%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP037^2 : 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 : n010.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 08:55:02 AM UTC 2026
% Result : Theorem 3.02s 1.67s
% Output : Refutation 3.02s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP037^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.34 % Computer : n010.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Tue Feb 24 08:24:03 EST 2026
% 0.18/0.34 % CPUTime :
% 0.18/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.46 ---- Original DTF file ---
% 0.20/0.47 thf(spec,logic,$$dhol).
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 % File : ITP037^2 : TPTP v9.2.1. Released v7.5.0.
% 0.20/0.47 % Domain : Interactive Theorem Proving
% 0.20/0.47 % Problem : Sledgehammer Coincidence problem prob_129__7211024_1
% 0.20/0.47 % Version : Especial.
% 0.20/0.47 % English :
% 0.20/0.47
% 0.20/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.20/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.20/0.47 % Source : [Des21]
% 0.20/0.47 % Names : Coincidence/prob_129__7211024_1 [Des21]
% 0.20/0.47
% 0.20/0.47 % Status : Theorem
% 0.20/0.47 % Rating : 0.33 v8.1.0, 0.25 v7.5.0
% 0.20/0.47 % Syntax : Number of formulae : 374 ( 148 unt; 86 typ; 0 def)
% 0.20/0.47 % Number of atoms : 791 ( 314 equ; 0 cnn)
% 0.20/0.47 % Maximal formula atoms : 15 ( 2 avg)
% 0.20/0.47 % Number of connectives : 5363 ( 112 ~; 11 |; 165 &;4764 @)
% 0.20/0.47 % ( 0 <=>; 311 =>; 0 <=; 0 <~>)
% 0.20/0.47 % Maximal formula depth : 34 ( 10 avg)
% 0.20/0.47 % Number of types : 9 ( 8 usr)
% 0.20/0.47 % Number of type conns : 253 ( 253 >; 0 *; 0 +; 0 <<)
% 0.20/0.47 % Number of symbols : 81 ( 78 usr; 10 con; 0-13 aty)
% 0.20/0.47 % Number of variables : 1660 ( 32 ^;1472 !; 35 ?;1660 :)
% 0.20/0.47 % ( 121 !>; 0 ?*; 0 @-; 0 @+)
% 0.20/0.47 % SPC : TH1_THM_EQU_NAR
% 0.20/0.47
% 0.20/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:22:14.005
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 % Could-be-implicit typings (18)
% 0.20/0.47 thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext,type,
% 0.20/0.47 denota1663640101rp_ext: $tType > $tType > $tType > $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_Frechet__Correctness_Oids_Ogood__interp,type,
% 0.20/0.47 frechet_good_interp: $tType > $tType > $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_Frechet__Correctness_Oids_Ostrm,type,
% 0.20/0.47 frechet_strm: $tType > $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_Finite__Cartesian__Product_Ovec,type,
% 0.20/0.47 finite_Cartesian_vec: $tType > $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_Product__Type_Ounit,type,
% 0.20/0.47 product_unit: $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_Product__Type_Oprod,type,
% 0.20/0.47 product_prod: $tType > $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_Syntax_Oformula,type,
% 0.20/0.47 formula: $tType > $tType > $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_Sum__Type_Osum,type,
% 0.20/0.47 sum_sum: $tType > $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_Syntax_Otrm,type,
% 0.20/0.47 trm: $tType > $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_Real_Oreal,type,
% 0.20/0.47 real: $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_List_Olist,type,
% 0.20/0.47 list: $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_Set_Oset,type,
% 0.20/0.47 set: $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_tf_sz,type,
% 0.20/0.47 sz: $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_tf_sf,type,
% 0.20/0.47 sf: $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_tf_sc,type,
% 0.20/0.47 sc: $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_tf_c,type,
% 0.20/0.47 c: $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_tf_b,type,
% 0.20/0.47 b: $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_tf_a,type,
% 0.20/0.47 a: $tType ).
% 0.20/0.47
% 0.20/0.47 % Explicit typings (68)
% 0.20/0.47 thf(sy_cl_Cardinality_OCARD__1,type,
% 0.20/0.47 cARD_1:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_HOL_Otype,type,
% 0.20/0.47 type:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Ozero,type,
% 0.20/0.47 zero:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Orderings_Otop,type,
% 0.20/0.47 top:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.20/0.47 finite_finite:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.20/0.47 linorder:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd,type,
% 0.20/0.47 euclid1678468529ng_gcd:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring,type,
% 0.20/0.47 euclid1155270486miring:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Denotational__Semantics_OVagree,type,
% 0.20/0.47 denotational_Vagree:
% 0.20/0.47 !>[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 ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Denotational__Semantics_Odirectional__derivative,type,
% 0.20/0.47 denota2078997598vative:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( trm @ A @ C ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > real ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Denotational__Semantics_Odterm__sem,type,
% 0.20/0.47 denota594965758rm_sem:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( trm @ A @ C ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > real ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Denotational__Semantics_Ofrechet,type,
% 0.20/0.47 denotational_frechet:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( trm @ A @ C ) > ( finite_Cartesian_vec @ real @ C ) > ( finite_Cartesian_vec @ real @ C ) > real ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Denotational__Semantics_Ois__interp,type,
% 0.20/0.47 denota2077489681interp:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Frechet__Correctness_Oids_Ocr__good__interp,type,
% 0.20/0.47 freche457001096interp:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( frechet_good_interp @ A @ B @ C ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Frechet__Correctness_Oids_Ocr__strm,type,
% 0.20/0.47 frechet_cr_strm:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( trm @ A @ B ) > ( frechet_strm @ A @ B ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Ogood__interp,type,
% 0.20/0.47 freche227871258interp:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( frechet_good_interp @ A @ B @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Oraw__interp,type,
% 0.20/0.47 freche229654227interp:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( frechet_good_interp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Frechet__Correctness_Oids_Ostrm_Oraw__term,type,
% 0.20/0.47 frechet_raw_term:
% 0.20/0.47 !>[A: $tType,C: $tType] : ( ( frechet_strm @ A @ C ) > ( trm @ A @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Frechet__Correctness_Oids_Ostrm_Osimple__term,type,
% 0.20/0.47 frechet_simple_term:
% 0.20/0.47 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( frechet_strm @ A @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.20/0.47 zero_zero:
% 0.20/0.47 !>[A: $tType] : A ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Ids_Oids,type,
% 0.20/0.47 ids:
% 0.20/0.47 !>[Sz: $tType,Sf: $tType,Sc: $tType] : ( Sz > Sz > Sz > Sf > Sf > Sf > Sc > Sc > Sc > Sc > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_If,type,
% 0.20/0.47 if:
% 0.20/0.47 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.20/0.47 top_top:
% 0.20/0.47 !>[A: $tType] : A ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_OPair,type,
% 0.20/0.47 product_Pair:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.20/0.47 produc2004651681e_prod:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_Omap__prod,type,
% 0.20/0.47 product_map_prod:
% 0.20/0.47 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.20/0.47 product_rec_prod:
% 0.20/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.20/0.47 product_fst:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.20/0.47 product_snd:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.20/0.47 product_swap:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Relation_ODomainp,type,
% 0.20/0.47 domainp:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_OCollect,type,
% 0.20/0.47 collect:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_Oimage,type,
% 0.20/0.47 image:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Static__Semantics_OFVDiff,type,
% 0.20/0.47 static_FVDiff:
% 0.20/0.47 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_OPredicational,type,
% 0.20/0.47 predicational:
% 0.20/0.47 !>[B: $tType,A: $tType,C: $tType] : ( B > ( formula @ A @ B @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Odfree,type,
% 0.20/0.47 dfree:
% 0.20/0.47 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Odsafe,type,
% 0.20/0.47 dsafe:
% 0.20/0.47 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Offree,type,
% 0.20/0.47 ffree:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Oformula_OProp,type,
% 0.20/0.47 prop:
% 0.20/0.47 !>[C: $tType,A: $tType,B: $tType] : ( C > ( C > ( trm @ A @ C ) ) > ( formula @ A @ B @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Oformula_Oset1__formula,type,
% 0.20/0.47 set1_formula:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Oids_OP,type,
% 0.20/0.47 p:
% 0.20/0.47 !>[Sc: $tType,Sf: $tType,Sz: $tType] : ( Sc > ( formula @ Sf @ Sc @ Sz ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Oids_Oempty,type,
% 0.20/0.47 empty:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( B > ( trm @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Oids_Of0,type,
% 0.20/0.47 f0:
% 0.20/0.47 !>[Sf: $tType,Sz: $tType] : ( Sf > ( trm @ Sf @ Sz ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Oids_Of1,type,
% 0.20/0.47 f1:
% 0.20/0.47 !>[Sz: $tType,Sf: $tType] : ( Sz > Sf > Sz > ( trm @ Sf @ Sz ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Oids_Op1,type,
% 0.20/0.47 p1:
% 0.20/0.47 !>[Sz: $tType,Sf: $tType,Sc: $tType] : ( Sz > Sz > Sz > ( formula @ Sf @ Sc @ Sz ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Oids_Osingleton,type,
% 0.20/0.47 singleton:
% 0.20/0.47 !>[Sz: $tType,A: $tType] : ( Sz > ( trm @ A @ Sz ) > Sz > ( trm @ A @ Sz ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Oids_Osingleton__rel,type,
% 0.20/0.47 singleton_rel:
% 0.20/0.47 !>[A: $tType,Sz: $tType] : ( ( product_prod @ ( trm @ A @ Sz ) @ Sz ) > ( product_prod @ ( trm @ A @ Sz ) @ Sz ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Otrm_OConst,type,
% 0.20/0.47 const:
% 0.20/0.47 !>[A: $tType,C: $tType] : ( real > ( trm @ A @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Otrm_ODiffVar,type,
% 0.20/0.47 diffVar:
% 0.20/0.47 !>[C: $tType,A: $tType] : ( C > ( trm @ A @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Otrm_ODifferential,type,
% 0.20/0.47 differential:
% 0.20/0.47 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( trm @ A @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Otrm_OFunction,type,
% 0.20/0.47 function:
% 0.20/0.47 !>[A: $tType,C: $tType] : ( A > ( C > ( trm @ A @ C ) ) > ( trm @ A @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Otrm_OPlus,type,
% 0.20/0.47 plus:
% 0.20/0.47 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( trm @ A @ C ) > ( trm @ A @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Otrm_OTimes,type,
% 0.20/0.47 times:
% 0.20/0.47 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( trm @ A @ C ) > ( trm @ A @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Otrm_OVar,type,
% 0.20/0.47 var:
% 0.20/0.47 !>[C: $tType,A: $tType] : ( C > ( trm @ A @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Syntax_Otrm_Oset__trm,type,
% 0.20/0.47 set_trm:
% 0.20/0.47 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Transfer_Oleft__unique,type,
% 0.20/0.47 left_unique:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Transfer_Oright__unique,type,
% 0.20/0.47 right_unique:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Wellfounded_Oaccp,type,
% 0.20/0.47 accp:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_member,type,
% 0.20/0.47 member:
% 0.20/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_I,type,
% 0.20/0.47 i: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v__092_060nu_062,type,
% 0.20/0.47 nu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v__092_060nu_062_H,type,
% 0.20/0.47 nu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_args____,type,
% 0.20/0.47 args: c > ( trm @ a @ c ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_i,type,
% 0.20/0.47 i2: c ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_var____,type,
% 0.20/0.47 var2: a ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_vid1,type,
% 0.20/0.47 vid1: sz ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_vid2,type,
% 0.20/0.47 vid2: sz ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_vid3,type,
% 0.20/0.47 vid3: sz ).
% 0.20/0.47
% 0.20/0.47 % Relevant facts (255)
% 0.20/0.47 thf(fact_0_free,axiom,
% 0.20/0.47 ! [I: c] : ( dfree @ a @ c @ ( args @ I ) ) ).
% 0.20/0.47
% 0.20/0.47 % free
% 0.20/0.47 thf(fact_1_raw__interp__inject,axiom,
% 0.20/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.20/0.47 ( ( ( finite_finite @ A )
% 0.20/0.47 & ( finite_finite @ B )
% 0.20/0.47 & ( finite_finite @ C ) )
% 0.20/0.47 => ! [X: frechet_good_interp @ A @ B @ C,Y: frechet_good_interp @ A @ B @ C] :
% 0.20/0.47 ( ( ( freche229654227interp @ A @ B @ C @ X )
% 0.20/0.47 = ( freche229654227interp @ A @ B @ C @ Y ) )
% 0.20/0.47 = ( X = Y ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % raw_interp_inject
% 0.20/0.47 thf(fact_2_raw__term__inject,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X: frechet_strm @ A @ C,Y: frechet_strm @ A @ C] :
% 0.20/0.47 ( ( ( frechet_raw_term @ A @ C @ X )
% 0.20/0.47 = ( frechet_raw_term @ A @ C @ Y ) )
% 0.20/0.47 = ( X = Y ) ) ).
% 0.20/0.47
% 0.20/0.47 % raw_term_inject
% 0.20/0.47 thf(fact_3_P__def,axiom,
% 0.20/0.47 ( ( p @ sc @ sf @ sz )
% 0.20/0.47 = ( predicational @ sc @ sf @ sz ) ) ).
% 0.20/0.47
% 0.20/0.47 % P_def
% 0.20/0.47 thf(fact_4_strm_Odomain,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType] :
% 0.20/0.47 ( ( domainp @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) )
% 0.20/0.47 = ( dfree @ A @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 % strm.domain
% 0.20/0.47 thf(fact_5_dfree__DiffVar__simps,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,X: B] :
% 0.20/0.47 ~ ( dfree @ A @ B @ ( diffVar @ B @ A @ X ) ) ).
% 0.20/0.47
% 0.20/0.47 % dfree_DiffVar_simps
% 0.20/0.47 thf(fact_6_seq__sem_Ocases,axiom,
% 0.20/0.47 ! [X: product_prod @ ( denota1663640101rp_ext @ sf @ sc @ sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) ) )] :
% 0.20/0.47 ~ ! [I2: denota1663640101rp_ext @ sf @ sc @ sz @ product_unit,S: product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) )] :
% 0.20/0.47 ( X
% 0.20/0.47 != ( product_Pair @ ( denota1663640101rp_ext @ sf @ sc @ sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) ) ) @ I2 @ S ) ) ).
% 0.20/0.47
% 0.20/0.47 % seq_sem.cases
% 0.20/0.47 thf(fact_7_singleton_Ocases,axiom,
% 0.20/0.47 ! [A: $tType,X: product_prod @ ( trm @ A @ sz ) @ sz] :
% 0.20/0.47 ~ ! [T2: trm @ A @ sz,I3: sz] :
% 0.20/0.47 ( X
% 0.20/0.47 != ( product_Pair @ ( trm @ A @ sz ) @ sz @ T2 @ I3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % singleton.cases
% 0.20/0.47 thf(fact_8_dfree__Differential__simps,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,X: trm @ A @ B] :
% 0.20/0.47 ~ ( dfree @ A @ B @ ( differential @ A @ B @ X ) ) ).
% 0.20/0.47
% 0.20/0.47 % dfree_Differential_simps
% 0.20/0.47 thf(fact_9_dfree__Var__simps,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,X: B] : ( dfree @ A @ B @ ( var @ B @ A @ X ) ) ).
% 0.20/0.47
% 0.20/0.47 % dfree_Var_simps
% 0.20/0.47 thf(fact_10_dfree__Times__simps,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A2: trm @ A @ B,B2: trm @ A @ B] :
% 0.20/0.47 ( ( dfree @ A @ B @ ( times @ A @ B @ A2 @ B2 ) )
% 0.20/0.47 = ( ( dfree @ A @ B @ A2 )
% 0.20/0.47 & ( dfree @ A @ B @ B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dfree_Times_simps
% 0.20/0.47 thf(fact_11_dfree__Plus__simps,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A2: trm @ A @ B,B2: trm @ A @ B] :
% 0.20/0.47 ( ( dfree @ A @ B @ ( plus @ A @ B @ A2 @ B2 ) )
% 0.20/0.47 = ( ( dfree @ A @ B @ A2 )
% 0.20/0.47 & ( dfree @ A @ B @ B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dfree_Plus_simps
% 0.20/0.47 thf(fact_12_dfree__Fun_Oprems,axiom,
% 0.20/0.47 denotational_Vagree @ c @ nu @ nu2 @ ( static_FVDiff @ a @ c @ ( function @ a @ c @ var2 @ args ) ) ).
% 0.20/0.47
% 0.20/0.47 % dfree_Fun.prems
% 0.20/0.47 thf(fact_13_trm_Oinject_I3_J,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,X31: A,X32: C > ( trm @ A @ C ),Y31: A,Y32: C > ( trm @ A @ C )] :
% 0.20/0.47 ( ( ( function @ A @ C @ X31 @ X32 )
% 0.20/0.47 = ( function @ A @ C @ Y31 @ Y32 ) )
% 0.20/0.47 = ( ( X31 = Y31 )
% 0.20/0.47 & ( X32 = Y32 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.inject(3)
% 0.20/0.47 thf(fact_14_trm_Oinject_I4_J,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X41: trm @ A @ C,X42: trm @ A @ C,Y41: trm @ A @ C,Y42: trm @ A @ C] :
% 0.20/0.47 ( ( ( plus @ A @ C @ X41 @ X42 )
% 0.20/0.47 = ( plus @ A @ C @ Y41 @ Y42 ) )
% 0.20/0.47 = ( ( X41 = Y41 )
% 0.20/0.47 & ( X42 = Y42 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.inject(4)
% 0.20/0.47 thf(fact_15_trm_Oinject_I5_J,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X51: trm @ A @ C,X52: trm @ A @ C,Y51: trm @ A @ C,Y52: trm @ A @ C] :
% 0.20/0.47 ( ( ( times @ A @ C @ X51 @ X52 )
% 0.20/0.47 = ( times @ A @ C @ Y51 @ Y52 ) )
% 0.20/0.47 = ( ( X51 = Y51 )
% 0.20/0.47 & ( X52 = Y52 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.inject(5)
% 0.20/0.47 thf(fact_16_trm_Oinject_I1_J,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,X1: C,Y1: C] :
% 0.20/0.47 ( ( ( var @ C @ A @ X1 )
% 0.20/0.47 = ( var @ C @ A @ Y1 ) )
% 0.20/0.47 = ( X1 = Y1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.inject(1)
% 0.20/0.47 thf(fact_17_trm_Oinject_I7_J,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X7: trm @ A @ C,Y7: trm @ A @ C] :
% 0.20/0.47 ( ( ( differential @ A @ C @ X7 )
% 0.20/0.47 = ( differential @ A @ C @ Y7 ) )
% 0.20/0.47 = ( X7 = Y7 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.inject(7)
% 0.20/0.47 thf(fact_18_trm_Oinject_I6_J,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,X6: C,Y6: C] :
% 0.20/0.47 ( ( ( diffVar @ C @ A @ X6 )
% 0.20/0.47 = ( diffVar @ C @ A @ Y6 ) )
% 0.20/0.47 = ( X6 = Y6 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.inject(6)
% 0.20/0.47 thf(fact_19_dfree__Fun__simps,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,I4: A,Args: B > ( trm @ A @ B )] :
% 0.20/0.47 ( ( dfree @ A @ B @ ( function @ A @ B @ I4 @ Args ) )
% 0.20/0.47 = ( ! [X2: B] : ( dfree @ A @ B @ ( Args @ X2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dfree_Fun_simps
% 0.20/0.47 thf(fact_20_trm_Odistinct_I41_J,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X6: C,X7: trm @ A @ C] :
% 0.20/0.47 ( ( diffVar @ C @ A @ X6 )
% 0.20/0.47 != ( differential @ A @ C @ X7 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(41)
% 0.20/0.47 thf(fact_21_trm_Odistinct_I39_J,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X51: trm @ A @ C,X52: trm @ A @ C,X7: trm @ A @ C] :
% 0.20/0.47 ( ( times @ A @ C @ X51 @ X52 )
% 0.20/0.47 != ( differential @ A @ C @ X7 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(39)
% 0.20/0.47 thf(fact_22_trm_Odistinct_I37_J,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,X51: trm @ A @ C,X52: trm @ A @ C,X6: C] :
% 0.20/0.47 ( ( times @ A @ C @ X51 @ X52 )
% 0.20/0.47 != ( diffVar @ C @ A @ X6 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(37)
% 0.20/0.47 thf(fact_23_trm_Odistinct_I35_J,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X41: trm @ A @ C,X42: trm @ A @ C,X7: trm @ A @ C] :
% 0.20/0.47 ( ( plus @ A @ C @ X41 @ X42 )
% 0.20/0.47 != ( differential @ A @ C @ X7 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(35)
% 0.20/0.47 thf(fact_24_trm_Odistinct_I33_J,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,X41: trm @ A @ C,X42: trm @ A @ C,X6: C] :
% 0.20/0.47 ( ( plus @ A @ C @ X41 @ X42 )
% 0.20/0.47 != ( diffVar @ C @ A @ X6 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(33)
% 0.20/0.47 thf(fact_25_trm_Odistinct_I31_J,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X41: trm @ A @ C,X42: trm @ A @ C,X51: trm @ A @ C,X52: trm @ A @ C] :
% 0.20/0.47 ( ( plus @ A @ C @ X41 @ X42 )
% 0.20/0.47 != ( times @ A @ C @ X51 @ X52 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(31)
% 0.20/0.47 thf(fact_26_trm_Odistinct_I29_J,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X31: A,X32: C > ( trm @ A @ C ),X7: trm @ A @ C] :
% 0.20/0.47 ( ( function @ A @ C @ X31 @ X32 )
% 0.20/0.47 != ( differential @ A @ C @ X7 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(29)
% 0.20/0.47 thf(fact_27_trm_Odistinct_I27_J,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,X31: A,X32: C > ( trm @ A @ C ),X6: C] :
% 0.20/0.47 ( ( function @ A @ C @ X31 @ X32 )
% 0.20/0.47 != ( diffVar @ C @ A @ X6 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(27)
% 0.20/0.47 thf(fact_28_trm_Odistinct_I25_J,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X31: A,X32: C > ( trm @ A @ C ),X51: trm @ A @ C,X52: trm @ A @ C] :
% 0.20/0.47 ( ( function @ A @ C @ X31 @ X32 )
% 0.20/0.47 != ( times @ A @ C @ X51 @ X52 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(25)
% 0.20/0.47 thf(fact_29_trm_Odistinct_I23_J,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X31: A,X32: C > ( trm @ A @ C ),X41: trm @ A @ C,X42: trm @ A @ C] :
% 0.20/0.47 ( ( function @ A @ C @ X31 @ X32 )
% 0.20/0.47 != ( plus @ A @ C @ X41 @ X42 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(23)
% 0.20/0.47 thf(fact_30_trm_Odistinct_I11_J,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X1: C,X7: trm @ A @ C] :
% 0.20/0.47 ( ( var @ C @ A @ X1 )
% 0.20/0.47 != ( differential @ A @ C @ X7 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(11)
% 0.20/0.47 thf(fact_31_trm_Odistinct_I9_J,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,X1: C,X6: C] :
% 0.20/0.47 ( ( var @ C @ A @ X1 )
% 0.20/0.47 != ( diffVar @ C @ A @ X6 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(9)
% 0.20/0.47 thf(fact_32_trm_Odistinct_I7_J,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X1: C,X51: trm @ A @ C,X52: trm @ A @ C] :
% 0.20/0.47 ( ( var @ C @ A @ X1 )
% 0.20/0.47 != ( times @ A @ C @ X51 @ X52 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(7)
% 0.20/0.47 thf(fact_33_trm_Odistinct_I5_J,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X1: C,X41: trm @ A @ C,X42: trm @ A @ C] :
% 0.20/0.47 ( ( var @ C @ A @ X1 )
% 0.20/0.47 != ( plus @ A @ C @ X41 @ X42 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(5)
% 0.20/0.47 thf(fact_34_trm_Odistinct_I3_J,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,X1: C,X31: A,X32: C > ( trm @ A @ C )] :
% 0.20/0.47 ( ( var @ C @ A @ X1 )
% 0.20/0.47 != ( function @ A @ C @ X31 @ X32 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(3)
% 0.20/0.47 thf(fact_35_dfree_Odfree__Fun,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,Args: C > ( trm @ A @ C ),I4: A] :
% 0.20/0.47 ( ! [I3: C] : ( dfree @ A @ C @ ( Args @ I3 ) )
% 0.20/0.47 => ( dfree @ A @ C @ ( function @ A @ C @ I4 @ Args ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dfree.dfree_Fun
% 0.20/0.47 thf(fact_36_dfree__Plus,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,Theta_1: trm @ A @ C,Theta_2: trm @ A @ C] :
% 0.20/0.47 ( ( dfree @ A @ C @ Theta_1 )
% 0.20/0.47 => ( ( dfree @ A @ C @ Theta_2 )
% 0.20/0.47 => ( dfree @ A @ C @ ( plus @ A @ C @ Theta_1 @ Theta_2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dfree_Plus
% 0.20/0.47 thf(fact_37_dfree__Times,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,Theta_1: trm @ A @ C,Theta_2: trm @ A @ C] :
% 0.20/0.47 ( ( dfree @ A @ C @ Theta_1 )
% 0.20/0.47 => ( ( dfree @ A @ C @ Theta_2 )
% 0.20/0.47 => ( dfree @ A @ C @ ( times @ A @ C @ Theta_1 @ Theta_2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dfree_Times
% 0.20/0.47 thf(fact_38_dfree__Var,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,I4: C] : ( dfree @ A @ C @ ( var @ C @ A @ I4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % dfree_Var
% 0.20/0.47 thf(fact_39_raw__interp__inverse,axiom,
% 0.20/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.20/0.47 ( ( ( finite_finite @ A )
% 0.20/0.47 & ( finite_finite @ B )
% 0.20/0.47 & ( finite_finite @ C ) )
% 0.20/0.47 => ! [X: frechet_good_interp @ A @ B @ C] :
% 0.20/0.47 ( ( freche227871258interp @ A @ B @ C @ ( freche229654227interp @ A @ B @ C @ X ) )
% 0.20/0.47 = X ) ) ).
% 0.20/0.47
% 0.20/0.47 % raw_interp_inverse
% 0.20/0.47 thf(fact_40_raw__term__inverse,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X: frechet_strm @ A @ C] :
% 0.20/0.47 ( ( frechet_simple_term @ A @ C @ ( frechet_raw_term @ A @ C @ X ) )
% 0.20/0.47 = X ) ).
% 0.20/0.47
% 0.20/0.47 % raw_term_inverse
% 0.20/0.47 thf(fact_41_agree__times1,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType] :
% 0.20/0.47 ( ( finite_finite @ A )
% 0.20/0.47 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),T1: trm @ B @ A,T22: trm @ B @ A] :
% 0.20/0.47 ( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( times @ B @ A @ T1 @ T22 ) ) )
% 0.20/0.47 => ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ T1 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % agree_times1
% 0.20/0.47 thf(fact_42_agree__times2,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType] :
% 0.20/0.47 ( ( finite_finite @ A )
% 0.20/0.47 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),T1: trm @ B @ A,T22: trm @ B @ A] :
% 0.20/0.47 ( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( times @ B @ A @ T1 @ T22 ) ) )
% 0.20/0.47 => ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ T22 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % agree_times2
% 0.20/0.47 thf(fact_43_agree__plus1,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType] :
% 0.20/0.47 ( ( finite_finite @ A )
% 0.20/0.47 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),T1: trm @ B @ A,T22: trm @ B @ A] :
% 0.20/0.47 ( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( plus @ B @ A @ T1 @ T22 ) ) )
% 0.20/0.47 => ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ T1 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % agree_plus1
% 0.20/0.47 thf(fact_44_agree__plus2,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType] :
% 0.20/0.47 ( ( finite_finite @ A )
% 0.20/0.47 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),T1: trm @ B @ A,T22: trm @ B @ A] :
% 0.20/0.47 ( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( plus @ B @ A @ T1 @ T22 ) ) )
% 0.20/0.47 => ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ T22 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % agree_plus2
% 0.20/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.20/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.20/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.20/0.47 = ( P @ A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % mem_Collect_eq
% 0.20/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.20/0.47 ! [A: $tType,A3: set @ A] :
% 0.20/0.47 ( ( collect @ A
% 0.20/0.47 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 0.20/0.47 = A3 ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_mem_eq
% 0.20/0.47 thf(fact_47_Collect__cong,axiom,
% 0.20/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.20/0.47 ( ! [X3: A] :
% 0.20/0.47 ( ( P @ X3 )
% 0.20/0.47 = ( Q @ X3 ) )
% 0.20/0.47 => ( ( collect @ A @ P )
% 0.20/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_cong
% 0.20/0.47 thf(fact_48_ext,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.20/0.47 ( ! [X3: A] :
% 0.20/0.47 ( ( F @ X3 )
% 0.20/0.47 = ( G @ X3 ) )
% 0.20/0.47 => ( F = G ) ) ).
% 0.20/0.47
% 0.20/0.47 % ext
% 0.20/0.47 thf(fact_49_agree__func,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( finite_finite @ A )
% 0.20/0.47 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Var: B,Args: A > ( trm @ B @ A ),I4: A] :
% 0.20/0.47 ( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( function @ B @ A @ Var @ Args ) ) )
% 0.20/0.47 => ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( Args @ I4 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % agree_func
% 0.20/0.47 thf(fact_50_prod_Oinject,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 0.20/0.47 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.20/0.47 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.20/0.47 = ( ( X1 = Y1 )
% 0.20/0.47 & ( X22 = Y2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod.inject
% 0.20/0.47 thf(fact_51_old_Oprod_Oinject,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B] :
% 0.20/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.20/0.47 = ( product_Pair @ A @ B @ A4 @ B3 ) )
% 0.20/0.47 = ( ( A2 = A4 )
% 0.20/0.47 & ( B2 = B3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.inject
% 0.20/0.47 thf(fact_52_trm_Oexhaust,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,Y: trm @ A @ C] :
% 0.20/0.47 ( ! [X12: C] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( var @ C @ A @ X12 ) )
% 0.20/0.47 => ( ! [X23: real] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( const @ A @ C @ X23 ) )
% 0.20/0.47 => ( ! [X312: A,X322: C > ( trm @ A @ C )] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( function @ A @ C @ X312 @ X322 ) )
% 0.20/0.47 => ( ! [X412: trm @ A @ C,X422: trm @ A @ C] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( plus @ A @ C @ X412 @ X422 ) )
% 0.20/0.47 => ( ! [X512: trm @ A @ C,X522: trm @ A @ C] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( times @ A @ C @ X512 @ X522 ) )
% 0.20/0.47 => ( ! [X62: C] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( diffVar @ C @ A @ X62 ) )
% 0.20/0.47 => ~ ! [X72: trm @ A @ C] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( differential @ A @ C @ X72 ) ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.exhaust
% 0.20/0.47 thf(fact_53_prod__cases3,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.20/0.47 ~ ! [A5: A,B4: B,C2: C] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases3
% 0.20/0.47 thf(fact_54_trm_Oinject_I2_J,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X22: real,Y2: real] :
% 0.20/0.47 ( ( ( const @ A @ C @ X22 )
% 0.20/0.47 = ( const @ A @ C @ Y2 ) )
% 0.20/0.47 = ( X22 = Y2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.inject(2)
% 0.20/0.47 thf(fact_55_dfree__Const__simps,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,R: real] : ( dfree @ A @ B @ ( const @ A @ B @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % dfree_Const_simps
% 0.20/0.47 thf(fact_56_dfree__Const,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,R: real] : ( dfree @ A @ C @ ( const @ A @ C @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % dfree_Const
% 0.20/0.47 thf(fact_57_trm_Odistinct_I13_J,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,X22: real,X31: A,X32: C > ( trm @ A @ C )] :
% 0.20/0.47 ( ( const @ A @ C @ X22 )
% 0.20/0.47 != ( function @ A @ C @ X31 @ X32 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(13)
% 0.20/0.47 thf(fact_58_trm_Odistinct_I15_J,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X22: real,X41: trm @ A @ C,X42: trm @ A @ C] :
% 0.20/0.47 ( ( const @ A @ C @ X22 )
% 0.20/0.47 != ( plus @ A @ C @ X41 @ X42 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(15)
% 0.20/0.47 thf(fact_59_trm_Odistinct_I17_J,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X22: real,X51: trm @ A @ C,X52: trm @ A @ C] :
% 0.20/0.47 ( ( const @ A @ C @ X22 )
% 0.20/0.47 != ( times @ A @ C @ X51 @ X52 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(17)
% 0.20/0.47 thf(fact_60_trm_Odistinct_I1_J,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,X1: C,X22: real] :
% 0.20/0.47 ( ( var @ C @ A @ X1 )
% 0.20/0.47 != ( const @ A @ C @ X22 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(1)
% 0.20/0.47 thf(fact_61_trm_Odistinct_I21_J,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X22: real,X7: trm @ A @ C] :
% 0.20/0.47 ( ( const @ A @ C @ X22 )
% 0.20/0.47 != ( differential @ A @ C @ X7 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(21)
% 0.20/0.47 thf(fact_62_trm_Odistinct_I19_J,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,X22: real,X6: C] :
% 0.20/0.47 ( ( const @ A @ C @ X22 )
% 0.20/0.47 != ( diffVar @ C @ A @ X6 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trm.distinct(19)
% 0.20/0.47 thf(fact_63_old_Oprod_Oinducts,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.20/0.47 ( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 0.20/0.47 => ( P @ Prod ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.inducts
% 0.20/0.47 thf(fact_64_old_Oprod_Oexhaust,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.20/0.47 ~ ! [A5: A,B4: B] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( product_Pair @ A @ B @ A5 @ B4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.exhaust
% 0.20/0.47 thf(fact_65_Pair__inject,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B] :
% 0.20/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.20/0.47 = ( product_Pair @ A @ B @ A4 @ B3 ) )
% 0.20/0.47 => ~ ( ( A2 = A4 )
% 0.20/0.47 => ( B2 != B3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Pair_inject
% 0.20/0.47 thf(fact_66_prod__cases,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.20/0.47 ( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 0.20/0.47 => ( P @ P2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases
% 0.20/0.47 thf(fact_67_surj__pair,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.20/0.47 ? [X3: A,Y3: B] :
% 0.20/0.47 ( P2
% 0.20/0.47 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % surj_pair
% 0.20/0.47 thf(fact_68_dfree_Oinducts,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,X: trm @ A @ C,P: ( trm @ A @ C ) > $o] :
% 0.20/0.47 ( ( dfree @ A @ C @ X )
% 0.20/0.47 => ( ! [I3: C] : ( P @ ( var @ C @ A @ I3 ) )
% 0.20/0.47 => ( ! [R2: real] : ( P @ ( const @ A @ C @ R2 ) )
% 0.20/0.47 => ( ! [Args2: C > ( trm @ A @ C ),I3: A] :
% 0.20/0.47 ( ! [Ia: C] : ( dfree @ A @ C @ ( Args2 @ Ia ) )
% 0.20/0.47 => ( ! [Ia: C] : ( P @ ( Args2 @ Ia ) )
% 0.20/0.47 => ( P @ ( function @ A @ C @ I3 @ Args2 ) ) ) )
% 0.20/0.47 => ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
% 0.20/0.47 ( ( dfree @ A @ C @ Theta_12 )
% 0.20/0.47 => ( ( P @ Theta_12 )
% 0.20/0.47 => ( ( dfree @ A @ C @ Theta_22 )
% 0.20/0.47 => ( ( P @ Theta_22 )
% 0.20/0.47 => ( P @ ( plus @ A @ C @ Theta_12 @ Theta_22 ) ) ) ) ) )
% 0.31/0.47 => ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
% 0.31/0.47 ( ( dfree @ A @ C @ Theta_12 )
% 0.31/0.47 => ( ( P @ Theta_12 )
% 0.31/0.47 => ( ( dfree @ A @ C @ Theta_22 )
% 0.31/0.47 => ( ( P @ Theta_22 )
% 0.31/0.47 => ( P @ ( times @ A @ C @ Theta_12 @ Theta_22 ) ) ) ) ) )
% 0.31/0.47 => ( P @ X ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dfree.inducts
% 0.31/0.47 thf(fact_69_dfree_Osimps,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType] :
% 0.31/0.47 ( ( dfree @ A @ C )
% 0.31/0.47 = ( ^ [A6: trm @ A @ C] :
% 0.31/0.47 ( ? [I5: C] :
% 0.31/0.47 ( A6
% 0.31/0.47 = ( var @ C @ A @ I5 ) )
% 0.31/0.47 | ? [R3: real] :
% 0.31/0.47 ( A6
% 0.31/0.47 = ( const @ A @ C @ R3 ) )
% 0.31/0.47 | ? [Args3: C > ( trm @ A @ C ),I5: A] :
% 0.31/0.47 ( ( A6
% 0.31/0.47 = ( function @ A @ C @ I5 @ Args3 ) )
% 0.31/0.47 & ! [X2: C] : ( dfree @ A @ C @ ( Args3 @ X2 ) ) )
% 0.31/0.47 | ? [Theta_13: trm @ A @ C,Theta_23: trm @ A @ C] :
% 0.31/0.47 ( ( A6
% 0.31/0.47 = ( plus @ A @ C @ Theta_13 @ Theta_23 ) )
% 0.31/0.47 & ( dfree @ A @ C @ Theta_13 )
% 0.31/0.47 & ( dfree @ A @ C @ Theta_23 ) )
% 0.31/0.47 | ? [Theta_13: trm @ A @ C,Theta_23: trm @ A @ C] :
% 0.31/0.47 ( ( A6
% 0.31/0.47 = ( times @ A @ C @ Theta_13 @ Theta_23 ) )
% 0.31/0.47 & ( dfree @ A @ C @ Theta_13 )
% 0.31/0.47 & ( dfree @ A @ C @ Theta_23 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dfree.simps
% 0.31/0.47 thf(fact_70_dfree_Ocases,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType,A2: trm @ A @ C] :
% 0.31/0.47 ( ( dfree @ A @ C @ A2 )
% 0.31/0.47 => ( ! [I3: C] :
% 0.31/0.47 ( A2
% 0.31/0.47 != ( var @ C @ A @ I3 ) )
% 0.31/0.47 => ( ! [R2: real] :
% 0.31/0.47 ( A2
% 0.31/0.47 != ( const @ A @ C @ R2 ) )
% 0.31/0.47 => ( ! [Args2: C > ( trm @ A @ C )] :
% 0.31/0.47 ( ? [I3: A] :
% 0.31/0.47 ( A2
% 0.31/0.47 = ( function @ A @ C @ I3 @ Args2 ) )
% 0.31/0.47 => ~ ! [I6: C] : ( dfree @ A @ C @ ( Args2 @ I6 ) ) )
% 0.31/0.47 => ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 = ( plus @ A @ C @ Theta_12 @ Theta_22 ) )
% 0.31/0.47 => ( ( dfree @ A @ C @ Theta_12 )
% 0.31/0.47 => ~ ( dfree @ A @ C @ Theta_22 ) ) )
% 0.31/0.47 => ~ ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 = ( times @ A @ C @ Theta_12 @ Theta_22 ) )
% 0.31/0.47 => ( ( dfree @ A @ C @ Theta_12 )
% 0.31/0.47 => ~ ( dfree @ A @ C @ Theta_22 ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dfree.cases
% 0.31/0.47 thf(fact_71_prod__induct7,axiom,
% 0.31/0.47 ! [G2: $tType,F2: $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 @ F2 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 0.31/0.47 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
% 0.31/0.47 => ( P @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_induct7
% 0.31/0.47 thf(fact_72_prod__induct6,axiom,
% 0.31/0.47 ! [F2: $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 @ F2 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.31/0.47 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
% 0.31/0.47 => ( P @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_induct6
% 0.31/0.47 thf(fact_73_prod__induct5,axiom,
% 0.31/0.47 ! [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 ) ) )] :
% 0.31/0.47 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( 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 ) ) ) ) )
% 0.31/0.47 => ( P @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_induct5
% 0.31/0.47 thf(fact_74_prod__induct4,axiom,
% 0.31/0.47 ! [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 ) )] :
% 0.31/0.47 ( ! [A5: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 0.31/0.47 => ( P @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_induct4
% 0.31/0.47 thf(fact_75_prod__induct3,axiom,
% 0.31/0.47 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.31/0.47 ( ! [A5: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 0.31/0.47 => ( P @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_induct3
% 0.31/0.47 thf(fact_76_prod__cases7,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 0.31/0.47 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2,G3: G2] :
% 0.31/0.47 ( Y
% 0.31/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_cases7
% 0.31/0.47 thf(fact_77_prod__cases6,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.31/0.47 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2] :
% 0.31/0.47 ( Y
% 0.31/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_cases6
% 0.31/0.47 thf(fact_78_prod__cases5,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.31/0.47 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E] :
% 0.31/0.47 ( Y
% 0.31/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_cases5
% 0.31/0.47 thf(fact_79_prod__cases4,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.31/0.47 ~ ! [A5: A,B4: B,C2: C,D2: D] :
% 0.31/0.47 ( Y
% 0.31/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_cases4
% 0.31/0.47 thf(fact_80_old_Oprod_Orec,axiom,
% 0.31/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.31/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.31/0.47 = ( F1 @ A2 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % old.prod.rec
% 0.31/0.47 thf(fact_81_f0__def,axiom,
% 0.31/0.47 ( ( f0 @ sf @ sz )
% 0.31/0.47 = ( ^ [F4: sf] : ( function @ sf @ sz @ F4 @ ( empty @ sz @ sf ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % f0_def
% 0.31/0.47 thf(fact_82_dsafe_Oinducts,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType,X: trm @ A @ C,P: ( trm @ A @ C ) > $o] :
% 0.31/0.47 ( ( dsafe @ A @ C @ X )
% 0.31/0.47 => ( ! [I3: C] : ( P @ ( var @ C @ A @ I3 ) )
% 0.31/0.47 => ( ! [R2: real] : ( P @ ( const @ A @ C @ R2 ) )
% 0.31/0.47 => ( ! [Args2: C > ( trm @ A @ C ),I3: A] :
% 0.31/0.47 ( ! [Ia: C] : ( dsafe @ A @ C @ ( Args2 @ Ia ) )
% 0.31/0.47 => ( ! [Ia: C] : ( P @ ( Args2 @ Ia ) )
% 0.31/0.47 => ( P @ ( function @ A @ C @ I3 @ Args2 ) ) ) )
% 0.31/0.47 => ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
% 0.31/0.47 ( ( dsafe @ A @ C @ Theta_12 )
% 0.31/0.47 => ( ( P @ Theta_12 )
% 0.31/0.47 => ( ( dsafe @ A @ C @ Theta_22 )
% 0.31/0.47 => ( ( P @ Theta_22 )
% 0.31/0.47 => ( P @ ( plus @ A @ C @ Theta_12 @ Theta_22 ) ) ) ) ) )
% 0.31/0.47 => ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
% 0.31/0.47 ( ( dsafe @ A @ C @ Theta_12 )
% 0.31/0.47 => ( ( P @ Theta_12 )
% 0.31/0.47 => ( ( dsafe @ A @ C @ Theta_22 )
% 0.31/0.47 => ( ( P @ Theta_22 )
% 0.31/0.47 => ( P @ ( times @ A @ C @ Theta_12 @ Theta_22 ) ) ) ) ) )
% 0.31/0.47 => ( ! [Theta: trm @ A @ C] :
% 0.31/0.47 ( ( dfree @ A @ C @ Theta )
% 0.31/0.47 => ( P @ ( differential @ A @ C @ Theta ) ) )
% 0.31/0.47 => ( ! [I3: C] : ( P @ ( diffVar @ C @ A @ I3 ) )
% 0.31/0.47 => ( P @ X ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dsafe.inducts
% 0.31/0.47 thf(fact_83_dsafe_Osimps,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType] :
% 0.31/0.47 ( ( dsafe @ A @ C )
% 0.31/0.47 = ( ^ [A6: trm @ A @ C] :
% 0.31/0.47 ( ? [I5: C] :
% 0.31/0.47 ( A6
% 0.31/0.47 = ( var @ C @ A @ I5 ) )
% 0.31/0.47 | ? [R3: real] :
% 0.31/0.47 ( A6
% 0.31/0.47 = ( const @ A @ C @ R3 ) )
% 0.31/0.47 | ? [Args3: C > ( trm @ A @ C ),I5: A] :
% 0.31/0.47 ( ( A6
% 0.31/0.47 = ( function @ A @ C @ I5 @ Args3 ) )
% 0.31/0.47 & ! [X2: C] : ( dsafe @ A @ C @ ( Args3 @ X2 ) ) )
% 0.31/0.47 | ? [Theta_13: trm @ A @ C,Theta_23: trm @ A @ C] :
% 0.31/0.47 ( ( A6
% 0.31/0.47 = ( plus @ A @ C @ Theta_13 @ Theta_23 ) )
% 0.31/0.47 & ( dsafe @ A @ C @ Theta_13 )
% 0.31/0.47 & ( dsafe @ A @ C @ Theta_23 ) )
% 0.31/0.47 | ? [Theta_13: trm @ A @ C,Theta_23: trm @ A @ C] :
% 0.31/0.47 ( ( A6
% 0.31/0.47 = ( times @ A @ C @ Theta_13 @ Theta_23 ) )
% 0.31/0.47 & ( dsafe @ A @ C @ Theta_13 )
% 0.31/0.47 & ( dsafe @ A @ C @ Theta_23 ) )
% 0.31/0.47 | ? [Theta2: trm @ A @ C] :
% 0.31/0.47 ( ( A6
% 0.31/0.47 = ( differential @ A @ C @ Theta2 ) )
% 0.31/0.47 & ( dfree @ A @ C @ Theta2 ) )
% 0.31/0.47 | ? [I5: C] :
% 0.31/0.47 ( A6
% 0.31/0.47 = ( diffVar @ C @ A @ I5 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dsafe.simps
% 0.31/0.47 thf(fact_84_dsafe_Ocases,axiom,
% 0.31/0.47 ! [A: $tType,C: $tType,A2: trm @ A @ C] :
% 0.31/0.47 ( ( dsafe @ A @ C @ A2 )
% 0.31/0.47 => ( ! [I3: C] :
% 0.31/0.47 ( A2
% 0.31/0.47 != ( var @ C @ A @ I3 ) )
% 0.31/0.47 => ( ! [R2: real] :
% 0.31/0.47 ( A2
% 0.31/0.47 != ( const @ A @ C @ R2 ) )
% 0.31/0.47 => ( ! [Args2: C > ( trm @ A @ C )] :
% 0.31/0.47 ( ? [I3: A] :
% 0.31/0.47 ( A2
% 0.31/0.47 = ( function @ A @ C @ I3 @ Args2 ) )
% 0.31/0.47 => ~ ! [I6: C] : ( dsafe @ A @ C @ ( Args2 @ I6 ) ) )
% 0.31/0.47 => ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 = ( plus @ A @ C @ Theta_12 @ Theta_22 ) )
% 0.31/0.47 => ( ( dsafe @ A @ C @ Theta_12 )
% 0.31/0.47 => ~ ( dsafe @ A @ C @ Theta_22 ) ) )
% 0.31/0.47 => ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 = ( times @ A @ C @ Theta_12 @ Theta_22 ) )
% 0.31/0.47 => ( ( dsafe @ A @ C @ Theta_12 )
% 0.31/0.47 => ~ ( dsafe @ A @ C @ Theta_22 ) ) )
% 0.31/0.47 => ( ! [Theta: trm @ A @ C] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 = ( differential @ A @ C @ Theta ) )
% 0.31/0.47 => ~ ( dfree @ A @ C @ Theta ) )
% 0.31/0.47 => ~ ! [I3: C] :
% 0.31/0.47 ( A2
% 0.31/0.47 != ( diffVar @ C @ A @ I3 ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dsafe.cases
% 0.31/0.47 thf(fact_85_euclid__ext__aux_Ocases,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( euclid1678468529ng_gcd @ A )
% 0.31/0.47 => ! [X: product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) )] :
% 0.31/0.47 ~ ! [S2: A,S3: A,T3: A,T2: A,R4: A,R2: A] :
% 0.31/0.47 ( X
% 0.31/0.47 != ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) ) @ S2 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) @ S3 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) @ T3 @ ( product_Pair @ A @ ( product_prod @ A @ A ) @ T2 @ ( product_Pair @ A @ A @ R4 @ R2 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % euclid_ext_aux.cases
% 0.31/0.47 thf(fact_86_internal__case__prod__conv,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 0.31/0.47 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.31/0.47 = ( C3 @ A2 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % internal_case_prod_conv
% 0.31/0.47 thf(fact_87_strm_Oright__unique,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType] : ( right_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ).
