (^ [I3:c]: (((((if @ real) @ (((member @ c) @ I3) @ (((static_FVO @ a) @ c) @ oDEa))) @ ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ I3)) @ ((((finite1433825200ec_nth @ real) @ c) @ ab) @ I3))))))) @ i2))))).
% 3.01/1.65 thf(conj_9, hypothesis, (((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ i2)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inl @ c) @ c)) @ (((((denota811733865E_vars @ a) @ b) @ c) @ i) @ oDEa))) => (((((finite1433825200ec_nth @ real) @ c) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ (sol @ x)))) @ i2) = ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ i2))) & ((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inl @ c) @ c) @ i2)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inr @ c) @ c)) @ (((((denota811733865E_vars @ a) @ b) @ c) @ i) @ oDEa))) => (((((finite1433825200ec_nth @ real) @ c) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ (sol @ x)))) @ i2) = ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ i2))))).
% 3.01/1.65 thf(conj_10, hypothesis, (((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inr @ c) @ c) @ i2)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inl @ c) @ c)) @ (((((denota811733865E_vars @ a) @ b) @ c) @ i) @ oDEa))) => (((((finite1433825200ec_nth @ real) @ c) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ (sol @ x)))) @ i2) = ((((finite1433825200ec_nth @ real) @ c) @ ((((((denotational_ODE_sem @ a) @ b) @ c) @ i) @ oDEa) @ (sol @ x))) @ i2))) & ((((member @ ((sum_sum @ c) @ c)) @ (((sum_Inr @ c) @ c) @ i2)) @ ((((image @ c) @ ((sum_sum @ c) @ c)) @ ((sum_Inr @ c) @ c)) @ (((((denota811733865E_vars @ a) @ b) @ c) @ i) @ oDEa))) => (((((finite1433825200ec_nth @ real) @ c) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ (((((((denotational_mk_v @ a) @ b) @ c) @ i) @ oDEa) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ ab) @ bb)) @ (sol @ x)))) @ i2) = ((((finite1433825200ec_nth @ real) @ c) @ ((((((denotational_ODE_sem @ a) @ b) @ c) @ i) @ oDEa) @ (sol @ x))) @ i2))))).
% 3.01/1.65 thf(conj_11, conjecture, (((((finite1433825200ec_nth @ real) @ c) @ ab) @ i2) = ((((finite1433825200ec_nth @ real) @ c) @ (sol @ x)) @ i2))).
% 3.01/1.65 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.iHbtgXCbd3/DTF2DTF15214.p
% 3.01/1.65 ------------------------
% 3.15/1.72 ---- Cleaned THF ---
% 3.15/1.72 thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext,type,
% 3.15/1.72 denota1663640101rp_ext: $tType > $tType > $tType > $tType > $tType ).
% 3.15/1.72
% 3.15/1.72 thf(ty_t_Finite__Cartesian__Product_Ovec,type,
% 3.15/1.72 finite_Cartesian_vec: $tType > $tType > $tType ).
% 3.15/1.72
% 3.15/1.72 thf(ty_t_Product__Type_Ounit,type,
% 3.15/1.72 product_unit: $tType ).
% 3.15/1.72
% 3.15/1.72 thf(ty_t_Product__Type_Oprod,type,
% 3.15/1.72 product_prod: $tType > $tType > $tType ).
% 3.15/1.72
% 3.15/1.72 thf(ty_t_Syntax_Oformula,type,
% 3.15/1.72 formula: $tType > $tType > $tType > $tType ).
% 3.15/1.72
% 3.15/1.72 thf(ty_t_Sum__Type_Osum,type,
% 3.15/1.72 sum_sum: $tType > $tType > $tType ).
% 3.15/1.72
% 3.15/1.72 thf(ty_t_Syntax_OODE,type,
% 3.15/1.72 ode: $tType > $tType > $tType ).
% 3.15/1.72
% 3.15/1.72 thf(ty_t_Real_Oreal,type,
% 3.15/1.72 real: $tType ).
% 3.15/1.72
% 3.15/1.72 thf(ty_t_Set_Oset,type,
% 3.15/1.72 set: $tType > $tType ).
% 3.15/1.72
% 3.15/1.72 thf(ty_tf_d,type,
% 3.15/1.72 d: $tType ).
% 3.15/1.72
% 3.15/1.72 thf(ty_tf_c,type,
% 3.15/1.72 c: $tType ).
% 3.15/1.72
% 3.15/1.72 thf(ty_tf_b,type,
% 3.15/1.72 b: $tType ).
% 3.15/1.72
% 3.15/1.72 thf(ty_tf_a,type,
% 3.15/1.72 a: $tType ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_Cardinality_OCARD__1,type,
% 3.15/1.72 cARD_1:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
% 3.15/1.72 ordere890947078_space:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_HOL_Otype,type,
% 3.15/1.72 type:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_Groups_Ozero,type,
% 3.15/1.72 zero:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_Groups_Ouminus,type,
% 3.15/1.72 uminus:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_Orderings_Oord,type,
% 3.15/1.72 ord:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_Orderings_Oorder,type,
% 3.15/1.72 order:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_Groups_Ogroup__add,type,
% 3.15/1.72 group_add:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_Finite__Set_Ofinite,type,
% 3.15/1.72 finite_finite:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_Orderings_Olinorder,type,
% 3.15/1.72 linorder:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_Orderings_Opreorder,type,
% 3.15/1.72 preorder:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 3.15/1.72 boolean_algebra:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 3.15/1.72 ordered_ab_group_add:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 3.15/1.72 linord219039673up_add:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 3.15/1.72 canoni770627133id_add:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_Lattice__Algebras_Olattice__ab__group__add,type,
% 3.15/1.72 lattic1601792062up_add:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd,type,
% 3.15/1.72 euclid1678468529ng_gcd:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring,type,
% 3.15/1.72 euclid1155270486miring:
% 3.15/1.72 !>[A: $tType] : $o ).
% 3.15/1.72
% 3.15/1.72 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__fml,type,
% 3.15/1.72 coinci1993344360de_fml:
% 3.15/1.72 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > $o ) ).
% 3.15/1.72
% 3.15/1.72 thf(sy_c_Denotational__Semantics_OODE__sem,type,
% 3.15/1.72 denotational_ODE_sem:
% 3.15/1.72 !>[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 ) ) ).
% 3.15/1.72
% 3.15/1.72 thf(sy_c_Denotational__Semantics_OODE__vars,type,
% 3.15/1.72 denota811733865E_vars:
% 3.15/1.72 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( set @ C ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Denotational__Semantics_OVSagree,type,
% 3.15/1.73 denotational_VSagree:
% 3.15/1.73 !>[C: $tType] : ( ( finite_Cartesian_vec @ real @ C ) > ( finite_Cartesian_vec @ real @ C ) > ( set @ C ) > $o ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Denotational__Semantics_OVagree,type,
% 3.15/1.73 denotational_Vagree:
% 3.15/1.73 !>[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 ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Denotational__Semantics_Oconcrete__v,type,
% 3.15/1.73 denota1738237502rete_v:
% 3.15/1.73 !>[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 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Denotational__Semantics_Oconcrete__v__rel,type,
% 3.15/1.73 denota2028625979_v_rel:
% 3.15/1.73 !>[A: $tType,B: $tType,C: $tType] : ( ( product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) > ( product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) > $o ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Denotational__Semantics_Ofml__sem,type,
% 3.15/1.73 denotational_fml_sem:
% 3.15/1.73 !>[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 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Denotational__Semantics_Omk__v,type,
% 3.15/1.73 denotational_mk_v:
% 3.15/1.73 !>[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 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Denotational__Semantics_Omk__xode,type,
% 3.15/1.73 denotational_mk_xode:
% 3.15/1.73 !>[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 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Denotational__Semantics_Orepd,type,
% 3.15/1.73 denotational_repd:
% 3.15/1.73 !>[C: $tType] : ( ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > C > real > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Denotational__Semantics_Orepv,type,
% 3.15/1.73 denotational_repv:
% 3.15/1.73 !>[C: $tType] : ( ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > C > real > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Denotational__Semantics_OsemBV,type,
% 3.15/1.73 denotational_semBV:
% 3.15/1.73 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Finite__Cartesian__Product_Omap__matrix,type,
% 3.15/1.73 finite1614961371matrix:
% 3.15/1.73 !>[A: $tType,B: $tType,I: $tType,J: $tType] : ( ( A > B ) > ( finite_Cartesian_vec @ ( finite_Cartesian_vec @ A @ I ) @ J ) > ( finite_Cartesian_vec @ ( finite_Cartesian_vec @ B @ I ) @ J ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__lambda,type,
% 3.15/1.73 finite1990238425lambda:
% 3.15/1.73 !>[B: $tType,A: $tType] : ( ( B > A ) > ( finite_Cartesian_vec @ A @ B ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__nth,type,
% 3.15/1.73 finite1433825200ec_nth:
% 3.15/1.73 !>[A: $tType,B: $tType] : ( ( finite_Cartesian_vec @ A @ B ) > B > A ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_FuncSet_OPi,type,
% 3.15/1.73 pi:
% 3.15/1.73 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( A > B ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 3.15/1.73 uminus_uminus:
% 3.15/1.73 !>[A: $tType] : ( A > A ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 3.15/1.73 zero_zero:
% 3.15/1.73 !>[A: $tType] : A ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_If,type,
% 3.15/1.73 if:
% 3.15/1.73 !>[A: $tType] : ( $o > A > A > A ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Initial__Value__Problem_Osolves__ode,type,
% 3.15/1.73 initia1685620758es_ode:
% 3.15/1.73 !>[A: $tType] : ( ( real > A ) > ( real > A > A ) > ( set @ real ) > ( set @ A ) > $o ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.15/1.73 ord_less_eq:
% 3.15/1.73 !>[A: $tType] : ( A > A > $o ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Product__Type_OPair,type,
% 3.15/1.73 product_Pair:
% 3.15/1.73 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 3.15/1.73 product_rec_prod:
% 3.15/1.73 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 3.15/1.73 product_fst:
% 3.15/1.73 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 3.15/1.73 product_snd:
% 3.15/1.73 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Set_OCollect,type,
% 3.15/1.73 collect:
% 3.15/1.73 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Set_Oimage,type,
% 3.15/1.73 image:
% 3.15/1.73 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
% 3.15/1.73 set_or331188842AtMost:
% 3.15/1.73 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Static__Semantics_OBVO,type,
% 3.15/1.73 static_BVO:
% 3.15/1.73 !>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Static__Semantics_OFVF,type,
% 3.15/1.73 static_FVF:
% 3.15/1.73 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Static__Semantics_OFVO,type,
% 3.15/1.73 static_FVO:
% 3.15/1.73 !>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > ( set @ C ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Sum__Type_OInl,type,
% 3.15/1.73 sum_Inl:
% 3.15/1.73 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Sum__Type_OInr,type,
% 3.15/1.73 sum_Inr:
% 3.15/1.73 !>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Syntax_OEquiv,type,
% 3.15/1.73 equiv:
% 3.15/1.73 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Syntax_Ofsafe,type,
% 3.15/1.73 fsafe:
% 3.15/1.73 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > $o ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Syntax_Oosafe,type,
% 3.15/1.73 osafe:
% 3.15/1.73 !>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > $o ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_Wellfounded_Oaccp,type,
% 3.15/1.73 accp:
% 3.15/1.73 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_c_member,type,
% 3.15/1.73 member:
% 3.15/1.73 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_v_I____,type,
% 3.15/1.73 i: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
% 3.15/1.73
% 3.15/1.73 thf(sy_v_ODE____,type,
% 3.15/1.73 ode2: ode @ a @ c ).
% 3.15/1.73
% 3.15/1.73 thf(sy_v_ODEa____,type,
% 3.15/1.73 oDEa: ode @ a @ c ).
% 3.15/1.73
% 3.15/1.73 thf(sy_v_P____,type,
% 3.15/1.73 p: formula @ a @ b @ c ).
% 3.15/1.73
% 3.15/1.73 thf(sy_v__092_060phi_062_H____,type,
% 3.15/1.73 phi: formula @ a @ b @ c ).
% 3.15/1.73
% 3.15/1.73 thf(sy_v_aa____,type,
% 3.15/1.73 aa: finite_Cartesian_vec @ real @ c ).
% 3.15/1.73
% 3.15/1.73 thf(sy_v_ab____,type,
% 3.15/1.73 ab: finite_Cartesian_vec @ real @ c ).
% 3.15/1.73
% 3.15/1.73 thf(sy_v_ba____,type,
% 3.15/1.73 ba: finite_Cartesian_vec @ real @ c ).
% 3.15/1.73
% 3.15/1.73 thf(sy_v_bb____,type,
% 3.15/1.73 bb: finite_Cartesian_vec @ real @ c ).
% 3.15/1.73
% 3.15/1.73 thf(sy_v_i____,type,
% 3.15/1.73 i2: c ).
% 3.15/1.73
% 3.15/1.73 thf(sy_v_sol____,type,
% 3.15/1.73 sol: real > ( finite_Cartesian_vec @ real @ c ) ).
% 3.15/1.73
% 3.15/1.73 thf(sy_v_t____,type,
% 3.15/1.73 t: real ).
% 3.15/1.73
% 3.15/1.73 thf(sy_v_x____,type,
% 3.15/1.73 x: real ).
% 3.15/1.73
% 3.15/1.73 thf(fact_0_ode__to__fvo,axiom,
% 3.15/1.73 ! [C: $tType,A: $tType,B: $tType,X: A,I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A] :
% 3.15/1.73 ( ( member @ A @ X @ ( denota811733865E_vars @ B @ C @ A @ I2 @ ODE ) )
% 3.15/1.73 => ( member @ A @ X @ ( static_FVO @ B @ A @ ODE ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_1_osafe,axiom,
% 3.15/1.73 osafe @ a @ c @ oDEa ).
% 3.15/1.73
% 3.15/1.73 thf(fact_2_eqP,axiom,
% 3.15/1.73 p = phi ).
% 3.15/1.73
% 3.15/1.73 thf(fact_3_x,axiom,
% 3.15/1.73 ord_less_eq @ real @ ( zero_zero @ real ) @ x ).
% 3.15/1.73
% 3.15/1.73 thf(fact_4_fsafe,axiom,
% 3.15/1.73 fsafe @ a @ b @ c @ phi ).
% 3.15/1.73
% 3.15/1.73 thf(fact_5__092_060open_062_IInl_Ai_A_092_060in_062_AsemBV_AI_AODE_J_A_061_A_IInr_Ai_A_092_060in_062_AsemBV_AI_AODE_J_092_060close_062,axiom,
% 3.15/1.73 ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) )
% 3.15/1.73 = ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_6_VA,axiom,
% 3.15/1.73 ! [X2: real] :
% 3.15/1.73 ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ X2 ) )
% 3.15/1.73 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ X2 ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
% 3.15/1.73 @ ( static_FVF @ a @ b @ c @ phi ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_7_prod_Ocollapse,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 3.15/1.73 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 3.15/1.73 = Prod ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_8_bvo__to__fvo,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,X: A,ODE: ode @ B @ A] :
% 3.15/1.73 ( ( member @ ( sum_sum @ A @ A ) @ ( sum_Inl @ A @ A @ X ) @ ( static_BVO @ B @ A @ ODE ) )
% 3.15/1.73 => ( member @ A @ X @ ( static_FVO @ B @ A @ ODE ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_9_vec__lambda__eta,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( finite_finite @ B )
% 3.15/1.73 => ! [G: finite_Cartesian_vec @ A @ B] :
% 3.15/1.73 ( ( finite1990238425lambda @ B @ A @ ( finite1433825200ec_nth @ A @ B @ G ) )
% 3.15/1.73 = G ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_10_Inl__Inr__False,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,X: A,Y: B] :
% 3.15/1.73 ( ( sum_Inl @ A @ B @ X )
% 3.15/1.73 != ( sum_Inr @ B @ A @ Y ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_11_Inr__Inl__False,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,X: B,Y: A] :
% 3.15/1.73 ( ( sum_Inr @ B @ A @ X )
% 3.15/1.73 != ( sum_Inl @ A @ B @ Y ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_12_vec__lambda__beta,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType] :
% 3.15/1.73 ( ( finite_finite @ B )
% 3.15/1.73 => ! [G: B > A,I4: B] :
% 3.15/1.73 ( ( finite1433825200ec_nth @ A @ B @ ( finite1990238425lambda @ B @ A @ G ) @ I4 )
% 3.15/1.73 = ( G @ I4 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_13_alt__sem__lemma,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,C: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ C )
% 3.15/1.73 & ( finite_finite @ A )
% 3.15/1.73 & ( finite_finite @ B ) )
% 3.15/1.73 => ! [ODE: ode @ A @ C,I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,Sol: real > ( finite_Cartesian_vec @ real @ C ),T2: real,Ab: finite_Cartesian_vec @ real @ C] :
% 3.15/1.73 ( ( osafe @ A @ C @ ODE )
% 3.15/1.73 => ( ( denotational_ODE_sem @ A @ B @ C @ I2 @ ODE @ ( Sol @ T2 ) )
% 3.15/1.73 = ( denotational_ODE_sem @ A @ B @ C @ I2 @ ODE
% 3.15/1.73 @ ( finite1990238425lambda @ C @ real
% 3.15/1.73 @ ^ [I3: C] : ( if @ real @ ( member @ C @ I3 @ ( static_FVO @ A @ C @ ODE ) ) @ ( finite1433825200ec_nth @ real @ C @ ( Sol @ T2 ) @ I3 ) @ ( finite1433825200ec_nth @ real @ C @ Ab @ I3 ) ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_14_snd__zero,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( ( zero @ A )
% 3.15/1.73 & ( zero @ B ) )
% 3.15/1.73 => ( ( product_snd @ B @ A @ ( zero_zero @ ( product_prod @ B @ A ) ) )
% 3.15/1.73 = ( zero_zero @ A ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_15_fst__zero,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( ( zero @ A )
% 3.15/1.73 & ( zero @ B ) )
% 3.15/1.73 => ( ( product_fst @ A @ B @ ( zero_zero @ ( product_prod @ A @ B ) ) )
% 3.15/1.73 = ( zero_zero @ A ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_16_zero__index,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( ( zero @ A )
% 3.15/1.73 & ( finite_finite @ B ) )
% 3.15/1.73 => ! [I4: B] :
% 3.15/1.73 ( ( finite1433825200ec_nth @ A @ B @ ( zero_zero @ ( finite_Cartesian_vec @ A @ B ) ) @ I4 )
% 3.15/1.73 = ( zero_zero @ A ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_17_hpsafe__Evolve_OIH,axiom,
% 3.15/1.73 coinci1993344360de_fml @ a @ b @ c @ p ).
% 3.15/1.73
% 3.15/1.73 thf(fact_18_hpsafe__Evolve_Ohyps_I1_J,axiom,
% 3.15/1.73 osafe @ a @ c @ ode2 ).
% 3.15/1.73
% 3.15/1.73 thf(fact_19_hpsafe__Evolve_Ohyps_I2_J,axiom,
% 3.15/1.73 fsafe @ a @ b @ c @ p ).