% 0.31/0.47
% 0.31/0.47 % strm.right_unique
% 0.31/0.47 thf(fact_88_dsafe__Const__simps,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,R: real] : ( dsafe @ A @ B @ ( const @ A @ B @ R ) ) ).
% 0.31/0.47
% 0.31/0.47 % dsafe_Const_simps
% 0.31/0.47 thf(fact_89_dsafe__Fun__simps,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,I4: A,Args: B > ( trm @ A @ B )] :
% 0.31/0.47 ( ( dsafe @ A @ B @ ( function @ A @ B @ I4 @ Args ) )
% 0.31/0.47 = ( ! [X2: B] : ( dsafe @ A @ B @ ( Args @ X2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dsafe_Fun_simps
% 0.31/0.47 thf(fact_90_dsafe__Plus__simps,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,A2: trm @ A @ B,B2: trm @ A @ B] :
% 0.31/0.47 ( ( dsafe @ A @ B @ ( plus @ A @ B @ A2 @ B2 ) )
% 0.31/0.47 = ( ( dsafe @ A @ B @ A2 )
% 0.31/0.47 & ( dsafe @ A @ B @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dsafe_Plus_simps
% 0.31/0.47 thf(fact_91_dsafe__Times__simps,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,A2: trm @ A @ B,B2: trm @ A @ B] :
% 0.31/0.47 ( ( dsafe @ A @ B @ ( times @ A @ B @ A2 @ B2 ) )
% 0.31/0.47 = ( ( dsafe @ A @ B @ A2 )
% 0.31/0.47 & ( dsafe @ A @ B @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dsafe_Times_simps
% 0.31/0.47 thf(fact_92_dsafe__Var__simps,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,X: B] : ( dsafe @ A @ B @ ( var @ B @ A @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % dsafe_Var_simps
% 0.31/0.47 thf(fact_93_dsafe__DiffVar__simps,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,X: B] : ( dsafe @ A @ B @ ( diffVar @ B @ A @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % dsafe_DiffVar_simps
% 0.31/0.47 thf(fact_94_dsafe__Diff__simps,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,A2: trm @ A @ B] :
% 0.31/0.47 ( ( dsafe @ A @ B @ ( differential @ A @ B @ A2 ) )
% 0.31/0.47 = ( dfree @ A @ B @ A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % dsafe_Diff_simps
% 0.31/0.47 thf(fact_95_dfree__is__dsafe,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,Theta3: trm @ A @ B] :
% 0.31/0.47 ( ( dfree @ A @ B @ Theta3 )
% 0.31/0.47 => ( dsafe @ A @ B @ Theta3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % dfree_is_dsafe
% 0.31/0.47 thf(fact_96_dsafe__Const,axiom,
% 0.31/0.47 ! [A: $tType,C: $tType,R: real] : ( dsafe @ A @ C @ ( const @ A @ C @ R ) ) ).
% 0.31/0.47
% 0.31/0.47 % dsafe_Const
% 0.31/0.47 thf(fact_97_dsafe__Fun,axiom,
% 0.31/0.47 ! [A: $tType,C: $tType,Args: C > ( trm @ A @ C ),I4: A] :
% 0.31/0.47 ( ! [I3: C] : ( dsafe @ A @ C @ ( Args @ I3 ) )
% 0.31/0.47 => ( dsafe @ A @ C @ ( function @ A @ C @ I4 @ Args ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dsafe_Fun
% 0.31/0.47 thf(fact_98_dsafe__Plus,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType,Theta_1: trm @ A @ C,Theta_2: trm @ A @ C] :
% 0.31/0.47 ( ( dsafe @ A @ C @ Theta_1 )
% 0.31/0.47 => ( ( dsafe @ A @ C @ Theta_2 )
% 0.31/0.47 => ( dsafe @ A @ C @ ( plus @ A @ C @ Theta_1 @ Theta_2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dsafe_Plus
% 0.31/0.47 thf(fact_99_dsafe__Times,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType,Theta_1: trm @ A @ C,Theta_2: trm @ A @ C] :
% 0.31/0.47 ( ( dsafe @ A @ C @ Theta_1 )
% 0.31/0.47 => ( ( dsafe @ A @ C @ Theta_2 )
% 0.31/0.47 => ( dsafe @ A @ C @ ( times @ A @ C @ Theta_1 @ Theta_2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dsafe_Times
% 0.31/0.47 thf(fact_100_dsafe__Var,axiom,
% 0.31/0.47 ! [A: $tType,C: $tType,I4: C] : ( dsafe @ A @ C @ ( var @ C @ A @ I4 ) ) ).
% 0.31/0.47
% 0.31/0.47 % dsafe_Var
% 0.31/0.47 thf(fact_101_dsafe__DiffVar,axiom,
% 0.31/0.47 ! [A: $tType,C: $tType,I4: C] : ( dsafe @ A @ C @ ( diffVar @ C @ A @ I4 ) ) ).
% 0.31/0.47
% 0.31/0.47 % dsafe_DiffVar
% 0.31/0.47 thf(fact_102_dsafe__Diff,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType,Theta3: trm @ A @ C] :
% 0.31/0.47 ( ( dfree @ A @ C @ Theta3 )
% 0.31/0.47 => ( dsafe @ A @ C @ ( differential @ A @ C @ Theta3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dsafe_Diff
% 0.31/0.47 thf(fact_103_gcd_Ocases,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( euclid1155270486miring @ A )
% 0.31/0.47 => ! [X: product_prod @ A @ A] :
% 0.31/0.47 ~ ! [A5: A,B4: A] :
% 0.31/0.47 ( X
% 0.31/0.47 != ( product_Pair @ A @ A @ A5 @ B4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % gcd.cases
% 0.31/0.47 thf(fact_104_local_Oempty__def,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( empty @ B @ A )
% 0.31/0.47 = ( ^ [I5: B] : ( const @ A @ B @ ( zero_zero @ real ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % local.empty_def
% 0.31/0.47 thf(fact_105_ids_Of0__def,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,F: Sf] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ( f0 @ Sf @ Sz @ F )
% 0.31/0.47 = ( function @ Sf @ Sz @ F @ ( empty @ Sz @ Sf ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.f0_def
% 0.31/0.47 thf(fact_106_dfree__Fun_OIH,axiom,
% 0.31/0.47 ! [I: c] :
% 0.31/0.47 ( ( denotational_Vagree @ c @ nu @ nu2 @ ( static_FVDiff @ a @ c @ ( args @ I ) ) )
% 0.31/0.47 => ( ( denotational_frechet @ a @ b @ c @ i @ ( args @ I ) @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu ) )
% 0.31/0.47 = ( denotational_frechet @ a @ b @ c @ i @ ( args @ I ) @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu2 ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dfree_Fun.IH
% 0.31/0.47 thf(fact_107_strm_Oleft__unique,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType] : ( left_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ).
% 0.31/0.47
% 0.31/0.47 % strm.left_unique
% 0.31/0.47 thf(fact_108_trm_Oinduct,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType,P: ( trm @ A @ C ) > $o,Trm: trm @ A @ C] :
% 0.31/0.47 ( ! [X3: C] : ( P @ ( var @ C @ A @ X3 ) )
% 0.31/0.47 => ( ! [X3: real] : ( P @ ( const @ A @ C @ X3 ) )
% 0.31/0.47 => ( ! [X1a: A,X2a: C > ( trm @ A @ C )] :
% 0.31/0.47 ( ! [X2aa: trm @ A @ C] :
% 0.31/0.47 ( ( member @ ( trm @ A @ C ) @ X2aa @ ( image @ C @ ( trm @ A @ C ) @ X2a @ ( top_top @ ( set @ C ) ) ) )
% 0.31/0.47 => ( P @ X2aa ) )
% 0.31/0.47 => ( P @ ( function @ A @ C @ X1a @ X2a ) ) )
% 0.31/0.47 => ( ! [X1a: trm @ A @ C,X2a: trm @ A @ C] :
% 0.31/0.47 ( ( P @ X1a )
% 0.31/0.47 => ( ( P @ X2a )
% 0.31/0.47 => ( P @ ( plus @ A @ C @ X1a @ X2a ) ) ) )
% 0.31/0.47 => ( ! [X1a: trm @ A @ C,X2a: trm @ A @ C] :
% 0.31/0.47 ( ( P @ X1a )
% 0.31/0.47 => ( ( P @ X2a )
% 0.31/0.47 => ( P @ ( times @ A @ C @ X1a @ X2a ) ) ) )
% 0.31/0.47 => ( ! [X3: C] : ( P @ ( diffVar @ C @ A @ X3 ) )
% 0.31/0.47 => ( ! [X3: trm @ A @ C] :
% 0.31/0.47 ( ( P @ X3 )
% 0.31/0.47 => ( P @ ( differential @ A @ C @ X3 ) ) )
% 0.31/0.47 => ( P @ Trm ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % trm.induct
% 0.31/0.47 thf(fact_109_agree__comm,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( finite_finite @ A )
% 0.31/0.47 => ! [A3: 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 ),V: set @ ( sum_sum @ A @ A )] :
% 0.31/0.47 ( ( denotational_Vagree @ A @ A3 @ B5 @ V )
% 0.31/0.47 => ( denotational_Vagree @ A @ B5 @ A3 @ V ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % agree_comm
% 0.31/0.47 thf(fact_110_prod_Ocollapse,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.31/0.47 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.31/0.47 = Prod ) ).
% 0.31/0.47
% 0.31/0.47 % prod.collapse
% 0.31/0.47 thf(fact_111_prod_Oexhaust__sel,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.31/0.47 ( Prod
% 0.31/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod.exhaust_sel
% 0.31/0.47 thf(fact_112_surjective__pairing,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,T4: product_prod @ A @ B] :
% 0.31/0.47 ( T4
% 0.31/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T4 ) @ ( product_snd @ A @ B @ T4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % surjective_pairing
% 0.31/0.47 thf(fact_113_agree__UNIV__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( finite_finite @ A )
% 0.31/0.47 => ! [Nu: 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 )] :
% 0.31/0.47 ( ( denotational_Vagree @ A @ Nu @ Omega @ ( top_top @ ( set @ ( sum_sum @ A @ A ) ) ) )
% 0.31/0.47 => ( Nu = Omega ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % agree_UNIV_eq
% 0.31/0.47 thf(fact_114_Vagree__univ,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( finite_finite @ A )
% 0.31/0.47 => ! [A2: finite_Cartesian_vec @ real @ A,B2: finite_Cartesian_vec @ real @ A,C3: finite_Cartesian_vec @ real @ A,D3: finite_Cartesian_vec @ real @ A] :
% 0.31/0.47 ( ( denotational_Vagree @ A @ ( product_Pair @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ A2 @ B2 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ C3 @ D3 ) @ ( top_top @ ( set @ ( sum_sum @ A @ A ) ) ) )
% 0.31/0.47 => ( ( A2 = C3 )
% 0.31/0.47 & ( B2 = D3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Vagree_univ
% 0.31/0.47 thf(fact_115_ids_Oseq__sem_Ocases,axiom,
% 0.31/0.47 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc )
% 0.31/0.47 & ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: product_prod @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) )] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ~ ! [I2: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,S: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
% 0.31/0.47 ( X
% 0.31/0.47 != ( product_Pair @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) @ I2 @ S ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.seq_sem.cases
% 0.31/0.47 thf(fact_116_ids_Osingleton_Oinduct,axiom,
% 0.31/0.47 ! [Sf: $tType,Sc: $tType,A: $tType,Sz: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sc )
% 0.31/0.47 & ( finite_finite @ Sf ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( trm @ A @ Sz ) > Sz > $o,A0: trm @ A @ Sz,A1: Sz] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ! [T2: trm @ A @ Sz,X_1: Sz] : ( P @ T2 @ X_1 )
% 0.31/0.47 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.singleton.induct
% 0.31/0.47 thf(fact_117_ids_Oseq__sem_Oinduct,axiom,
% 0.31/0.47 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc )
% 0.31/0.47 & ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) > ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) > $o,A0: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,A1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ! [I2: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,X_1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] : ( P @ I2 @ X_1 )
% 0.31/0.47 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.seq_sem.induct
% 0.31/0.47 thf(fact_118_frechet_Osimps_I5_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,C: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ C )
% 0.31/0.47 & ( finite_finite @ A )
% 0.31/0.47 & ( finite_finite @ B ) )
% 0.31/0.47 => ! [I7: denota1663640101rp_ext @ A @ B @ C @ product_unit,R: real,V2: finite_Cartesian_vec @ real @ C] :
% 0.31/0.47 ( ( denotational_frechet @ A @ B @ C @ I7 @ ( const @ A @ C @ R ) @ V2 )
% 0.31/0.47 = ( ^ [V3: finite_Cartesian_vec @ real @ C] : ( zero_zero @ real ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % frechet.simps(5)
% 0.31/0.47 thf(fact_119_prod__eqI,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 0.31/0.47 ( ( ( product_fst @ A @ B @ P2 )
% 0.31/0.47 = ( product_fst @ A @ B @ Q2 ) )
% 0.31/0.47 => ( ( ( product_snd @ A @ B @ P2 )
% 0.31/0.47 = ( product_snd @ A @ B @ Q2 ) )
% 0.31/0.47 => ( P2 = Q2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_eqI
% 0.31/0.47 thf(fact_120_prod_Oexpand,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 0.31/0.47 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.31/0.47 = ( product_fst @ A @ B @ Prod2 ) )
% 0.31/0.47 & ( ( product_snd @ A @ B @ Prod )
% 0.31/0.47 = ( product_snd @ A @ B @ Prod2 ) ) )
% 0.31/0.47 => ( Prod = Prod2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod.expand
% 0.31/0.47 thf(fact_121_prod__eq__iff,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ^ [Y4: product_prod @ A @ B,Z: product_prod @ A @ B] : ( Y4 = Z ) )
% 0.31/0.47 = ( ^ [S4: product_prod @ A @ B,T5: product_prod @ A @ B] :
% 0.31/0.47 ( ( ( product_fst @ A @ B @ S4 )
% 0.31/0.47 = ( product_fst @ A @ B @ T5 ) )
% 0.31/0.47 & ( ( product_snd @ A @ B @ S4 )
% 0.31/0.47 = ( product_snd @ A @ B @ T5 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_eq_iff
% 0.31/0.47 thf(fact_122_snd__conv,axiom,
% 0.31/0.47 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 0.31/0.47 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 0.31/0.47 = X22 ) ).
% 0.31/0.47
% 0.31/0.47 % snd_conv
% 0.31/0.47 thf(fact_123_snd__eqD,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
% 0.31/0.47 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.31/0.47 = A2 )
% 0.31/0.47 => ( Y = A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % snd_eqD
% 0.31/0.47 thf(fact_124_fst__conv,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 0.31/0.47 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.31/0.47 = X1 ) ).
% 0.31/0.47
% 0.31/0.47 % fst_conv
% 0.31/0.47 thf(fact_125_fst__eqD,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
% 0.31/0.47 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 0.31/0.47 = A2 )
% 0.31/0.47 => ( X = A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % fst_eqD
% 0.31/0.47 thf(fact_126_ids_Oempty__def,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,Sz: $tType,A: $tType,B: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ( empty @ B @ A )
% 0.31/0.47 = ( ^ [I5: B] : ( const @ A @ B @ ( zero_zero @ real ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.empty_def
% 0.31/0.47 thf(fact_127_ids_Osingleton_Ocases,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: product_prod @ ( trm @ A @ Sz ) @ Sz] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ~ ! [T2: trm @ A @ Sz,I3: Sz] :
% 0.31/0.47 ( X
% 0.31/0.47 != ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ T2 @ I3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.singleton.cases
% 0.31/0.47 thf(fact_128_ids_OP__def,axiom,
% 0.31/0.47 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sc )
% 0.31/0.47 & ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P2: Sc] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ( p @ Sc @ Sf @ Sz @ P2 )
% 0.31/0.47 = ( predicational @ Sc @ Sf @ Sz @ P2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.P_def
% 0.31/0.47 thf(fact_129_FunctionFrechet_Ocases,axiom,
% 0.31/0.47 ! [B: $tType,C: $tType,A: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A )
% 0.31/0.47 & ( finite_finite @ C )
% 0.31/0.47 & ( finite_finite @ B ) )
% 0.31/0.47 => ! [X: product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A] :
% 0.31/0.47 ~ ! [I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,I3: A] :
% 0.31/0.47 ( X
% 0.31/0.47 != ( product_Pair @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A @ I2 @ I3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % FunctionFrechet.cases
% 0.31/0.47 thf(fact_130_agree__refl,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( finite_finite @ A )
% 0.31/0.47 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A3: set @ ( sum_sum @ A @ A )] : ( denotational_Vagree @ A @ Nu @ Nu @ A3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % agree_refl
% 0.31/0.47 thf(fact_131_ids_Ostrm_Odomain,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ( domainp @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) )
% 0.31/0.47 = ( dfree @ A @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.strm.domain
% 0.31/0.47 thf(fact_132_snd__top,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( top @ A )
% 0.31/0.47 & ( top @ B ) )
% 0.31/0.47 => ( ( product_snd @ B @ A @ ( top_top @ ( product_prod @ B @ A ) ) )
% 0.31/0.47 = ( top_top @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % snd_top
% 0.31/0.47 thf(fact_133_fst__top,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( top @ A )
% 0.31/0.47 & ( top @ B ) )
% 0.31/0.47 => ( ( product_fst @ A @ B @ ( top_top @ ( product_prod @ A @ B ) ) )
% 0.31/0.47 = ( top_top @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % fst_top
% 0.31/0.47 thf(fact_134_snd__zero,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( zero @ A )
% 0.31/0.47 & ( zero @ B ) )
% 0.31/0.47 => ( ( product_snd @ B @ A @ ( zero_zero @ ( product_prod @ B @ A ) ) )
% 0.31/0.47 = ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % snd_zero
% 0.31/0.47 thf(fact_135_fst__zero,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( zero @ A )
% 0.31/0.47 & ( zero @ B ) )
% 0.31/0.47 => ( ( product_fst @ A @ B @ ( zero_zero @ ( product_prod @ A @ B ) ) )
% 0.31/0.47 = ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % fst_zero
% 0.31/0.47 thf(fact_136_zero__prod__def,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( ( zero @ B )
% 0.31/0.47 & ( zero @ A ) )
% 0.31/0.47 => ( ( zero_zero @ ( product_prod @ A @ B ) )
% 0.31/0.47 = ( product_Pair @ A @ B @ ( zero_zero @ A ) @ ( zero_zero @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_prod_def
% 0.31/0.47 thf(fact_137_top__prod__def,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( ( top @ B )
% 0.31/0.47 & ( top @ A ) )
% 0.31/0.47 => ( ( top_top @ ( product_prod @ A @ B ) )
% 0.31/0.47 = ( product_Pair @ A @ B @ ( top_top @ A ) @ ( top_top @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % top_prod_def
% 0.31/0.47 thf(fact_138_ids_Oraw__interp__inject,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A )
% 0.31/0.47 & ( finite_finite @ B )
% 0.31/0.47 & ( finite_finite @ C )
% 0.31/0.47 & ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_good_interp @ A @ B @ C,Y: frechet_good_interp @ A @ B @ C] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ( ( freche229654227interp @ A @ B @ C @ X )
% 0.31/0.47 = ( freche229654227interp @ A @ B @ C @ Y ) )
% 0.31/0.47 = ( X = Y ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.raw_interp_inject
% 0.31/0.47 thf(fact_139_ids_Oraw__term__inject,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_strm @ A @ C,Y: frechet_strm @ A @ C] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ( ( frechet_raw_term @ A @ C @ X )
% 0.31/0.47 = ( frechet_raw_term @ A @ C @ Y ) )
% 0.31/0.47 = ( X = Y ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.raw_term_inject
% 0.31/0.47 thf(fact_140_ids_Oraw__interp__inverse,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A )
% 0.31/0.47 & ( finite_finite @ B )
% 0.31/0.47 & ( finite_finite @ C )
% 0.31/0.47 & ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_good_interp @ A @ B @ C] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ( freche227871258interp @ A @ B @ C @ ( freche229654227interp @ A @ B @ C @ X ) )
% 0.31/0.47 = X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.raw_interp_inverse
% 0.31/0.47 thf(fact_141_ids_Oraw__term__inverse,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_strm @ A @ C] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ( frechet_simple_term @ A @ C @ ( frechet_raw_term @ A @ C @ X ) )
% 0.31/0.47 = X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.raw_term_inverse
% 0.31/0.47 thf(fact_142_ids_Ostrm_Oright__unique,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( right_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.strm.right_unique
% 0.31/0.47 thf(fact_143_ids_Ostrm_Oleft__unique,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( left_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.strm.left_unique
% 0.31/0.47 thf(fact_144_directional__derivative__def,axiom,
% 0.31/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A )
% 0.31/0.47 & ( finite_finite @ B )
% 0.31/0.47 & ( finite_finite @ C ) )
% 0.31/0.47 => ( ( denota2078997598vative @ A @ B @ C )
% 0.31/0.47 = ( ^ [I8: denota1663640101rp_ext @ A @ B @ C @ product_unit,T5: trm @ A @ C,V4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] : ( denotational_frechet @ A @ B @ C @ I8 @ T5 @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V4 ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % directional_derivative_def
% 0.31/0.47 thf(fact_145_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
% 0.31/0.47 ( ( P @ X @ Y )
% 0.31/0.47 => ( ( A2
% 0.31/0.47 = ( product_Pair @ A @ B @ X @ Y ) )
% 0.31/0.47 => ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % BNF_Greatest_Fixpoint.subst_Pair
% 0.31/0.47 thf(fact_146_conjI__realizer,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
% 0.31/0.47 ( ( P @ P2 )
% 0.31/0.47 => ( ( Q @ Q2 )
% 0.31/0.47 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
% 0.31/0.47 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % conjI_realizer
% 0.31/0.47 thf(fact_147_exI__realizer,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,P: A > B > $o,Y: A,X: B] :
% 0.31/0.47 ( ( P @ Y @ X )
% 0.31/0.47 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % exI_realizer
% 0.31/0.47 thf(fact_148_exE__realizer_H,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A] :
% 0.31/0.47 ( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
% 0.31/0.47 => ~ ! [X3: B,Y3: A] :
% 0.31/0.47 ~ ( P @ Y3 @ X3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % exE_realizer'
% 0.31/0.47 thf(fact_149_iso__tuple__UNIV__I,axiom,
% 0.31/0.47 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % iso_tuple_UNIV_I
% 0.31/0.47 thf(fact_150_UNIV__I,axiom,
% 0.31/0.47 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % UNIV_I
% 0.31/0.47 thf(fact_151_top__apply,axiom,
% 0.31/0.47 ! [C: $tType,D: $tType] :
% 0.31/0.47 ( ( top @ C )
% 0.31/0.47 => ( ( top_top @ ( D > C ) )
% 0.31/0.47 = ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % top_apply
% 0.31/0.47 thf(fact_152_image__eqI,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,B2: A,F: B > A,X: B,A3: set @ B] :
% 0.31/0.47 ( ( B2
% 0.31/0.47 = ( F @ X ) )
% 0.31/0.47 => ( ( member @ B @ X @ A3 )
% 0.31/0.47 => ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % image_eqI
% 0.31/0.47 thf(fact_153_imageI,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,X: A,A3: set @ A,F: A > B] :
% 0.31/0.47 ( ( member @ A @ X @ A3 )
% 0.31/0.47 => ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % imageI
% 0.31/0.47 thf(fact_154_image__iff,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,Z2: A,F: B > A,A3: set @ B] :
% 0.31/0.47 ( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A3 ) )
% 0.31/0.47 = ( ? [X2: B] :
% 0.31/0.47 ( ( member @ B @ X2 @ A3 )
% 0.31/0.47 & ( Z2
% 0.31/0.47 = ( F @ X2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % image_iff
% 0.31/0.47 thf(fact_155_bex__imageD,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
% 0.31/0.47 ( ? [X4: A] :
% 0.31/0.47 ( ( member @ A @ X4 @ ( image @ B @ A @ F @ A3 ) )
% 0.31/0.47 & ( P @ X4 ) )
% 0.31/0.47 => ? [X3: B] :
% 0.31/0.47 ( ( member @ B @ X3 @ A3 )
% 0.31/0.47 & ( P @ ( F @ X3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % bex_imageD
% 0.31/0.47 thf(fact_156_image__cong,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G: A > B] :
% 0.31/0.47 ( ( M = N )
% 0.31/0.47 => ( ! [X3: A] :
% 0.31/0.47 ( ( member @ A @ X3 @ N )
% 0.31/0.47 => ( ( F @ X3 )
% 0.31/0.47 = ( G @ X3 ) ) )
% 0.31/0.47 => ( ( image @ A @ B @ F @ M )
% 0.31/0.47 = ( image @ A @ B @ G @ N ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % image_cong
% 0.31/0.47 thf(fact_157_ball__imageD,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
% 0.31/0.47 ( ! [X3: A] :
% 0.31/0.47 ( ( member @ A @ X3 @ ( image @ B @ A @ F @ A3 ) )
% 0.31/0.47 => ( P @ X3 ) )
% 0.31/0.47 => ! [X4: B] :
% 0.31/0.47 ( ( member @ B @ X4 @ A3 )
% 0.31/0.47 => ( P @ ( F @ X4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ball_imageD
% 0.31/0.47 thf(fact_158_rev__image__eqI,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,X: A,A3: set @ A,B2: B,F: A > B] :
% 0.31/0.47 ( ( member @ A @ X @ A3 )
% 0.31/0.47 => ( ( B2
% 0.31/0.47 = ( F @ X ) )
% 0.31/0.47 => ( member @ B @ B2 @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % rev_image_eqI
% 0.31/0.47 thf(fact_159_top__set__def,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( top_top @ ( set @ A ) )
% 0.31/0.47 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % top_set_def
% 0.31/0.47 thf(fact_160_UNIV__eq__I,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A] :
% 0.31/0.47 ( ! [X3: A] : ( member @ A @ X3 @ A3 )
% 0.31/0.47 => ( ( top_top @ ( set @ A ) )
% 0.31/0.47 = A3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % UNIV_eq_I
% 0.31/0.47 thf(fact_161_UNIV__witness,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % UNIV_witness
% 0.31/0.47 thf(fact_162_rangeI,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % rangeI
% 0.31/0.47 thf(fact_163_range__eqI,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,B2: A,F: B > A,X: B] :
% 0.31/0.47 ( ( B2
% 0.31/0.47 = ( F @ X ) )
% 0.31/0.47 => ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % range_eqI
% 0.31/0.47 thf(fact_164_sndI,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
% 0.31/0.47 ( ( X
% 0.31/0.47 = ( product_Pair @ A @ B @ Y @ Z2 ) )
% 0.31/0.47 => ( ( product_snd @ A @ B @ X )
% 0.31/0.47 = Z2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sndI
% 0.31/0.47 thf(fact_165_eq__snd__iff,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
% 0.31/0.47 ( ( B2
% 0.31/0.47 = ( product_snd @ B @ A @ P2 ) )
% 0.31/0.47 = ( ? [A6: B] :
% 0.31/0.47 ( P2
% 0.31/0.47 = ( product_Pair @ B @ A @ A6 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % eq_snd_iff
% 0.31/0.47 thf(fact_166_top1I,axiom,
% 0.31/0.47 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 0.31/0.47
% 0.31/0.47 % top1I
% 0.31/0.47 thf(fact_167_zero__reorient,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( zero @ A )
% 0.31/0.47 => ! [X: A] :
% 0.31/0.47 ( ( ( zero_zero @ A )
% 0.31/0.47 = X )
% 0.31/0.47 = ( X
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_reorient
% 0.31/0.47 thf(fact_168_eq__fst__iff,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 = ( product_fst @ A @ B @ P2 ) )
% 0.31/0.47 = ( ? [B6: B] :
% 0.31/0.47 ( P2
% 0.31/0.47 = ( product_Pair @ A @ B @ A2 @ B6 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % eq_fst_iff
% 0.31/0.47 thf(fact_169_fstI,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
% 0.31/0.47 ( ( X
% 0.31/0.47 = ( product_Pair @ A @ B @ Y @ Z2 ) )
% 0.31/0.47 => ( ( product_fst @ A @ B @ X )
% 0.31/0.47 = Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % fstI
% 0.31/0.47 thf(fact_170_range__snd,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.31/0.47 = ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % range_snd
% 0.31/0.47 thf(fact_171_range__fst,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.31/0.47 = ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % range_fst
% 0.31/0.47 thf(fact_172_Domain__eq__top,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( domainp @ A @ A
% 0.31/0.47 @ ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.31/0.47 = ( top_top @ ( A > $o ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Domain_eq_top
% 0.31/0.47 thf(fact_173_DomainpE,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,R: A > B > $o,A2: A] :
% 0.31/0.47 ( ( domainp @ A @ B @ R @ A2 )
% 0.31/0.47 => ~ ! [B4: B] :
% 0.31/0.47 ~ ( R @ A2 @ B4 ) ) ).
% 0.31/0.47
% 0.31/0.47 % DomainpE
% 0.31/0.47 thf(fact_174_Domainp__iff,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( domainp @ A @ B )
% 0.31/0.47 = ( ^ [T6: A > B > $o,X2: A] :
% 0.31/0.47 ? [X5: B] : ( T6 @ X2 @ X5 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Domainp_iff
% 0.31/0.47 thf(fact_175_Domainp__refl,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( domainp @ A @ B )
% 0.31/0.47 = ( domainp @ A @ B ) ) ).
% 0.31/0.47
% 0.31/0.47 % Domainp_refl
% 0.31/0.47 thf(fact_176_Domainp_Ocases,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,R: A > B > $o,A2: A] :
% 0.31/0.47 ( ( domainp @ A @ B @ R @ A2 )
% 0.31/0.47 => ~ ! [B4: B] :
% 0.31/0.47 ~ ( R @ A2 @ B4 ) ) ).
% 0.31/0.47
% 0.31/0.47 % Domainp.cases
% 0.31/0.47 thf(fact_177_Domainp_Osimps,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( domainp @ A @ B )
% 0.31/0.47 = ( ^ [R3: A > B > $o,A6: A] :
% 0.31/0.47 ? [B6: A,C4: B] :
% 0.31/0.47 ( ( A6 = B6 )
% 0.31/0.47 & ( R3 @ B6 @ C4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Domainp.simps
% 0.31/0.47 thf(fact_178_Domainp_ODomainI,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R: A > B > $o,A2: A,B2: B] :
% 0.31/0.47 ( ( R @ A2 @ B2 )
% 0.31/0.47 => ( domainp @ A @ B @ R @ A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % Domainp.DomainI
% 0.31/0.47 thf(fact_179_Domainp_Oinducts,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R: A > B > $o,X: A,P: A > $o] :
% 0.31/0.47 ( ( domainp @ A @ B @ R @ X )
% 0.31/0.47 => ( ! [A5: A,B4: B] :
% 0.31/0.47 ( ( R @ A5 @ B4 )
% 0.31/0.47 => ( P @ A5 ) )
% 0.31/0.47 => ( P @ X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Domainp.inducts
% 0.31/0.47 thf(fact_180_right__uniqueD,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,A3: A > B > $o,X: A,Y: B,Z2: B] :
% 0.31/0.47 ( ( right_unique @ A @ B @ A3 )
% 0.31/0.47 => ( ( A3 @ X @ Y )
% 0.31/0.47 => ( ( A3 @ X @ Z2 )
% 0.31/0.47 => ( Y = Z2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % right_uniqueD
% 0.31/0.47 thf(fact_181_right__uniqueI,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,A3: A > B > $o] :
% 0.31/0.47 ( ! [X3: A,Y3: B,Z3: B] :
% 0.31/0.47 ( ( A3 @ X3 @ Y3 )
% 0.31/0.47 => ( ( A3 @ X3 @ Z3 )
% 0.31/0.47 => ( Y3 = Z3 ) ) )
% 0.31/0.47 => ( right_unique @ A @ B @ A3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % right_uniqueI
% 0.31/0.47 thf(fact_182_right__unique__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( right_unique @ A @ A
% 0.31/0.47 @ ^ [Y4: A,Z: A] : ( Y4 = Z ) ) ).
% 0.31/0.47
% 0.31/0.47 % right_unique_eq
% 0.31/0.47 thf(fact_183_right__unique__def,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( right_unique @ A @ B )
% 0.31/0.47 = ( ^ [R5: A > B > $o] :
% 0.31/0.47 ! [X2: A,Y5: B,Z4: B] :
% 0.31/0.47 ( ( R5 @ X2 @ Y5 )
% 0.31/0.47 => ( ( R5 @ X2 @ Z4 )
% 0.31/0.47 => ( Y5 = Z4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % right_unique_def
% 0.31/0.47 thf(fact_184_left__uniqueD,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,A3: A > B > $o,X: A,Z2: B,Y: A] :
% 0.31/0.47 ( ( left_unique @ A @ B @ A3 )
% 0.31/0.47 => ( ( A3 @ X @ Z2 )
% 0.31/0.47 => ( ( A3 @ Y @ Z2 )
% 0.31/0.47 => ( X = Y ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % left_uniqueD
% 0.31/0.47 thf(fact_185_left__uniqueI,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,A3: A > B > $o] :
% 0.31/0.47 ( ! [X3: A,Y3: A,Z3: B] :
% 0.31/0.47 ( ( A3 @ X3 @ Z3 )
% 0.31/0.47 => ( ( A3 @ Y3 @ Z3 )
% 0.31/0.47 => ( X3 = Y3 ) ) )
% 0.31/0.47 => ( left_unique @ A @ B @ A3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % left_uniqueI
% 0.31/0.47 thf(fact_186_left__unique__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( left_unique @ A @ A
% 0.31/0.47 @ ^ [Y4: A,Z: A] : ( Y4 = Z ) ) ).
% 0.31/0.47
% 0.31/0.47 % left_unique_eq
% 0.31/0.47 thf(fact_187_left__unique__def,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( left_unique @ A @ B )
% 0.31/0.47 = ( ^ [R5: A > B > $o] :
% 0.31/0.47 ! [X2: A,Y5: A,Z4: B] :
% 0.31/0.47 ( ( R5 @ X2 @ Z4 )
% 0.31/0.47 => ( ( R5 @ Y5 @ Z4 )
% 0.31/0.47 => ( X2 = Y5 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % left_unique_def
% 0.31/0.47 thf(fact_188_top__empty__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( top_top @ ( A > $o ) )
% 0.31/0.47 = ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % top_empty_eq
% 0.31/0.47 thf(fact_189_trm_Oset__cases,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType,E3: A,A2: trm @ A @ C] :
% 0.31/0.47 ( ( member @ A @ E3 @ ( set_trm @ A @ C @ A2 ) )
% 0.31/0.47 => ( ! [Z22: C > ( trm @ A @ C )] :
% 0.31/0.47 ( A2
% 0.31/0.47 != ( function @ A @ C @ E3 @ Z22 ) )
% 0.31/0.47 => ( ! [Z1: A,Z22: C > ( trm @ A @ C )] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 = ( function @ A @ C @ Z1 @ Z22 ) )
% 0.31/0.47 => ! [X3: trm @ A @ C] :
% 0.31/0.47 ( ( member @ ( trm @ A @ C ) @ X3 @ ( image @ C @ ( trm @ A @ C ) @ Z22 @ ( top_top @ ( set @ C ) ) ) )
% 0.31/0.47 => ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ X3 ) ) ) )
% 0.31/0.47 => ( ! [Z1: trm @ A @ C] :
% 0.31/0.47 ( ? [Z22: trm @ A @ C] :
% 0.31/0.47 ( A2
% 0.31/0.47 = ( plus @ A @ C @ Z1 @ Z22 ) )
% 0.31/0.47 => ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ Z1 ) ) )
% 0.31/0.47 => ( ! [Z1: trm @ A @ C,Z22: trm @ A @ C] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 = ( plus @ A @ C @ Z1 @ Z22 ) )
% 0.31/0.47 => ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ Z22 ) ) )
% 0.31/0.47 => ( ! [Z1: trm @ A @ C] :
% 0.31/0.47 ( ? [Z22: trm @ A @ C] :
% 0.31/0.47 ( A2
% 0.31/0.47 = ( times @ A @ C @ Z1 @ Z22 ) )
% 0.31/0.47 => ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ Z1 ) ) )
% 0.31/0.47 => ( ! [Z1: trm @ A @ C,Z22: trm @ A @ C] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 = ( times @ A @ C @ Z1 @ Z22 ) )
% 0.31/0.47 => ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ Z22 ) ) )
% 0.31/0.47 => ~ ! [Z3: trm @ A @ C] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 = ( differential @ A @ C @ Z3 ) )
% 0.31/0.47 => ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ Z3 ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % trm.set_cases
% 0.31/0.47 thf(fact_190_surjD,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 0.31/0.47 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.31/0.47 = ( top_top @ ( set @ A ) ) )
% 0.31/0.47 => ? [X3: B] :
% 0.31/0.47 ( Y
% 0.31/0.47 = ( F @ X3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % surjD
% 0.31/0.47 thf(fact_191_surjE,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 0.31/0.47 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.31/0.47 = ( top_top @ ( set @ A ) ) )
% 0.31/0.47 => ~ ! [X3: B] :
% 0.31/0.47 ( Y
% 0.31/0.47 != ( F @ X3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % surjE
% 0.31/0.47 thf(fact_192_trm_Oset__intros_I1_J,axiom,
% 0.31/0.47 ! [A: $tType,C: $tType,X31: A,X32: C > ( trm @ A @ C )] : ( member @ A @ X31 @ ( set_trm @ A @ C @ ( function @ A @ C @ X31 @ X32 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % trm.set_intros(1)
% 0.31/0.47 thf(fact_193_trm_Oset__intros_I4_J,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType,Yc: A,X42: trm @ A @ C,X41: trm @ A @ C] :
% 0.31/0.47 ( ( member @ A @ Yc @ ( set_trm @ A @ C @ X42 ) )
% 0.31/0.47 => ( member @ A @ Yc @ ( set_trm @ A @ C @ ( plus @ A @ C @ X41 @ X42 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % trm.set_intros(4)
% 0.31/0.47 thf(fact_194_trm_Oset__intros_I3_J,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType,Yb: A,X41: trm @ A @ C,X42: trm @ A @ C] :
% 0.31/0.47 ( ( member @ A @ Yb @ ( set_trm @ A @ C @ X41 ) )
% 0.31/0.47 => ( member @ A @ Yb @ ( set_trm @ A @ C @ ( plus @ A @ C @ X41 @ X42 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % trm.set_intros(3)
% 0.31/0.47 thf(fact_195_trm_Oset__intros_I6_J,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType,Ye: A,X52: trm @ A @ C,X51: trm @ A @ C] :
% 0.31/0.47 ( ( member @ A @ Ye @ ( set_trm @ A @ C @ X52 ) )
% 0.31/0.47 => ( member @ A @ Ye @ ( set_trm @ A @ C @ ( times @ A @ C @ X51 @ X52 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % trm.set_intros(6)
% 0.31/0.47 thf(fact_196_trm_Oset__intros_I5_J,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType,Yd: A,X51: trm @ A @ C,X52: trm @ A @ C] :
% 0.31/0.47 ( ( member @ A @ Yd @ ( set_trm @ A @ C @ X51 ) )
% 0.31/0.47 => ( member @ A @ Yd @ ( set_trm @ A @ C @ ( times @ A @ C @ X51 @ X52 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % trm.set_intros(5)
% 0.31/0.47 thf(fact_197_trm_Osimps_I126_J,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType,X7: trm @ A @ C] :
% 0.31/0.47 ( ( set_trm @ A @ C @ ( differential @ A @ C @ X7 ) )
% 0.31/0.47 = ( set_trm @ A @ C @ X7 ) ) ).
% 0.31/0.47
% 0.31/0.47 % trm.simps(126)
% 0.31/0.47 thf(fact_198_trm_Oset__intros_I7_J,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType,Yf: A,X7: trm @ A @ C] :
% 0.31/0.47 ( ( member @ A @ Yf @ ( set_trm @ A @ C @ X7 ) )
% 0.31/0.47 => ( member @ A @ Yf @ ( set_trm @ A @ C @ ( differential @ A @ C @ X7 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % trm.set_intros(7)
% 0.31/0.47 thf(fact_199_trm_Oset__intros_I2_J,axiom,
% 0.31/0.47 ! [A: $tType,C: $tType,Y: trm @ A @ C,X32: C > ( trm @ A @ C ),Ya: A,X31: A] :
% 0.31/0.47 ( ( member @ ( trm @ A @ C ) @ Y @ ( image @ C @ ( trm @ A @ C ) @ X32 @ ( top_top @ ( set @ C ) ) ) )
% 0.31/0.47 => ( ( member @ A @ Ya @ ( set_trm @ A @ C @ Y ) )
% 0.31/0.47 => ( member @ A @ Ya @ ( set_trm @ A @ C @ ( function @ A @ C @ X31 @ X32 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % trm.set_intros(2)
% 0.31/0.47 thf(fact_200_surj__def,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,F: B > A] :
% 0.31/0.47 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.31/0.47 = ( top_top @ ( set @ A ) ) )
% 0.31/0.47 = ( ! [Y5: A] :
% 0.31/0.47 ? [X2: B] :
% 0.31/0.47 ( Y5
% 0.31/0.47 = ( F @ X2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % surj_def
% 0.31/0.47 thf(fact_201_surjI,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 0.31/0.47 ( ! [X3: A] :
% 0.31/0.47 ( ( G @ ( F @ X3 ) )
% 0.31/0.47 = X3 )
% 0.31/0.47 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 0.31/0.47 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % surjI
% 0.31/0.47 thf(fact_202_ids_Oexpand__singleton,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,T4: trm @ A @ Sz] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ( singleton @ Sz @ A @ Vid1 @ T4 )
% 0.31/0.47 = ( ^ [I5: Sz] : ( if @ ( trm @ A @ Sz ) @ ( I5 = Vid1 ) @ T4 @ ( const @ A @ Sz @ ( zero_zero @ real ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.expand_singleton
% 0.31/0.47 thf(fact_203_ids_Osingleton_Osimps,axiom,
% 0.31/0.47 ! [Sf: $tType,Sc: $tType,A: $tType,Sz: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sc )
% 0.31/0.47 & ( finite_finite @ Sf ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,I4: Sz,T4: trm @ A @ Sz] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ( ( I4 = Vid1 )
% 0.31/0.47 => ( ( singleton @ Sz @ A @ Vid1 @ T4 @ I4 )
% 0.31/0.47 = T4 ) )
% 0.31/0.47 & ( ( I4 != Vid1 )
% 0.31/0.47 => ( ( singleton @ Sz @ A @ Vid1 @ T4 @ I4 )
% 0.31/0.47 = ( const @ A @ Sz @ ( zero_zero @ real ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.singleton.simps
% 0.31/0.47 thf(fact_204_ids_Oproj__sing1,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,Sz: $tType,A: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Theta3: trm @ A @ Sz] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ( singleton @ Sz @ A @ Vid1 @ Theta3 @ Vid1 )
% 0.31/0.47 = Theta3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.proj_sing1
% 0.31/0.47 thf(fact_205_ids_Osingleton_Ocong,axiom,
% 0.31/0.47 ! [A: $tType,Sz: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz ) )
% 0.31/0.47 => ( ( singleton @ Sz @ A )
% 0.31/0.47 = ( singleton @ Sz @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.singleton.cong
% 0.31/0.47 thf(fact_206_ids_Oproj__sing2,axiom,
% 0.31/0.47 ! [Sf: $tType,Sc: $tType,A: $tType,Sz: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sc )
% 0.31/0.47 & ( finite_finite @ Sf ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: Sz,Theta3: trm @ A @ Sz] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ( Vid1 != Y )
% 0.31/0.47 => ( ( singleton @ Sz @ A @ Vid1 @ Theta3 @ Y )
% 0.31/0.47 = ( const @ A @ Sz @ ( zero_zero @ real ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.proj_sing2
% 0.31/0.47 thf(fact_207_ids_Osingleton_Oelims,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: trm @ A @ Sz,Xa: Sz,Y: trm @ A @ Sz] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ( ( singleton @ Sz @ A @ Vid1 @ X @ Xa )
% 0.31/0.47 = Y )
% 0.31/0.47 => ( ( ( Xa = Vid1 )
% 0.31/0.47 => ( Y = X ) )
% 0.31/0.47 & ( ( Xa != Vid1 )
% 0.31/0.47 => ( Y
% 0.31/0.47 = ( const @ A @ Sz @ ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.singleton.elims
% 0.31/0.47 thf(fact_208_ids_Of1__def,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,F: Sf,X: Sz] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ( f1 @ Sz @ Sf @ Vid1 @ F @ X )
% 0.31/0.47 = ( function @ Sf @ Sz @ F @ ( singleton @ Sz @ Sf @ Vid1 @ ( var @ Sz @ Sf @ X ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.f1_def
% 0.31/0.47 thf(fact_209_ids_Osingleton_Opelims,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: trm @ A @ Sz,Xa: Sz,Y: trm @ A @ Sz] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ( ( singleton @ Sz @ A @ Vid1 @ X @ Xa )
% 0.31/0.47 = Y )
% 0.31/0.47 => ( ( accp @ ( product_prod @ ( trm @ A @ Sz ) @ Sz ) @ ( singleton_rel @ A @ Sz ) @ ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ X @ Xa ) )
% 0.31/0.47 => ~ ( ( ( ( Xa = Vid1 )
% 0.31/0.47 => ( Y = X ) )
% 0.31/0.47 & ( ( Xa != Vid1 )
% 0.31/0.47 => ( Y
% 0.31/0.47 = ( const @ A @ Sz @ ( zero_zero @ real ) ) ) ) )
% 0.31/0.47 => ~ ( accp @ ( product_prod @ ( trm @ A @ Sz ) @ Sz ) @ ( singleton_rel @ A @ Sz ) @ ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ X @ Xa ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.singleton.pelims
% 0.31/0.47 thf(fact_210_ids_Of1_Ocong,axiom,
% 0.31/0.47 ! [Sf: $tType,Sz: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf ) )
% 0.31/0.47 => ( ( f1 @ Sz @ Sf )
% 0.31/0.47 = ( f1 @ Sz @ Sf ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.f1.cong
% 0.31/0.47 thf(fact_211_singleton_Opelims,axiom,
% 0.31/0.47 ! [A: $tType,X: trm @ A @ sz,Xa: sz,Y: trm @ A @ sz] :
% 0.31/0.47 ( ( ( singleton @ sz @ A @ vid1 @ X @ Xa )
% 0.31/0.47 = Y )
% 0.31/0.47 => ( ( accp @ ( product_prod @ ( trm @ A @ sz ) @ sz ) @ ( singleton_rel @ A @ sz ) @ ( product_Pair @ ( trm @ A @ sz ) @ sz @ X @ Xa ) )
% 0.31/0.47 => ~ ( ( ( ( Xa = vid1 )
% 0.31/0.47 => ( Y = X ) )
% 0.31/0.47 & ( ( Xa != vid1 )
% 0.31/0.47 => ( Y
% 0.31/0.47 = ( const @ A @ sz @ ( zero_zero @ real ) ) ) ) )
% 0.31/0.47 => ~ ( accp @ ( product_prod @ ( trm @ A @ sz ) @ sz ) @ ( singleton_rel @ A @ sz ) @ ( product_Pair @ ( trm @ A @ sz ) @ sz @ X @ Xa ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % singleton.pelims
% 0.31/0.47 thf(fact_212_f1__def,axiom,
% 0.31/0.47 ! [F: sf,X: sz] :
% 0.31/0.47 ( ( f1 @ sz @ sf @ vid1 @ F @ X )
% 0.31/0.47 = ( function @ sf @ sz @ F @ ( singleton @ sz @ sf @ vid1 @ ( var @ sz @ sf @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % f1_def
% 0.31/0.47 thf(fact_213_proj__sing1,axiom,
% 0.31/0.47 ! [A: $tType,Theta3: trm @ A @ sz] :
% 0.31/0.47 ( ( singleton @ sz @ A @ vid1 @ Theta3 @ vid1 )
% 0.31/0.47 = Theta3 ) ).
% 0.31/0.47
% 0.31/0.47 % proj_sing1
% 0.31/0.47 thf(fact_214_expand__singleton,axiom,
% 0.31/0.47 ! [A: $tType,T4: trm @ A @ sz] :
% 0.31/0.47 ( ( singleton @ sz @ A @ vid1 @ T4 )
% 0.31/0.47 = ( ^ [I5: sz] : ( if @ ( trm @ A @ sz ) @ ( I5 = vid1 ) @ T4 @ ( const @ A @ sz @ ( zero_zero @ real ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % expand_singleton
% 0.31/0.47 thf(fact_215_proj__sing2,axiom,
% 0.31/0.47 ! [A: $tType,Y: sz,Theta3: trm @ A @ sz] :
% 0.31/0.47 ( ( vid1 != Y )
% 0.31/0.47 => ( ( singleton @ sz @ A @ vid1 @ Theta3 @ Y )
% 0.31/0.47 = ( const @ A @ sz @ ( zero_zero @ real ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % proj_sing2
% 0.31/0.47 thf(fact_216_singleton_Oelims,axiom,
% 0.31/0.47 ! [A: $tType,X: trm @ A @ sz,Xa: sz,Y: trm @ A @ sz] :
% 0.31/0.47 ( ( ( singleton @ sz @ A @ vid1 @ X @ Xa )
% 0.31/0.47 = Y )
% 0.31/0.47 => ( ( ( Xa = vid1 )
% 0.31/0.47 => ( Y = X ) )
% 0.31/0.47 & ( ( Xa != vid1 )
% 0.31/0.47 => ( Y
% 0.31/0.47 = ( const @ A @ sz @ ( zero_zero @ real ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % singleton.elims
% 0.31/0.47 thf(fact_217_singleton_Osimps,axiom,
% 0.31/0.47 ! [A: $tType,I4: sz,T4: trm @ A @ sz] :
% 0.31/0.47 ( ( ( I4 = vid1 )
% 0.31/0.47 => ( ( singleton @ sz @ A @ vid1 @ T4 @ I4 )
% 0.31/0.47 = T4 ) )
% 0.31/0.47 & ( ( I4 != vid1 )
% 0.31/0.47 => ( ( singleton @ sz @ A @ vid1 @ T4 @ I4 )
% 0.31/0.47 = ( const @ A @ sz @ ( zero_zero @ real ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % singleton.simps
% 0.31/0.47 thf(fact_218_local_Oid__simps_I15_J,axiom,
% 0.31/0.47 vid3 != vid1 ).
% 0.31/0.47
% 0.31/0.47 % local.id_simps(15)
% 0.31/0.47 thf(fact_219_local_Oid__simps_I13_J,axiom,
% 0.31/0.47 vid2 != vid1 ).
% 0.31/0.47
% 0.31/0.47 % local.id_simps(13)
% 0.31/0.47 thf(fact_220_vne13,axiom,
% 0.31/0.47 vid1 != vid3 ).
% 0.31/0.47
% 0.31/0.47 % vne13
% 0.31/0.47 thf(fact_221_vne12,axiom,
% 0.31/0.47 vid1 != vid2 ).
% 0.31/0.47
% 0.31/0.47 % vne12
% 0.31/0.47 thf(fact_222_local_Oid__simps_I1_J,axiom,
% 0.31/0.47 vid1 != vid2 ).
% 0.31/0.47
% 0.31/0.47 % local.id_simps(1)
% 0.31/0.47 thf(fact_223_local_Oid__simps_I3_J,axiom,
% 0.31/0.47 vid1 != vid3 ).
% 0.31/0.47
% 0.31/0.47 % local.id_simps(3)
% 0.31/0.47 thf(fact_224_vne23,axiom,
% 0.31/0.47 vid2 != vid3 ).
% 0.31/0.47
% 0.31/0.47 % vne23
% 0.31/0.47 thf(fact_225_local_Oid__simps_I2_J,axiom,
% 0.31/0.47 vid2 != vid3 ).
% 0.31/0.47
% 0.31/0.47 % local.id_simps(2)
% 0.31/0.47 thf(fact_226_local_Oid__simps_I14_J,axiom,
% 0.31/0.47 vid3 != vid2 ).
% 0.31/0.47
% 0.31/0.47 % local.id_simps(14)
% 0.31/0.47 thf(fact_227_p1__def,axiom,
% 0.31/0.47 ! [P2: sz,X: sz] :
% 0.31/0.47 ( ( p1 @ sz @ sf @ sc @ vid1 @ P2 @ X )
% 0.31/0.47 = ( prop @ sz @ sf @ sc @ P2 @ ( singleton @ sz @ sf @ vid1 @ ( var @ sz @ sf @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % p1_def
% 0.31/0.47 thf(fact_228_prod_Oswap__def,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( product_swap @ A @ B )
% 0.31/0.47 = ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod.swap_def
% 0.31/0.47 thf(fact_229_formula_Oinject_I2_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,C: $tType,X21: C,X222: C > ( trm @ A @ C ),Y21: C,Y22: C > ( trm @ A @ C )] :
% 0.31/0.47 ( ( ( prop @ C @ A @ B @ X21 @ X222 )
% 0.31/0.47 = ( prop @ C @ A @ B @ Y21 @ Y22 ) )
% 0.31/0.47 = ( ( X21 = Y21 )
% 0.31/0.47 & ( X222 = Y22 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % formula.inject(2)
% 0.31/0.47 thf(fact_230_swap__swap,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.31/0.47 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 0.31/0.47 = P2 ) ).
% 0.31/0.47
% 0.31/0.47 % swap_swap
% 0.31/0.47 thf(fact_231_pair__in__swap__image,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,Y: A,X: B,A3: set @ ( product_prod @ B @ A )] :
% 0.31/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A3 ) )
% 0.31/0.47 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % pair_in_swap_image
% 0.31/0.47 thf(fact_232_swap__simp,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,X: B,Y: A] :
% 0.31/0.47 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.31/0.47 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % swap_simp
% 0.31/0.47 thf(fact_233_surj__swap,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.31/0.47 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % surj_swap
% 0.31/0.47 thf(fact_234_snd__swap,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 0.31/0.47 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 0.31/0.47 = ( product_fst @ A @ B @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % snd_swap
% 0.31/0.47 thf(fact_235_fst__swap,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 0.31/0.47 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 0.31/0.47 = ( product_snd @ B @ A @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % fst_swap
% 0.31/0.47 thf(fact_236_ids_Op1_Ocong,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ( ( p1 @ Sz @ Sf @ Sc )
% 0.31/0.47 = ( p1 @ Sz @ Sf @ Sc ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.p1.cong
% 0.31/0.47 thf(fact_237_ids_Op1__def,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P2: Sz,X: Sz] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ( p1 @ Sz @ Sf @ Sc @ Vid1 @ P2 @ X )
% 0.31/0.47 = ( prop @ Sz @ Sf @ Sc @ P2 @ ( singleton @ Sz @ Sf @ Vid1 @ ( var @ Sz @ Sf @ X ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.p1_def
% 0.31/0.47 thf(fact_238_formula_Oset__intros_I3_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,C: $tType,Yb: trm @ A @ C,X222: C > ( trm @ A @ C ),Yc: A,X21: C] :
% 0.31/0.47 ( ( member @ ( trm @ A @ C ) @ Yb @ ( image @ C @ ( trm @ A @ C ) @ X222 @ ( top_top @ ( set @ C ) ) ) )
% 0.31/0.47 => ( ( member @ A @ Yc @ ( set_trm @ A @ C @ Yb ) )
% 0.31/0.47 => ( member @ A @ Yc @ ( set1_formula @ A @ B @ C @ ( prop @ C @ A @ B @ X21 @ X222 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % formula.set_intros(3)
% 0.31/0.47 thf(fact_239_hpfree__ffree_Ointros_I10_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,C: $tType,Args: C > ( trm @ A @ C ),P2: C] :
% 0.31/0.47 ( ! [Arg: trm @ A @ C] :
% 0.31/0.47 ( ( member @ ( trm @ A @ C ) @ Arg @ ( image @ C @ ( trm @ A @ C ) @ Args @ ( top_top @ ( set @ C ) ) ) )
% 0.31/0.47 => ( dfree @ A @ C @ Arg ) )
% 0.31/0.47 => ( ffree @ A @ B @ C @ ( prop @ C @ A @ B @ P2 @ Args ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % hpfree_ffree.intros(10)
% 0.31/0.47 thf(fact_240_hpfree__ffree_Ointros_I15_J,axiom,
% 0.31/0.47 ! [A: $tType,C: $tType,B: $tType,P: B] : ( ffree @ A @ B @ C @ ( predicational @ B @ A @ C @ P ) ) ).
% 0.31/0.47
% 0.31/0.47 % hpfree_ffree.intros(15)
% 0.31/0.47 thf(fact_241_map__prod__surj,axiom,
% 0.31/0.47 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G: C > D] :
% 0.31/0.47 ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.31/0.47 = ( top_top @ ( set @ B ) ) )
% 0.31/0.47 => ( ( ( image @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
% 0.31/0.47 = ( top_top @ ( set @ D ) ) )
% 0.31/0.47 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 0.31/0.47 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % map_prod_surj
% 0.31/0.47 thf(fact_242_good__interp_Oright__unique,axiom,
% 0.31/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A )
% 0.31/0.47 & ( finite_finite @ B )
% 0.31/0.47 & ( finite_finite @ C ) )
% 0.31/0.47 => ( right_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % good_interp.right_unique
% 0.31/0.47 thf(fact_243_good__interp_Oleft__unique,axiom,
% 0.31/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A )
% 0.31/0.47 & ( finite_finite @ B )
% 0.31/0.47 & ( finite_finite @ C ) )
% 0.31/0.47 => ( left_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % good_interp.left_unique
% 0.31/0.47 thf(fact_244_map__prod__simp,axiom,
% 0.31/0.47 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,A2: C,B2: D] :
% 0.31/0.47 ( ( product_map_prod @ C @ A @ D @ B @ F @ G @ ( product_Pair @ C @ D @ A2 @ B2 ) )
% 0.31/0.47 = ( product_Pair @ A @ B @ ( F @ A2 ) @ ( G @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % map_prod_simp
% 0.31/0.47 thf(fact_245_fst__map__prod,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
% 0.31/0.47 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X ) )
% 0.31/0.47 = ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % fst_map_prod
% 0.31/0.47 thf(fact_246_snd__map__prod,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X: product_prod @ C @ D] :
% 0.31/0.47 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X ) )
% 0.31/0.47 = ( G @ ( product_snd @ C @ D @ X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % snd_map_prod
% 0.31/0.47 thf(fact_247_map__prod__imageI,axiom,
% 0.31/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R6: set @ ( product_prod @ A @ B ),F: A > C,G: B > D] :
% 0.31/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R6 )
% 0.31/0.47 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A2 ) @ ( G @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) @ R6 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % map_prod_imageI
% 0.31/0.47 thf(fact_248_prod__fun__imageE,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F: C > A,G: D > B,R6: set @ ( product_prod @ C @ D )] :
% 0.31/0.47 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G ) @ R6 ) )
% 0.31/0.47 => ~ ! [X3: C,Y3: D] :
% 0.31/0.47 ( ( C3
% 0.31/0.47 = ( product_Pair @ A @ B @ ( F @ X3 ) @ ( G @ Y3 ) ) )
% 0.31/0.47 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R6 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_fun_imageE
% 0.31/0.47 thf(fact_249_ids_Ogood__interp_Oright__unique,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A )
% 0.31/0.47 & ( finite_finite @ B )
% 0.31/0.47 & ( finite_finite @ C )
% 0.31/0.47 & ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( right_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.good_interp.right_unique
% 0.31/0.47 thf(fact_250_ids_Ogood__interp_Oleft__unique,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A )
% 0.31/0.47 & ( finite_finite @ B )
% 0.31/0.47 & ( finite_finite @ C )
% 0.31/0.47 & ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( left_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.good_interp.left_unique
% 0.31/0.47 thf(fact_251_good__interp_Odomain,axiom,
% 0.31/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A )
% 0.31/0.47 & ( finite_finite @ B )
% 0.31/0.47 & ( finite_finite @ C ) )
% 0.31/0.47 => ( ( domainp @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) )
% 0.31/0.47 = ( denota2077489681interp @ A @ B @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % good_interp.domain
% 0.31/0.47 thf(fact_252_dterm__sem_Osimps_I6_J,axiom,
% 0.31/0.47 ! [B: $tType,C: $tType,A: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A )
% 0.31/0.47 & ( finite_finite @ C )
% 0.31/0.47 & ( finite_finite @ B ) )
% 0.31/0.47 => ! [I7: denota1663640101rp_ext @ A @ B @ C @ product_unit,T4: trm @ A @ C] :
% 0.31/0.47 ( ( denota594965758rm_sem @ A @ B @ C @ I7 @ ( differential @ A @ C @ T4 ) )
% 0.31/0.47 = ( denota2078997598vative @ A @ B @ C @ I7 @ T4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dterm_sem.simps(6)
% 0.31/0.47 thf(fact_253_ids_Ogood__interp_Odomain,axiom,
% 0.31/0.47 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A )
% 0.31/0.47 & ( finite_finite @ B )
% 0.31/0.47 & ( finite_finite @ C )
% 0.31/0.47 & ( finite_finite @ Sz )
% 0.31/0.47 & ( linorder @ Sz )
% 0.31/0.47 & ( finite_finite @ Sf )
% 0.31/0.47 & ( finite_finite @ Sc ) )
% 0.31/0.47 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.31/0.47 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.31/0.47 => ( ( domainp @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) )
% 0.31/0.47 = ( denota2077489681interp @ A @ B @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ids.good_interp.domain
% 0.31/0.47 thf(fact_254_dterm__sem_Osimps_I7_J,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,C: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ C )
% 0.31/0.47 & ( finite_finite @ B )
% 0.31/0.47 & ( finite_finite @ A ) )
% 0.31/0.47 => ! [I7: denota1663640101rp_ext @ A @ B @ C @ product_unit,C3: real] :
% 0.31/0.47 ( ( denota594965758rm_sem @ A @ B @ C @ I7 @ ( const @ A @ C @ C3 ) )
% 0.31/0.47 = ( ^ [V4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] : C3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dterm_sem.simps(7)
% 0.31/0.47
% 0.31/0.47 % Subclasses (2)
% 0.31/0.47 thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( finite_finite @ A )
% 0.31/0.47 => ( type @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A )
% 0.31/0.47 => ( type @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 % Type constructors (20)
% 0.31/0.47 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A7 )
% 0.31/0.47 & ( finite_finite @ A8 ) )
% 0.31/0.47 => ( finite_finite @ ( A7 > A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Otop,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( top @ A8 )
% 0.31/0.47 => ( top @ ( A7 > A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Finite__Set_Ofinite_1,axiom,
% 0.31/0.47 ! [A7: $tType] :
% 0.31/0.47 ( ( finite_finite @ A7 )
% 0.31/0.47 => ( finite_finite @ ( set @ A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Otop_2,axiom,
% 0.31/0.47 ! [A7: $tType] : ( top @ ( set @ A7 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Groups_Ozero,axiom,
% 0.31/0.47 ! [A7: $tType] :
% 0.31/0.47 ( ( zero @ A7 )
% 0.31/0.47 => ( zero @ ( set @ A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.31/0.47 linorder @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Finite__Set_Ofinite_3,axiom,
% 0.31/0.47 finite_finite @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Otop_4,axiom,
% 0.31/0.47 top @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Orderings_Olinorder_5,axiom,
% 0.31/0.47 linorder @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Groups_Ozero_6,axiom,
% 0.31/0.47 zero @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_7,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A7 )
% 0.31/0.47 & ( finite_finite @ A8 ) )
% 0.31/0.47 => ( finite_finite @ ( sum_sum @ A7 @ A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_8,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A7 )
% 0.31/0.47 & ( finite_finite @ A8 ) )
% 0.31/0.47 => ( finite_finite @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Oprod___Orderings_Otop_9,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( ( top @ A7 )
% 0.31/0.47 & ( top @ A8 ) )
% 0.31/0.47 => ( top @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Oprod___Groups_Ozero_10,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( ( zero @ A7 )
% 0.31/0.47 & ( zero @ A8 ) )
% 0.31/0.47 => ( zero @ ( product_prod @ A7 @ A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_11,axiom,
% 0.31/0.47 linorder @ product_unit ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_12,axiom,
% 0.31/0.47 finite_finite @ product_unit ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Ounit___Orderings_Otop_13,axiom,
% 0.31/0.47 top @ product_unit ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_14,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( ( linorder @ A7 )
% 0.31/0.47 & ( cARD_1 @ A8 ) )
% 0.31/0.47 => ( linorder @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_15,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A7 )
% 0.31/0.47 & ( finite_finite @ A8 ) )
% 0.31/0.47 => ( finite_finite @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_16,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( ( zero @ A7 )
% 0.31/0.47 & ( finite_finite @ A8 ) )
% 0.31/0.47 => ( zero @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Helper facts (3)
% 0.31/0.47 thf(help_If_3_1_T,axiom,
% 0.31/0.47 ! [P: $o] :
% 0.31/0.47 ( ( P = $true )
% 0.31/0.47 | ( P = $false ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(help_If_2_1_T,axiom,
% 0.31/0.47 ! [A: $tType,X: A,Y: A] :
% 0.31/0.47 ( ( if @ A @ $false @ X @ Y )
% 0.31/0.47 = Y ) ).
% 0.31/0.47
% 0.31/0.47 thf(help_If_1_1_T,axiom,
% 0.31/0.47 ! [A: $tType,X: A,Y: A] :
% 0.31/0.47 ( ( if @ A @ $true @ X @ Y )
% 0.31/0.47 = X ) ).
% 0.31/0.47
% 0.31/0.47 % Free types (7)
% 0.31/0.47 thf(tfree_0,hypothesis,
% 0.31/0.47 finite_finite @ a ).
% 0.31/0.47
% 0.31/0.47 thf(tfree_1,hypothesis,
% 0.31/0.47 finite_finite @ c ).
% 0.31/0.47
% 0.31/0.47 thf(tfree_2,hypothesis,
% 0.31/0.47 finite_finite @ sz ).
% 0.31/0.47
% 0.31/0.47 thf(tfree_3,hypothesis,
% 0.31/0.47 linorder @ sz ).
% 0.31/0.47
% 0.31/0.47 thf(tfree_4,hypothesis,
% 0.31/0.47 finite_finite @ sf ).
% 0.31/0.47
% 0.31/0.47 thf(tfree_5,hypothesis,
% 0.31/0.47 finite_finite @ sc ).
% 0.31/0.47
% 0.31/0.47 thf(tfree_6,hypothesis,
% 0.31/0.47 finite_finite @ b ).
% 0.31/0.47
% 0.31/0.47 % Conjectures (1)
% 0.31/0.47 thf(conj_0,conjecture,
% 0.31/0.47 dfree @ a @ c @ ( args @ i2 ) ).
% 0.31/0.47
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 ------------------------
% 2.86/1.55 ---- Embedded in THF ---
% 2.86/1.55 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.86/1.55 %%% Generated on Tue Feb 24 08:24:04 EST 2026
% 2.86/1.55 %%% using '$$dhol' embedding, version 1.3.0.
% 2.86/1.55 %%% Logic specification used:
% 2.86/1.55 %%% thf(spec, logic, $$dhol).
% 2.86/1.55
% 2.86/1.55 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.s6Ia5X0taI/DTF2DTF9156.p
% 2.86/1.55 thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext, type, denota1663640101rp_ext: ($tType > ($tType > ($tType > ($tType > $tType))))).
% 2.86/1.55 thf(ty_t_Frechet__Correctness_Oids_Ogood__interp, type, frechet_good_interp: ($tType > ($tType > ($tType > $tType)))).
% 2.86/1.55 thf(ty_t_Frechet__Correctness_Oids_Ostrm, type, frechet_strm: ($tType > ($tType > $tType))).
% 2.86/1.55 thf(ty_t_Finite__Cartesian__Product_Ovec, type, finite_Cartesian_vec: ($tType > ($tType > $tType))).
% 2.86/1.55 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 2.86/1.55 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.86/1.55 thf(ty_t_Syntax_Oformula, type, formula: ($tType > ($tType > ($tType > $tType)))).
% 2.86/1.55 thf(ty_t_Sum__Type_Osum, type, sum_sum: ($tType > ($tType > $tType))).
% 2.86/1.55 thf(ty_t_Syntax_Otrm, type, trm: ($tType > ($tType > $tType))).
% 2.86/1.55 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.86/1.55 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.86/1.55 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.86/1.55 thf(ty_tf_sz, type, sz: $tType).
% 2.86/1.55 thf(ty_tf_sf, type, sf: $tType).
% 2.86/1.55 thf(ty_tf_sc, type, sc: $tType).
% 2.86/1.55 thf(ty_tf_c, type, c: $tType).
% 2.86/1.55 thf(ty_tf_b, type, b: $tType).
% 2.86/1.55 thf(ty_tf_a, type, a: $tType).
% 2.86/1.55 thf(sy_cl_Cardinality_OCARD__1, type, cARD_1: (!> [A:$tType]: ($o))).
% 2.86/1.55 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.86/1.55 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.86/1.55 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 2.86/1.55 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 2.86/1.55 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.86/1.55 thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd, type, euclid1678468529ng_gcd: (!> [A:$tType]: ($o))).
% 2.86/1.55 thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring, type, euclid1155270486miring: (!> [A:$tType]: ($o))).
% 2.86/1.55 thf(sy_c_Denotational__Semantics_OVagree, type, denotational_Vagree: (!> [C:$tType]: ((((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) > (((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) > ((set @ ((sum_sum @ C) @ C)) > $o)))))).
% 2.86/1.55 thf(sy_c_Denotational__Semantics_Odirectional__derivative, type, denota2078997598vative: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((trm @ A) @ C) > (((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) > real)))))).
% 2.86/1.55 thf(sy_c_Denotational__Semantics_Odterm__sem, type, denota594965758rm_sem: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((trm @ A) @ C) > (((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) > real)))))).
% 2.86/1.55 thf(sy_c_Denotational__Semantics_Ofrechet, type, denotational_frechet: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((trm @ A) @ C) > (((finite_Cartesian_vec @ real) @ C) > (((finite_Cartesian_vec @ real) @ C) > real))))))).
% 2.86/1.55 thf(sy_c_Denotational__Semantics_Ois__interp, type, denota2077489681interp: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > $o)))).
% 2.86/1.55 thf(sy_c_Frechet__Correctness_Oids_Ocr__good__interp, type, freche457001096interp: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > ((((frechet_good_interp @ A) @ B) @ C) > $o))))).
% 2.86/1.55 thf(sy_c_Frechet__Correctness_Oids_Ocr__strm, type, frechet_cr_strm: (!> [A:$tType,B:$tType]: ((((trm @ A) @ B) > (((frechet_strm @ A) @ B) > $o))))).
% 2.86/1.55 thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Ogood__interp, type, freche227871258interp: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((frechet_good_interp @ A) @ B) @ C))))).
% 2.86/1.55 thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Oraw__interp, type, freche229654227interp: (!> [A:$tType,B:$tType,C:$tType]: (((((frechet_good_interp @ A) @ B) @ C) > ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit))))).
% 2.86/1.55 thf(sy_c_Frechet__Correctness_Oids_Ostrm_Oraw__term, type, frechet_raw_term: (!> [A:$tType,C:$tType]: ((((frechet_strm @ A) @ C) > ((trm @ A) @ C))))).
% 2.86/1.55 thf(sy_c_Frechet__Correctness_Oids_Ostrm_Osimple__term, type, frechet_simple_term: (!> [A:$tType,C:$tType]: ((((trm @ A) @ C) > ((frechet_strm @ A) @ C))))).
% 2.86/1.55 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.86/1.55 thf(sy_c_Ids_Oids, type, ids: (!> [Sz:$tType,Sf:$tType,Sc:$tType]: ((Sz > (Sz > (Sz > (Sf > (Sf > (Sf > (Sc > (Sc > (Sc > (Sc > $o))))))))))))).
% 2.86/1.55 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.86/1.55 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.86/1.55 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.86/1.55 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.86/1.55 thf(sy_c_Product__Type_Omap__prod, type, product_map_prod: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((product_prod @ A) @ B) > ((product_prod @ C) @ D))))))).
% 2.86/1.55 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.86/1.55 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 2.86/1.55 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 2.86/1.55 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 2.86/1.55 thf(sy_c_Relation_ODomainp, type, domainp: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > (A > $o))))).
% 2.86/1.55 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.86/1.55 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.86/1.55 thf(sy_c_Static__Semantics_OFVDiff, type, static_FVDiff: (!> [A:$tType,C:$tType]: ((((trm @ A) @ C) > (set @ ((sum_sum @ C) @ C)))))).
% 2.86/1.55 thf(sy_c_Syntax_OPredicational, type, predicational: (!> [B:$tType,A:$tType,C:$tType]: ((B > (((formula @ A) @ B) @ C))))).
% 2.86/1.55 thf(sy_c_Syntax_Odfree, type, dfree: (!> [A:$tType,C:$tType]: ((((trm @ A) @ C) > $o)))).
% 2.86/1.55 thf(sy_c_Syntax_Odsafe, type, dsafe: (!> [A:$tType,C:$tType]: ((((trm @ A) @ C) > $o)))).
% 2.86/1.55 thf(sy_c_Syntax_Offree, type, ffree: (!> [A:$tType,B:$tType,C:$tType]: (((((formula @ A) @ B) @ C) > $o)))).
% 2.86/1.55 thf(sy_c_Syntax_Oformula_OProp, type, prop: (!> [C:$tType,A:$tType,B:$tType]: ((C > ((C > ((trm @ A) @ C)) > (((formula @ A) @ B) @ C)))))).
% 2.86/1.55 thf(sy_c_Syntax_Oformula_Oset1__formula, type, set1_formula: (!> [A:$tType,B:$tType,C:$tType]: (((((formula @ A) @ B) @ C) > (set @ A))))).
% 2.86/1.55 thf(sy_c_Syntax_Oids_OP, type, p: (!> [Sc:$tType,Sf:$tType,Sz:$tType]: ((Sc > (((formula @ Sf) @ Sc) @ Sz))))).
% 2.86/1.55 thf(sy_c_Syntax_Oids_Oempty, type, empty: (!> [B:$tType,A:$tType]: ((B > ((trm @ A) @ B))))).
% 2.86/1.55 thf(sy_c_Syntax_Oids_Of0, type, f0: (!> [Sf:$tType,Sz:$tType]: ((Sf > ((trm @ Sf) @ Sz))))).
% 2.86/1.55 thf(sy_c_Syntax_Oids_Of1, type, f1: (!> [Sz:$tType,Sf:$tType]: ((Sz > (Sf > (Sz > ((trm @ Sf) @ Sz))))))).
% 2.86/1.55 thf(sy_c_Syntax_Oids_Op1, type, p1: (!> [Sz:$tType,Sf:$tType,Sc:$tType]: ((Sz > (Sz > (Sz > (((formula @ Sf) @ Sc) @ Sz))))))).
% 2.86/1.55 thf(sy_c_Syntax_Oids_Osingleton, type, singleton: (!> [Sz:$tType,A:$tType]: ((Sz > (((trm @ A) @ Sz) > (Sz > ((trm @ A) @ Sz))))))).
% 2.86/1.55 thf(sy_c_Syntax_Oids_Osingleton__rel, type, singleton_rel: (!> [A:$tType,Sz:$tType]: ((((product_prod @ ((trm @ A) @ Sz)) @ Sz) > (((product_prod @ ((trm @ A) @ Sz)) @ Sz) > $o))))).
% 2.86/1.55 thf(sy_c_Syntax_Otrm_OConst, type, const: (!> [A:$tType,C:$tType]: ((real > ((trm @ A) @ C))))).
% 2.86/1.55 thf(sy_c_Syntax_Otrm_ODiffVar, type, diffVar: (!> [C:$tType,A:$tType]: ((C > ((trm @ A) @ C))))).
% 2.86/1.55 thf(sy_c_Syntax_Otrm_ODifferential, type, differential: (!> [A:$tType,C:$tType]: ((((trm @ A) @ C) > ((trm @ A) @ C))))).
% 2.86/1.55 thf(sy_c_Syntax_Otrm_OFunction, type, function: (!> [A:$tType,C:$tType]: ((A > ((C > ((trm @ A) @ C)) > ((trm @ A) @ C)))))).
% 2.86/1.55 thf(sy_c_Syntax_Otrm_OPlus, type, plus: (!> [A:$tType,C:$tType]: ((((trm @ A) @ C) > (((trm @ A) @ C) > ((trm @ A) @ C)))))).
% 2.86/1.55 thf(sy_c_Syntax_Otrm_OTimes, type, times: (!> [A:$tType,C:$tType]: ((((trm @ A) @ C) > (((trm @ A) @ C) > ((trm @ A) @ C)))))).
% 2.86/1.55 thf(sy_c_Syntax_Otrm_OVar, type, var: (!> [C:$tType,A:$tType]: ((C > ((trm @ A) @ C))))).
% 2.86/1.55 thf(sy_c_Syntax_Otrm_Oset__trm, type, set_trm: (!> [A:$tType,C:$tType]: ((((trm @ A) @ C) > (set @ A))))).
% 2.86/1.55 thf(sy_c_Transfer_Oleft__unique, type, left_unique: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > $o)))).
% 2.86/1.55 thf(sy_c_Transfer_Oright__unique, type, right_unique: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > $o)))).
% 2.86/1.55 thf(sy_c_Wellfounded_Oaccp, type, accp: (!> [A:$tType]: (((A > (A > $o)) > (A > $o))))).
% 2.86/1.55 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.86/1.55 thf(sy_v_I, type, i: ((((denota1663640101rp_ext @ a) @ b) @ c) @ product_unit)).
% 2.86/1.55 thf(sy_v__092_060nu_062, type, nu: ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))).
% 2.86/1.55 thf(sy_v__092_060nu_062_H, type, nu2: ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))).
% 2.86/1.55 thf(sy_v_args____, type, args: (c > ((trm @ a) @ c))).
% 2.86/1.55 thf(sy_v_i, type, i2: c).
% 2.86/1.55 thf(sy_v_var____, type, var2: a).
% 2.86/1.55 thf(sy_v_vid1, type, vid1: sz).
% 2.86/1.55 thf(sy_v_vid2, type, vid2: sz).
% 2.86/1.55 thf(sy_v_vid3, type, vid3: sz).
% 2.86/1.55 thf(fact_0_free, axiom, (! [I:c]: ((((dfree @ a) @ c) @ (args @ I))))).
% 2.86/1.55 thf(fact_1_raw__interp__inject, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [X:(((frechet_good_interp @ A) @ B) @ C),Y:(((frechet_good_interp @ A) @ B) @ C)]: (((((((freche229654227interp @ A) @ B) @ C) @ X) = ((((freche229654227interp @ A) @ B) @ C) @ Y)) = (X = Y)))))))).
% 2.86/1.55 thf(fact_2_raw__term__inject, axiom, (! [C:$tType,A:$tType,X:((frechet_strm @ A) @ C),Y:((frechet_strm @ A) @ C)]: ((((((frechet_raw_term @ A) @ C) @ X) = (((frechet_raw_term @ A) @ C) @ Y)) = (X = Y))))).
% 2.86/1.55 thf(fact_3_P__def, axiom, ((((p @ sc) @ sf) @ sz) = (((predicational @ sc) @ sf) @ sz))).
% 2.86/1.55 thf(fact_4_strm_Odomain, axiom, (! [C:$tType,A:$tType]: (((((domainp @ ((trm @ A) @ C)) @ ((frechet_strm @ A) @ C)) @ ((frechet_cr_strm @ A) @ C)) = ((dfree @ A) @ C))))).
% 2.86/1.55 thf(fact_5_dfree__DiffVar__simps, axiom, (! [A:$tType,B:$tType,X:B]: ((~ ((((dfree @ A) @ B) @ (((diffVar @ B) @ A) @ X))))))).
% 2.86/1.55 thf(fact_6_seq__sem_Ocases, axiom, (! [X:((product_prod @ ((((denota1663640101rp_ext @ sf) @ sc) @ sz) @ product_unit)) @ ((product_prod @ (list @ (((formula @ sf) @ sc) @ sz))) @ (list @ (((formula @ sf) @ sc) @ sz))))]: ((~ ((! [I2:((((denota1663640101rp_ext @ sf) @ sc) @ sz) @ product_unit),S:((product_prod @ (list @ (((formula @ sf) @ sc) @ sz))) @ (list @ (((formula @ sf) @ sc) @ sz)))]: ((X != ((((product_Pair @ ((((denota1663640101rp_ext @ sf) @ sc) @ sz) @ product_unit)) @ ((product_prod @ (list @ (((formula @ sf) @ sc) @ sz))) @ (list @ (((formula @ sf) @ sc) @ sz)))) @ I2) @ S))))))))).
% 2.86/1.55 thf(fact_7_singleton_Ocases, axiom, (! [A:$tType,X:((product_prod @ ((trm @ A) @ sz)) @ sz)]: ((~ ((! [T2:((trm @ A) @ sz),I3:sz]: ((X != ((((product_Pair @ ((trm @ A) @ sz)) @ sz) @ T2) @ I3))))))))).
% 2.86/1.55 thf(fact_8_dfree__Differential__simps, axiom, (! [B:$tType,A:$tType,X:((trm @ A) @ B)]: ((~ ((((dfree @ A) @ B) @ (((differential @ A) @ B) @ X))))))).
% 2.86/1.55 thf(fact_9_dfree__Var__simps, axiom, (! [A:$tType,B:$tType,X:B]: ((((dfree @ A) @ B) @ (((var @ B) @ A) @ X))))).
% 2.86/1.55 thf(fact_10_dfree__Times__simps, axiom, (! [B:$tType,A:$tType,A2:((trm @ A) @ B),B2:((trm @ A) @ B)]: (((((dfree @ A) @ B) @ ((((times @ A) @ B) @ A2) @ B2)) = ((((dfree @ A) @ B) @ A2) & (((dfree @ A) @ B) @ B2)))))).