% 3.15/1.73
% 3.15/1.73 thf(fact_20_old_Oprod_Oinject,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 3.15/1.73 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 3.15/1.73 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 3.15/1.73 = ( ( A2 = A3 )
% 3.15/1.73 & ( B2 = B3 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_21_prod_Oinject,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 3.15/1.73 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 3.15/1.73 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 3.15/1.73 = ( ( X1 = Y1 )
% 3.15/1.73 & ( X22 = Y2 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_22_t,axiom,
% 3.15/1.73 ord_less_eq @ real @ x @ t ).
% 3.15/1.73
% 3.15/1.73 thf(fact_23_all,axiom,
% 3.15/1.73 ! [I5: c] :
% 3.15/1.73 ( ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ I5 ) @ ( static_BVO @ a @ c @ oDEa ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I5 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c @ ab @ I5 ) ) )
% 3.15/1.73 & ( ( member @ ( sum_sum @ c @ d ) @ ( sum_Inl @ c @ d @ I5 ) @ ( image @ c @ ( sum_sum @ c @ d ) @ ( sum_Inl @ c @ d ) @ ( static_FVO @ a @ c @ oDEa ) ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I5 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c @ ab @ I5 ) ) )
% 3.15/1.73 & ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ I5 ) @ ( static_FVF @ a @ b @ c @ phi ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I5 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c @ ab @ I5 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_24_less__eq__vec__def,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( ( ord @ A )
% 3.15/1.73 & ( finite_finite @ B ) )
% 3.15/1.73 => ( ( ord_less_eq @ ( finite_Cartesian_vec @ A @ B ) )
% 3.15/1.73 = ( ^ [X3: finite_Cartesian_vec @ A @ B,Y3: finite_Cartesian_vec @ A @ B] :
% 3.15/1.73 ! [I3: B] : ( ord_less_eq @ A @ ( finite1433825200ec_nth @ A @ B @ X3 @ I3 ) @ ( finite1433825200ec_nth @ A @ B @ Y3 @ I3 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_25_old_Oprod_Oinducts,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 3.15/1.73 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 3.15/1.73 => ( P @ Prod ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_26_old_Oprod_Oexhaust,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 3.15/1.73 ~ ! [A4: A,B4: B] :
% 3.15/1.73 ( Y
% 3.15/1.73 != ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_27_prod__induct7,axiom,
% 3.15/1.73 ! [G2: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) )] :
% 3.15/1.73 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G2 ) @ E2 @ ( product_Pair @ F @ G2 @ F2 @ G3 ) ) ) ) ) ) )
% 3.15/1.73 => ( P @ X ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_28_prod__induct6,axiom,
% 3.15/1.73 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 3.15/1.73 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( 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 ) ) ) ) ) )
% 3.15/1.73 => ( P @ X ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_29_prod__induct5,axiom,
% 3.15/1.73 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 3.15/1.73 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 3.15/1.73 => ( P @ X ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_30_prod__induct4,axiom,
% 3.15/1.73 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 3.15/1.73 ( ! [A4: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 3.15/1.73 => ( P @ X ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_31_prod__induct3,axiom,
% 3.15/1.73 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.15/1.73 ( ! [A4: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 3.15/1.73 => ( P @ X ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_32_prod__cases7,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) )] :
% 3.15/1.73 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G3: G2] :
% 3.15/1.73 ( Y
% 3.15/1.73 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G2 ) @ E2 @ ( product_Pair @ F @ G2 @ F2 @ G3 ) ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_33_prod__cases6,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 3.15/1.73 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
% 3.15/1.73 ( Y
% 3.15/1.73 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( 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 ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_34_prod__cases5,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 3.15/1.73 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
% 3.15/1.73 ( Y
% 3.15/1.73 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_35_prod__cases4,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 3.15/1.73 ~ ! [A4: A,B4: B,C2: C,D2: D] :
% 3.15/1.73 ( Y
% 3.15/1.73 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_36_prod__cases3,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.15/1.73 ~ ! [A4: A,B4: B,C2: C] :
% 3.15/1.73 ( Y
% 3.15/1.73 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_37_Pair__inject,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 3.15/1.73 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 3.15/1.73 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 3.15/1.73 => ~ ( ( A2 = A3 )
% 3.15/1.73 => ( B2 != B3 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_38_prod__cases,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 3.15/1.73 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 3.15/1.73 => ( P @ P2 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_39_surj__pair,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 3.15/1.73 ? [X4: A,Y4: B] :
% 3.15/1.73 ( P2
% 3.15/1.73 = ( product_Pair @ A @ B @ X4 @ Y4 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_40_vec__nth__inject,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( finite_finite @ B )
% 3.15/1.73 => ! [X: finite_Cartesian_vec @ A @ B,Y: finite_Cartesian_vec @ A @ B] :
% 3.15/1.73 ( ( ( finite1433825200ec_nth @ A @ B @ X )
% 3.15/1.73 = ( finite1433825200ec_nth @ A @ B @ Y ) )
% 3.15/1.73 = ( X = Y ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_41_cond__component,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType] :
% 3.15/1.73 ( ( finite_finite @ B )
% 3.15/1.73 => ! [B2: $o,X: finite_Cartesian_vec @ A @ B,Y: finite_Cartesian_vec @ A @ B,I4: B] :
% 3.15/1.73 ( ( B2
% 3.15/1.73 => ( ( finite1433825200ec_nth @ A @ B @ ( if @ ( finite_Cartesian_vec @ A @ B ) @ B2 @ X @ Y ) @ I4 )
% 3.15/1.73 = ( finite1433825200ec_nth @ A @ B @ X @ I4 ) ) )
% 3.15/1.73 & ( ~ B2
% 3.15/1.73 => ( ( finite1433825200ec_nth @ A @ B @ ( if @ ( finite_Cartesian_vec @ A @ B ) @ B2 @ X @ Y ) @ I4 )
% 3.15/1.73 = ( finite1433825200ec_nth @ A @ B @ Y @ I4 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_42_vec__eq__iff,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( finite_finite @ B )
% 3.15/1.73 => ( ( ^ [Y5: finite_Cartesian_vec @ A @ B,Z: finite_Cartesian_vec @ A @ B] : ( Y5 = Z ) )
% 3.15/1.73 = ( ^ [X3: finite_Cartesian_vec @ A @ B,Y3: finite_Cartesian_vec @ A @ B] :
% 3.15/1.73 ! [I3: B] :
% 3.15/1.73 ( ( finite1433825200ec_nth @ A @ B @ X3 @ I3 )
% 3.15/1.73 = ( finite1433825200ec_nth @ A @ B @ Y3 @ I3 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_43_fun__cong__unused__0,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,C: $tType] :
% 3.15/1.73 ( ( zero @ B )
% 3.15/1.73 => ! [F3: ( A > B ) > C,G: C] :
% 3.15/1.73 ( ( F3
% 3.15/1.73 = ( ^ [X3: A > B] : G ) )
% 3.15/1.73 => ( ( F3
% 3.15/1.73 @ ^ [X3: A] : ( zero_zero @ B ) )
% 3.15/1.73 = G ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_44_zero__prod__def,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType] :
% 3.15/1.73 ( ( ( zero @ B )
% 3.15/1.73 & ( zero @ A ) )
% 3.15/1.73 => ( ( zero_zero @ ( product_prod @ A @ B ) )
% 3.15/1.73 = ( product_Pair @ A @ B @ ( zero_zero @ A ) @ ( zero_zero @ B ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_45_mem__Collect__eq,axiom,
% 3.15/1.73 ! [A: $tType,A2: A,P: A > $o] :
% 3.15/1.73 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 3.15/1.73 = ( P @ A2 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_46_Collect__mem__eq,axiom,
% 3.15/1.73 ! [A: $tType,A5: set @ A] :
% 3.15/1.73 ( ( collect @ A
% 3.15/1.73 @ ^ [X3: A] : ( member @ A @ X3 @ A5 ) )
% 3.15/1.73 = A5 ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_47_Collect__cong,axiom,
% 3.15/1.73 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.15/1.73 ( ! [X4: A] :
% 3.15/1.73 ( ( P @ X4 )
% 3.15/1.73 = ( Q @ X4 ) )
% 3.15/1.73 => ( ( collect @ A @ P )
% 3.15/1.73 = ( collect @ A @ Q ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_48_ext,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,F3: A > B,G: A > B] :
% 3.15/1.73 ( ! [X4: A] :
% 3.15/1.73 ( ( F3 @ X4 )
% 3.15/1.73 = ( G @ X4 ) )
% 3.15/1.73 => ( F3 = G ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_49_fst__conv,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 3.15/1.73 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 3.15/1.73 = X1 ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_50_fst__eqD,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
% 3.15/1.73 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 3.15/1.73 = A2 )
% 3.15/1.73 => ( X = A2 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_51_snd__conv,axiom,
% 3.15/1.73 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 3.15/1.73 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 3.15/1.73 = X22 ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_52_snd__eqD,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
% 3.15/1.73 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 3.15/1.73 = A2 )
% 3.15/1.73 => ( Y = A2 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_53_prod__eq__iff,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( ^ [Y5: product_prod @ A @ B,Z: product_prod @ A @ B] : ( Y5 = Z ) )
% 3.15/1.73 = ( ^ [S: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 3.15/1.73 ( ( ( product_fst @ A @ B @ S )
% 3.15/1.73 = ( product_fst @ A @ B @ T3 ) )
% 3.15/1.73 & ( ( product_snd @ A @ B @ S )
% 3.15/1.73 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_54_prod_Oexpand,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 3.15/1.73 ( ( ( ( product_fst @ A @ B @ Prod )
% 3.15/1.73 = ( product_fst @ A @ B @ Prod2 ) )
% 3.15/1.73 & ( ( product_snd @ A @ B @ Prod )
% 3.15/1.73 = ( product_snd @ A @ B @ Prod2 ) ) )
% 3.15/1.73 => ( Prod = Prod2 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_55_prod__eqI,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 3.15/1.73 ( ( ( product_fst @ A @ B @ P2 )
% 3.15/1.73 = ( product_fst @ A @ B @ Q2 ) )
% 3.15/1.73 => ( ( ( product_snd @ A @ B @ P2 )
% 3.15/1.73 = ( product_snd @ A @ B @ Q2 ) )
% 3.15/1.73 => ( P2 = Q2 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_56_vec__nth__inverse,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( finite_finite @ B )
% 3.15/1.73 => ! [X: finite_Cartesian_vec @ A @ B] :
% 3.15/1.73 ( ( finite1990238425lambda @ B @ A @ ( finite1433825200ec_nth @ A @ B @ X ) )
% 3.15/1.73 = X ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_57_vec__lambda__unique,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType] :
% 3.15/1.73 ( ( finite_finite @ A )
% 3.15/1.73 => ! [F3: finite_Cartesian_vec @ B @ A,G: A > B] :
% 3.15/1.73 ( ( ! [I3: A] :
% 3.15/1.73 ( ( finite1433825200ec_nth @ B @ A @ F3 @ I3 )
% 3.15/1.73 = ( G @ I3 ) ) )
% 3.15/1.73 = ( ( finite1990238425lambda @ A @ B @ G )
% 3.15/1.73 = F3 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_58_obj__sumE__f,axiom,
% 3.15/1.73 ! [A: $tType,C: $tType,B: $tType,S2: B,F3: ( sum_sum @ A @ C ) > B,P: $o] :
% 3.15/1.73 ( ! [X4: A] :
% 3.15/1.73 ( ( S2
% 3.15/1.73 = ( F3 @ ( sum_Inl @ A @ C @ X4 ) ) )
% 3.15/1.73 => P )
% 3.15/1.73 => ( ! [X4: C] :
% 3.15/1.73 ( ( S2
% 3.15/1.73 = ( F3 @ ( sum_Inr @ C @ A @ X4 ) ) )
% 3.15/1.73 => P )
% 3.15/1.73 => ! [X5: sum_sum @ A @ C] :
% 3.15/1.73 ( ( S2
% 3.15/1.73 = ( F3 @ X5 ) )
% 3.15/1.73 => P ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_59_zero__vec__def,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ B )
% 3.15/1.73 & ( zero @ A ) )
% 3.15/1.73 => ( ( zero_zero @ ( finite_Cartesian_vec @ A @ B ) )
% 3.15/1.73 = ( finite1990238425lambda @ B @ A
% 3.15/1.73 @ ^ [I3: B] : ( zero_zero @ A ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_60_surjective__pairing,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 3.15/1.73 ( T2
% 3.15/1.73 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_61_prod_Oexhaust__sel,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 3.15/1.73 ( Prod
% 3.15/1.73 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_62_mk__v__concrete,axiom,
% 3.15/1.73 ! [A: $tType,C: $tType,B: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ B )
% 3.15/1.73 & ( finite_finite @ C )
% 3.15/1.73 & ( finite_finite @ A ) )
% 3.15/1.73 => ( ( denotational_mk_v @ B @ C @ A )
% 3.15/1.73 = ( ^ [I6: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE2: ode @ B @ A,Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Sol2: finite_Cartesian_vec @ real @ A] :
% 3.15/1.73 ( product_Pair @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )
% 3.15/1.73 @ ( finite1990238425lambda @ A @ real
% 3.15/1.73 @ ^ [I3: A] : ( finite1433825200ec_nth @ real @ A @ ( if @ ( finite_Cartesian_vec @ real @ A ) @ ( member @ ( sum_sum @ A @ A ) @ ( sum_Inl @ A @ A @ I3 ) @ ( denotational_semBV @ B @ C @ A @ I6 @ ODE2 ) ) @ Sol2 @ ( product_fst @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ Nu ) ) @ I3 ) )
% 3.15/1.73 @ ( finite1990238425lambda @ A @ real
% 3.15/1.73 @ ^ [I3: A] : ( finite1433825200ec_nth @ real @ A @ ( if @ ( finite_Cartesian_vec @ real @ A ) @ ( member @ ( sum_sum @ A @ A ) @ ( sum_Inr @ A @ A @ I3 ) @ ( denotational_semBV @ B @ C @ A @ I6 @ ODE2 ) ) @ ( denotational_ODE_sem @ B @ C @ A @ I6 @ ODE2 @ Sol2 ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ Nu ) ) @ I3 ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_63__092_060open_062Vagree_A_Imk__v_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_Ax_A_E_Ai_Aelse_Aab_A_E_Ai_J_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_N_AsemBV_AI_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_Ax_A_E_Ai_Aelse_Aab_A_E_Ai_J_J_A_Imk__xode_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_Ax_A_E_Ai_Aelse_Aab_A_E_Ai_J_J_A_IsemBV_AI_AODE_J_092_060close_062,axiom,
% 3.15/1.73 ( ( denotational_Vagree @ c
% 3.15/1.73 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 3.15/1.73 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 3.15/1.73 @ bb )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
% 3.15/1.73 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 3.15/1.73 @ bb )
% 3.15/1.73 @ ( uminus_uminus @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) ) )
% 3.15/1.73 & ( denotational_Vagree @ c
% 3.15/1.73 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 3.15/1.73 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 3.15/1.73 @ bb )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
% 3.15/1.73 @ ( denotational_mk_xode @ a @ b @ c @ i @ oDEa
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
% 3.15/1.73 @ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_64_Vagree__def,axiom,
% 3.15/1.73 ! [C: $tType] :
% 3.15/1.73 ( ( finite_finite @ C )
% 3.15/1.73 => ( ( denotational_Vagree @ C )
% 3.15/1.73 = ( ^ [Nu: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),V: set @ ( sum_sum @ C @ C )] :
% 3.15/1.73 ( ! [I3: C] :
% 3.15/1.73 ( ( member @ ( sum_sum @ C @ C ) @ ( sum_Inl @ C @ C @ I3 ) @ V )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ C @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) @ I3 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ C @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu2 ) @ I3 ) ) )
% 3.15/1.73 & ! [I3: C] :
% 3.15/1.73 ( ( member @ ( sum_sum @ C @ C ) @ ( sum_Inr @ C @ C @ I3 ) @ V )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ C @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) @ I3 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ C @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu2 ) @ I3 ) ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_65__092_060open_062Vagree_A_Imk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_Ax_J_J_A_Iab_M_Abb_J_A_I_N_AsemBV_AI_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_Ax_J_J_A_Imk__xode_AI_AODE_A_Isol_Ax_J_J_A_IsemBV_AI_AODE_J_092_060close_062,axiom,
% 3.15/1.73 ( ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( uminus_uminus @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) ) )
% 3.15/1.73 & ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) @ ( denotational_mk_xode @ a @ b @ c @ i @ oDEa @ ( sol @ x ) ) @ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_66_concrete__v_Osimps,axiom,
% 3.15/1.73 ! [C: $tType,B: $tType,A: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ A )
% 3.15/1.73 & ( finite_finite @ B )
% 3.15/1.73 & ( finite_finite @ C ) )
% 3.15/1.73 => ( ( denota1738237502rete_v @ A @ B @ C )
% 3.15/1.73 = ( ^ [I6: denota1663640101rp_ext @ A @ B @ C @ product_unit,ODE2: ode @ A @ C,Nu: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Sol2: finite_Cartesian_vec @ real @ C] :
% 3.15/1.73 ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )
% 3.15/1.73 @ ( finite1990238425lambda @ C @ real
% 3.15/1.73 @ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inl @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ I6 @ ODE2 ) ) @ Sol2 @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) ) @ I3 ) )
% 3.15/1.73 @ ( finite1990238425lambda @ C @ real
% 3.15/1.73 @ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inr @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ I6 @ ODE2 ) ) @ ( denotational_ODE_sem @ A @ B @ C @ I6 @ ODE2 @ Sol2 ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) ) @ I3 ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_67_concrete__v_Oelims,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,C: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ C )
% 3.15/1.73 & ( finite_finite @ A )
% 3.15/1.73 & ( finite_finite @ B ) )
% 3.15/1.73 => ! [X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Xa: ode @ A @ C,Xb: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Xc: finite_Cartesian_vec @ real @ C,Y: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 3.15/1.73 ( ( ( denota1738237502rete_v @ A @ B @ C @ X @ Xa @ Xb @ Xc )
% 3.15/1.73 = Y )
% 3.15/1.73 => ( Y
% 3.15/1.73 = ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )
% 3.15/1.73 @ ( finite1990238425lambda @ C @ real
% 3.15/1.73 @ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inl @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ X @ Xa ) ) @ Xc @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb ) ) @ I3 ) )
% 3.15/1.73 @ ( finite1990238425lambda @ C @ real
% 3.15/1.73 @ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inr @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ X @ Xa ) ) @ ( denotational_ODE_sem @ A @ B @ C @ X @ Xa @ Xc ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb ) ) @ I3 ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_68_Pair__le,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType] :
% 3.15/1.73 ( ( ( ord @ B )
% 3.15/1.73 & ( ord @ A ) )
% 3.15/1.73 => ! [A2: A,B2: B,C3: A,D3: B] :
% 3.15/1.73 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C3 @ D3 ) )
% 3.15/1.73 = ( ( ord_less_eq @ A @ A2 @ C3 )
% 3.15/1.73 & ( ord_less_eq @ B @ B2 @ D3 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_69_le__zero__eq,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( canoni770627133id_add @ A )
% 3.15/1.73 => ! [N: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 3.15/1.73 = ( N
% 3.15/1.73 = ( zero_zero @ A ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_70_sem,axiom,
% 3.15/1.73 ! [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 )] :
% 3.15/1.73 ( ( denotational_Vagree @ c @ Nu3 @ Nu4 @ ( static_FVF @ a @ b @ c @ p ) )
% 3.15/1.73 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu3 @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) )
% 3.15/1.73 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu4 @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_71_aaba,axiom,
% 3.15/1.73 ( ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba )
% 3.15/1.73 = ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ t ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_72_neg__equal__iff__equal,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( group_add @ A )
% 3.15/1.73 => ! [A2: A,B2: A] :
% 3.15/1.73 ( ( ( uminus_uminus @ A @ A2 )
% 3.15/1.73 = ( uminus_uminus @ A @ B2 ) )
% 3.15/1.73 = ( A2 = B2 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_73_add_Oinverse__inverse,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( group_add @ A )
% 3.15/1.73 => ! [A2: A] :
% 3.15/1.73 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 3.15/1.73 = A2 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_74_add_Oinverse__neutral,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( group_add @ A )
% 3.15/1.73 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 3.15/1.73 = ( zero_zero @ A ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_75_neg__0__equal__iff__equal,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( group_add @ A )
% 3.15/1.73 => ! [A2: A] :
% 3.15/1.73 ( ( ( zero_zero @ A )
% 3.15/1.73 = ( uminus_uminus @ A @ A2 ) )
% 3.15/1.73 = ( ( zero_zero @ A )
% 3.15/1.73 = A2 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_76_neg__equal__0__iff__equal,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( group_add @ A )
% 3.15/1.73 => ! [A2: A] :
% 3.15/1.73 ( ( ( uminus_uminus @ A @ A2 )
% 3.15/1.73 = ( zero_zero @ A ) )
% 3.15/1.73 = ( A2
% 3.15/1.73 = ( zero_zero @ A ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_77_equal__neg__zero,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( linord219039673up_add @ A )
% 3.15/1.73 => ! [A2: A] :
% 3.15/1.73 ( ( A2
% 3.15/1.73 = ( uminus_uminus @ A @ A2 ) )
% 3.15/1.73 = ( A2
% 3.15/1.73 = ( zero_zero @ A ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_78_neg__equal__zero,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( linord219039673up_add @ A )
% 3.15/1.73 => ! [A2: A] :
% 3.15/1.73 ( ( ( uminus_uminus @ A @ A2 )
% 3.15/1.73 = A2 )
% 3.15/1.73 = ( A2
% 3.15/1.73 = ( zero_zero @ A ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_79_neg__le__iff__le,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( ordered_ab_group_add @ A )
% 3.15/1.73 => ! [B2: A,A2: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 3.15/1.73 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_80_uminus__Pair,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType] :
% 3.15/1.73 ( ( ( uminus @ B )
% 3.15/1.73 & ( uminus @ A ) )
% 3.15/1.73 => ! [A2: A,B2: B] :
% 3.15/1.73 ( ( uminus_uminus @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 3.15/1.73 = ( product_Pair @ A @ B @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ B @ B2 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_81_vector__uminus__component,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ B )
% 3.15/1.73 & ( uminus @ A ) )
% 3.15/1.73 => ! [X: finite_Cartesian_vec @ A @ B,I4: B] :
% 3.15/1.73 ( ( finite1433825200ec_nth @ A @ B @ ( uminus_uminus @ ( finite_Cartesian_vec @ A @ B ) @ X ) @ I4 )
% 3.15/1.73 = ( uminus_uminus @ A @ ( finite1433825200ec_nth @ A @ B @ X @ I4 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_82_fst__uminus,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( ( uminus @ A )
% 3.15/1.73 & ( uminus @ B ) )
% 3.15/1.73 => ! [X: product_prod @ A @ B] :
% 3.15/1.73 ( ( product_fst @ A @ B @ ( uminus_uminus @ ( product_prod @ A @ B ) @ X ) )
% 3.15/1.73 = ( uminus_uminus @ A @ ( product_fst @ A @ B @ X ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_83_snd__uminus,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType] :
% 3.15/1.73 ( ( ( uminus @ B )
% 3.15/1.73 & ( uminus @ A ) )
% 3.15/1.73 => ! [X: product_prod @ B @ A] :
% 3.15/1.73 ( ( product_snd @ B @ A @ ( uminus_uminus @ ( product_prod @ B @ A ) @ X ) )
% 3.15/1.73 = ( uminus_uminus @ A @ ( product_snd @ B @ A @ X ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_84_neg__less__eq__nonneg,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( linord219039673up_add @ A )
% 3.15/1.73 => ! [A2: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 3.15/1.73 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_85_less__eq__neg__nonpos,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( linord219039673up_add @ A )
% 3.15/1.73 => ! [A2: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 3.15/1.73 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_86_neg__le__0__iff__le,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( ordered_ab_group_add @ A )
% 3.15/1.73 => ! [A2: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 3.15/1.73 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_87_neg__0__le__iff__le,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( ordered_ab_group_add @ A )
% 3.15/1.73 => ! [A2: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 3.15/1.73 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_88_allT,axiom,
% 3.15/1.73 ! [S3: real] :
% 3.15/1.73 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ S3 )
% 3.15/1.73 => ( ( ord_less_eq @ real @ S3 @ t )
% 3.15/1.73 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ S3 ) ) @ ( denotational_fml_sem @ a @ b @ c @ i @ phi ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_89_minus__equation__iff,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( group_add @ A )
% 3.15/1.73 => ! [A2: A,B2: A] :
% 3.15/1.73 ( ( ( uminus_uminus @ A @ A2 )
% 3.15/1.73 = B2 )
% 3.15/1.73 = ( ( uminus_uminus @ A @ B2 )
% 3.15/1.73 = A2 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_90_equation__minus__iff,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( group_add @ A )
% 3.15/1.73 => ! [A2: A,B2: A] :
% 3.15/1.73 ( ( A2
% 3.15/1.73 = ( uminus_uminus @ A @ B2 ) )
% 3.15/1.73 = ( B2
% 3.15/1.73 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_91_iff__to__impl,axiom,
% 3.15/1.73 ! [A: $tType,C: $tType,B: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ B )
% 3.15/1.73 & ( finite_finite @ C )
% 3.15/1.73 & ( finite_finite @ A ) )
% 3.15/1.73 => ! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,A5: formula @ B @ C @ A,B5: formula @ B @ C @ A] :
% 3.15/1.73 ( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A5 ) )
% 3.15/1.73 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B5 ) ) )
% 3.15/1.73 = ( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A5 ) )
% 3.15/1.73 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B5 ) ) )
% 3.15/1.73 & ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B5 ) )
% 3.15/1.73 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A5 ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_92_le__imp__neg__le,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( ordered_ab_group_add @ A )
% 3.15/1.73 => ! [A2: A,B2: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.15/1.73 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_93_minus__le__iff,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( ordered_ab_group_add @ A )
% 3.15/1.73 => ! [A2: A,B2: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 3.15/1.73 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_94_le__minus__iff,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( ordered_ab_group_add @ A )
% 3.15/1.73 => ! [A2: A,B2: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 3.15/1.73 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_95_agree__supset,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( finite_finite @ A )
% 3.15/1.73 => ! [B5: set @ ( sum_sum @ A @ A ),A5: set @ ( sum_sum @ A @ A ),Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu6: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ B5 @ A5 )
% 3.15/1.73 => ( ( denotational_Vagree @ A @ Nu5 @ Nu6 @ A5 )
% 3.15/1.73 => ( denotational_Vagree @ A @ Nu5 @ Nu6 @ B5 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_96_agree__sub,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( finite_finite @ A )
% 3.15/1.73 => ! [A5: set @ ( sum_sum @ A @ A ),B5: set @ ( sum_sum @ A @ A ),Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Omega: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ A5 @ B5 )
% 3.15/1.73 => ( ( denotational_Vagree @ A @ Nu5 @ Omega @ B5 )
% 3.15/1.73 => ( denotational_Vagree @ A @ Nu5 @ Omega @ A5 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_97_mk__v__exists,axiom,
% 3.15/1.73 ! [C: $tType,A: $tType,B: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ B )
% 3.15/1.73 & ( finite_finite @ A )
% 3.15/1.73 & ( finite_finite @ C ) )
% 3.15/1.73 => ! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A,Sol: finite_Cartesian_vec @ real @ A] :
% 3.15/1.73 ? [Omega2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
% 3.15/1.73 ( ( denotational_Vagree @ A @ Omega2 @ Nu5 @ ( uminus_uminus @ ( set @ ( sum_sum @ A @ A ) ) @ ( denotational_semBV @ B @ C @ A @ I2 @ ODE ) ) )
% 3.15/1.73 & ( denotational_Vagree @ A @ Omega2 @ ( denotational_mk_xode @ B @ C @ A @ I2 @ ODE @ Sol ) @ ( denotational_semBV @ B @ C @ A @ I2 @ ODE ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_98_uminus__vec__def,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( ( uminus @ A )
% 3.15/1.73 & ( finite_finite @ B ) )
% 3.15/1.73 => ( ( uminus_uminus @ ( finite_Cartesian_vec @ A @ B ) )
% 3.15/1.73 = ( ^ [X3: finite_Cartesian_vec @ A @ B] :
% 3.15/1.73 ( finite1990238425lambda @ B @ A
% 3.15/1.73 @ ^ [I3: B] : ( uminus_uminus @ A @ ( finite1433825200ec_nth @ A @ B @ X3 @ I3 ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_99_uminus__prod__def,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( ( uminus @ A )
% 3.15/1.73 & ( uminus @ B ) )
% 3.15/1.73 => ( ( uminus_uminus @ ( product_prod @ A @ B ) )
% 3.15/1.73 = ( ^ [X3: product_prod @ A @ B] : ( product_Pair @ A @ B @ ( uminus_uminus @ A @ ( product_fst @ A @ B @ X3 ) ) @ ( uminus_uminus @ B @ ( product_snd @ A @ B @ X3 ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_100_mk__v__agree,axiom,
% 3.15/1.73 ! [C: $tType,A: $tType,B: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ B )
% 3.15/1.73 & ( finite_finite @ A )
% 3.15/1.73 & ( finite_finite @ C ) )
% 3.15/1.73 => ! [I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A,Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Sol: finite_Cartesian_vec @ real @ A] :
% 3.15/1.73 ( ( denotational_Vagree @ A @ ( denotational_mk_v @ B @ C @ A @ I2 @ ODE @ Nu5 @ Sol ) @ Nu5 @ ( uminus_uminus @ ( set @ ( sum_sum @ A @ A ) ) @ ( denotational_semBV @ B @ C @ A @ I2 @ ODE ) ) )
% 3.15/1.73 & ( denotational_Vagree @ A @ ( denotational_mk_v @ B @ C @ A @ I2 @ ODE @ Nu5 @ Sol ) @ ( denotational_mk_xode @ B @ C @ A @ I2 @ ODE @ Sol ) @ ( denotational_semBV @ B @ C @ A @ I2 @ ODE ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_101_mk__xode_Oelims,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,C: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ C )
% 3.15/1.73 & ( finite_finite @ B )
% 3.15/1.73 & ( finite_finite @ A ) )
% 3.15/1.73 => ! [X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Xa: ode @ A @ C,Xb: finite_Cartesian_vec @ real @ C,Y: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 3.15/1.73 ( ( ( denotational_mk_xode @ A @ B @ C @ X @ Xa @ Xb )
% 3.15/1.73 = Y )
% 3.15/1.73 => ( Y
% 3.15/1.73 = ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb @ ( denotational_ODE_sem @ A @ B @ C @ X @ Xa @ Xb ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_102_mk__xode_Osimps,axiom,
% 3.15/1.73 ! [C: $tType,B: $tType,A: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ A )
% 3.15/1.73 & ( finite_finite @ B )
% 3.15/1.73 & ( finite_finite @ C ) )
% 3.15/1.73 => ( ( denotational_mk_xode @ A @ B @ C )
% 3.15/1.73 = ( ^ [I6: 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 @ I6 @ ODE2 @ Sol2 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_103_zero__reorient,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( zero @ A )
% 3.15/1.73 => ! [X: A] :
% 3.15/1.73 ( ( ( zero_zero @ A )
% 3.15/1.73 = X )
% 3.15/1.73 = ( X
% 3.15/1.73 = ( zero_zero @ A ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_104_agree__refl,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( finite_finite @ A )
% 3.15/1.73 => ! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A5: set @ ( sum_sum @ A @ A )] : ( denotational_Vagree @ A @ Nu5 @ Nu5 @ A5 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_105_agree__comm,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( finite_finite @ A )
% 3.15/1.73 => ! [A5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),B5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),V2: set @ ( sum_sum @ A @ A )] :
% 3.15/1.73 ( ( denotational_Vagree @ A @ A5 @ B5 @ V2 )
% 3.15/1.73 => ( denotational_Vagree @ A @ B5 @ A5 @ V2 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_106_zero__le,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( canoni770627133id_add @ A )
% 3.15/1.73 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_107_Pair__mono,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType] :
% 3.15/1.73 ( ( ( ord @ B )
% 3.15/1.73 & ( ord @ A ) )
% 3.15/1.73 => ! [X: A,X6: A,Y: B,Y6: B] :
% 3.15/1.73 ( ( ord_less_eq @ A @ X @ X6 )
% 3.15/1.73 => ( ( ord_less_eq @ B @ Y @ Y6 )
% 3.15/1.73 => ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( product_Pair @ A @ B @ X6 @ Y6 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_108_fst__mono,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( ( ord @ A )
% 3.15/1.73 & ( ord @ B ) )
% 3.15/1.73 => ! [X: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 3.15/1.73 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ X @ Y )
% 3.15/1.73 => ( ord_less_eq @ A @ ( product_fst @ A @ B @ X ) @ ( product_fst @ A @ B @ Y ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_109_snd__mono,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( ( ord @ A )
% 3.15/1.73 & ( ord @ B ) )
% 3.15/1.73 => ! [X: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 3.15/1.73 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ X @ Y )
% 3.15/1.73 => ( ord_less_eq @ B @ ( product_snd @ A @ B @ X ) @ ( product_snd @ A @ B @ Y ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_110_less__eq__prod__def,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( ( ord @ A )
% 3.15/1.73 & ( ord @ B ) )
% 3.15/1.73 => ( ( ord_less_eq @ ( product_prod @ A @ B ) )
% 3.15/1.73 = ( ^ [X3: product_prod @ A @ B,Y3: product_prod @ A @ B] :
% 3.15/1.73 ( ( ord_less_eq @ A @ ( product_fst @ A @ B @ X3 ) @ ( product_fst @ A @ B @ Y3 ) )
% 3.15/1.73 & ( ord_less_eq @ B @ ( product_snd @ A @ B @ X3 ) @ ( product_snd @ A @ B @ Y3 ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_111_ODE__vars__lr,axiom,
% 3.15/1.73 ! [Sc: $tType,Sz: $tType,Sf: $tType,X: Sz,I2: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,ODE: ode @ Sf @ Sz] :
% 3.15/1.73 ( ( member @ ( sum_sum @ Sz @ Sz ) @ ( sum_Inl @ Sz @ Sz @ X ) @ ( denotational_semBV @ Sf @ Sc @ Sz @ I2 @ ODE ) )
% 3.15/1.73 = ( member @ ( sum_sum @ Sz @ Sz ) @ ( sum_Inr @ Sz @ Sz @ X ) @ ( denotational_semBV @ Sf @ Sc @ Sz @ I2 @ ODE ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_112__092_060open_062_092_060lbrakk_062osafe_AODE_059_Afsafe_A_092_060phi_062_059_AODE_____A_061_AODE_059_AP_A_061_A_092_060phi_062_059_A0_A_092_060le_062_At_059_A_Iaa_M_Aba_J_A_061_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_059_A_Isol_Asolves__ode_A_I_092_060lambda_062a_O_AODE__sem_AI_AODE_J_J_A_1230_O_Ot_125_A_123x_O_Amk__v_AI_AODE_A_Iab_M_Abb_J_Ax_A_092_060in_062_Afml__sem_AI_A_092_060phi_062_125_059_AVSagree_A_Isol_A0_J_Aab_A_123uu___O_AInl_Auu___A_092_060in_062_ABVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AInl_A_096_AFVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AFVF_A_092_060phi_062_125_092_060rbrakk_062_A_092_060Longrightarrow_062_Aab_A_061_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_J_092_060close_062,axiom,
% 3.15/1.73 ( ( osafe @ a @ c @ oDEa )
% 3.15/1.73 => ( ( fsafe @ a @ b @ c @ phi )
% 3.15/1.73 => ( ( ode2 = oDEa )
% 3.15/1.73 => ( ( p = phi )
% 3.15/1.73 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ t )
% 3.15/1.73 => ( ( ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba )
% 3.15/1.73 = ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ t ) ) )
% 3.15/1.73 => ( ( initia1685620758es_ode @ ( finite_Cartesian_vec @ real @ c ) @ sol
% 3.15/1.73 @ ^ [X7: real] : ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa )
% 3.15/1.73 @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t )
% 3.15/1.73 @ ( collect @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ^ [X3: 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 @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ X3 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ phi ) ) ) )
% 3.15/1.73 => ( ( denotational_VSagree @ c @ ( sol @ ( zero_zero @ real ) ) @ ab
% 3.15/1.73 @ ( collect @ c
% 3.15/1.73 @ ^ [Uu: c] :
% 3.15/1.73 ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_BVO @ a @ c @ oDEa ) )
% 3.15/1.73 | ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( static_FVO @ a @ c @ oDEa ) ) )
% 3.15/1.73 | ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_FVF @ a @ b @ c @ phi ) ) ) ) )
% 3.15/1.73 => ( ab
% 3.15/1.73 = ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_113__092_060open_062_092_060lbrakk_062osafe_AODE_059_Afsafe_A_092_060phi_062_059_AODE_____A_061_AODE_059_AP_A_061_A_092_060phi_062_059_A0_A_092_060le_062_At_059_A_Iaa_M_Aba_J_A_061_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_059_A_Isol_Asolves__ode_A_I_092_060lambda_062a_O_AODE__sem_AI_AODE_J_J_A_1230_O_Ot_125_A_123x_O_Amk__v_AI_AODE_A_Iab_M_Abb_J_Ax_A_092_060in_062_Afml__sem_AI_A_092_060phi_062_125_059_AVSagree_A_Isol_A0_J_Aab_A_123uu___O_AInl_Auu___A_092_060in_062_ABVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AInl_A_096_AFVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AFVF_A_092_060phi_062_125_092_060rbrakk_062_A_092_060Longrightarrow_062_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_A_061_Amk__v_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_At_A_E_Ai_Aelse_Aab_A_E_Ai_J_092_060close_062,axiom,
% 3.15/1.73 ( ( osafe @ a @ c @ oDEa )
% 3.15/1.73 => ( ( fsafe @ a @ b @ c @ phi )
% 3.15/1.73 => ( ( ode2 = oDEa )
% 3.15/1.73 => ( ( p = phi )
% 3.15/1.73 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ t )
% 3.15/1.73 => ( ( ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba )
% 3.15/1.73 = ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ t ) ) )
% 3.15/1.73 => ( ( initia1685620758es_ode @ ( finite_Cartesian_vec @ real @ c ) @ sol
% 3.15/1.73 @ ^ [X7: real] : ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa )
% 3.15/1.73 @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t )
% 3.15/1.73 @ ( collect @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ^ [X3: 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 @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ X3 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ phi ) ) ) )
% 3.15/1.73 => ( ( denotational_VSagree @ c @ ( sol @ ( zero_zero @ real ) ) @ ab