% 2.86/1.55 thf(fact_11_dfree__Plus__simps, axiom, (! [B:$tType,A:$tType,A2:((trm @ A) @ B),B2:((trm @ A) @ B)]: (((((dfree @ A) @ B) @ ((((plus @ A) @ B) @ A2) @ B2)) = ((((dfree @ A) @ B) @ A2) & (((dfree @ A) @ B) @ B2)))))).
% 2.86/1.55 thf(fact_12_dfree__Fun_Oprems, axiom, ((((denotational_Vagree @ c) @ nu) @ nu2) @ (((static_FVDiff @ a) @ c) @ ((((function @ a) @ c) @ var2) @ args)))).
% 2.86/1.55 thf(fact_13_trm_Oinject_I3_J, axiom, (! [A:$tType,C:$tType,X31:A,X32:(C > ((trm @ A) @ C)),Y31:A,Y32:(C > ((trm @ A) @ C))]: (((((((function @ A) @ C) @ X31) @ X32) = ((((function @ A) @ C) @ Y31) @ Y32)) = ((X31 = Y31) & (X32 = Y32)))))).
% 2.86/1.55 thf(fact_14_trm_Oinject_I4_J, axiom, (! [C:$tType,A:$tType,X41:((trm @ A) @ C),X42:((trm @ A) @ C),Y41:((trm @ A) @ C),Y42:((trm @ A) @ C)]: (((((((plus @ A) @ C) @ X41) @ X42) = ((((plus @ A) @ C) @ Y41) @ Y42)) = ((X41 = Y41) & (X42 = Y42)))))).
% 2.86/1.55 thf(fact_15_trm_Oinject_I5_J, axiom, (! [C:$tType,A:$tType,X51:((trm @ A) @ C),X52:((trm @ A) @ C),Y51:((trm @ A) @ C),Y52:((trm @ A) @ C)]: (((((((times @ A) @ C) @ X51) @ X52) = ((((times @ A) @ C) @ Y51) @ Y52)) = ((X51 = Y51) & (X52 = Y52)))))).
% 2.86/1.55 thf(fact_16_trm_Oinject_I1_J, axiom, (! [A:$tType,C:$tType,X1:C,Y1:C]: ((((((var @ C) @ A) @ X1) = (((var @ C) @ A) @ Y1)) = (X1 = Y1))))).
% 2.86/1.55 thf(fact_17_trm_Oinject_I7_J, axiom, (! [C:$tType,A:$tType,X7:((trm @ A) @ C),Y7:((trm @ A) @ C)]: ((((((differential @ A) @ C) @ X7) = (((differential @ A) @ C) @ Y7)) = (X7 = Y7))))).
% 2.86/1.55 thf(fact_18_trm_Oinject_I6_J, axiom, (! [A:$tType,C:$tType,X6:C,Y6:C]: ((((((diffVar @ C) @ A) @ X6) = (((diffVar @ C) @ A) @ Y6)) = (X6 = Y6))))).
% 2.86/1.55 thf(fact_19_dfree__Fun__simps, axiom, (! [A:$tType,B:$tType,I4:A,Args:(B > ((trm @ A) @ B))]: (((((dfree @ A) @ B) @ ((((function @ A) @ B) @ I4) @ Args)) = (! [X2:B]: ((((dfree @ A) @ B) @ (Args @ X2)))))))).
% 2.86/1.55 thf(fact_20_trm_Odistinct_I41_J, axiom, (! [C:$tType,A:$tType,X6:C,X7:((trm @ A) @ C)]: (((((diffVar @ C) @ A) @ X6) != (((differential @ A) @ C) @ X7))))).
% 2.86/1.55 thf(fact_21_trm_Odistinct_I39_J, axiom, (! [C:$tType,A:$tType,X51:((trm @ A) @ C),X52:((trm @ A) @ C),X7:((trm @ A) @ C)]: ((((((times @ A) @ C) @ X51) @ X52) != (((differential @ A) @ C) @ X7))))).
% 2.86/1.55 thf(fact_22_trm_Odistinct_I37_J, axiom, (! [A:$tType,C:$tType,X51:((trm @ A) @ C),X52:((trm @ A) @ C),X6:C]: ((((((times @ A) @ C) @ X51) @ X52) != (((diffVar @ C) @ A) @ X6))))).
% 2.86/1.55 thf(fact_23_trm_Odistinct_I35_J, axiom, (! [C:$tType,A:$tType,X41:((trm @ A) @ C),X42:((trm @ A) @ C),X7:((trm @ A) @ C)]: ((((((plus @ A) @ C) @ X41) @ X42) != (((differential @ A) @ C) @ X7))))).
% 2.86/1.55 thf(fact_24_trm_Odistinct_I33_J, axiom, (! [A:$tType,C:$tType,X41:((trm @ A) @ C),X42:((trm @ A) @ C),X6:C]: ((((((plus @ A) @ C) @ X41) @ X42) != (((diffVar @ C) @ A) @ X6))))).
% 2.86/1.55 thf(fact_25_trm_Odistinct_I31_J, axiom, (! [C:$tType,A:$tType,X41:((trm @ A) @ C),X42:((trm @ A) @ C),X51:((trm @ A) @ C),X52:((trm @ A) @ C)]: ((((((plus @ A) @ C) @ X41) @ X42) != ((((times @ A) @ C) @ X51) @ X52))))).
% 2.86/1.55 thf(fact_26_trm_Odistinct_I29_J, axiom, (! [C:$tType,A:$tType,X31:A,X32:(C > ((trm @ A) @ C)),X7:((trm @ A) @ C)]: ((((((function @ A) @ C) @ X31) @ X32) != (((differential @ A) @ C) @ X7))))).
% 2.86/1.55 thf(fact_27_trm_Odistinct_I27_J, axiom, (! [A:$tType,C:$tType,X31:A,X32:(C > ((trm @ A) @ C)),X6:C]: ((((((function @ A) @ C) @ X31) @ X32) != (((diffVar @ C) @ A) @ X6))))).
% 2.86/1.55 thf(fact_28_trm_Odistinct_I25_J, axiom, (! [C:$tType,A:$tType,X31:A,X32:(C > ((trm @ A) @ C)),X51:((trm @ A) @ C),X52:((trm @ A) @ C)]: ((((((function @ A) @ C) @ X31) @ X32) != ((((times @ A) @ C) @ X51) @ X52))))).
% 2.86/1.55 thf(fact_29_trm_Odistinct_I23_J, axiom, (! [C:$tType,A:$tType,X31:A,X32:(C > ((trm @ A) @ C)),X41:((trm @ A) @ C),X42:((trm @ A) @ C)]: ((((((function @ A) @ C) @ X31) @ X32) != ((((plus @ A) @ C) @ X41) @ X42))))).
% 2.86/1.55 thf(fact_30_trm_Odistinct_I11_J, axiom, (! [C:$tType,A:$tType,X1:C,X7:((trm @ A) @ C)]: (((((var @ C) @ A) @ X1) != (((differential @ A) @ C) @ X7))))).
% 2.86/1.55 thf(fact_31_trm_Odistinct_I9_J, axiom, (! [A:$tType,C:$tType,X1:C,X6:C]: (((((var @ C) @ A) @ X1) != (((diffVar @ C) @ A) @ X6))))).
% 2.86/1.55 thf(fact_32_trm_Odistinct_I7_J, axiom, (! [C:$tType,A:$tType,X1:C,X51:((trm @ A) @ C),X52:((trm @ A) @ C)]: (((((var @ C) @ A) @ X1) != ((((times @ A) @ C) @ X51) @ X52))))).
% 2.86/1.55 thf(fact_33_trm_Odistinct_I5_J, axiom, (! [C:$tType,A:$tType,X1:C,X41:((trm @ A) @ C),X42:((trm @ A) @ C)]: (((((var @ C) @ A) @ X1) != ((((plus @ A) @ C) @ X41) @ X42))))).
% 2.86/1.55 thf(fact_34_trm_Odistinct_I3_J, axiom, (! [A:$tType,C:$tType,X1:C,X31:A,X32:(C > ((trm @ A) @ C))]: (((((var @ C) @ A) @ X1) != ((((function @ A) @ C) @ X31) @ X32))))).
% 2.86/1.55 thf(fact_35_dfree_Odfree__Fun, axiom, (! [A:$tType,C:$tType,Args:(C > ((trm @ A) @ C)),I4:A]: (((! [I3:C]: ((((dfree @ A) @ C) @ (Args @ I3)))) => (((dfree @ A) @ C) @ ((((function @ A) @ C) @ I4) @ Args)))))).
% 2.86/1.55 thf(fact_36_dfree__Plus, axiom, (! [C:$tType,A:$tType,Theta_1:((trm @ A) @ C),Theta_2:((trm @ A) @ C)]: (((((dfree @ A) @ C) @ Theta_1) => ((((dfree @ A) @ C) @ Theta_2) => (((dfree @ A) @ C) @ ((((plus @ A) @ C) @ Theta_1) @ Theta_2))))))).
% 2.86/1.55 thf(fact_37_dfree__Times, axiom, (! [C:$tType,A:$tType,Theta_1:((trm @ A) @ C),Theta_2:((trm @ A) @ C)]: (((((dfree @ A) @ C) @ Theta_1) => ((((dfree @ A) @ C) @ Theta_2) => (((dfree @ A) @ C) @ ((((times @ A) @ C) @ Theta_1) @ Theta_2))))))).
% 2.86/1.55 thf(fact_38_dfree__Var, axiom, (! [A:$tType,C:$tType,I4:C]: ((((dfree @ A) @ C) @ (((var @ C) @ A) @ I4))))).
% 2.86/1.55 thf(fact_39_raw__interp__inverse, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [X:(((frechet_good_interp @ A) @ B) @ C)]: ((((((freche227871258interp @ A) @ B) @ C) @ ((((freche229654227interp @ A) @ B) @ C) @ X)) = X))))))).
% 2.86/1.55 thf(fact_40_raw__term__inverse, axiom, (! [C:$tType,A:$tType,X:((frechet_strm @ A) @ C)]: (((((frechet_simple_term @ A) @ C) @ (((frechet_raw_term @ A) @ C) @ X)) = X)))).
% 2.86/1.55 thf(fact_41_agree__times1, axiom, (! [A:$tType,B:$tType]: (((finite_finite @ A) => (! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),Nu2:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),T1:((trm @ B) @ A),T22:((trm @ B) @ A)]: ((((((denotational_Vagree @ A) @ Nu) @ Nu2) @ (((static_FVDiff @ B) @ A) @ ((((times @ B) @ A) @ T1) @ T22))) => ((((denotational_Vagree @ A) @ Nu) @ Nu2) @ (((static_FVDiff @ B) @ A) @ T1))))))))).
% 2.86/1.55 thf(fact_42_agree__times2, axiom, (! [A:$tType,B:$tType]: (((finite_finite @ A) => (! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),Nu2:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),T1:((trm @ B) @ A),T22:((trm @ B) @ A)]: ((((((denotational_Vagree @ A) @ Nu) @ Nu2) @ (((static_FVDiff @ B) @ A) @ ((((times @ B) @ A) @ T1) @ T22))) => ((((denotational_Vagree @ A) @ Nu) @ Nu2) @ (((static_FVDiff @ B) @ A) @ T22))))))))).
% 2.86/1.55 thf(fact_43_agree__plus1, axiom, (! [A:$tType,B:$tType]: (((finite_finite @ A) => (! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),Nu2:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),T1:((trm @ B) @ A),T22:((trm @ B) @ A)]: ((((((denotational_Vagree @ A) @ Nu) @ Nu2) @ (((static_FVDiff @ B) @ A) @ ((((plus @ B) @ A) @ T1) @ T22))) => ((((denotational_Vagree @ A) @ Nu) @ Nu2) @ (((static_FVDiff @ B) @ A) @ T1))))))))).
% 2.86/1.55 thf(fact_44_agree__plus2, axiom, (! [A:$tType,B:$tType]: (((finite_finite @ A) => (! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),Nu2:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),T1:((trm @ B) @ A),T22:((trm @ B) @ A)]: ((((((denotational_Vagree @ A) @ Nu) @ Nu2) @ (((static_FVDiff @ B) @ A) @ ((((plus @ B) @ A) @ T1) @ T22))) => ((((denotational_Vagree @ A) @ Nu) @ Nu2) @ (((static_FVDiff @ B) @ A) @ T22))))))))).
% 2.86/1.55 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.86/1.55 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A3)))) = A3)))).
% 2.86/1.55 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.86/1.55 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.86/1.55 thf(fact_49_agree__func, axiom, (! [B:$tType,A:$tType]: (((finite_finite @ A) => (! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),Nu2:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),Var:B,Args:(A > ((trm @ B) @ A)),I4:A]: ((((((denotational_Vagree @ A) @ Nu) @ Nu2) @ (((static_FVDiff @ B) @ A) @ ((((function @ B) @ A) @ Var) @ Args))) => ((((denotational_Vagree @ A) @ Nu) @ Nu2) @ (((static_FVDiff @ B) @ A) @ (Args @ I4)))))))))).
% 2.86/1.55 thf(fact_50_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X22 = Y2)))))).
% 2.86/1.55 thf(fact_51_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A4:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A4) @ B3)) = ((A2 = A4) & (B2 = B3)))))).
% 2.86/1.55 thf(fact_52_trm_Oexhaust, axiom, (! [C:$tType,A:$tType,Y:((trm @ A) @ C)]: (((! [X12:C]: ((Y != (((var @ C) @ A) @ X12)))) => ((! [X23:real]: ((Y != (((const @ A) @ C) @ X23)))) => ((! [X312:A,X322:(C > ((trm @ A) @ C))]: ((Y != ((((function @ A) @ C) @ X312) @ X322)))) => ((! [X412:((trm @ A) @ C),X422:((trm @ A) @ C)]: ((Y != ((((plus @ A) @ C) @ X412) @ X422)))) => ((! [X512:((trm @ A) @ C),X522:((trm @ A) @ C)]: ((Y != ((((times @ A) @ C) @ X512) @ X522)))) => ((! [X62:C]: ((Y != (((diffVar @ C) @ A) @ X62)))) => (~ ((! [X72:((trm @ A) @ C)]: ((Y != (((differential @ A) @ C) @ X72))))))))))))))).
% 2.86/1.55 thf(fact_53_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B4:B,C2:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B4) @ C2)))))))))).
% 2.86/1.55 thf(fact_54_trm_Oinject_I2_J, axiom, (! [C:$tType,A:$tType,X22:real,Y2:real]: ((((((const @ A) @ C) @ X22) = (((const @ A) @ C) @ Y2)) = (X22 = Y2))))).
% 2.86/1.55 thf(fact_55_dfree__Const__simps, axiom, (! [A:$tType,B:$tType,R:real]: ((((dfree @ A) @ B) @ (((const @ A) @ B) @ R))))).
% 2.86/1.55 thf(fact_56_dfree__Const, axiom, (! [A:$tType,C:$tType,R:real]: ((((dfree @ A) @ C) @ (((const @ A) @ C) @ R))))).
% 2.86/1.55 thf(fact_57_trm_Odistinct_I13_J, axiom, (! [A:$tType,C:$tType,X22:real,X31:A,X32:(C > ((trm @ A) @ C))]: (((((const @ A) @ C) @ X22) != ((((function @ A) @ C) @ X31) @ X32))))).
% 2.86/1.55 thf(fact_58_trm_Odistinct_I15_J, axiom, (! [C:$tType,A:$tType,X22:real,X41:((trm @ A) @ C),X42:((trm @ A) @ C)]: (((((const @ A) @ C) @ X22) != ((((plus @ A) @ C) @ X41) @ X42))))).
% 2.86/1.55 thf(fact_59_trm_Odistinct_I17_J, axiom, (! [C:$tType,A:$tType,X22:real,X51:((trm @ A) @ C),X52:((trm @ A) @ C)]: (((((const @ A) @ C) @ X22) != ((((times @ A) @ C) @ X51) @ X52))))).
% 2.86/1.55 thf(fact_60_trm_Odistinct_I1_J, axiom, (! [A:$tType,C:$tType,X1:C,X22:real]: (((((var @ C) @ A) @ X1) != (((const @ A) @ C) @ X22))))).
% 2.86/1.55 thf(fact_61_trm_Odistinct_I21_J, axiom, (! [C:$tType,A:$tType,X22:real,X7:((trm @ A) @ C)]: (((((const @ A) @ C) @ X22) != (((differential @ A) @ C) @ X7))))).
% 2.86/1.55 thf(fact_62_trm_Odistinct_I19_J, axiom, (! [A:$tType,C:$tType,X22:real,X6:C]: (((((const @ A) @ C) @ X22) != (((diffVar @ C) @ A) @ X6))))).
% 2.86/1.55 thf(fact_63_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B4)))) => (P @ Prod))))).
% 2.86/1.55 thf(fact_64_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B4:B]: ((Y != ((((product_Pair @ A) @ B) @ A5) @ B4))))))))).
% 2.86/1.55 thf(fact_65_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A4:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A4) @ B3)) => (~ (((A2 = A4) => (B2 != B3)))))))).
% 2.86/1.55 thf(fact_66_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A5:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B4)))) => (P @ P2))))).
% 2.86/1.55 thf(fact_67_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X3:A,Y3:B]: ((P2 = ((((product_Pair @ A) @ B) @ X3) @ Y3))))))).
% 2.86/1.55 thf(fact_68_dfree_Oinducts, axiom, (! [C:$tType,A:$tType,X:((trm @ A) @ C),P:(((trm @ A) @ C) > $o)]: (((((dfree @ A) @ C) @ X) => ((! [I3:C]: ((P @ (((var @ C) @ A) @ I3)))) => ((! [R2:real]: ((P @ (((const @ A) @ C) @ R2)))) => ((! [Args2:(C > ((trm @ A) @ C)),I3:A]: (((! [Ia:C]: ((((dfree @ A) @ C) @ (Args2 @ Ia)))) => ((! [Ia:C]: ((P @ (Args2 @ Ia)))) => (P @ ((((function @ A) @ C) @ I3) @ Args2)))))) => ((! [Theta_12:((trm @ A) @ C),Theta_22:((trm @ A) @ C)]: (((((dfree @ A) @ C) @ Theta_12) => ((P @ Theta_12) => ((((dfree @ A) @ C) @ Theta_22) => ((P @ Theta_22) => (P @ ((((plus @ A) @ C) @ Theta_12) @ Theta_22)))))))) => ((! [Theta_12:((trm @ A) @ C),Theta_22:((trm @ A) @ C)]: (((((dfree @ A) @ C) @ Theta_12) => ((P @ Theta_12) => ((((dfree @ A) @ C) @ Theta_22) => ((P @ Theta_22) => (P @ ((((times @ A) @ C) @ Theta_12) @ Theta_22)))))))) => (P @ X)))))))))).
% 2.86/1.55 thf(fact_69_dfree_Osimps, axiom, (! [C:$tType,A:$tType]: ((((dfree @ A) @ C) = (^ [A6:((trm @ A) @ C)]: (((? [I5:C]: ((A6 = (((var @ C) @ A) @ I5)))) | ((? [R3:real]: ((A6 = (((const @ A) @ C) @ R3)))) | ((? [Args3:(C > ((trm @ A) @ C)),I5:A]: (((A6 = ((((function @ A) @ C) @ I5) @ Args3)) & (! [X2:C]: ((((dfree @ A) @ C) @ (Args3 @ X2))))))) | ((? [Theta_13:((trm @ A) @ C),Theta_23:((trm @ A) @ C)]: (((A6 = ((((plus @ A) @ C) @ Theta_13) @ Theta_23)) & ((((dfree @ A) @ C) @ Theta_13) & (((dfree @ A) @ C) @ Theta_23))))) | (? [Theta_13:((trm @ A) @ C),Theta_23:((trm @ A) @ C)]: (((A6 = ((((times @ A) @ C) @ Theta_13) @ Theta_23)) & ((((dfree @ A) @ C) @ Theta_13) & (((dfree @ A) @ C) @ Theta_23))))))))))))))).
% 2.86/1.55 thf(fact_70_dfree_Ocases, axiom, (! [C:$tType,A:$tType,A2:((trm @ A) @ C)]: (((((dfree @ A) @ C) @ A2) => ((! [I3:C]: ((A2 != (((var @ C) @ A) @ I3)))) => ((! [R2:real]: ((A2 != (((const @ A) @ C) @ R2)))) => ((! [Args2:(C > ((trm @ A) @ C))]: (((? [I3:A]: ((A2 = ((((function @ A) @ C) @ I3) @ Args2)))) => (~ ((! [I6:C]: ((((dfree @ A) @ C) @ (Args2 @ I6))))))))) => ((! [Theta_12:((trm @ A) @ C),Theta_22:((trm @ A) @ C)]: (((A2 = ((((plus @ A) @ C) @ Theta_12) @ Theta_22)) => ((((dfree @ A) @ C) @ Theta_12) => (~ ((((dfree @ A) @ C) @ Theta_22))))))) => (~ ((! [Theta_12:((trm @ A) @ C),Theta_22:((trm @ A) @ C)]: (((A2 = ((((times @ A) @ C) @ Theta_12) @ Theta_22)) => ((((dfree @ A) @ C) @ Theta_12) => (~ ((((dfree @ A) @ C) @ Theta_22))))))))))))))))).
% 2.86/1.55 thf(fact_71_prod__induct7, axiom, (! [G2:$tType,F2:$tType,E:$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 @ F2) @ G2)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))))]: (((! [A5:A,B4:B,C2:C,D2:D,E2:E,F3:F2,G3:G2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F2) @ G2)) @ E2) @ ((((product_Pair @ F2) @ G2) @ F3) @ G3))))))))) => (P @ X))))).
% 2.86/1.55 thf(fact_72_prod__induct6, axiom, (! [F2:$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) @ F2))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2)))))]: (((! [A5:A,B4:B,C2:C,D2:D,E2:E,F3:F2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F2)) @ D2) @ ((((product_Pair @ E) @ F2) @ E2) @ F3)))))))) => (P @ X))))).
% 2.86/1.55 thf(fact_73_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A5:A,B4:B,C2:C,D2:D,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P @ X))))).
% 2.86/1.55 thf(fact_74_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A5:A,B4:B,C2:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2)))))) => (P @ X))))).
% 2.86/1.55 thf(fact_75_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B4:B,C2:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B4) @ C2))))) => (P @ X))))).
% 2.86/1.55 thf(fact_76_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F2:$tType,G2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))))]: ((~ ((! [A5:A,B4:B,C2:C,D2:D,E2:E,F3:F2,G3:G2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F2) @ G2)) @ E2) @ ((((product_Pair @ F2) @ G2) @ F3) @ G3)))))))))))))).
% 2.86/1.55 thf(fact_77_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2)))))]: ((~ ((! [A5:A,B4:B,C2:C,D2:D,E2:E,F3:F2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F2)) @ D2) @ ((((product_Pair @ E) @ F2) @ E2) @ F3))))))))))))).
% 2.86/1.55 thf(fact_78_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A5:A,B4:B,C2:C,D2:D,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 2.86/1.55 thf(fact_79_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A5:A,B4:B,C2:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2))))))))))).
% 2.86/1.55 thf(fact_80_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 2.86/1.55 thf(fact_81_f0__def, axiom, (((f0 @ sf) @ sz) = (^ [F4:sf]: (((((function @ sf) @ sz) @ F4) @ ((empty @ sz) @ sf)))))).
% 2.86/1.55 thf(fact_82_dsafe_Oinducts, axiom, (! [C:$tType,A:$tType,X:((trm @ A) @ C),P:(((trm @ A) @ C) > $o)]: (((((dsafe @ A) @ C) @ X) => ((! [I3:C]: ((P @ (((var @ C) @ A) @ I3)))) => ((! [R2:real]: ((P @ (((const @ A) @ C) @ R2)))) => ((! [Args2:(C > ((trm @ A) @ C)),I3:A]: (((! [Ia:C]: ((((dsafe @ A) @ C) @ (Args2 @ Ia)))) => ((! [Ia:C]: ((P @ (Args2 @ Ia)))) => (P @ ((((function @ A) @ C) @ I3) @ Args2)))))) => ((! [Theta_12:((trm @ A) @ C),Theta_22:((trm @ A) @ C)]: (((((dsafe @ A) @ C) @ Theta_12) => ((P @ Theta_12) => ((((dsafe @ A) @ C) @ Theta_22) => ((P @ Theta_22) => (P @ ((((plus @ A) @ C) @ Theta_12) @ Theta_22)))))))) => ((! [Theta_12:((trm @ A) @ C),Theta_22:((trm @ A) @ C)]: (((((dsafe @ A) @ C) @ Theta_12) => ((P @ Theta_12) => ((((dsafe @ A) @ C) @ Theta_22) => ((P @ Theta_22) => (P @ ((((times @ A) @ C) @ Theta_12) @ Theta_22)))))))) => ((! [Theta:((trm @ A) @ C)]: (((((dfree @ A) @ C) @ Theta) => (P @ (((differential @ A) @ C) @ Theta))))) => ((! [I3:C]: ((P @ (((diffVar @ C) @ A) @ I3)))) => (P @ X)))))))))))).
% 2.86/1.55 thf(fact_83_dsafe_Osimps, axiom, (! [C:$tType,A:$tType]: ((((dsafe @ A) @ C) = (^ [A6:((trm @ A) @ C)]: (((? [I5:C]: ((A6 = (((var @ C) @ A) @ I5)))) | ((? [R3:real]: ((A6 = (((const @ A) @ C) @ R3)))) | ((? [Args3:(C > ((trm @ A) @ C)),I5:A]: (((A6 = ((((function @ A) @ C) @ I5) @ Args3)) & (! [X2:C]: ((((dsafe @ A) @ C) @ (Args3 @ X2))))))) | ((? [Theta_13:((trm @ A) @ C),Theta_23:((trm @ A) @ C)]: (((A6 = ((((plus @ A) @ C) @ Theta_13) @ Theta_23)) & ((((dsafe @ A) @ C) @ Theta_13) & (((dsafe @ A) @ C) @ Theta_23))))) | ((? [Theta_13:((trm @ A) @ C),Theta_23:((trm @ A) @ C)]: (((A6 = ((((times @ A) @ C) @ Theta_13) @ Theta_23)) & ((((dsafe @ A) @ C) @ Theta_13) & (((dsafe @ A) @ C) @ Theta_23))))) | ((? [Theta2:((trm @ A) @ C)]: (((A6 = (((differential @ A) @ C) @ Theta2)) & (((dfree @ A) @ C) @ Theta2)))) | (? [I5:C]: ((A6 = (((diffVar @ C) @ A) @ I5)))))))))))))))).
% 2.86/1.55 thf(fact_84_dsafe_Ocases, axiom, (! [A:$tType,C:$tType,A2:((trm @ A) @ C)]: (((((dsafe @ A) @ C) @ A2) => ((! [I3:C]: ((A2 != (((var @ C) @ A) @ I3)))) => ((! [R2:real]: ((A2 != (((const @ A) @ C) @ R2)))) => ((! [Args2:(C > ((trm @ A) @ C))]: (((? [I3:A]: ((A2 = ((((function @ A) @ C) @ I3) @ Args2)))) => (~ ((! [I6:C]: ((((dsafe @ A) @ C) @ (Args2 @ I6))))))))) => ((! [Theta_12:((trm @ A) @ C),Theta_22:((trm @ A) @ C)]: (((A2 = ((((plus @ A) @ C) @ Theta_12) @ Theta_22)) => ((((dsafe @ A) @ C) @ Theta_12) => (~ ((((dsafe @ A) @ C) @ Theta_22))))))) => ((! [Theta_12:((trm @ A) @ C),Theta_22:((trm @ A) @ C)]: (((A2 = ((((times @ A) @ C) @ Theta_12) @ Theta_22)) => ((((dsafe @ A) @ C) @ Theta_12) => (~ ((((dsafe @ A) @ C) @ Theta_22))))))) => ((! [Theta:((trm @ A) @ C)]: (((A2 = (((differential @ A) @ C) @ Theta)) => (~ ((((dfree @ A) @ C) @ Theta)))))) => (~ ((! [I3:C]: ((A2 != (((diffVar @ C) @ A) @ I3)))))))))))))))).
% 2.86/1.55 thf(fact_85_euclid__ext__aux_Ocases, axiom, (! [A:$tType]: (((euclid1678468529ng_gcd @ A) => (! [X:((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A)))))]: ((~ ((! [S2:A,S3:A,T3:A,T2:A,R4:A,R2:A]: ((X != ((((product_Pair @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A))))) @ S2) @ ((((product_Pair @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A)))) @ S3) @ ((((product_Pair @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A))) @ T3) @ ((((product_Pair @ A) @ ((product_prod @ A) @ A)) @ T2) @ ((((product_Pair @ A) @ A) @ R4) @ R2)))))))))))))))).
% 2.86/1.55 thf(fact_86_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(B > (C > A)),A2:B,B2:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C3) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((C3 @ A2) @ B2))))).
% 2.86/1.55 thf(fact_87_strm_Oright__unique, axiom, (! [C:$tType,A:$tType]: ((((right_unique @ ((trm @ A) @ C)) @ ((frechet_strm @ A) @ C)) @ ((frechet_cr_strm @ A) @ C))))).
% 2.86/1.55 thf(fact_88_dsafe__Const__simps, axiom, (! [A:$tType,B:$tType,R:real]: ((((dsafe @ A) @ B) @ (((const @ A) @ B) @ R))))).
% 2.86/1.55 thf(fact_89_dsafe__Fun__simps, axiom, (! [A:$tType,B:$tType,I4:A,Args:(B > ((trm @ A) @ B))]: (((((dsafe @ A) @ B) @ ((((function @ A) @ B) @ I4) @ Args)) = (! [X2:B]: ((((dsafe @ A) @ B) @ (Args @ X2)))))))).
% 2.86/1.55 thf(fact_90_dsafe__Plus__simps, axiom, (! [B:$tType,A:$tType,A2:((trm @ A) @ B),B2:((trm @ A) @ B)]: (((((dsafe @ A) @ B) @ ((((plus @ A) @ B) @ A2) @ B2)) = ((((dsafe @ A) @ B) @ A2) & (((dsafe @ A) @ B) @ B2)))))).
% 2.86/1.55 thf(fact_91_dsafe__Times__simps, axiom, (! [B:$tType,A:$tType,A2:((trm @ A) @ B),B2:((trm @ A) @ B)]: (((((dsafe @ A) @ B) @ ((((times @ A) @ B) @ A2) @ B2)) = ((((dsafe @ A) @ B) @ A2) & (((dsafe @ A) @ B) @ B2)))))).
% 2.86/1.55 thf(fact_92_dsafe__Var__simps, axiom, (! [A:$tType,B:$tType,X:B]: ((((dsafe @ A) @ B) @ (((var @ B) @ A) @ X))))).
% 2.86/1.55 thf(fact_93_dsafe__DiffVar__simps, axiom, (! [A:$tType,B:$tType,X:B]: ((((dsafe @ A) @ B) @ (((diffVar @ B) @ A) @ X))))).
% 2.86/1.55 thf(fact_94_dsafe__Diff__simps, axiom, (! [B:$tType,A:$tType,A2:((trm @ A) @ B)]: (((((dsafe @ A) @ B) @ (((differential @ A) @ B) @ A2)) = (((dfree @ A) @ B) @ A2))))).
% 2.86/1.55 thf(fact_95_dfree__is__dsafe, axiom, (! [B:$tType,A:$tType,Theta3:((trm @ A) @ B)]: (((((dfree @ A) @ B) @ Theta3) => (((dsafe @ A) @ B) @ Theta3))))).
% 2.86/1.55 thf(fact_96_dsafe__Const, axiom, (! [A:$tType,C:$tType,R:real]: ((((dsafe @ A) @ C) @ (((const @ A) @ C) @ R))))).
% 2.86/1.55 thf(fact_97_dsafe__Fun, axiom, (! [A:$tType,C:$tType,Args:(C > ((trm @ A) @ C)),I4:A]: (((! [I3:C]: ((((dsafe @ A) @ C) @ (Args @ I3)))) => (((dsafe @ A) @ C) @ ((((function @ A) @ C) @ I4) @ Args)))))).
% 2.86/1.55 thf(fact_98_dsafe__Plus, axiom, (! [C:$tType,A:$tType,Theta_1:((trm @ A) @ C),Theta_2:((trm @ A) @ C)]: (((((dsafe @ A) @ C) @ Theta_1) => ((((dsafe @ A) @ C) @ Theta_2) => (((dsafe @ A) @ C) @ ((((plus @ A) @ C) @ Theta_1) @ Theta_2))))))).
% 2.86/1.55 thf(fact_99_dsafe__Times, axiom, (! [C:$tType,A:$tType,Theta_1:((trm @ A) @ C),Theta_2:((trm @ A) @ C)]: (((((dsafe @ A) @ C) @ Theta_1) => ((((dsafe @ A) @ C) @ Theta_2) => (((dsafe @ A) @ C) @ ((((times @ A) @ C) @ Theta_1) @ Theta_2))))))).
% 2.86/1.55 thf(fact_100_dsafe__Var, axiom, (! [A:$tType,C:$tType,I4:C]: ((((dsafe @ A) @ C) @ (((var @ C) @ A) @ I4))))).
% 2.86/1.55 thf(fact_101_dsafe__DiffVar, axiom, (! [A:$tType,C:$tType,I4:C]: ((((dsafe @ A) @ C) @ (((diffVar @ C) @ A) @ I4))))).
% 2.86/1.55 thf(fact_102_dsafe__Diff, axiom, (! [C:$tType,A:$tType,Theta3:((trm @ A) @ C)]: (((((dfree @ A) @ C) @ Theta3) => (((dsafe @ A) @ C) @ (((differential @ A) @ C) @ Theta3)))))).
% 2.86/1.55 thf(fact_103_gcd_Ocases, axiom, (! [A:$tType]: (((euclid1155270486miring @ A) => (! [X:((product_prod @ A) @ A)]: ((~ ((! [A5:A,B4:A]: ((X != ((((product_Pair @ A) @ A) @ A5) @ B4)))))))))))).
% 2.86/1.55 thf(fact_104_local_Oempty__def, axiom, (! [A:$tType,B:$tType]: ((((empty @ B) @ A) = (^ [I5:B]: ((((const @ A) @ B) @ (zero_zero @ real)))))))).
% 2.86/1.55 thf(fact_105_ids_Of0__def, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,F:Sf]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((f0 @ Sf) @ Sz) @ F) = ((((function @ Sf) @ Sz) @ F) @ ((empty @ Sz) @ Sf)))))))))).
% 2.86/1.55 thf(fact_106_dfree__Fun_OIH, axiom, (! [I:c]: ((((((denotational_Vagree @ c) @ nu) @ nu2) @ (((static_FVDiff @ a) @ c) @ (args @ I))) => ((((((((denotational_frechet @ a) @ b) @ c) @ i) @ (args @ I)) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ nu)) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ nu)) = (((((((denotational_frechet @ a) @ b) @ c) @ i) @ (args @ I)) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ nu2)) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ nu2))))))).
% 2.86/1.55 thf(fact_107_strm_Oleft__unique, axiom, (! [C:$tType,A:$tType]: ((((left_unique @ ((trm @ A) @ C)) @ ((frechet_strm @ A) @ C)) @ ((frechet_cr_strm @ A) @ C))))).
% 2.86/1.55 thf(fact_108_trm_Oinduct, axiom, (! [C:$tType,A:$tType,P:(((trm @ A) @ C) > $o),Trm:((trm @ A) @ C)]: (((! [X3:C]: ((P @ (((var @ C) @ A) @ X3)))) => ((! [X3:real]: ((P @ (((const @ A) @ C) @ X3)))) => ((! [X1a:A,X2a:(C > ((trm @ A) @ C))]: (((! [X2aa:((trm @ A) @ C)]: (((((member @ ((trm @ A) @ C)) @ X2aa) @ ((((image @ C) @ ((trm @ A) @ C)) @ X2a) @ (top_top @ (set @ C)))) => (P @ X2aa)))) => (P @ ((((function @ A) @ C) @ X1a) @ X2a))))) => ((! [X1a:((trm @ A) @ C),X2a:((trm @ A) @ C)]: (((P @ X1a) => ((P @ X2a) => (P @ ((((plus @ A) @ C) @ X1a) @ X2a)))))) => ((! [X1a:((trm @ A) @ C),X2a:((trm @ A) @ C)]: (((P @ X1a) => ((P @ X2a) => (P @ ((((times @ A) @ C) @ X1a) @ X2a)))))) => ((! [X3:C]: ((P @ (((diffVar @ C) @ A) @ X3)))) => ((! [X3:((trm @ A) @ C)]: (((P @ X3) => (P @ (((differential @ A) @ C) @ X3))))) => (P @ Trm))))))))))).
% 2.86/1.55 thf(fact_109_agree__comm, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A3:((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)),V:(set @ ((sum_sum @ A) @ A))]: ((((((denotational_Vagree @ A) @ A3) @ B5) @ V) => ((((denotational_Vagree @ A) @ B5) @ A3) @ V)))))))).
% 2.86/1.55 thf(fact_110_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)))).
% 2.86/1.55 thf(fact_111_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)))))).
% 2.86/1.55 thf(fact_112_surjective__pairing, axiom, (! [B:$tType,A:$tType,T4:((product_prod @ A) @ B)]: ((T4 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T4)) @ (((product_snd @ A) @ B) @ T4)))))).
% 2.86/1.55 thf(fact_113_agree__UNIV__eq, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),Omega:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))]: ((((((denotational_Vagree @ A) @ Nu) @ Omega) @ (top_top @ (set @ ((sum_sum @ A) @ A)))) => (Nu = Omega)))))))).
% 2.86/1.55 thf(fact_114_Vagree__univ, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A2:((finite_Cartesian_vec @ real) @ A),B2:((finite_Cartesian_vec @ real) @ A),C3:((finite_Cartesian_vec @ real) @ A),D3:((finite_Cartesian_vec @ real) @ A)]: ((((((denotational_Vagree @ A) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)) @ A2) @ B2)) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)) @ C3) @ D3)) @ (top_top @ (set @ ((sum_sum @ A) @ A)))) => ((A2 = C3) & (B2 = D3))))))))).
% 2.86/1.55 thf(fact_115_ids_Oseq__sem_Ocases, axiom, (! [Sz:$tType,Sc:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sc) & ((finite_finite @ Sz) & (linorder @ Sz)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X:((product_prod @ ((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit)) @ ((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz))))]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (~ ((! [I2:((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit),S:((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz)))]: ((X != ((((product_Pair @ ((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit)) @ ((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz)))) @ I2) @ S))))))))))))).
% 2.86/1.55 thf(fact_116_ids_Osingleton_Oinduct, axiom, (! [Sf:$tType,Sc:$tType,A:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sc) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,P:(((trm @ A) @ Sz) > (Sz > $o)),A0:((trm @ A) @ Sz),A1:Sz]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((! [T2:((trm @ A) @ Sz),X_1:Sz]: (((P @ T2) @ X_1))) => ((P @ A0) @ A1))))))))).
% 2.86/1.55 thf(fact_117_ids_Oseq__sem_Oinduct, axiom, (! [Sz:$tType,Sc:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sc) & ((finite_finite @ Sz) & (linorder @ Sz)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,P:(((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit) > (((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz))) > $o)),A0:((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit),A1:((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz)))]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((! [I2:((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit),X_1:((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz)))]: (((P @ I2) @ X_1))) => ((P @ A0) @ A1))))))))).
% 2.86/1.55 thf(fact_118_frechet_Osimps_I5_J, axiom, (! [B:$tType,A:$tType,C:$tType]: ((((finite_finite @ C) & ((finite_finite @ A) & (finite_finite @ B))) => (! [I7:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),R:real,V2:((finite_Cartesian_vec @ real) @ C)]: ((((((((denotational_frechet @ A) @ B) @ C) @ I7) @ (((const @ A) @ C) @ R)) @ V2) = (^ [V3:((finite_Cartesian_vec @ real) @ C)]: ((zero_zero @ real)))))))))).
% 2.86/1.55 thf(fact_119_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)))))).
% 2.86/1.55 thf(fact_120_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))))).
% 2.86/1.55 thf(fact_121_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y4:((product_prod @ A) @ B),Z:((product_prod @ A) @ B)]: ((Y4 = Z))) = (^ [S4:((product_prod @ A) @ B),T5:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S4) = (((product_fst @ A) @ B) @ T5)) & ((((product_snd @ A) @ B) @ S4) = (((product_snd @ A) @ B) @ T5))))))))).
% 2.86/1.55 thf(fact_122_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X22:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X22)) = X22)))).
% 2.86/1.55 thf(fact_123_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))))).
% 2.86/1.55 thf(fact_124_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X22:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = X1)))).
% 2.86/1.55 thf(fact_125_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))))).
% 2.86/1.55 thf(fact_126_ids_Oempty__def, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,A:$tType,B:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((empty @ B) @ A) = (^ [I5:B]: ((((const @ A) @ B) @ (zero_zero @ real)))))))))))).
% 2.86/1.55 thf(fact_127_ids_Osingleton_Ocases, axiom, (! [Sc:$tType,Sf:$tType,A:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X:((product_prod @ ((trm @ A) @ Sz)) @ Sz)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (~ ((! [T2:((trm @ A) @ Sz),I3:Sz]: ((X != ((((product_Pair @ ((trm @ A) @ Sz)) @ Sz) @ T2) @ I3))))))))))))).
% 2.86/1.55 thf(fact_128_ids_OP__def, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,P2:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((((p @ Sc) @ Sf) @ Sz) @ P2) = ((((predicational @ Sc) @ Sf) @ Sz) @ P2))))))))).
% 2.86/1.55 thf(fact_129_FunctionFrechet_Ocases, axiom, (! [B:$tType,C:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ C) & (finite_finite @ B))) => (! [X:((product_prod @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ A)]: ((~ ((! [I2:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),I3:A]: ((X != ((((product_Pair @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ A) @ I2) @ I3)))))))))))).
% 2.86/1.55 thf(fact_130_agree__refl, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),A3:(set @ ((sum_sum @ A) @ A))]: (((((denotational_Vagree @ A) @ Nu) @ Nu) @ A3))))))).
% 2.86/1.55 thf(fact_131_ids_Ostrm_Odomain, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,A:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((domainp @ ((trm @ A) @ C)) @ ((frechet_strm @ A) @ C)) @ ((frechet_cr_strm @ A) @ C)) = ((dfree @ A) @ C))))))))).
% 2.86/1.55 thf(fact_132_snd__top, axiom, (! [B:$tType,A:$tType]: ((((top @ A) & (top @ B)) => ((((product_snd @ B) @ A) @ (top_top @ ((product_prod @ B) @ A))) = (top_top @ A)))))).
% 2.86/1.55 thf(fact_133_fst__top, axiom, (! [B:$tType,A:$tType]: ((((top @ A) & (top @ B)) => ((((product_fst @ A) @ B) @ (top_top @ ((product_prod @ A) @ B))) = (top_top @ A)))))).
% 2.86/1.55 thf(fact_134_snd__zero, axiom, (! [B:$tType,A:$tType]: ((((zero @ A) & (zero @ B)) => ((((product_snd @ B) @ A) @ (zero_zero @ ((product_prod @ B) @ A))) = (zero_zero @ A)))))).
% 2.86/1.55 thf(fact_135_fst__zero, axiom, (! [B:$tType,A:$tType]: ((((zero @ A) & (zero @ B)) => ((((product_fst @ A) @ B) @ (zero_zero @ ((product_prod @ A) @ B))) = (zero_zero @ A)))))).
% 2.86/1.55 thf(fact_136_zero__prod__def, axiom, (! [A:$tType,B:$tType]: ((((zero @ B) & (zero @ A)) => ((zero_zero @ ((product_prod @ A) @ B)) = ((((product_Pair @ A) @ B) @ (zero_zero @ A)) @ (zero_zero @ B))))))).
% 2.86/1.55 thf(fact_137_top__prod__def, axiom, (! [A:$tType,B:$tType]: ((((top @ B) & (top @ A)) => ((top_top @ ((product_prod @ A) @ B)) = ((((product_Pair @ A) @ B) @ (top_top @ A)) @ (top_top @ B))))))).
% 2.86/1.55 thf(fact_138_ids_Oraw__interp__inject, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ C) & ((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X:(((frechet_good_interp @ A) @ B) @ C),Y:(((frechet_good_interp @ A) @ B) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((((freche229654227interp @ A) @ B) @ C) @ X) = ((((freche229654227interp @ A) @ B) @ C) @ Y)) = (X = Y))))))))).
% 2.86/1.55 thf(fact_139_ids_Oraw__term__inject, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,A:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X:((frechet_strm @ A) @ C),Y:((frechet_strm @ A) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((((frechet_raw_term @ A) @ C) @ X) = (((frechet_raw_term @ A) @ C) @ Y)) = (X = Y))))))))).
% 2.86/1.55 thf(fact_140_ids_Oraw__interp__inverse, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ C) & ((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X:(((frechet_good_interp @ A) @ B) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((((freche227871258interp @ A) @ B) @ C) @ ((((freche229654227interp @ A) @ B) @ C) @ X)) = X)))))))).
% 2.86/1.55 thf(fact_141_ids_Oraw__term__inverse, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,A:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X:((frechet_strm @ A) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((frechet_simple_term @ A) @ C) @ (((frechet_raw_term @ A) @ C) @ X)) = X)))))))).
% 2.86/1.55 thf(fact_142_ids_Ostrm_Oright__unique, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,A:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((right_unique @ ((trm @ A) @ C)) @ ((frechet_strm @ A) @ C)) @ ((frechet_cr_strm @ A) @ C))))))))).
% 2.86/1.55 thf(fact_143_ids_Ostrm_Oleft__unique, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,A:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((left_unique @ ((trm @ A) @ C)) @ ((frechet_strm @ A) @ C)) @ ((frechet_cr_strm @ A) @ C))))))))).
% 2.86/1.55 thf(fact_144_directional__derivative__def, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => ((((denota2078997598vative @ A) @ B) @ C) = (^ [I8:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),T5:((trm @ A) @ C),V4:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))]: ((((((((denotational_frechet @ A) @ B) @ C) @ I8) @ T5) @ (((product_fst @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ V4)) @ (((product_snd @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) @ V4))))))))).
% 2.86/1.55 thf(fact_145_BNF__Greatest__Fixpoint_Osubst__Pair, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),X:A,Y:B,A2:((product_prod @ A) @ B)]: ((((P @ X) @ Y) => ((A2 = ((((product_Pair @ A) @ B) @ X) @ Y)) => ((P @ (((product_fst @ A) @ B) @ A2)) @ (((product_snd @ A) @ B) @ A2))))))).
% 2.86/1.55 thf(fact_146_conjI__realizer, axiom, (! [A:$tType,B:$tType,P:(A > $o),P2:A,Q:(B > $o),Q2:B]: (((P @ P2) => ((Q @ Q2) => ((P @ (((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q2))) & (Q @ (((product_snd @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q2))))))))).
% 2.86/1.55 thf(fact_147_exI__realizer, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Y:A,X:B]: ((((P @ Y) @ X) => ((P @ (((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y))) @ (((product_fst @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y))))))).
% 2.86/1.55 thf(fact_148_exE__realizer_H, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),P2:((product_prod @ B) @ A)]: ((((P @ (((product_snd @ B) @ A) @ P2)) @ (((product_fst @ B) @ A) @ P2)) => (~ ((! [X3:B,Y3:A]: ((~ (((P @ Y3) @ X3))))))))))).
% 2.86/1.55 thf(fact_149_iso__tuple__UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.86/1.55 thf(fact_150_UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.86/1.55 thf(fact_151_top__apply, axiom, (! [C:$tType,D:$tType]: (((top @ C) => ((top_top @ (D > C)) = (^ [X2:D]: ((top_top @ C)))))))).
% 2.86/1.55 thf(fact_152_image__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A),X:B,A3:(set @ B)]: (((B2 = (F @ X)) => ((((member @ B) @ X) @ A3) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ A3))))))).
% 2.86/1.55 thf(fact_153_imageI, axiom, (! [B:$tType,A:$tType,X:A,A3:(set @ A),F:(A > B)]: (((((member @ A) @ X) @ A3) => (((member @ B) @ (F @ X)) @ ((((image @ A) @ B) @ F) @ A3)))))).
% 2.86/1.55 thf(fact_154_image__iff, axiom, (! [A:$tType,B:$tType,Z2:A,F:(B > A),A3:(set @ B)]: (((((member @ A) @ Z2) @ ((((image @ B) @ A) @ F) @ A3)) = (? [X2:B]: (((((member @ B) @ X2) @ A3) & (Z2 = (F @ X2))))))))).
% 2.86/1.55 thf(fact_155_bex__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A3:(set @ B),P:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ ((((image @ B) @ A) @ F) @ A3)) & (P @ X4)))) => (? [X3:B]: (((((member @ B) @ X3) @ A3) & (P @ (F @ X3))))))))).
% 2.86/1.55 thf(fact_156_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N:(set @ A),F:(A > B),G:(A > B)]: (((M = N) => ((! [X3:A]: (((((member @ A) @ X3) @ N) => ((F @ X3) = (G @ X3))))) => (((((image @ A) @ B) @ F) @ M) = ((((image @ A) @ B) @ G) @ N))))))).
% 2.86/1.55 thf(fact_157_ball__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A3:(set @ B),P:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((((image @ B) @ A) @ F) @ A3)) => (P @ X3)))) => (! [X4:B]: (((((member @ B) @ X4) @ A3) => (P @ (F @ X4))))))))).
% 2.86/1.55 thf(fact_158_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X:A,A3:(set @ A),B2:B,F:(A > B)]: (((((member @ A) @ X) @ A3) => ((B2 = (F @ X)) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ F) @ A3))))))).
% 2.86/1.55 thf(fact_159_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.86/1.55 thf(fact_160_UNIV__eq__I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X3:A]: ((((member @ A) @ X3) @ A3))) => ((top_top @ (set @ A)) = A3))))).
% 2.86/1.55 thf(fact_161_UNIV__witness, axiom, (! [A:$tType]: ((? [X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))))).
% 2.86/1.55 thf(fact_162_rangeI, axiom, (! [A:$tType,B:$tType,F:(B > A),X:B]: ((((member @ A) @ (F @ X)) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B))))))).
% 2.86/1.55 thf(fact_163_range__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A),X:B]: (((B2 = (F @ X)) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))))))).
% 2.86/1.55 thf(fact_164_sndI, axiom, (! [A:$tType,B:$tType,X:((product_prod @ A) @ B),Y:A,Z2:B]: (((X = ((((product_Pair @ A) @ B) @ Y) @ Z2)) => ((((product_snd @ A) @ B) @ X) = Z2))))).
% 2.86/1.55 thf(fact_165_eq__snd__iff, axiom, (! [A:$tType,B:$tType,B2:A,P2:((product_prod @ B) @ A)]: (((B2 = (((product_snd @ B) @ A) @ P2)) = (? [A6:B]: ((P2 = ((((product_Pair @ B) @ A) @ A6) @ B2)))))))).
% 2.86/1.55 thf(fact_166_top1I, axiom, (! [A:$tType,X:A]: (((top_top @ (A > $o)) @ X)))).
% 2.86/1.55 thf(fact_167_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 2.86/1.55 thf(fact_168_eq__fst__iff, axiom, (! [A:$tType,B:$tType,A2:A,P2:((product_prod @ A) @ B)]: (((A2 = (((product_fst @ A) @ B) @ P2)) = (? [B6:B]: ((P2 = ((((product_Pair @ A) @ B) @ A2) @ B6)))))))).
% 2.86/1.55 thf(fact_169_fstI, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),Y:A,Z2:B]: (((X = ((((product_Pair @ A) @ B) @ Y) @ Z2)) => ((((product_fst @ A) @ B) @ X) = Y))))).
% 2.86/1.55 thf(fact_170_range__snd, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ A)))))).
% 2.86/1.55 thf(fact_171_range__fst, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = (top_top @ (set @ A)))))).
% 2.86/1.55 thf(fact_172_Domain__eq__top, axiom, (! [A:$tType]: (((((domainp @ A) @ A) @ (^ [Y4:A,Z:A]: ((Y4 = Z)))) = (top_top @ (A > $o)))))).
% 2.86/1.55 thf(fact_173_DomainpE, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),A2:A]: ((((((domainp @ A) @ B) @ R) @ A2) => (~ ((! [B4:B]: ((~ (((R @ A2) @ B4))))))))))).
% 2.86/1.55 thf(fact_174_Domainp__iff, axiom, (! [B:$tType,A:$tType]: ((((domainp @ A) @ B) = (^ [T6:(A > (B > $o)),X2:A]: ((? [X5:B]: (((T6 @ X2) @ X5))))))))).
% 2.86/1.55 thf(fact_175_Domainp__refl, axiom, (! [B:$tType,A:$tType]: ((((domainp @ A) @ B) = ((domainp @ A) @ B))))).
% 2.86/1.55 thf(fact_176_Domainp_Ocases, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),A2:A]: ((((((domainp @ A) @ B) @ R) @ A2) => (~ ((! [B4:B]: ((~ (((R @ A2) @ B4))))))))))).
% 2.86/1.55 thf(fact_177_Domainp_Osimps, axiom, (! [B:$tType,A:$tType]: ((((domainp @ A) @ B) = (^ [R3:(A > (B > $o)),A6:A]: ((? [B6:A,C4:B]: (((A6 = B6) & ((R3 @ B6) @ C4)))))))))).
% 2.86/1.55 thf(fact_178_Domainp_ODomainI, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o)),A2:A,B2:B]: ((((R @ A2) @ B2) => ((((domainp @ A) @ B) @ R) @ A2))))).
% 2.86/1.55 thf(fact_179_Domainp_Oinducts, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o)),X:A,P:(A > $o)]: ((((((domainp @ A) @ B) @ R) @ X) => ((! [A5:A,B4:B]: ((((R @ A5) @ B4) => (P @ A5)))) => (P @ X)))))).
% 2.86/1.55 thf(fact_180_right__uniqueD, axiom, (! [A:$tType,B:$tType,A3:(A > (B > $o)),X:A,Y:B,Z2:B]: (((((right_unique @ A) @ B) @ A3) => (((A3 @ X) @ Y) => (((A3 @ X) @ Z2) => (Y = Z2))))))).
% 2.86/1.55 thf(fact_181_right__uniqueI, axiom, (! [B:$tType,A:$tType,A3:(A > (B > $o))]: (((! [X3:A,Y3:B,Z3:B]: ((((A3 @ X3) @ Y3) => (((A3 @ X3) @ Z3) => (Y3 = Z3))))) => (((right_unique @ A) @ B) @ A3))))).
% 2.86/1.55 thf(fact_182_right__unique__eq, axiom, (! [A:$tType]: ((((right_unique @ A) @ A) @ (^ [Y4:A,Z:A]: ((Y4 = Z))))))).
% 2.86/1.55 thf(fact_183_right__unique__def, axiom, (! [B:$tType,A:$tType]: ((((right_unique @ A) @ B) = (^ [R5:(A > (B > $o))]: ((! [X2:A,Y5:B,Z4:B]: ((((R5 @ X2) @ Y5) => (((R5 @ X2) @ Z4) => (Y5 = Z4))))))))))).
% 2.86/1.55 thf(fact_184_left__uniqueD, axiom, (! [B:$tType,A:$tType,A3:(A > (B > $o)),X:A,Z2:B,Y:A]: (((((left_unique @ A) @ B) @ A3) => (((A3 @ X) @ Z2) => (((A3 @ Y) @ Z2) => (X = Y))))))).
% 2.86/1.55 thf(fact_185_left__uniqueI, axiom, (! [B:$tType,A:$tType,A3:(A > (B > $o))]: (((! [X3:A,Y3:A,Z3:B]: ((((A3 @ X3) @ Z3) => (((A3 @ Y3) @ Z3) => (X3 = Y3))))) => (((left_unique @ A) @ B) @ A3))))).
% 2.86/1.55 thf(fact_186_left__unique__eq, axiom, (! [A:$tType]: ((((left_unique @ A) @ A) @ (^ [Y4:A,Z:A]: ((Y4 = Z))))))).
% 2.86/1.55 thf(fact_187_left__unique__def, axiom, (! [B:$tType,A:$tType]: ((((left_unique @ A) @ B) = (^ [R5:(A > (B > $o))]: ((! [X2:A,Y5:A,Z4:B]: ((((R5 @ X2) @ Z4) => (((R5 @ Y5) @ Z4) => (X2 = Y5))))))))))).
% 2.86/1.55 thf(fact_188_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A))))))))).
% 2.86/1.55 thf(fact_189_trm_Oset__cases, axiom, (! [C:$tType,A:$tType,E3:A,A2:((trm @ A) @ C)]: (((((member @ A) @ E3) @ (((set_trm @ A) @ C) @ A2)) => ((! [Z22:(C > ((trm @ A) @ C))]: ((A2 != ((((function @ A) @ C) @ E3) @ Z22)))) => ((! [Z1:A,Z22:(C > ((trm @ A) @ C))]: (((A2 = ((((function @ A) @ C) @ Z1) @ Z22)) => (! [X3:((trm @ A) @ C)]: (((((member @ ((trm @ A) @ C)) @ X3) @ ((((image @ C) @ ((trm @ A) @ C)) @ Z22) @ (top_top @ (set @ C)))) => (~ ((((member @ A) @ E3) @ (((set_trm @ A) @ C) @ X3)))))))))) => ((! [Z1:((trm @ A) @ C)]: (((? [Z22:((trm @ A) @ C)]: ((A2 = ((((plus @ A) @ C) @ Z1) @ Z22)))) => (~ ((((member @ A) @ E3) @ (((set_trm @ A) @ C) @ Z1))))))) => ((! [Z1:((trm @ A) @ C),Z22:((trm @ A) @ C)]: (((A2 = ((((plus @ A) @ C) @ Z1) @ Z22)) => (~ ((((member @ A) @ E3) @ (((set_trm @ A) @ C) @ Z22))))))) => ((! [Z1:((trm @ A) @ C)]: (((? [Z22:((trm @ A) @ C)]: ((A2 = ((((times @ A) @ C) @ Z1) @ Z22)))) => (~ ((((member @ A) @ E3) @ (((set_trm @ A) @ C) @ Z1))))))) => ((! [Z1:((trm @ A) @ C),Z22:((trm @ A) @ C)]: (((A2 = ((((times @ A) @ C) @ Z1) @ Z22)) => (~ ((((member @ A) @ E3) @ (((set_trm @ A) @ C) @ Z22))))))) => (~ ((! [Z3:((trm @ A) @ C)]: (((A2 = (((differential @ A) @ C) @ Z3)) => (~ ((((member @ A) @ E3) @ (((set_trm @ A) @ C) @ Z3))))))))))))))))))).
% 2.86/1.55 thf(fact_190_surjD, axiom, (! [A:$tType,B:$tType,F:(B > A),Y:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X3:B]: ((Y = (F @ X3)))))))).
% 2.86/1.55 thf(fact_191_surjE, axiom, (! [A:$tType,B:$tType,F:(B > A),Y:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X3:B]: ((Y != (F @ X3)))))))))).
% 2.86/1.55 thf(fact_192_trm_Oset__intros_I1_J, axiom, (! [A:$tType,C:$tType,X31:A,X32:(C > ((trm @ A) @ C))]: ((((member @ A) @ X31) @ (((set_trm @ A) @ C) @ ((((function @ A) @ C) @ X31) @ X32)))))).
% 2.86/1.55 thf(fact_193_trm_Oset__intros_I4_J, axiom, (! [C:$tType,A:$tType,Yc:A,X42:((trm @ A) @ C),X41:((trm @ A) @ C)]: (((((member @ A) @ Yc) @ (((set_trm @ A) @ C) @ X42)) => (((member @ A) @ Yc) @ (((set_trm @ A) @ C) @ ((((plus @ A) @ C) @ X41) @ X42))))))).
% 2.86/1.55 thf(fact_194_trm_Oset__intros_I3_J, axiom, (! [C:$tType,A:$tType,Yb:A,X41:((trm @ A) @ C),X42:((trm @ A) @ C)]: (((((member @ A) @ Yb) @ (((set_trm @ A) @ C) @ X41)) => (((member @ A) @ Yb) @ (((set_trm @ A) @ C) @ ((((plus @ A) @ C) @ X41) @ X42))))))).
% 2.86/1.55 thf(fact_195_trm_Oset__intros_I6_J, axiom, (! [C:$tType,A:$tType,Ye:A,X52:((trm @ A) @ C),X51:((trm @ A) @ C)]: (((((member @ A) @ Ye) @ (((set_trm @ A) @ C) @ X52)) => (((member @ A) @ Ye) @ (((set_trm @ A) @ C) @ ((((times @ A) @ C) @ X51) @ X52))))))).
% 2.86/1.55 thf(fact_196_trm_Oset__intros_I5_J, axiom, (! [C:$tType,A:$tType,Yd:A,X51:((trm @ A) @ C),X52:((trm @ A) @ C)]: (((((member @ A) @ Yd) @ (((set_trm @ A) @ C) @ X51)) => (((member @ A) @ Yd) @ (((set_trm @ A) @ C) @ ((((times @ A) @ C) @ X51) @ X52))))))).
% 2.86/1.55 thf(fact_197_trm_Osimps_I126_J, axiom, (! [C:$tType,A:$tType,X7:((trm @ A) @ C)]: (((((set_trm @ A) @ C) @ (((differential @ A) @ C) @ X7)) = (((set_trm @ A) @ C) @ X7))))).
% 2.86/1.55 thf(fact_198_trm_Oset__intros_I7_J, axiom, (! [C:$tType,A:$tType,Yf:A,X7:((trm @ A) @ C)]: (((((member @ A) @ Yf) @ (((set_trm @ A) @ C) @ X7)) => (((member @ A) @ Yf) @ (((set_trm @ A) @ C) @ (((differential @ A) @ C) @ X7))))))).
% 2.86/1.55 thf(fact_199_trm_Oset__intros_I2_J, axiom, (! [A:$tType,C:$tType,Y:((trm @ A) @ C),X32:(C > ((trm @ A) @ C)),Ya:A,X31:A]: (((((member @ ((trm @ A) @ C)) @ Y) @ ((((image @ C) @ ((trm @ A) @ C)) @ X32) @ (top_top @ (set @ C)))) => ((((member @ A) @ Ya) @ (((set_trm @ A) @ C) @ Y)) => (((member @ A) @ Ya) @ (((set_trm @ A) @ C) @ ((((function @ A) @ C) @ X31) @ X32)))))))).
% 2.86/1.55 thf(fact_200_surj__def, axiom, (! [B:$tType,A:$tType,F:(B > A)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y5:A]: ((? [X2:B]: ((Y5 = (F @ X2)))))))))).
% 2.86/1.55 thf(fact_201_surjI, axiom, (! [B:$tType,A:$tType,G:(B > A),F:(A > B)]: (((! [X3:A]: (((G @ (F @ X3)) = X3))) => (((((image @ B) @ A) @ G) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.86/1.55 thf(fact_202_ids_Oexpand__singleton, axiom, (! [Sc:$tType,Sf:$tType,A:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,T4:((trm @ A) @ Sz)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((((singleton @ Sz) @ A) @ Vid1) @ T4) = (^ [I5:Sz]: (((((if @ ((trm @ A) @ Sz)) @ (I5 = Vid1)) @ T4) @ (((const @ A) @ Sz) @ (zero_zero @ real))))))))))))).
% 2.86/1.55 thf(fact_203_ids_Osingleton_Osimps, axiom, (! [Sf:$tType,Sc:$tType,A:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sc) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,I4:Sz,T4:((trm @ A) @ Sz)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((I4 = Vid1) => ((((((singleton @ Sz) @ A) @ Vid1) @ T4) @ I4) = T4)) & ((I4 != Vid1) => ((((((singleton @ Sz) @ A) @ Vid1) @ T4) @ I4) = (((const @ A) @ Sz) @ (zero_zero @ real)))))))))))).
% 2.86/1.55 thf(fact_204_ids_Oproj__sing1, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,A:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,Theta3:((trm @ A) @ Sz)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((((singleton @ Sz) @ A) @ Vid1) @ Theta3) @ Vid1) = Theta3)))))))).
% 2.86/1.55 thf(fact_205_ids_Osingleton_Ocong, axiom, (! [A:$tType,Sz:$tType]: ((((finite_finite @ Sz) & (linorder @ Sz)) => (((singleton @ Sz) @ A) = ((singleton @ Sz) @ A)))))).
% 2.86/1.55 thf(fact_206_ids_Oproj__sing2, axiom, (! [Sf:$tType,Sc:$tType,A:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sc) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,Y:Sz,Theta3:((trm @ A) @ Sz)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((Vid1 != Y) => ((((((singleton @ Sz) @ A) @ Vid1) @ Theta3) @ Y) = (((const @ A) @ Sz) @ (zero_zero @ real))))))))))).
% 2.86/1.55 thf(fact_207_ids_Osingleton_Oelims, axiom, (! [Sc:$tType,Sf:$tType,A:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X:((trm @ A) @ Sz),Xa:Sz,Y:((trm @ A) @ Sz)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((((((singleton @ Sz) @ A) @ Vid1) @ X) @ Xa) = Y) => (((Xa = Vid1) => (Y = X)) & ((Xa != Vid1) => (Y = (((const @ A) @ Sz) @ (zero_zero @ real))))))))))))).
% 2.86/1.55 thf(fact_208_ids_Of1__def, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,F:Sf,X:Sz]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((((f1 @ Sz) @ Sf) @ Vid1) @ F) @ X) = ((((function @ Sf) @ Sz) @ F) @ ((((singleton @ Sz) @ Sf) @ Vid1) @ (((var @ Sz) @ Sf) @ X))))))))))).
% 2.86/1.55 thf(fact_209_ids_Osingleton_Opelims, axiom, (! [Sc:$tType,Sf:$tType,A:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X:((trm @ A) @ Sz),Xa:Sz,Y:((trm @ A) @ Sz)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((((((singleton @ Sz) @ A) @ Vid1) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ ((trm @ A) @ Sz)) @ Sz)) @ ((singleton_rel @ A) @ Sz)) @ ((((product_Pair @ ((trm @ A) @ Sz)) @ Sz) @ X) @ Xa)) => (~ (((((Xa = Vid1) => (Y = X)) & ((Xa != Vid1) => (Y = (((const @ A) @ Sz) @ (zero_zero @ real))))) => (~ ((((accp @ ((product_prod @ ((trm @ A) @ Sz)) @ Sz)) @ ((singleton_rel @ A) @ Sz)) @ ((((product_Pair @ ((trm @ A) @ Sz)) @ Sz) @ X) @ Xa)))))))))))))))).
% 2.86/1.55 thf(fact_210_ids_Of1_Ocong, axiom, (! [Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf))) => (((f1 @ Sz) @ Sf) = ((f1 @ Sz) @ Sf)))))).
% 2.86/1.55 thf(fact_211_singleton_Opelims, axiom, (! [A:$tType,X:((trm @ A) @ sz),Xa:sz,Y:((trm @ A) @ sz)]: ((((((((singleton @ sz) @ A) @ vid1) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ ((trm @ A) @ sz)) @ sz)) @ ((singleton_rel @ A) @ sz)) @ ((((product_Pair @ ((trm @ A) @ sz)) @ sz) @ X) @ Xa)) => (~ (((((Xa = vid1) => (Y = X)) & ((Xa != vid1) => (Y = (((const @ A) @ sz) @ (zero_zero @ real))))) => (~ ((((accp @ ((product_prod @ ((trm @ A) @ sz)) @ sz)) @ ((singleton_rel @ A) @ sz)) @ ((((product_Pair @ ((trm @ A) @ sz)) @ sz) @ X) @ Xa)))))))))))).
% 2.86/1.55 thf(fact_212_f1__def, axiom, (! [F:sf,X:sz]: (((((((f1 @ sz) @ sf) @ vid1) @ F) @ X) = ((((function @ sf) @ sz) @ F) @ ((((singleton @ sz) @ sf) @ vid1) @ (((var @ sz) @ sf) @ X))))))).
% 2.86/1.55 thf(fact_213_proj__sing1, axiom, (! [A:$tType,Theta3:((trm @ A) @ sz)]: (((((((singleton @ sz) @ A) @ vid1) @ Theta3) @ vid1) = Theta3)))).
% 2.86/1.55 thf(fact_214_expand__singleton, axiom, (! [A:$tType,T4:((trm @ A) @ sz)]: ((((((singleton @ sz) @ A) @ vid1) @ T4) = (^ [I5:sz]: (((((if @ ((trm @ A) @ sz)) @ (I5 = vid1)) @ T4) @ (((const @ A) @ sz) @ (zero_zero @ real))))))))).
% 2.86/1.55 thf(fact_215_proj__sing2, axiom, (! [A:$tType,Y:sz,Theta3:((trm @ A) @ sz)]: (((vid1 != Y) => ((((((singleton @ sz) @ A) @ vid1) @ Theta3) @ Y) = (((const @ A) @ sz) @ (zero_zero @ real))))))).
% 2.86/1.55 thf(fact_216_singleton_Oelims, axiom, (! [A:$tType,X:((trm @ A) @ sz),Xa:sz,Y:((trm @ A) @ sz)]: ((((((((singleton @ sz) @ A) @ vid1) @ X) @ Xa) = Y) => (((Xa = vid1) => (Y = X)) & ((Xa != vid1) => (Y = (((const @ A) @ sz) @ (zero_zero @ real))))))))).
% 2.86/1.55 thf(fact_217_singleton_Osimps, axiom, (! [A:$tType,I4:sz,T4:((trm @ A) @ sz)]: ((((I4 = vid1) => ((((((singleton @ sz) @ A) @ vid1) @ T4) @ I4) = T4)) & ((I4 != vid1) => ((((((singleton @ sz) @ A) @ vid1) @ T4) @ I4) = (((const @ A) @ sz) @ (zero_zero @ real)))))))).
% 2.86/1.55 thf(fact_218_local_Oid__simps_I15_J, axiom, (vid3 != vid1)).
% 2.86/1.55 thf(fact_219_local_Oid__simps_I13_J, axiom, (vid2 != vid1)).
% 2.86/1.55 thf(fact_220_vne13, axiom, (vid1 != vid3)).
% 2.86/1.55 thf(fact_221_vne12, axiom, (vid1 != vid2)).
% 2.86/1.55 thf(fact_222_local_Oid__simps_I1_J, axiom, (vid1 != vid2)).
% 2.86/1.55 thf(fact_223_local_Oid__simps_I3_J, axiom, (vid1 != vid3)).
% 2.86/1.55 thf(fact_224_vne23, axiom, (vid2 != vid3)).
% 2.86/1.55 thf(fact_225_local_Oid__simps_I2_J, axiom, (vid2 != vid3)).
% 2.86/1.55 thf(fact_226_local_Oid__simps_I14_J, axiom, (vid3 != vid2)).
% 2.86/1.55 thf(fact_227_p1__def, axiom, (! [P2:sz,X:sz]: ((((((((p1 @ sz) @ sf) @ sc) @ vid1) @ P2) @ X) = (((((prop @ sz) @ sf) @ sc) @ P2) @ ((((singleton @ sz) @ sf) @ vid1) @ (((var @ sz) @ sf) @ X))))))).
% 2.86/1.55 thf(fact_228_prod_Oswap__def, axiom, (! [B:$tType,A:$tType]: ((((product_swap @ A) @ B) = (^ [P3:((product_prod @ A) @ B)]: (((((product_Pair @ B) @ A) @ (((product_snd @ A) @ B) @ P3)) @ (((product_fst @ A) @ B) @ P3)))))))).
% 2.86/1.55 thf(fact_229_formula_Oinject_I2_J, axiom, (! [B:$tType,A:$tType,C:$tType,X21:C,X222:(C > ((trm @ A) @ C)),Y21:C,Y22:(C > ((trm @ A) @ C))]: ((((((((prop @ C) @ A) @ B) @ X21) @ X222) = (((((prop @ C) @ A) @ B) @ Y21) @ Y22)) = ((X21 = Y21) & (X222 = Y22)))))).
% 2.86/1.55 thf(fact_230_swap__swap, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P2)) = P2)))).
% 2.86/1.55 thf(fact_231_pair__in__swap__image, axiom, (! [A:$tType,B:$tType,Y:A,X:B,A3:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y) @ X)) @ ((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ A3)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X) @ Y)) @ A3))))).
% 2.86/1.55 thf(fact_232_swap__simp, axiom, (! [A:$tType,B:$tType,X:B,Y:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ Y) @ X))))).
% 2.86/1.55 thf(fact_233_surj__swap, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ ((product_prod @ A) @ B))))))).
% 2.86/1.55 thf(fact_234_snd__swap, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B)]: (((((product_snd @ B) @ A) @ (((product_swap @ A) @ B) @ X)) = (((product_fst @ A) @ B) @ X))))).
% 2.86/1.55 thf(fact_235_fst__swap, axiom, (! [A:$tType,B:$tType,X:((product_prod @ B) @ A)]: (((((product_fst @ A) @ B) @ (((product_swap @ B) @ A) @ X)) = (((product_snd @ B) @ A) @ X))))).
% 2.86/1.55 thf(fact_236_ids_Op1_Ocong, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => ((((p1 @ Sz) @ Sf) @ Sc) = (((p1 @ Sz) @ Sf) @ Sc)))))).
% 2.86/1.55 thf(fact_237_ids_Op1__def, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,P2:Sz,X:Sz]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((((((p1 @ Sz) @ Sf) @ Sc) @ Vid1) @ P2) @ X) = (((((prop @ Sz) @ Sf) @ Sc) @ P2) @ ((((singleton @ Sz) @ Sf) @ Vid1) @ (((var @ Sz) @ Sf) @ X))))))))))).
% 2.86/1.55 thf(fact_238_formula_Oset__intros_I3_J, axiom, (! [B:$tType,A:$tType,C:$tType,Yb:((trm @ A) @ C),X222:(C > ((trm @ A) @ C)),Yc:A,X21:C]: (((((member @ ((trm @ A) @ C)) @ Yb) @ ((((image @ C) @ ((trm @ A) @ C)) @ X222) @ (top_top @ (set @ C)))) => ((((member @ A) @ Yc) @ (((set_trm @ A) @ C) @ Yb)) => (((member @ A) @ Yc) @ ((((set1_formula @ A) @ B) @ C) @ (((((prop @ C) @ A) @ B) @ X21) @ X222)))))))).
% 2.86/1.55 thf(fact_239_hpfree__ffree_Ointros_I10_J, axiom, (! [B:$tType,A:$tType,C:$tType,Args:(C > ((trm @ A) @ C)),P2:C]: (((! [Arg:((trm @ A) @ C)]: (((((member @ ((trm @ A) @ C)) @ Arg) @ ((((image @ C) @ ((trm @ A) @ C)) @ Args) @ (top_top @ (set @ C)))) => (((dfree @ A) @ C) @ Arg)))) => ((((ffree @ A) @ B) @ C) @ (((((prop @ C) @ A) @ B) @ P2) @ Args)))))).
% 2.86/1.55 thf(fact_240_hpfree__ffree_Ointros_I15_J, axiom, (! [A:$tType,C:$tType,B:$tType,P:B]: (((((ffree @ A) @ B) @ C) @ ((((predicational @ B) @ A) @ C) @ P))))).
% 2.86/1.55 thf(fact_241_map__prod__surj, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F:(A > B),G:(C > D)]: (((((((image @ A) @ B) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((((((image @ C) @ D) @ G) @ (top_top @ (set @ C))) = (top_top @ (set @ D))) => (((((image @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)) @ ((((((product_map_prod @ A) @ B) @ C) @ D) @ F) @ G)) @ (top_top @ (set @ ((product_prod @ A) @ C)))) = (top_top @ (set @ ((product_prod @ B) @ D))))))))).
% 2.86/1.55 thf(fact_242_good__interp_Oright__unique, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (((right_unique @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((frechet_good_interp @ A) @ B) @ C)) @ (((freche457001096interp @ A) @ B) @ C)))))).
% 2.86/1.55 thf(fact_243_good__interp_Oleft__unique, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (((left_unique @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((frechet_good_interp @ A) @ B) @ C)) @ (((freche457001096interp @ A) @ B) @ C)))))).
% 2.86/1.55 thf(fact_244_map__prod__simp, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F:(C > A),G:(D > B),A2:C,B2:D]: (((((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G) @ ((((product_Pair @ C) @ D) @ A2) @ B2)) = ((((product_Pair @ A) @ B) @ (F @ A2)) @ (G @ B2)))))).
% 2.86/1.55 thf(fact_245_fst__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(C > A),G:(D > B),X:((product_prod @ C) @ D)]: (((((product_fst @ A) @ B) @ (((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G) @ X)) = (F @ (((product_fst @ C) @ D) @ X)))))).
% 2.86/1.55 thf(fact_246_snd__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(C > B),G:(D > A),X:((product_prod @ C) @ D)]: (((((product_snd @ B) @ A) @ (((((((product_map_prod @ C) @ B) @ D) @ A) @ F) @ G) @ X)) = (G @ (((product_snd @ C) @ D) @ X)))))).
% 2.86/1.55 thf(fact_247_map__prod__imageI, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,A2:A,B2:B,R6:(set @ ((product_prod @ A) @ B)),F:(A > C),G:(B > D)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R6) => (((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ (F @ A2)) @ (G @ B2))) @ ((((image @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F) @ G)) @ R6)))))).
% 2.86/1.55 thf(fact_248_prod__fun__imageE, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,C3:((product_prod @ A) @ B),F:(C > A),G:(D > B),R6:(set @ ((product_prod @ C) @ D))]: (((((member @ ((product_prod @ A) @ B)) @ C3) @ ((((image @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G)) @ R6)) => (~ ((! [X3:C,Y3:D]: (((C3 = ((((product_Pair @ A) @ B) @ (F @ X3)) @ (G @ Y3))) => (~ ((((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ X3) @ Y3)) @ R6)))))))))))).
% 2.86/1.55 thf(fact_249_ids_Ogood__interp_Oright__unique, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ C) & ((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((right_unique @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((frechet_good_interp @ A) @ B) @ C)) @ (((freche457001096interp @ A) @ B) @ C))))))))).
% 2.86/1.55 thf(fact_250_ids_Ogood__interp_Oleft__unique, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ C) & ((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((left_unique @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((frechet_good_interp @ A) @ B) @ C)) @ (((freche457001096interp @ A) @ B) @ C))))))))).
% 2.86/1.55 thf(fact_251_good__interp_Odomain, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => ((((domainp @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((frechet_good_interp @ A) @ B) @ C)) @ (((freche457001096interp @ A) @ B) @ C)) = (((denota2077489681interp @ A) @ B) @ C)))))).
% 2.86/1.55 thf(fact_252_dterm__sem_Osimps_I6_J, axiom, (! [B:$tType,C:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ C) & (finite_finite @ B))) => (! [I7:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),T4:((trm @ A) @ C)]: (((((((denota594965758rm_sem @ A) @ B) @ C) @ I7) @ (((differential @ A) @ C) @ T4)) = (((((denota2078997598vative @ A) @ B) @ C) @ I7) @ T4)))))))).
% 2.86/1.55 thf(fact_253_ids_Ogood__interp_Odomain, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ C) & ((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((domainp @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((frechet_good_interp @ A) @ B) @ C)) @ (((freche457001096interp @ A) @ B) @ C)) = (((denota2077489681interp @ A) @ B) @ C))))))))).
% 2.86/1.55 thf(fact_254_dterm__sem_Osimps_I7_J, axiom, (! [A:$tType,B:$tType,C:$tType]: ((((finite_finite @ C) & ((finite_finite @ B) & (finite_finite @ A))) => (! [I7:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),C3:real]: (((((((denota594965758rm_sem @ A) @ B) @ C) @ I7) @ (((const @ A) @ C) @ C3)) = (^ [V4:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))]: (C3))))))))).
% 2.86/1.55 thf(subcl_Finite__Set_Ofinite___HOL_Otype, axiom, (! [A:$tType]: (((finite_finite @ A) => (type @ A))))).
% 2.86/1.55 thf(subcl_Orderings_Olinorder___HOL_Otype, axiom, (! [A:$tType]: (((linorder @ A) => (type @ A))))).
% 2.86/1.55 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A7:$tType,A8:$tType]: ((((finite_finite @ A7) & (finite_finite @ A8)) => (finite_finite @ (A7 > A8)))))).
% 2.86/1.55 thf(tcon_fun___Orderings_Otop, axiom, (! [A7:$tType,A8:$tType]: (((top @ A8) => (top @ (A7 > A8)))))).
% 2.86/1.55 thf(tcon_Set_Oset___Finite__Set_Ofinite_1, axiom, (! [A7:$tType]: (((finite_finite @ A7) => (finite_finite @ (set @ A7)))))).
% 2.86/1.55 thf(tcon_Set_Oset___Orderings_Otop_2, axiom, (! [A7:$tType]: ((top @ (set @ A7))))).
% 2.86/1.55 thf(tcon_Set_Oset___Groups_Ozero, axiom, (! [A7:$tType]: (((zero @ A7) => (zero @ (set @ A7)))))).
% 2.86/1.55 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 2.86/1.55 thf(tcon_HOL_Obool___Finite__Set_Ofinite_3, axiom, (finite_finite @ $o)).
% 2.86/1.55 thf(tcon_HOL_Obool___Orderings_Otop_4, axiom, (top @ $o)).
% 2.86/1.55 thf(tcon_Real_Oreal___Orderings_Olinorder_5, axiom, (linorder @ real)).
% 2.86/1.55 thf(tcon_Real_Oreal___Groups_Ozero_6, axiom, (zero @ real)).
% 2.86/1.55 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_7, axiom, (! [A7:$tType,A8:$tType]: ((((finite_finite @ A7) & (finite_finite @ A8)) => (finite_finite @ ((sum_sum @ A7) @ A8)))))).
% 2.86/1.55 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_8, axiom, (! [A7:$tType,A8:$tType]: ((((finite_finite @ A7) & (finite_finite @ A8)) => (finite_finite @ ((product_prod @ A7) @ A8)))))).
% 2.86/1.55 thf(tcon_Product__Type_Oprod___Orderings_Otop_9, axiom, (! [A7:$tType,A8:$tType]: ((((top @ A7) & (top @ A8)) => (top @ ((product_prod @ A7) @ A8)))))).
% 2.86/1.55 thf(tcon_Product__Type_Oprod___Groups_Ozero_10, axiom, (! [A7:$tType,A8:$tType]: ((((zero @ A7) & (zero @ A8)) => (zero @ ((product_prod @ A7) @ A8)))))).
% 2.86/1.55 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_11, axiom, (linorder @ product_unit)).
% 2.86/1.55 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_12, axiom, (finite_finite @ product_unit)).
% 2.86/1.55 thf(tcon_Product__Type_Ounit___Orderings_Otop_13, axiom, (top @ product_unit)).
% 2.86/1.55 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_14, axiom, (! [A7:$tType,A8:$tType]: ((((linorder @ A7) & (cARD_1 @ A8)) => (linorder @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.86/1.55 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_15, axiom, (! [A7:$tType,A8:$tType]: ((((finite_finite @ A7) & (finite_finite @ A8)) => (finite_finite @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.86/1.55 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_16, axiom, (! [A7:$tType,A8:$tType]: ((((zero @ A7) & (finite_finite @ A8)) => (zero @ ((finite_Cartesian_vec @ A7) @ A8)))))).
% 2.86/1.55 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.86/1.55 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.86/1.55 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.86/1.55 thf(tfree_0, hypothesis, (finite_finite @ a)).
% 2.86/1.55 thf(tfree_1, hypothesis, (finite_finite @ c)).
% 2.86/1.55 thf(tfree_2, hypothesis, (finite_finite @ sz)).
% 2.86/1.55 thf(tfree_3, hypothesis, (linorder @ sz)).
% 2.86/1.55 thf(tfree_4, hypothesis, (finite_finite @ sf)).
% 2.86/1.55 thf(tfree_5, hypothesis, (finite_finite @ sc)).
% 2.86/1.55 thf(tfree_6, hypothesis, (finite_finite @ b)).
% 2.86/1.55 thf(conj_0, conjecture, (((dfree @ a) @ c) @ (args @ i2))).
% 2.86/1.55 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.s6Ia5X0taI/DTF2DTF9156.p
% 2.86/1.55 ------------------------
% 2.86/1.59 ---- Cleaned THF ---
% 2.86/1.59 thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext,type,
% 2.86/1.59 denota1663640101rp_ext: $tType > $tType > $tType > $tType > $tType ).
% 2.86/1.59
% 2.86/1.59 thf(ty_t_Frechet__Correctness_Oids_Ogood__interp,type,
% 2.86/1.59 frechet_good_interp: $tType > $tType > $tType > $tType ).
% 2.86/1.59
% 2.86/1.59 thf(ty_t_Frechet__Correctness_Oids_Ostrm,type,
% 2.86/1.59 frechet_strm: $tType > $tType > $tType ).
% 2.86/1.59
% 2.86/1.59 thf(ty_t_Finite__Cartesian__Product_Ovec,type,
% 2.86/1.59 finite_Cartesian_vec: $tType > $tType > $tType ).
% 2.86/1.59
% 2.86/1.59 thf(ty_t_Product__Type_Ounit,type,
% 2.86/1.59 product_unit: $tType ).
% 2.86/1.59
% 2.86/1.59 thf(ty_t_Product__Type_Oprod,type,
% 2.86/1.59 product_prod: $tType > $tType > $tType ).
% 2.86/1.59
% 2.86/1.59 thf(ty_t_Syntax_Oformula,type,
% 2.86/1.59 formula: $tType > $tType > $tType > $tType ).
% 2.86/1.59
% 2.86/1.59 thf(ty_t_Sum__Type_Osum,type,
% 2.86/1.59 sum_sum: $tType > $tType > $tType ).
% 2.86/1.59
% 2.86/1.59 thf(ty_t_Syntax_Otrm,type,
% 2.86/1.59 trm: $tType > $tType > $tType ).
% 2.86/1.59
% 2.86/1.59 thf(ty_t_Real_Oreal,type,
% 2.86/1.59 real: $tType ).
% 2.86/1.59
% 2.86/1.59 thf(ty_t_List_Olist,type,
% 2.86/1.59 list: $tType > $tType ).
% 2.86/1.59
% 2.86/1.59 thf(ty_t_Set_Oset,type,
% 2.86/1.59 set: $tType > $tType ).
% 2.86/1.59
% 2.86/1.59 thf(ty_tf_sz,type,
% 2.86/1.59 sz: $tType ).
% 2.86/1.59
% 2.86/1.59 thf(ty_tf_sf,type,
% 2.86/1.59 sf: $tType ).
% 2.86/1.59
% 2.86/1.59 thf(ty_tf_sc,type,
% 2.86/1.59 sc: $tType ).
% 2.86/1.59
% 2.86/1.59 thf(ty_tf_c,type,
% 2.86/1.59 c: $tType ).