% 3.15/1.73 @ ( collect @ c
% 3.15/1.73 @ ^ [Uu: c] :
% 3.15/1.73 ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_BVO @ a @ c @ oDEa ) )
% 3.15/1.73 | ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( static_FVO @ a @ c @ oDEa ) ) )
% 3.15/1.73 | ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_FVF @ a @ b @ c @ phi ) ) ) ) )
% 3.15/1.73 => ( ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ t ) )
% 3.15/1.73 = ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 3.15/1.73 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 3.15/1.73 @ bb )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ t ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_114__092_060open_062_092_060lbrakk_062osafe_AODE_059_Afsafe_A_092_060phi_062_059_AODE_____A_061_AODE_059_AP_A_061_A_092_060phi_062_059_A0_A_092_060le_062_At_059_A_Iaa_M_Aba_J_A_061_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_059_A_Isol_Asolves__ode_A_I_092_060lambda_062a_O_AODE__sem_AI_AODE_J_J_A_1230_O_Ot_125_A_123x_O_Amk__v_AI_AODE_A_Iab_M_Abb_J_Ax_A_092_060in_062_Afml__sem_AI_A_092_060phi_062_125_059_AVSagree_A_Isol_A0_J_Aab_A_123uu___O_AInl_Auu___A_092_060in_062_ABVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AInl_A_096_AFVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AFVF_A_092_060phi_062_125_092_060rbrakk_062_A_092_060Longrightarrow_062_A0_A_092_060le_062_At_092_060close_062,axiom,
% 3.15/1.73 ( ( osafe @ a @ c @ oDEa )
% 3.15/1.73 => ( ( fsafe @ a @ b @ c @ phi )
% 3.15/1.73 => ( ( ode2 = oDEa )
% 3.15/1.73 => ( ( p = phi )
% 3.15/1.73 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ t )
% 3.15/1.73 => ( ( ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba )
% 3.15/1.73 = ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ t ) ) )
% 3.15/1.73 => ( ( initia1685620758es_ode @ ( finite_Cartesian_vec @ real @ c ) @ sol
% 3.15/1.73 @ ^ [X7: real] : ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa )
% 3.15/1.73 @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t )
% 3.15/1.73 @ ( collect @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ^ [X3: 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 @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ X3 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ phi ) ) ) )
% 3.15/1.73 => ( ( denotational_VSagree @ c @ ( sol @ ( zero_zero @ real ) ) @ ab
% 3.15/1.73 @ ( collect @ c
% 3.15/1.73 @ ^ [Uu: c] :
% 3.15/1.73 ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_BVO @ a @ c @ oDEa ) )
% 3.15/1.73 | ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( static_FVO @ a @ c @ oDEa ) ) )
% 3.15/1.73 | ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_FVF @ a @ b @ c @ phi ) ) ) ) )
% 3.15/1.73 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ t ) ) ) ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_115_mkV,axiom,
% 3.15/1.73 ( member @ ( real > ( finite_Cartesian_vec @ real @ c ) ) @ sol
% 3.15/1.73 @ ( pi @ real @ ( finite_Cartesian_vec @ real @ c ) @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t )
% 3.15/1.73 @ ^ [Uu: real] :
% 3.15/1.73 ( collect @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ^ [X3: 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 @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ X3 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ phi ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_116_euclid__ext__aux_Ocases,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( euclid1678468529ng_gcd @ A )
% 3.15/1.73 => ! [X: product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) )] :
% 3.15/1.73 ~ ! [S4: A,S5: A,T4: A,T5: A,R: A,R2: A] :
% 3.15/1.73 ( X
% 3.15/1.73 != ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) ) @ S4 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) @ S5 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) @ T4 @ ( product_Pair @ A @ ( product_prod @ A @ A ) @ T5 @ ( product_Pair @ A @ A @ R @ R2 ) ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_117_compl__le__compl__iff,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( boolean_algebra @ A )
% 3.15/1.73 => ! [X: A,Y: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
% 3.15/1.73 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_118_iff__sem,axiom,
% 3.15/1.73 ! [A: $tType,C: $tType,B: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ B )
% 3.15/1.73 & ( finite_finite @ C )
% 3.15/1.73 & ( finite_finite @ A ) )
% 3.15/1.73 => ! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,A5: formula @ B @ C @ A,B5: formula @ B @ C @ A] :
% 3.15/1.73 ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ ( equiv @ B @ C @ A @ A5 @ B5 ) ) )
% 3.15/1.73 = ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A5 ) )
% 3.15/1.73 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B5 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_119_uminus__apply,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( uminus @ B )
% 3.15/1.73 => ( ( uminus_uminus @ ( A > B ) )
% 3.15/1.73 = ( ^ [A6: A > B,X3: A] : ( uminus_uminus @ B @ ( A6 @ X3 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_120_double__compl,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( boolean_algebra @ A )
% 3.15/1.73 => ! [X: A] :
% 3.15/1.73 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
% 3.15/1.73 = X ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_121_compl__eq__compl__iff,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( boolean_algebra @ A )
% 3.15/1.73 => ! [X: A,Y: A] :
% 3.15/1.73 ( ( ( uminus_uminus @ A @ X )
% 3.15/1.73 = ( uminus_uminus @ A @ Y ) )
% 3.15/1.73 = ( X = Y ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_122_VSagree__sub,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( finite_finite @ A )
% 3.15/1.73 => ! [A5: set @ A,B5: set @ A,Nu5: finite_Cartesian_vec @ real @ A,Omega: finite_Cartesian_vec @ real @ A] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 3.15/1.73 => ( ( denotational_VSagree @ A @ Nu5 @ Omega @ B5 )
% 3.15/1.73 => ( denotational_VSagree @ A @ Nu5 @ Omega @ A5 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_123_VSagree__supset,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( finite_finite @ A )
% 3.15/1.73 => ! [B5: set @ A,A5: set @ A,Nu5: finite_Cartesian_vec @ real @ A,Nu6: finite_Cartesian_vec @ real @ A] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
% 3.15/1.73 => ( ( denotational_VSagree @ A @ Nu5 @ Nu6 @ A5 )
% 3.15/1.73 => ( denotational_VSagree @ A @ Nu5 @ Nu6 @ B5 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_124_VSagree__refl,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( finite_finite @ A )
% 3.15/1.73 => ! [Nu5: finite_Cartesian_vec @ real @ A,A5: set @ A] : ( denotational_VSagree @ A @ Nu5 @ Nu5 @ A5 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_125_VSagree__def,axiom,
% 3.15/1.73 ! [C: $tType] :
% 3.15/1.73 ( ( finite_finite @ C )
% 3.15/1.73 => ( ( denotational_VSagree @ C )
% 3.15/1.73 = ( ^ [Nu: finite_Cartesian_vec @ real @ C,Nu2: finite_Cartesian_vec @ real @ C,V: set @ C] :
% 3.15/1.73 ! [X3: C] :
% 3.15/1.73 ( ( member @ C @ X3 @ V )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ C @ Nu @ X3 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ C @ Nu2 @ X3 ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_126_gcd_Ocases,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( euclid1155270486miring @ A )
% 3.15/1.73 => ! [X: product_prod @ A @ A] :
% 3.15/1.73 ~ ! [A4: A,B4: A] :
% 3.15/1.73 ( X
% 3.15/1.73 != ( product_Pair @ A @ A @ A4 @ B4 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_127_fun__Compl__def,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( uminus @ B )
% 3.15/1.73 => ( ( uminus_uminus @ ( A > B ) )
% 3.15/1.73 = ( ^ [A6: A > B,X3: A] : ( uminus_uminus @ B @ ( A6 @ X3 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_128_compl__mono,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( boolean_algebra @ A )
% 3.15/1.73 => ! [X: A,Y: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ X @ Y )
% 3.15/1.73 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_129_compl__le__swap1,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( boolean_algebra @ A )
% 3.15/1.73 => ! [Y: A,X: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
% 3.15/1.73 => ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_130_compl__le__swap2,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( boolean_algebra @ A )
% 3.15/1.73 => ! [Y: A,X: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
% 3.15/1.73 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_131_image__uminus__atLeastAtMost,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( ordered_ab_group_add @ A )
% 3.15/1.73 => ! [X: A,Y: A] :
% 3.15/1.73 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or331188842AtMost @ A @ X @ Y ) )
% 3.15/1.73 = ( set_or331188842AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_132_atLeastatMost__subset__iff,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( preorder @ A )
% 3.15/1.73 => ! [A2: A,B2: A,C3: A,D3: A] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A2 @ B2 ) @ ( set_or331188842AtMost @ A @ C3 @ D3 ) )
% 3.15/1.73 = ( ~ ( ord_less_eq @ A @ A2 @ B2 )
% 3.15/1.73 | ( ( ord_less_eq @ A @ C3 @ A2 )
% 3.15/1.73 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_133_Compl__subset__Compl__iff,axiom,
% 3.15/1.73 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) )
% 3.15/1.73 = ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_134_Compl__anti__mono,axiom,
% 3.15/1.73 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 3.15/1.73 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_135_image__eqI,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,B2: A,F3: B > A,X: B,A5: set @ B] :
% 3.15/1.73 ( ( B2
% 3.15/1.73 = ( F3 @ X ) )
% 3.15/1.73 => ( ( member @ B @ X @ A5 )
% 3.15/1.73 => ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A5 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_136_subset__antisym,axiom,
% 3.15/1.73 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 3.15/1.73 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
% 3.15/1.73 => ( A5 = B5 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_137_subsetI,axiom,
% 3.15/1.73 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 3.15/1.73 ( ! [X4: A] :
% 3.15/1.73 ( ( member @ A @ X4 @ A5 )
% 3.15/1.73 => ( member @ A @ X4 @ B5 ) )
% 3.15/1.73 => ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_138_Compl__eq__Compl__iff,axiom,
% 3.15/1.73 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 3.15/1.73 ( ( ( uminus_uminus @ ( set @ A ) @ A5 )
% 3.15/1.73 = ( uminus_uminus @ ( set @ A ) @ B5 ) )
% 3.15/1.73 = ( A5 = B5 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_139_Compl__iff,axiom,
% 3.15/1.73 ! [A: $tType,C3: A,A5: set @ A] :
% 3.15/1.73 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
% 3.15/1.73 = ( ~ ( member @ A @ C3 @ A5 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_140_ComplI,axiom,
% 3.15/1.73 ! [A: $tType,C3: A,A5: set @ A] :
% 3.15/1.73 ( ~ ( member @ A @ C3 @ A5 )
% 3.15/1.73 => ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_141_image__ident,axiom,
% 3.15/1.73 ! [A: $tType,Y7: set @ A] :
% 3.15/1.73 ( ( image @ A @ A
% 3.15/1.73 @ ^ [X3: A] : X3
% 3.15/1.73 @ Y7 )
% 3.15/1.73 = Y7 ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_142_Icc__eq__Icc,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( order @ A )
% 3.15/1.73 => ! [L: A,H: A,L2: A,H2: A] :
% 3.15/1.73 ( ( ( set_or331188842AtMost @ A @ L @ H )
% 3.15/1.73 = ( set_or331188842AtMost @ A @ L2 @ H2 ) )
% 3.15/1.73 = ( ( ( L = L2 )
% 3.15/1.73 & ( H = H2 ) )
% 3.15/1.73 | ( ~ ( ord_less_eq @ A @ L @ H )
% 3.15/1.73 & ~ ( ord_less_eq @ A @ L2 @ H2 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_143_atLeastAtMost__iff,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( ord @ A )
% 3.15/1.73 => ! [I4: A,L: A,U: A] :
% 3.15/1.73 ( ( member @ A @ I4 @ ( set_or331188842AtMost @ A @ L @ U ) )
% 3.15/1.73 = ( ( ord_less_eq @ A @ L @ I4 )
% 3.15/1.73 & ( ord_less_eq @ A @ I4 @ U ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_144_imageI,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,X: A,A5: set @ A,F3: A > B] :
% 3.15/1.73 ( ( member @ A @ X @ A5 )
% 3.15/1.73 => ( member @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ A5 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_145_image__iff,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,Z2: A,F3: B > A,A5: set @ B] :
% 3.15/1.73 ( ( member @ A @ Z2 @ ( image @ B @ A @ F3 @ A5 ) )
% 3.15/1.73 = ( ? [X3: B] :
% 3.15/1.73 ( ( member @ B @ X3 @ A5 )
% 3.15/1.73 & ( Z2
% 3.15/1.73 = ( F3 @ X3 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_146_bex__imageD,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P: A > $o] :
% 3.15/1.73 ( ? [X5: A] :
% 3.15/1.73 ( ( member @ A @ X5 @ ( image @ B @ A @ F3 @ A5 ) )
% 3.15/1.73 & ( P @ X5 ) )
% 3.15/1.73 => ? [X4: B] :
% 3.15/1.73 ( ( member @ B @ X4 @ A5 )
% 3.15/1.73 & ( P @ ( F3 @ X4 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_147_image__cong,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,M: set @ A,N2: set @ A,F3: A > B,G: A > B] :
% 3.15/1.73 ( ( M = N2 )
% 3.15/1.73 => ( ! [X4: A] :
% 3.15/1.73 ( ( member @ A @ X4 @ N2 )
% 3.15/1.73 => ( ( F3 @ X4 )
% 3.15/1.73 = ( G @ X4 ) ) )
% 3.15/1.73 => ( ( image @ A @ B @ F3 @ M )
% 3.15/1.73 = ( image @ A @ B @ G @ N2 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_148_ball__imageD,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P: A > $o] :
% 3.15/1.73 ( ! [X4: A] :
% 3.15/1.73 ( ( member @ A @ X4 @ ( image @ B @ A @ F3 @ A5 ) )
% 3.15/1.73 => ( P @ X4 ) )
% 3.15/1.73 => ! [X5: B] :
% 3.15/1.73 ( ( member @ B @ X5 @ A5 )
% 3.15/1.73 => ( P @ ( F3 @ X5 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_149_rev__image__eqI,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,X: A,A5: set @ A,B2: B,F3: A > B] :
% 3.15/1.73 ( ( member @ A @ X @ A5 )
% 3.15/1.73 => ( ( B2
% 3.15/1.73 = ( F3 @ X ) )
% 3.15/1.73 => ( member @ B @ B2 @ ( image @ A @ B @ F3 @ A5 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_150_Collect__mono__iff,axiom,
% 3.15/1.73 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 3.15/1.73 = ( ! [X3: A] :
% 3.15/1.73 ( ( P @ X3 )
% 3.15/1.73 => ( Q @ X3 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_151_set__eq__subset,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( ^ [Y5: set @ A,Z: set @ A] : ( Y5 = Z ) )
% 3.15/1.73 = ( ^ [A6: set @ A,B6: set @ A] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.15/1.73 & ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_152_subset__trans,axiom,
% 3.15/1.73 ! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 3.15/1.73 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 3.15/1.73 => ( ord_less_eq @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_153_Collect__mono,axiom,
% 3.15/1.73 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.15/1.73 ( ! [X4: A] :
% 3.15/1.73 ( ( P @ X4 )
% 3.15/1.73 => ( Q @ X4 ) )
% 3.15/1.73 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_154_subset__refl,axiom,
% 3.15/1.73 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_155_subset__iff,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) )
% 3.15/1.73 = ( ^ [A6: set @ A,B6: set @ A] :
% 3.15/1.73 ! [T3: A] :
% 3.15/1.73 ( ( member @ A @ T3 @ A6 )
% 3.15/1.73 => ( member @ A @ T3 @ B6 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_156_equalityD2,axiom,
% 3.15/1.73 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 3.15/1.73 ( ( A5 = B5 )
% 3.15/1.73 => ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_157_equalityD1,axiom,
% 3.15/1.73 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 3.15/1.73 ( ( A5 = B5 )
% 3.15/1.73 => ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_158_subset__eq,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) )
% 3.15/1.73 = ( ^ [A6: set @ A,B6: set @ A] :
% 3.15/1.73 ! [X3: A] :
% 3.15/1.73 ( ( member @ A @ X3 @ A6 )
% 3.15/1.73 => ( member @ A @ X3 @ B6 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_159_equalityE,axiom,
% 3.15/1.73 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 3.15/1.73 ( ( A5 = B5 )
% 3.15/1.73 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 3.15/1.73 => ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_160_subsetD,axiom,
% 3.15/1.73 ! [A: $tType,A5: set @ A,B5: set @ A,C3: A] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 3.15/1.73 => ( ( member @ A @ C3 @ A5 )
% 3.15/1.73 => ( member @ A @ C3 @ B5 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_161_in__mono,axiom,
% 3.15/1.73 ! [A: $tType,A5: set @ A,B5: set @ A,X: A] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 3.15/1.73 => ( ( member @ A @ X @ A5 )
% 3.15/1.73 => ( member @ A @ X @ B5 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_162_double__complement,axiom,
% 3.15/1.73 ! [A: $tType,A5: set @ A] :
% 3.15/1.73 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
% 3.15/1.73 = A5 ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_163_ComplD,axiom,
% 3.15/1.73 ! [A: $tType,C3: A,A5: set @ A] :
% 3.15/1.73 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
% 3.15/1.73 => ~ ( member @ A @ C3 @ A5 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_164_imageE,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,B2: A,F3: B > A,A5: set @ B] :
% 3.15/1.73 ( ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A5 ) )
% 3.15/1.73 => ~ ! [X4: B] :
% 3.15/1.73 ( ( B2
% 3.15/1.73 = ( F3 @ X4 ) )
% 3.15/1.73 => ~ ( member @ B @ X4 @ A5 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_165_image__image,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,C: $tType,F3: B > A,G: C > B,A5: set @ C] :
% 3.15/1.73 ( ( image @ B @ A @ F3 @ ( image @ C @ B @ G @ A5 ) )
% 3.15/1.73 = ( image @ C @ A
% 3.15/1.73 @ ^ [X3: C] : ( F3 @ ( G @ X3 ) )
% 3.15/1.73 @ A5 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_166_Compr__image__eq,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P: A > $o] :
% 3.15/1.73 ( ( collect @ A
% 3.15/1.73 @ ^ [X3: A] :
% 3.15/1.73 ( ( member @ A @ X3 @ ( image @ B @ A @ F3 @ A5 ) )
% 3.15/1.73 & ( P @ X3 ) ) )
% 3.15/1.73 = ( image @ B @ A @ F3
% 3.15/1.73 @ ( collect @ B
% 3.15/1.73 @ ^ [X3: B] :
% 3.15/1.73 ( ( member @ B @ X3 @ A5 )
% 3.15/1.73 & ( P @ ( F3 @ X3 ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_167_less__eq__set__def,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) )
% 3.15/1.73 = ( ^ [A6: set @ A,B6: set @ A] :
% 3.15/1.73 ( ord_less_eq @ ( A > $o )
% 3.15/1.73 @ ^ [X3: A] : ( member @ A @ X3 @ A6 )
% 3.15/1.73 @ ^ [X3: A] : ( member @ A @ X3 @ B6 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_168_Collect__subset,axiom,
% 3.15/1.73 ! [A: $tType,A5: set @ A,P: A > $o] :
% 3.15/1.73 ( ord_less_eq @ ( set @ A )
% 3.15/1.73 @ ( collect @ A
% 3.15/1.73 @ ^ [X3: A] :
% 3.15/1.73 ( ( member @ A @ X3 @ A5 )
% 3.15/1.73 & ( P @ X3 ) ) )
% 3.15/1.73 @ A5 ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_169_Collect__neg__eq,axiom,
% 3.15/1.73 ! [A: $tType,P: A > $o] :
% 3.15/1.73 ( ( collect @ A
% 3.15/1.73 @ ^ [X3: A] :
% 3.15/1.73 ~ ( P @ X3 ) )
% 3.15/1.73 = ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_170_Compl__eq,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( uminus_uminus @ ( set @ A ) )
% 3.15/1.73 = ( ^ [A6: set @ A] :
% 3.15/1.73 ( collect @ A
% 3.15/1.73 @ ^ [X3: A] :
% 3.15/1.73 ~ ( member @ A @ X3 @ A6 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_171_uminus__set__def,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( uminus_uminus @ ( set @ A ) )
% 3.15/1.73 = ( ^ [A6: set @ A] :
% 3.15/1.73 ( collect @ A
% 3.15/1.73 @ ( uminus_uminus @ ( A > $o )
% 3.15/1.73 @ ^ [X3: A] : ( member @ A @ X3 @ A6 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_172_image__mono,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,A5: set @ A,B5: set @ A,F3: A > B] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 3.15/1.73 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ ( image @ A @ B @ F3 @ B5 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_173_image__subsetI,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,A5: set @ A,F3: A > B,B5: set @ B] :
% 3.15/1.73 ( ! [X4: A] :
% 3.15/1.73 ( ( member @ A @ X4 @ A5 )
% 3.15/1.73 => ( member @ B @ ( F3 @ X4 ) @ B5 ) )
% 3.15/1.73 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ B5 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_174_subset__imageE,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A5: set @ B] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F3 @ A5 ) )
% 3.15/1.73 => ~ ! [C5: set @ B] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A5 )
% 3.15/1.73 => ( B5
% 3.15/1.73 != ( image @ B @ A @ F3 @ C5 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_175_image__subset__iff,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,F3: B > A,A5: set @ B,B5: set @ A] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ A5 ) @ B5 )
% 3.15/1.73 = ( ! [X3: B] :
% 3.15/1.73 ( ( member @ B @ X3 @ A5 )
% 3.15/1.73 => ( member @ A @ ( F3 @ X3 ) @ B5 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_176_subset__image__iff,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A5: set @ B] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F3 @ A5 ) )
% 3.15/1.73 = ( ? [AA: set @ B] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ B ) @ AA @ A5 )
% 3.15/1.73 & ( B5
% 3.15/1.73 = ( image @ B @ A @ F3 @ AA ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_177_repd_Oelims,axiom,
% 3.15/1.73 ! [C: $tType] :
% 3.15/1.73 ( ( finite_finite @ C )
% 3.15/1.73 => ! [X: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Xa: C,Xb: real,Y: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 3.15/1.73 ( ( ( denotational_repd @ C @ X @ Xa @ Xb )
% 3.15/1.73 = Y )
% 3.15/1.73 => ( Y
% 3.15/1.73 = ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ X )
% 3.15/1.73 @ ( finite1990238425lambda @ C @ real
% 3.15/1.73 @ ^ [Y3: C] : ( if @ real @ ( Xa = Y3 ) @ Xb @ ( finite1433825200ec_nth @ real @ C @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ X ) @ Y3 ) ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_178_repd_Osimps,axiom,
% 3.15/1.73 ! [C: $tType] :
% 3.15/1.73 ( ( finite_finite @ C )
% 3.15/1.73 => ( ( denotational_repd @ C )
% 3.15/1.73 = ( ^ [V3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),X3: C,R3: real] :
% 3.15/1.73 ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V3 )
% 3.15/1.73 @ ( finite1990238425lambda @ C @ real
% 3.15/1.73 @ ^ [Y3: C] : ( if @ real @ ( X3 = Y3 ) @ R3 @ ( finite1433825200ec_nth @ real @ C @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V3 ) @ Y3 ) ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_179_nth__map__matrix,axiom,
% 3.15/1.73 ! [A: $tType,C: $tType,D: $tType,B: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ B )
% 3.15/1.73 & ( finite_finite @ C ) )
% 3.15/1.73 => ! [F3: D > A,X: finite_Cartesian_vec @ ( finite_Cartesian_vec @ D @ B ) @ C,I4: C,J2: B] :
% 3.15/1.73 ( ( finite1433825200ec_nth @ A @ B @ ( finite1433825200ec_nth @ ( finite_Cartesian_vec @ A @ B ) @ C @ ( finite1614961371matrix @ D @ A @ B @ C @ F3 @ X ) @ I4 ) @ J2 )
% 3.15/1.73 = ( F3 @ ( finite1433825200ec_nth @ D @ B @ ( finite1433825200ec_nth @ ( finite_Cartesian_vec @ D @ B ) @ C @ X @ I4 ) @ J2 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_180_image__subset__iff__funcset,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,F4: B > A,A5: set @ B,B5: set @ A] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A5 ) @ B5 )
% 3.15/1.73 = ( member @ ( B > A ) @ F4
% 3.15/1.73 @ ( pi @ B @ A @ A5
% 3.15/1.73 @ ^ [Uu: B] : B5 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_181_Pi__I,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,A5: set @ A,F3: A > B,B5: A > ( set @ B )] :
% 3.15/1.73 ( ! [X4: A] :
% 3.15/1.73 ( ( member @ A @ X4 @ A5 )
% 3.15/1.73 => ( member @ B @ ( F3 @ X4 ) @ ( B5 @ X4 ) ) )
% 3.15/1.73 => ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ A5 @ B5 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_182_PiE,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B5: A > ( set @ B ),X: A] :
% 3.15/1.73 ( ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ A5 @ B5 ) )
% 3.15/1.73 => ( ~ ( member @ B @ ( F3 @ X ) @ ( B5 @ X ) )
% 3.15/1.73 => ~ ( member @ A @ X @ A5 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_183_Pi__I_H,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,A5: set @ A,F3: A > B,B5: A > ( set @ B )] :
% 3.15/1.73 ( ! [X4: A] :
% 3.15/1.73 ( ( member @ A @ X4 @ A5 )
% 3.15/1.73 => ( member @ B @ ( F3 @ X4 ) @ ( B5 @ X4 ) ) )
% 3.15/1.73 => ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ A5 @ B5 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_184_Pi__iff,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,F3: A > B,I2: set @ A,X8: A > ( set @ B )] :
% 3.15/1.73 ( ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ I2 @ X8 ) )
% 3.15/1.73 = ( ! [X3: A] :
% 3.15/1.73 ( ( member @ A @ X3 @ I2 )
% 3.15/1.73 => ( member @ B @ ( F3 @ X3 ) @ ( X8 @ X3 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_185_Pi__mem,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B5: A > ( set @ B ),X: A] :
% 3.15/1.73 ( ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ A5 @ B5 ) )
% 3.15/1.73 => ( ( member @ A @ X @ A5 )
% 3.15/1.73 => ( member @ B @ ( F3 @ X ) @ ( B5 @ X ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_186_Pi__cong,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,A5: set @ A,F3: A > B,G: A > B,B5: A > ( set @ B )] :
% 3.15/1.73 ( ! [W: A] :
% 3.15/1.73 ( ( member @ A @ W @ A5 )
% 3.15/1.73 => ( ( F3 @ W )
% 3.15/1.73 = ( G @ W ) ) )
% 3.15/1.73 => ( ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ A5 @ B5 ) )
% 3.15/1.73 = ( member @ ( A > B ) @ G @ ( pi @ A @ B @ A5 @ B5 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_187_funcsetI,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,A5: set @ A,F3: A > B,B5: set @ B] :
% 3.15/1.73 ( ! [X4: A] :
% 3.15/1.73 ( ( member @ A @ X4 @ A5 )
% 3.15/1.73 => ( member @ B @ ( F3 @ X4 ) @ B5 ) )
% 3.15/1.73 => ( member @ ( A > B ) @ F3
% 3.15/1.73 @ ( pi @ A @ B @ A5
% 3.15/1.73 @ ^ [Uu: A] : B5 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_188_funcset__id,axiom,
% 3.15/1.73 ! [A: $tType,A5: set @ A] :
% 3.15/1.73 ( member @ ( A > A )
% 3.15/1.73 @ ^ [X3: A] : X3
% 3.15/1.73 @ ( pi @ A @ A @ A5
% 3.15/1.73 @ ^ [Uu: A] : A5 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_189_funcset__mem,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B5: set @ B,X: A] :
% 3.15/1.73 ( ( member @ ( A > B ) @ F3
% 3.15/1.73 @ ( pi @ A @ B @ A5
% 3.15/1.73 @ ^ [Uu: A] : B5 ) )
% 3.15/1.73 => ( ( member @ A @ X @ A5 )
% 3.15/1.73 => ( member @ B @ ( F3 @ X ) @ B5 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_190_Pi__mono,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,A5: set @ A,B5: A > ( set @ B ),C4: A > ( set @ B )] :
% 3.15/1.73 ( ! [X4: A] :
% 3.15/1.73 ( ( member @ A @ X4 @ A5 )
% 3.15/1.73 => ( ord_less_eq @ ( set @ B ) @ ( B5 @ X4 ) @ ( C4 @ X4 ) ) )
% 3.15/1.73 => ( ord_less_eq @ ( set @ ( A > B ) ) @ ( pi @ A @ B @ A5 @ B5 ) @ ( pi @ A @ B @ A5 @ C4 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_191_Pi__anti__mono,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,A7: set @ A,A5: set @ A,B5: A > ( set @ B )] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ A7 @ A5 )
% 3.15/1.73 => ( ord_less_eq @ ( set @ ( A > B ) ) @ ( pi @ A @ B @ A5 @ B5 ) @ ( pi @ A @ B @ A7 @ B5 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_192_map__matrix__def,axiom,
% 3.15/1.73 ! [I: $tType,J: $tType,B: $tType,A: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ J )
% 3.15/1.73 & ( finite_finite @ I ) )
% 3.15/1.73 => ( ( finite1614961371matrix @ A @ B @ I @ J )
% 3.15/1.73 = ( ^ [F5: A > B,X3: finite_Cartesian_vec @ ( finite_Cartesian_vec @ A @ I ) @ J] :
% 3.15/1.73 ( finite1990238425lambda @ J @ ( finite_Cartesian_vec @ B @ I )
% 3.15/1.73 @ ^ [I3: J] :
% 3.15/1.73 ( finite1990238425lambda @ I @ B
% 3.15/1.73 @ ^ [J3: I] : ( F5 @ ( finite1433825200ec_nth @ A @ I @ ( finite1433825200ec_nth @ ( finite_Cartesian_vec @ A @ I ) @ J @ X3 @ I3 ) @ J3 ) ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_193_funcset__image,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B5: set @ B] :
% 3.15/1.73 ( ( member @ ( A > B ) @ F3
% 3.15/1.73 @ ( pi @ A @ B @ A5
% 3.15/1.73 @ ^ [Uu: A] : B5 ) )
% 3.15/1.73 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ B5 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_194_old_Oprod_Orec,axiom,
% 3.15/1.73 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 3.15/1.73 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 3.15/1.73 = ( F1 @ A2 @ B2 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_195_repv_Osimps,axiom,
% 3.15/1.73 ! [C: $tType] :
% 3.15/1.73 ( ( finite_finite @ C )
% 3.15/1.73 => ( ( denotational_repv @ C )
% 3.15/1.73 = ( ^ [V3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),X3: C,R3: real] :
% 3.15/1.73 ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )
% 3.15/1.73 @ ( finite1990238425lambda @ C @ real
% 3.15/1.73 @ ^ [Y3: C] : ( if @ real @ ( X3 = Y3 ) @ R3 @ ( finite1433825200ec_nth @ real @ C @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V3 ) @ Y3 ) ) )
% 3.15/1.73 @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V3 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_196_repv_Oelims,axiom,
% 3.15/1.73 ! [C: $tType] :
% 3.15/1.73 ( ( finite_finite @ C )
% 3.15/1.73 => ! [X: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Xa: C,Xb: real,Y: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 3.15/1.73 ( ( ( denotational_repv @ C @ X @ Xa @ Xb )
% 3.15/1.73 = Y )
% 3.15/1.73 => ( Y
% 3.15/1.73 = ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )
% 3.15/1.73 @ ( finite1990238425lambda @ C @ real
% 3.15/1.73 @ ^ [Y3: C] : ( if @ real @ ( Xa = Y3 ) @ Xb @ ( finite1433825200ec_nth @ real @ C @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ X ) @ Y3 ) ) )
% 3.15/1.73 @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ X ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_197_exI__realizer,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,P: A > B > $o,Y: A,X: B] :
% 3.15/1.73 ( ( P @ Y @ X )
% 3.15/1.73 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_198_conjI__realizer,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
% 3.15/1.73 ( ( P @ P2 )
% 3.15/1.73 => ( ( Q @ Q2 )
% 3.15/1.73 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
% 3.15/1.73 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_199_exE__realizer_H,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A] :
% 3.15/1.73 ( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
% 3.15/1.73 => ~ ! [X4: B,Y4: A] :
% 3.15/1.73 ~ ( P @ Y4 @ X4 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_200_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
% 3.15/1.73 ( ( P @ X @ Y )
% 3.15/1.73 => ( ( A2
% 3.15/1.73 = ( product_Pair @ A @ B @ X @ Y ) )
% 3.15/1.73 => ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_201_concrete__v_Opelims,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,C: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ C )
% 3.15/1.73 & ( finite_finite @ A )
% 3.15/1.73 & ( finite_finite @ B ) )
% 3.15/1.73 => ! [X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Xa: ode @ A @ C,Xb: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Xc: finite_Cartesian_vec @ real @ C,Y: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 3.15/1.73 ( ( ( denota1738237502rete_v @ A @ B @ C @ X @ Xa @ Xb @ Xc )
% 3.15/1.73 = Y )
% 3.15/1.73 => ( ( accp @ ( product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) @ ( denota2028625979_v_rel @ A @ B @ C ) @ ( product_Pair @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ X @ ( product_Pair @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Xa @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb @ Xc ) ) ) )
% 3.15/1.73 => ~ ( ( Y
% 3.15/1.73 = ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )
% 3.15/1.73 @ ( finite1990238425lambda @ C @ real
% 3.15/1.73 @ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inl @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ X @ Xa ) ) @ Xc @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb ) ) @ I3 ) )
% 3.15/1.73 @ ( finite1990238425lambda @ C @ real
% 3.15/1.73 @ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inr @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ X @ Xa ) ) @ ( denotational_ODE_sem @ A @ B @ C @ X @ Xa @ Xc ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb ) ) @ I3 ) ) ) )
% 3.15/1.73 => ~ ( accp @ ( product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) @ ( denota2028625979_v_rel @ A @ B @ C ) @ ( product_Pair @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ X @ ( product_Pair @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Xa @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb @ Xc ) ) ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_202_not__arg__cong__Inr,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,X: A,Y: A] :
% 3.15/1.73 ( ( X != Y )
% 3.15/1.73 => ( ( sum_Inr @ A @ B @ X )
% 3.15/1.73 != ( sum_Inr @ A @ B @ Y ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_203_subset__Collect__iff,axiom,
% 3.15/1.73 ! [A: $tType,B5: set @ A,A5: set @ A,P: A > $o] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
% 3.15/1.73 => ( ( ord_less_eq @ ( set @ A ) @ B5
% 3.15/1.73 @ ( collect @ A
% 3.15/1.73 @ ^ [X3: A] :
% 3.15/1.73 ( ( member @ A @ X3 @ A5 )
% 3.15/1.73 & ( P @ X3 ) ) ) )
% 3.15/1.73 = ( ! [X3: A] :
% 3.15/1.73 ( ( member @ A @ X3 @ B5 )
% 3.15/1.73 => ( P @ X3 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_204_subset__CollectI,axiom,
% 3.15/1.73 ! [A: $tType,B5: set @ A,A5: set @ A,Q: A > $o,P: A > $o] :
% 3.15/1.73 ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
% 3.15/1.73 => ( ! [X4: A] :
% 3.15/1.73 ( ( member @ A @ X4 @ B5 )
% 3.15/1.73 => ( ( Q @ X4 )
% 3.15/1.73 => ( P @ X4 ) ) )
% 3.15/1.73 => ( ord_less_eq @ ( set @ A )
% 3.15/1.73 @ ( collect @ A
% 3.15/1.73 @ ^ [X3: A] :
% 3.15/1.73 ( ( member @ A @ X3 @ B5 )
% 3.15/1.73 & ( Q @ X3 ) ) )
% 3.15/1.73 @ ( collect @ A
% 3.15/1.73 @ ^ [X3: A] :
% 3.15/1.73 ( ( member @ A @ X3 @ A5 )
% 3.15/1.73 & ( P @ X3 ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_205_obj__sumE,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
% 3.15/1.73 ( ! [X4: A] :
% 3.15/1.73 ( S2
% 3.15/1.73 != ( sum_Inl @ A @ B @ X4 ) )
% 3.15/1.73 => ~ ! [X4: B] :
% 3.15/1.73 ( S2
% 3.15/1.73 != ( sum_Inr @ B @ A @ X4 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_206_predicate1I,axiom,
% 3.15/1.73 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.15/1.73 ( ! [X4: A] :
% 3.15/1.73 ( ( P @ X4 )
% 3.15/1.73 => ( Q @ X4 ) )
% 3.15/1.73 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_207_minus__le__self__iff,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( lattic1601792062up_add @ A )
% 3.15/1.73 => ! [A2: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 3.15/1.73 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_208_order__refl,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( preorder @ A )
% 3.15/1.73 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_209_order__subst1,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType] :
% 3.15/1.73 ( ( ( order @ B )
% 3.15/1.73 & ( order @ A ) )
% 3.15/1.73 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 3.15/1.73 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
% 3.15/1.73 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 3.15/1.73 => ( ! [X4: B,Y4: B] :
% 3.15/1.73 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 3.15/1.73 => ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) )
% 3.15/1.73 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_210_order__subst2,axiom,
% 3.15/1.73 ! [A: $tType,C: $tType] :
% 3.15/1.73 ( ( ( order @ C )
% 3.15/1.73 & ( order @ A ) )
% 3.15/1.73 => ! [A2: A,B2: A,F3: A > C,C3: C] :
% 3.15/1.73 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.15/1.73 => ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
% 3.15/1.73 => ( ! [X4: A,Y4: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 3.15/1.73 => ( ord_less_eq @ C @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) )
% 3.15/1.73 => ( ord_less_eq @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_211_ord__eq__le__subst,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType] :
% 3.15/1.73 ( ( ( ord @ B )
% 3.15/1.73 & ( ord @ A ) )
% 3.15/1.73 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 3.15/1.73 ( ( A2
% 3.15/1.73 = ( F3 @ B2 ) )
% 3.15/1.73 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 3.15/1.73 => ( ! [X4: B,Y4: B] :
% 3.15/1.73 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 3.15/1.73 => ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) )
% 3.15/1.73 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_212_ord__le__eq__subst,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType] :
% 3.15/1.73 ( ( ( ord @ B )
% 3.15/1.73 & ( ord @ A ) )
% 3.15/1.73 => ! [A2: A,B2: A,F3: A > B,C3: B] :
% 3.15/1.73 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.15/1.73 => ( ( ( F3 @ B2 )
% 3.15/1.73 = C3 )
% 3.15/1.73 => ( ! [X4: A,Y4: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 3.15/1.73 => ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) )
% 3.15/1.73 => ( ord_less_eq @ B @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_213_eq__iff,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( order @ A )
% 3.15/1.73 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 3.15/1.73 = ( ^ [X3: A,Y3: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.15/1.73 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_214_antisym,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( order @ A )
% 3.15/1.73 => ! [X: A,Y: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ X @ Y )
% 3.15/1.73 => ( ( ord_less_eq @ A @ Y @ X )
% 3.15/1.73 => ( X = Y ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_215_linear,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( linorder @ A )
% 3.15/1.73 => ! [X: A,Y: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ X @ Y )
% 3.15/1.73 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_216_eq__refl,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( preorder @ A )
% 3.15/1.73 => ! [X: A,Y: A] :
% 3.15/1.73 ( ( X = Y )
% 3.15/1.73 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_217_le__cases,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( linorder @ A )
% 3.15/1.73 => ! [X: A,Y: A] :
% 3.15/1.73 ( ~ ( ord_less_eq @ A @ X @ Y )
% 3.15/1.73 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_218_order_Otrans,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( order @ A )
% 3.15/1.73 => ! [A2: A,B2: A,C3: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.15/1.73 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 3.15/1.73 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_219_le__cases3,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( linorder @ A )
% 3.15/1.73 => ! [X: A,Y: A,Z2: A] :
% 3.15/1.73 ( ( ( ord_less_eq @ A @ X @ Y )
% 3.15/1.73 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 3.15/1.73 => ( ( ( ord_less_eq @ A @ Y @ X )