% 2.86/1.59
% 2.86/1.59 thf(ty_tf_b,type,
% 2.86/1.59 b: $tType ).
% 2.86/1.59
% 2.86/1.59 thf(ty_tf_a,type,
% 2.86/1.59 a: $tType ).
% 2.86/1.59
% 2.86/1.59 thf(sy_cl_Cardinality_OCARD__1,type,
% 2.86/1.59 cARD_1:
% 2.86/1.59 !>[A: $tType] : $o ).
% 2.86/1.59
% 2.86/1.59 thf(sy_cl_HOL_Otype,type,
% 2.86/1.59 type:
% 2.86/1.59 !>[A: $tType] : $o ).
% 2.86/1.59
% 2.86/1.59 thf(sy_cl_Groups_Ozero,type,
% 2.86/1.59 zero:
% 2.86/1.59 !>[A: $tType] : $o ).
% 2.86/1.59
% 2.86/1.59 thf(sy_cl_Orderings_Otop,type,
% 2.86/1.59 top:
% 2.86/1.59 !>[A: $tType] : $o ).
% 2.86/1.59
% 2.86/1.59 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.86/1.59 finite_finite:
% 2.86/1.59 !>[A: $tType] : $o ).
% 2.86/1.59
% 2.86/1.59 thf(sy_cl_Orderings_Olinorder,type,
% 2.86/1.59 linorder:
% 2.86/1.59 !>[A: $tType] : $o ).
% 2.86/1.59
% 2.86/1.59 thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd,type,
% 2.86/1.59 euclid1678468529ng_gcd:
% 2.86/1.59 !>[A: $tType] : $o ).
% 2.86/1.59
% 2.86/1.59 thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring,type,
% 2.86/1.59 euclid1155270486miring:
% 2.86/1.59 !>[A: $tType] : $o ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Denotational__Semantics_OVagree,type,
% 2.86/1.59 denotational_Vagree:
% 2.86/1.59 !>[C: $tType] : ( ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( set @ ( sum_sum @ C @ C ) ) > $o ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Denotational__Semantics_Odirectional__derivative,type,
% 2.86/1.59 denota2078997598vative:
% 2.86/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( trm @ A @ C ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > real ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Denotational__Semantics_Odterm__sem,type,
% 2.86/1.59 denota594965758rm_sem:
% 2.86/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( trm @ A @ C ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > real ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Denotational__Semantics_Ofrechet,type,
% 2.86/1.59 denotational_frechet:
% 2.86/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( trm @ A @ C ) > ( finite_Cartesian_vec @ real @ C ) > ( finite_Cartesian_vec @ real @ C ) > real ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Denotational__Semantics_Ois__interp,type,
% 2.86/1.59 denota2077489681interp:
% 2.86/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Frechet__Correctness_Oids_Ocr__good__interp,type,
% 2.86/1.59 freche457001096interp:
% 2.86/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( frechet_good_interp @ A @ B @ C ) > $o ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Frechet__Correctness_Oids_Ocr__strm,type,
% 2.86/1.59 frechet_cr_strm:
% 2.86/1.59 !>[A: $tType,B: $tType] : ( ( trm @ A @ B ) > ( frechet_strm @ A @ B ) > $o ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Ogood__interp,type,
% 2.86/1.59 freche227871258interp:
% 2.86/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( frechet_good_interp @ A @ B @ C ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Oraw__interp,type,
% 2.86/1.59 freche229654227interp:
% 2.86/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( frechet_good_interp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Frechet__Correctness_Oids_Ostrm_Oraw__term,type,
% 2.86/1.59 frechet_raw_term:
% 2.86/1.59 !>[A: $tType,C: $tType] : ( ( frechet_strm @ A @ C ) > ( trm @ A @ C ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Frechet__Correctness_Oids_Ostrm_Osimple__term,type,
% 2.86/1.59 frechet_simple_term:
% 2.86/1.59 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( frechet_strm @ A @ C ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.86/1.59 zero_zero:
% 2.86/1.59 !>[A: $tType] : A ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Ids_Oids,type,
% 2.86/1.59 ids:
% 2.86/1.59 !>[Sz: $tType,Sf: $tType,Sc: $tType] : ( Sz > Sz > Sz > Sf > Sf > Sf > Sc > Sc > Sc > Sc > $o ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_If,type,
% 2.86/1.59 if:
% 2.86/1.59 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.86/1.59 top_top:
% 2.86/1.59 !>[A: $tType] : A ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Product__Type_OPair,type,
% 2.86/1.59 product_Pair:
% 2.86/1.59 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 2.86/1.59 produc2004651681e_prod:
% 2.86/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Product__Type_Omap__prod,type,
% 2.86/1.59 product_map_prod:
% 2.86/1.59 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.86/1.59 product_rec_prod:
% 2.86/1.59 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 2.86/1.59 product_fst:
% 2.86/1.59 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 2.86/1.59 product_snd:
% 2.86/1.59 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 2.86/1.59 product_swap:
% 2.86/1.59 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Relation_ODomainp,type,
% 2.86/1.59 domainp:
% 2.86/1.59 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Set_OCollect,type,
% 2.86/1.59 collect:
% 2.86/1.59 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Set_Oimage,type,
% 2.86/1.59 image:
% 2.86/1.59 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Static__Semantics_OFVDiff,type,
% 2.86/1.59 static_FVDiff:
% 2.86/1.59 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_OPredicational,type,
% 2.86/1.59 predicational:
% 2.86/1.59 !>[B: $tType,A: $tType,C: $tType] : ( B > ( formula @ A @ B @ C ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Odfree,type,
% 2.86/1.59 dfree:
% 2.86/1.59 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > $o ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Odsafe,type,
% 2.86/1.59 dsafe:
% 2.86/1.59 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > $o ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Offree,type,
% 2.86/1.59 ffree:
% 2.86/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > $o ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Oformula_OProp,type,
% 2.86/1.59 prop:
% 2.86/1.59 !>[C: $tType,A: $tType,B: $tType] : ( C > ( C > ( trm @ A @ C ) ) > ( formula @ A @ B @ C ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Oformula_Oset1__formula,type,
% 2.86/1.59 set1_formula:
% 2.86/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( set @ A ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Oids_OP,type,
% 2.86/1.59 p:
% 2.86/1.59 !>[Sc: $tType,Sf: $tType,Sz: $tType] : ( Sc > ( formula @ Sf @ Sc @ Sz ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Oids_Oempty,type,
% 2.86/1.59 empty:
% 2.86/1.59 !>[B: $tType,A: $tType] : ( B > ( trm @ A @ B ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Oids_Of0,type,
% 2.86/1.59 f0:
% 2.86/1.59 !>[Sf: $tType,Sz: $tType] : ( Sf > ( trm @ Sf @ Sz ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Oids_Of1,type,
% 2.86/1.59 f1:
% 2.86/1.59 !>[Sz: $tType,Sf: $tType] : ( Sz > Sf > Sz > ( trm @ Sf @ Sz ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Oids_Op1,type,
% 2.86/1.59 p1:
% 2.86/1.59 !>[Sz: $tType,Sf: $tType,Sc: $tType] : ( Sz > Sz > Sz > ( formula @ Sf @ Sc @ Sz ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Oids_Osingleton,type,
% 2.86/1.59 singleton:
% 2.86/1.59 !>[Sz: $tType,A: $tType] : ( Sz > ( trm @ A @ Sz ) > Sz > ( trm @ A @ Sz ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Oids_Osingleton__rel,type,
% 2.86/1.59 singleton_rel:
% 2.86/1.59 !>[A: $tType,Sz: $tType] : ( ( product_prod @ ( trm @ A @ Sz ) @ Sz ) > ( product_prod @ ( trm @ A @ Sz ) @ Sz ) > $o ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Otrm_OConst,type,
% 2.86/1.59 const:
% 2.86/1.59 !>[A: $tType,C: $tType] : ( real > ( trm @ A @ C ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Otrm_ODiffVar,type,
% 2.86/1.59 diffVar:
% 2.86/1.59 !>[C: $tType,A: $tType] : ( C > ( trm @ A @ C ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Otrm_ODifferential,type,
% 2.86/1.59 differential:
% 2.86/1.59 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( trm @ A @ C ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Otrm_OFunction,type,
% 2.86/1.59 function:
% 2.86/1.59 !>[A: $tType,C: $tType] : ( A > ( C > ( trm @ A @ C ) ) > ( trm @ A @ C ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Otrm_OPlus,type,
% 2.86/1.59 plus:
% 2.86/1.59 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( trm @ A @ C ) > ( trm @ A @ C ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Otrm_OTimes,type,
% 2.86/1.59 times:
% 2.86/1.59 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( trm @ A @ C ) > ( trm @ A @ C ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Otrm_OVar,type,
% 2.86/1.59 var:
% 2.86/1.59 !>[C: $tType,A: $tType] : ( C > ( trm @ A @ C ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Syntax_Otrm_Oset__trm,type,
% 2.86/1.59 set_trm:
% 2.86/1.59 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( set @ A ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Transfer_Oleft__unique,type,
% 2.86/1.59 left_unique:
% 2.86/1.59 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Transfer_Oright__unique,type,
% 2.86/1.59 right_unique:
% 2.86/1.59 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_Wellfounded_Oaccp,type,
% 2.86/1.59 accp:
% 2.86/1.59 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_c_member,type,
% 2.86/1.59 member:
% 2.86/1.59 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_v_I,type,
% 2.86/1.59 i: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
% 2.86/1.59
% 2.86/1.59 thf(sy_v__092_060nu_062,type,
% 2.86/1.59 nu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_v__092_060nu_062_H,type,
% 2.86/1.59 nu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_v_args____,type,
% 2.86/1.59 args: c > ( trm @ a @ c ) ).
% 2.86/1.59
% 2.86/1.59 thf(sy_v_i,type,
% 2.86/1.59 i2: c ).
% 2.86/1.59
% 2.86/1.59 thf(sy_v_var____,type,
% 2.86/1.59 var2: a ).
% 2.86/1.59
% 2.86/1.59 thf(sy_v_vid1,type,
% 2.86/1.59 vid1: sz ).
% 2.86/1.59
% 2.86/1.59 thf(sy_v_vid2,type,
% 2.86/1.59 vid2: sz ).
% 2.86/1.59
% 2.86/1.59 thf(sy_v_vid3,type,
% 2.86/1.59 vid3: sz ).
% 2.86/1.59
% 2.86/1.59 thf(fact_0_free,axiom,
% 2.86/1.59 ! [I: c] : ( dfree @ a @ c @ ( args @ I ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_1_raw__interp__inject,axiom,
% 2.86/1.59 ! [C: $tType,B: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ A )
% 2.86/1.59 & ( finite_finite @ B )
% 2.86/1.59 & ( finite_finite @ C ) )
% 2.86/1.59 => ! [X: frechet_good_interp @ A @ B @ C,Y: frechet_good_interp @ A @ B @ C] :
% 2.86/1.59 ( ( ( freche229654227interp @ A @ B @ C @ X )
% 2.86/1.59 = ( freche229654227interp @ A @ B @ C @ Y ) )
% 2.86/1.59 = ( X = Y ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_2_raw__term__inject,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X: frechet_strm @ A @ C,Y: frechet_strm @ A @ C] :
% 2.86/1.59 ( ( ( frechet_raw_term @ A @ C @ X )
% 2.86/1.59 = ( frechet_raw_term @ A @ C @ Y ) )
% 2.86/1.59 = ( X = Y ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_3_P__def,axiom,
% 2.86/1.59 ( ( p @ sc @ sf @ sz )
% 2.86/1.59 = ( predicational @ sc @ sf @ sz ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_4_strm_Odomain,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType] :
% 2.86/1.59 ( ( domainp @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) )
% 2.86/1.59 = ( dfree @ A @ C ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_5_dfree__DiffVar__simps,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,X: B] :
% 2.86/1.59 ~ ( dfree @ A @ B @ ( diffVar @ B @ A @ X ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_6_seq__sem_Ocases,axiom,
% 2.86/1.59 ! [X: product_prod @ ( denota1663640101rp_ext @ sf @ sc @ sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) ) )] :
% 2.86/1.59 ~ ! [I2: denota1663640101rp_ext @ sf @ sc @ sz @ product_unit,S: product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) )] :
% 2.86/1.59 ( X
% 2.86/1.59 != ( product_Pair @ ( denota1663640101rp_ext @ sf @ sc @ sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) ) ) @ I2 @ S ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_7_singleton_Ocases,axiom,
% 2.86/1.59 ! [A: $tType,X: product_prod @ ( trm @ A @ sz ) @ sz] :
% 2.86/1.59 ~ ! [T2: trm @ A @ sz,I3: sz] :
% 2.86/1.59 ( X
% 2.86/1.59 != ( product_Pair @ ( trm @ A @ sz ) @ sz @ T2 @ I3 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_8_dfree__Differential__simps,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,X: trm @ A @ B] :
% 2.86/1.59 ~ ( dfree @ A @ B @ ( differential @ A @ B @ X ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_9_dfree__Var__simps,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,X: B] : ( dfree @ A @ B @ ( var @ B @ A @ X ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_10_dfree__Times__simps,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,A2: trm @ A @ B,B2: trm @ A @ B] :
% 2.86/1.59 ( ( dfree @ A @ B @ ( times @ A @ B @ A2 @ B2 ) )
% 2.86/1.59 = ( ( dfree @ A @ B @ A2 )
% 2.86/1.59 & ( dfree @ A @ B @ B2 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_11_dfree__Plus__simps,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,A2: trm @ A @ B,B2: trm @ A @ B] :
% 2.86/1.59 ( ( dfree @ A @ B @ ( plus @ A @ B @ A2 @ B2 ) )
% 2.86/1.59 = ( ( dfree @ A @ B @ A2 )
% 2.86/1.59 & ( dfree @ A @ B @ B2 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_12_dfree__Fun_Oprems,axiom,
% 2.86/1.59 denotational_Vagree @ c @ nu @ nu2 @ ( static_FVDiff @ a @ c @ ( function @ a @ c @ var2 @ args ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_13_trm_Oinject_I3_J,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,X31: A,X32: C > ( trm @ A @ C ),Y31: A,Y32: C > ( trm @ A @ C )] :
% 2.86/1.59 ( ( ( function @ A @ C @ X31 @ X32 )
% 2.86/1.59 = ( function @ A @ C @ Y31 @ Y32 ) )
% 2.86/1.59 = ( ( X31 = Y31 )
% 2.86/1.59 & ( X32 = Y32 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_14_trm_Oinject_I4_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X41: trm @ A @ C,X42: trm @ A @ C,Y41: trm @ A @ C,Y42: trm @ A @ C] :
% 2.86/1.59 ( ( ( plus @ A @ C @ X41 @ X42 )
% 2.86/1.59 = ( plus @ A @ C @ Y41 @ Y42 ) )
% 2.86/1.59 = ( ( X41 = Y41 )
% 2.86/1.59 & ( X42 = Y42 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_15_trm_Oinject_I5_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X51: trm @ A @ C,X52: trm @ A @ C,Y51: trm @ A @ C,Y52: trm @ A @ C] :
% 2.86/1.59 ( ( ( times @ A @ C @ X51 @ X52 )
% 2.86/1.59 = ( times @ A @ C @ Y51 @ Y52 ) )
% 2.86/1.59 = ( ( X51 = Y51 )
% 2.86/1.59 & ( X52 = Y52 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_16_trm_Oinject_I1_J,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,X1: C,Y1: C] :
% 2.86/1.59 ( ( ( var @ C @ A @ X1 )
% 2.86/1.59 = ( var @ C @ A @ Y1 ) )
% 2.86/1.59 = ( X1 = Y1 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_17_trm_Oinject_I7_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X7: trm @ A @ C,Y7: trm @ A @ C] :
% 2.86/1.59 ( ( ( differential @ A @ C @ X7 )
% 2.86/1.59 = ( differential @ A @ C @ Y7 ) )
% 2.86/1.59 = ( X7 = Y7 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_18_trm_Oinject_I6_J,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,X6: C,Y6: C] :
% 2.86/1.59 ( ( ( diffVar @ C @ A @ X6 )
% 2.86/1.59 = ( diffVar @ C @ A @ Y6 ) )
% 2.86/1.59 = ( X6 = Y6 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_19_dfree__Fun__simps,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,I4: A,Args: B > ( trm @ A @ B )] :
% 2.86/1.59 ( ( dfree @ A @ B @ ( function @ A @ B @ I4 @ Args ) )
% 2.86/1.59 = ( ! [X2: B] : ( dfree @ A @ B @ ( Args @ X2 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_20_trm_Odistinct_I41_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X6: C,X7: trm @ A @ C] :
% 2.86/1.59 ( ( diffVar @ C @ A @ X6 )
% 2.86/1.59 != ( differential @ A @ C @ X7 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_21_trm_Odistinct_I39_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X51: trm @ A @ C,X52: trm @ A @ C,X7: trm @ A @ C] :
% 2.86/1.59 ( ( times @ A @ C @ X51 @ X52 )
% 2.86/1.59 != ( differential @ A @ C @ X7 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_22_trm_Odistinct_I37_J,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,X51: trm @ A @ C,X52: trm @ A @ C,X6: C] :
% 2.86/1.59 ( ( times @ A @ C @ X51 @ X52 )
% 2.86/1.59 != ( diffVar @ C @ A @ X6 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_23_trm_Odistinct_I35_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X41: trm @ A @ C,X42: trm @ A @ C,X7: trm @ A @ C] :
% 2.86/1.59 ( ( plus @ A @ C @ X41 @ X42 )
% 2.86/1.59 != ( differential @ A @ C @ X7 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_24_trm_Odistinct_I33_J,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,X41: trm @ A @ C,X42: trm @ A @ C,X6: C] :
% 2.86/1.59 ( ( plus @ A @ C @ X41 @ X42 )
% 2.86/1.59 != ( diffVar @ C @ A @ X6 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_25_trm_Odistinct_I31_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X41: trm @ A @ C,X42: trm @ A @ C,X51: trm @ A @ C,X52: trm @ A @ C] :
% 2.86/1.59 ( ( plus @ A @ C @ X41 @ X42 )
% 2.86/1.59 != ( times @ A @ C @ X51 @ X52 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_26_trm_Odistinct_I29_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X31: A,X32: C > ( trm @ A @ C ),X7: trm @ A @ C] :
% 2.86/1.59 ( ( function @ A @ C @ X31 @ X32 )
% 2.86/1.59 != ( differential @ A @ C @ X7 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_27_trm_Odistinct_I27_J,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,X31: A,X32: C > ( trm @ A @ C ),X6: C] :
% 2.86/1.59 ( ( function @ A @ C @ X31 @ X32 )
% 2.86/1.59 != ( diffVar @ C @ A @ X6 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_28_trm_Odistinct_I25_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X31: A,X32: C > ( trm @ A @ C ),X51: trm @ A @ C,X52: trm @ A @ C] :
% 2.86/1.59 ( ( function @ A @ C @ X31 @ X32 )
% 2.86/1.59 != ( times @ A @ C @ X51 @ X52 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_29_trm_Odistinct_I23_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X31: A,X32: C > ( trm @ A @ C ),X41: trm @ A @ C,X42: trm @ A @ C] :
% 2.86/1.59 ( ( function @ A @ C @ X31 @ X32 )
% 2.86/1.59 != ( plus @ A @ C @ X41 @ X42 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_30_trm_Odistinct_I11_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X1: C,X7: trm @ A @ C] :
% 2.86/1.59 ( ( var @ C @ A @ X1 )
% 2.86/1.59 != ( differential @ A @ C @ X7 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_31_trm_Odistinct_I9_J,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,X1: C,X6: C] :
% 2.86/1.59 ( ( var @ C @ A @ X1 )
% 2.86/1.59 != ( diffVar @ C @ A @ X6 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_32_trm_Odistinct_I7_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X1: C,X51: trm @ A @ C,X52: trm @ A @ C] :
% 2.86/1.59 ( ( var @ C @ A @ X1 )
% 2.86/1.59 != ( times @ A @ C @ X51 @ X52 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_33_trm_Odistinct_I5_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X1: C,X41: trm @ A @ C,X42: trm @ A @ C] :
% 2.86/1.59 ( ( var @ C @ A @ X1 )
% 2.86/1.59 != ( plus @ A @ C @ X41 @ X42 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_34_trm_Odistinct_I3_J,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,X1: C,X31: A,X32: C > ( trm @ A @ C )] :
% 2.86/1.59 ( ( var @ C @ A @ X1 )
% 2.86/1.59 != ( function @ A @ C @ X31 @ X32 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_35_dfree_Odfree__Fun,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,Args: C > ( trm @ A @ C ),I4: A] :
% 2.86/1.59 ( ! [I3: C] : ( dfree @ A @ C @ ( Args @ I3 ) )
% 2.86/1.59 => ( dfree @ A @ C @ ( function @ A @ C @ I4 @ Args ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_36_dfree__Plus,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,Theta_1: trm @ A @ C,Theta_2: trm @ A @ C] :
% 2.86/1.59 ( ( dfree @ A @ C @ Theta_1 )
% 2.86/1.59 => ( ( dfree @ A @ C @ Theta_2 )
% 2.86/1.59 => ( dfree @ A @ C @ ( plus @ A @ C @ Theta_1 @ Theta_2 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_37_dfree__Times,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,Theta_1: trm @ A @ C,Theta_2: trm @ A @ C] :
% 2.86/1.59 ( ( dfree @ A @ C @ Theta_1 )
% 2.86/1.59 => ( ( dfree @ A @ C @ Theta_2 )
% 2.86/1.59 => ( dfree @ A @ C @ ( times @ A @ C @ Theta_1 @ Theta_2 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_38_dfree__Var,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,I4: C] : ( dfree @ A @ C @ ( var @ C @ A @ I4 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_39_raw__interp__inverse,axiom,
% 2.86/1.59 ! [C: $tType,B: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ A )
% 2.86/1.59 & ( finite_finite @ B )
% 2.86/1.59 & ( finite_finite @ C ) )
% 2.86/1.59 => ! [X: frechet_good_interp @ A @ B @ C] :
% 2.86/1.59 ( ( freche227871258interp @ A @ B @ C @ ( freche229654227interp @ A @ B @ C @ X ) )
% 2.86/1.59 = X ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_40_raw__term__inverse,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X: frechet_strm @ A @ C] :
% 2.86/1.59 ( ( frechet_simple_term @ A @ C @ ( frechet_raw_term @ A @ C @ X ) )
% 2.86/1.59 = X ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_41_agree__times1,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType] :
% 2.86/1.59 ( ( finite_finite @ A )
% 2.86/1.59 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),T1: trm @ B @ A,T22: trm @ B @ A] :
% 2.86/1.59 ( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( times @ B @ A @ T1 @ T22 ) ) )
% 2.86/1.59 => ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ T1 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_42_agree__times2,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType] :
% 2.86/1.59 ( ( finite_finite @ A )
% 2.86/1.59 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),T1: trm @ B @ A,T22: trm @ B @ A] :
% 2.86/1.59 ( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( times @ B @ A @ T1 @ T22 ) ) )
% 2.86/1.59 => ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ T22 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_43_agree__plus1,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType] :
% 2.86/1.59 ( ( finite_finite @ A )
% 2.86/1.59 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),T1: trm @ B @ A,T22: trm @ B @ A] :
% 2.86/1.59 ( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( plus @ B @ A @ T1 @ T22 ) ) )
% 2.86/1.59 => ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ T1 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_44_agree__plus2,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType] :
% 2.86/1.59 ( ( finite_finite @ A )
% 2.86/1.59 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),T1: trm @ B @ A,T22: trm @ B @ A] :
% 2.86/1.59 ( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( plus @ B @ A @ T1 @ T22 ) ) )
% 2.86/1.59 => ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ T22 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_45_mem__Collect__eq,axiom,
% 2.86/1.59 ! [A: $tType,A2: A,P: A > $o] :
% 2.86/1.59 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.86/1.59 = ( P @ A2 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_46_Collect__mem__eq,axiom,
% 2.86/1.59 ! [A: $tType,A3: set @ A] :
% 2.86/1.59 ( ( collect @ A
% 2.86/1.59 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 2.86/1.59 = A3 ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_47_Collect__cong,axiom,
% 2.86/1.59 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.86/1.59 ( ! [X3: A] :
% 2.86/1.59 ( ( P @ X3 )
% 2.86/1.59 = ( Q @ X3 ) )
% 2.86/1.59 => ( ( collect @ A @ P )
% 2.86/1.59 = ( collect @ A @ Q ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_48_ext,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.86/1.59 ( ! [X3: A] :
% 2.86/1.59 ( ( F @ X3 )
% 2.86/1.59 = ( G @ X3 ) )
% 2.86/1.59 => ( F = G ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_49_agree__func,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType] :
% 2.86/1.59 ( ( finite_finite @ A )
% 2.86/1.59 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Var: B,Args: A > ( trm @ B @ A ),I4: A] :
% 2.86/1.59 ( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( function @ B @ A @ Var @ Args ) ) )
% 2.86/1.59 => ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( Args @ I4 ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_50_prod_Oinject,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 2.86/1.59 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 2.86/1.59 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 2.86/1.59 = ( ( X1 = Y1 )
% 2.86/1.59 & ( X22 = Y2 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_51_old_Oprod_Oinject,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B] :
% 2.86/1.59 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.86/1.59 = ( product_Pair @ A @ B @ A4 @ B3 ) )
% 2.86/1.59 = ( ( A2 = A4 )
% 2.86/1.59 & ( B2 = B3 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_52_trm_Oexhaust,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,Y: trm @ A @ C] :
% 2.86/1.59 ( ! [X12: C] :
% 2.86/1.59 ( Y
% 2.86/1.59 != ( var @ C @ A @ X12 ) )
% 2.86/1.59 => ( ! [X23: real] :
% 2.86/1.59 ( Y
% 2.86/1.59 != ( const @ A @ C @ X23 ) )
% 2.86/1.59 => ( ! [X312: A,X322: C > ( trm @ A @ C )] :
% 2.86/1.59 ( Y
% 2.86/1.59 != ( function @ A @ C @ X312 @ X322 ) )
% 2.86/1.59 => ( ! [X412: trm @ A @ C,X422: trm @ A @ C] :
% 2.86/1.59 ( Y
% 2.86/1.59 != ( plus @ A @ C @ X412 @ X422 ) )
% 2.86/1.59 => ( ! [X512: trm @ A @ C,X522: trm @ A @ C] :
% 2.86/1.59 ( Y
% 2.86/1.59 != ( times @ A @ C @ X512 @ X522 ) )
% 2.86/1.59 => ( ! [X62: C] :
% 2.86/1.59 ( Y
% 2.86/1.59 != ( diffVar @ C @ A @ X62 ) )
% 2.86/1.59 => ~ ! [X72: trm @ A @ C] :
% 2.86/1.59 ( Y
% 2.86/1.59 != ( differential @ A @ C @ X72 ) ) ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_53_prod__cases3,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.86/1.59 ~ ! [A5: A,B4: B,C2: C] :
% 2.86/1.59 ( Y
% 2.86/1.59 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_54_trm_Oinject_I2_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X22: real,Y2: real] :
% 2.86/1.59 ( ( ( const @ A @ C @ X22 )
% 2.86/1.59 = ( const @ A @ C @ Y2 ) )
% 2.86/1.59 = ( X22 = Y2 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_55_dfree__Const__simps,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,R: real] : ( dfree @ A @ B @ ( const @ A @ B @ R ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_56_dfree__Const,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,R: real] : ( dfree @ A @ C @ ( const @ A @ C @ R ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_57_trm_Odistinct_I13_J,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,X22: real,X31: A,X32: C > ( trm @ A @ C )] :
% 2.86/1.59 ( ( const @ A @ C @ X22 )
% 2.86/1.59 != ( function @ A @ C @ X31 @ X32 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_58_trm_Odistinct_I15_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X22: real,X41: trm @ A @ C,X42: trm @ A @ C] :
% 2.86/1.59 ( ( const @ A @ C @ X22 )
% 2.86/1.59 != ( plus @ A @ C @ X41 @ X42 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_59_trm_Odistinct_I17_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X22: real,X51: trm @ A @ C,X52: trm @ A @ C] :
% 2.86/1.59 ( ( const @ A @ C @ X22 )
% 2.86/1.59 != ( times @ A @ C @ X51 @ X52 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_60_trm_Odistinct_I1_J,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,X1: C,X22: real] :
% 2.86/1.59 ( ( var @ C @ A @ X1 )
% 2.86/1.59 != ( const @ A @ C @ X22 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_61_trm_Odistinct_I21_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X22: real,X7: trm @ A @ C] :
% 2.86/1.59 ( ( const @ A @ C @ X22 )
% 2.86/1.59 != ( differential @ A @ C @ X7 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_62_trm_Odistinct_I19_J,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,X22: real,X6: C] :
% 2.86/1.59 ( ( const @ A @ C @ X22 )
% 2.86/1.59 != ( diffVar @ C @ A @ X6 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_63_old_Oprod_Oinducts,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.86/1.59 ( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 2.86/1.59 => ( P @ Prod ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_64_old_Oprod_Oexhaust,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 2.86/1.59 ~ ! [A5: A,B4: B] :
% 2.86/1.59 ( Y
% 2.86/1.59 != ( product_Pair @ A @ B @ A5 @ B4 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_65_Pair__inject,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B] :
% 2.86/1.59 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.86/1.59 = ( product_Pair @ A @ B @ A4 @ B3 ) )
% 2.86/1.59 => ~ ( ( A2 = A4 )
% 2.86/1.59 => ( B2 != B3 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_66_prod__cases,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 2.86/1.59 ( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 2.86/1.59 => ( P @ P2 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_67_surj__pair,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 2.86/1.59 ? [X3: A,Y3: B] :
% 2.86/1.59 ( P2
% 2.86/1.59 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_68_dfree_Oinducts,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X: trm @ A @ C,P: ( trm @ A @ C ) > $o] :
% 2.86/1.59 ( ( dfree @ A @ C @ X )
% 2.86/1.59 => ( ! [I3: C] : ( P @ ( var @ C @ A @ I3 ) )
% 2.86/1.59 => ( ! [R2: real] : ( P @ ( const @ A @ C @ R2 ) )
% 2.86/1.59 => ( ! [Args2: C > ( trm @ A @ C ),I3: A] :
% 2.86/1.59 ( ! [Ia: C] : ( dfree @ A @ C @ ( Args2 @ Ia ) )
% 2.86/1.59 => ( ! [Ia: C] : ( P @ ( Args2 @ Ia ) )
% 2.86/1.59 => ( P @ ( function @ A @ C @ I3 @ Args2 ) ) ) )
% 2.86/1.59 => ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
% 2.86/1.59 ( ( dfree @ A @ C @ Theta_12 )
% 2.86/1.59 => ( ( P @ Theta_12 )
% 2.86/1.59 => ( ( dfree @ A @ C @ Theta_22 )
% 2.86/1.59 => ( ( P @ Theta_22 )
% 2.86/1.59 => ( P @ ( plus @ A @ C @ Theta_12 @ Theta_22 ) ) ) ) ) )
% 2.86/1.59 => ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
% 2.86/1.59 ( ( dfree @ A @ C @ Theta_12 )
% 2.86/1.59 => ( ( P @ Theta_12 )
% 2.86/1.59 => ( ( dfree @ A @ C @ Theta_22 )
% 2.86/1.59 => ( ( P @ Theta_22 )
% 2.86/1.59 => ( P @ ( times @ A @ C @ Theta_12 @ Theta_22 ) ) ) ) ) )
% 2.86/1.59 => ( P @ X ) ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_69_dfree_Osimps,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType] :
% 2.86/1.59 ( ( dfree @ A @ C )
% 2.86/1.59 = ( ^ [A6: trm @ A @ C] :
% 2.86/1.59 ( ? [I5: C] :
% 2.86/1.59 ( A6
% 2.86/1.59 = ( var @ C @ A @ I5 ) )
% 2.86/1.59 | ? [R3: real] :
% 2.86/1.59 ( A6
% 2.86/1.59 = ( const @ A @ C @ R3 ) )
% 2.86/1.59 | ? [Args3: C > ( trm @ A @ C ),I5: A] :
% 2.86/1.59 ( ( A6
% 2.86/1.59 = ( function @ A @ C @ I5 @ Args3 ) )
% 2.86/1.59 & ! [X2: C] : ( dfree @ A @ C @ ( Args3 @ X2 ) ) )
% 2.86/1.59 | ? [Theta_13: trm @ A @ C,Theta_23: trm @ A @ C] :
% 2.86/1.59 ( ( A6
% 2.86/1.59 = ( plus @ A @ C @ Theta_13 @ Theta_23 ) )
% 2.86/1.59 & ( dfree @ A @ C @ Theta_13 )
% 2.86/1.59 & ( dfree @ A @ C @ Theta_23 ) )
% 2.86/1.59 | ? [Theta_13: trm @ A @ C,Theta_23: trm @ A @ C] :
% 2.86/1.59 ( ( A6
% 2.86/1.59 = ( times @ A @ C @ Theta_13 @ Theta_23 ) )
% 2.86/1.59 & ( dfree @ A @ C @ Theta_13 )
% 2.86/1.59 & ( dfree @ A @ C @ Theta_23 ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_70_dfree_Ocases,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,A2: trm @ A @ C] :
% 2.86/1.59 ( ( dfree @ A @ C @ A2 )
% 2.86/1.59 => ( ! [I3: C] :
% 2.86/1.59 ( A2
% 2.86/1.59 != ( var @ C @ A @ I3 ) )
% 2.86/1.59 => ( ! [R2: real] :
% 2.86/1.59 ( A2
% 2.86/1.59 != ( const @ A @ C @ R2 ) )
% 2.86/1.59 => ( ! [Args2: C > ( trm @ A @ C )] :
% 2.86/1.59 ( ? [I3: A] :
% 2.86/1.59 ( A2
% 2.86/1.59 = ( function @ A @ C @ I3 @ Args2 ) )
% 2.86/1.59 => ~ ! [I6: C] : ( dfree @ A @ C @ ( Args2 @ I6 ) ) )
% 2.86/1.59 => ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
% 2.86/1.59 ( ( A2
% 2.86/1.59 = ( plus @ A @ C @ Theta_12 @ Theta_22 ) )
% 2.86/1.59 => ( ( dfree @ A @ C @ Theta_12 )
% 2.86/1.59 => ~ ( dfree @ A @ C @ Theta_22 ) ) )
% 2.86/1.59 => ~ ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
% 2.86/1.59 ( ( A2
% 2.86/1.59 = ( times @ A @ C @ Theta_12 @ Theta_22 ) )
% 2.86/1.59 => ( ( dfree @ A @ C @ Theta_12 )
% 2.86/1.59 => ~ ( dfree @ A @ C @ Theta_22 ) ) ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_71_prod__induct7,axiom,
% 2.86/1.59 ! [G2: $tType,F2: $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 @ F2 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 2.86/1.59 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
% 2.86/1.59 => ( P @ X ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_72_prod__induct6,axiom,
% 2.86/1.59 ! [F2: $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 @ F2 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
% 2.86/1.59 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
% 2.86/1.59 => ( P @ X ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_73_prod__induct5,axiom,
% 2.86/1.59 ! [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 ) ) )] :
% 2.86/1.59 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( 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 ) ) ) ) )
% 2.86/1.59 => ( P @ X ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_74_prod__induct4,axiom,
% 2.86/1.59 ! [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 ) )] :
% 2.86/1.59 ( ! [A5: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 2.86/1.59 => ( P @ X ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_75_prod__induct3,axiom,
% 2.86/1.59 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.86/1.59 ( ! [A5: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 2.86/1.59 => ( P @ X ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_76_prod__cases7,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 2.86/1.59 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2,G3: G2] :
% 2.86/1.59 ( Y
% 2.86/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_77_prod__cases6,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
% 2.86/1.59 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2] :
% 2.86/1.59 ( Y
% 2.86/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_78_prod__cases5,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.86/1.59 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E] :
% 2.86/1.59 ( Y
% 2.86/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( 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 ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_79_prod__cases4,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.86/1.59 ~ ! [A5: A,B4: B,C2: C,D2: D] :
% 2.86/1.59 ( Y
% 2.86/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_80_old_Oprod_Orec,axiom,
% 2.86/1.59 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 2.86/1.59 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.86/1.59 = ( F1 @ A2 @ B2 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_81_f0__def,axiom,
% 2.86/1.59 ( ( f0 @ sf @ sz )
% 2.86/1.59 = ( ^ [F4: sf] : ( function @ sf @ sz @ F4 @ ( empty @ sz @ sf ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_82_dsafe_Oinducts,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X: trm @ A @ C,P: ( trm @ A @ C ) > $o] :
% 2.86/1.59 ( ( dsafe @ A @ C @ X )
% 2.86/1.59 => ( ! [I3: C] : ( P @ ( var @ C @ A @ I3 ) )
% 2.86/1.59 => ( ! [R2: real] : ( P @ ( const @ A @ C @ R2 ) )
% 2.86/1.59 => ( ! [Args2: C > ( trm @ A @ C ),I3: A] :
% 2.86/1.59 ( ! [Ia: C] : ( dsafe @ A @ C @ ( Args2 @ Ia ) )
% 2.86/1.59 => ( ! [Ia: C] : ( P @ ( Args2 @ Ia ) )
% 2.86/1.59 => ( P @ ( function @ A @ C @ I3 @ Args2 ) ) ) )
% 2.86/1.59 => ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
% 2.86/1.59 ( ( dsafe @ A @ C @ Theta_12 )
% 2.86/1.59 => ( ( P @ Theta_12 )
% 2.86/1.59 => ( ( dsafe @ A @ C @ Theta_22 )
% 2.86/1.59 => ( ( P @ Theta_22 )
% 2.86/1.59 => ( P @ ( plus @ A @ C @ Theta_12 @ Theta_22 ) ) ) ) ) )
% 2.86/1.59 => ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
% 2.86/1.59 ( ( dsafe @ A @ C @ Theta_12 )
% 2.86/1.59 => ( ( P @ Theta_12 )
% 2.86/1.59 => ( ( dsafe @ A @ C @ Theta_22 )
% 2.86/1.59 => ( ( P @ Theta_22 )
% 2.86/1.59 => ( P @ ( times @ A @ C @ Theta_12 @ Theta_22 ) ) ) ) ) )
% 2.86/1.59 => ( ! [Theta: trm @ A @ C] :
% 2.86/1.59 ( ( dfree @ A @ C @ Theta )
% 2.86/1.59 => ( P @ ( differential @ A @ C @ Theta ) ) )
% 2.86/1.59 => ( ! [I3: C] : ( P @ ( diffVar @ C @ A @ I3 ) )
% 2.86/1.59 => ( P @ X ) ) ) ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_83_dsafe_Osimps,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType] :
% 2.86/1.59 ( ( dsafe @ A @ C )
% 2.86/1.59 = ( ^ [A6: trm @ A @ C] :
% 2.86/1.59 ( ? [I5: C] :
% 2.86/1.59 ( A6
% 2.86/1.59 = ( var @ C @ A @ I5 ) )
% 2.86/1.59 | ? [R3: real] :
% 2.86/1.59 ( A6
% 2.86/1.59 = ( const @ A @ C @ R3 ) )
% 2.86/1.59 | ? [Args3: C > ( trm @ A @ C ),I5: A] :
% 2.86/1.59 ( ( A6
% 2.86/1.59 = ( function @ A @ C @ I5 @ Args3 ) )
% 2.86/1.59 & ! [X2: C] : ( dsafe @ A @ C @ ( Args3 @ X2 ) ) )
% 2.86/1.59 | ? [Theta_13: trm @ A @ C,Theta_23: trm @ A @ C] :
% 2.86/1.59 ( ( A6
% 2.86/1.59 = ( plus @ A @ C @ Theta_13 @ Theta_23 ) )
% 2.86/1.59 & ( dsafe @ A @ C @ Theta_13 )
% 2.86/1.59 & ( dsafe @ A @ C @ Theta_23 ) )
% 2.86/1.59 | ? [Theta_13: trm @ A @ C,Theta_23: trm @ A @ C] :
% 2.86/1.59 ( ( A6
% 2.86/1.59 = ( times @ A @ C @ Theta_13 @ Theta_23 ) )
% 2.86/1.59 & ( dsafe @ A @ C @ Theta_13 )
% 2.86/1.59 & ( dsafe @ A @ C @ Theta_23 ) )
% 2.86/1.59 | ? [Theta2: trm @ A @ C] :
% 2.86/1.59 ( ( A6
% 2.86/1.59 = ( differential @ A @ C @ Theta2 ) )
% 2.86/1.59 & ( dfree @ A @ C @ Theta2 ) )
% 2.86/1.59 | ? [I5: C] :
% 2.86/1.59 ( A6
% 2.86/1.59 = ( diffVar @ C @ A @ I5 ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_84_dsafe_Ocases,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,A2: trm @ A @ C] :
% 2.86/1.59 ( ( dsafe @ A @ C @ A2 )
% 2.86/1.59 => ( ! [I3: C] :
% 2.86/1.59 ( A2
% 2.86/1.59 != ( var @ C @ A @ I3 ) )
% 2.86/1.59 => ( ! [R2: real] :
% 2.86/1.59 ( A2
% 2.86/1.59 != ( const @ A @ C @ R2 ) )
% 2.86/1.59 => ( ! [Args2: C > ( trm @ A @ C )] :
% 2.86/1.59 ( ? [I3: A] :
% 2.86/1.59 ( A2
% 2.86/1.59 = ( function @ A @ C @ I3 @ Args2 ) )
% 2.86/1.59 => ~ ! [I6: C] : ( dsafe @ A @ C @ ( Args2 @ I6 ) ) )
% 2.86/1.59 => ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
% 2.86/1.59 ( ( A2
% 2.86/1.59 = ( plus @ A @ C @ Theta_12 @ Theta_22 ) )
% 2.86/1.59 => ( ( dsafe @ A @ C @ Theta_12 )
% 2.86/1.59 => ~ ( dsafe @ A @ C @ Theta_22 ) ) )
% 2.86/1.59 => ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
% 2.86/1.59 ( ( A2
% 2.86/1.59 = ( times @ A @ C @ Theta_12 @ Theta_22 ) )
% 2.86/1.59 => ( ( dsafe @ A @ C @ Theta_12 )
% 2.86/1.59 => ~ ( dsafe @ A @ C @ Theta_22 ) ) )
% 2.86/1.59 => ( ! [Theta: trm @ A @ C] :
% 2.86/1.59 ( ( A2
% 2.86/1.59 = ( differential @ A @ C @ Theta ) )
% 2.86/1.59 => ~ ( dfree @ A @ C @ Theta ) )
% 2.86/1.59 => ~ ! [I3: C] :
% 2.86/1.59 ( A2
% 2.86/1.59 != ( diffVar @ C @ A @ I3 ) ) ) ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_85_euclid__ext__aux_Ocases,axiom,
% 2.86/1.59 ! [A: $tType] :
% 2.86/1.59 ( ( euclid1678468529ng_gcd @ A )
% 2.86/1.59 => ! [X: product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) )] :
% 2.86/1.59 ~ ! [S2: A,S3: A,T3: A,T2: A,R4: A,R2: A] :
% 2.86/1.59 ( X
% 2.86/1.59 != ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) ) @ S2 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) @ S3 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) @ T3 @ ( product_Pair @ A @ ( product_prod @ A @ A ) @ T2 @ ( product_Pair @ A @ A @ R4 @ R2 ) ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_86_internal__case__prod__conv,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 2.86/1.59 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 2.86/1.59 = ( C3 @ A2 @ B2 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_87_strm_Oright__unique,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType] : ( right_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_88_dsafe__Const__simps,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,R: real] : ( dsafe @ A @ B @ ( const @ A @ B @ R ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_89_dsafe__Fun__simps,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,I4: A,Args: B > ( trm @ A @ B )] :
% 2.86/1.59 ( ( dsafe @ A @ B @ ( function @ A @ B @ I4 @ Args ) )
% 2.86/1.59 = ( ! [X2: B] : ( dsafe @ A @ B @ ( Args @ X2 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_90_dsafe__Plus__simps,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,A2: trm @ A @ B,B2: trm @ A @ B] :
% 2.86/1.59 ( ( dsafe @ A @ B @ ( plus @ A @ B @ A2 @ B2 ) )
% 2.86/1.59 = ( ( dsafe @ A @ B @ A2 )
% 2.86/1.59 & ( dsafe @ A @ B @ B2 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_91_dsafe__Times__simps,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,A2: trm @ A @ B,B2: trm @ A @ B] :
% 2.86/1.59 ( ( dsafe @ A @ B @ ( times @ A @ B @ A2 @ B2 ) )
% 2.86/1.59 = ( ( dsafe @ A @ B @ A2 )
% 2.86/1.59 & ( dsafe @ A @ B @ B2 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_92_dsafe__Var__simps,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,X: B] : ( dsafe @ A @ B @ ( var @ B @ A @ X ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_93_dsafe__DiffVar__simps,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,X: B] : ( dsafe @ A @ B @ ( diffVar @ B @ A @ X ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_94_dsafe__Diff__simps,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,A2: trm @ A @ B] :
% 2.86/1.59 ( ( dsafe @ A @ B @ ( differential @ A @ B @ A2 ) )
% 2.86/1.59 = ( dfree @ A @ B @ A2 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_95_dfree__is__dsafe,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,Theta3: trm @ A @ B] :
% 2.86/1.59 ( ( dfree @ A @ B @ Theta3 )
% 2.86/1.59 => ( dsafe @ A @ B @ Theta3 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_96_dsafe__Const,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,R: real] : ( dsafe @ A @ C @ ( const @ A @ C @ R ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_97_dsafe__Fun,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,Args: C > ( trm @ A @ C ),I4: A] :
% 2.86/1.59 ( ! [I3: C] : ( dsafe @ A @ C @ ( Args @ I3 ) )
% 2.86/1.59 => ( dsafe @ A @ C @ ( function @ A @ C @ I4 @ Args ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_98_dsafe__Plus,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,Theta_1: trm @ A @ C,Theta_2: trm @ A @ C] :
% 2.86/1.59 ( ( dsafe @ A @ C @ Theta_1 )
% 2.86/1.59 => ( ( dsafe @ A @ C @ Theta_2 )
% 2.86/1.59 => ( dsafe @ A @ C @ ( plus @ A @ C @ Theta_1 @ Theta_2 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_99_dsafe__Times,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,Theta_1: trm @ A @ C,Theta_2: trm @ A @ C] :
% 2.86/1.59 ( ( dsafe @ A @ C @ Theta_1 )
% 2.86/1.59 => ( ( dsafe @ A @ C @ Theta_2 )
% 2.86/1.59 => ( dsafe @ A @ C @ ( times @ A @ C @ Theta_1 @ Theta_2 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_100_dsafe__Var,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,I4: C] : ( dsafe @ A @ C @ ( var @ C @ A @ I4 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_101_dsafe__DiffVar,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,I4: C] : ( dsafe @ A @ C @ ( diffVar @ C @ A @ I4 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_102_dsafe__Diff,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,Theta3: trm @ A @ C] :
% 2.86/1.59 ( ( dfree @ A @ C @ Theta3 )
% 2.86/1.59 => ( dsafe @ A @ C @ ( differential @ A @ C @ Theta3 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_103_gcd_Ocases,axiom,
% 2.86/1.59 ! [A: $tType] :
% 2.86/1.59 ( ( euclid1155270486miring @ A )
% 2.86/1.59 => ! [X: product_prod @ A @ A] :
% 2.86/1.59 ~ ! [A5: A,B4: A] :
% 2.86/1.59 ( X
% 2.86/1.59 != ( product_Pair @ A @ A @ A5 @ B4 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_104_local_Oempty__def,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType] :
% 2.86/1.59 ( ( empty @ B @ A )
% 2.86/1.59 = ( ^ [I5: B] : ( const @ A @ B @ ( zero_zero @ real ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_105_ids_Of0__def,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,F: Sf] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ( f0 @ Sf @ Sz @ F )
% 2.86/1.59 = ( function @ Sf @ Sz @ F @ ( empty @ Sz @ Sf ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_106_dfree__Fun_OIH,axiom,
% 2.86/1.59 ! [I: c] :
% 2.86/1.59 ( ( denotational_Vagree @ c @ nu @ nu2 @ ( static_FVDiff @ a @ c @ ( args @ I ) ) )
% 2.86/1.59 => ( ( denotational_frechet @ a @ b @ c @ i @ ( args @ I ) @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu ) )
% 2.86/1.59 = ( denotational_frechet @ a @ b @ c @ i @ ( args @ I ) @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu2 ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu2 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_107_strm_Oleft__unique,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType] : ( left_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_108_trm_Oinduct,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,P: ( trm @ A @ C ) > $o,Trm: trm @ A @ C] :
% 2.86/1.59 ( ! [X3: C] : ( P @ ( var @ C @ A @ X3 ) )
% 2.86/1.59 => ( ! [X3: real] : ( P @ ( const @ A @ C @ X3 ) )
% 2.86/1.59 => ( ! [X1a: A,X2a: C > ( trm @ A @ C )] :
% 2.86/1.59 ( ! [X2aa: trm @ A @ C] :
% 2.86/1.59 ( ( member @ ( trm @ A @ C ) @ X2aa @ ( image @ C @ ( trm @ A @ C ) @ X2a @ ( top_top @ ( set @ C ) ) ) )
% 2.86/1.59 => ( P @ X2aa ) )
% 2.86/1.59 => ( P @ ( function @ A @ C @ X1a @ X2a ) ) )
% 2.86/1.59 => ( ! [X1a: trm @ A @ C,X2a: trm @ A @ C] :
% 2.86/1.59 ( ( P @ X1a )
% 2.86/1.59 => ( ( P @ X2a )
% 2.86/1.59 => ( P @ ( plus @ A @ C @ X1a @ X2a ) ) ) )
% 2.86/1.59 => ( ! [X1a: trm @ A @ C,X2a: trm @ A @ C] :
% 2.86/1.59 ( ( P @ X1a )
% 2.86/1.59 => ( ( P @ X2a )
% 2.86/1.59 => ( P @ ( times @ A @ C @ X1a @ X2a ) ) ) )
% 2.86/1.59 => ( ! [X3: C] : ( P @ ( diffVar @ C @ A @ X3 ) )
% 2.86/1.59 => ( ! [X3: trm @ A @ C] :
% 2.86/1.59 ( ( P @ X3 )
% 2.86/1.59 => ( P @ ( differential @ A @ C @ X3 ) ) )
% 2.86/1.59 => ( P @ Trm ) ) ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_109_agree__comm,axiom,
% 2.86/1.59 ! [A: $tType] :
% 2.86/1.59 ( ( finite_finite @ A )
% 2.86/1.59 => ! [A3: 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 ),V: set @ ( sum_sum @ A @ A )] :
% 2.86/1.59 ( ( denotational_Vagree @ A @ A3 @ B5 @ V )
% 2.86/1.59 => ( denotational_Vagree @ A @ B5 @ A3 @ V ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_110_prod_Ocollapse,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.86/1.59 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 2.86/1.59 = Prod ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_111_prod_Oexhaust__sel,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.86/1.59 ( Prod
% 2.86/1.59 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_112_surjective__pairing,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,T4: product_prod @ A @ B] :
% 2.86/1.59 ( T4
% 2.86/1.59 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T4 ) @ ( product_snd @ A @ B @ T4 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_113_agree__UNIV__eq,axiom,
% 2.86/1.59 ! [A: $tType] :
% 2.86/1.59 ( ( finite_finite @ A )
% 2.86/1.59 => ! [Nu: 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 )] :
% 2.86/1.59 ( ( denotational_Vagree @ A @ Nu @ Omega @ ( top_top @ ( set @ ( sum_sum @ A @ A ) ) ) )
% 2.86/1.59 => ( Nu = Omega ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_114_Vagree__univ,axiom,
% 2.86/1.59 ! [A: $tType] :
% 2.86/1.59 ( ( finite_finite @ A )
% 2.86/1.59 => ! [A2: finite_Cartesian_vec @ real @ A,B2: finite_Cartesian_vec @ real @ A,C3: finite_Cartesian_vec @ real @ A,D3: finite_Cartesian_vec @ real @ A] :
% 2.86/1.59 ( ( denotational_Vagree @ A @ ( product_Pair @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ A2 @ B2 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ C3 @ D3 ) @ ( top_top @ ( set @ ( sum_sum @ A @ A ) ) ) )
% 2.86/1.59 => ( ( A2 = C3 )
% 2.86/1.59 & ( B2 = D3 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_115_ids_Oseq__sem_Ocases,axiom,
% 2.86/1.59 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc )
% 2.86/1.59 & ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: product_prod @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) )] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ~ ! [I2: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,S: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
% 2.86/1.59 ( X
% 2.86/1.59 != ( product_Pair @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) @ I2 @ S ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_116_ids_Osingleton_Oinduct,axiom,
% 2.86/1.59 ! [Sf: $tType,Sc: $tType,A: $tType,Sz: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sc )
% 2.86/1.59 & ( finite_finite @ Sf ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( trm @ A @ Sz ) > Sz > $o,A0: trm @ A @ Sz,A1: Sz] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ! [T2: trm @ A @ Sz,X_1: Sz] : ( P @ T2 @ X_1 )
% 2.86/1.59 => ( P @ A0 @ A1 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_117_ids_Oseq__sem_Oinduct,axiom,
% 2.86/1.59 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc )
% 2.86/1.59 & ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) > ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) > $o,A0: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,A1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ! [I2: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,X_1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] : ( P @ I2 @ X_1 )
% 2.86/1.59 => ( P @ A0 @ A1 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_118_frechet_Osimps_I5_J,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,C: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ C )
% 2.86/1.59 & ( finite_finite @ A )
% 2.86/1.59 & ( finite_finite @ B ) )
% 2.86/1.59 => ! [I7: denota1663640101rp_ext @ A @ B @ C @ product_unit,R: real,V2: finite_Cartesian_vec @ real @ C] :
% 2.86/1.59 ( ( denotational_frechet @ A @ B @ C @ I7 @ ( const @ A @ C @ R ) @ V2 )
% 2.86/1.59 = ( ^ [V3: finite_Cartesian_vec @ real @ C] : ( zero_zero @ real ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_119_prod__eqI,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 2.86/1.59 ( ( ( product_fst @ A @ B @ P2 )
% 2.86/1.59 = ( product_fst @ A @ B @ Q2 ) )
% 2.86/1.59 => ( ( ( product_snd @ A @ B @ P2 )
% 2.86/1.59 = ( product_snd @ A @ B @ Q2 ) )
% 2.86/1.59 => ( P2 = Q2 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_120_prod_Oexpand,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 2.86/1.59 ( ( ( ( product_fst @ A @ B @ Prod )
% 2.86/1.59 = ( product_fst @ A @ B @ Prod2 ) )
% 2.86/1.59 & ( ( product_snd @ A @ B @ Prod )
% 2.86/1.59 = ( product_snd @ A @ B @ Prod2 ) ) )
% 2.86/1.59 => ( Prod = Prod2 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_121_prod__eq__iff,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType] :
% 2.86/1.59 ( ( ^ [Y4: product_prod @ A @ B,Z: product_prod @ A @ B] : ( Y4 = Z ) )
% 2.86/1.59 = ( ^ [S4: product_prod @ A @ B,T5: product_prod @ A @ B] :
% 2.86/1.59 ( ( ( product_fst @ A @ B @ S4 )
% 2.86/1.59 = ( product_fst @ A @ B @ T5 ) )
% 2.86/1.59 & ( ( product_snd @ A @ B @ S4 )
% 2.86/1.59 = ( product_snd @ A @ B @ T5 ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_122_snd__conv,axiom,
% 2.86/1.59 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 2.86/1.59 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 2.86/1.59 = X22 ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_123_snd__eqD,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
% 2.86/1.59 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 2.86/1.59 = A2 )
% 2.86/1.59 => ( Y = A2 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_124_fst__conv,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 2.86/1.59 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 2.86/1.59 = X1 ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_125_fst__eqD,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
% 2.86/1.59 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 2.86/1.59 = A2 )
% 2.86/1.59 => ( X = A2 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_126_ids_Oempty__def,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,Sz: $tType,A: $tType,B: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ( empty @ B @ A )
% 2.86/1.59 = ( ^ [I5: B] : ( const @ A @ B @ ( zero_zero @ real ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_127_ids_Osingleton_Ocases,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: product_prod @ ( trm @ A @ Sz ) @ Sz] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ~ ! [T2: trm @ A @ Sz,I3: Sz] :
% 2.86/1.59 ( X
% 2.86/1.59 != ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ T2 @ I3 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_128_ids_OP__def,axiom,
% 2.86/1.59 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sc )
% 2.86/1.59 & ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P2: Sc] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ( p @ Sc @ Sf @ Sz @ P2 )
% 2.86/1.59 = ( predicational @ Sc @ Sf @ Sz @ P2 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_129_FunctionFrechet_Ocases,axiom,
% 2.86/1.59 ! [B: $tType,C: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ A )
% 2.86/1.59 & ( finite_finite @ C )
% 2.86/1.59 & ( finite_finite @ B ) )
% 2.86/1.59 => ! [X: product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A] :
% 2.86/1.59 ~ ! [I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,I3: A] :
% 2.86/1.59 ( X
% 2.86/1.59 != ( product_Pair @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A @ I2 @ I3 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_130_agree__refl,axiom,
% 2.86/1.59 ! [A: $tType] :
% 2.86/1.59 ( ( finite_finite @ A )
% 2.86/1.59 => ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A3: set @ ( sum_sum @ A @ A )] : ( denotational_Vagree @ A @ Nu @ Nu @ A3 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_131_ids_Ostrm_Odomain,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ( domainp @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) )
% 2.86/1.59 = ( dfree @ A @ C ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_132_snd__top,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType] :
% 2.86/1.59 ( ( ( top @ A )
% 2.86/1.59 & ( top @ B ) )
% 2.86/1.59 => ( ( product_snd @ B @ A @ ( top_top @ ( product_prod @ B @ A ) ) )
% 2.86/1.59 = ( top_top @ A ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_133_fst__top,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType] :
% 2.86/1.59 ( ( ( top @ A )
% 2.86/1.59 & ( top @ B ) )
% 2.86/1.59 => ( ( product_fst @ A @ B @ ( top_top @ ( product_prod @ A @ B ) ) )
% 2.86/1.59 = ( top_top @ A ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_134_snd__zero,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType] :
% 2.86/1.59 ( ( ( zero @ A )
% 2.86/1.59 & ( zero @ B ) )
% 2.86/1.59 => ( ( product_snd @ B @ A @ ( zero_zero @ ( product_prod @ B @ A ) ) )
% 2.86/1.59 = ( zero_zero @ A ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_135_fst__zero,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType] :
% 2.86/1.59 ( ( ( zero @ A )
% 2.86/1.59 & ( zero @ B ) )
% 2.86/1.59 => ( ( product_fst @ A @ B @ ( zero_zero @ ( product_prod @ A @ B ) ) )
% 2.86/1.59 = ( zero_zero @ A ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_136_zero__prod__def,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType] :
% 2.86/1.59 ( ( ( zero @ B )
% 2.86/1.59 & ( zero @ A ) )
% 2.86/1.59 => ( ( zero_zero @ ( product_prod @ A @ B ) )
% 2.86/1.59 = ( product_Pair @ A @ B @ ( zero_zero @ A ) @ ( zero_zero @ B ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_137_top__prod__def,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType] :
% 2.86/1.59 ( ( ( top @ B )
% 2.86/1.59 & ( top @ A ) )
% 2.86/1.59 => ( ( top_top @ ( product_prod @ A @ B ) )
% 2.86/1.59 = ( product_Pair @ A @ B @ ( top_top @ A ) @ ( top_top @ B ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_138_ids_Oraw__interp__inject,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ A )
% 2.86/1.59 & ( finite_finite @ B )
% 2.86/1.59 & ( finite_finite @ C )
% 2.86/1.59 & ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_good_interp @ A @ B @ C,Y: frechet_good_interp @ A @ B @ C] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ( ( freche229654227interp @ A @ B @ C @ X )
% 2.86/1.59 = ( freche229654227interp @ A @ B @ C @ Y ) )
% 2.86/1.59 = ( X = Y ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_139_ids_Oraw__term__inject,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_strm @ A @ C,Y: frechet_strm @ A @ C] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ( ( frechet_raw_term @ A @ C @ X )
% 2.86/1.59 = ( frechet_raw_term @ A @ C @ Y ) )
% 2.86/1.59 = ( X = Y ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_140_ids_Oraw__interp__inverse,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ A )
% 2.86/1.59 & ( finite_finite @ B )
% 2.86/1.59 & ( finite_finite @ C )
% 2.86/1.59 & ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_good_interp @ A @ B @ C] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ( freche227871258interp @ A @ B @ C @ ( freche229654227interp @ A @ B @ C @ X ) )
% 2.86/1.59 = X ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_141_ids_Oraw__term__inverse,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_strm @ A @ C] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ( frechet_simple_term @ A @ C @ ( frechet_raw_term @ A @ C @ X ) )
% 2.86/1.59 = X ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_142_ids_Ostrm_Oright__unique,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( right_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_143_ids_Ostrm_Oleft__unique,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( left_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_144_directional__derivative__def,axiom,
% 2.86/1.59 ! [C: $tType,B: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ A )
% 2.86/1.59 & ( finite_finite @ B )
% 2.86/1.59 & ( finite_finite @ C ) )
% 2.86/1.59 => ( ( denota2078997598vative @ A @ B @ C )
% 2.86/1.59 = ( ^ [I8: denota1663640101rp_ext @ A @ B @ C @ product_unit,T5: trm @ A @ C,V4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] : ( denotational_frechet @ A @ B @ C @ I8 @ T5 @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V4 ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V4 ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_145_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
% 2.86/1.59 ( ( P @ X @ Y )
% 2.86/1.59 => ( ( A2
% 2.86/1.59 = ( product_Pair @ A @ B @ X @ Y ) )
% 2.86/1.59 => ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_146_conjI__realizer,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
% 2.86/1.59 ( ( P @ P2 )
% 2.86/1.59 => ( ( Q @ Q2 )
% 2.86/1.59 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
% 2.86/1.59 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_147_exI__realizer,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,P: A > B > $o,Y: A,X: B] :
% 2.86/1.59 ( ( P @ Y @ X )
% 2.86/1.59 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_148_exE__realizer_H,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A] :
% 2.86/1.59 ( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
% 2.86/1.59 => ~ ! [X3: B,Y3: A] :
% 2.86/1.59 ~ ( P @ Y3 @ X3 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_149_iso__tuple__UNIV__I,axiom,
% 2.86/1.59 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_150_UNIV__I,axiom,
% 2.86/1.59 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_151_top__apply,axiom,
% 2.86/1.59 ! [C: $tType,D: $tType] :
% 2.86/1.59 ( ( top @ C )
% 2.86/1.59 => ( ( top_top @ ( D > C ) )
% 2.86/1.59 = ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_152_image__eqI,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,B2: A,F: B > A,X: B,A3: set @ B] :
% 2.86/1.59 ( ( B2
% 2.86/1.59 = ( F @ X ) )
% 2.86/1.59 => ( ( member @ B @ X @ A3 )
% 2.86/1.59 => ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_153_imageI,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,X: A,A3: set @ A,F: A > B] :
% 2.86/1.59 ( ( member @ A @ X @ A3 )
% 2.86/1.59 => ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_154_image__iff,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,Z2: A,F: B > A,A3: set @ B] :
% 2.86/1.59 ( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A3 ) )
% 2.86/1.59 = ( ? [X2: B] :
% 2.86/1.59 ( ( member @ B @ X2 @ A3 )
% 2.86/1.59 & ( Z2
% 2.86/1.59 = ( F @ X2 ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_155_bex__imageD,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
% 2.86/1.59 ( ? [X4: A] :
% 2.86/1.59 ( ( member @ A @ X4 @ ( image @ B @ A @ F @ A3 ) )
% 2.86/1.59 & ( P @ X4 ) )
% 2.86/1.59 => ? [X3: B] :
% 2.86/1.59 ( ( member @ B @ X3 @ A3 )
% 2.86/1.59 & ( P @ ( F @ X3 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_156_image__cong,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G: A > B] :
% 2.86/1.59 ( ( M = N )
% 2.86/1.59 => ( ! [X3: A] :
% 2.86/1.59 ( ( member @ A @ X3 @ N )
% 2.86/1.59 => ( ( F @ X3 )
% 2.86/1.59 = ( G @ X3 ) ) )
% 2.86/1.59 => ( ( image @ A @ B @ F @ M )
% 2.86/1.59 = ( image @ A @ B @ G @ N ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_157_ball__imageD,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
% 2.86/1.59 ( ! [X3: A] :
% 2.86/1.59 ( ( member @ A @ X3 @ ( image @ B @ A @ F @ A3 ) )
% 2.86/1.59 => ( P @ X3 ) )
% 2.86/1.59 => ! [X4: B] :
% 2.86/1.59 ( ( member @ B @ X4 @ A3 )
% 2.86/1.59 => ( P @ ( F @ X4 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_158_rev__image__eqI,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,X: A,A3: set @ A,B2: B,F: A > B] :
% 2.86/1.59 ( ( member @ A @ X @ A3 )
% 2.86/1.59 => ( ( B2
% 2.86/1.59 = ( F @ X ) )
% 2.86/1.59 => ( member @ B @ B2 @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_159_top__set__def,axiom,
% 2.86/1.59 ! [A: $tType] :
% 2.86/1.59 ( ( top_top @ ( set @ A ) )
% 2.86/1.59 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_160_UNIV__eq__I,axiom,
% 2.86/1.59 ! [A: $tType,A3: set @ A] :
% 2.86/1.59 ( ! [X3: A] : ( member @ A @ X3 @ A3 )
% 2.86/1.59 => ( ( top_top @ ( set @ A ) )
% 2.86/1.59 = A3 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_161_UNIV__witness,axiom,
% 2.86/1.59 ! [A: $tType] :
% 2.86/1.59 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_162_rangeI,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_163_range__eqI,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,B2: A,F: B > A,X: B] :
% 2.86/1.59 ( ( B2
% 2.86/1.59 = ( F @ X ) )
% 2.86/1.59 => ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_164_sndI,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
% 2.86/1.59 ( ( X
% 2.86/1.59 = ( product_Pair @ A @ B @ Y @ Z2 ) )
% 2.86/1.59 => ( ( product_snd @ A @ B @ X )
% 2.86/1.59 = Z2 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_165_eq__snd__iff,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
% 2.86/1.59 ( ( B2
% 2.86/1.59 = ( product_snd @ B @ A @ P2 ) )
% 2.86/1.59 = ( ? [A6: B] :
% 2.86/1.59 ( P2
% 2.86/1.59 = ( product_Pair @ B @ A @ A6 @ B2 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_166_top1I,axiom,
% 2.86/1.59 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_167_zero__reorient,axiom,
% 2.86/1.59 ! [A: $tType] :
% 2.86/1.59 ( ( zero @ A )
% 2.86/1.59 => ! [X: A] :
% 2.86/1.59 ( ( ( zero_zero @ A )
% 2.86/1.59 = X )
% 2.86/1.59 = ( X
% 2.86/1.59 = ( zero_zero @ A ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_168_eq__fst__iff,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
% 2.86/1.59 ( ( A2
% 2.86/1.59 = ( product_fst @ A @ B @ P2 ) )
% 2.86/1.59 = ( ? [B6: B] :
% 2.86/1.59 ( P2
% 2.86/1.59 = ( product_Pair @ A @ B @ A2 @ B6 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_169_fstI,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
% 2.86/1.59 ( ( X
% 2.86/1.59 = ( product_Pair @ A @ B @ Y @ Z2 ) )
% 2.86/1.59 => ( ( product_fst @ A @ B @ X )
% 2.86/1.59 = Y ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_170_range__snd,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType] :
% 2.86/1.59 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.86/1.59 = ( top_top @ ( set @ A ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_171_range__fst,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType] :
% 2.86/1.59 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.86/1.59 = ( top_top @ ( set @ A ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_172_Domain__eq__top,axiom,
% 2.86/1.59 ! [A: $tType] :
% 2.86/1.59 ( ( domainp @ A @ A
% 2.86/1.59 @ ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 2.86/1.59 = ( top_top @ ( A > $o ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_173_DomainpE,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,R: A > B > $o,A2: A] :
% 2.86/1.59 ( ( domainp @ A @ B @ R @ A2 )
% 2.86/1.59 => ~ ! [B4: B] :
% 2.86/1.59 ~ ( R @ A2 @ B4 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_174_Domainp__iff,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType] :
% 2.86/1.59 ( ( domainp @ A @ B )
% 2.86/1.59 = ( ^ [T6: A > B > $o,X2: A] :
% 2.86/1.59 ? [X5: B] : ( T6 @ X2 @ X5 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_175_Domainp__refl,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType] :
% 2.86/1.59 ( ( domainp @ A @ B )
% 2.86/1.59 = ( domainp @ A @ B ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_176_Domainp_Ocases,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,R: A > B > $o,A2: A] :
% 2.86/1.59 ( ( domainp @ A @ B @ R @ A2 )
% 2.86/1.59 => ~ ! [B4: B] :
% 2.86/1.59 ~ ( R @ A2 @ B4 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_177_Domainp_Osimps,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType] :
% 2.86/1.59 ( ( domainp @ A @ B )
% 2.86/1.59 = ( ^ [R3: A > B > $o,A6: A] :
% 2.86/1.59 ? [B6: A,C4: B] :
% 2.86/1.59 ( ( A6 = B6 )
% 2.86/1.59 & ( R3 @ B6 @ C4 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_178_Domainp_ODomainI,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,R: A > B > $o,A2: A,B2: B] :
% 2.86/1.59 ( ( R @ A2 @ B2 )
% 2.86/1.59 => ( domainp @ A @ B @ R @ A2 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_179_Domainp_Oinducts,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,R: A > B > $o,X: A,P: A > $o] :
% 2.86/1.59 ( ( domainp @ A @ B @ R @ X )
% 2.86/1.59 => ( ! [A5: A,B4: B] :
% 2.86/1.59 ( ( R @ A5 @ B4 )
% 2.86/1.59 => ( P @ A5 ) )
% 2.86/1.59 => ( P @ X ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_180_right__uniqueD,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,A3: A > B > $o,X: A,Y: B,Z2: B] :
% 2.86/1.59 ( ( right_unique @ A @ B @ A3 )
% 2.86/1.59 => ( ( A3 @ X @ Y )
% 2.86/1.59 => ( ( A3 @ X @ Z2 )
% 2.86/1.59 => ( Y = Z2 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_181_right__uniqueI,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,A3: A > B > $o] :
% 2.86/1.59 ( ! [X3: A,Y3: B,Z3: B] :
% 2.86/1.59 ( ( A3 @ X3 @ Y3 )
% 2.86/1.59 => ( ( A3 @ X3 @ Z3 )
% 2.86/1.59 => ( Y3 = Z3 ) ) )
% 2.86/1.59 => ( right_unique @ A @ B @ A3 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_182_right__unique__eq,axiom,
% 2.86/1.59 ! [A: $tType] :
% 2.86/1.59 ( right_unique @ A @ A
% 2.86/1.59 @ ^ [Y4: A,Z: A] : ( Y4 = Z ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_183_right__unique__def,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType] :
% 2.86/1.59 ( ( right_unique @ A @ B )
% 2.86/1.59 = ( ^ [R5: A > B > $o] :
% 2.86/1.59 ! [X2: A,Y5: B,Z4: B] :
% 2.86/1.59 ( ( R5 @ X2 @ Y5 )
% 2.86/1.59 => ( ( R5 @ X2 @ Z4 )
% 2.86/1.59 => ( Y5 = Z4 ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_184_left__uniqueD,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,A3: A > B > $o,X: A,Z2: B,Y: A] :
% 2.86/1.59 ( ( left_unique @ A @ B @ A3 )
% 2.86/1.59 => ( ( A3 @ X @ Z2 )
% 2.86/1.59 => ( ( A3 @ Y @ Z2 )
% 2.86/1.59 => ( X = Y ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_185_left__uniqueI,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,A3: A > B > $o] :
% 2.86/1.59 ( ! [X3: A,Y3: A,Z3: B] :
% 2.86/1.59 ( ( A3 @ X3 @ Z3 )
% 2.86/1.59 => ( ( A3 @ Y3 @ Z3 )
% 2.86/1.59 => ( X3 = Y3 ) ) )
% 2.86/1.59 => ( left_unique @ A @ B @ A3 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_186_left__unique__eq,axiom,
% 2.86/1.59 ! [A: $tType] :
% 2.86/1.59 ( left_unique @ A @ A
% 2.86/1.59 @ ^ [Y4: A,Z: A] : ( Y4 = Z ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_187_left__unique__def,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType] :
% 2.86/1.59 ( ( left_unique @ A @ B )
% 2.86/1.59 = ( ^ [R5: A > B > $o] :
% 2.86/1.59 ! [X2: A,Y5: A,Z4: B] :
% 2.86/1.59 ( ( R5 @ X2 @ Z4 )
% 2.86/1.59 => ( ( R5 @ Y5 @ Z4 )
% 2.86/1.59 => ( X2 = Y5 ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_188_top__empty__eq,axiom,
% 2.86/1.59 ! [A: $tType] :
% 2.86/1.59 ( ( top_top @ ( A > $o ) )
% 2.86/1.59 = ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_189_trm_Oset__cases,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,E3: A,A2: trm @ A @ C] :
% 2.86/1.59 ( ( member @ A @ E3 @ ( set_trm @ A @ C @ A2 ) )
% 2.86/1.59 => ( ! [Z22: C > ( trm @ A @ C )] :
% 2.86/1.59 ( A2
% 2.86/1.59 != ( function @ A @ C @ E3 @ Z22 ) )
% 2.86/1.59 => ( ! [Z1: A,Z22: C > ( trm @ A @ C )] :
% 2.86/1.59 ( ( A2
% 2.86/1.59 = ( function @ A @ C @ Z1 @ Z22 ) )
% 2.86/1.59 => ! [X3: trm @ A @ C] :
% 2.86/1.59 ( ( member @ ( trm @ A @ C ) @ X3 @ ( image @ C @ ( trm @ A @ C ) @ Z22 @ ( top_top @ ( set @ C ) ) ) )
% 2.86/1.59 => ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ X3 ) ) ) )
% 2.86/1.59 => ( ! [Z1: trm @ A @ C] :
% 2.86/1.59 ( ? [Z22: trm @ A @ C] :
% 2.86/1.59 ( A2
% 2.86/1.59 = ( plus @ A @ C @ Z1 @ Z22 ) )
% 2.86/1.59 => ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ Z1 ) ) )
% 2.86/1.59 => ( ! [Z1: trm @ A @ C,Z22: trm @ A @ C] :
% 2.86/1.59 ( ( A2
% 2.86/1.59 = ( plus @ A @ C @ Z1 @ Z22 ) )
% 2.86/1.59 => ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ Z22 ) ) )
% 2.86/1.59 => ( ! [Z1: trm @ A @ C] :
% 2.86/1.59 ( ? [Z22: trm @ A @ C] :
% 2.86/1.59 ( A2
% 2.86/1.59 = ( times @ A @ C @ Z1 @ Z22 ) )
% 2.86/1.59 => ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ Z1 ) ) )
% 2.86/1.59 => ( ! [Z1: trm @ A @ C,Z22: trm @ A @ C] :
% 2.86/1.59 ( ( A2
% 2.86/1.59 = ( times @ A @ C @ Z1 @ Z22 ) )
% 2.86/1.59 => ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ Z22 ) ) )
% 2.86/1.59 => ~ ! [Z3: trm @ A @ C] :
% 2.86/1.59 ( ( A2
% 2.86/1.59 = ( differential @ A @ C @ Z3 ) )
% 2.86/1.59 => ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ Z3 ) ) ) ) ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_190_surjD,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 2.86/1.59 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.86/1.59 = ( top_top @ ( set @ A ) ) )
% 2.86/1.59 => ? [X3: B] :
% 2.86/1.59 ( Y
% 2.86/1.59 = ( F @ X3 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_191_surjE,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 2.86/1.59 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.86/1.59 = ( top_top @ ( set @ A ) ) )
% 2.86/1.59 => ~ ! [X3: B] :
% 2.86/1.59 ( Y
% 2.86/1.59 != ( F @ X3 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_192_trm_Oset__intros_I1_J,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,X31: A,X32: C > ( trm @ A @ C )] : ( member @ A @ X31 @ ( set_trm @ A @ C @ ( function @ A @ C @ X31 @ X32 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_193_trm_Oset__intros_I4_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,Yc: A,X42: trm @ A @ C,X41: trm @ A @ C] :
% 2.86/1.59 ( ( member @ A @ Yc @ ( set_trm @ A @ C @ X42 ) )
% 2.86/1.59 => ( member @ A @ Yc @ ( set_trm @ A @ C @ ( plus @ A @ C @ X41 @ X42 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_194_trm_Oset__intros_I3_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,Yb: A,X41: trm @ A @ C,X42: trm @ A @ C] :
% 2.86/1.59 ( ( member @ A @ Yb @ ( set_trm @ A @ C @ X41 ) )
% 2.86/1.59 => ( member @ A @ Yb @ ( set_trm @ A @ C @ ( plus @ A @ C @ X41 @ X42 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_195_trm_Oset__intros_I6_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,Ye: A,X52: trm @ A @ C,X51: trm @ A @ C] :
% 2.86/1.59 ( ( member @ A @ Ye @ ( set_trm @ A @ C @ X52 ) )
% 2.86/1.59 => ( member @ A @ Ye @ ( set_trm @ A @ C @ ( times @ A @ C @ X51 @ X52 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_196_trm_Oset__intros_I5_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,Yd: A,X51: trm @ A @ C,X52: trm @ A @ C] :
% 2.86/1.59 ( ( member @ A @ Yd @ ( set_trm @ A @ C @ X51 ) )
% 2.86/1.59 => ( member @ A @ Yd @ ( set_trm @ A @ C @ ( times @ A @ C @ X51 @ X52 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_197_trm_Osimps_I126_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,X7: trm @ A @ C] :
% 2.86/1.59 ( ( set_trm @ A @ C @ ( differential @ A @ C @ X7 ) )
% 2.86/1.59 = ( set_trm @ A @ C @ X7 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_198_trm_Oset__intros_I7_J,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,Yf: A,X7: trm @ A @ C] :
% 2.86/1.59 ( ( member @ A @ Yf @ ( set_trm @ A @ C @ X7 ) )
% 2.86/1.59 => ( member @ A @ Yf @ ( set_trm @ A @ C @ ( differential @ A @ C @ X7 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_199_trm_Oset__intros_I2_J,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,Y: trm @ A @ C,X32: C > ( trm @ A @ C ),Ya: A,X31: A] :
% 2.86/1.59 ( ( member @ ( trm @ A @ C ) @ Y @ ( image @ C @ ( trm @ A @ C ) @ X32 @ ( top_top @ ( set @ C ) ) ) )
% 2.86/1.59 => ( ( member @ A @ Ya @ ( set_trm @ A @ C @ Y ) )
% 2.86/1.59 => ( member @ A @ Ya @ ( set_trm @ A @ C @ ( function @ A @ C @ X31 @ X32 ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_200_surj__def,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,F: B > A] :
% 2.86/1.59 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.86/1.59 = ( top_top @ ( set @ A ) ) )
% 2.86/1.59 = ( ! [Y5: A] :
% 2.86/1.59 ? [X2: B] :
% 2.86/1.59 ( Y5
% 2.86/1.59 = ( F @ X2 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_201_surjI,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 2.86/1.59 ( ! [X3: A] :
% 2.86/1.59 ( ( G @ ( F @ X3 ) )
% 2.86/1.59 = X3 )
% 2.86/1.59 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 2.86/1.59 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_202_ids_Oexpand__singleton,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,T4: trm @ A @ Sz] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ( singleton @ Sz @ A @ Vid1 @ T4 )
% 2.86/1.59 = ( ^ [I5: Sz] : ( if @ ( trm @ A @ Sz ) @ ( I5 = Vid1 ) @ T4 @ ( const @ A @ Sz @ ( zero_zero @ real ) ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_203_ids_Osingleton_Osimps,axiom,
% 2.86/1.59 ! [Sf: $tType,Sc: $tType,A: $tType,Sz: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sc )
% 2.86/1.59 & ( finite_finite @ Sf ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,I4: Sz,T4: trm @ A @ Sz] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ( ( I4 = Vid1 )
% 2.86/1.59 => ( ( singleton @ Sz @ A @ Vid1 @ T4 @ I4 )
% 2.86/1.59 = T4 ) )
% 2.86/1.59 & ( ( I4 != Vid1 )
% 2.86/1.59 => ( ( singleton @ Sz @ A @ Vid1 @ T4 @ I4 )
% 2.86/1.59 = ( const @ A @ Sz @ ( zero_zero @ real ) ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_204_ids_Oproj__sing1,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,Sz: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Theta3: trm @ A @ Sz] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ( singleton @ Sz @ A @ Vid1 @ Theta3 @ Vid1 )
% 2.86/1.59 = Theta3 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_205_ids_Osingleton_Ocong,axiom,
% 2.86/1.59 ! [A: $tType,Sz: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz ) )
% 2.86/1.59 => ( ( singleton @ Sz @ A )
% 2.86/1.59 = ( singleton @ Sz @ A ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_206_ids_Oproj__sing2,axiom,
% 2.86/1.59 ! [Sf: $tType,Sc: $tType,A: $tType,Sz: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sc )
% 2.86/1.59 & ( finite_finite @ Sf ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: Sz,Theta3: trm @ A @ Sz] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ( Vid1 != Y )
% 2.86/1.59 => ( ( singleton @ Sz @ A @ Vid1 @ Theta3 @ Y )
% 2.86/1.59 = ( const @ A @ Sz @ ( zero_zero @ real ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_207_ids_Osingleton_Oelims,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: trm @ A @ Sz,Xa: Sz,Y: trm @ A @ Sz] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ( ( singleton @ Sz @ A @ Vid1 @ X @ Xa )
% 2.86/1.59 = Y )
% 2.86/1.59 => ( ( ( Xa = Vid1 )
% 2.86/1.59 => ( Y = X ) )
% 2.86/1.59 & ( ( Xa != Vid1 )
% 2.86/1.59 => ( Y
% 2.86/1.59 = ( const @ A @ Sz @ ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_208_ids_Of1__def,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,F: Sf,X: Sz] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ( f1 @ Sz @ Sf @ Vid1 @ F @ X )
% 2.86/1.59 = ( function @ Sf @ Sz @ F @ ( singleton @ Sz @ Sf @ Vid1 @ ( var @ Sz @ Sf @ X ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_209_ids_Osingleton_Opelims,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: trm @ A @ Sz,Xa: Sz,Y: trm @ A @ Sz] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ( ( singleton @ Sz @ A @ Vid1 @ X @ Xa )
% 2.86/1.59 = Y )
% 2.86/1.59 => ( ( accp @ ( product_prod @ ( trm @ A @ Sz ) @ Sz ) @ ( singleton_rel @ A @ Sz ) @ ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ X @ Xa ) )
% 2.86/1.59 => ~ ( ( ( ( Xa = Vid1 )
% 2.86/1.59 => ( Y = X ) )
% 2.86/1.59 & ( ( Xa != Vid1 )
% 2.86/1.59 => ( Y
% 2.86/1.59 = ( const @ A @ Sz @ ( zero_zero @ real ) ) ) ) )
% 2.86/1.59 => ~ ( accp @ ( product_prod @ ( trm @ A @ Sz ) @ Sz ) @ ( singleton_rel @ A @ Sz ) @ ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ X @ Xa ) ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_210_ids_Of1_Ocong,axiom,
% 2.86/1.59 ! [Sf: $tType,Sz: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf ) )
% 2.86/1.59 => ( ( f1 @ Sz @ Sf )
% 2.86/1.59 = ( f1 @ Sz @ Sf ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_211_singleton_Opelims,axiom,
% 2.86/1.59 ! [A: $tType,X: trm @ A @ sz,Xa: sz,Y: trm @ A @ sz] :
% 2.86/1.59 ( ( ( singleton @ sz @ A @ vid1 @ X @ Xa )
% 2.86/1.59 = Y )
% 2.86/1.59 => ( ( accp @ ( product_prod @ ( trm @ A @ sz ) @ sz ) @ ( singleton_rel @ A @ sz ) @ ( product_Pair @ ( trm @ A @ sz ) @ sz @ X @ Xa ) )
% 2.86/1.59 => ~ ( ( ( ( Xa = vid1 )
% 2.86/1.59 => ( Y = X ) )
% 2.86/1.59 & ( ( Xa != vid1 )
% 2.86/1.59 => ( Y
% 2.86/1.59 = ( const @ A @ sz @ ( zero_zero @ real ) ) ) ) )
% 2.86/1.59 => ~ ( accp @ ( product_prod @ ( trm @ A @ sz ) @ sz ) @ ( singleton_rel @ A @ sz ) @ ( product_Pair @ ( trm @ A @ sz ) @ sz @ X @ Xa ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_212_f1__def,axiom,
% 2.86/1.59 ! [F: sf,X: sz] :
% 2.86/1.59 ( ( f1 @ sz @ sf @ vid1 @ F @ X )
% 2.86/1.59 = ( function @ sf @ sz @ F @ ( singleton @ sz @ sf @ vid1 @ ( var @ sz @ sf @ X ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_213_proj__sing1,axiom,
% 2.86/1.59 ! [A: $tType,Theta3: trm @ A @ sz] :
% 2.86/1.59 ( ( singleton @ sz @ A @ vid1 @ Theta3 @ vid1 )
% 2.86/1.59 = Theta3 ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_214_expand__singleton,axiom,
% 2.86/1.59 ! [A: $tType,T4: trm @ A @ sz] :
% 2.86/1.59 ( ( singleton @ sz @ A @ vid1 @ T4 )
% 2.86/1.59 = ( ^ [I5: sz] : ( if @ ( trm @ A @ sz ) @ ( I5 = vid1 ) @ T4 @ ( const @ A @ sz @ ( zero_zero @ real ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_215_proj__sing2,axiom,
% 2.86/1.59 ! [A: $tType,Y: sz,Theta3: trm @ A @ sz] :
% 2.86/1.59 ( ( vid1 != Y )
% 2.86/1.59 => ( ( singleton @ sz @ A @ vid1 @ Theta3 @ Y )
% 2.86/1.59 = ( const @ A @ sz @ ( zero_zero @ real ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_216_singleton_Oelims,axiom,
% 2.86/1.59 ! [A: $tType,X: trm @ A @ sz,Xa: sz,Y: trm @ A @ sz] :
% 2.86/1.59 ( ( ( singleton @ sz @ A @ vid1 @ X @ Xa )
% 2.86/1.59 = Y )
% 2.86/1.59 => ( ( ( Xa = vid1 )
% 2.86/1.59 => ( Y = X ) )
% 2.86/1.59 & ( ( Xa != vid1 )
% 2.86/1.59 => ( Y
% 2.86/1.59 = ( const @ A @ sz @ ( zero_zero @ real ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_217_singleton_Osimps,axiom,
% 2.86/1.59 ! [A: $tType,I4: sz,T4: trm @ A @ sz] :
% 2.86/1.59 ( ( ( I4 = vid1 )
% 2.86/1.59 => ( ( singleton @ sz @ A @ vid1 @ T4 @ I4 )
% 2.86/1.59 = T4 ) )
% 2.86/1.59 & ( ( I4 != vid1 )
% 2.86/1.59 => ( ( singleton @ sz @ A @ vid1 @ T4 @ I4 )
% 2.86/1.59 = ( const @ A @ sz @ ( zero_zero @ real ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_218_local_Oid__simps_I15_J,axiom,
% 2.86/1.59 vid3 != vid1 ).