% 3.15/1.73 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 3.15/1.73 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 3.15/1.73 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 3.15/1.73 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 3.15/1.73 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 3.15/1.73 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 3.15/1.73 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 3.15/1.73 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 3.15/1.73 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_220_antisym__conv,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( order @ A )
% 3.15/1.73 => ! [Y: A,X: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ Y @ X )
% 3.15/1.73 => ( ( ord_less_eq @ A @ X @ Y )
% 3.15/1.73 = ( X = Y ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_221_order__class_Oorder_Oeq__iff,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( order @ A )
% 3.15/1.73 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 3.15/1.73 = ( ^ [A8: A,B7: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ A8 @ B7 )
% 3.15/1.73 & ( ord_less_eq @ A @ B7 @ A8 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_222_ord__eq__le__trans,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( ord @ A )
% 3.15/1.73 => ! [A2: A,B2: A,C3: A] :
% 3.15/1.73 ( ( A2 = B2 )
% 3.15/1.73 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 3.15/1.73 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_223_ord__le__eq__trans,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( ord @ A )
% 3.15/1.73 => ! [A2: A,B2: A,C3: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.15/1.73 => ( ( B2 = C3 )
% 3.15/1.73 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_224_order__class_Oorder_Oantisym,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( order @ A )
% 3.15/1.73 => ! [A2: A,B2: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.15/1.73 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.15/1.73 => ( A2 = B2 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_225_order__trans,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( preorder @ A )
% 3.15/1.73 => ! [X: A,Y: A,Z2: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ X @ Y )
% 3.15/1.73 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 3.15/1.73 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_226_dual__order_Orefl,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( order @ A )
% 3.15/1.73 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_227_linorder__wlog,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( linorder @ A )
% 3.15/1.73 => ! [P: A > A > $o,A2: A,B2: A] :
% 3.15/1.73 ( ! [A4: A,B4: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.15/1.73 => ( P @ A4 @ B4 ) )
% 3.15/1.73 => ( ! [A4: A,B4: A] :
% 3.15/1.73 ( ( P @ B4 @ A4 )
% 3.15/1.73 => ( P @ A4 @ B4 ) )
% 3.15/1.73 => ( P @ A2 @ B2 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_228_dual__order_Otrans,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( order @ A )
% 3.15/1.73 => ! [B2: A,A2: A,C3: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.15/1.73 => ( ( ord_less_eq @ A @ C3 @ B2 )
% 3.15/1.73 => ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_229_dual__order_Oeq__iff,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( order @ A )
% 3.15/1.73 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 3.15/1.73 = ( ^ [A8: A,B7: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ B7 @ A8 )
% 3.15/1.73 & ( ord_less_eq @ A @ A8 @ B7 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_230_dual__order_Oantisym,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( order @ A )
% 3.15/1.73 => ! [B2: A,A2: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.15/1.73 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.15/1.73 => ( A2 = B2 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_231_predicate1D,axiom,
% 3.15/1.73 ! [A: $tType,P: A > $o,Q: A > $o,X: A] :
% 3.15/1.73 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 3.15/1.73 => ( ( P @ X )
% 3.15/1.73 => ( Q @ X ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_232_rev__predicate1D,axiom,
% 3.15/1.73 ! [A: $tType,P: A > $o,X: A,Q: A > $o] :
% 3.15/1.73 ( ( P @ X )
% 3.15/1.73 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 3.15/1.73 => ( Q @ X ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_233_le__funD,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( ord @ B )
% 3.15/1.73 => ! [F3: A > B,G: A > B,X: A] :
% 3.15/1.73 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 3.15/1.73 => ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_234_le__funE,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( ord @ B )
% 3.15/1.73 => ! [F3: A > B,G: A > B,X: A] :
% 3.15/1.73 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 3.15/1.73 => ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_235_le__funI,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( ord @ B )
% 3.15/1.73 => ! [F3: A > B,G: A > B] :
% 3.15/1.73 ( ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
% 3.15/1.73 => ( ord_less_eq @ ( A > B ) @ F3 @ G ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_236_le__fun__def,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( ord @ B )
% 3.15/1.73 => ( ( ord_less_eq @ ( A > B ) )
% 3.15/1.73 = ( ^ [F5: A > B,G4: A > B] :
% 3.15/1.73 ! [X3: A] : ( ord_less_eq @ B @ ( F5 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_237_le__minus__self__iff,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( lattic1601792062up_add @ A )
% 3.15/1.73 => ! [A2: A] :
% 3.15/1.73 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 3.15/1.73 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_238_old_Osum_Oinject_I1_J,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,A2: A,A3: A] :
% 3.15/1.73 ( ( ( sum_Inl @ A @ B @ A2 )
% 3.15/1.73 = ( sum_Inl @ A @ B @ A3 ) )
% 3.15/1.73 = ( A2 = A3 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_239_sum_Oinject_I1_J,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,X1: A,Y1: A] :
% 3.15/1.73 ( ( ( sum_Inl @ A @ B @ X1 )
% 3.15/1.73 = ( sum_Inl @ A @ B @ Y1 ) )
% 3.15/1.73 = ( X1 = Y1 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_240_sum_Oinject_I2_J,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,X22: B,Y2: B] :
% 3.15/1.73 ( ( ( sum_Inr @ B @ A @ X22 )
% 3.15/1.73 = ( sum_Inr @ B @ A @ Y2 ) )
% 3.15/1.73 = ( X22 = Y2 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_241_old_Osum_Oinject_I2_J,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,B2: B,B3: B] :
% 3.15/1.73 ( ( ( sum_Inr @ B @ A @ B2 )
% 3.15/1.73 = ( sum_Inr @ B @ A @ B3 ) )
% 3.15/1.73 = ( B2 = B3 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_242_Inr__inject,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,X: B,Y: B] :
% 3.15/1.73 ( ( ( sum_Inr @ B @ A @ X )
% 3.15/1.73 = ( sum_Inr @ B @ A @ Y ) )
% 3.15/1.73 => ( X = Y ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_243_Inl__inject,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,X: A,Y: A] :
% 3.15/1.73 ( ( ( sum_Inl @ A @ B @ X )
% 3.15/1.73 = ( sum_Inl @ A @ B @ Y ) )
% 3.15/1.73 => ( X = Y ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_244_sum_Odistinct_I1_J,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,X1: A,X22: B] :
% 3.15/1.73 ( ( sum_Inl @ A @ B @ X1 )
% 3.15/1.73 != ( sum_Inr @ B @ A @ X22 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_245_old_Osum_Odistinct_I2_J,axiom,
% 3.15/1.73 ! [B8: $tType,A9: $tType,B9: B8,A10: A9] :
% 3.15/1.73 ( ( sum_Inr @ B8 @ A9 @ B9 )
% 3.15/1.73 != ( sum_Inl @ A9 @ B8 @ A10 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_246_old_Osum_Odistinct_I1_J,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,A2: A,B3: B] :
% 3.15/1.73 ( ( sum_Inl @ A @ B @ A2 )
% 3.15/1.73 != ( sum_Inr @ B @ A @ B3 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_247_sumE,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
% 3.15/1.73 ( ! [X4: A] :
% 3.15/1.73 ( S2
% 3.15/1.73 != ( sum_Inl @ A @ B @ X4 ) )
% 3.15/1.73 => ~ ! [Y4: B] :
% 3.15/1.73 ( S2
% 3.15/1.73 != ( sum_Inr @ B @ A @ Y4 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_248_Inr__not__Inl,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,B2: B,A2: A] :
% 3.15/1.73 ( ( sum_Inr @ B @ A @ B2 )
% 3.15/1.73 != ( sum_Inl @ A @ B @ A2 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_249_split__sum__ex,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 3.15/1.73 ? [X9: sum_sum @ A @ B] : ( P3 @ X9 ) )
% 3.15/1.73 = ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
% 3.15/1.73 ( ? [X3: A] : ( P4 @ ( sum_Inl @ A @ B @ X3 ) )
% 3.15/1.73 | ? [X3: B] : ( P4 @ ( sum_Inr @ B @ A @ X3 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_250_split__sum__all,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType] :
% 3.15/1.73 ( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 3.15/1.73 ! [X9: sum_sum @ A @ B] : ( P3 @ X9 ) )
% 3.15/1.73 = ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
% 3.15/1.73 ( ! [X3: A] : ( P4 @ ( sum_Inl @ A @ B @ X3 ) )
% 3.15/1.73 & ! [X3: B] : ( P4 @ ( sum_Inr @ B @ A @ X3 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_251_old_Osum_Oexhaust,axiom,
% 3.15/1.73 ! [A: $tType,B: $tType,Y: sum_sum @ A @ B] :
% 3.15/1.73 ( ! [A4: A] :
% 3.15/1.73 ( Y
% 3.15/1.73 != ( sum_Inl @ A @ B @ A4 ) )
% 3.15/1.73 => ~ ! [B4: B] :
% 3.15/1.73 ( Y
% 3.15/1.73 != ( sum_Inr @ B @ A @ B4 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_252_old_Osum_Oinducts,axiom,
% 3.15/1.73 ! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
% 3.15/1.73 ( ! [A4: A] : ( P @ ( sum_Inl @ A @ B @ A4 ) )
% 3.15/1.73 => ( ! [B4: B] : ( P @ ( sum_Inr @ B @ A @ B4 ) )
% 3.15/1.73 => ( P @ Sum ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_253_pred__subset__eq,axiom,
% 3.15/1.73 ! [A: $tType,R4: set @ A,S6: set @ A] :
% 3.15/1.73 ( ( ord_less_eq @ ( A > $o )
% 3.15/1.73 @ ^ [X3: A] : ( member @ A @ X3 @ R4 )
% 3.15/1.73 @ ^ [X3: A] : ( member @ A @ X3 @ S6 ) )
% 3.15/1.73 = ( ord_less_eq @ ( set @ A ) @ R4 @ S6 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(fact_254_verit__minus__simplify_I4_J,axiom,
% 3.15/1.73 ! [B: $tType] :
% 3.15/1.73 ( ( group_add @ B )
% 3.15/1.73 => ! [B2: B] :
% 3.15/1.73 ( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
% 3.15/1.73 = B2 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
% 3.15/1.73 ! [A: $tType] :
% 3.15/1.73 ( ( finite_finite @ A )
% 3.15/1.73 => ( type @ A ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( ordere890947078_space @ A9 )
% 3.15/1.73 & ( finite_finite @ A11 ) )
% 3.15/1.73 => ( ordere890947078_space @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space_1,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( ordere890947078_space @ A9 )
% 3.15/1.73 & ( ordere890947078_space @ A11 ) )
% 3.15/1.73 => ( ordere890947078_space @ ( product_prod @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_2,axiom,
% 3.15/1.73 ordere890947078_space @ real ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( boolean_algebra @ A11 )
% 3.15/1.73 => ( boolean_algebra @ ( A9 > A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( preorder @ A11 )
% 3.15/1.73 => ( preorder @ ( A9 > A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ A9 )
% 3.15/1.73 & ( finite_finite @ A11 ) )
% 3.15/1.73 => ( finite_finite @ ( A9 > A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( order @ A11 )
% 3.15/1.73 => ( order @ ( A9 > A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_fun___Orderings_Oord,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ord @ A11 )
% 3.15/1.73 => ( ord @ ( A9 > A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_fun___Groups_Ouminus,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( uminus @ A11 )
% 3.15/1.73 => ( uminus @ ( A9 > A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_3,axiom,
% 3.15/1.73 ! [A9: $tType] : ( boolean_algebra @ ( set @ A9 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 3.15/1.73 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
% 3.15/1.73 ! [A9: $tType] :
% 3.15/1.73 ( ( finite_finite @ A9 )
% 3.15/1.73 => ( finite_finite @ ( set @ A9 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 3.15/1.73 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 3.15/1.73 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Set_Oset___Groups_Ouminus_8,axiom,
% 3.15/1.73 ! [A9: $tType] : ( uminus @ ( set @ A9 ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Set_Oset___Groups_Ozero,axiom,
% 3.15/1.73 ! [A9: $tType] :
% 3.15/1.73 ( ( zero @ A9 )
% 3.15/1.73 => ( zero @ ( set @ A9 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_9,axiom,
% 3.15/1.73 boolean_algebra @ $o ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
% 3.15/1.73 preorder @ $o ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 3.15/1.73 linorder @ $o ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_HOL_Obool___Finite__Set_Ofinite_11,axiom,
% 3.15/1.73 finite_finite @ $o ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
% 3.15/1.73 order @ $o ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
% 3.15/1.73 ord @ $o ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_HOL_Obool___Groups_Ouminus_14,axiom,
% 3.15/1.73 uminus @ $o ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Real_Oreal___Lattice__Algebras_Olattice__ab__group__add,axiom,
% 3.15/1.73 lattic1601792062up_add @ real ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
% 3.15/1.73 linord219039673up_add @ real ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
% 3.15/1.73 ordered_ab_group_add @ real ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Real_Oreal___Orderings_Opreorder_15,axiom,
% 3.15/1.73 preorder @ real ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Real_Oreal___Orderings_Olinorder_16,axiom,
% 3.15/1.73 linorder @ real ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 3.15/1.73 group_add @ real ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Real_Oreal___Orderings_Oorder_17,axiom,
% 3.15/1.73 order @ real ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Real_Oreal___Orderings_Oord_18,axiom,
% 3.15/1.73 ord @ real ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Real_Oreal___Groups_Ouminus_19,axiom,
% 3.15/1.73 uminus @ real ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Real_Oreal___Groups_Ozero_20,axiom,
% 3.15/1.73 zero @ real ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_21,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ A9 )
% 3.15/1.73 & ( finite_finite @ A11 ) )
% 3.15/1.73 => ( finite_finite @ ( sum_sum @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__group__add_22,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( ordere890947078_space @ A9 )
% 3.15/1.73 & ( ordere890947078_space @ A11 ) )
% 3.15/1.73 => ( ordered_ab_group_add @ ( product_prod @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Product__Type_Oprod___Lattices_Oboolean__algebra_23,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( boolean_algebra @ A9 )
% 3.15/1.73 & ( boolean_algebra @ A11 ) )
% 3.15/1.73 => ( boolean_algebra @ ( product_prod @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_24,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( preorder @ A9 )
% 3.15/1.73 & ( preorder @ A11 ) )
% 3.15/1.73 => ( preorder @ ( product_prod @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_25,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ A9 )
% 3.15/1.73 & ( finite_finite @ A11 ) )
% 3.15/1.73 => ( finite_finite @ ( product_prod @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Product__Type_Oprod___Groups_Ogroup__add_26,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( group_add @ A9 )
% 3.15/1.73 & ( group_add @ A11 ) )
% 3.15/1.73 => ( group_add @ ( product_prod @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Product__Type_Oprod___Orderings_Oorder_27,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( order @ A9 )
% 3.15/1.73 & ( order @ A11 ) )
% 3.15/1.73 => ( order @ ( product_prod @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Product__Type_Oprod___Orderings_Oord_28,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( ord @ A9 )
% 3.15/1.73 & ( ord @ A11 ) )
% 3.15/1.73 => ( ord @ ( product_prod @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Product__Type_Oprod___Groups_Ouminus_29,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( uminus @ A9 )
% 3.15/1.73 & ( uminus @ A11 ) )
% 3.15/1.73 => ( uminus @ ( product_prod @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Product__Type_Oprod___Groups_Ozero_30,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( zero @ A9 )
% 3.15/1.73 & ( zero @ A11 ) )
% 3.15/1.73 => ( zero @ ( product_prod @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Product__Type_Ounit___Lattices_Oboolean__algebra_31,axiom,
% 3.15/1.73 boolean_algebra @ product_unit ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_32,axiom,
% 3.15/1.73 preorder @ product_unit ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_33,axiom,
% 3.15/1.73 linorder @ product_unit ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_34,axiom,
% 3.15/1.73 finite_finite @ product_unit ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Product__Type_Ounit___Orderings_Oorder_35,axiom,
% 3.15/1.73 order @ product_unit ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Product__Type_Ounit___Orderings_Oord_36,axiom,
% 3.15/1.73 ord @ product_unit ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Product__Type_Ounit___Groups_Ouminus_37,axiom,
% 3.15/1.73 uminus @ product_unit ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__group__add_38,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( ordere890947078_space @ A9 )
% 3.15/1.73 & ( finite_finite @ A11 ) )
% 3.15/1.73 => ( ordered_ab_group_add @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Opreorder_39,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( order @ A9 )
% 3.15/1.73 & ( finite_finite @ A11 ) )
% 3.15/1.73 => ( preorder @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_40,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( linorder @ A9 )
% 3.15/1.73 & ( cARD_1 @ A11 ) )
% 3.15/1.73 => ( linorder @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_41,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( finite_finite @ A9 )
% 3.15/1.73 & ( finite_finite @ A11 ) )
% 3.15/1.73 => ( finite_finite @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ogroup__add_42,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( group_add @ A9 )
% 3.15/1.73 & ( finite_finite @ A11 ) )
% 3.15/1.73 => ( group_add @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oorder_43,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( order @ A9 )
% 3.15/1.73 & ( finite_finite @ A11 ) )
% 3.15/1.73 => ( order @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oord_44,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( ord @ A9 )
% 3.15/1.73 & ( finite_finite @ A11 ) )
% 3.15/1.73 => ( ord @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ouminus_45,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( uminus @ A9 )
% 3.15/1.73 & ( finite_finite @ A11 ) )
% 3.15/1.73 => ( uminus @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_46,axiom,
% 3.15/1.73 ! [A9: $tType,A11: $tType] :
% 3.15/1.73 ( ( ( zero @ A9 )
% 3.15/1.73 & ( finite_finite @ A11 ) )
% 3.15/1.73 => ( zero @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(help_If_3_1_T,axiom,
% 3.15/1.73 ! [P: $o] :
% 3.15/1.73 ( ( P = $true )
% 3.15/1.73 | ( P = $false ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(help_If_2_1_T,axiom,
% 3.15/1.73 ! [A: $tType,X: A,Y: A] :
% 3.15/1.73 ( ( if @ A @ $false @ X @ Y )
% 3.15/1.73 = Y ) ).
% 3.15/1.73
% 3.15/1.73 thf(help_If_1_1_T,axiom,
% 3.15/1.73 ! [A: $tType,X: A,Y: A] :
% 3.15/1.73 ( ( if @ A @ $true @ X @ Y )
% 3.15/1.73 = X ) ).
% 3.15/1.73
% 3.15/1.73 thf(tfree_0,hypothesis,
% 3.15/1.73 finite_finite @ c ).
% 3.15/1.73
% 3.15/1.73 thf(tfree_1,hypothesis,
% 3.15/1.73 finite_finite @ a ).
% 3.15/1.73
% 3.15/1.73 thf(tfree_2,hypothesis,
% 3.15/1.73 finite_finite @ b ).
% 3.15/1.73
% 3.15/1.73 thf(conj_0,hypothesis,
% 3.15/1.73 member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( static_FVF @ a @ b @ c @ phi ) ).
% 3.15/1.73
% 3.15/1.73 thf(conj_1,hypothesis,
% 3.15/1.73 member @ c @ i2 @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ).
% 3.15/1.73
% 3.15/1.73 thf(conj_2,hypothesis,
% 3.15/1.73 ~ ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(conj_3,hypothesis,
% 3.15/1.73 ( ( ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 3.15/1.73 & ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) ) )
% 3.15/1.73 => ( ( ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c
% 3.15/1.73 @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 3.15/1.73 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 3.15/1.73 @ bb )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 3.15/1.73 @ i2 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ i2 ) ) )
% 3.15/1.73 & ( ~ ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c
% 3.15/1.73 @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 3.15/1.73 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 3.15/1.73 @ bb )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 3.15/1.73 @ i2 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c @ ab @ i2 ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(conj_4,hypothesis,
% 3.15/1.73 ( ( ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 3.15/1.73 & ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c @ ab @ i2 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(conj_5,hypothesis,
% 3.15/1.73 ( ( ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 3.15/1.73 & ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c
% 3.15/1.73 @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 3.15/1.73 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 3.15/1.73 @ bb )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 3.15/1.73 @ i2 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c @ bb @ i2 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(conj_6,hypothesis,
% 3.15/1.73 ( ( ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 3.15/1.73 & ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c @ bb @ i2 ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(conj_7,hypothesis,
% 3.15/1.73 ( ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 3.15/1.73 => ( ( ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c
% 3.15/1.73 @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 3.15/1.73 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 3.15/1.73 @ bb )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 3.15/1.73 @ i2 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ i2 ) ) )
% 3.15/1.73 & ( ~ ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c
% 3.15/1.73 @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 3.15/1.73 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 3.15/1.73 @ bb )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 3.15/1.73 @ i2 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c @ ab @ i2 ) ) ) ) )
% 3.15/1.73 & ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 3.15/1.73 => ( ( ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c
% 3.15/1.73 @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 3.15/1.73 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 3.15/1.73 @ bb )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 3.15/1.73 @ i2 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ i2 ) ) )
% 3.15/1.73 & ( ~ ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c
% 3.15/1.73 @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 3.15/1.73 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 3.15/1.73 @ bb )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 3.15/1.73 @ i2 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c @ ab @ i2 ) ) ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(conj_8,hypothesis,
% 3.15/1.73 ( ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c
% 3.15/1.73 @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 3.15/1.73 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 3.15/1.73 @ bb )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 3.15/1.73 @ i2 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c
% 3.15/1.73 @ ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
% 3.15/1.73 @ i2 ) ) )
% 3.15/1.73 & ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c
% 3.15/1.73 @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
% 3.15/1.73 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
% 3.15/1.73 @ bb )
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
% 3.15/1.73 @ i2 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c
% 3.15/1.73 @ ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa
% 3.15/1.73 @ ( finite1990238425lambda @ c @ real
% 3.15/1.73 @ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
% 3.15/1.73 @ i2 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(conj_9,hypothesis,
% 3.15/1.73 ( ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ i2 ) ) )
% 3.15/1.73 & ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ i2 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(conj_10,hypothesis,
% 3.15/1.73 ( ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c @ ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa @ ( sol @ x ) ) @ i2 ) ) )
% 3.15/1.73 & ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
% 3.15/1.73 => ( ( finite1433825200ec_nth @ real @ c @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c @ ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa @ ( sol @ x ) ) @ i2 ) ) ) ) ).
% 3.15/1.73
% 3.15/1.73 thf(conj_11,conjecture,
% 3.15/1.73 ( ( finite1433825200ec_nth @ real @ c @ ab @ i2 )
% 3.15/1.73 = ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ i2 ) ) ).
% 3.15/1.73 ------------------------
% 3.15/1.76 % (15325)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_15214 for (2999ds/275Mi)
% 3.15/1.76 % (15326)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_15214 for (2999ds/18Mi)
% 3.15/1.77 % (15326)Instruction limit reached!
% 3.15/1.77 % (15326)------------------------------
% 3.15/1.77 % (15326)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.77 % (15326)Termination reason: Unknown
% 3.15/1.77 % (15326)Termination phase: shuffling
% 3.15/1.77
% 3.15/1.77 % (15326)Memory used [KB]: 1791
% 3.15/1.77 % (15326)Time elapsed: 0.007 s
% 3.15/1.77 % (15326)Instructions burned: 18 (million)
% 3.15/1.77 % (15326)------------------------------
% 3.15/1.77 % (15326)------------------------------
% 3.15/1.77 % (15321)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_15214 for (2999ds/4Mi)
% 3.15/1.77 % (15320)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_15214 for (2999ds/183Mi)
% 3.15/1.77 % (15322)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_15214 for (2999ds/27Mi)
% 3.15/1.77 % (15323)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_15214 for (2999ds/2Mi)
% 3.15/1.77 % (15324)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_15214 for (2999ds/2Mi)
% 3.15/1.77 % (15323)Instruction limit reached!
% 3.15/1.77 % (15323)------------------------------
% 3.15/1.77 % (15323)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.77 % (15323)Termination reason: Unknown
% 3.15/1.77 % (15324)Instruction limit reached!
% 3.15/1.77 % (15324)------------------------------
% 3.15/1.77 % (15324)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.77 % (15324)Termination reason: Unknown
% 3.15/1.77 % (15324)Termination phase: shuffling
% 3.15/1.77
% 3.15/1.77 % (15324)Memory used [KB]: 1535
% 3.15/1.77 % (15324)Time elapsed: 0.004 s
% 3.15/1.77 % (15324)Instructions burned: 3 (million)
% 3.15/1.77 % (15324)------------------------------
% 3.15/1.77 % (15324)------------------------------
% 3.15/1.77 % (15323)Termination phase: shuffling
% 3.15/1.77
% 3.15/1.77 % (15323)Memory used [KB]: 1535
% 3.15/1.77 % (15323)Time elapsed: 0.004 s
% 3.15/1.77 % (15323)Instructions burned: 3 (million)
% 3.15/1.77 % (15323)------------------------------
% 3.15/1.77 % (15323)------------------------------
% 3.15/1.77 % (15321)Instruction limit reached!
% 3.15/1.77 % (15321)------------------------------
% 3.15/1.77 % (15321)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.77 % (15321)Termination reason: Unknown
% 3.15/1.77 % (15321)Termination phase: shuffling
% 3.15/1.77
% 3.15/1.77 % (15321)Memory used [KB]: 1663
% 3.15/1.77 % (15321)Time elapsed: 0.005 s
% 3.15/1.77 % (15321)Instructions burned: 5 (million)
% 3.15/1.77 % (15321)------------------------------
% 3.15/1.77 % (15321)------------------------------
% 3.15/1.78 % (15327)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_15214 for (2999ds/3Mi)
% 3.15/1.78 % (15327)Instruction limit reached!
% 3.15/1.78 % (15327)------------------------------
% 3.15/1.78 % (15327)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.78 % (15327)Termination reason: Unknown
% 3.15/1.78 % (15327)Termination phase: shuffling
% 3.15/1.78
% 3.15/1.78 % (15327)Memory used [KB]: 1535
% 3.15/1.78 % (15327)Time elapsed: 0.003 s
% 3.15/1.78 % (15327)Instructions burned: 3 (million)
% 3.15/1.78 % (15327)------------------------------
% 3.15/1.78 % (15327)------------------------------
% 3.15/1.78 % (15322)Instruction limit reached!
% 3.15/1.78 % (15322)------------------------------
% 3.15/1.78 % (15322)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.78 % (15322)Termination reason: Unknown
% 3.15/1.78 % (15322)Termination phase: shuffling
% 3.15/1.78
% 3.15/1.78 % (15322)Memory used [KB]: 1918
% 3.15/1.78 % (15322)Time elapsed: 0.018 s
% 3.15/1.78 % (15322)Instructions burned: 27 (million)
% 3.15/1.78 % (15322)------------------------------
% 3.15/1.78 % (15322)------------------------------
% 3.15/1.79 % (15329)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_15214 for (2999ds/15Mi)
% 3.15/1.79 % (15328)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_15214 for (2999ds/37Mi)
% 3.15/1.79 % (15330)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_15214 for (2999ds/3Mi)
% 3.15/1.79 % (15331)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_15214 for (2999ds/1041Mi)
% 3.15/1.79 % (15330)Instruction limit reached!
% 3.15/1.79 % (15330)------------------------------
% 3.15/1.79 % (15330)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.79 % (15330)Termination reason: Unknown
% 3.15/1.79 % (15330)Termination phase: shuffling
% 3.15/1.79
% 3.15/1.79 % (15330)Memory used [KB]: 1535
% 3.15/1.79 % (15330)Time elapsed: 0.004 s
% 3.15/1.79 % (15330)Instructions burned: 3 (million)
% 3.15/1.79 % (15330)------------------------------
% 3.15/1.79 % (15330)------------------------------
% 3.15/1.79 % (15329)Instruction limit reached!
% 3.15/1.79 % (15329)------------------------------
% 3.15/1.79 % (15329)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.79 % (15329)Termination reason: Unknown
% 3.15/1.79 % (15329)Termination phase: shuffling
% 3.15/1.79
% 3.15/1.79 % (15329)Memory used [KB]: 1918
% 3.15/1.79 % (15329)Time elapsed: 0.009 s
% 3.15/1.79 % (15329)Instructions burned: 17 (million)
% 3.15/1.79 % (15329)------------------------------
% 3.15/1.79 % (15329)------------------------------
% 3.24/1.80 % (15332)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_15214 for (2999ds/7Mi)
% 3.24/1.80 % (15333)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_15214 for (2999ds/16Mi)
% 3.24/1.80 % (15332)Instruction limit reached!
% 3.24/1.80 % (15332)------------------------------
% 3.24/1.80 % (15332)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.80 % (15332)Termination reason: Unknown
% 3.24/1.80 % (15332)Termination phase: shuffling
% 3.24/1.80
% 3.24/1.80 % (15332)Memory used [KB]: 1535
% 3.24/1.80 % (15332)Time elapsed: 0.006 s
% 3.24/1.80 % (15332)Instructions burned: 7 (million)
% 3.24/1.80 % (15332)------------------------------
% 3.24/1.80 % (15332)------------------------------
% 3.24/1.81 % (15334)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_15214 for (2999ds/3Mi)
% 3.24/1.81 % (15334)Instruction limit reached!
% 3.24/1.81 % (15334)------------------------------
% 3.24/1.81 % (15334)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.81 % (15334)Termination reason: Unknown
% 3.24/1.81 % (15334)Termination phase: shuffling
% 3.24/1.81
% 3.24/1.81 % (15334)Memory used [KB]: 1535
% 3.24/1.81 % (15334)Time elapsed: 0.002 s
% 3.24/1.81 % (15334)Instructions burned: 3 (million)
% 3.24/1.81 % (15334)------------------------------
% 3.24/1.81 % (15334)------------------------------
% 3.24/1.81 % (15325)First to succeed.
% 3.24/1.81 % (15328)Instruction limit reached!
% 3.24/1.81 % (15328)------------------------------
% 3.24/1.81 % (15328)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.81 % (15328)Termination reason: Unknown
% 3.24/1.81 % (15328)Termination phase: shuffling
% 3.24/1.81
% 3.24/1.81 % (15328)Memory used [KB]: 2302
% 3.24/1.81 % (15328)Time elapsed: 0.022 s
% 3.24/1.81 % (15328)Instructions burned: 38 (million)
% 3.24/1.81 % (15328)------------------------------
% 3.24/1.81 % (15328)------------------------------
% 3.24/1.81 % (15333)Instruction limit reached!
% 3.24/1.81 % (15333)------------------------------
% 3.24/1.81 % (15333)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.81 % (15333)Termination reason: Unknown
% 3.24/1.81 % (15333)Termination phase: shuffling
% 3.24/1.81
% 3.24/1.81 % (15333)Memory used [KB]: 1791
% 3.24/1.81 % (15333)Time elapsed: 0.007 s
% 3.24/1.81 % (15333)Instructions burned: 17 (million)
% 3.24/1.81 % (15333)------------------------------
% 3.24/1.81 % (15333)------------------------------
% 3.24/1.81 % (15325)Refutation found. Thanks to Tanya!
% 3.24/1.81 % SZS status Theorem for DTF2THF_15214
% 3.24/1.81 % SZS output start Proof for DTF2THF_15214
% 3.24/1.81 thf(type_def_5, type, formula: ($tType * $tType * $tType) > $tType).
% 3.24/1.81 thf(type_def_7, type, product_unit: $tType).
% 3.24/1.81 thf(type_def_8, type, denota1663640101rp_ext: ($tType * $tType * $tType * $tType) > $tType).
% 3.24/1.81 thf(type_def_9, type, ode: ($tType * $tType) > $tType).
% 3.24/1.81 thf(type_def_10, type, real: $tType).
% 3.24/1.81 thf(type_def_11, type, finite_Cartesian_vec: ($tType * $tType) > $tType).
% 3.24/1.81 thf(type_def_12, type, set: ($tType) > $tType).
% 3.24/1.81 thf(type_def_13, type, product_prod: ($tType * $tType) > $tType).
% 3.24/1.81 thf(type_def_14, type, sum_sum: ($tType * $tType) > $tType).
% 3.24/1.81 thf(type_def_15, type, a: $tType).
% 3.24/1.81 thf(type_def_16, type, b: $tType).
% 3.24/1.81 thf(type_def_17, type, c: $tType).
% 3.24/1.81 thf(type_def_18, type, d: $tType).
% 3.24/1.81 thf(func_def_0, type, denota1663640101rp_ext: ($tType * $tType * $tType * $tType) > $tType).
% 3.24/1.81 thf(func_def_1, type, finite_Cartesian_vec: ($tType * $tType) > $tType).
% 3.24/1.81 thf(func_def_2, type, product_unit: $tType).
% 3.24/1.81 thf(func_def_3, type, product_prod: ($tType * $tType) > $tType).
% 3.24/1.81 thf(func_def_4, type, formula: ($tType * $tType * $tType) > $tType).
% 3.24/1.81 thf(func_def_5, type, sum_sum: ($tType * $tType) > $tType).
% 3.24/1.81 thf(func_def_6, type, ode: ($tType * $tType) > $tType).
% 3.24/1.81 thf(func_def_7, type, real: $tType).
% 3.24/1.81 thf(func_def_8, type, set: ($tType) > $tType).
% 3.24/1.81 thf(func_def_9, type, d: $tType).
% 3.24/1.81 thf(func_def_10, type, c: $tType).
% 3.24/1.81 thf(func_def_11, type, b: $tType).
% 3.24/1.81 thf(func_def_12, type, a: $tType).
% 3.24/1.81 thf(func_def_13, type, cARD_1: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_14, type, ordere890947078_space: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_15, type, type: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_16, type, zero: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_17, type, uminus: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_18, type, ord: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_19, type, order: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_20, type, group_add: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_21, type, finite_finite: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_22, type, linorder: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_23, type, preorder: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_24, type, boolean_algebra: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_25, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_26, type, linord219039673up_add: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_27, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_28, type, lattic1601792062up_add: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_29, type, euclid1678468529ng_gcd: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_30, type, euclid1155270486miring: !>[X0: $tType]:($o)).
% 3.24/1.81 thf(func_def_31, type, coinci1993344360de_fml: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > $o)).
% 3.24/1.81 thf(func_def_32, 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)).
% 3.24/1.81 thf(func_def_33, type, denota811733865E_vars: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > ode @ X0 @ X2 > set @ X2)).
% 3.24/1.81 thf(func_def_34, type, denotational_VSagree: !>[X0: $tType]:(finite_Cartesian_vec @ real @ X0 > finite_Cartesian_vec @ real @ X0 > set @ X0 > $o)).
% 3.24/1.81 thf(func_def_35, 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)).
% 3.24/1.81 thf(func_def_36, type, denota1738237502rete_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)).
% 3.24/1.81 thf(func_def_37, type, denota2028625979_v_rel: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit @ product_prod @ ode @ X0 @ X2 @ product_prod @ product_prod @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2 > product_prod @ denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit @ product_prod @ ode @ X0 @ X2 @ product_prod @ product_prod @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2 > $o)).
% 3.24/1.81 thf(func_def_38, 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)).
% 3.24/1.81 thf(func_def_39, 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)).
% 3.24/1.81 thf(func_def_40, 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)).
% 3.24/1.81 thf(func_def_41, type, denotational_repd: !>[X0: $tType]:(product_prod @ finite_Cartesian_vec @ real @ X0 @ finite_Cartesian_vec @ real @ X0 > X0 > real > product_prod @ finite_Cartesian_vec @ real @ X0 @ finite_Cartesian_vec @ real @ X0)).