% 2.86/1.59
% 2.86/1.59 thf(fact_219_local_Oid__simps_I13_J,axiom,
% 2.86/1.59 vid2 != vid1 ).
% 2.86/1.59
% 2.86/1.59 thf(fact_220_vne13,axiom,
% 2.86/1.59 vid1 != vid3 ).
% 2.86/1.59
% 2.86/1.59 thf(fact_221_vne12,axiom,
% 2.86/1.59 vid1 != vid2 ).
% 2.86/1.59
% 2.86/1.59 thf(fact_222_local_Oid__simps_I1_J,axiom,
% 2.86/1.59 vid1 != vid2 ).
% 2.86/1.59
% 2.86/1.59 thf(fact_223_local_Oid__simps_I3_J,axiom,
% 2.86/1.59 vid1 != vid3 ).
% 2.86/1.59
% 2.86/1.59 thf(fact_224_vne23,axiom,
% 2.86/1.59 vid2 != vid3 ).
% 2.86/1.59
% 2.86/1.59 thf(fact_225_local_Oid__simps_I2_J,axiom,
% 2.86/1.59 vid2 != vid3 ).
% 2.86/1.59
% 2.86/1.59 thf(fact_226_local_Oid__simps_I14_J,axiom,
% 2.86/1.59 vid3 != vid2 ).
% 2.86/1.59
% 2.86/1.59 thf(fact_227_p1__def,axiom,
% 2.86/1.59 ! [P2: sz,X: sz] :
% 2.86/1.59 ( ( p1 @ sz @ sf @ sc @ vid1 @ P2 @ X )
% 2.86/1.59 = ( prop @ sz @ sf @ sc @ P2 @ ( singleton @ sz @ sf @ vid1 @ ( var @ sz @ sf @ X ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_228_prod_Oswap__def,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType] :
% 2.86/1.59 ( ( product_swap @ A @ B )
% 2.86/1.59 = ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_229_formula_Oinject_I2_J,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,C: $tType,X21: C,X222: C > ( trm @ A @ C ),Y21: C,Y22: C > ( trm @ A @ C )] :
% 2.86/1.59 ( ( ( prop @ C @ A @ B @ X21 @ X222 )
% 2.86/1.59 = ( prop @ C @ A @ B @ Y21 @ Y22 ) )
% 2.86/1.59 = ( ( X21 = Y21 )
% 2.86/1.59 & ( X222 = Y22 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_230_swap__swap,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 2.86/1.59 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 2.86/1.59 = P2 ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_231_pair__in__swap__image,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,Y: A,X: B,A3: set @ ( product_prod @ B @ A )] :
% 2.86/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A3 ) )
% 2.86/1.59 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A3 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_232_swap__simp,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,X: B,Y: A] :
% 2.86/1.59 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 2.86/1.59 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_233_surj__swap,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType] :
% 2.86/1.59 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.86/1.59 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_234_snd__swap,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 2.86/1.59 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 2.86/1.59 = ( product_fst @ A @ B @ X ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_235_fst__swap,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 2.86/1.59 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 2.86/1.59 = ( product_snd @ B @ A @ X ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_236_ids_Op1_Ocong,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ( ( p1 @ Sz @ Sf @ Sc )
% 2.86/1.59 = ( p1 @ Sz @ Sf @ Sc ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_237_ids_Op1__def,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P2: Sz,X: Sz] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ( p1 @ Sz @ Sf @ Sc @ Vid1 @ P2 @ X )
% 2.86/1.59 = ( prop @ Sz @ Sf @ Sc @ P2 @ ( singleton @ Sz @ Sf @ Vid1 @ ( var @ Sz @ Sf @ X ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_238_formula_Oset__intros_I3_J,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,C: $tType,Yb: trm @ A @ C,X222: C > ( trm @ A @ C ),Yc: A,X21: C] :
% 2.86/1.59 ( ( member @ ( trm @ A @ C ) @ Yb @ ( image @ C @ ( trm @ A @ C ) @ X222 @ ( top_top @ ( set @ C ) ) ) )
% 2.86/1.59 => ( ( member @ A @ Yc @ ( set_trm @ A @ C @ Yb ) )
% 2.86/1.59 => ( member @ A @ Yc @ ( set1_formula @ A @ B @ C @ ( prop @ C @ A @ B @ X21 @ X222 ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_239_hpfree__ffree_Ointros_I10_J,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,C: $tType,Args: C > ( trm @ A @ C ),P2: C] :
% 2.86/1.59 ( ! [Arg: trm @ A @ C] :
% 2.86/1.59 ( ( member @ ( trm @ A @ C ) @ Arg @ ( image @ C @ ( trm @ A @ C ) @ Args @ ( top_top @ ( set @ C ) ) ) )
% 2.86/1.59 => ( dfree @ A @ C @ Arg ) )
% 2.86/1.59 => ( ffree @ A @ B @ C @ ( prop @ C @ A @ B @ P2 @ Args ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_240_hpfree__ffree_Ointros_I15_J,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,B: $tType,P: B] : ( ffree @ A @ B @ C @ ( predicational @ B @ A @ C @ P ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_241_map__prod__surj,axiom,
% 2.86/1.59 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G: C > D] :
% 2.86/1.59 ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.86/1.59 = ( top_top @ ( set @ B ) ) )
% 2.86/1.59 => ( ( ( image @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
% 2.86/1.59 = ( top_top @ ( set @ D ) ) )
% 2.86/1.59 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 2.86/1.59 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_242_good__interp_Oright__unique,axiom,
% 2.86/1.59 ! [C: $tType,B: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ A )
% 2.86/1.59 & ( finite_finite @ B )
% 2.86/1.59 & ( finite_finite @ C ) )
% 2.86/1.59 => ( right_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_243_good__interp_Oleft__unique,axiom,
% 2.86/1.59 ! [C: $tType,B: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ A )
% 2.86/1.59 & ( finite_finite @ B )
% 2.86/1.59 & ( finite_finite @ C ) )
% 2.86/1.59 => ( left_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_244_map__prod__simp,axiom,
% 2.86/1.59 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,A2: C,B2: D] :
% 2.86/1.59 ( ( product_map_prod @ C @ A @ D @ B @ F @ G @ ( product_Pair @ C @ D @ A2 @ B2 ) )
% 2.86/1.59 = ( product_Pair @ A @ B @ ( F @ A2 ) @ ( G @ B2 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_245_fst__map__prod,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
% 2.86/1.59 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X ) )
% 2.86/1.59 = ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_246_snd__map__prod,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X: product_prod @ C @ D] :
% 2.86/1.59 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X ) )
% 2.86/1.59 = ( G @ ( product_snd @ C @ D @ X ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_247_map__prod__imageI,axiom,
% 2.86/1.59 ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R6: set @ ( product_prod @ A @ B ),F: A > C,G: B > D] :
% 2.86/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R6 )
% 2.86/1.59 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A2 ) @ ( G @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) @ R6 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_248_prod__fun__imageE,axiom,
% 2.86/1.59 ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F: C > A,G: D > B,R6: set @ ( product_prod @ C @ D )] :
% 2.86/1.59 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G ) @ R6 ) )
% 2.86/1.59 => ~ ! [X3: C,Y3: D] :
% 2.86/1.59 ( ( C3
% 2.86/1.59 = ( product_Pair @ A @ B @ ( F @ X3 ) @ ( G @ Y3 ) ) )
% 2.86/1.59 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R6 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_249_ids_Ogood__interp_Oright__unique,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ A )
% 2.86/1.59 & ( finite_finite @ B )
% 2.86/1.59 & ( finite_finite @ C )
% 2.86/1.59 & ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( right_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_250_ids_Ogood__interp_Oleft__unique,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ A )
% 2.86/1.59 & ( finite_finite @ B )
% 2.86/1.59 & ( finite_finite @ C )
% 2.86/1.59 & ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( left_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_251_good__interp_Odomain,axiom,
% 2.86/1.59 ! [C: $tType,B: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ A )
% 2.86/1.59 & ( finite_finite @ B )
% 2.86/1.59 & ( finite_finite @ C ) )
% 2.86/1.59 => ( ( domainp @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) )
% 2.86/1.59 = ( denota2077489681interp @ A @ B @ C ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_252_dterm__sem_Osimps_I6_J,axiom,
% 2.86/1.59 ! [B: $tType,C: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ A )
% 2.86/1.59 & ( finite_finite @ C )
% 2.86/1.59 & ( finite_finite @ B ) )
% 2.86/1.59 => ! [I7: denota1663640101rp_ext @ A @ B @ C @ product_unit,T4: trm @ A @ C] :
% 2.86/1.59 ( ( denota594965758rm_sem @ A @ B @ C @ I7 @ ( differential @ A @ C @ T4 ) )
% 2.86/1.59 = ( denota2078997598vative @ A @ B @ C @ I7 @ T4 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_253_ids_Ogood__interp_Odomain,axiom,
% 2.86/1.59 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ A )
% 2.86/1.59 & ( finite_finite @ B )
% 2.86/1.59 & ( finite_finite @ C )
% 2.86/1.59 & ( finite_finite @ Sz )
% 2.86/1.59 & ( linorder @ Sz )
% 2.86/1.59 & ( finite_finite @ Sf )
% 2.86/1.59 & ( finite_finite @ Sc ) )
% 2.86/1.59 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.86/1.59 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.86/1.59 => ( ( domainp @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) )
% 2.86/1.59 = ( denota2077489681interp @ A @ B @ C ) ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(fact_254_dterm__sem_Osimps_I7_J,axiom,
% 2.86/1.59 ! [A: $tType,B: $tType,C: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ C )
% 2.86/1.59 & ( finite_finite @ B )
% 2.86/1.59 & ( finite_finite @ A ) )
% 2.86/1.59 => ! [I7: denota1663640101rp_ext @ A @ B @ C @ product_unit,C3: real] :
% 2.86/1.59 ( ( denota594965758rm_sem @ A @ B @ C @ I7 @ ( const @ A @ C @ C3 ) )
% 2.86/1.59 = ( ^ [V4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] : C3 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
% 2.86/1.59 ! [A: $tType] :
% 2.86/1.59 ( ( finite_finite @ A )
% 2.86/1.59 => ( type @ A ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
% 2.86/1.59 ! [A: $tType] :
% 2.86/1.59 ( ( linorder @ A )
% 2.86/1.59 => ( type @ A ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 2.86/1.59 ! [A7: $tType,A8: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ A7 )
% 2.86/1.59 & ( finite_finite @ A8 ) )
% 2.86/1.59 => ( finite_finite @ ( A7 > A8 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_fun___Orderings_Otop,axiom,
% 2.86/1.59 ! [A7: $tType,A8: $tType] :
% 2.86/1.59 ( ( top @ A8 )
% 2.86/1.59 => ( top @ ( A7 > A8 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_Set_Oset___Finite__Set_Ofinite_1,axiom,
% 2.86/1.59 ! [A7: $tType] :
% 2.86/1.59 ( ( finite_finite @ A7 )
% 2.86/1.59 => ( finite_finite @ ( set @ A7 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_Set_Oset___Orderings_Otop_2,axiom,
% 2.86/1.59 ! [A7: $tType] : ( top @ ( set @ A7 ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_Set_Oset___Groups_Ozero,axiom,
% 2.86/1.59 ! [A7: $tType] :
% 2.86/1.59 ( ( zero @ A7 )
% 2.86/1.59 => ( zero @ ( set @ A7 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.86/1.59 linorder @ $o ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_HOL_Obool___Finite__Set_Ofinite_3,axiom,
% 2.86/1.59 finite_finite @ $o ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_HOL_Obool___Orderings_Otop_4,axiom,
% 2.86/1.59 top @ $o ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_Real_Oreal___Orderings_Olinorder_5,axiom,
% 2.86/1.59 linorder @ real ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_Real_Oreal___Groups_Ozero_6,axiom,
% 2.86/1.59 zero @ real ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_7,axiom,
% 2.86/1.59 ! [A7: $tType,A8: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ A7 )
% 2.86/1.59 & ( finite_finite @ A8 ) )
% 2.86/1.59 => ( finite_finite @ ( sum_sum @ A7 @ A8 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_8,axiom,
% 2.86/1.59 ! [A7: $tType,A8: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ A7 )
% 2.86/1.59 & ( finite_finite @ A8 ) )
% 2.86/1.59 => ( finite_finite @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_Product__Type_Oprod___Orderings_Otop_9,axiom,
% 2.86/1.59 ! [A7: $tType,A8: $tType] :
% 2.86/1.59 ( ( ( top @ A7 )
% 2.86/1.59 & ( top @ A8 ) )
% 2.86/1.59 => ( top @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_Product__Type_Oprod___Groups_Ozero_10,axiom,
% 2.86/1.59 ! [A7: $tType,A8: $tType] :
% 2.86/1.59 ( ( ( zero @ A7 )
% 2.86/1.59 & ( zero @ A8 ) )
% 2.86/1.59 => ( zero @ ( product_prod @ A7 @ A8 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_11,axiom,
% 2.86/1.59 linorder @ product_unit ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_12,axiom,
% 2.86/1.59 finite_finite @ product_unit ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_Product__Type_Ounit___Orderings_Otop_13,axiom,
% 2.86/1.59 top @ product_unit ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_14,axiom,
% 2.86/1.59 ! [A7: $tType,A8: $tType] :
% 2.86/1.59 ( ( ( linorder @ A7 )
% 2.86/1.59 & ( cARD_1 @ A8 ) )
% 2.86/1.59 => ( linorder @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_15,axiom,
% 2.86/1.59 ! [A7: $tType,A8: $tType] :
% 2.86/1.59 ( ( ( finite_finite @ A7 )
% 2.86/1.59 & ( finite_finite @ A8 ) )
% 2.86/1.59 => ( finite_finite @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_16,axiom,
% 2.86/1.59 ! [A7: $tType,A8: $tType] :
% 2.86/1.59 ( ( ( zero @ A7 )
% 2.86/1.59 & ( finite_finite @ A8 ) )
% 2.86/1.59 => ( zero @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(help_If_3_1_T,axiom,
% 2.86/1.59 ! [P: $o] :
% 2.86/1.59 ( ( P = $true )
% 2.86/1.59 | ( P = $false ) ) ).
% 2.86/1.59
% 2.86/1.59 thf(help_If_2_1_T,axiom,
% 2.86/1.59 ! [A: $tType,X: A,Y: A] :
% 2.86/1.59 ( ( if @ A @ $false @ X @ Y )
% 2.86/1.59 = Y ) ).
% 2.86/1.59
% 2.86/1.59 thf(help_If_1_1_T,axiom,
% 2.86/1.59 ! [A: $tType,X: A,Y: A] :
% 2.86/1.59 ( ( if @ A @ $true @ X @ Y )
% 2.86/1.59 = X ) ).
% 2.86/1.59
% 2.86/1.59 thf(tfree_0,hypothesis,
% 2.86/1.59 finite_finite @ a ).
% 2.86/1.59
% 2.86/1.59 thf(tfree_1,hypothesis,
% 2.86/1.59 finite_finite @ c ).
% 2.86/1.59
% 2.86/1.59 thf(tfree_2,hypothesis,
% 2.86/1.59 finite_finite @ sz ).
% 2.86/1.59
% 2.86/1.59 thf(tfree_3,hypothesis,
% 2.86/1.59 linorder @ sz ).
% 2.86/1.59
% 2.86/1.59 thf(tfree_4,hypothesis,
% 2.86/1.59 finite_finite @ sf ).
% 2.86/1.59
% 2.86/1.59 thf(tfree_5,hypothesis,
% 2.86/1.59 finite_finite @ sc ).
% 2.86/1.59
% 2.86/1.59 thf(tfree_6,hypothesis,
% 2.86/1.59 finite_finite @ b ).
% 2.86/1.59
% 2.86/1.59 thf(conj_0,conjecture,
% 2.86/1.59 dfree @ a @ c @ ( args @ i2 ) ).
% 2.86/1.59 ------------------------
% 3.02/1.62 % (9485)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_9156 for (2999ds/183Mi)
% 3.02/1.62 % (9487)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_9156 for (2999ds/27Mi)
% 3.02/1.63 % (9486)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_9156 for (2999ds/4Mi)
% 3.02/1.63 % (9488)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_9156 for (2999ds/2Mi)
% 3.02/1.63 % (9489)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_9156 for (2999ds/2Mi)
% 3.02/1.63 % (9490)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_9156 for (2999ds/275Mi)
% 3.02/1.63 % (9491)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_9156 for (2999ds/18Mi)
% 3.02/1.63 % (9487)Instruction limit reached!
% 3.02/1.63 % (9487)------------------------------
% 3.02/1.63 % (9487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.63 % (9487)Termination reason: Unknown
% 3.02/1.63 % (9487)Termination phase: shuffling
% 3.02/1.63
% 3.02/1.63 % (9487)Memory used [KB]: 2046
% 3.02/1.63 % (9487)Time elapsed: 0.011 s
% 3.02/1.63 % (9487)Instructions burned: 29 (million)
% 3.02/1.63 % (9487)------------------------------
% 3.02/1.63 % (9487)------------------------------
% 3.02/1.63 % (9488)Instruction limit reached!
% 3.02/1.63 % (9488)------------------------------
% 3.02/1.63 % (9488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.63 % (9488)Termination reason: Unknown
% 3.02/1.63 % (9488)Termination phase: shuffling
% 3.02/1.63
% 3.02/1.63 % (9488)Memory used [KB]: 1407
% 3.02/1.63 % (9488)Time elapsed: 0.004 s
% 3.02/1.63 % (9488)Instructions burned: 3 (million)
% 3.02/1.63 % (9488)------------------------------
% 3.02/1.63 % (9488)------------------------------
% 3.02/1.63 % (9489)Instruction limit reached!
% 3.02/1.63 % (9489)------------------------------
% 3.02/1.63 % (9489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.63 % (9489)Termination reason: Unknown
% 3.02/1.63 % (9489)Termination phase: shuffling
% 3.02/1.63
% 3.02/1.63 % (9489)Memory used [KB]: 1407
% 3.02/1.63 % (9489)Time elapsed: 0.004 s
% 3.02/1.63 % (9489)Instructions burned: 3 (million)
% 3.02/1.63 % (9489)------------------------------
% 3.02/1.63 % (9489)------------------------------
% 3.02/1.63 % (9486)Instruction limit reached!
% 3.02/1.63 % (9486)------------------------------
% 3.02/1.63 % (9486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.63 % (9486)Termination reason: Unknown
% 3.02/1.63 % (9486)Termination phase: shuffling
% 3.02/1.63
% 3.02/1.63 % (9486)Memory used [KB]: 1535
% 3.02/1.63 % (9486)Time elapsed: 0.005 s
% 3.02/1.63 % (9486)Instructions burned: 5 (million)
% 3.02/1.63 % (9486)------------------------------
% 3.02/1.63 % (9486)------------------------------
% 3.02/1.64 % (9492)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9156 for (2999ds/3Mi)
% 3.02/1.64 % (9491)Instruction limit reached!
% 3.02/1.64 % (9491)------------------------------
% 3.02/1.64 % (9491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.64 % (9491)Termination reason: Unknown
% 3.02/1.64 % (9491)Termination phase: shuffling
% 3.02/1.64
% 3.02/1.64 % (9491)Memory used [KB]: 1918
% 3.02/1.64 % (9491)Time elapsed: 0.014 s
% 3.02/1.64 % (9491)Instructions burned: 19 (million)
% 3.02/1.64 % (9491)------------------------------
% 3.02/1.64 % (9491)------------------------------
% 3.02/1.64 % (9492)Instruction limit reached!
% 3.02/1.64 % (9492)------------------------------
% 3.02/1.64 % (9492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.64 % (9492)Termination reason: Unknown
% 3.02/1.64 % (9492)Termination phase: shuffling
% 3.02/1.64
% 3.02/1.64 % (9492)Memory used [KB]: 1407
% 3.02/1.64 % (9492)Time elapsed: 0.002 s
% 3.02/1.64 % (9492)Instructions burned: 3 (million)
% 3.02/1.64 % (9492)------------------------------
% 3.02/1.64 % (9492)------------------------------
% 3.02/1.65 % (9493)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_9156 for (2999ds/37Mi)
% 3.02/1.65 % (9494)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_9156 for (2999ds/15Mi)
% 3.02/1.65 % (9495)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_9156 for (2999ds/3Mi)
% 3.02/1.65 % (9495)Instruction limit reached!
% 3.02/1.65 % (9495)------------------------------
% 3.02/1.65 % (9495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.65 % (9495)Termination reason: Unknown
% 3.02/1.65 % (9495)Termination phase: shuffling
% 3.02/1.65
% 3.02/1.65 % (9495)Memory used [KB]: 1407
% 3.02/1.65 % (9495)Time elapsed: 0.003 s
% 3.02/1.65 % (9495)Instructions burned: 3 (million)
% 3.02/1.65 % (9495)------------------------------
% 3.02/1.65 % (9495)------------------------------
% 3.02/1.65 % (9496)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_9156 for (2999ds/1041Mi)
% 3.02/1.66 % (9494)Instruction limit reached!
% 3.02/1.66 % (9494)------------------------------
% 3.02/1.66 % (9494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.66 % (9494)Termination reason: Unknown
% 3.02/1.66 % (9494)Termination phase: shuffling
% 3.02/1.66
% 3.02/1.66 % (9494)Memory used [KB]: 1791
% 3.02/1.66 % (9494)Time elapsed: 0.012 s
% 3.02/1.66 % (9494)Instructions burned: 16 (million)
% 3.02/1.66 % (9494)------------------------------
% 3.02/1.66 % (9494)------------------------------
% 3.02/1.66 % (9497)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_9156 for (2999ds/7Mi)
% 3.02/1.67 % (9497)Instruction limit reached!
% 3.02/1.67 % (9497)------------------------------
% 3.02/1.67 % (9497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.67 % (9497)Termination reason: Unknown
% 3.02/1.67 % (9497)Termination phase: shuffling
% 3.02/1.67
% 3.02/1.67 % (9497)Memory used [KB]: 1535
% 3.02/1.67 % (9497)Time elapsed: 0.006 s
% 3.02/1.67 % (9497)Instructions burned: 7 (million)
% 3.02/1.67 % (9497)------------------------------
% 3.02/1.67 % (9497)------------------------------
% 3.02/1.67 % (9498)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_9156 for (2999ds/16Mi)
% 3.02/1.67 % (9490)First to succeed.
% 3.02/1.67 % (9493)Instruction limit reached!
% 3.02/1.67 % (9493)------------------------------
% 3.02/1.67 % (9493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.67 % (9493)Termination reason: Unknown
% 3.02/1.67 % (9493)Termination phase: shuffling
% 3.02/1.67
% 3.02/1.67 % (9493)Memory used [KB]: 2302
% 3.02/1.67 % (9493)Time elapsed: 0.022 s
% 3.02/1.67 % (9493)Instructions burned: 38 (million)
% 3.02/1.67 % (9493)------------------------------
% 3.02/1.67 % (9493)------------------------------
% 3.02/1.67 % (9498)Instruction limit reached!
% 3.02/1.67 % (9498)------------------------------
% 3.02/1.67 % (9498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.67 % (9498)Termination reason: Unknown
% 3.02/1.67 % (9498)Termination phase: shuffling
% 3.02/1.67
% 3.02/1.67 % (9498)Memory used [KB]: 1918
% 3.02/1.67 % (9498)Time elapsed: 0.007 s
% 3.02/1.67 % (9498)Instructions burned: 16 (million)
% 3.02/1.67 % (9498)------------------------------
% 3.02/1.67 % (9498)------------------------------
% 3.02/1.67 % (9490)Refutation found. Thanks to Tanya!
% 3.02/1.67 % SZS status Theorem for DTF2THF_9156
% 3.02/1.67 % SZS output start Proof for DTF2THF_9156
% 3.02/1.67 thf(type_def_5, type, real: $tType).
% 3.02/1.67 thf(type_def_6, type, finite_Cartesian_vec: ($tType * $tType) > $tType).
% 3.02/1.67 thf(type_def_7, type, product_prod: ($tType * $tType) > $tType).
% 3.02/1.67 thf(type_def_8, type, sum_sum: ($tType * $tType) > $tType).
% 3.02/1.67 thf(type_def_9, type, set: ($tType) > $tType).
% 3.02/1.67 thf(type_def_11, type, product_unit: $tType).
% 3.02/1.67 thf(type_def_12, type, denota1663640101rp_ext: ($tType * $tType * $tType * $tType) > $tType).
% 3.02/1.67 thf(type_def_13, type, trm: ($tType * $tType) > $tType).
% 3.02/1.67 thf(type_def_14, type, frechet_good_interp: ($tType * $tType * $tType) > $tType).
% 3.02/1.67 thf(type_def_15, type, frechet_strm: ($tType * $tType) > $tType).
% 3.02/1.67 thf(type_def_16, type, formula: ($tType * $tType * $tType) > $tType).
% 3.02/1.67 thf(type_def_17, type, a: $tType).
% 3.02/1.67 thf(type_def_18, type, b: $tType).
% 3.02/1.67 thf(type_def_19, type, c: $tType).
% 3.02/1.67 thf(type_def_20, type, sz: $tType).
% 3.02/1.67 thf(type_def_21, type, sc: $tType).
% 3.02/1.67 thf(type_def_22, type, sf: $tType).
% 3.02/1.67 thf(type_def_23, type, list: ($tType) > $tType).
% 3.02/1.67 thf(func_def_0, type, denota1663640101rp_ext: ($tType * $tType * $tType * $tType) > $tType).
% 3.02/1.67 thf(func_def_1, type, frechet_good_interp: ($tType * $tType * $tType) > $tType).
% 3.02/1.67 thf(func_def_2, type, frechet_strm: ($tType * $tType) > $tType).
% 3.02/1.67 thf(func_def_3, type, finite_Cartesian_vec: ($tType * $tType) > $tType).
% 3.02/1.67 thf(func_def_4, type, product_unit: $tType).
% 3.02/1.67 thf(func_def_5, type, product_prod: ($tType * $tType) > $tType).
% 3.02/1.67 thf(func_def_6, type, formula: ($tType * $tType * $tType) > $tType).
% 3.02/1.67 thf(func_def_7, type, sum_sum: ($tType * $tType) > $tType).
% 3.02/1.67 thf(func_def_8, type, trm: ($tType * $tType) > $tType).
% 3.02/1.67 thf(func_def_9, type, real: $tType).
% 3.02/1.67 thf(func_def_10, type, list: ($tType) > $tType).
% 3.02/1.67 thf(func_def_11, type, set: ($tType) > $tType).
% 3.02/1.67 thf(func_def_12, type, sz: $tType).
% 3.02/1.67 thf(func_def_13, type, sf: $tType).
% 3.02/1.67 thf(func_def_14, type, sc: $tType).
% 3.02/1.67 thf(func_def_15, type, c: $tType).
% 3.02/1.67 thf(func_def_16, type, b: $tType).
% 3.02/1.67 thf(func_def_17, type, a: $tType).
% 3.02/1.67 thf(func_def_18, type, cARD_1: !>[X0: $tType]:($o)).
% 3.02/1.67 thf(func_def_19, type, type: !>[X0: $tType]:($o)).
% 3.02/1.67 thf(func_def_20, type, zero: !>[X0: $tType]:($o)).
% 3.02/1.67 thf(func_def_21, type, top: !>[X0: $tType]:($o)).
% 3.02/1.67 thf(func_def_22, type, finite_finite: !>[X0: $tType]:($o)).
% 3.02/1.67 thf(func_def_23, type, linorder: !>[X0: $tType]:($o)).
% 3.02/1.67 thf(func_def_24, type, euclid1678468529ng_gcd: !>[X0: $tType]:($o)).
% 3.02/1.67 thf(func_def_25, type, euclid1155270486miring: !>[X0: $tType]:($o)).
% 3.02/1.67 thf(func_def_26, 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)).
% 3.02/1.67 thf(func_def_27, type, denota2078997598vative: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > trm @ X0 @ X2 > product_prod @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2 > real)).
% 3.02/1.67 thf(func_def_28, type, denota594965758rm_sem: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > trm @ X0 @ X2 > product_prod @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2 > real)).
% 3.02/1.67 thf(func_def_29, type, denotational_frechet: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > trm @ X0 @ X2 > finite_Cartesian_vec @ real @ X2 > finite_Cartesian_vec @ real @ X2 > real)).
% 3.02/1.67 thf(func_def_30, type, denota2077489681interp: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > $o)).
% 3.02/1.67 thf(func_def_31, type, freche457001096interp: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > frechet_good_interp @ X0 @ X1 @ X2 > $o)).
% 3.02/1.67 thf(func_def_32, type, frechet_cr_strm: !>[X0: $tType, X1: $tType]:(trm @ X0 @ X1 > frechet_strm @ X0 @ X1 > $o)).
% 3.02/1.67 thf(func_def_33, type, freche227871258interp: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > frechet_good_interp @ X0 @ X1 @ X2)).
% 3.02/1.67 thf(func_def_34, type, freche229654227interp: !>[X0: $tType, X1: $tType, X2: $tType]:(frechet_good_interp @ X0 @ X1 @ X2 > denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit)).
% 3.02/1.67 thf(func_def_35, type, frechet_raw_term: !>[X0: $tType, X1: $tType]:(frechet_strm @ X0 @ X1 > trm @ X0 @ X1)).
% 3.02/1.67 thf(func_def_36, type, frechet_simple_term: !>[X0: $tType, X1: $tType]:(trm @ X0 @ X1 > frechet_strm @ X0 @ X1)).
% 3.02/1.67 thf(func_def_37, type, zero_zero: !>[X0: $tType]:(X0)).
% 3.02/1.67 thf(func_def_38, type, ids: !>[X0: $tType, X1: $tType, X2: $tType]:(X0 > X0 > X0 > X1 > X1 > X1 > X2 > X2 > X2 > X2 > $o)).
% 3.02/1.67 thf(func_def_39, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.02/1.67 thf(func_def_40, type, top_top: !>[X0: $tType]:(X0)).
% 3.02/1.67 thf(func_def_41, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 3.02/1.67 thf(func_def_42, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.02/1.67 thf(func_def_43, type, product_map_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3)).
% 3.02/1.67 thf(func_def_44, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.02/1.67 thf(func_def_45, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.02/1.67 thf(func_def_46, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.02/1.67 thf(func_def_47, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 3.02/1.67 thf(func_def_48, type, domainp: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X0 > $o)).
% 3.02/1.67 thf(func_def_49, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.02/1.67 thf(func_def_50, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 3.02/1.67 thf(func_def_51, type, static_FVDiff: !>[X0: $tType, X1: $tType]:(trm @ X0 @ X1 > set @ sum_sum @ X1 @ X1)).
% 3.02/1.67 thf(func_def_52, type, predicational: !>[X0: $tType, X1: $tType, X2: $tType]:(X0 > formula @ X1 @ X0 @ X2)).
% 3.02/1.67 thf(func_def_53, type, dfree: !>[X0: $tType, X1: $tType]:(trm @ X0 @ X1 > $o)).
% 3.02/1.67 thf(func_def_54, type, dsafe: !>[X0: $tType, X1: $tType]:(trm @ X0 @ X1 > $o)).
% 3.02/1.67 thf(func_def_55, type, ffree: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > $o)).
% 3.02/1.67 thf(func_def_56, type, prop: !>[X0: $tType, X1: $tType, X2: $tType]:(X0 > (X0 > trm @ X1 @ X0) > formula @ X1 @ X2 @ X0)).
% 3.02/1.67 thf(func_def_57, type, set1_formula: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > set @ X0)).
% 3.02/1.67 thf(func_def_58, type, p: !>[X0: $tType, X1: $tType, X2: $tType]:(X0 > formula @ X1 @ X0 @ X2)).
% 3.02/1.67 thf(func_def_59, type, empty: !>[X0: $tType, X1: $tType]:(X0 > trm @ X1 @ X0)).
% 3.02/1.67 thf(func_def_60, type, f0: !>[X0: $tType, X1: $tType]:(X0 > trm @ X0 @ X1)).
% 3.02/1.67 thf(func_def_61, type, f1: !>[X0: $tType, X1: $tType]:(X0 > X1 > X0 > trm @ X1 @ X0)).
% 3.02/1.67 thf(func_def_62, type, p1: !>[X0: $tType, X1: $tType, X2: $tType]:(X0 > X0 > X0 > formula @ X1 @ X2 @ X0)).
% 3.02/1.67 thf(func_def_63, type, singleton: !>[X0: $tType, X1: $tType]:(X0 > trm @ X1 @ X0 > X0 > trm @ X1 @ X0)).
% 3.02/1.67 thf(func_def_64, type, singleton_rel: !>[X0: $tType, X1: $tType]:(product_prod @ trm @ X0 @ X1 @ X1 > product_prod @ trm @ X0 @ X1 @ X1 > $o)).
% 3.02/1.67 thf(func_def_65, type, const: !>[X0: $tType, X1: $tType]:(real > trm @ X0 @ X1)).
% 3.02/1.67 thf(func_def_66, type, diffVar: !>[X0: $tType, X1: $tType]:(X0 > trm @ X1 @ X0)).
% 3.02/1.67 thf(func_def_67, type, differential: !>[X0: $tType, X1: $tType]:(trm @ X0 @ X1 > trm @ X0 @ X1)).
% 3.02/1.67 thf(func_def_68, type, function: !>[X0: $tType, X1: $tType]:(X0 > (X1 > trm @ X0 @ X1) > trm @ X0 @ X1)).
% 3.02/1.67 thf(func_def_69, type, plus: !>[X0: $tType, X1: $tType]:(trm @ X0 @ X1 > trm @ X0 @ X1 > trm @ X0 @ X1)).
% 3.02/1.67 thf(func_def_70, type, times: !>[X0: $tType, X1: $tType]:(trm @ X0 @ X1 > trm @ X0 @ X1 > trm @ X0 @ X1)).
% 3.02/1.67 thf(func_def_71, type, var: !>[X0: $tType, X1: $tType]:(X0 > trm @ X1 @ X0)).
% 3.02/1.67 thf(func_def_72, type, set_trm: !>[X0: $tType, X1: $tType]:(trm @ X0 @ X1 > set @ X0)).
% 3.02/1.67 thf(func_def_73, type, left_unique: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > $o)).
% 3.02/1.67 thf(func_def_74, type, right_unique: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > $o)).
% 3.02/1.67 thf(func_def_75, type, accp: !>[X0: $tType]:((X0 > X0 > $o) > X0 > $o)).
% 3.02/1.67 thf(func_def_76, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.02/1.67 thf(func_def_77, type, i: denota1663640101rp_ext @ a @ b @ c @ product_unit).
% 3.02/1.67 thf(func_def_78, type, nu: product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c).
% 3.02/1.67 thf(func_def_79, type, nu2: product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c).
% 3.02/1.67 thf(func_def_80, type, args: c > trm @ a @ c).
% 3.02/1.67 thf(func_def_81, type, i2: c).
% 3.02/1.67 thf(func_def_82, type, var2: a).
% 3.02/1.67 thf(func_def_83, type, vid1: sz).
% 3.02/1.67 thf(func_def_84, type, vid2: sz).
% 3.02/1.67 thf(func_def_85, type, vid3: sz).
% 3.02/1.67 thf(func_def_100, type, sK0: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.02/1.67 thf(func_def_101, type, sK1: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.02/1.67 thf(func_def_102, type, sK2: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.02/1.67 thf(func_def_103, type, sK3: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.02/1.67 thf(func_def_104, type, sK4: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.02/1.67 thf(func_def_105, type, sK5: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.02/1.67 thf(func_def_106, type, sK6: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.02/1.67 thf(func_def_107, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 3.02/1.67 thf(func_def_108, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 3.02/1.67 thf(func_def_109, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 3.02/1.67 thf(func_def_110, type, sK10: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 3.02/1.67 thf(func_def_111, type, sK11: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.02/1.67 thf(func_def_112, type, sK12: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.02/1.67 thf(func_def_113, type, sK13: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.02/1.67 thf(func_def_114, type, sK14: !>[X0: $tType, X1: $tType]:((X1 > trm @ X0 @ X1) > X1)).
% 3.02/1.67 thf(func_def_115, type, sK15: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.02/1.67 thf(func_def_116, type, sK16: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.02/1.67 thf(func_def_117, type, sK17: !>[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)).
% 3.02/1.67 thf(func_def_118, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.02/1.67 thf(func_def_119, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.02/1.67 thf(func_def_120, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.02/1.67 thf(func_def_121, type, sK21: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.02/1.67 thf(func_def_122, type, sK22: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.02/1.67 thf(func_def_123, type, sK23: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.02/1.67 thf(func_def_124, type, sK24: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.02/1.67 thf(func_def_125, type, sK25: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.02/1.67 thf(func_def_126, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 3.02/1.67 thf(func_def_127, type, sK27: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 3.02/1.67 thf(func_def_128, type, sK28: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 3.02/1.67 thf(func_def_129, type, sK29: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 3.02/1.67 thf(func_def_130, type, sK30: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 3.02/1.67 thf(func_def_131, type, sK31: !>[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)).
% 3.02/1.67 thf(func_def_132, 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 > X1)).
% 3.02/1.67 thf(func_def_133, 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 > X2)).
% 3.02/1.67 thf(func_def_134, 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 > X3)).
% 3.02/1.67 thf(func_def_135, 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 > X4)).
% 3.02/1.67 thf(func_def_136, 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 > X5)).
% 3.02/1.67 thf(func_def_137, 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 > X6)).
% 3.02/1.67 thf(func_def_138, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 3.02/1.67 thf(func_def_139, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 3.02/1.67 thf(func_def_140, type, sK40: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 3.02/1.67 thf(func_def_141, type, sK41: !>[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)).
% 3.02/1.67 thf(func_def_142, type, sK42: !>[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)).
% 3.02/1.67 thf(func_def_143, type, sK43: !>[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)).
% 3.02/1.67 thf(func_def_144, type, sK44: !>[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)).
% 3.02/1.67 thf(func_def_145, type, sK45: !>[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)).
% 3.02/1.67 thf(func_def_146, type, sK46: !>[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)).
% 3.02/1.67 thf(func_def_147, type, sK47: !>[X0: $tType]:(product_prod @ trm @ X0 @ sz @ sz > trm @ X0 @ sz)).
% 3.02/1.67 thf(func_def_148, type, sK48: !>[X0: $tType]:(product_prod @ trm @ X0 @ sz @ sz > sz)).
% 3.02/1.67 thf(func_def_149, type, sK49: !>[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)).
% 3.02/1.67 thf(func_def_150, type, sK50: !>[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)).
% 3.02/1.67 thf(func_def_151, type, sK51: !>[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)).
% 3.02/1.67 thf(func_def_152, type, sK52: !>[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)).
% 3.02/1.67 thf(func_def_153, type, sK53: !>[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)).
% 3.02/1.67 thf(func_def_154, type, sK54: !>[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)).
% 3.02/1.67 thf(func_def_155, type, sK55: !>[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)).
% 3.02/1.67 thf(func_def_156, type, sK56: !>[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)).
% 3.02/1.67 thf(func_def_157, type, sK57: !>[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)).
% 3.02/1.67 thf(func_def_158, type, sK58: !>[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)).
% 3.02/1.67 thf(func_def_159, type, sK59: !>[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)).
% 3.02/1.67 thf(func_def_160, type, sK60: !>[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)).
% 3.02/1.67 thf(func_def_161, type, sK61: !>[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)).
% 3.02/1.67 thf(func_def_162, type, sK62: !>[X0: $tType, X1: $tType]:((X1 > trm @ X0 @ X1) > X1)).
% 3.02/1.67 thf(func_def_164, type, ph64: !>[X0: $tType]:(X0)).
% 3.02/1.67 thf(f873,plain,(
% 3.02/1.67 $false),
% 3.02/1.67 inference(subsumption_resolution,[status(thm)],[f856,f835])).
% 3.02/1.67 thf(f835,plain,(
% 3.02/1.67 ( ! [X0 : c] : (((dfree @ a @ c @ (args @ X0)) = $true)) )),
% 3.02/1.67 inference(cnf_transformation,[status(thm)],[f614])).
% 3.02/1.67 thf(f614,plain,(
% 3.02/1.67 ! [X0 : c] : ((dfree @ a @ c @ (args @ X0)) = $true)),
% 3.02/1.67 inference(fool_elimination,[status(thm)],[f613])).
% 3.02/1.67 thf(f613,plain,(
% 3.02/1.67 ! [X0 : c] : (dfree @ a @ c @ (args @ X0))),
% 3.02/1.67 inference(rectify,[status(thm)],[f1])).
% 3.02/1.67 thf(f1,axiom,(
% 3.02/1.67 ! [X8 : c] : (dfree @ a @ c @ (args @ X8))),
% 3.02/1.67 file('/export/starexec/sandbox/tmp/tmp.s6Ia5X0taI/DTF2THF_9156.p',fact_0_free)).
% 3.02/1.67 thf(f856,plain,(
% 3.02/1.67 ((dfree @ a @ c @ (args @ i2)) != $true)),
% 3.02/1.67 inference(cnf_transformation,[status(thm)],[f697])).
% 3.02/1.67 thf(f697,plain,(
% 3.02/1.67 ((dfree @ a @ c @ (args @ i2)) != $true)),
% 3.02/1.67 inference(flattening,[status(thm)],[f458])).
% 3.02/1.67 thf(f458,plain,(
% 3.02/1.67 ~((dfree @ a @ c @ (args @ i2)) = $true)),
% 3.02/1.67 inference(fool_elimination,[status(thm)],[f457])).
% 3.02/1.67 thf(f457,plain,(
% 3.02/1.67 ~(dfree @ a @ c @ (args @ i2))),
% 3.02/1.67 inference(rectify,[status(thm)],[f289])).
% 3.02/1.67 thf(f289,negated_conjecture,(
% 3.02/1.67 ~(dfree @ a @ c @ (args @ i2))),
% 3.02/1.67 inference(negated_conjecture,[status(cth)],[f288])).
% 3.02/1.67 thf(f288,conjecture,(
% 3.02/1.67 (dfree @ a @ c @ (args @ i2))),
% 3.02/1.67 file('/export/starexec/sandbox/tmp/tmp.s6Ia5X0taI/DTF2THF_9156.p',conj_0)).
% 3.02/1.67 % SZS output end Proof for DTF2THF_9156
% 3.02/1.67 % (9490)------------------------------
% 3.02/1.67 % (9490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.67 % (9490)Termination reason: Refutation
% 3.02/1.67
% 3.02/1.67 % (9490)Memory used [KB]: 6908
% 3.02/1.67 % (9490)Time elapsed: 0.041 s
% 3.02/1.67 % (9490)Instructions burned: 64 (million)
% 3.02/1.67 % (9490)------------------------------
% 3.02/1.67 % (9490)------------------------------
% 3.02/1.67 % (9484)Success in time 0.076 s
%------------------------------------------------------------------------------