% 3.24/1.81 thf(func_def_42, type, denotational_repv: !>[X0: $tType]:(product_prod @ finite_Cartesian_vec @ real @ X0 @ finite_Cartesian_vec @ real @ X0 > X0 > real > product_prod @ finite_Cartesian_vec @ real @ X0 @ finite_Cartesian_vec @ real @ X0)).
% 3.24/1.81 thf(func_def_43, type, denotational_semBV: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > ode @ X0 @ X2 > set @ sum_sum @ X2 @ X2)).
% 3.24/1.81 thf(func_def_44, type, finite1614961371matrix: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > finite_Cartesian_vec @ finite_Cartesian_vec @ X0 @ X2 @ X3 > finite_Cartesian_vec @ finite_Cartesian_vec @ X1 @ X2 @ X3)).
% 3.24/1.81 thf(func_def_45, type, finite1990238425lambda: !>[X0: $tType, X1: $tType]:((X0 > X1) > finite_Cartesian_vec @ X1 @ X0)).
% 3.24/1.81 thf(func_def_46, type, finite1433825200ec_nth: !>[X0: $tType, X1: $tType]:(finite_Cartesian_vec @ X0 @ X1 > X1 > X0)).
% 3.24/1.81 thf(func_def_47, type, pi: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > set @ X1) > set @ (X0 > X1))).
% 3.24/1.81 thf(func_def_48, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 3.24/1.81 thf(func_def_49, type, zero_zero: !>[X0: $tType]:(X0)).
% 3.24/1.81 thf(func_def_50, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.24/1.81 thf(func_def_51, type, initia1685620758es_ode: !>[X0: $tType]:((real > X0) > (real > X0 > X0) > set @ real > set @ X0 > $o)).
% 3.24/1.81 thf(func_def_52, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.24/1.81 thf(func_def_53, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 3.24/1.81 thf(func_def_54, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.24/1.81 thf(func_def_55, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.24/1.81 thf(func_def_56, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.24/1.81 thf(func_def_57, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.24/1.81 thf(func_def_58, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 3.24/1.81 thf(func_def_59, type, set_or331188842AtMost: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 3.24/1.81 thf(func_def_60, type, static_BVO: !>[X0: $tType, X1: $tType]:(ode @ X0 @ X1 > set @ sum_sum @ X1 @ X1)).
% 3.24/1.81 thf(func_def_61, type, static_FVF: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > set @ sum_sum @ X2 @ X2)).
% 3.24/1.81 thf(func_def_62, type, static_FVO: !>[X0: $tType, X1: $tType]:(ode @ X0 @ X1 > set @ X1)).
% 3.24/1.81 thf(func_def_63, type, sum_Inl: !>[X0: $tType, X1: $tType]:(X0 > sum_sum @ X0 @ X1)).
% 3.24/1.81 thf(func_def_64, type, sum_Inr: !>[X0: $tType, X1: $tType]:(X0 > sum_sum @ X1 @ X0)).
% 3.24/1.81 thf(func_def_65, type, equiv: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > formula @ X0 @ X1 @ X2 > formula @ X0 @ X1 @ X2)).
% 3.24/1.81 thf(func_def_66, type, fsafe: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > $o)).
% 3.24/1.81 thf(func_def_67, type, osafe: !>[X0: $tType, X1: $tType]:(ode @ X0 @ X1 > $o)).
% 3.24/1.81 thf(func_def_68, type, accp: !>[X0: $tType]:((X0 > X0 > $o) > X0 > $o)).
% 3.24/1.81 thf(func_def_69, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.24/1.81 thf(func_def_70, type, i: denota1663640101rp_ext @ a @ b @ c @ product_unit).
% 3.24/1.81 thf(func_def_71, type, ode2: ode @ a @ c).
% 3.24/1.81 thf(func_def_72, type, oDEa: ode @ a @ c).
% 3.24/1.81 thf(func_def_73, type, p: formula @ a @ b @ c).
% 3.24/1.81 thf(func_def_74, type, phi: formula @ a @ b @ c).
% 3.24/1.81 thf(func_def_75, type, aa: finite_Cartesian_vec @ real @ c).
% 3.24/1.81 thf(func_def_76, type, ab: finite_Cartesian_vec @ real @ c).
% 3.24/1.81 thf(func_def_77, type, ba: finite_Cartesian_vec @ real @ c).
% 3.24/1.81 thf(func_def_78, type, bb: finite_Cartesian_vec @ real @ c).
% 3.24/1.81 thf(func_def_79, type, i2: c).
% 3.24/1.81 thf(func_def_80, type, sol: real > finite_Cartesian_vec @ real @ c).
% 3.24/1.81 thf(func_def_81, type, t: real).
% 3.24/1.81 thf(func_def_82, type, x: real).
% 3.24/1.81 thf(func_def_99, type, sK0: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.24/1.81 thf(func_def_100, type, sK1: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.24/1.81 thf(func_def_101, type, sK2: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > set @ X0 > set @ X1)).
% 3.24/1.81 thf(func_def_102, type, sK3: !>[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)).
% 3.24/1.81 thf(func_def_103, type, sK4: !>[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)).
% 3.24/1.81 thf(func_def_104, type, sK5: !>[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)).
% 3.24/1.81 thf(func_def_105, type, sK6: !>[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)).
% 3.24/1.81 thf(func_def_106, type, sK7: !>[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)).
% 3.24/1.81 thf(func_def_107, type, sK8: !>[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)).
% 3.24/1.81 thf(func_def_108, type, sK9: !>[X0: $tType, X1: $tType]:((sum_sum @ X1 @ X0 > $o) > X0)).
% 3.24/1.81 thf(func_def_109, type, sK10: !>[X0: $tType, X1: $tType]:((sum_sum @ X1 @ X0 > $o) > X1)).
% 3.24/1.81 thf(func_def_110, type, sK11: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > (X1 > X0) > X1)).
% 3.24/1.81 thf(func_def_111, type, sK12: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X0)).
% 3.24/1.81 thf(func_def_112, type, sK13: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X1)).
% 3.24/1.81 thf(func_def_113, type, sK14: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X1)).
% 3.24/1.81 thf(func_def_114, type, sK15: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X0)).
% 3.24/1.81 thf(func_def_115, type, sK16: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.24/1.81 thf(func_def_116, type, sK17: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.24/1.81 thf(func_def_117, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 3.24/1.81 thf(func_def_118, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 3.24/1.81 thf(func_def_119, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 3.24/1.81 thf(func_def_120, type, sK21: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 3.24/1.81 thf(func_def_121, type, sK22: !>[X0: $tType, X1: $tType]:((X0 > X1) > finite_Cartesian_vec @ X1 @ X0 > X0)).
% 3.24/1.81 thf(func_def_122, type, sK23: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.24/1.81 thf(func_def_123, type, sK24: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.24/1.81 thf(func_def_124, type, sK25: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 3.24/1.81 thf(func_def_125, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 3.24/1.81 thf(func_def_126, type, sK27: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 3.24/1.81 thf(func_def_127, type, sK28: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 3.24/1.81 thf(func_def_128, type, sK29: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > X0 > X1)).
% 3.24/1.81 thf(func_def_129, type, sK30: !>[X0: $tType, X1: $tType, X2: $tType]:($o > (sum_sum @ X0 @ X1 > X2) > X2 > X1)).
% 3.24/1.81 thf(func_def_130, type, sK31: !>[X0: $tType, X1: $tType, X2: $tType]:($o > (sum_sum @ X0 @ X1 > X2) > X2 > X0)).
% 3.24/1.81 thf(func_def_131, type, sK32: !>[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)).
% 3.24/1.81 thf(func_def_132, type, sK33: !>[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)).
% 3.24/1.81 thf(func_def_133, type, sK34: !>[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)).
% 3.24/1.81 thf(func_def_134, type, sK35: !>[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)).
% 3.24/1.81 thf(func_def_135, type, sK36: !>[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)).
% 3.24/1.81 thf(func_def_136, type, sK37: !>[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)).
% 3.24/1.81 thf(func_def_137, type, sK38: !>[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)).
% 3.24/1.81 thf(func_def_138, type, sK39: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.24/1.81 thf(func_def_139, 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 > X0)).
% 3.24/1.81 thf(func_def_140, 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 > X1)).
% 3.24/1.81 thf(func_def_141, 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 > X2)).
% 3.24/1.81 thf(func_def_142, type, sK43: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 3.24/1.81 thf(func_def_143, type, sK44: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 3.24/1.81 thf(func_def_144, type, sK45: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X1)).
% 3.24/1.81 thf(func_def_145, type, sK46: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X0)).
% 3.24/1.81 thf(func_def_146, type, sK47: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.24/1.81 thf(func_def_147, type, sK48: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.24/1.81 thf(func_def_148, type, sK49: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.24/1.81 thf(func_def_149, 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) > X5)).
% 3.24/1.81 thf(func_def_150, 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) > X4)).
% 3.24/1.81 thf(func_def_151, type, sK52: !>[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)).
% 3.24/1.81 thf(func_def_152, type, sK53: !>[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)).
% 3.24/1.81 thf(func_def_153, type, sK54: !>[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)).
% 3.24/1.81 thf(func_def_154, type, sK55: !>[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)).
% 3.24/1.81 thf(func_def_155, type, sK56: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.24/1.81 thf(func_def_156, type, sK57: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.24/1.81 thf(func_def_157, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.24/1.81 thf(func_def_158, type, sK59: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X1)).
% 3.24/1.81 thf(func_def_159, type, sK60: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X0)).
% 3.24/1.81 thf(func_def_160, type, sK61: !>[X0: $tType, X1: $tType]:(set @ X1 > (X0 > X1) > set @ X0 > X0)).
% 3.24/1.81 thf(func_def_161, type, sK62: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X1)).
% 3.24/1.81 thf(func_def_162, type, sK63: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X0)).
% 3.24/1.81 thf(func_def_163, type, sK64: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 3.24/1.81 thf(func_def_164, type, sK65: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > set @ X0 > set @ X1)).
% 3.24/1.81 thf(func_def_165, type, sK66: !>[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)).
% 3.24/1.81 thf(func_def_166, type, sK67: !>[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)).
% 3.24/1.81 thf(func_def_167, type, sK68: !>[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)).
% 3.24/1.81 thf(func_def_168, type, sK69: !>[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)).
% 3.24/1.81 thf(func_def_169, type, sK70: !>[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)).
% 3.24/1.81 thf(func_def_170, type, sK71: !>[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)).
% 3.24/1.81 thf(func_def_171, type, sK72: !>[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)).
% 3.24/1.81 thf(func_def_172, type, sK73: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.24/1.81 thf(func_def_173, type, sK74: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.24/1.81 thf(func_def_174, type, sK75: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.24/1.81 thf(func_def_175, type, sK76: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.24/1.81 thf(func_def_176, type, sK77: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > X0 > X1)).
% 3.24/1.81 thf(func_def_177, type, sK78: !>[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)).
% 3.24/1.81 thf(func_def_178, type, sK79: !>[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)).
% 3.24/1.81 thf(func_def_179, type, sK80: !>[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)).
% 3.24/1.81 thf(func_def_180, type, sK81: !>[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)).
% 3.24/1.81 thf(func_def_181, type, sK82: !>[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)).
% 3.24/1.81 thf(func_def_183, type, ph84: !>[X0: $tType]:(X0)).
% 3.24/1.81 thf(f1438,plain,(
% 3.24/1.81 $false),
% 3.24/1.81 inference(avatar_sat_refutation,[status(thm)],[f1423,f1437])).
% 3.24/1.81 thf(f1437,plain,(
% 3.24/1.81 spl83_5),
% 3.24/1.81 inference(avatar_contradiction_clause,[status(thm)],[f1436])).
% 3.24/1.81 thf(f1436,plain,(
% 3.24/1.81 $false | spl83_5),
% 3.24/1.81 inference(subsumption_resolution,[status(thm)],[f1435,f1378])).
% 3.24/1.81 thf(f1378,plain,(
% 3.24/1.81 ((member @ c @ i2 @ (static_FVO @ a @ c @ oDEa)) != $true) | spl83_5),
% 3.24/1.81 inference(avatar_component_clause,[status(thm)],[f1376])).
% 3.24/1.81 thf(f1376,definition,(
% 3.24/1.81 spl83_5 <=> ((member @ c @ i2 @ (static_FVO @ a @ c @ oDEa)) = $true)),
% 3.24/1.81 introduced(definition,[new_symbols(naming,[spl83_5])],[avatar_definition])).
% 3.24/1.81 thf(f1435,plain,(
% 3.24/1.81 ((member @ c @ i2 @ (static_FVO @ a @ c @ oDEa)) = $true)),
% 3.24/1.81 inference(trivial_inequality_removal,[status(thm)],[f1430])).
% 3.24/1.81 thf(f1430,plain,(
% 3.24/1.81 ($true != $true) | ((member @ c @ i2 @ (static_FVO @ a @ c @ oDEa)) = $true)),
% 3.24/1.81 inference(superposition,[status(thm)],[f1252,f1263])).
% 3.24/1.81 thf(f1263,plain,(
% 3.24/1.81 ((member @ c @ i2 @ (denota811733865E_vars @ a @ b @ c @ i @ oDEa)) = $true)),
% 3.24/1.81 inference(cnf_transformation,[status(thm)],[f679])).
% 3.24/1.81 thf(f679,plain,(
% 3.24/1.81 ((member @ c @ i2 @ (denota811733865E_vars @ a @ b @ c @ i @ oDEa)) = $true)),
% 3.24/1.81 inference(fool_elimination,[status(thm)],[f678])).
% 3.24/1.81 thf(f678,plain,(
% 3.24/1.81 (member @ c @ i2 @ (denota811733865E_vars @ a @ b @ c @ i @ oDEa))),
% 3.24/1.81 inference(rectify,[status(thm)],[f323])).
% 3.24/1.81 thf(f323,axiom,(
% 3.24/1.81 (member @ c @ i2 @ (denota811733865E_vars @ a @ b @ c @ i @ oDEa))),
% 3.24/1.81 file('/export/starexec/sandbox/tmp/tmp.iHbtgXCbd3/DTF2THF_15214.p',conj_1)).
% 3.24/1.81 thf(f1252,plain,(
% 3.24/1.81 ( ! [X1 : $tType,X0 : $tType,X2 : $tType,X3 : X1,X4 : denota1663640101rp_ext @ X2 @ X0 @ X1 @ product_unit,X5 : ode @ X2 @ X1] : (($true != (member @ X1 @ X3 @ (denota811733865E_vars @ X2 @ X0 @ X1 @ X4 @ X5))) | ($true = (member @ X1 @ X3 @ (static_FVO @ X2 @ X1 @ X5)))) )),
% 3.24/1.81 inference(cnf_transformation,[status(thm)],[f1018])).
% 3.24/1.81 thf(f1018,plain,(
% 3.24/1.81 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : X1,X4 : denota1663640101rp_ext @ X2 @ X0 @ X1 @ product_unit,X5 : ode @ X2 @ X1] : (($true = (member @ X1 @ X3 @ (static_FVO @ X2 @ X1 @ X5))) | ($true != (member @ X1 @ X3 @ (denota811733865E_vars @ X2 @ X0 @ X1 @ X4 @ X5))))),
% 3.24/1.81 inference(ennf_transformation,[status(thm)],[f765])).
% 3.24/1.81 thf(f765,plain,(
% 3.24/1.81 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : X1,X4 : denota1663640101rp_ext @ X2 @ X0 @ X1 @ product_unit,X5 : ode @ X2 @ X1] : (($true = (member @ X1 @ X3 @ (denota811733865E_vars @ X2 @ X0 @ X1 @ X4 @ X5))) => ($true = (member @ X1 @ X3 @ (static_FVO @ X2 @ X1 @ X5))))),
% 3.24/1.81 inference(fool_elimination,[status(thm)],[f764])).
% 3.24/1.81 thf(f764,plain,(
% 3.24/1.81 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : X1,X4 : denota1663640101rp_ext @ X2 @ X0 @ X1 @ product_unit,X5 : ode @ X2 @ X1] : ((member @ X1 @ X3 @ (denota811733865E_vars @ X2 @ X0 @ X1 @ X4 @ X5)) => (member @ X1 @ X3 @ (static_FVO @ X2 @ X1 @ X5)))),
% 3.24/1.81 inference(rectify,[status(thm)],[f1])).
% 3.24/1.81 thf(f1,axiom,(
% 3.24/1.81 ! [X2 : $tType,X0 : $tType,X1 : $tType,X6 : X0,X7 : denota1663640101rp_ext @ X1 @ X2 @ X0 @ product_unit,X8 : ode @ X1 @ X0] : ((member @ X0 @ X6 @ (denota811733865E_vars @ X1 @ X2 @ X0 @ X7 @ X8)) => (member @ X0 @ X6 @ (static_FVO @ X1 @ X0 @ X8)))),
% 3.24/1.81 file('/export/starexec/sandbox/tmp/tmp.iHbtgXCbd3/DTF2THF_15214.p',fact_0_ode__to__fvo)).
% 3.24/1.81 thf(f1423,plain,(
% 3.24/1.81 ~spl83_5),
% 3.24/1.81 inference(avatar_split_clause,[status(thm)],[f1216,f1376])).
% 3.24/1.81 thf(f1216,plain,(
% 3.24/1.81 ((member @ c @ i2 @ (static_FVO @ a @ c @ oDEa)) != $true)),
% 3.24/1.81 inference(cnf_transformation,[status(thm)],[f927])).
% 3.24/1.81 thf(f927,plain,(
% 3.24/1.81 ((member @ c @ i2 @ (static_FVO @ a @ c @ oDEa)) != $true)),
% 3.24/1.81 inference(flattening,[status(thm)],[f709])).
% 3.24/1.81 thf(f709,plain,(
% 3.24/1.81 ~((member @ c @ i2 @ (static_FVO @ a @ c @ oDEa)) = $true)),
% 3.24/1.81 inference(fool_elimination,[status(thm)],[f708])).
% 3.24/1.81 thf(f708,plain,(
% 3.24/1.81 ~(member @ c @ i2 @ (static_FVO @ a @ c @ oDEa))),
% 3.24/1.81 inference(rectify,[status(thm)],[f324])).
% 3.24/1.81 thf(f324,axiom,(
% 3.24/1.81 ~(member @ c @ i2 @ (static_FVO @ a @ c @ oDEa))),
% 3.24/1.81 file('/export/starexec/sandbox/tmp/tmp.iHbtgXCbd3/DTF2THF_15214.p',conj_2)).
% 3.24/1.81 % SZS output end Proof for DTF2THF_15214
% 3.24/1.81 % (15325)------------------------------
% 3.24/1.81 % (15325)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.81 % (15325)Termination reason: Refutation
% 3.24/1.81
% 3.24/1.81 % (15325)Memory used [KB]: 7419
% 3.24/1.81 % (15325)Time elapsed: 0.049 s
% 3.24/1.81 % (15325)Instructions burned: 157 (million)
% 3.24/1.81 % (15325)------------------------------
% 3.24/1.81 % (15325)------------------------------
% 3.24/1.81 % (15319)Success in time 0.055 s
%------------------------------------------------------------------------------